Physics Reports vol.380

Instructions to authors Aims and scope Physics Reports keeps the active physicist up-to-date on developments in a wide r...

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Instructions to authors Aims and scope Physics Reports keeps the active physicist up-to-date on developments in a wide range of topics by publishing timely reviews which are more extensive than just literature surveys but normally less than a full monograph. Each Report deals with one specific subject. These reviews are specialist in nature but contain enough introductory material to make the main points intelligible to a non-specialist. The reader will not only be able to distinguish important developments and trends but will also find a sufficient number of references to the original literature. Submission In principle, papers are written and submitted on the invitation of one of the Editors, although the Editors would be glad to receive suggestions. Proposals for review articles (approximately 500–1000 words) should be sent by the authors to one of the Editors listed below. The Editor will evaluate proposals on the basis of timeliness and relevance and inform the authors as soon as possible. All submitted papers are subject to a refereeing process. Editors J.V. ALLABY (Experimental high-energy physics), EP Division, CERN, CH-1211 Geneva 23, Switzerland. E-mail: [email protected] D.D. AWSCHALOM (Experimental condensed matter physics), Department of Physics, University of California, Santa Barbara, CA 93106, USA. E-mail: [email protected] J.A. BAGGER (High-energy physics), Department of Physics & Astronomy, The Johns Hopkins University, 3400 North Charles Street, Baltimore MD 21218, USA. E-mail: [email protected] C.W.J. BEENAKKER (Mesoscopic physics), Instituut–Lorentz, Universiteit Leiden, P.O. Box 9506, 2300 RA Leiden, The Netherlands. E-mail: [email protected] G.E. BROWN (Nuclear physics), Institute for Theoretical Physics, State University of New York at Stony Brook, Stony Brook, NY 11974, USA. E-mail: [email protected] D.K. CAMPBELL (Non-linear dynamics), Dean, College of Engineering, Boston University, 44 Cummington Street, Boston, MA 02215, USA. E-mail: [email protected] G. COMSA (Surfaces and thin films), Institut fur . Physikalische und Theoretische Chemie, Universit.at Bonn, Wegelerstrasse 12, D-53115 Bonn, Germany. E-mail: [email protected] J. EICHLER (Atomic and molecular physics), Hahn-Meitner-Institut Berlin, Abteilung Theoretische Physik, Glienicker Strasse 100, 14109 Berlin, Germany. E-mail: [email protected] M.P. KAMIONKOWSKI (Astrophysics), Theoretical Astrophysics 130-33, California Institute of Technology, 1200 East California Blvd., Pasadena, CA 91125, USA. E-mail: [email protected] M.L. KLEIN (Soft condensed matter physics), Department of Chemistry, University of Pennsylvania, Philadelphia, PA 19104-6323, USA. E-mail: [email protected]

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Instructions to authors

A.A. MARADUDIN (Condensed matter physics), Department of Physics and Astronomy, University of California, Irvine, CA 92697-4575, USA. E-mail: [email protected] D.L. MILLS (Condensed matter physics), Department of Physics and Astronomy, University of California, Irvine, CA 92697-4575, USA. E-mail: [email protected] H. ORLAND (Statistical physics and field theory), Service de Physique Theorique, CE-Saclay, CEA, 91191 Gif-sur-Yvette Cedex, France. E-mail: [email protected] R. PETRONZIO (High-energy physics), Dipartimento di Fisica, Universita" di Roma – Tor Vergata, Via della Ricerca Scientifica, 1, I-00133 Rome, Italy. E-mail: [email protected] S. PEYERIMHOFF (Molecular physics), Institute of Physical and Theoretical Chemistry, Wegelerstrasse 12, D-53115 Bonn, Germany. E-mail: [email protected] I. PROCACCIA (Statistical mechanics), Department of Chemical Physics, Weizmann Institute of Science, Rehovot 76100, Israel. E-mail: [email protected] E. SACKMANN (Biological physics), Physik-Department E22 (Biophysics Lab.), Technische Universit.at Munchen, . D-85747 Garching, Germany. E-mail: [email protected] A. SCHWIMMER (High-energy physics), Department of Physics of Complex Systems, Weizmann Institute of Science, Rehovot 76100, Israel. E-mail: [email protected] R.N. SUDAN (Plasma physics), Laboratory of Plasma Studies, Cornell University, 369 Upson Hall, Ithaca, NY 14853-7501, USA. E-mail: [email protected] W. WEISE (Physics of hadrons and nuclei), Institut fur . Theoretische Physik, Physik Department, Technische Universit.at Munchen, . James Franck Strae, D-85748 Garching, Germany. E-mail: [email protected] Manuscript style guidelines Papers should be written in correct English. Authors with insufficient command of the English language should seek linguistic advice. Manuscripts should be typed on one side of the paper, with double line spacing and a wide margin. The character size should be sufficiently large that all subscripts and superscripts in mathematical expressions are clearly legible. Please note that manuscripts should be accompanied by separate sheets containing: the title, authors’ names and addresses, abstract, PACS codes and keywords, a table of contents, and a list of figure captions and tables. – Address: The name, complete postal address, e-mail address, telephone and fax number of the corresponding author should be indicated on the manuscript. – Abstract: A short informative abstract not exceeding approximately 150 words is required. – PACS codes/keywords: Please supply one or more PACS-1999 classification codes and up to 4 keywords of your own choice for indexing purposes. PACS is available online from our homepage (http://www.elsevier.com/locate/physrep). References. The list of references may be organized according to the number system or the nameyear (Harvard) system. Number system: [1] M.J. Ablowitz, D.J. Kaup, A.C. Newell and H. Segur, The inverse scattering transform – Fourier analysis for nonlinear problems, Studies in Applied Mathematics 53 (1974) 249–315. [2] M. Abramowitz and I. Stegun, Handbook of Mathematical Functions (Dover, New York, 1965).


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[3] B. Ziegler, in: New Vistas in Electro-nuclear Physics, eds E.L. Tomusiak, H.S. Kaplan and E.T. Dressler (Plenum, New York, 1986) p. 293. A reference should not contain more than one article. Harvard system:

Ablowitz, M.J., D.J. Kaup, A.C. Newell and H. Segur, 1974. The inverse scattering transform – Fourier analysis for nonlinear problems, Studies in Applied Mathematics 53, 249–315. Abramowitz, M. and I. Stegun, 1965, Handbook of Mathematical Functions (Dover, New York). Ziegler, B., 1986, in: New Vistas in Electro-nuclear Physics, eds E.L. Tomusiak, H.S. Kaplan and E.T. Dressler (Plenum, New York) p. 293. Ranking of references. The references in Physics Reports are ranked: crucial references are indicated by three asterisks, very important ones with two, and important references with one. Please indicate in your final version the ranking of the references with the asterisk system. Please use the asterisks sparingly: certainly not more than 15% of all references should be placed in either of the three categories. Formulas. Formulas should be typed or unambiguously written. Special care should be taken of those symbols which might cause confusion. Unusual symbols should be identified in the margin the first time they occur.

Equations should be numbered consecutively throughout the paper or per section, e.g., Eq. (15) or Eq. (2.5). Equations which are referred to should have a number; it is not necessary to number all equations. Figures and tables may be numbered the same way. Footnotes. Footnotes may be typed at the foot of the page where they are alluded to, or collected at the end of the paper on a separate sheet. Please do not mix footnotes with references. Figures. Each figure should be submitted on a separate sheet labeled with the figure number. Line diagrams should be original drawings or laser prints. Photographs should be contrasted originals, or high-resolution laserprints on glossy paper. Photocopies usually do not give good results. The size of the lettering should be proportionate to the details of the figure so as to be legible after reduction. Original figures will be returned to the author only if this is explicitly requested. Colour illustrations. Colour illustrations will be accepted if the use of colour is judged by the Editor to be essential for the presentation. Upon acceptance, the author will be asked to bear part of the extra cost involved in colour reproduction and printing. After acceptance – Proofs: Proofs will be sent to the author by e-mail, 6–8 weeks after receipt of the manuscript. Please note that the proofs have been proofread by the Publisher and only a cursory check by the author is needed; we are unable to accept changes in, or additions to, the edited manuscript at this stage. Your proof corrections should be returned within two days of receipt by fax, courier or airmail. The Publisher may proceed with publication of no response is received. – Copyright transfer: The author(s) will receive a form with which they can transfer copyright of the article to the Publisher. This transfer will ensure the widest possible dissemination of information. LaTeX manuscripts The Publisher welcomes the receipt of an electronic version of your accepted manuscript (encoded in LATEX). If you have not already supplied the final, revised version of your article (on diskette) to the Journal Editor, you are requested herewith to send a file with the text of the manuscript (after acceptance) by e-mail to the address provided by the Publisher. Please note that no deviations

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Physics Reports 380 (2003) 1 – 95 www.elsevier.com/locate/physrep

Neutral and charged polymers at interfaces Roland R. Netza; b , David Andelmanc;∗ a

Max-Planck Institute for Colloids and Interfaces, D-14424 Potsdam, Germany Sektion Physik, Ludwig-Maximilians-Universit$at, Theresienstr. 37, 80333 M$unchen, Germany c School of Physics and Astronomy, Raymond and Beverly Sackler, Faculty of Exact Sciences Tel Aviv University, Ramat Aviv, Tel Aviv 69978, Israel b

Accepted 27 February 2003 editor: E. Sackmann

Abstract Chain-like macromolecules (polymers) show characteristicadsorption properties due to their 1exibility and internal degrees of freedom, when attracted to surfaces and interfaces. In this review we discuss concepts and features that are relevant to the adsorption of neutral and charged polymers at equilibrium, including the type of polymer/surface interaction, the solvent quality, the characteristics of the surface, and the polymer structure. We pay special attention to the case of charged polymers (polyelectrolytes) that have a special importance due to their water solubility. We present a summary of recent progress in this rapidly evolving 6eld. Because many experimental studies are performed with rather sti8 biopolymers, we discuss in detail the case of semi-1exible polymers in addition to 1exible ones. We 6rst review the behavior of neutral and charged chains in solution. Then, the adsorption of a single polymer chain is considered. Next, the adsorption and depletion processes in the many-chain case are reviewed. Pro6les, changes in the surface tension and polymer surface excess are presented. Mean-6eld and corrections due to 1uctuations and lateral correlations are discussed. The force of interaction between two adsorbed layers, which is important in understanding colloidal stability, is characterized. The behavior of grafted polymers is also reviewed, both for neutral and charged polymer brushes. c 2003 Elsevier Science B.V. All rights reserved. PACS: 61.25.Hq Contents 1. Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4 1.1. Types of polymers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5 2. Neutral polymer chains . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7 2.1. Flexible chains . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7 2.2. Chain swelling and chain collapse: Flory theory and blob formation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11 2.3. Semi-1exible chains . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12 ∗

Corresponding author. E-mail address: [email protected] (D. Andelman).

c 2003 Elsevier Science B.V. All rights reserved. 0370-1573/03/$ - see front matter doi:10.1016/S0370-1573(03)00118-2

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R.R. Netz, D. Andelman / Physics Reports 380 (2003) 1 – 95

2.4. Dilute, semi-dilute and concentrated solutions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3. Charged polymer chains . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.1. Interactions between charged objects . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.2. Isolated polyelectrolyte chains . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.3. Manning condensation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.4. Self-avoidance and polyelectrolyte chain conformations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.5. Dilute polyelectrolyte solutions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.6. Semi-dilute polyelectrolyte solutions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4. General considerations on adsorption . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.1. Adsorption and depletion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.2. Surface characteristics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.3. Surface–polymer interactions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5. Adsorption of a single neutral chain . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.1. Mean-6eld regime . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.2. Fluctuation dominated regime . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6. Adsorption of a single polyelectrolyte chain . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7. Neutral polymer adsorption from solution . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7.1. The mean-6eld approach: ground state dominance . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7.1.1. The adsorption case . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7.1.2. The depletion case . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7.2. Beyond mean-6eld theory: scaling arguments for good solvents . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7.2.1. Scaling for polymer adsorption . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7.2.2. Scaling for polymer depletion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7.3. Proximal region corrections . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7.4. Loops and tails . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8. Adsorption of polyelectrolytes—mean 6eld . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8.1. Mean-6eld theory and its pro6le equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8.2. Constant Us : the low-salt limit . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8.2.1. Numerical solutions of mean-6eld equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8.2.2. Scaling arguments . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8.3. Adsorption behavior in the presence of 6nite salt . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8.4. Adsorption–depletion crossover in high-salt conditions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8.5. Adsorption of PEs for constant surface charge and its overcompensation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8.5.1. Low-salt limit: D−1 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8.5.2. High-salt limit: D ¿ −1 and depletion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9. Lateral correlation e8ects in polyelectrolyte adsorption . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10. Interaction between two adsorbed layers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10.1. Non-adsorbing polymers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10.2. Adsorbing neutral polymers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10.3. Adsorbing charged polymers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11. Polymer adsorption on heterogeneous surfaces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12. Polymer adsorption on curved and 1uctuating interfaces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12.1. Neutral polymers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12.2. Charged polymers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13. Grafted polymer chains . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13.1. Neutral grafted polymers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13.2. Solvent and substrate e8ects on polymer grafting . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13.3. Charged grafted polymers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14. Concluding remarks . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Acknowledgements . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

14 15 15 17 23 24 25 25 29 29 30 32 33 35 37 40 44 44 47 49 50 50 51 52 53 54 55 57 57 57 60 61 63 63 64 65 70 70 71 72 73 74 74 74 76 77 81 82 86 87 87


Nomenclature a b cm cm (x) cmb cm∗ csalt c± (x) d D e f fˆ F F g h(x) H kB T L Lel Lsw ‘B ‘0 ‘OSF ‘e8 N R Rel S(q) S0 (q) U (x) Us u(x) us = u(0) v2 v˜2 v(r) vDH (r) z

Kuhn length or e8ective monomer size monomer size monomer concentration (per unit volume) monomer density pro6le at distance x from the surface bulk monomer concentration in semi-dilute solutions. overlap concentration of bulk polymer solution salt concentration in the solution pro6les of ± ions polymer diameter (or cross-section) adsorption layer thickness, height of brush electronic unit charge fractional charge of the chain 0 ¡ f ¡ 1 force rescaled by kB T intensive free energy in units of kB T (per unit area or unit volume) extensive free energy in units of kB T number of monomers per blob dimensionless PE adsorption pro6le height of counterion cloud (PE brush case) thermal energy contour length of a chain chain length inside one electrostatic blob chain length inside one swollen blob Bjerrum length (=e2 =kB T ) bare (mechanical) persistence length electrostatic contribution to persistence length (Odijk, Skolnick, and Fixman length) e8ective persistence length polymerization index end-to-end polymer chain radius radius of one electrostatic blob structure factor (or scattering function) of a PE solution form factor of a single chain electrostatic potential at point x from the surface surface potential at x = 0 dimensionless potential pro6le (=eU (x)=kB T ) rescaled surface potential 2nd virial coeIcient of monomers in solution. v2 ¿ 0 for good solvents dimensionless 2nd virial coeIcient of monomers in solution (=v2 =a3 ) Coulomb interaction between two ions in units of kB T (=e2 =kB Tr) Debye–HKuckel interaction (=v(r) exp(−r)) valency of the ions (= ± 1; ±2; : : :)

3

4


# −1 salt $± $p & 'b ; 's ( ) * L* + ,(x) ,b ,s (x) b

polymer surface excess per unit area dielectric constant of the medium. = 80 for water Debye–HKuckel screening length salt contribution to chemical potential of ± ions chemical potential of polymer Flory exponent for the polymer size correlation length (mesh size) of semi-dilute polymer solution in bulk and at surface osmotic pressure in units of kB T grafting density of a polymer brush surface charge density (in units of e) at x = 0 overcharging parameter (=f# − *) linear charge density on the chain (=f=b) monomer volume fraction (dimensionless) at distance x from the surface (=a3 cm (x)) bulk value of , (=a3 cmb ) surface value of , polymer order parameter (= ,(x)) bulk value of polymer order parameter

1. Introduction Polymers are long chain molecules which play important roles in industrial applications and in biological processes. On a more fundamental level, polymers exhibit interesting behavior which can be derived from the knowledge of their statistical mechanics properties. We review the basic mechanisms underlying the equilibrium properties of these macromolecules in solution and, in particular, their behavior at surfaces and interfaces. The understanding of polymer systems progressed tremendously from the late 1960s because of innovation in experimental techniques such as X-ray and neutron di8raction and light scattering. Some techniques like ellipsometry, second harmonics generation (SHG), Brewster angle microscopy, surface force apparatus, atomic force microscopy (AFM) and X-ray or neutron re1ectivity are especially appropriate to study polymers at interfaces. Of equal merit was the advancement in theoretical methods ranging from 6eld theoretical methods and scaling arguments to numerical simulations. The major progress in the 6eld of polymer adsorption at liquid interfaces and solid surfaces is even more recent. Even though several excellent books [1,2] and review articles [3–6] exist, we feel that the present review is timely because we address recent progress in the 6eld of chains at interfaces, paying particular attention to charged chains. Charged polymers are interesting from the application point of view, since they allow for a number of water-based formulations which are advantageous for economical and ecological reasons. Recent years have seen a tremendous research activity on charged polymers in bulk and at interfaces. Likewise, adsorption of biopolymers such as DNA at planar or spherical substrates is an intermediate step in the fabrication of gene-technology related structures, and therefore of great current interest. In addition to being charged, DNA is rather


5

sti8 on the nanoscopic length scale. On intermediate length scales, it can be well described as a semi-1exible polymer, in contrast to most synthetic polymers, which are well represented by 1exible polymer models. Accordingly, we discuss the complexity of charged and semi-1exible chains in addition to neutral and 1exible ones. We also contrast the situation of physical adsorption of chains with that of terminally attached chains (neutral or charged) to surfaces. This review is focused on physical aspects of polymer adsorption at thermodynamical equilibrium and summarizes the main theoretical and recent progress. We only outline theoretical calculations and do not explain in detail theoretical and experimental techniques. Whenever possible we try to explain principal concepts in simple terms. Experimental results are mentioned when they are of direct relevance but this review should not be considered as an exhaustive review of various experimental techniques and data. The review starts by explaining well known facts about conformations of a single ideal chain as well as self-avoiding chain and their behavior in solution (Section 2). We then examine the e8ect of charges on the statistics of an isolated chain and of multi-chains in solution (Section 3). The rest of the paper deals with adsorption in several distinct situations: a general introduction to adsorption processes (Section 4), adsorption of a single neutral chain (Section 5) and of a single polyelectrolyte chain (Section 6), mean 6eld theories for adsorption of neutral (Section 7) and charged (Section 8) chains. Corrections to mean-6eld theories are considered in Sections 7 and 9. In Section 10 the interaction between two adsorption layers is presented, while adsorption on more complicated substrates such as heterogeneous and curved interfaces are brie1y discussed in Sections 11 and 12. Finally, chains that are terminally anchored to the surface are mentioned in Section 13. These polymer brushes are discussed both for neutral and charged chains. Although this review is written as one coherent manuscript, expert readers can skip the 6rst three sections and concentrate on adsorption of neutral chains (Sections 4, 5, 7, 10, 12), adsorption of charged chains (Sections 6, 8–12) and grafted polymer layers (brushes) (Section 13). 1.1. Types of polymers The polymers considered here are taken as linear and long chains, as is schematically depicted in Fig. 1a. We brie1y mention other, more complex, chain architectures. For example, branched chains [7], Fig. 1b, appear in many applications. One special type of branched structures, Fig. 1f, is a chain having a backbone (main chain) with repeated side branches. The chemical nature of the side and main chain can be di8erent. This demonstrates the di8erence between homopolymers, formed from a single repeat unit (monomer) and heteropolymers, formed from several chemical di8erent monomers. The heteropolymer can be statistical, e.g. DNA, where the di8erent units repeat in a non-periodic or random fashion, Fig. 1d. Another case is that of block copolymers built from several blocks each being a homopolymer by itself. For example, an A–B–A–C block copolymer is a chain composed of an A, a B, an A and a C block linked serially to form a quarto-block chain, Fig. 1e. Synthetic polymers such as polystyrene and polyethylene are composed of 1exible chains which can be solubilized in a variety of organic solvents like toluene, cyclohexane, etc. These polymers are highly insoluble in water. Another class of polymers are water soluble ones. They either have strong dipolar groups which are compatible with the strong polarizability of the aqueous media (e.g., polyethylene oxide) or they carry charged groups.

6


(a)

(b) + +

+

+

+ +

+

+

+

+

+

(c) B B

A

C A

B C

A

C

C A

B

C

(d)

A

B

A

C

(e)

(f)

Fig. 1. Schematic view of di8erent polymer types: (a) linear homopolymers that are the main subject of this review; (b) branched polymers; (c) charged polymers or polyelectrolytes (PEs), with a certain fraction of charged monomers; (d) a disordered (hetero) copolymer with no speci6c order of the di8erent monomers: A, B, C, etc.; (e) a block copolymer. For example, a quatro-block A–B–A–C is drawn, where each of the blocks is a homopolymer by itself; (f) a copolymer composed of a backbone (dashed line) and side chains (solid line) of di8erent chemical nature. The backbone could for example be hydrophilic and make the polymer water-soluble as a whole, while the side chain might be hydrophobic and attract other hydrophobic solutes in the solution.

Charged polymers, also known as polyelectrolytes (PE), are shown schematically in Fig. 1c. They are extensively studied not only because of their numerous industrial applications, but also from a pure scienti6c interest [8–11]. One of the most important properties of PEs is their water solubility


7

giving rise to a wide range of non-toxic, environmentally friendly and cheap formulations. On the theoretical side, the physics of PEs combines the 6eld of statistical mechanics of charged systems with the 6eld of polymer science and o8ers quite a number of surprises and challenges. Two other concepts associated with PEs and water soluble polymers are associating polymers (not discussed in this review) and the 1exibility of the polymer chain. In cases when the copolymers have both hydrophobic and hydrophilic groups (similarly to short-chain amphiphiles), they will self-assemble in solution to form meso-structures such as lamellae, cylinders and spheres dispersed in solution. The inside of these structures is where the hydrophobic chain sections are packed, away from the water environment. In other cases, association of hydrophobic groups may lead to inter-chain networking and drastically modify the visco-elasticity of the solution. Another concept discussed at large in this review is the chain 1exibility. The chains considered here are either 1exible or semi-1exible. Flexible chains are chains where it does not cost energy to bend them, while the sti8ness of semi-;exible chains is an important property. For PEs the charge groups contribute substantially to the chain sti8ness, and the chain conformational degrees of freedom are coupled with the electrostatic ones.

2. Neutral polymer chains 2.1. Flexible chains The statistical thermodynamics of 1exible chains is well developed and the theoretical concepts can be applied with a considerable degree of con6dence [7,12–15]. In contrast to other molecules or particles, polymer chains contain not only translational and rotational degrees of freedom, but also a vast number of conformational degrees of freedom. This fact plays a crucial role in determining their behavior in solution and at surfaces. When 1exible chains adsorb on surfaces they form di=usive adsorption layers extending away from the surface into the solution. This is in contrast to semi-1exible or rigid chains, which can form dense and compact adsorption layers. From the experimental point of view, the main parameters used to describe a polymer chain are the polymerization index N , which counts the number of repeat units or monomers along the chain, and the monomer size b, being the size of one monomer or the distance between two neighboring monomers. The monomer size ranges from a few Angstroms for synthetic polymers to a few nanometers for biopolymers [12]. The simplest theoretical description of 1exible chain conformations is achieved with the so-called freely jointed chain (FJC) model, where a polymer consisting of N + 1 monomers is represented by N bonds de6ned by bond vectors rj with j = 1; : : : ; N . Each bond vector has a 6xed length |rj | = a corresponding to the Kuhn length, but otherwise is allowed to rotate freely, as is schematically shown in Fig. 2a. This model of course only gives a coarse-grained description of real polymer chains, but we will later see that by a careful adjustment of the Kuhn length a (which is related but not identical to the monomer size b), an accurate description of the large-scale properties of real polymer chains is possible. The main advantage is that due to the simplicity of the FJC model, all interesting observables (such as chain size or distribution functions) can be calculated with relative ease. Fixing one of the chain ends at the origin, the position of the (k + 1)th monomer is given by

8


a

(a)

b ϑ

(b)

(c)

Fig. 2. (a) Freely jointed chain (FJC) model, where N bonds of length a are connected to form a 1exible chain. (b) Freely rotating chain (FRC) model, which describes a polymer chain with a saturated carbon backbone. It consists of a chain of N bonds of length b, with 6xed bond angles #, re1ecting the chemical bond structure, but with freely rotating torsional angles. (c) The simpli6ed model, appropriate for more advanced theoretical calculations, consists of a structureless line, governed by some bending rigidity or line tension. This continuous model can be used when the relevant length scales are much larger than the monomer size.

the vectorial sum Rk =

k

rj :

(2.1)

j=1

Because two arbitrary bond vectors are uncorrelated in this simple model, the thermal average over the scalar product of two di8erent bond vectors vanishes, rj · rk = 0 for j = k, while the mean squared bond vector length is simply given by rj2 = a2 . It follows that the mean squared end-to-end radius R2 is proportional to the number of monomers, R2 ≡ RN2 = Na2 = La ;

(2.2)

where the contour length of the chain is given by L = Na. The same result is obtained for the mean quadratic displacement of a freely di8using particle and alludes to the same underlying physical principle, namely the statistics of Markov processes.


9

Fig. 3. Snapshot of a Monte-Carlo simulation of a neutral freely jointed chain (FJC) consisting of N = 100 monomers with a diameter corresponding to the Kuhn length a. The theoretical end-to-end radius, R = 10a, is indicated by the upper horizontal bar.

In Fig. 3 we show a snapshot of a Monte-Carlo simulation of a freely jointed chain consisting of 100 non-interacting monomers, each being represented by a sphere of diameter a. The horizontal bar has a length of 10a, which according to Eq. (2.2) is the average distance between the chain ends. As can be seen in the 6gure, the end-to-end radius gives a good idea of the typical chain size. In the so-called freely rotating chain (FRC) model, di8erent chain conformations are produced by torsional rotations of the polymer backbone bonds of length b at 6xed bond angle #, as shown schematically in Fig. 2b. This model is closer to real synthetic polymers than the FJC model, but is also more complicated to calculate. In contrast to the FJC model, the correlation between two neighboring bond vectors does not vanish and is given by rj · rj+1 = b2 cos #. Correlations between further-nearest neighbors are transmitted through the backbone and one thus obtains for the bond-vector correlation function [7] rj · rk = b2 (cos #)|j−k | :

(2.3)

The mean-squared end-to-end radius is for this model in the limit of long chains (N → ∞) given by [7] R2 Nb2

1 + cos # : 1 − cos #

(2.4)

We will now demonstrate that the simple result for the FJC model, Eq. (2.2), applies on length scales which are large compared with the microscopic chain details also to the more complicated FRC

10


model (which takes the detailed microscopic chain structure into account). To make the connection between the two models, we observe that the FRC contour length is L = Nb cos(#=2). Using the scaling relation R2 = aL (which we established for the FJC model) as a de6nition for the Kuhn length a, we obtain for the FRC model a=b

1 + cos # ; cos(#=2)(1 − cos #)

(2.5)

where the Kuhn length a is now interpreted as an e8ective monomer size. For a typical saturated carbon backbone one 6nds a bond angle # ≈ 70◦ and thus obtains for the relation between the Kuhn length and the monomer size a ≈ 2:5b. With a typical bond length of b ≈ 0:15 nm this results in a Kuhn length of a ≈ 0:38 nm. Clearly, the Kuhn length a is always larger than the monomer size b. We have thus shown that it is possible to use the simple FJC model also for more detailed chain models if one interprets the Kuhn length a as an e8ective length which takes correlations between chemical bonds into account. In the remainder of this review, we will in most cases use a 1exible chain model characterized by the Kuhn length a. Only in cases where the microscopic structure of the polymer chains matters will we use more detailed models (and then have to distinguish between the Kuhn length a, characterizing the large-scale properties of a chain, and the monomer size b). In many theoretical calculations aimed at elucidating large-scale properties, the simpli6cation is carried even a step further and a continuous model is used, as schematically shown in Fig. 2c. In such models the polymer backbone is replaced by a continuous line and all microscopic details are neglected. The models discussed so far describe ideal Gaussian chains and do not account for interactions between monomers which are not necessarily close neighbors along the backbone. Including these interactions will give a di8erent scaling behavior for long polymer chains. The end-to-end radius, R = R2N , can be written more generally for N 1 as R aN & :

(2.6)

For an ideal polymer chain (no interactions between monomers), Eq. (2.2) implies &=1=2. This holds only for polymers where the attraction between monomers (as compared with the monomer–solvent interaction) cancels the steric repulsion (which is due to the fact that the monomers cannot penetrate each other). This situation can be achieved in the condition of “theta” solvents. More generally, polymers in solution can experience three types of solvent conditions, with theta solvent condition being intermediate between “good” and “bad” solvent conditions. The solvent quality depends mainly on the speci6c chemistry determining the interaction between the solvent molecules and monomers. It can be changed by varying the temperature. In good solvents the monomer–solvent interaction is more favorable than the monomer–monomer one. Single polymer chains in good solvents have “swollen” spatial con6gurations dominated by the steric repulsion, characterized by an exponent & 3=5 [12]. This spatial size of a polymer coil is much smaller than the extended contour length L = aN but larger than the size of an ideal chain aN 1=2 . The reason for this peculiar behavior is entropy combined with the favorable interaction between monomers and solvent molecules in good solvents, as we will see in the following section. Similarly, for adsorption of polymer chains on solid substrates, the conformational degrees of freedom of polymer coils lead to salient di8erences between the adsorption of polymers and small molecules.


11

In the opposite case of “bad” (sometimes called “poor”) solvent conditions, the e8ective interaction between monomers is attractive, leading to collapse of the chains and to their precipitation from solution (phase separation between the polymer and the solvent). In this case, the polymer size, like any space 6lling object embedded in three-dimensional space, scales as N ∼ R3 , yielding & = 1=3. 2.2. Chain swelling and chain collapse: Flory theory and blob formation The standard way of taking into account interactions between monomers is the Flory theory, which treats these interactions on a mean-6eld level [7,12–15]. Let us 6rst consider the case of repulsive interactions between monomers, which can be described by a positive second-virial coeIcient v2 . This corresponds to the aforementioned good-solvent condition. For pure hard-core interactions and with no additional attractions between monomers, the second virial coeIcient (which corresponds to the excluded volume) is of the order of a3 , the monomer volume. The repulsive interaction between monomers, which tends to swell the chain, is counteracted and balanced by the ideal chain elasticity, which is brought about by the entropy loss associated with stretching the chain. The analogy with an external stretching force is helpful: For a freely jointed chain, the stretching response due to an ˆ for weak forces f1=a ˆ external force fˆ (rescaled by the thermal energy kB T ) is R a2 N f=3 [14]. Hence, a freely jointed chain acts like an ideal spring with a spring constant (rescaled by kB T ) of 3=(2a2 N ). The temperature dependence of the spring constant tells us that the chain elasticity is purely entropic. The origin is that the number of polymer con6gurations having an end-to-end radius of the order of the unperturbed end-to-end radius is large. These con6gurations are entropically favored over con6gurations characterized by a large end-to-end radius, for which the number of possible polymer conformations is drastically reduced. The standard Flory theory [12] for a 1exible chain of radius R is based on writing the free energy F (in units of the thermal energy kB T ) as a sum of two terms (omitting numerical prefactors) 2 N R2 3 ; (2.7) F 2 + v2 R aN R3 where the 6rst term is the entropic elastic energy associated with swelling a polymer chain to a radius R, proportional to the e8ective spring constant of an ideal chain, and the second term is the second-virial repulsive energy proportional to the coeIcient v2 , and the segment density squared. It is integrated over the volume R3 . The optimal radius R is calculated by minimizing this free energy and gives the swollen radius R ∼ a(v2 =a3 )1=5 N &

(2.8) 3

&

3

with & = 3=5. For purely steric interactions with v2 a we obtain R ∼ aN . For v2 ¡ a one 6nds that the swollen radius Eq. (2.8) is only realized above a minimal monomer number Nsw (v2 =a3 )−2 below which the chain statistics is unperturbed by the interaction and the scaling of the chain radius is Gaussian and given by Eq. (2.2). A di8erent way of looking at this crossover from Gaussian to swollen behavior is to denote a Gaussian coil of monomer number Nsw as a blob with size 1=2 Rsw = aNsw a4 =v2 , after which the swollen radius Eq. (2.8) can be rewritten as R ∼ Rsw (N=Nsw )& :

(2.9)

The swollen chain can be viewed as chain of N=Nsw impenetrable blobs, each with a spatial size Rsw [14].

12


In the opposite limit of negative second virial coeIcient, corresponding to the bad or poor solvent regime, the polymer coil will be collapsed due to the attractions between monomers. In this case, the attraction term in the free energy is balanced by the third-virial term in a low-density expansion (where we assume that v3 ¿ 0), 2 3 N N 3 3 F v2 R + v3 R : (2.10) R3 R3 Minimizing this free energy with respect to the chain radius one obtains R (v3 =|v2 |)1=3 N & :

(2.11)

with & = 1=3. This indicates the formation of a compact globule, since the monomer density inside the globule, cm ∼ N=R3 , is independent of the chain length. The minimal chain length to observe a collapse behavior is Ncol ∼ (v3 =a3 v2 )2 , and the chain radius Eq. (2.11) can be rewritten as R ∼ 1=2 Rcol (N=Ncol )1=3 , where the size of a Gaussian blob is Rcol ∼ aNcol . For not too long chains and a second virial coeIcient not too much di8ering from zero, the interaction is irrelevant and one obtains e8ective Gaussian or ideal behavior. It should be noted, however, that even small deviations from the exact theta conditions (de6ned by strictly v2 = 0) will lead to chain collapse or swelling for very long chains. 2.3. Semi-;exible chains The freely rotating chain model exhibits orientational correlations between bonds that are not too far from each other, see Eq. (2.3). These correlations give rise to a certain chain sti8ness, which plays an important role for the local structure of polymers, and leads to more rigid structures. For synthetic polymers with bond torsional degrees of freedom, this sti8ness is due to 6xed bond angles and is further enhanced by the hindered rotations around individual back-bone bonds [12], as schematically shown in Fig. 2b. This e8ect is even more pronounced for polymers with bulky side chains, where, because of steric constraints, the persistence length can be of the order of a few nanometers [12]. This sti8ness can be conveniently characterized by the persistence length ‘0 , de6ned as the length over which the normalized bond (tangent) vectors at di8erent locations on the chain are correlated. In other words, the persistence length gives an estimate for the typical radius of curvature, while taking into account thermal 1uctuations. For the FRC model, the persistence length ‘0 is de6ned by rj · rk = b2 e−|j−k |b cos(#=2)=‘0 : With the result Eq. (2.3), one obtains for the FRC model the persistence length b cos(#=2) : ‘0 = |ln cos #|

(2.12)

(2.13)

For typical saturated carbon backbones with # ≈ 70◦ one obtains a persistence length of ‘0 ≈ 0:8b which is thus of the order of the bond length. Clearly, as the bond angle goes down, the persistence length increases dramatically. Biopolymers with a more complex structure on the molecular level tend to be sti8er than simple synthetic polymers. Some typical persistence lengths encountered in biological systems are


13

‘0 ≈ 5 mm for tubulin [16], ‘0 ≈ 20 m for actin [17,18], and ‘0 ≈ 50 nm for double-stranded DNA [19]. Because some of these biopolymer are charged, we will discuss in Section 3.2 at length the dependence of the persistence length on the electrostatic conditions. In some cases the main contribution to the persistence length comes from the repulsion between charged monomers. In these cases, it is important to include the e8ect of sti8ness into the theoretical description, even if the bare or mechanical sti8ness is only slightly larger than the monomer size. To describe the bending rigidity of neutral polymers, it is easier to use a continuum model, where one neglects the discrete nature of monomers, as shown in Fig. 2c. In this approach the bending energy (rescaled by the thermal energy, kB T ) of a sti8 or semi-1exible polymer of contour length L, which is parameterized by the space curve r(s), is given by [7] 2 2 ‘0 L d r(s) ds ; (2.14) 2 0 ds2 where d 2 r(s)=ds2 is the local curvature of the polymer. We assume here that the polymer segments ˙ = dr(s)=ds are always normalized, |r(s)| ˙ = 1. Clearly, are non-expendable, i.e. the tangent vectors r(s) this continuum description will only be good if the persistence length is larger than the monomer size b. For the semi-1exible polymer model, the correlations between tangent vectors exhibit a purely exponential decay, ˙ · r(s ˙ ) = e−|s−s |=‘0 : r(s)

(2.15)

From this result, the mean-squared end-to-end radius of a semi-1exible chain, described by the bending energy Eq. (2.14), can be calculated and reads [7] R2 = 2‘0 L + 2‘02 (e−L=‘0 − 1) ;

(2.16)

where the persistence length is ‘0 and the total contour length of a chain is L. Two limiting behaviors are obtained for R from Eq. (2.16): for long chains L‘0 , the chain behaves as a 1exible one, R2 2‘0 L; while for rather short chains, L‘0 , the chain behaves as a rigid rod, R L. Comparison with the scaling of the freely jointed chain model (having no persistence length, ‘0 = 0), Eq. (2.2), shows that a semi-1exible chain can, for L‘0 , be described by a freely jointed chain model with an e8ective Kuhn length of a = 2‘0 ;

(2.17)

and an e8ective number of segments L N= : (2.18) 2‘0 In this case the Kuhn length takes into account the chain sti8ness. In Fig. 4 we show snapshots taken from a Monte-Carlo simulation of a semi-1exible chain consisting of 100 polymer beads of diameter b. The persistence length is varied from ‘0 = 2b (Fig. 4a), over ‘0 = 10b (Fig. 4b), to ‘0 =100b (Fig. 4c). Comparison with the freely jointed chain model is given in Fig. 3 (a=b, ‘0 =0). It is seen that as the persistence length is increased, the chain structure becomes more expanded. The average end-to-end radius R, Eq. (2.16), is shown as the bar on the 6gure and gives a good estimate on typical sizes of semi-1exible polymers. The main point here is that even though the semi-1exible polymer model describes biopolymers much better than the freely jointed model does, on large length scales both models coincide if the

14


(a)

(b)

(c)

Fig. 4. Snapshots of Monte-Carlo simulations of a neutral and semi-1exible chain consisting of N = 100 monomers with a diameter b. The theoretical end-to-end radius R is indicated by a straight bar. The persistence lengths used in the simulations are: (a) ‘0 =b = 2, leading according to Eq. (2.16) to R=b = 19:8, (b) ‘0 =b = 10, leading to R=b = 42:4, (c) ‘0 =b = 100, leading to R=b = 85:8.

Kuhn length a of the freely jointed chain model is the e8ective length which is extracted from the scaling of the end-to-end radius in the semi-1exible model, Eq. (2.16). When the small-scale behavior is probed, as for example in the case of polymer adsorption with short-ranged potentials, see Section 6, the di8erence between the models matters and one has to use the semi-1exible model. On the other hand, it should be kept in mind that the semi-1exible polymer model is an idealization, which neglects the detailed architecture of the polymer at the molecular level. For synthetic polymers, a freely rotating chain model with a bond length b and a bond angle # as shown in Fig. 2b is closer to reality but is more complicated to handle theoretically [7]. 2.4. Dilute, semi-dilute and concentrated solutions It is natural to generalize the discussion of single chain behavior to that of many chains for dilute monomer concentrations. The dilute regime is de6ned by cm ¡ cm∗ , for which cm denotes the monomer concentration (per unit volume) and cm∗ is the concentration where individual chains start to overlap. Clearly, the overlap concentration is reached when the average bulk monomer concentration exceeds the monomer concentration inside a polymer coil. To estimate the overlap concentration cm∗ , we simply note that the average monomer concentration inside a coil with radius R ∼ aN & is given by N (2.19) cm∗ 3 ∼ N 1−3& a−3 : R For ideal chains with & = 1=2 the overlap concentration scales as a3 cm∗ ∼ N −1=2 and thus decreases slowly as the polymerization index N increases. For swollen chains with & = 3=5, on the other hand,


15

the overlap concentration scales as a3 cm∗ ∼ N −4=5 and thus decreases more rapidly with increasing chain length. The crossover to the concentrated or melt-like regime occurs when the monomer concentration in the solution reaches the local monomer concentration inside a Gaussian blob, which is for good solvent conditions given by (see the discussion before Eq. (2.9)) cm∗∗ Nsw =R3sw v2 =a6 :

(2.20) c∗

¡ c∗∗ ,

It is seen that the semi-dilute regime, obtained for concentrations m ¡ cm spans for long m chains and under good solvent conditions a rather wide range of concentrations and is thus important for typical applications. For chains characterized by a negative second virial coeIcient, attractions between collapsed single-chain globules lead to phase separation between a very dilute solution of single-polymer globules and a dense melt-like phase of entangled polymer coils [14]. 3. Charged polymer chains 3.1. Interactions between charged objects A polyelectrolyte (PE) is a polymer where a fraction f of its monomers are charged. When this fraction is small, f1, the PE is weakly charged, whereas when f is close to unity, the polyelectrolyte is strongly charged. There are two common ways to control f [11]. One way is to polymerize a heteropolymer using strongly acidic and neutral monomers as building blocks. Upon contact with water, the acidic groups dissociate into positively charged protons (H+ ) that bind immediately to water molecules, and negatively charged monomers. Although this process e8ectively charges the polymer molecules, the counterions make the PE solution electro-neutral on larger length scales. The charge distribution along the chain is quenched (“frozen”) during the polymerization stage, and it is characterized by the fraction of charged monomers on the chain, f. In the second way, the PE is a weak polyacid or polybase. The e8ective charge of each monomer is controlled by the pH of the solution. Moreover, this annealed fraction depends on the local electric potential. This is in particular important for adsorption processes since the local electric 6eld close to a strongly charged surface can be very di8erent from its value in the bulk solution. The counterions are attracted to the charged polymers via long-ranged Coulomb interactions, but this physical association typically only leads to a rather loosely bound counterion cloud around the PE chain. Because PEs are present in a background of a polarizable and di8usive counterion cloud, there is a strong in1uence of the counterion distribution on the PE structure, as will be discussed at length in this section. Counterions contribute signi6cantly towards bulk properties, such as the osmotic pressure, and their translational entropy is responsible for the generally good water solubility of charged polymers. In addition, the statistics of PE chain conformation is governed by intra-chain Coulombic repulsion between charged monomers, resulting in a more extended and swollen conformation of PEs as compared to neutral polymers. For polyelectrolytes, electrostatic interactions provide the driving force for their salient features and have to be included in any theoretical description. The reduced electrostatic interaction between two point-like charges can be written as z1 z2 v(r) where v(r) = ‘B =r

(3.1)

16


is the Coulomb interaction between two elementary charges in units of kB T and z1 and z2 are the valencies (or the reduced charges in units of the elementary charge e). The Bjerrum length ‘B is de6ned as e2 ; (3.2) ‘B = kB T where is the medium dielectric constant. It denotes the distance at which the Coulombic interaction between two unit charges in a dielectric medium is equal to thermal energy (kB T ). It is a measure of the distance below which the Coulomb energy is strong enough to compete with the thermal 1uctuations; in water at room temperatures, one 6nds ‘B ≈ 0:7 nm. The electrostatic interaction in a homogeneous medium depends only on the distance r between the charges. The total electrostatic energy of a given distribution of charges is obtained from adding up all pairwise interactions between charges according to Eq. (3.1). In principle, the equilibrium behavior of an ensemble of charged particles (e.g. a salt solution) follows from the partition function, i.e. the weighted sum over all di8erent microscopic con6gurations, which—via the Boltzmann factor— depends on the electrostatic energy of each con6guration. In practice, however, this route is very complicated for several reasons: (i) The Coulomb interaction, Eq. (3.1), is long-ranged and couples many charged particles. Electrostatic problems are typically many-body problems, even for low densities. (ii) Charged objects in most cases are dissolved in water. Like any material, water is polarizable and reacts to the presence of a charge with polarization charges. In addition, and this is by far a more important e8ect, water molecules carry a permanent dipole moment that partially orients in the vicinity of charged objects. Within linearized response theory, these polarization e8ects can be incorporated by the dielectric constant of water, a procedure which of course neglects non-local and non-linear e8ects. Note that for water, ≈ 80, so that electrostatic interactions and self energies are much weaker in water than in air (where ≈ 1) or some other low-dielectric solvents. Still, the electrostatic interactions are especially important in polar solvents because in these solvents, charges dissociate more easily than in apolar solvents. (iii) In biological systems and most industrial applications, the aqueous solution contains mobile salt ions. Salt ions of opposite charge are drawn to the charged object and form a loosely bound counterion cloud around it. They e8ectively reduce or screen the charge of the object. The e8ective (screened) electrostatic interaction between two charges z1 e and z2 e in the presence of salt ions and a polarizable solvent can be written as z1 z2 vDH (r), with the Debye–HKuckel (DH) potential vDH (r) given (in units of kB T ) by ‘B −r e : (3.3) vDH (r) = r The exponential decay is characterized by the screening length −1 , which is related to the salt concentration csalt by 2 = 80z 2 ‘B csalt ;

(3.4)

where z denotes the valency of z : z salt. At physiological conditions the salt concentration is csalt ≈ 0:1 M and for monovalent ions (z = 1) this leads to −1 ≈ 1 nm. This means that although the Coulombic interactions are long-ranged, in physiological conditions they are highly screened above length scales of a few nanometers, which results from multi-body correlations between ions in a salt solution.


17

The Debye–HKuckel potential in Eq. (3.3) results from a linearized mean-6eld procedure, and becomes inaccurate when (i) the number of correlated ions is small and (ii) when the typical interaction between ions exceeds the thermal energy. In the following we estimate the validity of the DH approximation using simple scaling arguments: The number of ions which are correlated in a salt solution with concentration csalt is of the order of n ∼ −3 csalt , where one employs the screening length −1 as the scale over which ions are correlated. Using the de6nition 2 = 80z 2 ‘B csalt , one 1=3 −3=2 −1=3 obtains n ∼ (z 2 ‘B csalt ) . The average distance between ions is roughly rsalt ∼ csalt . The typical electrostatic interaction between two ions in the solution, rescaled by the thermal energy, thus is 1=3 Wel ∼ z 2 ‘B =rsalt ∼ z 2 ‘B csalt and we obtain Wel ∼ n−2=3 . Using these scaling arguments one obtains that either (i) many ions are weakly coupled together (i.e. n1 and Wel 1), or (ii) a few ions interact strongly with each other (n Wel 1). In the 6rst case, and in the absence of external 6elds, the approximations leading to the Debye–HKuckel approximation, Eq. (3.3), are valid. In the second case, correlation e8ects and non-linear e8ects become important, as will be discussed at various points in this review. 3.2. Isolated polyelectrolyte chains We discuss now the scaling behavior of a single semi-1exible PE in the bulk, including chain sti8ness and electrostatic repulsion between monomers. For charged polymers, the e8ective persistence length is increased due to electrostatic repulsion between monomers. This e8ect modi6es considerably not only the PE behavior in solution but also their adsorption characteristics. The scaling analysis is a simple extension of previous calculations for 1exible (Gaussian) PE’s [20–23]. The semi-1exible polymer chain is characterized by a bare persistence length ‘0 and a linear charge density +. Using the monomer length b and the fraction of charged monomers f as parameters, the linear charge density can be expressed as + = f=b. Note that in the limit where the persistence length is small and comparable to a monomer size, only a single length scale remains, ‘0 a b. Many interesting e8ects, however, are obtained in the general case treating the persistence length ‘0 and the monomer size b as two independent parameters. In the regime where the electrostatic energy is weak, and for long enough contour length L, L‘0 , a polymer coil will be formed with a radius R unperturbed by the electrostatic repulsion between monomers. According to Eq. (2.16) we get R2 2‘0 L. To estimate when the electrostatic interaction will be suIciently strong to swell the polymer coil we recall that the electrostatic energy (rescaled by the thermal energy kB T ) of a homogeneously charged sphere of total charge Z (in units of the elementary charge e) and radius R is 3‘B Z 2 : (3.5) 5R The exact charge distribution inside the sphere only changes the prefactor of order unity and is not important for the scaling arguments. For a polymer of length L and line charge density + the total charge is Z = +L. The electrostatic energy of a (roughly spherical) polymer coil is then Wel =

Wel ‘B +2 L3=2 ‘0−1=2 :

(3.6)

The polymer length Lel at which the electrostatic self energy is of order kB T , i.e. Wel 1, is then Lel ‘0 (‘B ‘0 +2 )−2=3 ;

(3.7)

18


(a)

(b)

(c)

(d)

Fig. 5. Snapshots of Monte-Carlo simulations of a PE chain of N = 100 monomers of size b. In all simulations the bare persistence length is 6xed at ‘0 =b = 1, and the screening length and the charge interactions are tuned such that the electrostatic √ persistence length is constant and√‘OSF =b = 100 according to Eq. (3.11). √ The parameters used are: (a) −1 =b = √ 50 and +2 ‘B ‘0 = 8, (b) −1 =b = 200 and +2 ‘B ‘0 = 2, (c) −1 =b = 800 and +2 ‘B ‘0 = 1=2, and (d) −1 =b = 3200 and +2 ‘B ‘0 = 1=8. Noticeably, the weakly charged chains crumple at small length scales and show a tendency to form electrostatic blobs.

and de6nes the electrostatic blob size or electrostatic polymer length. We expect a locally crumpled polymer con6guration if Lel ¿ ‘0 , i.e. if + ‘ B ‘0 ¡ 1 ; (3.8) because the electrostatic repulsion between two segments of length ‘0 is smaller than the thermal energy and is not suIcient to align the two segments. This is in accord with more detailed calculations by Joanny and Barrat [22]. A recent general Gaussian variational calculation con6rms this scaling result and in addition yields logarithmic corrections [23]. Conversely, for + ‘ B ‘0 ¿ 1 ; (3.9) electrostatic chain–chain repulsion is already relevant on length scales comparable to the persistence length. The chain is expected to have a conformation characterized by an e8ective persistence length ‘e8 , larger than the bare persistence length ‘0 , i.e. one expects ‘e8 ¿ ‘0 . This e8ect is visualized in Fig. 5, where we show snapshots of Monte-Carlo simulations for charged chains consisting of 100 monomers of size b. The monomers are interacting solely via


19

screened DH potentials as de6ned in Eq. (3.3). In all simulations the bare persistence length equals the monomer size, ‘0 = b. The screening length −1 and the linear charge density + are varied such that the ratio += is the same for all four simulations. The number of persistent segments in an electrostatic blob can be written according to Eq. (3.7) as Lel =‘0 = (+2 ‘B ‘0 )−2=3 and yields for Fig. 5a Lel =‘0 = 0:25, for Fig. 5b Lel =‘0 = 0:63, for Fig. 5c Lel =‘0 = 1:6, and for Fig. 5d Lel =‘0 = 4. In other words, in Fig. 5d the electrostatic blobs consist of four persistent segments, and indeed this weakly charged chain crumples at small length scales. On the other hand, in Fig. 5a the persistence length is four times larger than the electrostatic blob length and therefore the chain is straight locally. A typical linear charge density reached with synthetic PEs is one charge per two carbon bonds (or, equivalently, one charge per monomer), and it corresponds to + ≈ 4 nm−1 . Since for these highly 1exible synthetic PEs the bare persistence length is of the order of the monomer size, ‘0 ≈ b ≈ 0:25 nm, the typical value of +2 ‘B ‘0 is roughly +2 ‘B ‘0 ≈ 3, and thus intermediate between the values in Fig. 5a and b. Smaller linear charge densities can always be obtained by replacing some of the charged monomers on the polymer backbone with neutral ones. In this case the crumpling observed in Fig. 5d becomes relevant. On the other hand, increasing the bare sti8ness ‘0 , for example by adding bulky side chains to a synthetic PE backbone, increases the value of +2 ‘B ‘0 and, therefore, increases the electrostatic sti8ening of the backbone. This is an interesting illustration of the fact that electrostatic interactions and chain architecture (embodied via the persistence length) combine to control the polymer con6gurational behavior. The question now arises as to what are the typical chain conformations at much larger length scales. Clearly, they will be in1uenced √ by the electrostatic repulsions between monomers. Indeed, in the persistent regime, obtained for + ‘B ‘0 ¿ 1, the polymer remains locally sti8 even for contour lengths larger than the bare persistence length ‘0 and the e8ective persistence length is given by ‘e8 ‘0 + ‘OSF :

(3.10)

The electrostatic contribution to the e8ective persistence length, 6rst derived by Odijk and independently by Skolnick and Fixman, reads [24,25] ‘B +2 ‘OSF = ; (3.11) 42 and is calculated from the electrostatic energy of a slightly bent polymer using the linearized Debye– HKuckel approximation, Eq. (3.3). It is valid only for polymer conformations which do not deviate too much from the rod-like reference state and for weakly charged polymers (two conditions that are often not simultaneously satis6ed in practice and therefore have led to criticism of the OSF result, as will be detailed below). The electrostatic persistence length gives a sizable contribution to the e8ective persistence length only for ‘OSF ¿ ‘0 . This is equivalent to the condition + ‘B ‘0 ¿ ‘0 : (3.12) The persistent regime is obtained for parameters satisfying both conditions (3.9) and (3.12) and exhibits chains that do not crumple locally and are sti8ened√ electrostatically. Another regime called the Gaussian regime is obtained in the opposite limit of + ‘B ‘0 ¡ ‘0 and does not exhibit chain sti8ening due to electrostatic monomer–monomer repulsions. The e8ects of the electrostatic persistence length are visualized in Fig. 6, where we present snapshots of a Monte-Carlo simulation of a charged chain consisting of 100 monomers of size b. The bare persistence length is 6xed at ‘0 = b, and the charge-interaction parameter is chosen to be

20


(a)

(b)

(c)

Fig. 6. Snapshots of Monte-Carlo simulations of a PE chain consisting of N = 100 monomers of size b. In all simulations, the bare persistence length is 6xed at ‘0 =b = 1, and the charge-interaction parameter is chosen to be +2 ‘B ‘0 = 2. The √ −1 snapshots correspond to varying screening length of: (a) =b = 2, leading to an electrostatic contribution to the √ √ persistence length of ‘OSF =b = 1, (b) −1 =b = 18, leading to ‘OSF =b = 9, and (c) −1 =b = 200, leading to ‘OSF =b = 100. According to the simple scaling principle, Eq. (3.10), the e8ective persistence length in the snapshots (a) – (c) should be similar to the bare persistence length in Fig. 4(a) – (c).

+2 ‘B b = 2 for all three simulations, close to the typical charge density obtained with fully charged synthetic PEs. In Fig. √ 6 we show con6gurations for three di8erent values of the screening length, 1 namely (a) − =b= 2, leading to an electrostatic contribution to the persistence length of ‘OSF =b=1; √ √ (b) −1 =b = 18, or ‘OSF =b = 9; and (c) −1 =b = 200, equivalent to an electrostatic persistence length of ‘OSF =b=100. According to the simple scaling principle, Eq. (3.10), the e8ective persistence length in the snapshots, Fig. 6a–c, should be similar to the bare persistence length in Fig. 4a–c, and indeed, the chain structures in Figs. 6c and 4c are very similar. Figs. 6a and 4a are clearly di8erent, although the e8ective persistence length is predicted to be quite similar. This deviation is mostly due to self-avoidance e8ects which are present in charged chains and which will be discussed in detail in Section 3.4. For the case where√the polymer crumples on length scales larger than the bare persistence length, i.e. for Lel ¿ ‘0 or + ‘B ‘0 ¡ 1, the electrostatic repulsion between polymer segments is not strong enough to prevent crumpling on length scales comparable to ‘0 , but can give rise to a chain sti8ening


21

Gaussian

persistent

Gaussian

swollen

3/5

Fig. 7. Schematic view of the four scaling ranges in the Gaussian-persistent regime. On spatial scales smaller than Rel the chain behavior is Gaussian; on length scales larger than Rel but smaller than ‘KK the Gaussian blobs are aligned linearly. On larger length scales the chain is isotropically swollen with an exponent & = 1=2, and on even larger length scales self-avoidance e8ects become important and the exponent changes to & = 3=5.

on larger length scales, as explained by Khokhlov and Khachaturian [21] and con6rmed by Gaussian variational methods [23]. Fig. 7 schematically shows the PE structure in this Gaussian-persistent regime, where the chain on small scales consists of Gaussian blobs of size Rel , each containing a chain segment of length Lel . Within these blobs electrostatic interactions are not important. On larger length scales electrostatic repulsion leads to a chain sti8ening, so that the PE forms a linear array of electrostatic blobs. To quantify this e8ect, one de6nes an e8ective line charge density +˜ of a linear √ array of electrostatic blobs with blob size Rel ‘0 Lel , 1=2 +Lel Lel +˜ + : (3.13) Rel ‘0

22


Gaussianpersistent 2/3 persistent

1 Gaussian 1

1

Fig. 8. Behavior diagram of a single semi-1exible PE in bulk solution with bare persistence length ‘0 and line charge density +, exhibiting various scaling regimes. High-salt concentration and small + correspond to the Gaussian regime, where the electrostatic interactions are irrelevant. In the persistent regime, the polymer persistence length is increased, and in the Gaussian-persistent regime the polymer forms a persistent chain of Gaussian blobs as indicated in Fig. 7. The broken line indicates the Manning condensation, at which counterions condense on the polymer and reduce the e8ective polymer line charge density. We use a log–log plot, and the various power-law exponents for the crossover boundaries are denoted by numbers.

Combining Eqs. (3.13) and (3.11) gives the e8ective electrostatic persistence length for a string of electrostatic blobs, ‘KK

‘B1=3 +2=3 ‘02=3 2

:

(3.14)

This electrostatic sti8ening is only relevant for the so-called Gaussian-persistent regime valid for ‘KK ¿ Rel , or equivalently + ‘B ‘0 ¿ (‘0 )3=2 : (3.15) When this inequality is inverted the Gaussian persistence regime crosses over to the Gaussian one. The crossover boundaries (3.9), (3.12), (3.15) between the various scaling regimes are summa√ rized in Fig. 8. We obtain three distinct regimes. In the persistent regime, for + ‘ ‘ B 0 ¿ ‘0 and √ + ‘B ‘0 ¿ 1, the polymer takes on a rod-like structure with an e8ective persistence length given by the OSF expression, and larger√than the bare persistence length Eq. (3.11). In the Gaussian-persistent √ regime, for + ‘B ‘0 ¡ 1 and + ‘B ‘0 ¿ (‘0 )3=2 , the polymer consists of a linear array of Gaussian electrostatic blobs, as shown in Fig. 7, with an e8ective persistence length ‘KK larger √ than the electrostatic blob size and given by Eq. (3.14). Finally, in the Gaussian regime, for + ‘B ‘0 ¡ (‘0 )3=2 √ and + ‘B ‘0 ¡ ‘0 , the electrostatic repulsion does not lead to sti8ening e8ects at any length scale (though the chain will be non-ideal). The persistence length ‘KK was also obtained from Monte-Carlo simulations with parameters similar to the ones used for the snapshot shown in Fig. 5d, where chain crumpling at small length scales and chain sti8ening at large length scales occur simultaneously [26–29]. However, extremely


23

long chains are needed in order to obtain reliable results for the persistence length, since the sti8ening occurs only at intermediate length scales and, therefore, 6tting of the tangent–tangent correlation function is non-trivial. Whereas previous simulations for rather short chains point to a di8erent scaling than in Eq. (3.14), with a dependence on the screening length closer to a linear one, in qualitative agreement with experimental results [10], more recent simulations for very long chains exhibit a persistence length in agreement with Eq. (3.14) [30,31]. The situation is complicated by the fact that recent theories for the single PE chain make con1icting predictions, some con6rming the simple scaling results described in Eqs. (3.11) and (3.14) [23,32,33], while others con6rming Eq. (3.11) but disagreeing with Eq. (3.14) [22,34,35]. This issue is not resolved and still under intense current investigation. For multivalent counterions 1uctuation e8ects can even give rise to a PE collapse purely due to electrostatic interactions [36–41], which is accompanied by a negative contribution to the e8ective persistence length [42–46]. A related issue is the e8ective interaction between highly charged parallel rods, which has been shown to become attractive in the presence of multivalent counterions [47–51]. 3.3. Manning condensation A peculiar phenomenon occurs for highly charged PEs and is known as the Manning condensation of counterions [52–55]. Strictly speaking, this phenomenon constitutes a true phase transition only in the absence of any added salt ions. For a single rigid PE chain represented by an in6nitely long and straight cylinder with a linear charge density larger than ‘B +z = 1 ;

(3.16)

where z is the counterion valency, it was shown that counterions condense on the oppositely charged cylinder in the limit of in6nite solvent dilution. Namely, in the limit where the inter-chain distance tends to in6nity. This is an e8ect which is not captured by the linear Debye–HKuckel theory used in the last section to calculate the electrostatic persistence length Eq. (3.11). A simple heuristic way to incorporate the non-linear e8ect of Manning condensation is to replace the bare linear charge density + by the renormalized one +renorm =1=(z‘B ) whenever ‘B +z ¿ 1 holds. This procedure, however, is not totally satisfactory at high-salt concentrations [56,57]. Also, real polymers have a 6nite length, and are neither completely straight nor in the in6nite dilution limit [58–60]. Still, Manning condensation has an experimental signi6cance for polymer solutions [61–63] because thermodynamic quantities, such as counterion activities [64] and osmotic coeIcients [65], show a pronounced signature of Manning condensation. Locally, polymer segments can be considered as straight over length scales comparable to the persistence length. The Manning condition Eq. (3.16) usually denotes a region where the binding of counterions to charged chain sections begins to deplete the solution from free counterions. Within the scaling diagram of Fig. 8, the Manning threshold (denoted by a vertical broken line) is reached typically for charge densities larger than the one needed to straighten out the chain. This holds for monovalent ions provided ‘0 ¿ ‘B , as is almost always the case. The Manning condensation of counterions will therefore not have a profound in1uence on the local chain structure since the chain is rather straight already due to monomer–monomer repulsion. A more complete description of various scaling regimes related to Manning condensation, chain collapse and chain swelling has recently been given in Ref. [66].

24


3.4. Self-avoidance and polyelectrolyte chain conformations Let us now consider how the √ self-avoidance of PE chains comes into play, concentrating on the persistent regime de6ned by + ‘B ‘0 ¿ 1. The end-to-end radius R of a strongly charged PE chain shows three distinct scaling ranges. For a chain length L smaller than the e8ective persistence length ‘e8 , which according to Eq. (3.10) is the sum of the bare and electrostatic persistence lengths, R grows linearly with the length, R ∼ L. Self-avoidance plays no role in this case, because the chain is too short to fold back on itself. For much longer chains, L‘e8 , we envision a single polymer coil as a solution of separate polymer pieces of length ‘e8 , and treat their interactions using a virial expansion. The second 2 virial coeIcient v2 of a rod of length ‘e8 and diameter d scales as v2 ∼ ‘e8 d [67,68]. The chain connectivity is taken into account by adding the entropic chain elasticity as a separate term. The standard Flory theory [12] (see Section 2.2) modi6ed to apply to a semi-1exible chain is based on writing the free energy F (in units of kB T ) as a sum of two terms R2 L=‘e8 2 3 + v2 R F ; (3.17) ‘e8 L R3 where the 6rst term is the entropic elastic energy associated with swelling a semi-1exible polymer chain to a radius R and the second term is the second-virial repulsive energy proportional to the coeIcient v2 and the segment density squared. It is integrated over the volume R3 . The optimal radius R is calculated by minimizing this free energy and gives the swollen radius R ∼ (v2 =‘e8 )1=5 L& ;

(3.18)

with & = 3=5 which is the semi-1exible analogue of Eq. (2.8). This radius is only realized above a 7 3 =v22 ∼ ‘e8 =d2 . For elongated segments with ‘e8 d, or, equivaminimal chain length L ¿ Lsw ‘e8 lently, for a highly charged PE, we obtain an intermediate range of chain lengths ‘e8 ¡ L ¡ Lsw for which the chain is predicted to be Gaussian and the chain radius scales as 1=2 1=2 R ∼ ‘e8 L :

(3.19)

For charged chains, the e8ective rod diameter d is given in low-salt concentrations by the screening length, i.e. d ∼ −1 plus logarithmic corrections [67,68]. √ The condition to have a Gaussian scaling regime, Eq. (3.19), thus becomes ‘e8 −1 . For the case + ‘B ‘0 ¡ 1, where the chain crumples and locally forms Gaussian blobs, a similar calculation to the one outlined here leads to the condition ‘KK ¿ −1 in order to see a Gaussian regime between the persistent and the swollen one. Within the persistent and the Gaussian-persistent scaling regimes depicted in Fig. 8 the e8ective persistence length is dominated by the electrostatic contribution and given by Eqs. (3.11) and (3.14), respectively, which in turn are always larger than the screening length −1 . It follows that a Gaussian scaling regime, Eq. (3.19), always exists between the persistent regime where R ∼ L and the asymptotically swollen scaling regime, Eq. (3.18). This situation is depicted in Fig. 7 for the Gaussian-persistent scaling regime, where the chain shows two distinct Gaussian scaling regimes at the small and large length scales. This multi-hierarchical scaling structure is only one of the many problems one faces when trying to understand the behavior of PE chains, be it experimentally, theoretically, or by simulations. A di8erent situation occurs when the polymer backbone is under bad-solvent conditions, in which case an intricate interplay between electrostatic chain swelling and short-range collapse occurs [69].


25

Quite recently, this interplay was theoretically shown to lead to a Rayleigh instability in the form of a necklace structure consisting of compact globules connected by stretched chain segments [70–74]. Small-angle X-ray scattering on solvophobic PEs in a series of polar organic solvents of various solvent quality could qualitatively con6rm these theoretical predictions [75]. 3.5. Dilute polyelectrolyte solutions In accordance with our discussion for neutral chains in Section 2.4, the dilute regime is de6ned by cm ¡ cm∗ , where cm denotes the monomer concentration (per unit volume) and cm∗ is the concentration where individual chains start to overlap. Using Eq. (2.19), for rigid polymers with & = 1 the overlap concentration scales as cm∗ ∼ a−3 N −2 and decreases strongly as N increases. This means that the dilute regime for semi-1exible PE chains corresponds to extremely low monomer concentrations. For example taking a Kuhn length a ≈ 0:25 nm (corresponding to the projected length of two carbon bonds) and a polymerization index of N = 104 , the overlap concentration becomes cm∗ ≈ 6 × 10−7 nm−3 ≈ 10−3 mM, which is a very small concentration. The osmotic pressure (rescaled by kB T ) in the dilute regime in the limit cm → 0 is given by fcm cm + ; (3.20) z N and consists of the ideal pressure of non-interacting counterions (6rst term) and polymer coils (second term). Note that since the second term scales as N −1 , it is quite small for large N and can be neglected. Hence, the main contribution to the osmotic pressure comes from the counterion entropy. This entropic term explains also why charged polymers can be dissolved in water even when their backbone is quite hydrophobic. Precipitation of the PE chains will also mean that the counterions are con6ned within the precipitate. The entropy loss associated with this con6nement is too large and keeps the polymers dispersed in solution. In contrast, for neutral polymers there are no counterions in solution. Only the second term in the osmotic pressure exists and contributes to the low osmotic pressure of these polymer solutions. In addition, this explains the trend towards precipitation even for very small attractive interactions between neutral polymers: The entropic pressure scale as cm =N , while the enthalpic pressure which favors precipitation scales as −cm2 with no additional N dependence, thus dominating the entropic term for large N [14]. (=

3.6. Semi-dilute polyelectrolyte solutions In the semi-dilute concentration regime, cm ¿ cm∗ , di8erent polymer coils are strongly overlapping, but the polymer solution is still far from being concentrated. This means that the volume fraction of the monomers in solution is much smaller than unity, a3 cm 1. In this concentration range, the statistics of counterions and polymer 1uctuations are intimately connected. One example where this feature is particularly prominent is furnished by neutron and X-ray scattering from semi-dilute PE solutions [76–82]. The structure factor S(q) shows a pronounced peak, which results from a competition between the connectivity of polymer chains and the electrostatic repulsion between charged monomers, as will be discussed below. An important length scale, schematically indicated in Fig. 9, is the mesh-size or correlation length 'b , which measures the length below which entanglement e8ects between di8erent chains are unimportant. The mesh size can be viewed as the polymer (blob)


+ +

-

+

-

-

+

-

+

+

-

-

-

+

+

+

26

-

+ +

+

-

-

-

-

-

-

-

+

-

+

-

+

+

-

-

+

+

+

+

+ +

+

+

-

+

+

+

+

+

-

-

+

-

+

-

+ +

+ +

+

+

+

+

+

+

+

+

+

+

-

+

+

-

sd OSF

Fig. 9. Schematic view of the PE chain structure in the semi-dilute concentration range. The mesh size 'b is about equal sd to the persistence length ‘OSF and to the screening length −1 (if no salt is added to the system).

scale below which single-chain statistics are valid. A semi-dilute solution can be roughly thought of as being composed of a close-packed array of polymer blobs of size 'b . The starting point for the present discussion is the screened interaction between two charges immersed in a semi-dilute PE solution containing charged polymers, their counterions and, possibly, additional salt ions. Screening in this case is produced not only by the ions, but also by the charged chain segments which can be easily polarized and shield any free charges. Using the random-phase approximation (RPA), the e8ective Debye–HKuckel (DH) interaction can be written in Fourier space as [83,84] vRPA (q) =

cm

f2 S

1 + v2 cm S0 (q) ; −1 −1 0 (q) + vDH (q) + v2 cm vDH (q)S0 (q)

(3.21)

recalling that cm is the average density of monomers in solution and f is the fraction of charged monomers on the PE chains. The second virial coeIcient of non-electrostatic monomer–monomer interactions is v2 and the single-chain form factor (discussed below) is denoted by S0 (q). In the case where no chains are present, cm = 0, the RPA expression reduces to vRPA (q) = vDH (q), the Fourier-transform of the Debye–HKuckel potential of Eq. (3.3), given by vDH (q) =

40‘B : + 2

q2

(3.22)


27

2 As before, −1 is the DH screening length, which is due to all mobile ions. We can write 2 =salt + 2 2 40‘B fcm , where salt = 80z ‘B csalt describes the screening due to added z:z salt of concentration csalt , and the second term describes the screening due to the counterions of the PE monomers. Within the same RPA approximation the monomer–monomer structure factor S(q) of a polymer solution with monomer density cm is given by [83,84]

S −1 (q) = f2 vDH (q) + S0−1 (q)=cm + v2 :

(3.23)

The structure factor (or scattering function) depends only on the form factor of an isolated, noninteracting polymer chain, S0 (q), the second virial coeIcient v2 , the fraction f of charged monomers, and the interaction between monomers, which in the present case is taken to be the Debye–HKuckel potential vDH (q). The structure factor of a non-interacting semi-1exible polymer is characterized, in addition to the monomer length b, by its persistence length ‘e8 . In general, this form factor is a complicated function which cannot be written down in closed form [15,85]. However, one can separate between three di8erent ranges of wavenumbers q, and within each range the form factor shows a rather simple scaling behavior, namely  −1 N for q2 ¡ 6=Nb‘e8 ;    2 S0−1 (q) q2 b‘e8 =6 for 6=Nb‘e8 ¡ q2 ¡ 36=02 ‘e8 (3.24) ;    2 qb=0 for 36=02 ‘e8 ¡ q2 : For small wavenumbers the polymer acts like a point scatterer, while in the intermediate wavenumber regime the polymer behaves like a 1exible, Gaussian polymer, and for the largest wavenumbers the polymer can be viewed as a sti8 rod. One of the most interesting features of semi-dilute PE solutions is the fact that the structure factor S(q) shows a pronounced peak. For weakly charged PEs, the peak position scales as q ∼ cm1=4 with the monomer density [79], in agreement with the above random-phase approximation (RPA) [83,84] and other theoretical approaches [86,87]. We now discuss the scaling of the characteristic scattering peak within the present formalism. The position of the peak follows from the inverse structure factor, Eq. (3.23), via 9S −1 (q)=9q = 0, which leads to the equation 1=2 80q‘B f2 cm 2 2 : (3.25) q + salt + 40‘B fcm = 9S0−1 (q)=9q In principle, there are two distinct scaling behaviors possible for the peak, depending on whether the chain form factor of Eq. (3.24) exhibits 1exible-like or rigid-like scaling [88]. We concentrate now on the 1exible case, i.e. the intermediate q-range in Eq. (3.24). A peak is only obtained if the 2 left-hand side of Eq. (3.25) is dominated by the q-dependent part, i.e. if q2 ¿ salt + 40‘B fcm . In this case, the peak of S(q) scales as 1=4 240‘B f2 cm ∗ ; (3.26) q b‘e8 in agreement with experimental results. In Fig. 10a we show density-normalized scattering curves for a PE solution characterized by the persistence length ‘e8 = 1 nm (taken to be constant and thus independent of the monomer concentration), with monomer length b=0:38 nm (as appropriate for Poly-DADMAC), polymerization

28

R.R. Netz, D. Andelman / Physics Reports 380 (2003) 1 – 95 30

S(q)/cm

20

10

0 0

1

2

3

4

5

3

4

5

q [nm-1]

(a) 20

qS(q)/cm

15

10

5 0 0 (b)

1

2

q

[nm-1]

Fig. 10. (a) RPA prediction for the rescaled structure factor S(q)=cm of a semi-dilute PE solution with persistence length ‘e8 = 1 nm, monomer length b = 0:38 nm, polymerization index N = 500 and charge fraction f = 0:5 in the salt-free case. The monomer densities are (from bottom to top) cm = 1 M; 0:3 M; 10 mM; 3 mM; 1 mM; 0:3 mM. (b) For the same series of cm values as in (a) the structure factor is multiplied by the wavenumber q. The semi-1exibility becomes more apparent because for large q the curves tend towards a constant.

index N = 500, charge fraction f = 0:5 and with no added salt. As the monomer density decreases (bottom to top in the 6gure), the peak moves to smaller wavenumbers and sharpens, in agreement with previous implementations of the RPA. In Fig. 10b we show the same data in a di8erent representation. Here we clearly demonstrate that the large-q region already is dominated by the 1=q behavior of the single-chain structure factor, S0 (q). Since neutron scattering data easily extend to wavenumbers as high as q ∼ 5 nm−1 , the sti8-rod like behavior in the high q-limit, exhibited on such a plot, will be important in interpreting and 6tting experimental data even at lower q-values. In a semi-dilute solution there are three di8erent, and in principle, independent length scales: The mesh size 'b , the screening length −1 , and the persistence length ‘e8 . In the absence of added salt, the screening length scales as −1 ∼ (‘B fcm )−1=2 :

(3.27)

Assuming that the persistence length is larger or of the same order of magnitude as the mesh size, as is depicted in Fig. 9, the polymer chains can be thought of as straight segments between di8erent cross-links. Denoting the number of monomers inside a correlation blob as g, this means


29

that 'b ∼ bg. The average monomer concentration scales as cm ∼ g='3b , from which we conclude that 'b ∼ (bcm )−1=2 :

(3.28)

Finally, the persistence length within a semi-dilute PE solution can be calculated by considering the electrostatic energy cost for slightly bending a charged rod. In PE solutions, it is important to include in addition to the screening by salt ions also the screening due to charged chain segments. This can be calculated by using the RPA interaction, Eq. (3.21). Since the screening due to polymer chains is scale dependent and increases for large separations, a q-dependent instability is encountered and leads to a persistence length [88] sd ‘OSF ∼ (bcm )−1=2 ;

(3.29)

where the ‘sd’ superscript stands for ‘semi-dilute’. This result is a generalization of the OSF result for a single chain, Eq. (3.11), and applies to semi-dilute solutions. Comparing the three lengths, we see that

‘B f − 1 sd : 'b ∼ ‘OSF ∼ (3.30) b Since the prefactor ‘B f=b for synthetic fully charged polymers is roughly of order unity, one 6nds that for salt-free semi-dilute PE solutions, all three length-scales scale in the same way with cm , namely as ∼ cm−1=2 . This scaling relation has been found 6rst in experiments [76–78] and was later con6rmed by theoretical calculations [89,90]. The screening e8ects due to neighboring PE chains, which form the basis for the reduction of the electrostatic PE sti8ness in a semi-dilute solution, have also been observed in computer simulations [91–94]. 4. General considerations on adsorption 4.1. Adsorption and depletion Polymers can adsorb spontaneously from solution onto surfaces if the interaction between the polymer and the surface is more favorable than that of the solvent with the surface. For example, a charged polymer like poly-styrene-sulfonate (PSS) is soluble in water but will adsorb on various hydrophobic surfaces and on the water/air interface [95]. This is the case of equilibrium adsorption where the concentration of the polymer monomers increases close to the surface with respect to their concentration in the bulk solution. We discuss this phenomenon at length both on the level of a single polymer chain (valid only for extremely dilute polymer solutions), Sections 5 and 6, and for polymers adsorbing from (semi-dilute) solutions, Sections 7 and 8. In Fig. 11a we show schematically the volume fraction pro6le ,(x) of monomers as a function of the distance x from the adsorbing substrate. In the bulk, namely far away from the substrate surface, the volume fraction of the monomers is ,b , whereas at the surface, the corresponding value is ,s ¿ ,b . The theoretical models address questions in relation to the polymer conformations at the interface, the local concentration of polymer in the vicinity of the surface and the total amount of adsorbing polymer chains. In turn, the knowledge of the polymer interfacial behavior is used to calculate thermodynamical properties like the surface tension in the presence of polymer adsorption.

30

R.R. Netz, D. Andelman / Physics Reports 380 (2003) 1 – 95 φ(x) φs

φb x (a)

φ(x)

φb

φs x (b)

Fig. 11. Schematic pro6le of the monomer volume fraction ,(x) as a function of the distance x from a 1at substrate as appropriate (a) for the case of adsorption, where the substrate attracts monomers, leading to an increase of the polymer concentration close to the surface; and, (b) for the case of depletion, where the substrate repels the monomers leading to a depression of the polymer concentration close to the surface. The bulk volume fraction, i.e., the monomer volume fraction in6nitely far away from the surface is denoted by ,b , and ,s denotes the surface volume fraction right at the substrate surface.

The opposite case of depletion can occur when the monomer–surface interaction is less favorable than the solvent–surface interaction, as entropy of mixing will always disfavor adsorption. This is, e.g., the case for polystyrene in toluene which is depleted from a mica substrate [96]. The depletion layer is de6ned as the layer adjacent to the surface from which the polymer is depleted. The concentration in the vicinity of the surface is lower than the bulk value, as shown schematically in Fig. 11b. 4.2. Surface characteristics Clearly, any adsorption process will be sensitive to the type of surface and its internal structure. As a starting point for adsorption problems we assume that the solid surface is atomically smooth,


31

(a)

(b)

(c)

(d) oil water

(e) water membrane (f)

water

Fig. 12. Di8erent possibilities of substrates: (a) the prototype, a 1at, homogeneous substrate; (b) a corrugated, rough substrate. Note that experimentally, every substrate exhibits a certain degree of roughness on some length scale; (c) a spherical adsorption substrate, such as a colloidal particle. If the colloidal radius is much larger than the polymer size, curvature e8ects (which means the deviation from the planar geometry) can be neglected; (d) a 1at but chemically heterogeneous substrate; (e) a liquid/liquid “soft” interface. For example between water and oil; (f) a lipid bilayer (membrane) which can have both shape undulations and lateral composition variations.

1at, and homogeneous, as shown in Fig. 12a. This ideal solid surface is impenetrable to the chains and imposes on them a surface interaction. The surface potential can be either short-ranged, affecting only monomers which are in direct contact with the substrate or in close vicinity of the surface. The surface can also have a longer range e8ect, like van der Waals, or electrostatic interactions if it is charged. Interesting extensions beyond ideal surface conditions are expected in several cases: (i) rough or corrugated surfaces, such as depicted in Fig. 12b; (ii) surfaces that are curved, e.g., adsorption on spherical colloidal particles, see Fig. 12c; (iii) substrates which are chemically inhomogeneous, i.e., which show some lateral organization, as shown schematically in Fig. 12d; (iv) polymer adsorbing on “soft” and “1exible” interfaces between two immiscible 1uids or at the liquid/air surface, Fig. 12e; and, (v) surfaces that have internal degrees of freedom like surfactant monolayers or amphiphilic bilayer (membrane), Fig. 12f. We brie1y mention those situations in Sections 11 and 12.

32


(b)

(a)

tail

(c)

tail

B

loop A

train

Fig. 13. The di8erent adsorption mechanisms discussed in this chapter: (a) adsorption of a homopolymer, where each monomer has the same interaction with the substrate. The ‘tail’, ‘train’ and ‘loop’ sections of the adsorbing chain are shown; (b) grafting of an end-functionalized polymer via a chemical or a physical bond, and; (c) adsorption of a diblock copolymer where one of the two block is attached to the substrate surface, while the other is not.

4.3. Surface–polymer interactions Equilibrium adsorption of polymers is only one of the methods used to create a change in the polymer concentration close to a surface. For an adsorbed polymer, it is interesting to look at the detailed conformation of a single polymer chain at the substrate. One distinguishes polymer sections that are bound to the surface (trains), sections that form loops, and end sections that can form dangling tails. This is schematically depicted in Fig. 13a. We mention two other methods to produce polymer layers at surfaces for polymers which do not adsorb spontaneously on a given surface. (i) In the 6rst method, the polymer is chemically attached (grafted) to the surface by one of the chain ends, as shown in Fig. 13b. In good solvent conditions the polymer chains look like “mushrooms” on the surface when the distance between grafting points is larger than the typical size of the chains. In some cases, it is possible to induce a much higher grafting density, resulting in a polymer “brush” extending in the perpendicular direction from the surface, as is discussed in detail in Section 13. (ii) A variant on the grafting method is to use a diblock copolymer made out of two distinct blocks, as shown in Fig. 13c. The 6rst block is insoluble and is attracted to the substrate. Thus, it acts as an “anchor” 6xing the chain to the surface; it is drawn as a thick line in Fig. 13c. It should be long enough to cause irreversible 6xation on the surface. The other block is a soluble one (the “buoy”), forming the brush layer (or “mushroom”). For example, 6xation on hydrophobic surfaces from a water solution can be made using a polystyrene–polyethylene oxide (PS–PEO) diblock copolymer. The PS block is insoluble in water and attracted towards the substrate, whereas the PEO forms the brush layer. The process of diblock copolymer 6xation has a complex dynamics during the formation stage but is very useful in applications [97]. A related application is to employ a polyethylene glycol (PEG) polymer connected to a lipid (PEG–lipid) chain and use the lipid to anchor the PEG chain onto a lipid membrane [98]. There are a variety of other adsorption phenomena not discussed in this review. For example the in1uence of di8erent polymer topologies on the adsorption characteristics. In Ref. [99] the adsorption of star polymers, where a number of polymer chains are connected to one center, is discussed. The adsorption of ring polymers has also received considerable attention [100,101]. Another important


33

class of polymers is made up of random copolymers, which are used to manipulate the interfacial properties of a variety of systems. The adsorption of such random copolymers has been studied at solid substrates [102–104] and at penetrable interfaces [105–109].

5. Adsorption of a single neutral chain Let us consider now the interaction of a single polymer chain with a solid substrate. The main e8ects particular to the adsorption of polymers (as opposed to the adsorption of simple molecules) are due to the reduction of conformational states of the polymer at the substrate, which is caused by the impenetrability of the substrate for monomers [110–115]. The second factor determining the adsorption behavior is the substrate–monomer interaction. Typically, for the case of an adsorbing substrate, the interaction potential V (x) (measured in units of kB T ) between the substrate and a single monomer has a form similar to the one shown in Fig. 14, where x measures the distance of the monomer from the substrate surface,  ∞ for x ¡ 0 ;    for 0 ¡ x ¡ B ; V (x) −V0 (5.1)    −wx−+ for x ¿ B : The separation of V (x) into three parts is done for convenience. It consists of a hard wall at x = 0, which embodies the impenetrability of the substrate, i.e., V (x) = ∞ for x ¡ 0. For positive x we V(x)

B

x

-wx −τ -V 0

Fig. 14. A typical surface potential felt by a monomer as a function of the distance x from an adsorbing surface. First the surface is impenetrable. Then, the attraction is of strength V0 and range B. For separations larger than B, typically a long-ranged tail exists and is modelled by −wx−+ .

34


B≈D (a)

D B (b)

Fig. 15. Schematic drawing of single-chain adsorption. (a) In the limit of strong coupling, the polymer decorrelates into a number of blobs (shown as dotted circles) and the chain is con6ned to a layer thickness D, of the same order of magnitude as the potential range B; (b) in the case of weak coupling, the width of the polymer layer D is much larger than the interaction range B and the polymer forms large blobs, within which the polymer is not perturbed by the surface.

assume the potential to be given by an attractive well of depth V0 and width B. At large distances, x ¿ B, the potential can be modelled by a long-ranged attractive tail decaying as V (x) ∼ −wx−+ . For the important case of (non-retarded) van-der-Waals interactions between the substrate and the polymer monomers, the potential shows a decay governed by the exponent + = 3 and can be attractive or repulsive, depending on the solvent, the chemical nature of the monomers and the substrate material. The decay power + = 3 follows from the van-der-Waals pair interaction, which decays as the inverse sixth power with distance, by integrating over the three spatial dimensions of the substrate, which is supposed to be a semi-in6nite half-space [116]. The strength of the potential well is measured by V0 , i.e., by comparing the potential depth with the thermal energy kB T . For strongly attractive potentials, i.e., for V0 large or, equivalently, for low temperatures, the polymer is strongly adsorbed and the thickness of the adsorbed layer, D, approximately equals the potential range B. The resulting polymer structure is shown in Fig. 15a, where the width of the potential well, B, is denoted by a broken line. For weakly attractive potentials, or for high temperatures, we anticipate a weakly adsorbed polymer layer, with a di8use layer thickness D much larger than the potential range B. This structure is depicted in Fig. 15b. For both cases shown in Fig. 15, the polymer conformations are unperturbed on a spatial scale of the order of D; on larger length scales, the polymer is broken up into decorrelated polymer blobs [14,15], which are denoted by dotted circles in Fig. 15. The idea of introducing polymer blobs is related to the fact that very long and 1exible chains have di8erent spatial arrangement at small and large length scales. Within each blob the short range interaction is irrelevant, and the polymer structure inside the blob is similar to the structure of an unperturbed polymer far from the surface. Since all monomers are connected, the blobs themselves are linearly connected and their spatial arrangement represents the behavior on large length scales. In the adsorbed state, the formation of each blob leads to an entropy loss of the order of one kB T (with a numerical prefactor of order unity that is neglected in this scaling argument), so the total entropy loss of a chain of N monomers is Frep ∼ N=g in units of kB T , where g denotes the number of monomers inside each blob. Using the scaling relation D ag& for the blob size dependence on the number of monomers g, Eq. (2.8), the entropy penalty for the con6nement of a polymer chain to a width D above the


surface can be written as [117]: a 1=& Frep N : D

35

(5.2)

The adsorption behavior of a polymer chain results from a competition between the attractive potential V (x), which tries to bind the monomers to the substrate, and the entropic repulsion Frep , which tries to maximize entropy, and favors a delocalized state where a large fraction of monomers is located farther away from the surface. It is of interest to compare the adsorption of long-chain polymers with the adsorption of small molecular solutes. Small molecules adsorb onto a surface only if there is a bulk reservoir with non-zero concentration in equilibrium with the surface. An in6nite polymer chain N → ∞ behaves di8erently as it remains adsorbed also in the limit of zero bulk concentration. This corresponds to a true thermodynamic phase transition in the limit N → ∞ [118]. For 6nite polymer length, however, the equilibrium adsorption resembles that of small molecules. Only a non-zero bulk polymer concentration will lead to adsorption of 6nite-length polymer chains on the substrate. Indeed, as all real polymers are of 6nite length, the adsorption of single polymers is never observed in practice. However, for fairly long polymers, the desorption of a single polymer is almost a ‘true’ phase transition, and corrections due to 6nite (but long) polymer length are often below experimental resolution. 5.1. Mean-?eld regime Fluctuations of the local monomer concentration are of importance for polymers at surfaces because of the large number of possible chain conformations. These 1uctuations are treated theoretically using 6eld-theoretic or transfer-matrix techniques. In a 6eld-theoretic formalism, the problem of accounting for di8erent polymer conformations is converted into a functional integral over di8erent monomer-concentration pro6les [15]. Within transfer-matrix techniques, the Markov-chain property of ideal polymers is exploited to re-express the conformational polymer 1uctuations as a product of matrices [119]. However, there are cases where 1uctuations in the local monomer concentration become unimportant. Then, the adsorption behavior of a single polymer chain is obtained using simple mean-?eld theory arguments. Mean-6eld theory is a very useful approximation applicable in many branches of physics, including polymer physics. In a nutshell, each monomer is placed in a “6eld”, generated by the external potential plus the averaged interaction with all the other monomers. The mean-6eld theory can be justi6ed for two cases: (i) a strongly adsorbed polymer chain, i.e., a polymer chain which is entirely con6ned inside the potential well; and, (ii) the case of long-ranged attractive surface potentials. To proceed, we assume that the adsorbed polymer layer is con6ned with an average thickness D, as depicted in Fig. 15a or b. Within mean-6eld theory, the polymer chain feels an average of the surface potential, V (x), which is replaced by the potential evaluated at the average distance from the surface, x D=2. Therefore, V (x) V (D=2). Further stringent conditions when such a mean-6eld theory is valid are detailed below. The full free energy of one chain, F, of polymerization index N , can be expressed as the sum of the repulsive entropic term,

36


Eq. (5.2), and the average potential a 1=& F N + NV (D=2) : (5.3) D Let us consider 6rst the case of a strongly adsorbed polymer, con6ned to a potential well of depth ∼ V0 . In this case the potential energy per monomer becomes V (D=2) −V0 . Comparing the repulsive entropic term with the potential term, we 6nd the two terms to be of equal strength for a well depth V0∗ (a=D)1=& . Hence, the strongly adsorbed state, which is depicted in Fig. 15a, should be realized for a high attraction strength V0 ¿ V0∗ . For intermediate attraction strength, V0 ≈ V0∗ , the adsorbed chain will actually be adsorbed in a layer of width D larger than the potential width B, as shown in Fig. 15b, which will be discussed further below. Since the potential depth V0 is measured in units of kB T , it follows that at high temperatures it becomes increasingly diIcult to con6ne the chain. This can be seen from expressing the bare potential depth as V˜ 0 = kB TV0 , so that the critical potential depth becomes V˜ ∗0 kB T (a=D)1=& and thus increases linearly with temperature. In fact, for an ideal chain, with & = 1=2, the resulting scaling relation for the critical well depth, V0∗ ∼ (a=D)2 , agrees with exact transfer-matrix predictions for the adsorption threshold in a squarewell potential [120]. We turn now to the case of a weakly adsorbed polymer layer. The potential depth is smaller than the threshold, i.e., V0 ¡ V0∗ , and the stability of the weakly adsorbed polymer chain, depicted in Fig. 15b, has to be examined. The thickness D of this polymer layer follows from the minimization of the free energy, Eq. (5.3), with respect to D, where we use the asymptotic form of the surface potential, Eq. (5.1), for large separations. The result is 1=& &=(1−&+) a : (5.4) D w Under which circumstances is the prediction Eq. (5.4) correct, at least on a qualitative level? It turns out that the prediction for D, Eq. (5.4), obtained within the simple mean-6eld theory, is correct if the attractive tail of the substrate potential in Eq. (5.1) decays for large values of x slower than the entropic repulsion in Eq. (5.2) [121]. In other words, the mean-6eld theory is valid for weakly adsorbed polymers only for + ¡ 1=&. This can already be guessed from the functional form of the layer thickness, Eq. (5.4), because for + ¿ 1=& the layer thickness D goes to zero as w diminishes. Clearly an unphysical result. For ideal polymers (theta solvent, & = 1=2), the validity condition is + ¡ 2, whereas for swollen polymers (good solvent conditions, & = 3=5), it is + ¡ 5=3. For most interactions (including van der Waals interactions with + = 3) this condition on + is not satis6ed, and 1uctuations are in fact important, as is discussed in the next section. There are two notable exceptions. The 6rst is for charged polymers close to an oppositely charged surface, in the absence of salt ions. Since the attraction of the polymer to an in6nite, planar and charged surface is linear in x, the interaction is described by Eq. (5.1) with an exponent + = −1, and the inequality + ¡ 1=& is satis6ed. For charged surfaces, Eq. (5.4) predicts the thickness D to increase to in6nity as the temperature increases or as the attraction strength w (proportional to the surface charge density) decreases. The resultant exponents for the scaling of D follow from Eq. (5.4) and are D ∼ w−1=3 for ideal chains, and D ∼ w−3=8 for swollen chains [122,123]. This case will be considered in more detail in Section 6. A second example where the mean-6eld theory can be used is the adsorption of polyampholytes on charged surfaces [124,125]. Polyampholytes are polymers consisting of negatively and positively


37

charged monomers. In cases where the total charge on such a polymer adds up to zero, it might seem that the interaction with a charged surface should vanish. However, it turns out that local charge 1uctuations (i.e., local spontaneous dipole moments) lead to a strong attraction of polyampholytes to charged substrates. In the absence of salt this attractive interaction has an algebraic decay with an exponent + = 2 [124]. On the other hand, in the presence of salt, the e8ective interaction is exponentially screened, yielding a decay faster than the 1uctuation repulsion, Eq. (5.2). Nevertheless, the mean-6eld theory, embodied in the free energy expression Eq. (5.3), can be used to predict the adsorption phase behavior within the strongly adsorbed case (i.e., far from any desorption transition) [126–129]. 5.2. Fluctuation dominated regime Here we consider the weakly adsorbed case for substrate potentials which decay (for large separations from the surface) faster than the entropic repulsion Eq. (5.2), i.e., + ¿ 1=&. This applies, e.g., to van-der-Waals attractive interaction between the substrate and monomers, screened electrostatic interactions, or any other short-ranged potential. In this case, 1uctuations play a decisive role. In fact, for ideal chains, it can be rigorously proven (using transfer-matrix techniques) that all potentials decaying faster than x−2 for large x have a continuous adsorption transition at a 6nite critical temperature T ∗ [121]. This means that the thickness of the adsorbed polymer layer diverges as D ∼ (T ∗ − T )−1

(5.5)

for T → T ∗ [130]. The power law divergence of D is universal. Namely, it does not depend on the speci6c functional form and strength of the potential as long as they satisfy the above condition. The case of non-ideal chains is much more complicated [131]. First progress has been made by de Gennes who recognized the analogy between the partition function of a self-avoiding chain and the correlation function of an n-component spin model in the zero-component (n → 0) limit [132]. The adsorption behavior of non-ideal chains has been treated by 6eld-theoretic methods using the analogy to surface critical behavior of magnets (again in the n → 0 limit) [2,133]. The resulting behavior is similar to the ideal-chain case and shows an adsorption transition at a 6nite temperature, and a continuous increase towards in6nite layer thickness characterized by a power law divergence as function of T − T ∗ [133]. The complete behavior for ideal and swollen chains can be described using scaling ideas in the following way. The entropic loss due to the con6nement of the chain to a region of thickness D close to the surface is again given by Eq. (5.2). Assuming that the adsorption layer is much thicker than the range of the attractive potential V (x), the attractive potential can be assumed to be localized at the substrate surface V (x) V (0). The attractive free energy of the chain due to the substrate surface can then be written as [134] (T ∗ − T ) Nf1 = −71 a2 Nf1 ; (5.6) T where f1 is the probability to 6nd a monomer at the substrate surface and 7˜ is a dimensionless interaction parameter. Two surface excess energies are typically being used: 71 = 7(T ˜ ∗ − T )=Ta2 is 2 the excess energy per unit area, while 71 a is the (dimensionless) excess energy per monomer at the surface. Both are positive for the attractive case (adsorption) and negative for the depletion case. Fatt −7˜

38


The dependence of 71 on T in Eq. (5.6) causes the attraction to vanish at a critical temperature, T = T ∗ , in accord with our expectations. The contact probability for a swollen chain with the surface, f1 , can be calculated as follows [135]. In order to force the chain of polymerization index N to be in contact with the surface, one of the chain ends is pinned to the substrate. The number of monomers which are in contact with the surface can be calculated using 6eld-theoretic methods and is given by N ’ , where ’ is called the surface crossover exponent [2,133]. The fraction of bound monomers follows to be f1 ∼ N ’−1 , and thus goes to zero as the polymer length increases, for ’ ¡ 1. Now instead of speaking of the entire chain, we refer to a ‘chain of blobs’ (see Fig. 15) adsorbing on the surface, each blob consisting of g monomers. We proceed by assuming that the size of an adsorbed blob D scales with the number of monomers per blob g similarly as in the bulk, D ∼ ag& , as is indeed con6rmed by 6eld theoretic calculations. The fraction of bound monomers can be expressed in terms of D and is given by (’−1)=& D f1 ∼ : a

(5.7)

Combining the entropic repulsion, Eq. (5.2), and the substrate attraction, Eqs. (5.6)–(5.7), the total free energy is given by F N

a 1=& D

−N

7(T ˜ ∗ − T) T

(’−1)=& D : a

(5.8)

Minimization with respect to D leads to the 6nal result

D a

7(T ˜ ∗ − T) T

−&=’

−&=’ a a2 71 :

(5.9)

For ideal chains, one has ’ = & = 1=2, and thus we recover the prediction from the transfer-matrix calculations, Eq. (5.5). For non-ideal chains, the crossover exponent ’ is in general di8erent from the swelling exponent &. However, extensive Monte Carlo computer simulations [133] and recent 6eld-theoretic calculations [136] point to a value for ’ close to &, such that the adsorption exponent &=’ appearing in Eq. (5.9) is close to unity, for polymers embedded in three-dimensional space. A further point which has been calculated using 6eld theory is the behavior of the monomer volume fraction ,(x) close to the substrate. Rather general arguments borrowed from the theory of critical phenomena suggest a power-law behavior for ,(x) at suIciently small distances from the substrate [133,135,137] ,(x) (x=a)−m ,s ;

(5.10)

recalling that the monomer density is related to ,(x) by cm (x) = ,(x)=a3 . In the following, we relate the so-called proximal exponent m with the two other exponents introduced above, & and ’. First note that the surface value of the monomer volume fraction, ,s = ,(x ≈ a), for one adsorbed blob follows from the number of monomers at the surface per blob, which is given by f1 g, and the cross-section area of a blob, which is of the order of D2 . The


39

surface volume fraction is given by f1 ga2 ∼ g’−2& : (5.11) D2 Using the scaling prediction Eq. (5.10), we see that the monomer volume fraction at the blob center, x D=2, is given by ,(D=2) ∼ g’−2& (D=a)−m , which (again using D ∼ ag& ) can be rewritten as ,(D=2) g’−2&−m& . On the other hand, at a distance D=2 from the surface, the monomer volume fraction should have decayed to the average monomer volume fraction a3 g=D3 ∼ g1−3& inside the blob since the statistics of the chain inside the blob is like for a chain in the bulk. By direct comparison of the two volume fractions, we see that the exponents ’ − 2& − m& and 1 − 3& have to match in order to have a consistent result, yielding ’+&−1 : (5.12) m= & For ideal chain (theta solvents), one has ’ = & = 1=2. Hence, the proximal exponent vanishes, m = 0. This means that the proximal exponent has no mean-6eld analog, explaining why it was discovered only within 6eld-theoretic calculations [2,133]. In the presence of correlations (good solvent conditions) one has ’ & 3=5 and thus m 1=3. Using D ag& and Eq. (5.9), the surface volume fraction, Eq. (5.11), can be rewritten as (’−2&)=& (2&−’)=’ D ,s ∼ ∼ a 2 71 a2 71 ; (5.13) a ,s ∼

where in the last approximation appearing in Eq. (5.13) we used the fact that ’ &. The last result shows that the surface volume fraction within one blob can become large if the adsorption energy per monomer, a2 71 , measured in units of kB T , is of order unity. Experimentally, this is often the case, and additional interactions (such as multi-body interactions) between monomers at the surface have to be taken into account. Note that the polymer concentration in the adsorbed layer can become so high that a transition into a glassy state is induced. This glassy state depends on the details of the molecular interaction, which are not considered here. It should be kept in mind that such high-concentration e8ects can slow down considerably the adsorption dynamics while prolonging equilibration times [138]. After having discussed the adsorption behavior of a single chain, a word of caution is in order. Experimentally, one never looks at single chains adsorbed to a surface. First, this is due to the fact that one always works with polymer solutions, where there is a large number of polymer chains contained in the bulk reservoir, even when the bulk monomer (or polymer) concentration is quite low. Second, even if the bulk polymer concentration is very low, and in fact so low that polymers in solution barely interact with each other, the surface concentration of polymer is enhanced relative to that in the bulk. Hence, adsorbed polymers at the surface usually do interact with neighboring chains, due to the higher polymer concentration at the surface [137]. Nevertheless, the adsorption behavior of a single chain serves as a basis and guideline for the more complicated adsorption scenarios involving many-chain e8ects. It will turn out that the scaling of the adsorption layer thickness D and the proximal volume fraction pro6le, Eqs. (5.9) and (5.10), are not a8ected by the presence of other chains. This 6nding as well as other many-chain e8ects on polymer adsorption is the subject of Section 7.

40


6. Adsorption of a single polyelectrolyte chain After reviewing bulk properties of PE solutions we address the complete adsorption diagram of a single semi-1exible PE on an oppositely charged substrate. In contrast to the adsorption of neutral polymers, the resulting phase diagram shows a large region where the adsorbed polymer is 1attened out on the substrate and creates a dense adsorption layer. The results on single PE adsorption summarized in this section are most relevant to the adsorption of highly charged synthetic PEs from dilute solutions [139–144] or the adsorption of rather sti8 charged biopolymers such as DNA [145–147]. In all these experiments, the adsorbed phases can be quite dilute, and the description of a single adsorbing polymer is a good starting point. Repeated adsorption of anionic and cationic PEs can lead to well characterized multilayers on planar [148–151] and spherical substrates [152,153]. The adsorption of a single PE chain has been treated theoretically employing a variety of methods [122,154–156]. The adsorption process results from a subtle balance between electrostatic repulsion between charged monomers, leading to chain sti8ening, and electrostatic attraction between the substrate and the polymer chain. It poses a much more complicated problem than the corresponding adsorption of neutral polymers. The adsorption of a single semi-1exible and charged chain on an oppositely charged plane [157] can be treated as a generalization of the adsorption of 1exible polymers [122]. A PE characterized by a linear charge density +, is subject to an electrostatic potential created by *, the homogeneous surface charge density (per unit area). Because this potential is attractive for an oppositely charged substrate, we consider it as the driving force for the adsorption. E8ects due to bad solvent [158] and more complex interactions are neglected. One example for the latter are interactions due to the dielectric discontinuity at the substrate surface and to the impenetrability of the substrate for salt ions. 1 Within the linearized DH theory, the electrostatic potential of a homogeneously charged plane is in units of kB T Vplane (x) = 40‘B *−1 e−x :

(6.1)

Assuming that the polymer is adsorbed over a layer of width D smaller than the screening length −1 , the electrostatic attraction force per monomer unit length can be written as (6.2) fˆ att = −40‘B *+ : For simplicity, we neglect non-linear e8ects due to counterion condensation on the PE (as obtained by the Manning theory, see Section 3.3) and on the surface (as obtained within the Gouy–Chapman theory). Although these e8ects are important for highly charged system [159], most of the important features of single PE adsorption already appear on the linearized Debye–HKuckel level. Because of the con6nement in the adsorbed layer, the polymer feels an entropic repulsion. If the layer thickness D is much smaller than the e8ective persistence length of the polymer, ‘e8 , as depicted in Fig. 16a, a new length scale, the so-called de1ection length 9, enters the description of the polymer statistics. The de1ection length 9 measures the average distance between two contact 1

An ion in solution has a repulsive interaction from the surface when the solution dielectric constant is higher than that of the substrate. This e8ect can lead to desorption for highly charged PE chains. On the contrary, when the substrate is a metal there is a possibility to induce PE adsorption on non-charged substrates or on substrates bearing charges of the same sign as the PE. See Ref. [157] for more details.


41

λ D

(a)

D

(b)

Fig. 16. (a) Schematic picture of the adsorbed polymer layer when the e8ective persistence length is larger than the layer thickness, ‘e8 ¿ D. The distance between two contacts of the polymer with the substrate, the so-called de1ection length, 1=3 scales as 9 ∼ D2=3 ‘e8 . (b) Adsorbed layer for the case when the persistence length is smaller than the layer thickness, ‘e8 ¡ D. In this case the polymer forms a random coil with many loops and a description in terms of a 1exible polymer model becomes appropriate.

points of the polymer chain with the substrate. As shown by Odijk, the de1ection length scales as 1=3 9 ∼ D2=3 ‘e8 and is larger than the layer thickness D but smaller than the persistence length ‘e8 [160]. The entropic repulsion follows in a simple manner from the de1ection length by assuming that the polymer loses roughly a free energy of one kB T per de1ection length. On the other hand, if D ¿ ‘e8 , as shown in Fig. 16b, the polymer forms a random coil with many loops within the adsorbed layer. The chain can be viewed as an assembly of decorrelated blobs, each of a chain length of L ∼ D2 =‘e8 , within which the polymer obeys Gaussian statistics. The decorrelation into blobs has an entropic cost of roughly one kB T per blob. The entropic repulsion force per polymer unit length is thus [160] −5=3 −1=3 D ‘e8 for D‘e8 ; fˆ rep ∼ (6.3) −3 for D‘e8 ; ‘e8 D where we neglected a logarithmic correction factor which is not important for our scaling arguments. As shown in the preceding section, the e8ective persistence length ‘e8 depends on the screening length and the line charge density; in essence, one has to keep in mind that ‘e8 is larger than ‘0 for a wide range of parameters because of electrostatic sti8ening e8ects. 2 The equilibrium layer thickness follows from equating the attractive and repulsive forces, Eqs. (6.2) and (6.3). For rather sti8 polymers and small layer thickness, D ¡ −1 ¡ ‘e8 , we obtain −3=5 1=3 : (6.4) D ∼ ‘B *+‘e8 2

The situation is complicated by the fact that the electrostatic contribution to the persistence length is scale dependent and decreases as the chain is bent at length scales smaller than the screening length. This leads to modi6cations of the entropic con6nement force, Eq. (6.3), if the de1ection length is smaller than the screening length. As can be checked explicitly, all results reported here are not changed by these modi6cations.

42


For a layer thickness corresponding to the screening length, D ≈ −1 , scaling arguments predict a rather abrupt desorption transition [157]. This is in accord with previous calculations [161–165] and simulations [166] for a semi-1exible polymer bound by short-ranged (square-well) potentials. Setting D ∼ −1 in Eq. (6.4), we obtain an expression for the adsorption threshold (for ‘e8 ¿ 1) *∗ ∼

5=3 1=3 +‘B ‘e8

:

(6.5)

For * ¿ *∗ the polymer is adsorbed and localized over a layer with a width smaller than the screening length (and with the condition ‘e8 ¿ −1 also satisfying D ¡ ‘e8 , indicative of a 1at layer). As * is decreased, the polymer abruptly desorbs at the threshold * =*∗ . In the Gaussian regime, the e8ective persistence length ‘e8 is given by the bare persistence length ‘0 and the desorption threshold is obtained by replacing ‘e8 by ‘0 in Eq. (6.5), i.e. *∗ ∼

5=3 +‘B ‘01=3

:

(6.6)

In the persistent regime, we have ‘e8 ∼ ‘OSF with ‘OSF given by Eq. (3.11). The adsorption threshold follows from Eq. (6.5) as *∗ ∼

7=3 +5=3 ‘B4=3

:

(6.7)

Finally, in the Gaussian-persistent regime, we have an e8ective line charge density from Eq. (3.13) and a modi6ed persistence length, Eq. (3.14). For the adsorption threshold we obtain from Eq. (6.5) ∗

* ∼

7=3 ‘05=9 +5=9 ‘B7=9

:

(6.8)

Let us now consider the opposite limit, ‘e8 ¡ −1 . 3 If the layer thickness is larger than the persistence length but smaller than the screening length, ‘e8 ¡ D ¡ −1 , the prediction for D obtained from balancing Eqs. (6.2) and (6.3) becomes ‘e8 1=3 D∼ ; (6.9) ‘B *+ which is in accord with our mean-6eld result in Eq. (5.4) for a linear potential characterized by + = −1 and an ideal polymer chain with & = 1=2. From the expression Eq. (6.9) we see that D has the same size as the screening length −1 for *∗ ∼

‘e8 3 : +‘B

(6.10)

This in fact denotes the location of a continuous adsorption transition at which the layer grows to in6nity [157]. The scaling results for the adsorption behavior of a 1exible polymer, Eqs. (6.9) and (6.10), are in agreement with previous results [155]. 3

2 From Eq. (6.4) we see that the layer thickness D is of the same order as ‘e8 for ‘B *+‘e8 ∼ 1, at which point the condition D‘e8 used in deriving Eq. (6.4) breaks down.


43

-1

-1

0

adsorbed coiled

adsorbed flat

1/3 1

-1 0

-2/3

Σ

-1

1

adsorbedcoiled

Σ

3/2

1/2 1/2 B 0

adsorbed-flat 5/21

1/3

desorbed

1

(b)

5/7

Σ

(a) -1

3/5

desorbed

Σ-2/3

Σ

1

5/7

1/2 1/2 B 0

Fig. 17. Adsorption scaling diagram shown on a log–log plot for (a) strongly charged surfaces, : = *‘03=2 ‘B1=2 ¿ 1 and for (b) weakly charged surfaces : ¡ 1. We 6nd a desorbed regime, an adsorbed phase where the polymer is 1at and dense, and an adsorbed phase where the polymer shows loops. It is seen that a fully charged PE is expected to adsorb as a 1at layer, whereas charge-diluted PEs can form coiled layers with loops and dangling ends. The broken lines denote the scaling boundaries of PE chains in the bulk as shown in Fig. 8. The numbers on the lines indicate the power law exponents of the crossover boundaries between the regimes.

In Fig. 17 we show the desorption transitions and the line at which the adsorbed layer crosses over from being 1at, D ¡ ‘e8 , to being crumpled or coiled, D ¿ ‘e8 . The underlying PE behavior in the bulk, as shown in Fig. 8, is denoted by broken lines. We obtain two di8erent phase diagrams, depending on the value of the parameter : = *‘03=2 ‘B1=2 :

(6.11)

For strongly charged surfaces, : ¿ 1, we obtain the phase diagram as in Fig. 17a, and √for weakly charged surfaces, : ¡ 1, as in Fig. 17b. We see that strongly charged PEs, obeying + ‘0 ‘B ¿ 1, always adsorb in 1at layers. The scaling of the desorption transitions is in general agreement with recent computer simulations of charged PEs [167]. Assuming an image-charge repulsion between

44


the charged monomers and the substrate, as relevant for low-dielectric substrates, some of the phase boundaries in Fig. 17 are eliminated, as explained in Ref. [157]. However, note that not all substrates are low-dielectric materials, so that the full phase structure in Fig. 17 might be relevant to some experiments. 7. Neutral polymer adsorption from solution 7.1. The mean-?eld approach: ground state dominance In this section we look at the equilibrium behavior of many chains adsorbing on (or equivalently depleting from) a surface in contact with a bulk reservoir of chains at equilibrium. The polymer chains in the reservoir are assumed to be in a semi-dilute concentration regime de6ned by cm ¿ cm∗ , where cm denotes the monomer concentration (per unit volume) and cm∗ is the overlap-concentration Eq. (2.19). As in the previous section, the adsorbing surface is taken as an ideal and smooth plane. Neglecting lateral concentration 1uctuations (which will be considered in Section 9), one can reduce the problem to an e8ective one-dimensional problem, where the monomer concentration depends only on the distance x from the surface, cm = cm (x). The two boundary values are: cmb = cm (x → ∞) in the bulk, while cms = cm (x = 0) on the surface. In addition to the monomer concentration cm , it is more convenient to work with the monomer volume fraction: ,(x)=a3 cm (x) where a is the Kuhn length which characterizes the e8ective monomer size. While the bulk value (far away from the surface) is 6xed by the concentration in the reservoir, the value on the surface at x = 0 is self-adjusting in response to a given surface interaction. The simplest phenomenological surface interaction is linear in the surface polymer concentration. The resulting contribution to the surface free energy (per unit area) is Fs = −71 ,s ;

(7.1)

˜ − T ∗ )=Ta2 , de6ned in Eq. (5.6), where ,s = a3 cms and a positive (negative) value of 71 = 7(T enhances adsorption (depletion) of the chains on (from) the surface. However, Fs represents only the local reduction in the interfacial free energy due to the adsorption. In order to calculate the full interfacial free energy, it is important to note that monomers adsorbing on the surface are connected to other monomers belonging to the same polymer chain. The latter accumulate in the vicinity of the surface. Hence, the interfacial free energy does not only depend on the surface concentration of the monomers but also on their concentration in the vicinity of the surface. Due to the polymer 1exibility and connectivity, the entire adsorbing layer can have a considerable width. The total interfacial free energy of the polymer chains will depend on this width and is quite di8erent from the interfacial free energy for simple molecular liquids. There are several theoretical frameworks to treat this polymer adsorption. One of the simplest methods which yet gives reasonable qualitative results is the Cahn–de Gennes approach [168,169]. In this approach, it is possible to write down a continuum functional which describes the contribution to the free energy of the polymer chains in the solution. This procedure was introduced by Edwards in the 1960s [112] and was applied to polymers at interfaces by de Gennes [169]. Below we present such a continuum version which can be studied analytically. Another approach is a discrete


45

one, where the monomers and solvent molecules are put on a lattice. The latter approach is quite useful in computer simulations and numerical self consistent 6eld (SCF) studies and is reviewed elsewhere [1]. In the continuum approach and using a mean-6eld theory, the bulk contribution to the adsorption free energy is written in terms of the local monomer volume fraction ,(x), neglecting all kinds of monomer–monomer correlations. The total reduction in the surface tension LF (interfacial free energy per unit area and in units of kB T ) is then 2 ∞ d, LF = −71 ,s + d x G(,) + Fb (,) − Fb (,b ) + $p (, − ,b ) ; (7.2) dx 0 where 71 was de6ned in Eq. (7.1). The sti8ness function G(,) represents the energy cost of local concentration 1uctuations and its form is speci6c to long polymer chains. For low polymer concentration it can be written as [14]: 2 1 a G(,) = 3 : (7.3) a 24, The other terms in Eq. (7.2) come from the Cahn–Hilliard free energy of mixing of the polymer solution, $p being the polymer chemical potential, and [12] 1 1 , 1 2 3 log , + v˜2 , + v˜3 , + · · · ; Fb (,) = 3 (7.4) a N 2 6 where N is the polymerization index. In the following, we neglect the 6rst term in Eq. (7.4) (translational entropy), as can be justi6ed in the long chain limit, N 1. The second and third dimensionless virial coeIcients are v˜2 =v2 =a3 and v˜3 =v3 =a6 , respectively. Good, bad and theta solvent conditions are achieved, respectively, for positive, negative or zero v˜2 . We concentrate hereafter only on good solvent conditions, v˜2 ¿ 0, in which case the higher order v˜3 -term can be safely neglected. In addition, the local monomer density is assumed to be small enough, in order to justify the omission of higher virial coeIcients. Note that for small molecules the translational entropy always acts in favor of desorbing from the surface. As was discussed in the Section 1, the vanishing small translational entropy for polymers results in a stronger adsorption (as compared with small solutes) and makes the polymer adsorption much more of an irreversible process. The key feature in obtaining Eq. (7.2) is the so-called ground state dominance, where for long enough chains N 1, only the lowest energy eigenstate (ground state) of a di8usion-like equation is taken into account. This approximation gives us the leading behavior in the N → ∞ limit [118]. It is based on the fact that the weight of the 6rst excited eigenstate is smaller than that of the ground state by an exponential factor: exp(−N LE) where LE = E1 − E0 ¿ 0 is the di8erence in the eigenvalues between the two eigenstates. Clearly, close to the surface more details on the polymer conformations can be important. The adsorbing chains have tails (end-sections of the chains that are connected to the surface by only one end), loops (mid-sections of the chains that are connected to the surface by both ends), and trains (sections of the chains that are adsorbed on the surface), as depicted in Fig. 13a. To some extent it is possible to get pro6les of the various chain segments even within mean-6eld theory, if the ground state dominance condition is relaxed as is discussed further below.

46


Taking into account all those simplifying assumptions and conditions, the mean-6eld theory for the interfacial free energy can be written as 2 d, 1 ∞ a2 1 (7.5) dx + v˜2 (,(x) − ,b )2 ; LF = −71 ,s + 3 a 0 24, d x 2 where the monomer bulk chemical potential $p is given by $p = 9f(,)=9,|b = v˜2 ,b . It is also useful to de6ne the total amount of monomers per unit area which take part in the adsorption layer. This is the so-called surface excess #; it is measured experimentally using, e.g., ellipsometry, and is de6ned as 1 ∞ d x[,(x) − ,b ] : (7.6) #= 3 a 0 (A di8erent quantity, not used in our review, is the so-called adsorbed amount, which measures the total amount of polymers per unit area that have at least one monomer in contact with the substrate.) The next step is to minimize the free energy functional (7.5) with respect to both ,(x) and ,s = ,(0). For the following algebraic manipulations, it is more convenient to re-express Eq. (7.5) in terms of the square root of the monomer volume fraction, (x) = ,1=2 (x) and s = ,s1=2 ∞ 2 2 d 1 a 1 (7.7) LF = −71 s2 + 3 dx + v˜2 ( 2 (x) − b2 )2 : a 0 6 dx 2 Minimization of Eq. (7.7) with respect to boundary condition a2 d 2 = v˜2 ( 2 − b2 ) ; 6 d x2 1 1 d = −6a71 = − : 2D s dx s

(x) and

s

leads to the following pro6le equation and

(7.8)

The second equation sets a boundary condition on the logarithmic derivative of the monomer volume fraction, d log ,=d x|s = 2 −1 d =d x|s = −1=D, where the strength of the surface interaction 71 can be expressed in terms of a length D ≡ 1=(12a71 ). Note that exactly the same scaling of D on 71 =T is obtained in Eq. (5.9) for the single chain behavior if one sets & = ’ = 1=2 (ideal chain exponents). This is strictly valid at the upper critical dimension (d = 4) and is a very good approximation in three dimensions. The pro6le equation (7.8) can be integrated once, yielding a2 d 2 1 = v˜2 ( 2 − b2 )2 : (7.9) 2 6 dx The above di8erential equation can now be solved analytically for adsorption (71 ¿ 0) and depletion (71 ¡ 0). We 6rst present the results in more detail for polymer adsorption and then repeat the main 6ndings for polymer depletion.


47

7.1.1. The adsorption case Setting 71 ¿ 0 as is applicable for the adsorption case, the 6rst-order di8erential equation (7.9) can be integrated and together with the boundary condition Eq. (7.8) yields x + x0 2 ; (7.10) ,(x) = ,b coth 'b where the length 'b = a(3v˜2 ,b )−1=2 is the Edwards correlation length characterizing the exponential decay of concentration 1uctuations in the bulk [14,112]. (See also the discussion in Section 7.2). The length x0 is not an independent length since it depends on D and 'b , as can be seen from the boundary condition Eq. (7.8) 4D 'b = 'b arccoth( ,s =,b ) : (7.11) x0 = arcsinh 2 'b Furthermore, ,s can be directly related to the surface interaction 71 and the bulk value ,b 6a2 71 'b ,b ,s = = −1 : 2D ,s ,b 3v˜2 ,b

(7.12)

In order to be consistent with the semi-dilute concentration regime, the correlation length 'b should be smaller than the size of a single chain, R = aN & , where & = 3=5 is the Flory exponent in good solvent conditions. This sets a lower bound on the polymer concentration in the bulk, cm ¿ cm∗ . So far three length scales have been introduced: the Kuhn length or monomer size a, the adsorbedlayer width D, and the bulk correlation length 'b . It is more convenient for the discussion to consider the case where those three length scales are quite separated: aD'b . Two conditions must be satis6ed. On one hand, the adsorption parameter is not large, 12a2 71 1 inorder to have Da. On the other hand, the adsorption energy is large enough to satisfy 12a2 71 3v˜2 ,b in order to have D'b . The latter inequality can be regarded also as a condition for the polymer bulk concentration. The bulk correlation length is large enough if indeed the bulk concentration (assumed to be in the semi-dilute concentration range) is not too large. Roughly, let us assume in a typical case that the three length scales are well separated: a is of the order of a few Angstroms, D of the order of a few dozens of Angstroms, and 'b of the order of a few hundred Angstroms. When the above two inequalities are satis6ed, three spatial regions of adsorption can be di8erentiated: the proximal, central, and distal regions, as is outlined below. In addition, as soon as 'b D, x0 2D, as follows from Eq. (7.11). (i) Close enough to the surface, x ∼ a, the adsorption pro6le depends on the details of the short range interactions between the surface and monomers. Hence, this region is not universal. In the proximal region, for axD, corrections to the mean-6eld theory analysis (which assumes the concentration to be constant) are presented below similarly to the treatment of the single chain section. These corrections reveal a new scaling exponent characterizing the concentration pro6le. They are of particular importance close to the adsorption/desorption transition. (ii) In the distal region, x'b , the excess polymer concentration decays exponentially to its bulk value ,(x) − ,b 4,b e−2x='b ;

(7.13)

48

R.R. Netz, D. Andelman / Physics Reports 380 (2003) 1 – 95 φ(x)

φs

φb (a)

a

D

ξb

x

φ(x)

φb

φs ξb

x

(b)

Fig. 18. (a) The schematic monomer volume fraction pro6le for the case of adsorption from a semi-dilute solution; we distinguish a layer of molecular thickness x ∼ a where the polymer density depends on details of the interaction with the substrate and the monomer size, the proximal region a ¡ x ¡ D where the decay of the density is governed by a universal power law (which cannot be obtained within mean-6eld theory), the central region for D ¡ x ¡ 'b with a self-similar pro6le, and the distal region for 'b ¡ x, where the monomer volume fraction relaxes exponentially towards its bulk value ,b . (b) The density pro6le for the case of depletion, where the concentration close to the surface is ,s and relaxes to its bulk value, ,b , at a distance of the order of the bulk correlation length 'b .

as follows from Eq. (7.10). This behavior is very similar to the decay of 1uctuations in the bulk with 'b being the correlation length. (iii) Finally, in the central region (and with the assumption that 'b is the largest length scale in the problem), Dx'b , the pro6le is universal and from Eq. (7.10) it can be shown to decay with a power law 2 a 1 : (7.14) ,(x) = 3v˜2 x + 2D A sketch of the di8erent scaling regions in the adsorption pro6le is given in Fig. 18a. Included in this 6gure are corrections in the proximal region, which is discussed further below. A special consideration should be given to the formal limit of setting the bulk concentration to zero, ,b → 0 (and equivalently 'b → ∞), which denotes the limit of an adsorbing layer in contact


49

with a polymer reservoir of vanishing concentration. It should be emphasized that this limit is not consistent with the assumption of a semi-dilute polymer solution in the bulk. Still, some information on the polymer density pro6le close to the adsorbing surface, where the polymer solution is locally semi-dilute [137], can be obtained. Formally, we take the limit 'b → ∞ in Eq. (7.10), and the limiting expression, given by Eq. (7.14), does not depend on 'b . The pro6le in the central region decays algebraically. In the case of zero polymer concentration in the bulk, the natural cuto8 is not 'b but rather R, the coil size of a single polymer in solution. Hence, the distal region loses its meaning and is replaced by a more complicated scaling regime [170]. The length D can be regarded as the layer thickness in the 'b → ∞ limit in the sense that a 6nite fraction of all the monomers are located in this layer of thickness D from the surface. Another observation is that ,(x) ∼ 1=x2 for xD. This power law is a result of the mean-6eld theory and its modi6cation is discussed below. It is now possible to calculate within the mean-6eld theory the two physical quantities that are measured in many experiments: the surface tension reduction LF and the surface excess #. The surface excess, de6ned in Eq. (7.6), can be calculated in a close form by inserting Eq. (7.10) into Eq. (7.6), 1 ' , , b b s #= √ (,1=2 − ,b1=2 ) = 3 −1 : (7.15) a ,b 3v˜2 a2 s For strong adsorption, we obtain from Eq. (7.12) that ,s (a=2D)2 =3v˜2 ,b , and Eq. (7.15) reduces to 1 a ∼ 71 ; (7.16) #= 3v˜2 a2 D while the surface volume fraction scales as ,s ∼ 721 . As can be seen from Eqs. (7.16) and (7.14), the surface excess as well as the entire pro6le does not depend (to leading order) on the bulk concentration ,b . We note again that the strong adsorption condition is always satis6ed in the ,b → 0 limit. Hence, Eq. (7.16) can be obtained directly by integrating the pro6le in the central region, Eq. (7.14). Finally, let us calculate the reduction in surface tension for the adsorbing case. Inserting the variational equations (7.8) in Eq. (7.5) yields √ 3=2 ,b ,b 3v˜2 3=2 : (7.17) LF = −71 ,s + +2 ,s 1 − 3 2 9a ,s ,s The surface term in Eq. (7.17) is negative while the second term is positive. For strong adsorption this reduction of LF does not depend on ,b and reduces to 1 + O(714=3 ) ; a2 where the leading term is just the contribution of the surface monomers. LF ∼ −(a2 71 )3

(7.18)

7.1.2. The depletion case We highlight the main di8erences between the polymer adsorption and polymer depletion. Keeping in mind that 71 ¡ 0 for depletion, the solution of the same pro6le equation (7.9), with the appropriate

50


boundary condition results in x + x0 2 ; ,(x) = ,b tanh 'b

(7.19)

which is schematically plotted in Fig. 18b. The limit ,b → 0 cannot be taken in the depletion case since depletion with respect to a null reservoir has no meaning. However, we can, alternatively, look at the strong depletion limit, de6ned by the condition ,s ,b . Here we 6nd x + 2D 2 2 ,(x) = 3v˜2 ,b : (7.20) a 2 In the same limit, we 6nd for the surface volume fraction ,s ∼ ,2b 7− 1 , and the exact expression for the surface excess Eq. (7.15) reduces to 1 ,b 'b ,b #=− 2 − 3 : (7.21) a 3v˜2 a

The negative surface excess can be directly estimated from a pro6le varying from ,b to zero over a length scale of order 'b . The dominating behavior for the surface tension can be calculated from Eq. (7.5) where both terms are now positive. For the strong depletion case we get 3 1 a LF 2 ∼ ,b3=2 : (7.22) a 'b 7.2. Beyond mean-?eld theory: scaling arguments for good solvents One of the mean-6eld theory results that should be corrected is the scaling of the correlation length with ,b . In the semi-dilute regime, the correlation length can be regarded as the average mesh size created by the overlapping chains. It can be estimated using very simple scaling arguments [14] similar to our derivation of the overlap concentration in Eq. (2.19). The volume fraction of monomers inside a coil formed by a subchain consisting of g monomers embedded in d dimensional space is ,b ∼ g1−d& where & is the Flory exponent. The spatial scale of this subchain is given by 'b ∼ ag& . Combining these two relations we obtain the general scaling of the correlation length −d&) ; 'b a,&=(1 b

(7.23)

and for good solvent condition and d = 3 'b a,b−3=4 :

(7.24)

This relation corrects the mean-6eld theory result 'b ∼ ,b−1=2 which can be obtained from, e.g., Eq. (7.5), and also directly from Eq. (7.23) by setting d = 4 and inserting the Gaussian exponent & = 1=2. 7.2.1. Scaling for polymer adsorption We repeat here an argument due to de Gennes [169]. The main idea is to assume that the relation Eq. (7.23) holds locally: ,(x) = ['(x)=a]−4=3 , where '(x) is the local “mesh size” of the semi-dilute polymer solution. Since there is no other length scale in the problem beside the distance from the


51

surface, x, the correlation length '(x) should scale as the distance x itself, '(x) x leading to the pro6le a 4=3 : (7.25) ,(x) x We note that this argument holds only in the central region Dx'b . It has been con6rmed experimentally using neutron scattering [171] and neutron re1ectivity [172]. Eq. (7.25) satis6es the distal boundary condition: x → 'b , ,(x) → ,b , but for x ¿ 'b we expect the regular exponential decay behavior of the distal region, Eq. (7.13). De Gennes also proposed (without a rigorous proof) a convenient expression for ,(x), which has the correct crossover from the central to the mean-6eld proximal region [169] 4=3 4=3 4 D a 3 ,(x) = ,s : (7.26) x + 43 D x + 43 D Note that the above equation reduces to Eq. (7.25) for xD. The extrapolation of Eq. (7.26) also gives the correct de6nition of D: D−1 = − d log ,=d x|s . In addition, ,s is obtained from the extrapolation to x = 0 and scales as a 4=3 : (7.27) ,s = ,(x = 0) = D For strong adsorption (,s ,b ), we have a 4=3 ∼ 721 ; ,s D 1 3=2 D a ∼ 71−3=2 ; 2 a 71 # a−2 (a2 71 )1=2 ∼ 711=2 ; LF −

1 3=2 , ∼ −731 : a2 s

(7.28)

It is interesting to note that although D and # have di8erent scaling with the surface interaction 71 in the mean-6eld theory and scaling approaches, ,s and LF have the same scaling using both approaches. This is a result of the same scaling ,s ∼ 721 , which, in turn, leads to LF 71 ,s ∼ 731 . 7.2.2. Scaling for polymer depletion For polymer depletion similar arguments led de Gennes [169] to propose the following scaling form for the central and mean-6eld proximal regions, a ¡ x ¡ 'b , 5=3 x + 53 D ; (7.29) ,(x) = ,b 'b

52


where the depletion thickness is 'b − D whereas in the strong depletion regime (,s ,b ) 5=3 D ∼ ,b9=4 71−5=2 ; ,s ,b 'b 2 −3=2 ; D = a a 71 # −,b a−3 ('b − D) ∼ ,b1=4 ; 1 3=2 , : (7.30) a2 b Note that the above scaling of the surface tension with the bulk concentration, ,b is the same as that obtained by the mean-6eld theory approach in Section 7.1.2, Eq. (7.22). LF

7.3. Proximal region corrections So far we did not address any corrections in the proximal region: a ¡ x ¡ D for the many chain adsorption. In the mean-6eld theory picture the pro6le in the proximal region is featureless and saturates smoothly to its extrapolated surface value, ,s ¿ 0. However, in relation to surface critical phenomena which is in particular relevant close to the adsorption–desorption phase transition (the so-called ‘special’ transition), the polymer pro6le in the proximal region has a scaling form with another exponent m. a m ,(x) ,s ; (7.31) x where m = (’ + & − 1)=& is the proximal exponent, Eq. (5.12). This is similar to the single chain treatment in Section 5. For good solvents, one has m 1=3, as was derived using analogies with surface critical phenomena, exact enumeration of polymer con6gurations, and Monte-Carlo simulations [133]. It is di8erent from the exponent 4=3 of the central region. With the proximal region correction, the polymer pro6le can be written as [135]  ,s for 0 ¡ x ¡ a ;      a 1=3  for a ¡ x ¡ D ; ,s (7.32) ,(x) x    1=3 D a    ,s for D ¡ x ¡ 'b ; x x+D where

a : (7.33) D The complete adsorption pro6le is shown schematically in Fig. 18a. By minimization of the free energy with respect to the layer thickness D it is possible to show that D is proportional to 1=71 ,s =

1 ; D ∼ 7− 1

in accord with the exact 6eld-theoretic results for a single chain as discussed in Section 5.

(7.34)


53

The surface concentration, surface excess and surface tension have the following scaling [135]: a ∼ 71 ; ,s D a 4=3 # a− 3 D ∼ 711=3 ; D LF −721 a2 ∼ 721 :

(7.35)

Note the di8erences in the scaling of the surface tension and surface excess in Eq. (7.35) as compared with the results obtained with no proximity exponent (m = 0) in the previous section, Eq. (7.28). At the end of our discussion of polymer adsorption from solutions, we would like to add that for the case of adsorption from dilute solutions, there is an intricate crossover from the single-chain adsorption behavior, as discussed in Section 5, to the adsorption from semi-dilute polymer solutions, as discussed in this section [137]. Since the two-dimensional adsorbed layer has a higher local polymer concentration than the bulk, it is possible that the adsorbed layer forms a two-dimensional semi-dilute state, while the bulk is a truly dilute polymer solution. Only for extremely low bulk concentration or for very weak adsorption energies the adsorbed layer has a single-chain structure with no chain crossings between di8erent polymer chains. 7.4. Loops and tails It has been realized quite some time ago that the so-called central region of an adsorbed polymer layer is characterized by a rather broad distribution of loop and tail sizes [1,173,174]. A loop is de6ned as a chain region located between two points of contact with the adsorbing surface, and a tail is de6ned as the chain region between the free end and the closest contact point to the surface, while a train denotes a chain section which is tightly bound to the substrate (see Fig. 13a). The relative statistical weight of loops and tails in the adsorbed layer is clearly of importance to applications. For example, it is expected that polymer loops which are bound at both ends to the substrate are more prone to entanglements with free polymers than tails and, thus, lead to enhanced friction e8ects. It was found in detailed numerical mean-6eld theory calculations that the external part of the adsorbed layer is dominated by dangling tails, while the inner part is mostly built up by loops [1,173]. Recently, an analytical theory was formulated which correctly takes into account the separate contributions of loops and tails and which thus goes beyond the ground state dominance assumption made in ordinary mean-6eld theories. The theory predicts that a crossover between tail-dominated and loop-dominated regions occurs at some distance x∗ aN 1=(d−1) [175] from the surface, where d is the dimension of the embedding space. It is well known that mean-6eld theory behavior can formally be obtained by setting the embedding dimensionality equal to the upper critical dimension, which is for self-avoiding polymers given by d = 4 [15]. Hence, the above expression predicts a crossover in the adsorption behavior at a distance x∗ aN 1=3 . For good-solvent conditions in three dimensions (d = 3); x∗ aN 1=2 . In both cases, the crossover occurs at a separation much smaller than the size of a free polymer R ∼ aN & where, according to the classical Flory argument [12], & = 3=(d + 2). A further rather subtle result of these improved mean-6eld theories is the occurrence of a depletion hole, i.e., a region at a certain separation from the adsorbing surface where the monomer

54


concentration is smaller than the bulk concentration [175]. This depletion hole results from an interplay between the depletion of free polymers from the adsorbed layer and the slowly decaying density pro6le due to dangling tails. It occurs at a distance from the surface comparable with the radius of gyration of a free polymer, but also shows some dependence on the bulk polymer concentration. In a di8erent formulation, the interfacial free energy of an adsorbed layer consisting of 6nite-length polymers has been calculated for the full concentration range from dilute to dense solutions [176]. These and other e8ects, related to the occurrence of loops and tails in the adsorbed layer, have been recently reviewed [177]. 8. Adsorption of polyelectrolytes—mean $eld In Section 6 we have been reviewing the behavior of single PE chains close to a charged wall (or surface). This will be now extended to include adsorption of PE from bulk (semi-dilute) solutions having a bulk concentration cmb . As before the chains are assumed to have a fraction f of charged monomers, each carrying a charge e resulting in a linear charge density, + = f=b. The interesting case of polyampholytes having negative and positive charges is not considered in this section. The solution can also contain salt (small ions) of concentration csalt which is directly related to the Debye–HKuckel screening length, −1 . For simplicity, the salt is assumed throughout this section to be monovalent (z = 1). We will consider adsorption only onto a single 1at and charged surface. Clearly the most important quantity is the pro6le of the polymer concentration, cm (x) = ,(x)=a3 , as function of x, the distance from the surface. Another useful quantity mentioned already in Section 7 is the polymer surface excess (per unit area) ∞ 1 ∞ b #= [cm (x) − cm ] d x = 3 [,(x) − ,b ] d x : (8.1) a 0 0 Related to the surface excess # is the amount of charges (in units of e) carried by the adsorbing PE chains, f#. In some cases the adsorbed polymer layer carries a higher charge (per unit area) than the charged surface itself, f# ¿ *, and the surface charge is overcompensated by the PE as we will see later. This does not violate global charge neutrality in the system because of the presence of counterions in solution. In many experiments, the total amount of polymer surface excess # is measured as a function of the bulk polymer concentration, pH and/or ionic strength of the bulk solution [178–185]. (For reviews see, e.g. Refs. [1,186–188].) More recently, spectroscopy [180] and ellipsometry [184] have been used to measure the width of the adsorbed PE layer. Other techniques such as neutron scattering can be employed to measure the entire pro6le cm (x) of the adsorbed layer [172,189]. Electrostatic interactions play a crucial role in the adsorption of PEs [1,186,187]. Besides the fraction f of charged monomers, the important parameters are the surface charge density (or surface potential in case of conducting surfaces), the amount of salt (ionic strength of low molecular weight electrolyte) in solution and, in some cases, the solution pH. For PEs the electrostatic interactions between the monomers themselves (same charges) are always repulsive, leading to an e8ective sti8ening of the chain [22,23]. Hence, this interaction will favor the adsorption of single polymer chains, because their con6gurations are already quite extended [157],


55

but it will oppose the formation of dense adsorption layers close to the surface [1]. If the PE chains and the surface are oppositely charged, the electrostatic interactions between them will enhance the adsorption. In addition, the presence of salt has a subtle e8ect. It simultaneously screens the monomer–monomer repulsive interactions as well as the attractive interactions between the oppositely charged surface and polymer. Presence of multivalent salt ions (not considered in this section) makes the PE adsorption even more complex. Two limiting adsorbing cases can be discussed separately: (i) a non-charged surface on which the chains like to adsorb due to, e.g., van-der-Waals attraction. Here the interaction between the surface and the chain does not have an electrostatic component. However, as the salt screens the monomer–monomer electrostatic repulsion, it leads to enhancement of the adsorption. (ii) The surface is charged but does not interact with the polymer besides the electrostatic interaction. This is called the pure electro-sorption case. At low-salt concentration, the polymer charge completely compensates the surface charge. At high-salt concentration some of the compensation is done by the salt, leading to a decrease in the amount of adsorbed polymer. In some cases, over-compensation of the surface charges by the polymer charges can also occur (as is reviewed below in Section 8.5), where the PE chains form a condensed layer and reverse the sign of the total surface charge. This is used, e.g., to build a multi-layered structure of cationic and anionic PEs—a process that can be continued for few dozen or even few hundred times [150,153]. The phenomenon of over-compensation is discussed in Refs. [157,190–193] but is still not well understood. In practice, electrostatic and other types of interactions with the surface can occur in parallel, making the analysis more complex. In spite of the diIculties to treat theoretically PE’s in solution because of the delicate interplay between the chain connectivity and the long range nature of electrostatic interactions [8,14,194,195], several simple approaches treating adsorption exist. One approach is a discrete multi-Stern layer model [196–200], where the system is placed on a lattice whose sites can be occupied by a monomer, a solvent molecule or a small ion. The electrostatic potential is determined self-consistently (mean-6eld theory) together with the concentration pro6les of the polymer and the small ions. In another approach, the electrostatic potential and the PE concentration are treated as continuous functions [155,191,201–206]. These quantities are obtained from two coupled di8erential equations derived from the total free energy of the system. In some cases the salt concentration is considered explicitly [202,203], while in other works, (e.g., in Refs. [155,157]) it induces a screened Coulombic interaction between the monomers and the substrate. We will review the main results of the continuum approach, presenting numerical solutions of the mean 6eld equations and scaling arguments. 8.1. Mean-?eld theory and its pro?le equations The charge density on the polymer chains is assumed to be continuous and uniformly distributed along the chains. Further treatments of the polymer charge distribution (annealed and quenched models) can be found in Refs. [203,205]. Within mean-6eld approximation, the free energy of the system can be expressed in terms of the local electrostatic potential U (r), the local monomer concentration cm (r) and the local concentration of positive and negative ions c± (r). The mean-6eld approximation means that the in1uence of the charged surface and the inter and intra-chain interactions can be expressed in term of an external potential which will determine the local concentration of the monomers, cm (r). This external potential depends both on the electrostatic potential and on

56


the excluded volume interactions between the monomers and the solvent molecules. The excess free energy with respect to the bulk can then be calculated using another important approximation, the ground state dominance. This approximation is used often for neutral polymers [14] (see Section 7) and is valid for very long polymer chains, before, we introduce the (dimensionless) polyN 1. As mer order parameter (r), where (r) = ,(r) = a3 cm (r), and express the adsorption free energy F (in units of kB T ) in terms of and U [201–206] F = dr{Fpol (r) + Fions (r) + Fel (r)} : (8.2) The polymer contribution is Fpol (r) =

a2 1 |∇ |2 + v˜2 ( 6 2

4

−

4 b)

− $p (

2

−

2 b)

;

(8.3)

where the 6rst term is the polymer elastic energy. Throughout this section we restrict ourselves to 1exible chains described by a Kuhn length a. The second term is the excluded volume contribution where the dimensionless second virial coeIcient v˜2 is positive and of order unity. The case of negative virial coeIcients (bad-solvent condition) has been treated in Ref. [207]. The last term couples the system to a polymer reservoir via a chemical potential $p , and b = ,b is related to the bulk monomer concentration, cmb = ,b =a3 . The entropic contribution of the small (monovalent) ions is Fions (r) = [ci ln ci − ci − csalt ln csalt + csalt ] − $i (ci − csalt ) ; (8.4) i=±

where ci (r) and $i are, respectively, the local concentration and the chemical potential of the i = ± ions, while csalt is the bulk concentration of salt. Finally, the electrostatic contributions (per kB T ) are |∇U |2 kB T : Fel (r) = fe 2 U + ec+ U − ec− U − (8.5) 80 The 6rst three terms are the electrostatic energies of the monomers (carrying f fractional charge per monomer), the positive ions and the negative ions, respectively. The last term is the self-energy of the electric 6eld, where is the dielectric constant of the solution. Note that the electrostatic contribution, Eq. (8.5), is equivalent to the well known result: (=80kB T ) dr|∇U |2 plus surface terms. This can be seen by substituting the Poisson–Boltzmann equation (as obtained below) into Eq. (8.5) and then integrating by parts. Minimization of the free energy Eqs. (8.2)–(8.5) with respect to c± ; and U yields a Boltzmann distribution for the density of the small ions, c± (r)=csalt exp(∓eU=kB T ), and two coupled di8erential equations for and U : ∇2 U (r) =

80e 40e csalt sinh(eU=kB T ) − (f

a2 2 ∇ (r) = v˜2 ( 6

3

−

2 b

) + f eU=kB T :

2

−f

2 eU=kB T ) be

;

(8.6) (8.7)


57

Eq. (8.6) is a generalized Poisson–Boltzmann equation including the free ions as well as the charged polymers. The 6rst term represents the salt contribution and the second term is due to the charged monomers and their counterions. Eq. (8.7) is a generalization of the self-consistent 6eld equation of neutral polymers Eq. (7.8) [14]. In the bulk, the above equations are satis6ed by setting U → 0 and → b. 8.2. Constant Us : the low-salt limit 8.2.1. Numerical solutions of mean-?eld equations When the surface is taken as ideal, i.e., 1at and homogeneous, the solutions of the mean-6eld equations depend only on the distance x from the surface. The surface imposes boundary conditions on the polymer order parameter (x) and the electrostatic potential U (x). Due to global electroneutrality, all charge carriers in solution should exactly balance the surface charges. The Poisson–Boltzmann equation (8.6), the self-consistent 6eld equation (8.7) and the boundary conditions uniquely determine the polymer concentration pro6le and the electrostatic potential. In all cases of interest, these two coupled non-linear equations can only be solved numerically. We present now numerical pro6les obtained for surfaces with a constant potential Us : U |x=0 = Us :

(8.8)

The boundary conditions for (x) depend on the nature of the short range non-electrostatic interaction of the monomers and the surface. For simplicity, we take a non-adsorbing surface and require that the monomer concentration will vanish there: |x=0 = 0 :

(8.9)

We note that the boundary conditions chosen in Eqs. (8.8) and (8.9) model the particular situation of electrostatic attraction at constant surface potential in competition with a steric (short range) repulsion of non-electrostatic origin. Possible variations of these boundary conditions include surfaces with a constant surface charge (discussed below) and surfaces with a non-electrostatic short range attractive (or repulsive) interaction with the polymer [190,209]. Far from the surface (x → ∞) both U and reach their bulk values and their derivatives vanish: U |x→∞ = 0 and |x→∞ = 0. The numerical solutions of the mean-6eld equations (8.6), (8.7) together with the boundary conditions discussed above are presented in Fig. 19, for several di8erent physical parameters in the low-salt limit. The polymer is positively charged and is attracted to the non-adsorbing surface held at a constant negative potential. The aqueous solution contains a small amount of monovalent salt (csalt = 0:1 mM). The reduced concentration pro6le ,(x)=,b is plotted as a function of the distance from the surface x. Di8erent curves correspond to di8erent values of the reduced surface potential us ≡ eUs =kB T , the charge fraction f and the Kuhn length a. Although the spatial variation of the pro6les di8ers in detail, they all have a single peak which can be characterized by its height and width. This observation serves as a motivation to using scaling arguments. 8.2.2. Scaling arguments The numerical pro6les of the previous section indicate that it may be possible to obtain simple analytical results for the PE adsorption by assuming that the adsorption is characterized by one


φ(x)/φb

58

x [Å] Fig. 19. Adsorption pro6les obtained by numerical solutions of Eqs. (8.6), (8.7) for several sets of physical parameters in the low-salt limit. The monomer volume fraction ,(x) scaled by its bulk value ,b = b2 is plotted as a function of the [ and us = eUs =kB T = −0:5 (solid curve); distance x from the surface. The di8erent curves correspond to: f = 1, a = 5 A [ and us = −0:5 (dots); f = 1, a = 5 A [ and us = −1:0 (short dashes); f = 1, a = 10 A [ and us = −0:5 f = 0:1, a = 5 A [ and us = −1:0 (dot-dash line). For all cases cmb = ,b a3 = 10−6 A [ 3 =a3 , [ −3 , v˜2 = 50 A (long dashes); and f = 0:1, a = 5 A = 80, T = 300 K and csalt = 0:1 mM. Adapted from Ref. [206].

dominant length scale D. Hence, we write the polymer order parameter pro6le in the form (x) = ,M h(x=D) ;

(8.10)

where h(x=D) is a dimensionless function normalized to unity at its maximum and ,M sets the scale of polymer adsorption, such that (D) = ,M . The free energy can now be expressed in terms of D and ,M , while the exact form of h(x=D) a8ects only the numerical prefactors. As discussed below, the scaling form Eq. (8.10), which describes the density pro6le as a function of a single scaling variable x=D, is only valid as long as −1 and D are not of the same order of magnitude. Otherwise, the scaling function h should be a function of both x and x=D. We concentrate now on the limiting low-salt regime, D−1 , where Eq. (8.10) can be justi6ed. In the other extreme of high-salt, D−1 , the adsorption crosses over to a depletion, as is discussed below (Sections 8.3 and 8.4). Note that the latter limit is in agreement with the single-chain adsorption (Section 6), where in the high-salt limit and for weakly charged chains, the PE desorbs from the wall. In the low-salt regime the e8ect of the small ions can be neglected and the free energy (per unit surface area), Eqs. (8.2)–(8.5), can be evaluated using the scaling form Eq. (8.10) and turns out to be given by (see also Refs. [202,206]) F

1 D D3 1 D ,M − f|us |,M 3 + 40lB f2 ,2M 6 + v˜2 ,2M 3 : 6aD a a 2 a

(8.11)


59

In what follows we drop prefactors of order unity from the various terms. The 6rst term of Eq. (8.11) is the elastic energy characterizing the response of the polymer to concentration inhomogeneities. The second term accounts for the electrostatic attraction of the polymers to the charged surface. The third term represents the Coulomb repulsion between adsorbed monomers. The last term represents the excluded volume repulsion between adsorbed monomers, where we assume that the monomer concentration near the surface is much larger than the bulk concentration ,M ,b . (The opposite limit, ,M 6 ,b , is consistent with depletion and will be discussed separately in the high-salt regime.) In the low-salt regime and for strong enough PEs the electrostatic interactions are much stronger than the excluded volume ones. Neglecting the latter interactions and minimizing the free energy with respect to D and ,M gives 1 a2 ∼ (8.12) D2 f|us | f|Us | and ,M

a|us |2 ∼ |Us |2 ; 40lB

(8.13)

recalling that us = eUs =kB T . These expressions are valid as long as (i) D−1 and (ii) the excluded volume term in Eq. (8.11) is negligible. Condition (i) translates into csalt f|us |=(80lB a2 ). For [ and lB = 7 A [ this limits the salt concentration to csalt =f0:4 M. Condition (ii) on |us | 1; a = 5 A the magnitude of the excluded volume term can be shown to be equivalent to fv˜2 a|us |=lB . These requirements are consistent with the numerical data presented in Fig. 19. We recall that the pro6les presented in Fig. 19 were obtained from the numerical solution of Eqs. (8.6) and (8.7), including the e8ect of small ions and excluded volume. The scaling relations are veri6ed by plotting in Fig. 20 the same sets of data as in Fig. 19 using rescaled variables as de6ned in Eqs. (8.12), (8.13). Namely, the rescaled electrostatic potential U (x)=us and polymer concentration ,(x)=,M ∼ ,(x)=|us |2 are plotted as functions of the rescaled distance x=D ∼ xf1=2 |us |1=2 =a. The di8erent numerical data roughly collapse on the same curve, which demonstrates that the scaling results in Eqs. (8.12), (8.13) are valid for a whole range of parameters in the low-salt regime. In many experiments the total amount of adsorbed polymer per unit area (surface excess) # is measured as function of the physical characteristics of the system such as the charge fraction f, the pH of the solution or the salt concentration csalt (see, e.g. Refs. [178–185]). While in the next section we give general predictions for a wide range of salt concentration, we comment here on the low-salt limit, where the scaling expressions, Eqs. (8.10), (8.12) and (8.13), yield D |us |3=2 |Us |3=2 1 ∞ [,(x) − ,b ] d x 3 ,M ∼ : (8.14) #= 3 a 0 a lB af1=2 f1=2 This scaling prediction for the adsorbed amount #(f) compares favorably with the numerical results shown in Fig. 21a, adapted from Ref. [208], for the low-salt limit (solid line corresponds to csalt = 1:0 mM, and dashed line to 10 mM). As a consequence of Eq. (8.14), # decreases with increasing charge fraction f. Similar behavior was also reported in experiments [181]. This e8ect is at 6rst glance quite puzzling because as the polymer charge increases, the chains are subject to a stronger attraction to the surface. On the other hand, the monomer–monomer repulsion is stronger and indeed, in this regime, the monomer–monomer Coulomb repulsion scales as (f,M )2 , and dominates over the adsorption energy that scales as f,M .


U(x)/|Us|

60

x/D

φ(x)/φM

(a)

(b)

x/D

Fig. 20. Rescaled pro6les of PE adsorption in the low-salt regime con6rming the scaling relations, Eqs. (8.12), (8.13). (a) The rescaled electrostatic potential U (x)=|Us | as a function of the rescaled distance x=D. (b) The rescaled monomer volume fraction ,(x)=,M as a function of the same rescaled distance. The pro6les are taken from Fig. 19 (with the same notation). The numerical prefactors of a piecewise linear h(x=D) pro6le were used in the calculation of D and ,M . Adapted from Ref. [206].

8.3. Adsorption behavior in the presence of ?nite salt The full dependence of # on csalt and f, as obtained from the numerical solutions of the mean-6eld equations with 6xed Us boundary condition [208], is presented in Fig. 21. Our results are in agreement with numerical solutions of discrete lattice models (the multi-Stern layer theory)


61

8

8

6

6

4

4

2

2

0

0 0

(a)

0.5 f

1

4

10

-3

10

(b)

-2

-1

10 10 csalt[M]

Γ [Å-2]

Γ [Å-2]

x 10 -3

0

10

Fig. 21. (a) Surface excess of polyelectrolyte adsorption, #, as function of the chain charged fraction f, for constant surface potential and for several csalt concentrations: 1:0 mM (solid line), 10 mM (dashed line), 0:1 M (dash-dot line), 0:5 M (dots). As the salt concentration increases, the peak in # shifts to higher f values and disappears for csalt = 0:5 M. The depletion–adsorption transition is de6ned to occur for # = 0. (b) Surface excess as function of salt concentration, csalt , for constant surface potential and for several f values: f = 0:03 (dots), 0.1 (dashes), 0.3 (dot-dash), 1.0 (solid line). # is almost independent of csalt for low-salt concentrations in the adsorption region. It is then followed by a strong descent [ 3 , a = 5 A, [ [ −3 , v2 = v˜2 a3 = 50 A into a depletion region. The other parameters used here are: us = −1:0, cmb = ,b a3 = 10−6 A T = 300 K, = 80. Adapted from Ref. [208].

[1,186,187,196–200]. In Fig. 21a the dependence of # on f is shown for several salt concentrations ranging from low-salt conditions, csalt = 1:0 mM, all the way to high salt, csalt = 0:5 M. For low enough f; # ¡ 0 indicates depletion (as is discussed below). As f increases, a crossover to the adsorption region, # ¿ 0, is seen. In the adsorption region, a peak in #(f) signals the maximum adsorption amount at constant csalt . As f increases further, beyond the peak, # decreases as f−1=2 for low-salt concentrations, in agreement with Eq. (8.14). Looking at the variation of # with salt, as csalt increases, the peak in #(f) decreases and shifts to higher values of f. For very large amount of salt, e.g., csalt = 0:5 M, the peak occurs in the limit f → 1, and only an increase in #(f) is seen from the negative depletion values (small f) towards the peak at f → 1. In Fig. 21b, we plot #(csalt ) for several f values: 0.03, 0.1, 0.3 and 1.0. For low enough salt condition, the surface excess is almost independent of csalt . In this adsorption regime, the surface excess is well characterized by the scaling result of the previous section, Eq. (8.14), # ∼ f−1=2 . As the amount of salt increases above some threshold, the adsorption regime crosses over to depletion quite sharply, signaling the adsorption–depletion transition. The salt concentration at the transition, ∗ , increases with the charge fraction f. csalt 8.4. Adsorption–depletion crossover in high-salt conditions In the scaling discussion in Section 8.2.2, it was assumed implicitly that the PE chains are adsorbing to the surface. Namely, the electrostatic interaction with the surface is strong enough so that

62

R.R. Netz, D. Andelman / Physics Reports 380 (2003) 1 – 95 7 6

φ(x)/φ

b

5 4 3 2 1 0 0

50

100

150

x [Å]

Fig. 22. Numerical monomer density pro6les exhibiting the transition from adsorption to depletion. The dashed line corresponds to f = 0:12, the dot-dash line to f = 0:1, the solid line to f = 0:09, and the dotted line to f = 0:08. From the ∗ condition # = 0 the adsorption–depletion transition is found to occur for f 0:09, corresponding to csalt 0:16|us |f=lB a2 . b 3 −6 [ −3 3 3 [ [ All pro6les have us = −0:5, csalt = 70 mM, cm = ,b a = 10 A , v2 = v˜2 a = 50 A , a = 5 A; T = 300 K, = 80. Adapted from Ref. [208].

it overcomes any compression and entropy loss of the polymers in the adsorbing layer. This is not correct for highly screened systems (high salt) and weakly charged PEs. The numerical PE pro6les obtained from solving Eqs. (8.6)–(8.7) [208] demonstrating the adsorption–depletion transition (which is not a sharp transition but rather a crossover) are presented in Fig. 22. The pro6les were obtained by solving numerically the di8erential equations for several values of f in a range including the adsorption–depletion transition. For salt concentration ∗ 0:16u f=(l a2 ) (solid line in Fig. 22 with f = 0:09), the 6gure demonstrates the of about csalt s B disappearance of the peak in the concentration pro6le. Our way of identifying this crossover is by looking at the surface excess, #. The place where # = 0 indicates an adsorption–depletion transition, separating positive # in the adsorption regime from negative ones in the depletion regime. The numerical phase diagrams displaying the adsorption–depletion transition are presented in Fig. 23, where the line of vanishing surface excess, # = 0, is located in the (f; csalt ) plane while 6xing us (Fig. 23a), and in the (|us |; csalt ) plane while 6xing f (Fig. 23b). From the 6gure it is apparent that the adsorption–depletion transition line 6ts quite well a line of slope 1.0 in both ∗ ∼ f for 6xed u , and c∗ ∼ u for 6xed f. Fig. 23a and b plotted on a log–log scale. Namely, csalt s s salt ∗ These scaling forms of csalt at the adsorption–depletion transition can be reproduced by using simpli6ed scaling arguments, similar to the single-polyelectrolyte adsorption situation in Section 6. There we found that the exact scaling of the desorption transition is recovered by de6ning desorption to occur when the prediction for the adsorption layer thickness D reaches the screening length −1 . The condition for adsorption is thus D ¡ 1. Using the scaling for D, Eq. (8.12), and the de6nition of , we 6nd the adsorption–depletion transition to occur at the salt concentration us f ∗ csalt : (8.15) lB a2


-1

0

Depletion

Depletion

10

csalt [M]

csalt [M]

10

-1

10

Adsorption

-2

10

-2

10 (a)

63

-1

10

f

Adsorption 0

10

10 (b)

-2

10 0 10

-1

|us|

Fig. 23. Numerically calculated crossover diagram from adsorption to depletion on a log–log scale for constant surface potential conditions. In (a) the (f; csalt ) parameter plane is presented for constant us = −1:0. The least-mean-square 6t has ∗ a slope of 1:00 ± 0:02, in excellent agreement with the scaling arguments, csalt ∼ f. In (b) the (us ; csalt ) parameter plane is presented, for constant f = 0:1. The least-mean-square 6t has a slope of 1:04 ± 0:02, in good agreement with scaling ∗ [ 3 , a = 5 A; [ T = 300 K, = 80. Adapted [ −3 , v2 = v˜2 a3 = 50 A arguments, csalt ∼ us . All pro6les have cmb = ,b a3 = 10−6 A from Ref. [208].

in the case of a 6xed surface potential. This explains the numerical results of Fig. 23a and b. We mention the analogous results for 6xed surface charge as well as the phenomenon of overcompensation in the next subsection. 8.5. Adsorption of PEs for constant surface charge and its overcompensation We turn now to a di8erent electrostatic boundary condition of constant surface charge density and look at the interesting phenomenon of charge overcompensation by the PE chains in relation to experiments for PE adsorption on 1at surfaces, as well as on charged colloidal particles [150,152,153]. What was observed in experiments is that PEs adsorbing on an oppositely charged surface can overcompensate the original surface charge. Because the PEs create a thin layer close to the surface, they can act as an e8ective absorbing surface to a second layer of PEs having an opposite charge compared to the 6rst layer. Repeating the adsorption of alternating positively and negatively charged PEs, it is possible to create a multilayer structure of PEs at the surface. Although many experiments and potential applications for PE multilayers exist, the theory of PE overcompensation is only starting to be developed [157,190,191,193,205,206,209]. The scaling laws presented for constant Us can be used also for the case of constant surface charge. A surface held at a constant potential Us will induce a surface charge density * (in units of e). The two quantities are related by: dU=d x = −40*e= at x = 0. We will now consider separately the two limits: low salt D−1 , and high salt D ¿ −1 . As will be explained in Section 9, an alternative mechanism for overcharging is produced by lateral correlations between adsorbed PEs, which in conjunction with screening by salt ions leads to strongly overcharged surfaces [157,193]. 8.5.1. Low-salt limit: D−1 Assuming that there is only one length scale characterizing the potential behavior in the vicinity of the surface, as demonstrated in Fig. 20, the surface potential Us and the surface charge * are

64


related by Us ∼ *eD=. In the low-salt limit we 6nd from Eq. (8.12) D ∼ (f*lB )−1=3

(8.16)

in accord with the single-chain result Eq. (6.9). Let us de6ne two related concepts via the e8ective surface charge density de6ned as L* ≡ f# − *, which is sum of the adsorbed polymer charge density f# and the charge density * of the bare substrate. For L* = 0 the adsorbed polymer charge exactly compensates the substrate charge. If L* is positive the PE overcompensates the substrate charge, more polymer adsorbs than is needed to exactly cancel the substrate charge. If L* is positive and reaches the value L* = * it means that the PE charge is f# = 2* and leads to an exact charge inversion of the substrate charge. In this case, the e8ective surface charge consisting of the substrate charge plus the PE layer has a charge density which is exactly opposite to the original substrate charge density *. Do we obtain overcompensation or even charge inversion in the low-salt limit within mean-6eld theory? Using scaling arguments this is not clear since one 6nds that L* ∼ f# ∼ *. Namely each of the two terms in L* scales linearly with *, and the occurrence of overcompensation or charge inversion will depend on numerical prefactors (which are diIcult to obtain using scaling arguments) determining the relative sign of the two opposing terms. However, if we look on the numerical solution for the mean-6eld electrostatic potential, Fig. 20, we see indeed that all plotted pro6les have a maximum of U (x) as function of x. An extremum in U means a zero local electric 6eld. Or equivalently, using Gauss law, this means that the integrated charge density from the surface to this special extremum point (including surface charges) is exactly zero. At this point the charges in solution exactly compensate the surface charges. For larger distances from the surface, the adsorption layer overcompensates the substrate charge. 8.5.2. High-salt limit: D ¿ −1 and depletion When we include salt in the solution and look at the high-salt limit, the only length characterizing the exponential decay of U close to the surface is the Debye–HKuckel screening length. Hence, using dU=d x|s ∼ −*e= yields Us ∼ *e= or us ∼ *‘B =. Inserting this relation into the adsorption threshold for constant surface potential, Eq. (8.15), we obtain for the crossover between adsorption and depletion ∗ csalt *2=3 f2=3 lB−1=3 a−4=3 ∼ *2=3 f2=3 ;

(8.17)

in accord with Refs. [154,155,195] and as con6rmed by the numerical studies of Eqs. (8.6) and (8.7) with constant * boundary conditions. More details can be found in Ref. [208]. We note that the same threshold is obtained by equating the adsorption layer thickness in the constant-surface-charge ensemble, Eq. (8.16), with the screening length −1 . We end this section with a short comment on the relation between the semi-dilute and single-chain adsorption behaviors. By construction of the scaling argument, the desorption threshold obtained here in the semi-dilute regime for 6xed surface charge, Eq. (8.17), is the same as the single-chain desorption transition, Eq. (6.10). It is important to point out that this equivalence is perfectly con6rmed by our numerical solutions of the full mean-6eld equations. Therefore, it follows that multi-chain e8ects (within mean-6eld level) do not modify the location of the single-polyelectrolyte chain adsorption transition.


65

ξs

(a)

ξs

(b)

ξs

(c)

Fig. 24. Schematic top views of the di8erent adsorbed surface phases considered. (a) Disordered uncrumpled phase, with an average mesh size 's smaller than the persistence length, exhibiting an average density of chain crossings of ∼ 1='2s . (b) Disordered crumpled phase, with a mesh size 's larger than the persistence length. (c) Lamellar phase, with a lamellar spacing 's smaller than the persistence length.

9. Lateral correlation e'ects in polyelectrolyte adsorption In this section we go beyond the mean-6eld approach by considering lateral correlation e8ects (for recent reviews on related subjects see [210,211]). The mean-6eld theories discussed before average the polymer pro6le in the lateral direction and only consider a spatially varying pro6le in the direction perpendicular to the substrate. Although mean-6eld equations can in principle be formulated which take also lateral order into account, this would be very involved and complicated. In this section we generalize the discussion of the single PE chain adsorption from Section 6 and consider the e8ect of interactions between di8erent adsorbed polymers on a simple scaling level. In order to do so, we assume that the adsorption energy is strong enough such that the polymers essentially lie 1at on the substrate. Lateral correlations are large enough to locally induce the polymers to form some type of ordered lattice. Due to the formation of two-dimensionally ordered adsorbed layers, the local chain structure becomes important and we therefore describe the PE chains as semi-1exible polymers in this section. We follow here the original ideas of Ref. [157], which were subsequently elaborated by Nguyen et al. [193]. To understand the idea, consider Fig. 24, where schematic top views of di8erent adsorbed phases are shown. A strongly adsorbed, 1at polymer phase can form a disordered surface pattern with many chain crossings, characterized by a certain mesh size 's which corresponds to the average distance between chain crossings. We distinguish two di8erent cases: if the e8ective persistence length ‘e8 is larger than the mesh size, we obtain a disordered uncrumpled phase, as depicted

66


schematically in Fig. 24a; if the e8ective persistence length is smaller than the mesh size, we expect a phase which is crumpled between consecutive chain crossings, as depicted in Fig. 24b. We also anticipate a lamellar phase where di8erent polymer strands are parallel locally, characterized by an average lamellar spacing 's , as shown in Fig. 24c. The lamellar phase is stabilized either by steric or by electrostatic repulsions between neighboring polymers; we will in fact encounter both stabilization mechanisms for di8erent values of the parameters. We now calculate the free energy and other characteristics of these adsorbed phases. In all the following calculations, we assume that we are inside the adsorbed regime of a single polymer, as discussed in Section 6. We basically assume, later on, that the desorption transitions obtained for the single-chain case also apply to the case of many-chain adsorption. As was shown in Ref. [157], to obtain the complete phase diagram it is suIcient to consider the lamellar phase depicted in Fig. 24c, since the other phase morphologies are metastable or degenerate. We assume that the distance between neighboring polymer strands, 's , is much smaller than the e8ective persistence length, 's ¡ ‘e8 (this assumption is checked self-consistently at the end). Since the possible conformations of the adsorbed polymers are severely restricted in the lateral directions, we have to include, in addition to the electrostatic interactions, a repulsive free energy contribution coming from steric interactions between sti8 polymers [160]. This is the same type of entropic repulsion that was used in Section 6 to estimate the pressure inducing desorption from a substrate. The total free energy density is given by 1 20‘B *+ ‘e8 Flam − + 1=3 5=3 ln + Frep ; (9.1) 's 's ‘e8 's where the 6rst term comes from the electrostatic attraction to the oppositely charged surface (which for consistency is taken to be penetrable to ions), the second term is the Odijk entropic repulsion [160] and Frep is the electrostatic repulsion of a lamellar array. To obtain the electrostatic repulsive energy, we 6rst note that the reduced potential created by a charged line with line charge density + = f=b at a distance 's is within the Debye–HKuckel approximation given by ∞ Vline ('s ) = + ds vDH ( '2s + s2 ) = 2‘B +K0 ['s ] ; (9.2) −∞

with the Debye–HKuckel potential vDH de6ned in Eq. (3.3). K0 denotes the modi6ed Bessel function. The repulsive electrostatic free energy density of an array of parallel lines with a nearest-neighbor distance of 's and line charge density + can thus be written as ∞ 2‘B +2 K0 [j's ] : Frep = 's j=1

(9.3)

This expression is also accurate for rods of 6nite radius d as long as d's holds. In the limit 's 1, when the distance between strands is much smaller than the screening length, the sum can be transformed into an integral and we obtain 2‘B +2 ∞ 0‘B +2 : (9.4) Frep ds K0 [s's ] = 2 's 's 0


67

This expression neglects e8ects due to the presence of a solid substrate. For example, and as discussed in Ref. [157], for a low-dielectric substrate the electrostatic interactions are enhanced by a factor of two close to the substrate surface, a rather small e8ect which will be neglected in the following. Since the average adsorbed surface charge density is given by *ads = +='s , it follows that the self 2 energy Eq. (9.4) in the limit 's 1 is given by Frep 0‘B *ads −1 and thus is identical to the self energy of a totally smeared-out charge distribution [157]. In this case, lateral correlations therefore do not matter. In the opposite limit, 's 1, when the polymers are much farther apart than the screening length, the sum in Eq. (9.3) is dominated by the 6rst term and (using the asymptotic expansion of the Bessel function) the free energy density (in units of kB T ) becomes √ 20‘B +2 e−'s Frep : (9.5) 's3=2 1=2 In this limit, it is important to note that the smeared-out repulsive energy Eq. (9.4) is much larger and thus considerably overestimates the actual electrostatic repulsion between polymer strands. Conversely, this reduction of the electrostatic repulsion between polymers results in an enormous overcharging of the substrate, as we will see shortly. In order to determine the equilibrium distance between the polymer strands, we balance the electrostatic attraction term, the 6rst term in Eq. (9.1), with the appropriate repulsion term. There are three choices to do this. For d ¡ −1 ¡ '∗s ¡ 's (with some crossover length '∗s to be determined later on), the electrostatic repulsion between the polymers is irrelevant (i.e. the last term in Eq. (9.1) can be neglected), and the lamellar phase is sterically stabilized in this case. The equilibrium lamellar spacing is given by 3=2 +*‘B ‘e8 's ∼ ln : (9.6) 1=3 +*‘B ‘e8 In all what follows, we neglect the logarithmic cofactor. For d ¡ −1 ¡ 's ¡ '∗s , the steric repulsion between the polymers is irrelevant (i.e. the second term in Eq. (9.1) can be neglected). The free energy is minimized by balancing the electrostatic adsorption term, the 6rst term in Eq. (9.1), with the electrostatic repulsion term appropriate for the case 's ¿ 1, Eq. (9.5), which leads to the electrostatically stabilized lamellar spacing + : (9.7) 's ∼ −1 ln * The adsorbed charge density then follows from *ads ∼ +='s as *ads ∼ *

+*−1 ln(+*−1 )

(9.8)

(note that in the previous section the adsorbed charge density was obtained as the product of the surface amount # and the charged-monomer fraction f, *ads = f#). Therefore, the electrostatically stabilized lamellar phase shows charge reversal as long as the spacing 's is larger than the screening length. As we will see, this is always the case. The crossover between the sterically stabilized lamellar phase, described by Eq. (9.6), and the lamellar phase which is stabilized by electrostatic

68


repulsion, Eq. (9.7), occurs when the predictions for 's become simultaneously equal to the crossover spacing '∗s , leading to a crossover for a surface charge density of (without logarithmic cofactors) *∼

5=3 1=3 +‘e8 ‘B

:

(9.9)

For * larger than the crossover value in Eq. (9.9) the distance between neighboring polymer strands is smaller than '∗s and the electrostatic stabilization mechanism is at work, for * smaller than the crossover value in Eq. (9.9) the lamellar spacing 's is larger than the characteristic crossover length '∗s and the Odijk repulsion dominates. One notes that the transition Eq. (9.9) is, on the scaling level, the same as the adsorption threshold in Eq. (6.5) and it is therefore not clear a priori whether the sterically stabilized lamellar phase exits. However, we note that additional non-electrostatic adsorption forces will stabilize the sterically stabilized lamellar phase which should therefore occur in a 6nite range of parameters [157]. The electrostatically stabilized lamellar phase crosses over to the charge-compensated phase when 's as given by Eq. (9.7) becomes of the order of the screening length −1 . In the charge-compensated phase, the lamellar spacing is obtained by balancing the electrostatic adsorption energy with the repulsion in the smeared-out limit Eq. (9.4) and is given by + 's : (9.10) * In this case the adsorbed surface charge density *ads = +='s exactly neutralizes the substrate charge density, *ads ∼ * :

(9.11)

The crossover between the charged-reversed phase and charge-compensated phase is obtained by matching Eqs. (9.7) and (9.10), leading to a threshold surface charge density of * ∼ + :

(9.12)

Finally, taking into account that the polymers have some width d, there is an upper limit for the amount of polymer that can be adsorbed in a single layer. Clearly, the lateral distance between polymers in the full phase is given by 's d

(9.13)

and thus the adsorbed surface charge density in the full phase reads + *ads = : (9.14) d The crossover between the full phase and the compensated phase is obtained by comparing Eqs. (9.10) and (9.13), leading to * ∼ +=d :

√

(9.15)

In Fig. 25 we show the adsorption diagram, for strongly charged polymers, de6ned by + ‘B ‘0 ¿ 1, as a function of the substrate charge density * and the inverse screening length . The electrostatically stabilized lamellar phase shows strong charge reversal as described by Eq. (9.8). At slightly larger surface charge densities we predict a charge-compensated phase which is not full (i.e. 's ¡ d) for a range of surface charge densities as determined by Eqs. (9.12) and (9.15). At even larger substrate charge density, the adsorbed polymer phase becomes close packed, i.e. 's = d. We note


69

σ full τ/d

compensated

σ∗ steric lamellar electr. lamellar

desorbed

d-1

κ

Fig. 25. Complete adsorption phase diagram as a function of the substrate charge density * and the inverse screening length . Note that we use logarithmic scales on both axes. We 6nd a desorbed regime, an adsorbed lamellar phase stabilized by electrostatic repulsions (which is strongly overcharged) and a lamellar phase which is stabilized by steric repulsion between polymer strands, an adsorbed charge-compensated phase, and a full phase, where the substrate charge cannot be compensated with a single adsorption layer because the layer is close-packed.

that since the full phase is not charge reversed, the full phase can consist of a second adsorbed layer (or even more layers). It should however be clear that close to charge compensation the e8ective substrate charge density an additional layer feels is so small that the condition for adsorption is not met. At low substrate charge densities the distance between adsorbed polymer strands becomes so large that the entropic repulsion between polymers dominates the electrostatic repulsion, and 6nally, at even lower charge densities, the polymers desorb. One notes that the transition between the electrostatically and sterically stabilized adsorbed phases, Eq. (9.9), has the same scaling form (disregarding logarithmic factors) as the desorption transition of semi-1exible polymers, Eq. (6.5). We have shifted the desorption transition to the right, though, because typically there are attractive non-electrostatic interactions as well, which tend to stabilize adsorbed phases. This is also motivated by the fact that the sterically stabilized phase has been seen in experiments on DNA adsorption, as will be discussed below. The critical charge density *∗ where the full phase, the electrostatically 1=3 and the sterically stabilized phases meet at one point, is given by *∗ = 1=(d5=3 ‘e8 +‘B ). In the phase diagram we have assumed that the charge density threshold for the full phase, * ∼ +=d, satis6es the inequality +=d ¿ *∗ , which for a fully charged PE at the Manning threshold, + = 1=‘B , amounts to the condition ‘e8 ¿ ‘B3 =d2 , which is true for a large class of PEs. The most important result of our discussion is that in the electrostatically stabilized phase the substrate charge is strongly reversed by the adsorbed polymer layer. This can give rise to a chargeoscillating multilayer formation if the adsorption of oppositely charged polymer is done in a second step. The general trend that emerges is that charge reversal is more likely to occur for intermediate salt concentrations and rather low substrate charge density. For too high-salt concentration and too low substrate charge density, on the other hand, the polymer does not adsorb at all. In essence, the salt concentration and the substrate charge density have to be tuned to intermediate values in order to create charge multilayers.

70


In experiments on DNA adsorbed on oppositely charged substrates one typically observes a lamellar phase [145,146]. In one experiment, the spacing between DNA strands was found to increase with increasing salt concentration [145]. One theoretical explanation invokes an e8ective interaction between neighboring DNA strands mediated by elastic deformations of the membrane, which forms the substrate in these experiments [212]. In the sterically stabilized regime, the distance between adsorbed polymers increases as 's ∼ 3=2 with the salt concentration, see Eq. (9.6), which o8ers an alternative explanation for the experimental 6ndings. It would be interesting to redo DNA adsorption experiments on rigid substrates, where the elastic coupling to the membrane is absent. For high enough substrate charge densities and by varying the salt concentration one should be able to see the crossover from the electrostatically stabilized phase, Eq. (9.7), where the DNA spacing decreases with added salt, to the sterically stabilized phase, Eq. (9.6), where the DNA spacing increases with added salt. Between the two limiting cases, di8usive mean-6eld adsorption pro6le with no lateral correlations (as treated in Section 8), and a 1at, two-dimensional adsorption layer with short-ranged lateral correlations (as discussed in this section), there clearly exists a continuous crossover. 10. Interaction between two adsorbed layers One of the many applications of polymers lies in their in1uence on the interaction between colloidal particles suspended in a solvent [97]. Depending on the details of substrate–polymer interactions and properties of polymers in solution, the e8ective interaction between colloids in a polymer solution can be attractive or repulsive, explaining why polymers are widely used as 1occulants and stabilizers in industrial processes [97]. The various regimes and e8ects obtained for the interaction of polymer solutions between two surfaces have recently been reviewed [213]. It transpires that force-microscope experiments done on adsorbed polymer layers form an ideal tool for investigating the basic mechanisms of polymer adsorption, colloidal stabilization and 1occulation. 10.1. Non-adsorbing polymers Let us 6rst discuss brie1y the relatively simple case when the polymers do not adsorb on the surface of the colloidal particles but are repelled from it. For low concentration of polymer, i.e. below the overlap concentration cm∗ , the depletion of polymer around the colloidal particles induces a strong attraction between the colloidal particles. The range of this attraction is about the same as the radius of an isolated polymer and can lead to polymer-induced 1occulation [214,215]. The e8ects of polymer excluded volume can be taken into account in analytical theories [216,217], while Monte-Carlo simulations in the grand-canonical ensemble con6rm the existence and characteristics of these depletion-induced attractive forces [218]. At polymer concentration higher than the overlap concentration, the depletion zones around the particles become of the order of the mesh-size in the solution. The attraction in this case is predicted to set in at separations equal to or smaller than the mesh-size [219]. The force apparatus was used to measure the interaction between depletion layers [96], as realized with polystyrene in toluene, which is a good solvent for polystyrene but does not favor the adsorption of polystyrene on mica surfaces. Surprisingly, the resultant depletion force is too weak to be detected.


71

10.2. Adsorbing neutral polymers (i) Equilibrium adsorption in good solvents: The case when polymers adsorb on the colloidal surface is much more complicated, and many cases have to be distinguished. If the polymer concentration is rather high and under good-solvent conditions, polymers show the experimentally well-known tendency to stabilize colloids against 1occulation, i.e., to induce an e8ective repulsion between the colloidal particles and to hinder them from coming close enough to each other so that van-der-Waals attractions will induce 1occulation [97]. We should also mention that in other applications, small polymer concentrations and high-molecular weight polymers are used in the opposite sense as 1occulants, to induce binding between unwanted sub-micron particles and, thereby, removing them from solution. It follows that adsorbing polymers can have di8erent e8ects on the stability of colloidal particles, depending on the detailed parameters. Hereafter, we assume that the polymers form an adsorbed layer around the colloidal particles, with a typical thickness much smaller than the particle radius, and curvature e8ects can be neglected. In that case, the e8ective interaction between the colloidal particles with adsorbed polymer layers can be traced back to the interaction energy between two planar substrates covered with polymer adsorption layers. In the case when the approach of the two particles is slow and the adsorbed polymer chains are in full equilibrium with the chains in solution, the interaction between two opposing adsorbed layers is predominantly attractive [220,221], mainly because polymers form bridges between the two surfaces. Recently, it has been shown that there is a small repulsive component to the interaction at large separations [222,223]. For the case of diblock copolymers, the force between two surfaces depends in a subtle way on the relative aInities of the blocks to the surfaces [224]. The typical equilibration times of polymers are extremely long. This holds in particular for adsorption and desorption processes, and is due to the slow di8usion of polymers and their rather high adsorption energies. Note that the adsorption energy of a polymer can be much higher than kB T even if the adsorption energy of a single monomer is small because many monomers of a single chain can be attached to the surface. Therefore, for the typical time scales of colloid contacts, the adsorbed polymers are not in equilibrium with the polymer solution. (ii) Constrained equilibrium: This is also the case for most of the experiments done with a surface-force apparatus, where two polymer layers adsorbed on crossed mica cylinders are brought in contact. In all these cases one has a constrained equilibrium situation, where the polymer con6gurations and thus the density pro6le can adjust only with the constraint that the total adsorbed polymer excess stays constant. This case has been 6rst considered by de Gennes [220] who found that two fully saturated adsorbed layers will strongly repel each other if the total adsorbed amount of polymer is not allowed to decrease. The repulsion is mostly due to osmotic pressure and originates from the steric interaction between the two opposing adsorption layers. It was experimentally veri6ed in a series of force-microscope experiments on polyethylene-oxide layers in water (which is a good solvent for PEO) [225]. (iii) Undersaturated layers: In other experiments, the formation of the adsorption layer is stopped before the layer is fully saturated. The resulting adsorption layer is called undersaturated. If two of those undersaturated adsorption layers approach each other, a strong attraction develops, which only at smaller separation changes to an osmotic repulsion [226]. The theory developed for such non-equilibrium conditions predicts that any surface excess lower than the one corresponding to full equilibrium will lead to attraction at large separations [227,228]. Similar mechanisms are valid for

72


colloidal suspensions, if the total surface available for polymer adsorption is large compared to the total amount of polymer in solution. In this case, the adsorption layers are undersaturated, and the resulting attraction is utilized in applications of polymers as 1occulation agents [97]. (iv) Bad solvent conditions: Another distinct mechanism leading to attractive forces between adsorption layers was investigated in experiments with dilute polymer solutions in bad solvents. As an example we mention polystyrene in cyclohexane below the theta temperature [229]. The subsequently developed theory [230] showed that the adsorption layers attract each other since the local concentration in the outer part of the adsorption layers is enhanced over the dilute solution and lies in the unstable two-phase region of the bulk phase diagram. Similar experiments have been repeated at the theta temperature [231]. (v) Dynamic e=ects: Additional e8ects that have been considered are the dynamical approach between two surfaces bearing adsorbed polymer layers, which is controlled by the 1ow of solvent through the polymer network aIxed to the surfaces [232]. 10.3. Adsorbing charged polymers More complicated e8ects are obtained for the interaction between two charged surfaces in the presence of oppositely charged PEs. Experimentally, this situation is encountered when one tries to 1occulate or stabilize charge-stabilized dispersions by the addition of oppositely charged PEs [97]. In the absence of added PEs, two similarly charged surfaces repel each other over a range of the order of the screening length in the case of added salt. This can be calculated on the mean-6eld level [191] and agrees quantitatively with Monte-Carlo simulations and experimental results for monovalent salt [233,234]. For divalent or trivalent salt mean-6eld theory becomes inaccurate and attractive forces are generated by ion–ion correlations [233,234]. Attractive forces between the surfaces can result, at some separation range, even on a mean 6eld level [191] from a combination of electrostatic interactions between all charged species and the adsorption energies of PE chains on the surfaces. In simulations [235,236] and mean-6eld theories [191,235,237,238] it has been found that the predominant e8ect of added PEs is an attraction between the surfaces, due to bridging between the surfaces and screening of the surface repulsion. Like in the case of neutral polymers between adsorbing surfaces, the force between the surfaces depends on the adsorbed amount. Salt can be used to control the amount of adsorbed polymers, and it also has an important e8ect on the net force. Since the adsorbed amount for highly charged PEs increases with added salt, the force becomes less attractive in this case and, for large salt concentration, is purely repulsive. For small salt concentrations, on the other hand, the attraction is strongest. Clearly, in the case of constrained equilibrium, i.e. when the amount of adsorbed polymer is 6xed as the plate separation changes, the force acquires an additional repulsive component as the plates approach each other, due to the force needed to compress the polymer layer. For larger separations and for undersaturated polymer layers, on the other hand, the forces are attractive. The precise crossover between attraction due to undersaturation (at large separation) to repulsion due to oversaturation (at small separations) depends on the adsorbed amount. This can be experimentally controlled for example by the total amount of added PE. Measurements of the disjoining pressure in thin liquid 6lms of PE solutions as a function of 6lm thickness demonstrated an oscillatory pressure [95,239–241] with a period of the oscillation of the order of the peak position in the bulk structure factor (which was discussed in Section 3.6).


73

Theoretically, those oscillations have been seen in mean-6eld calculations [204] as well as more elaborate integral-equation calculations [242]. An e8ect which is missed by mean-6eld theories is the so-called mosaic-binding of charged surfaces in the presence of a very low concentration of oppositely charged PE [97]. In this case the adsorbed layers of the separate surfaces are very undersaturated. Individual polymer coils form isolated patches on the substrate, where the local surface charge is reversed. The substrate shows a mosaic pattern of oscillating charge patches. If two of those patterned surfaces approach each other, the patterns will readjust in order to match oppositely charged patches, resulting in a very strong, irreversible binding [97]. 11. Polymer adsorption on heterogeneous surfaces Polymer adsorption can be coupled in a subtle way with lateral changes in the chemical composition or density of the surface. Such a surface undergoing lateral rearrangements at thermodynamical equilibrium is called an annealed surface [243,244]. A Langmuir monolayer of insoluble surfactant monolayers at the air/water interface is an example of such an annealed surface. As function of the temperature change, a Langmuir monolayer can undergo a phase transition from a high-temperature homogeneous state to a low-temperature demixed state, where domains of dilute and dense regions coexist. Alternatively, the transition from a dilute phase to a dense one may be induced by compressing the monolayer at constant temperature, in which case the adsorbed polymer layer contributes to the pressure [245]. The domain boundary between the dilute and dense phases can act as nucleation site for adsorption of bulky molecules [246]. The case where the insoluble surfactant monolayer interacts with a semi-dilute polymer solution solubilized in the water subphase was considered in some detail. The phase diagrams of the mixed surfactant/polymer system were investigated within the framework of mean-6eld theory [247]. The polymer enhances the 1uctuations of the monolayer and induces an upward shift of the critical temperature. The critical concentration is increased if the monomers are more attracted (or at least less repelled) by the surfactant molecules than by the bare water/air interface. In the case where the monomers are repelled by the bare interface but attracted by the surfactant molecules (or vice versa), the phase diagram may have a triple point. The location of the polymer desorption transition line (i.e., where the substrate–polymer interaction changes from being repulsive to being attractive) appears to have a big e8ect on the phase diagram of the surfactant monolayer [247]. A similar e8ect is seen with DNA which adsorbs on a mixed lipid bilayer consisting of cationic and neutral lipid molecules [147]. Experimentally, it is seen that the negatively charged DNA attracts the positively charged lipid molecules and leads to a local demixing of the membrane [147]. Theoretically, this can be studied by formulating the Poisson–Boltzmann theory for a single charged cylinder (which models the rigid DNA molecule) at some distance from a surface with mobile charged lipids of a given density and size [248]. For low-salt concentrations, the charged DNA leads to a strong accumulation of cationic lipids in its vicinity. Depending on the size of the lipid heads, this lipid concentration pro6le can extend far away from the cylinder. For high-salt concentrations on the other hand, this accumulation e8ect is much weaker due to screening. Similar e8ects have been studied for periodic arrays of adsorbed DNA cylinders [249,250] which describe experimental results for bulk DNA–cationic lipid complexes [146].

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The adsorption of DNA on laterally structured substrates was recently characterized by direct AFM visualization [251]. Patches of positively charged lipids were embedded in a matrix of negative surface potential, and the size of the cationic surface patches was varied from the micrometer down to the nanometer scale. DNA adsorption was found to depend both on the average surface charge density and on the size of positively charged patches. Similar phenomena were studied theoretically using o8-lattice Monte-Carlo simulations [252,253]. 12. Polymer adsorption on curved and ,uctuating interfaces 12.1. Neutral polymers The adsorption of polymers on rough substrates is of high interest to applications. One example is the reinforcement of rubbers by 6ller particles such as carbon black or silica particles [254]. Theoretical models considered sinusoidal surfaces [255], rough and corrugated substrates [256,257]. In all cases, enhanced adsorption was found and rationalized in terms of the excess surface available for adsorption. The adsorption on macroscopically curved bodies, such as spheres and cylinders, leads to modi6ed adsorption pro6les [258]. Of considerable interest is the e8ective interaction between two colloidal particles covered by adsorption layers [259]. Another application is obtained for the adsorption of polymers on 1exible interfaces or membranes [243,260,261]. Here one interesting aspect concerns the polymer-induced contribution to the elastic bending moduli of the 1exible surface. The elastic energy of a liquid-like membrane can be expressed in terms of two bending moduli, and G . The elastic energy (per unit area) is (12.1) (c1 + c2 − 2c0 )2 + G c1 c2 ; 2 where and G are the elastic and Gaussian bending moduli, respectively. The principle curvatures of the surface are given by c1 and c2 , and c0 is the spontaneous curvature. Quite generally, in presence of adsorbing polymers G turns out to be positive and thus favors the formation of surfaces with negative Gaussian curvature. An ‘egg-carton’ structure is an example to such a multi-saddle surface. On the other hand, the e8ective is reduced, leading to a more deformable and 1exible surface due to the adsorbed polymer layer [243,262,263]. The spontaneous curvature c0 is only non-zero if the adsorption pro6le is di8erent on both sides of the membrane [260]. This can be achieved, for example, by incubating vesicle solutions with polymers, so that the vesicle interior is devoid of polymers (neglecting polymer translocation through the membrane which is indeed a rather slow process). If the polymers do not adsorb on the membrane, the spontaneous curvature is such that the membrane bends towards the polymer solution [216,217]. If, on the other hand, the polymers do adsorb on the membrane, the membrane bends away from the polymer solution with a continuous crossover between the two cases as the adsorption strength is varied [264]. 12.2. Charged polymers Of particular interest is the adsorption of strongly charged polymers on oppositely charged cylinders [265–267] and spheres [268–273], because these are geometries encountered in many colloidal


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150 wrapped

Z

100

50

unwrapped 0 0

2

4

6

8

10

κ [nm-1]

Fig. 26. Numerically determined adsorption diagram for a negatively charged semi-1exible polymer of length L = 50 nm, linear charge density + = 6 nm−1 , persistence length ‘0 = 30 nm, interacting with an oppositely charged sphere of radius Rsp = 5 nm. Shown is the main transition from the unwrapped con6guration (at the bottom) to the wrapped con6guration (at the top) as a function of sphere charge Z and inverse Debye–HKuckel screening length . Wrapping is favored at intermediate salt concentrations. The parameters are chosen for the problem of DNA–histone complexation. Adapted from Ref. [275].

science applications and in bio-cellular processes. When the curvature of the small colloidal particles is large enough, it can lead to a much more pronounced e8ect for PE adsorption as compared with neutral polymer. This is mainly due to the fact that the electrostatic energy of the adsorbed PE layer depends sensitively on curvature [269,272–274]. Bending a charged polymer around a small sphere costs a large amount of electrostatic energy, which will disfavor adsorption of long, strongly charged PE at too low-salt concentration. In Fig. 26 we show the adsorption phase diagram of a single sti8 PE of 6nite length which interacts with an oppositely charged sphere of charge Z (in units of e). The speci6c parameters were chosen as appropriate for the complexation of DNA (a negatively charged, relatively sti8 biopolymer) with positively charged histone proteins, corresponding to a DNA length of L = 50 nm, a chain persistence length of ‘0 = 30 nm, and a sphere radius of Rsp = 5 nm. The phase diagram was obtained by minimization of the total energy including bending energy of the DNA, electrostatic attraction between the sphere and the DNA, and electrostatic repulsion between the DNA segments [275]. All interactions are represented by screened Debye–HKuckel potentials of the form of Eq. (3.3). Fluctuations of the DNA shape are unimportant for such sti8 polymers. Therefore, the ground-state analysis performed is an acceptable approximation. We show in Fig. 26 the main transition between an unwrapped state, at low sphere charge Z, and the wrapped state, at large sphere charge Z. It is seen that at values of the sphere charge between Z = 10 and 130 the wrapping only occurs at intermediate values of the inverse screening length 1=2 ∼ csalt . At low-salt concentrations (lower left corner in the phase diagram), the self-repulsion between DNA segments prevents wrapping, while at large salt concentrations (lower right corner in the diagram), the electrostatic attraction is not strong enough to overcome the mechanical bending energy of the DNA molecule. These results are in good agreement with experiments on DNA/histone complexes [276]. Interestingly, the optimal salt concentration, where a minimal sphere charge is needed to wrap the DNA, occurs at physiological salt concentration, for −1 ≈ 1 nm. For colloidal

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particles of larger size and for 1exible synthetic polymers, con6gurational 1uctuations become important. They have been treated using a mean-6eld description in terms of the average monomer density pro6le around the sphere [268,270]. 13. Grafted polymer chains The discussion so far assumed that all monomers of a polymer are alike, showing the same tendency to adsorb to the substrate surface. For industrial and technological applications, one is often interested in end-functionalized polymers. These are polymers which attach with one end only to the substrate, as is depicted in Fig. 13b, while the rest of the polymer is not particularly attracted to (or even repelled from) the grafting surface. Hence, it attains a random-coil structure in the vicinity of the surface. Another possibility of block copolymer grafting, as shown in Fig. 13c, will be brie1y discussed below as well. The motivation to study such terminally attached polymers lies in their enhanced power to stabilize particles and surfaces against 1occulation. Attaching a polymer by its end to the surface opens up a much more e8ective route to stable surfaces. Bridging and creation of polymer loops on the same surface, as encountered in the case of homopolymer adsorption (and leading to attraction between two particle surfaces and destabilization, see Section 10), do not occur if the main-polymer section is chosen such that it does not adsorb to the surface. Experimentally, the end-adsorbed polymer layer can be built in several di8erent ways, depending on the application in mind. First, one of the polymer ends can be chemically bound to the grafting surface, leading to a tight and irreversible attachment [189] shown schematically in Fig. 13b. The second possibility consists of physical adsorption of a specialized end-group which favors interaction with the substrate. For example, polystyrene chains have been used which contain a zwitterionic end group that adsorbs strongly on mica sheets [277]. Physical grafting is also possible with a suitably chosen diblock copolymer (Fig. 13c), e.g., a polystyrene–polyvinylpyridine (PS–PVP) diblock in the solvent toluene at a quartz substrate [278]. Toluene is a selective solvent for this diblock. The PVP (polyvinylpyridine) block is strongly adsorbed to the quartz substrate and forms a collapsed anchor, while the PS (polystyrene) block is under good-solvent conditions. It does not adsorb to the substrate and remains solubilized in the solvent. General adsorption scenarios for diblock copolymers have been theoretically discussed, both for selective and non-selective solvents, with special consideration to the case when the asymmetry of the diblock copolymer, i.e., the length di8erence between the two blocks, is large [279]. Another experimental realization is possible with diblock copolymers which are anchored at the liquid–air [280] or at a liquid–liquid interface of two immiscible liquids [281]. This scenario o8ers the advantage that the surface pressure can be directly measured. A well studied example is that of a diblock copolymer of polystyrene–polyethylene oxide (PS–PEO). The PS block is shorter and functions as an anchor at the air/water interface because it is immiscible in water. The PEO block is miscible in water but because of attractive interaction with the air/water interface it forms a quasi-two dimensional layer at very low surface coverage. As the surface pressure increases and the area per polymer decreases, the PEO block is expelled from the surface and forms a quasi polymer ‘brush’. In the following we simplify the discussion by assuming that the polymers are irreversibly grafted at one of their chain ends to the substrate. We limit the discussion to good solvent conditions and


77

R ρ-1/2

(a)

D

(b)

ρ-1/2

Fig. 27. For grafted chains, one distinguishes between: (a) the mushroom regime, where the distance between chains, )−1=2 , is larger than the size of a polymer coil; and, (b) the brush regime, where the distance between chains is smaller than the unperturbed coil size. Here, the chains are stretched away from the surface due to repulsive interactions between monomers. The brush height D scales linearly with the polymerization index, D ∼ N , and thus is larger than the unperturbed coil radius R ∼ aN & .

absence of any attractive interactions between the polymer chains and the surface. The important new system parameter is the grafting density (or area per chain) ), which is the inverse of the average area available for each polymer at the surface. For small grafting densities, ) ¡ )∗ , the polymer chains will be far apart from each other and hardly interact, as schematically shown in Fig. 27a. The overlap grafting density for chains in good solvent conditions (swollen chains) is )∗ ∼ a−2 N −6=5 , where N is the polymerization index [282]. For large grafting densities, ) ¿ )∗ , the chains begin to overlap. Since we assume the solvent to be good, monomers repel each other. The lateral separation between the polymer coils is 6xed by the grafting density, so that the polymers extend away from the grafting surface in order to avoid each other, as depicted in Fig. 27b. The resulting structure is called a polymer ‘brush’, with a vertical height D which greatly exceeds the unperturbed coil radius [282,283]. Similar stretched structures occur in many other situations, such as diblock copolymer melts in the strong segregation regime, or star polymers under good solvent conditions [284]. The universal occurrence of stretched polymer con6gurations in several seemingly unconnected situations warrants a detailed discussion. Below, this discussion is separated for neutral and charged grafted chains. 13.1. Neutral grafted polymers The scaling behavior of the brush height D can be analyzed using a Flory-like mean-6eld theory, which is a simpli6ed version of the original Alexander theory [283] for polymer brushes. The stretching of the chain leads to an entropic free energy loss of D2 =(a2 N ) per chain, and the repulsive energy density due to unfavorable monomer–monomer contacts is proportional to the squared monomer density times the excluded-volume parameter v2 (introduced in Section 2.2). The free

78


energy per chain (and in units of kB T ) is then D2 )N 2 D : F 2 + v2 aN D )

(13.1)

The equilibrium height is obtained by minimizing Eq. (13.1) with respect to D, and the result is Deq = N (2v2 a2 )=3)1=3 ;

(13.2)

where the numerical constants have been added for numerical convenience in the following considerations. The vertical size of the brush scales linearly with the polymerization index N , a clear signature of the strong stretching of the polymer chains, as was originally obtained by Alexander [283]. At the overlap threshold, )∗ ∼ a−2 N −6=5 , the height scales as Deq ∼ N 3=5 , and thus agrees with the scaling of an unperturbed chain radius in a good solvent, Eq. (2.8), as it should. The simple scaling calculation predicts the brush height D correctly in the asymptotic limit of long chains and strong overlap. It has been con6rmed by experiments [189,277,278] and computer simulations [285,286]. The above scaling result assumes that all chains are stretched to exactly the same height, leading to a step-like shape of the density pro6le. Monte-Carlo and numerical mean-6eld calculations con6rm the general scaling of the brush height, but exhibit a more rounded monomer density pro6le which goes continuously to zero at the outer perimeter [285]. A big step towards a better understanding of stretched polymer systems was made by Semenov [287], who recognized the importance of classical paths for such systems. The classical polymer path is de6ned as the path which minimizes the free energy, for a given start and end positions, and thus corresponds to the most likely path a polymer can take. The name follows from the analogy with quantum mechanics, where the classical motion of a particle is given by the quantum path with maximal probability. Since for strongly stretched polymers the 1uctuations around the classical path are weak, it is expected that a theory that takes into account only classical paths, is a good approximation in the strong-stretching limit. To quantify the stretching of the brush, let us introduce the (dimensionless) stretching parameter A, de6ned as A≡N

3v22 )2 2a2

1=3

3 = 2

Deq aN 1=2

2

;

(13.3)

where Deq is the brush height according to Alexander’s theory, compare Eq. (13.2). The parameter A is proportional to the square of the ratio of the Alexander prediction for the brush height, Deq , and the unperturbed Gaussian chain radius R ∼ aN 1=2 , and, therefore, is a measure of the stretching of the brush. Constructing a classical theory in the in6nite-stretching limit, de6ned as the limit A → ∞, it was shown independently by Milner et al. [288] and Skvortsov et al. [289] that the resulting monomer volume-fraction pro6le depends only on the vertical distance from the grafting surface and has in fact a parabolic pro6le. Normalized to unity, the density pro6le is given by ,(x) =

30 4

2=3

−

0x 2Deq

2

:

(13.4)


79

β=100

1.75 β=10

φ

1.25

β=1

0.75

β=0.1

0.25 0

0

0.4

0.8

1.2

x/D

Fig. 28. Results for the density pro6le (normalized to unity) of a strongly compressed brush, as obtained within mean-6eld theory. As the compression increases, described by the stretching parameter A, which varies from 0.1 (dots) to 1 (dash-dots), 10 (dashes), and 100 (solid line), the density pro6le approaches the parabolic pro6le (shown as a thick, dashed line) obtained within a classical-path analysis (adapted from Ref. [290]).

The brush height, i.e., the value of x for which the monomer density becomes zero, is given by x∗ = (6=02 )1=3 Deq and is thus proportional to the scaling prediction for the brush height, Eq. (13.2). The parabolic brush pro6le has subsequently been con6rmed in computer simulations [285,286] and experiments [189] as the limiting density pro6le in the strong-stretching limit, and constitutes one of the cornerstones in this 6eld. Intimately connected with the density pro6le is the distribution of polymer end points, which is non-zero everywhere inside the brush, in contrast with the original scaling description leading to Eq. (13.2). However, deviations from the parabolic pro6le become progressively important as the length of the polymers N or the grafting density ) decreases. In a systematic derivation of the mean-6eld theory for Gaussian brushes [290] it was shown that the mean-6eld theory is characterized by a single parameter, namely the stretching parameter A. In the limit A → ∞, the di8erence between the classical approximation and the mean-6eld theory vanishes, and one obtains the parabolic density pro6le. For 6nite A the full mean-6eld theory and the classical approximation lead to di8erent results and both show deviations from the parabolic pro6le. In Fig. 28 we show the density pro6les (normalized to unity) for four di8erent values of A, obtained with the full mean-6eld theory [290]. The parameter values used are A = 100 (solid line), A = 10 (thin dashed line), A = 1 ( dotted-dashed line), and A = 0:1 (dotted line). For comparison, we also show the asymptotic result according to Eq. (13.4) as a thick dashed line. In contrast to earlier numerical implementations [1], the self-consistent mean-6eld equations were solved in the continuum limit, where the results depend only on the single parameter A and direct comparison with other continuum theories becomes possible. Already for A = 100 the density pro6le obtained within mean-6eld theory is almost indistinguishable from the parabolic pro6le denoted by a thick dashed line. Experimentally, the highest achievable A values are in the range of A 20. Namely, deviations from the asymptotic parabolic pro6le are important. For moderately large values of A ¿ 10, the classical approximation (not shown here), derived from the mean-6eld theory by taking into account only one polymer path per end-point position, is still a good approximation, as judged by comparing

80


density pro6les obtained from both theories [290], except very close to the surface. Unlike mean-6eld theory, the classical theory misses completely the depletion e8ects at the substrate. Depletion e8ects at the substrate lead to a pronounced density depression close to the grafting surface, as is clearly visible in Fig. 28. A further interesting question concerns the behavior of individual polymer paths. As was already discussed for the in6nite-stretching theories (A → ∞), polymers paths can end at any distance from the surface. Analyzing the polymer paths which end at a common distance from the surface, two rather unexpected features are obtained: (i) free polymer ends, in general, are stretched; and, (ii) the end-points lying close to the substrate are pointing towards the surface (such that the polymer path 6rst turns away from the grafting surface before moving back towards it). In contrast, end-points lying beyond a certain distance from the substrate, point away from the surface (such that the paths move monotonously towards the surface). We should point out that these two features have been recently con6rmed in molecular-dynamics simulations [291]. They are not an artifact of the continuous self-consistent theory used in Ref. [290] nor are they due to the neglect of 1uctuations. These are interesting results, especially since it has been long assumed that free polymer ends are unstretched, based on the assumption that no forces act on free polymer ends. Let us now turn to the thermodynamic behavior of a polymer brush. Using the Alexander description, we can calculate the free energy per chain by putting the result for the optimal brush height, Eq. (13.2), into the free-energy expression, Eq. (13.1): F ∼ N (v2 )=a)2=3 :

(13.5)

In the presence of excluded-volume correlations, i.e., when the chain overlap is rather moderate, the brush height D is still correctly predicted by the Alexander calculation, but the prediction for the free energy is in error. Including correlations [283], the free energy is predicted to scale as F ∼ N)5=6 . The osmotic surface pressure ( is related to the free energy per chain by ( = )2

9F ; 9)

(13.6)

and should thus scale as ( ∼ )5=3 in the absence of correlations, and as ( ∼ )11=6 in the presence of correlations. However, these theoretical predictions do not compare well with all experimental results for the surface pressure of a compressed brush [280]. Currently there is still some debate about the cause for this discrepancy. An alternative theoretical method to study tethered chains is the so-called single-chain mean-6eld method [292], where the statistical mechanics of a single chain is treated exactly, and the interactions with the other chains are taken into account on a mean-6eld level. This method is especially useful for short chains, where 1uctuation e8ects are important, and for dense systems, where excluded volume interactions play a role. The calculated pro6les and brush heights agree very well with experiments and computer simulations. Moreover, these calculations explain the pressure isotherms measured experimentally [280] and in molecular-dynamics simulations [293]. As we described earlier, the main interest in end-adsorbed or grafted polymer layers stems from their ability to stabilize surfaces against van-der-Waals attraction. The force between colloids with grafted polymers is repulsive if the polymers do not adsorb on the grafting substrates [294]. This is in accord with our discussion of the interaction between adsorption layers, where attraction was found to be caused mainly by bridging and creation of polymer loops, which of course are absent for non-adsorbing brushes. A stringent test of brush theories was possible with accurate experimental


81

measurements of the repulsive interaction between two opposing grafted polymer layers using a surface force apparatus [277]. The resultant force could be 6tted very nicely by the in6nite-stretching theory of Milner et al. [295]. It was also shown that polydispersity e8ects, as appear in experiments, have to be taken into account theoretically in order to obtain a good 6t of the data [296]. 13.2. Solvent and substrate e=ects on polymer grafting So far we assumed that the polymer grafted layer is in contact with a good solvent. In this case, the grafted polymers try to minimize their mutual contacts by stretching out into the solvent. If the solvent is bad, the monomers try to avoid the solvent by forming a collapsed brush, the height of which is considerably reduced with respect to the good-solvent case. It turns out that the collapse transition, which leads to phase separation in the bulk, is smeared out for the grafted layer and does not correspond to a true phase transition [297]. The height of the collapsed layer scales linearly in )N , which re1ects the constant density within the brush, in agreement with experiments [298]. Some interesting e8ects have been described theoretically [299] and experimentally [298] for brushes in mixtures of good and bad solvent, which can be rationalized in terms of a partial solvent demixing. For a theta solvent (v2 =0) the relevant interaction is described by the third-virial coeIcient; using a simple Alexander approach similar to the one leading to Eq. (13.2), the brush height is predicted to vary with the grafting density as D ∼ )1=2 , in agreement with computer simulations [300]. Up to now we discussed planar grafting layers. It is of much interest to consider the case where polymers are grafted to curved surfaces. The 6rst study taking into account curvature e8ects of stretched and tethered polymers was done in the context of star polymers [301]. It was found that chain tethering in the spherical geometry leads to a universal density pro6le, showing a densely packed core, an intermediate region where correlation e8ects are negligible and the density decays as ,(r) ∼ 1=r, and an outside region where correlations are important and the density decays as , ∼ r −4=3 . These considerations were extended using the in6nite-stretching theory of Milner et al. [302], self-consistent mean-6eld theories [303], and molecular-dynamics simulations [304]. Of particular interest is the behavior of the bending rigidity of a polymer brush, which can be calculated from the free energy of a cylindrical and a spherical brush and forms a conceptually simple model for the bending rigidity of a lipid bilayer [305]. A di8erent scenario is obtained with special functionalized lipids linked to the polymer chain. If such lipids are incorporated into lipid vesicles, the water-soluble polymers (typically one uses PEG (poly-ethylene glycol) for its non-toxic properties) form well-separated mushrooms, or, at higher concentration of PEG lipid, a dense brush. These modi6ed vesicles are very interesting in the context of drug delivery, because they show prolonged circulation times in vivo [306]. This is probably due to a steric serum-protein-binding inhibition by the hydrophilic brush coat consisting of the PEG lipids. Since the lipid bilayer is rather 1exible and undergoes thermal bending 1uctuations, there is an interesting coupling between the polymer density distribution and the membrane shape [98,307,308]. For non-adsorbing, anchored polymers, the membrane will bend away from the polymer due to steric repulsion [309–311], but for adsorbing anchored polymer the membrane will bend towards the anchored polymer [312,313]. The behavior of a polymer brush in contact with a solvent, which is by itself also a polymer, consisting of chemically identical but somewhat shorter chains than the brush, had been 6rst considered by de Gennes [282]. A complete scaling description has been given only recently [314]. One

82


distinguishes di8erent regimes where the polymer solvent is expelled to various degrees from the brush. A somewhat related question concerns the behavior of two opposing brushes brought closely together, and separated by a solvent consisting of a polymer solution [315]. Here one distinguishes a regime where the polymer solution leads to a strong attraction between the surfaces via the ordinary depletion interaction (compare to Ref. [219]), but also a high polymer concentration regime where the attraction is not strong enough to induce colloidal 1occulation. This phenomenon is called colloidal restabilization [315]. Considering a mixed brush made of mutually incompatible grafted chains, a novel transition to a brush characterized by a lateral composition modulation was found [316]. Even more complicated spatial structures are obtained with grafted diblock copolymers [317]. Finally, we would like to mention in passing that these static brush phenomena have interesting consequences on dynamic properties of polymer brushes [318]. 13.3. Charged grafted polymers Brushes can also be formed by charged polymers which are densely end-grafted to a surface; they are called polyelectrolyte or charged brushes. They have been the focus of numerous theoretical [319–327] and experimental [328–330] studies. In addition to the basic interest, charged brushes are considered for their applications because they serve as an eIcient mean for preventing colloids in polar media (such as aqueous solutions) from 1occulating and precipitating out of solution [97]. This stabilization arises from steric (entropic) as well as electrostatic repulsion. A strongly charged brush is able to trap its own counterions and generates a layer of locally enhanced salt concentration [321]. It is thus less sensitive to the salinity of the surrounding aqueous medium than a stabilization mechanism based on pure electrostatics (i.e. without polymers). Little is known from experiments on the scaling behavior of PE brushes, as compared to neutral brushes. The thickness of the brush layer has been calculated from neutron-scattering experiments on end-grafted polymers [328] and charged diblock-copolymers at the air–water interface [330]. Theoretical work on PE brushes was initiated by the works of Miklavic and Marcelja [319] and Misra et al. [320]. In 1991, Pincus [321] and Borisov et al. [322] presented scaling theories for charged brushes in the so-called osmotic regime, where the brush height results from the balance between the chain elasticity (which tends to decrease the brush height) and the repulsive osmotic counterion pressure (which tends to increase the brush height). In later studies, these works have been generalized to poor solvents [323] and to the regime where excluded volume e8ects become important, the so-called quasi-neutral or Alexander regime [326]. In what follows we assume that the charged brush is characterized by two length scales: the average vertical extension of polymer chains from the surface D, and the typical extent of the counterion cloud, denoted by H . We neglect the presence of additional salt, which has been discussed extensively in the original literature, and only consider screening e8ects due to the counterions of the charged brush. Two di8erent scenarios emerge, as is schematically presented in Fig. 29. The counterions can either extend outside the brush, H D, as shown in (a), or be con6ned inside the brush, H ≈ D as shown in (b). As we show now, case (b) is indicative of strongly charged brushes, while case (a) is typical for weakly charged brushes. The free energy density per unit area and in units of kB T contains several contributions, which we now calculate one by one. We recall that the grafting density of PEs is denoted by ), z is the

R.R. Netz, D. Andelman / Physics Reports 380 (2003) 1 – 95 D

H +

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H D

Fig. 29. Schematic PE brush structure. In (a) we show the weak-charge limit where the counterion cloud has a thickness H larger than the thickness of the brush layer, D. In (b) we show the opposite case of the strong-charge limit, where all counterions are contained inside the brush and a single length scale D ≈ H exists.

counterion valency, N the polymerization index of grafted chains, and f the charge fraction. The osmotic free energy, Fos , associated with the ideal entropy cost of con6ning the counterions to a layer of thickness H is given by Nf) Nf) : (13.7) Fos ln z zH Fv2 is the second virial contribution to the free energy, arising from steric repulsion between the monomers (contributions due to counter ions are neglected). Throughout this section, the polymers are assumed to be in a good solvent (positive second virial coeIcient v2 ¿ 0). The contribution thus reads 2 N) 1 Fv2 Dv2 : (13.8) 2 D

84


Finally, a direct electrostatic contribution Fel occurs if the PE brush is not locally electro-neutral throughout the system, as for example is depicted in Fig. 29a. This energy is given by Fel =

20‘B (Nf))2 (H − D)2 : 3 H

(13.9)

This situation arises in the limit of low charge, when the counterion density pro6le extends beyond the brush layer, i.e. H ¿ D. The last contribution is the stretching energy of the chains which is Fst =

3D2 ) : 2Na2

(13.10)

Here, a is the Kuhn length of the polymer, implying that we neglect any chain sti8ness for the brush problem. The di8erent free energy contributions lead, upon minimization with respect to the two length scales H and D, to di8erent behaviors. Let us 6rst consider the weak charging limit, i.e. the situation where the counterions leave the brush, H ¿ D. In this case, minimization of Fos + Fel with respect to the counterion height H leads to H∼

1 z‘B Nf)

(13.11)

which has the same scaling as the Gouy–Chapman length for z-valent counterions at a surface of surface charge density *=Nf). Balancing now the polymer stretching energy Fst and the electrostatic energy Fel one obtains the so-called Pincus brush height D N 3 )a2 ‘B f2 ;

(13.12)

which results from the electrostatic attraction between the counterions and the charged monomers. One notes the peculiar dependence on the polymerization index N . In the limit of H ≈ D where D given by Eq. (13.12), the PE brush can be considered as neutral and the electrostatic energy vanishes. There are two ways of balancing the remaining free energy contributions. The 6rst is obtained by comparing the osmotic energy of counterion con6nement, Fos , with the polymer stretching term, Fst , leading to the height D∼

Naf1=2 ; z 1=2

(13.13)

constituting the so-called osmotic brush regime. Finally comparing the second-virial free energy, Fv2 , with the polymer stretching energy, Fst , one obtains D ∼ Na(v2 )=a)1=3 ;

(13.14)

and the PE brush is found to have the same scaling behavior as the neutral brush [283,282], compare Eq. (13.2). Comparing the brush heights in all three regimes we arrive at the phase diagram shown in Fig. 30. The three scaling regimes coexist at the characteristic charge fraction 1=3 zv2 ; (13.15) fco ∼ N 2 a 2 ‘B


85

ρ 3/2

ρco

osmotic brush

-3/2 neutral brush

Pincus brush

-3 fco

f

Fig. 30. Scaling diagram for PE brushes on a log–log plot as a function of the grafting density ) and the fraction of charged monomers f. Featured are the Pincus-brush regime, where the counterion layer thickness is much larger than the brush thickness, the osmotic-brush regime, where all counterions are inside the brush and the brush height is determined by an equilibrium between the counterion osmotic pressure and the PE stretching energy, and the neutral-brush regime, where charge e8ects are not important and the brush height results from a balance of PE stretching energy and second-virial repulsion. The power law exponents of the various lines are denoted by numbers.

and the characteristic grafting density )co ∼

1 N‘B1=2 v21=2

:

(13.16)

For large values of the charge fraction f and the grafting density ) it has been found numerically that the brush height does not follow any of the scaling laws discussed here [331]. This has been recently rationalized in terms of another scaling regime, the collapsed regime. In this regime one 6nds that correlation and 1uctuation e8ects, which are neglected in the discussion in this section, lead to a net attraction between charged monomers and counterions [332]. Similarly, two charged surfaces, one decorated with a charged brush, the other one neutralized by counter ions, attract each other at large enough grafting densities [333]. Another way of creating a charged brush is to dissolve a diblock copolymer consisting of a hydrophobic and a charged block in water. The diblocks associate to form a hydrophobic core, thereby minimizing the unfavorable interaction with water, while the charged blocks form a highly charged corona or brush [334]. The micelle morphology depends on di8erent parameters. Most importantly, it can be shown that salt acts as a morphology switch, giving rise to the sequence spherical, cylindrical, to planar micellar morphology as the salt concentration is increased [334]. Theoretically, this can be explained by the entropy cost of counterion con6nement in the charged corona [335]. The charged corona can be studied by neutron scattering [336] or atomic-force microscopy [337] and gives information on the behavior of highly curved charged brushes.

86


14. Concluding remarks We reviewed simple physical concepts underlying the main theories which deal with equilibrium and static properties of neutral and charged polymers adsorbed or grafted to substrates. Most of the review dealt with somewhat ideal situations: smooth and 1at surfaces which are chemically homogeneous; long and linear homopolymer chains where chemical properties can be averaged on; simple phenomenological type of interactions between the monomers and the substrate as well as between the monomers and the solvent. Even with all the simplifying assumptions, the emerging physical picture is quite rich and robust. Adsorption of polymers from dilute solutions can be understood in terms of single-chain adsorption on the substrate. Mean-6eld theory is quite successful but in some cases 1uctuations in the local monomer concentration play an important role. Adsorption from more concentrated solutions results in even more complex density pro6les, with several regimes (proximal, central, distal). Each regime is characterized by a di8erent physical behavior. We reviewed the principle theories used to model the polymer behavior. We also mentioned brie1y more recent ideas about the statistics of polymer loops and tails. For charged polymers, the structure of the adsorbed layer is in part controlled by the counterion distribution which is coupled to the polymer layer. The second part of this review is about neutral and charged polymers which are terminally grafted on one end to the surface and are called polymer brushes. The theories here are quite di8erent since the statistics of the grafted layer depends crucially on the fact that the chain is not attracted to the surface but is forced to be in contact to the surface since one of its ends is chemically or physically bonded to the surface. Here as well we review the classical mean-6eld theory and more advanced theories giving the concentration pro6les of the entire polymer layer as well as that of the polymer free ends. In general, the theory for neutral polymers is more advanced than the one for charged polymers, partly because charged polymers became the target for theoretical modelling fairly recently. In addition, due to the long-range interactions between charged monomers, and due to a number of additional relevant parameters (such as salt concentration, pH), the resultant behavior for charged polymers is more complex. We have introduced some of the basic concepts of charged polymers, such as the Manning condensation of counterions and the electrostatic chain sti8ening. Due to this increased sti8ness of polyelectrolytes, their chain statistics is described by semi-1exible models. We have, therefore, introduced such models in some detail and also demonstrated some e8ects speci6c to semi-1exible charged polymers. At present, studies of polyelectrolytes in solutions and at surfaces is shifting more towards biological systems. We mentioned in this review the complexation of DNA and histones. This is only one of many examples of interest where charged biopolymers, receptors, proteins and DNA molecules interact with each other or with other cellular components. The challenge for future fundamental research will be to try to understand the role of electrostatic interactions combined with speci6c biological (lock–key) mechanisms and to infer on biological functionality of such interactions. In this review, we also discussed additional factors that have an e8ect on the polymer adsorption and grafted layers: the quality of the solvent, undulating and 1exible substrates such as 1uid/1uid interfaces or lipid membranes; adsorption and grafting on curved surfaces such as spherical colloidal particles.


87

Although our main aim was to review the theoretical progress in this 6eld, we mentioned many relevant experiments. In this active 6eld several advanced experimental techniques are used to probe adsorbed or grafted polymer layers: neutron scattering and high-resolution X-ray re1ectivity, light scattering using 1uorescent probes, ellipsometry, surface isotherms as well as the surface force apparatus for the force measurement between two surfaces. This paper should be viewed as a general introduction to adsorption phenomena involving charged and neutral chains and can serve as a starting point to understand more complex systems as encountered in applications and current experiments. Acknowledgements It is a pleasure to thank our collaborators G. Ariel, I. Borukhov, M. Breidenich, Y. Burak, H. Diamant, H. Gaub, J.-F. Joanny, K. Kunze, L. Leibler, R. Lipowsky, A. Moreira, H. Orland, M. Schick, C. Seidel, A. Sha6r and Y. Tsori with whom we have been working on polymers and polyelectrolytes. One of us (DA) would like to acknowledge partial support from the Israel Science Foundation, Centers of Excellence Program and under grant no. 210/02, the Israel–US Binational Science Foundation (BSF) under grant no. 98-00429, and the Alexander von Humboldt Foundation for a research award. RRN acknowledges 6nancial support by Deutsche Forschungsgemeinschaft (DFG, SFB 486 and German–French Network) and the Fonds der Chemischen Industrie. References [1] G.J. Fleer, M.A. Cohen Stuart, J.M.H.M. Scheutjens, T. Cosgrove, B. Vincent, Polymers at Interfaces, Chapman & Hall, London, 1993. [2] E. Eisenriegler, Polymers Near Surfaces, World Scienti6c, Singapore, 1993. [3] M.A. Cohen Stuart, T. Cosgrove, B. Vincent, Adv. Colloid Interface Sci. 24 (1986) 143. [4] P.G. de Gennes, Adv. Colloid Interface Sci. 27 (1987) 189. [5] I. Szleifer, Curr. Opin. Colloid Interface Sci. 1 (1996) 416. [6] G.J. Fleer, F.A.M. Leermakers, Curr. Opin. Colloid Interface Sci. 2 (1997) 308. [7] A.Yu. Grosberg, A.R. Khokhlov, Statistical Physics of Macromolecules, AIP Press, New York, 1994. [8] F. Oosawa, Polyelectrolytes, Dekker, New York, 1971. [9] H. Dautzenberg, W. Jaeger, B.P.J. KKotz, C. Seidel, D. Stscherbina, Polyelectrolytes: Formation, Characterization and Application, Hanser Press, Munich, 1994. [10] S. FKorster, M. Schmidt, Adv. Polym. Sci. 120 (1995) 50. [11] J.-L. Barrat, J.-F. Joanny, Adv. Chem. Phys. 94 (1996) 1. [12] P.J. Flory, Principles of Polymer Chemistry, Cornell University, Ithaca, 1953. [13] H. Yamakawa, Modern Theory of Polymer Solutions, Harper & Row, New York, 1971. [14] P.G. de Gennes, Scaling Concepts in Polymer Physics, Cornell University, Ithaca, 1979. [15] J. des Cloizeaux, J. Jannink, Polymers in Solution, Oxford University, Oxford, 1990. [16] F. Gittes, B. Mickey, J. Nettleton, J. Howard, J. Cell Biol. 120 (1993) 923. [17] J. KKas, H. Strey, M. BKarmann, E. Sackmann, Europhys. Lett. 21 (1993) 865. [18] A. Ott, M. Magnasco, A. Simon, A. Libchaber, Phys. Rev. E 48 (1993) R1642. [19] C. Frontale, E. Dore, A. Ferrauto, E. Gratton, Biopolymers 18 (1979) 1353. [20] P.G. de Gennes, P.A. Pincus, R.M. Velasco, F. Brochard, J. Phys. (France) 37 (1976) 1461. [21] A.R. Khokhlov, K.A. Khachaturian, Polymer 23 (1982) 1742. [22] J.-L. Barrat, J.-F. Joanny, Europhys. Lett. 24 (1993) 333.

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Erratum

Erratum to “The Alpha Magnetic Spectrometer (AMS) on the International Space Station: Part I—results from the test *ight on the space shuttle” [Physics Reports 366 (2002) 331– 405] AMS Collaboration J.V. Allaby∗ EP Division, CERN, CH-1211, Geneva 23, Switzerland

The Acknowledgements section on page 401 should read as follows: Acknowledgements The success of the 8rst AMS mission is due to many individuals and organizations. The support of NASA and the U.S. Department of Energy was vital to the inception, development and operation of the experiment. The interest and support of Mr. Daniel S. Goldin, then NASA Administrator, is gratefully acknowledged. The dedication of Dr. John O’Fallon, Dr. Peter Rosen and Dr. P.K. Williams of U.S. DOE, as well as of Dr. Douglas P. Blanchard, Mr. Mark J. Sistilli, Mr. James R. Bates, Mr. Kenneth Bollweg and the NASA and Lockheed-Martin Mission Management team is acknowledged. The support of the Max-Planck Institute for Extraterrestrial Physics, the support of the space agencies from Germany (DLR), Italy (ASI), France (CNES), Spain (CDTI) and China (CALT) and the support of CSIST, Taiwan, made it possible to complete this experiment on time. The support of CERN and GSI-Darmstadt, particularly of Professor Hans Specht and Dr. Reinhard Simon made it possible for us to calibrate the detector after the shuttle returned from orbit. The support of INFN of Italy, IN2P3 of France, CIEMAT and CICYT of Spain, LIP of Portugal, CHEP of Korea, the Chinese Academy of Sciences, Academia Sinica of Taiwan, M.I.T., ETH-ZKurich, the University of Geneva, National Central University of Taiwan, the RWTH-Aachen

PII of the original article S0370-1573(02)00013-3 Corresponding author on behalf of the AMS Collaboration. E-mail address: [email protected] (J.V. Allaby).

∗


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J.V. Allaby / Physics Reports 380 (2003) 97 – 98

of Germany, the University of Turku and the University of Technology of Helsinki, Finland, is gratefully acknowledged. We are most grateful to the STS-91 astronauts, particularly to Dr. Franklin Chang-Diaz who provided vital help to AMS during the *ight. We thank Professors S. Ahlen, C. Canizares, A. De Rujula, J. Ellis, A. Guth, M. Jacob, L. Maiani, R. Mewaldt, R. Orava, J.F. Ormes and M. Salamon for helping us to initiate this experiment.



Cosmological consequences of MSSM %at directions Kari Enqvista; b;∗ , Anupam Mazumdarc;1 a

Department of Physical Sciences, University of Helsinki, P.O. Box 64, FIN-0014, Helsinki, Finland b Helsinki Institute of Physics, University of Helsinki, P.O. Box 9, FIN-00014 Helsinki, Finland c The Abdus Salam International Centre for Theoretical Physics, Strada Costiera-11, 34100 Trieste, Italy Accepted 1 March 2003 editor: M.P. Kamionkowski

Abstract We review the cosmological implications of the %at directions of the minimally supersymmetric standard model (MSSM). We describe how 1eld condensates are created along the %at directions because of in%ationary %uctuations. The post-in%ationary dynamical evolution of the 1eld condensate can charge up the condensate with B or L in a process known as A4eck–Dine baryogenesis. Condensate %uctuations can give rise to both adiabatic and isocurvature density perturbations and could be observable in future cosmic microwave experiments. In many cases the condensate is however not the state of lowest energy but fragments, with many interesting cosmological consequences. Fragmentation is triggered by in%ation-induced perturbations and the condensate lumps will eventually form non-topological solitons, known as Q-balls. Their properties depend on how supersymmetry breaking is transmitted to the MSSM; if by gravity, then the Q-balls are semi-stable but long-lived and can be the source of all the baryons and LSP dark matter; if by gauge interactions, the Q-balls can be absolutely stable and form dark matter that can be searched for directly. We also discuss some cosmological applications of generic %at directions and Q-balls in the context of self-interacting dark matter, in%atonic solitons and extra dimensions. c 2003 Elsevier Science B.V. All rights reserved. PACS: 98.80.−k; 12.60.Jv

Contents 1. Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 102 2. Baryogenesis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 105 ∗ Corresponding author. Department of Physical Sciences, University of Helsinki, P.O. Box 64, FIN-0014 Helsinki, Finland. E-mail addresses: [email protected], [email protected] (K. Enqvist), [email protected] (A. Mazumdar). 1 Present address: Physics Department, McGill University, Montreal, QC, H3A 2T8, Canada.


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2.1. Baryon asymmetric Universe . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.2. Thermal history of the Universe . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.2.1. Expanding Universe . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.2.2. Entropy . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.2.3. Nucleosynthesis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.3. Requirements for baryogenesis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.3.1. Non-conservation of baryonic charge . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.3.2. C and CP violation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.3.3. Departure from thermal equilibrium . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.3.4. Sphalerons . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.3.5. Washing out B + L . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.4. Alternatives for baryogenesis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.4.1. GUT-baryogenesis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.4.2. Electroweak baryogenesis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.4.3. Electroweak baryogenesis in MSSM . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.4.4. Leptogenesis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.4.5. Baryogenesis through 1eld condensate decay . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.5. Old A4eck–Dine baryogenesis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.5.1. Classical motion of the order parameter . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.5.2. Condensate decay . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3. Field %uctuations during in%ation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.1. Fluctuation spectrum in de Sitter space . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.2. Slow roll in%ation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.3. Primordial density perturbations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.3.1. Adiabatic perturbations and the Sachs–Wolfe eJect . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.3.2. Spectrum of adiabatic perturbations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.3.3. Gravitational waves . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.4. Multi-1eld perturbations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.4.1. Adiabatic and isocurvature conditions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.4.2. Adiabatic perturbations due to multi-1eld . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.4.3. Isocurvature perturbations and CMB . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.5. In%ation models . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.5.1. Non-supersymmetric in%ation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.5.2. F-term in%ation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.5.3. D-term in%ation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.5.4. Supergravity corrections . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.6. Reheating of the Universe . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.6.1. Perturbative in%aton decay . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.6.2. Non-perturbative in%aton decay . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.6.3. Gravitino and in%atino problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4. Flat directions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.1. Degenerate vacua . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.2. MSSM and its potential . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.2.1. F- and D-renormalizable %at directions of MSSM . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.2.2. An example of F- and D-%at direction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.3. Lifting the %at direction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.3.1. Lifting by non-renormalizable operators . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.3.2. Lifting by soft supersymmetry breaking . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.4. Supersymmetry breaking in the MSSM . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.4.1. Soft supersymmetry breaking Lagrangian . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.4.2. Gravity mediated supersymmetry breaking . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

105 106 106 108 109 110 110 111 111 112 114 114 114 115 117 117 119 119 119 121 121 122 123 124 124 126 128 128 129 129 130 130 130 131 133 134 134 134 135 136 137 137 138 139 140 141 141 143 143 143 144

K. Enqvist, A. Mazumdar / Physics Reports 380 (2003) 99 – 234 4.4.3. Gauge mediated supersymmetry breaking . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.5. Supersymmetry breaking in the early Universe . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.5.1. In%aton-induced terms . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.5.2. Supergravity corrections . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.6. The potential for %at direction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.6.1. F-term in%ation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.6.2. D-term in%ation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5. Dynamics of %at directions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.1. Running of the couplings . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.1.1. Running of gravitational coupling . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.1.2. Renormalization group equations in the MSSM . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.2. Hubble induced radiative corrections . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.2.1. The case with CI ≈ −1 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.2.2. The case with CI ≈ +1 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.2.3. Running of the %at direction 1eld in no-scale supergravity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.3. Post-in%ationary running of the %at direction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.3.1. Gravity mediated supersymmetry breaking . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.3.2. Gauge mediated supersymmetry breaking . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.4. Density perturbations from the %at direction condensate . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.4.1. Energetics of %at direction and the in%aton 1eld . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.4.2. Adiabatic perturbations during D-term in%ation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.4.3. Adiabatic perturbations during F-term in%ation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.4.4. Isocurvature %uctuations in D-term in%ation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.4.5. Isocurvature %uctuations in F-term in%ation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.4.6. MSSM %at directions as a source for curvature perturbations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.5. Baryon number asymmetry . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.6. Thermal eJects . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.6.1. Thermal corrections to the %at direction potential . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.6.2. Thermal evaporation of the %at direction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.7. Baryogenesis and neutrino mass . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.8. Trajectory of a %at direction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.9. Instability of the coherent condensate . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.9.1. Negative pressure . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.9.2. Growth of perturbations in the AD condensate . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.9.3. The true ground state . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.10. Numerical studies of fragmentation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.10.1. Perturbation theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.10.2. Lattice simulations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.11. Equilibrium ensembles . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6. Q-balls . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.1. Q-ball as a non-topological soliton . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.1.1. Proofs of existence . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.1.2. Beyond thin wall solution . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.2. Varieties of Q-balls . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.2.1. Thin wall Q-balls . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.2.2. Thick wall Q-balls in the gauge mediated case . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.2.3. Thick wall Q-balls in the gravity mediated case . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.2.4. Hybrid case: gauge and gravity mediated Q-ball . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.2.5. EJect of gravity on Q-balls . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.2.6. Q-balls and local gauge invariance . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.3. Q-ball decay . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

101 145 146 146 147 149 149 149 150 150 150 151 152 153 155 156 156 157 158 159 159 161 162 163 165 166 166 169 169 170 170 171 172 172 174 175 176 176 178 182 183 183 183 184 185 185 185 186 187 188 188 189

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6.3.1. Surface evaporation to fermions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.3.2. The decay temperature . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.3.3. Q-ball decay into a pair of bosons . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.4. Cosmological formation of Q-balls . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.4.1. In gravity mediated case . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.4.2. In gauge mediated case . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.5. Q-ball collisions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.6. Q-balls in a thermal bath . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.6.1. Dissociation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.6.2. DiJusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.6.3. Evaporation at 1nite T . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7. Cosmological consequences of Q-balls . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7.1. L-ball cosmology . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7.2. B-ball cosmology . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7.3. B-balls in gravity mediated supersymmetry breaking . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7.3.1. B-ball baryogenesis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7.3.2. LSP dark matter from B-ball decay . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7.3.3. The LSP abundance . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7.3.4. Which direction? . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7.3.5. Direct LSP searches and B-balls . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7.4. Q-balls and gauge mediated supersymmetry breaking . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7.4.1. Baryogenesis and gauge mediation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7.4.2. Generic gauge mediated models . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7.4.3. Late formation of gauged Q-balls . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7.5. Q-balls as self-interacting dark matter . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7.6. Direct searches for gauge mediated Q-balls . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8. Flat directions other than MSSM . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8.1. Fragmentation of the in%aton condensate . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8.1.1. Reheating as a surface eJect . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8.1.2. Q-balls from the in%aton condensate . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8.2. A4eck–Dine baryogenesis without MSSM %at directions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8.2.1. Leptogenesis with sneutrino . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8.2.2. AD baryogenesis in theories with low scale quantum gravity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8.3. Solitosynthesis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8.3.1. Accretion of charge by Q-balls . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8.3.2. Phase transition via solitogenesis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Acknowledgements . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

189 190 191 193 193 193 194 195 196 197 197 198 199 199 201 201 202 203 205 206 206 207 208 209 210 210 212 213 213 214 216 216 217 218 218 220 220 220

1. Introduction The interplay between particle physics and cosmology plays an increasing role in understanding the physics beyond the standard model (SM) [1] and the early Universe before the era of Big Bang Nucleosynthesis (BBN) [2,3]. On both fronts we currently lack hard data. Above the electroweak scale E ∼ O(100) GeV, the particle content is largely unknown, while beyond the BBN scale T ∼ O(1) MeV, there is no direct information about the thermal history of the Universe. However, there are some observational hints, as well as a number of theoretical considerations, which seem to be pointing towards a wealth of new physics both at small distances and in the very early Universe.


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Perhaps most importantly, new data is expected soon from accelerator experiments such as LHC and from cosmological measurements carried out by satellites such as MAP [4] and Planck [5]. In cosmology the recent observations of the cosmic microwave background (CMB) radiation, which has a temperature ∼ 2:728 ± 0:004 K [6], have given rise to an era of precision cosmology. The Cosmic Background Explorer (COBE) satellite [7] 1rst detected in a full-sky map a temperature perturbation of one part in 105 at scales larger than 7 degrees [8]. The irregularities are present at a scale larger than the size of the horizon at the time when the microwave photons were generated and cannot be explained within the traditional hot Big Bang model [9]. The recent balloon experiments BOOMERANG [10] and MAXIMA [11], together with the ground-based DASI [12] experiment have established the existence of the 1rst few acoustic peaks in the positions predicted by cosmic in%ation [13–15,9]. In%ation, a period of exponential expansion in the very early Universe, is a direct link to physics at energy scales that will not be accessible to Earth-bound experiments for any foreseeable future. In%ation could occur because a slowly rolling scalar 1eld, the in%aton, dynamically gives rise to an epoch dominated by a false vacuum. During in%ation quantum %uctuation are imprinted on space–time as energy perturbations which then are stretched outside the causal horizon. These primordial %uctuations eventually re-enter our horizon, whence their form can be extracted from the CMB (for a review, see [16,9]). In%ation can be considered as a model for the origin of matter since all matter arises from the vacuum energy stored in the in%aton 1eld. However the present models do not give clear predictions as to what sort of matter there is to be found in the Universe. From observations we know that baryons constitute about 3% of the total mass [3], whereas relic diJuse cosmic ray background virtually excludes any domains of anti-baryons in the visible Universe [17]. Almost 30% of the total energy density is in non-luminous, non-baryonic dark matter [18]. Its origin and nature is unknown, although various simulations of large scale structure formation suggest that there must be at least some cold dark matter (CDM), comprising of particles with negligible velocity, although there may also be a component of hot dark matter (HDM), comprising of particles with relativistic velocities [19]. The rest of the energy density is in the form of dark energy [20,21]. The striking asymmetry in the baryonic matter has existed at least since the time of BBN and plays an important role in providing the right abundances for the light elements. The present helium (3 He), deuterium (D) and lithium abundances suggest a baryon density and an asymmetry relative to photon density of order 10−10 [3]. Such an asymmetry is larger by a factor of 109 than what it should have been by merely assuming a initially baryon symmetric hot Big Bang [22]. Therefore baryon asymmetry must have been created dynamically in the early Universe. The origin of baryon asymmetry and dark matter bring cosmology and particle physics together. Within SM all the three Sakharov conditions for baryogenesis [23] are in principle met; there is baryon number violation, C and CP violation, and an out-of-equilibrium environment during a 1rst-order electroweak phase transition. However, it has turned out that within SM the electroweak phase transition is not strong enough [24–27], and therefore the existence of baryons requires new physics. Regarding HDM, light neutrinos are a possible candidate [19,28], but there is no candidate for CDM in the SM. HDM alone cannot lead a successful structure formation because of HDM free streaming length [29,19]. Therefore one must resort to physics beyond the SM also to 1nd a candidate for CDM [18]. The tangible evidence for small but non-vanishing neutrino masses as indicated by the neutrino oscillations observed by the Super-Kamiokande [30] and SNO collaborations [31] is de1nitely another

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indication for new physics beyond the SM. The main sources of neutrino mass could be either Dirac or Majorana. A Dirac neutrino would require a large 1ne tuning in the Yukawa sector (one part in 1011 ) while a Majorana mass would appear to require a scale much above the electroweak scale together with an extension of the SM gauge group SU (3)C × SU (2)L × U (1)Y . In the Majorana case the lightness of the neutrino could be explained via the see-saw mechanism [32,33]. A theoretical conundrum is that the mass scale of SM is ∼ O(100) GeV, much lower that the scale of gravity MP = (8GN )−1=2 = 2:436 × 1018 GeV, and not protected from quantum corrections. The most popular remedy is of course supersymmetry (for a review, see [35–37]), despite the fact that so far supersymmetry has evaded all observations [38]. The minimal supersymmetric extension of the SM is called the MSSM. Supersymmetry must be broken at a scale ∼ O(1) TeV, presumably in some hidden sector from which breaking is transmitted to the MSSM, e.g., by gravitational [34,35] or gauge interactions [39]. In the MSSM the number of degrees of freedom are increased by virtue of the supersymmetric counterparts of the SM bosons and fermions. One of them, known as the lightest supersymmetric particle (LSP), could be absolutely stable with a mass of the order of supersymmetry breaking scale. LSP would be a natural candidate for CDM (see e.g. [18]). In addition, because of the larger parameter space, electroweak baryogenesis in MSSM in principle has a much better chance to succeed. However, there are a number of important constraints, and lately Higgs searches at LEP have narrowed down the parameter space to the point where it has all but disappeared [40–42]. Electroweak baryogenesis within MSSM thus appears to be heading towards deep trouble. Moreover, although MSSM can provide CDM, there is no connection between dark matter and electroweak baryogenesis. On the other hand, by virtue of supersymmetry, MSSM has the intriguing feature that there are directions in the 1eld space which have virtually no potential. They are usually known as 9at directions, which are made up of squarks and sleptons and therefore carry baryon number and/or lepton number. The MSSM %at directions have been all classi1ed [43]. Because it does not cost anything in energy, during in%ation squarks and sleptons are free to %uctuate along the %at directions and form scalar condensates. Because in%ation smoothes out all gradients, only the homogeneous condensate mode survives. However, like any massless scalar 1eld, the condensate is subject to in%aton-induced zero point %uctuations which impart a small, and in in%ation models a calculable, spectrum of perturbations on the condensate. After in%ation the dynamical evolution of the condensate can charge the condensate up with a large baryon or lepton number, which can then released into the Universe when the condensate decays, as was 1rst discussed by A4eck and Dine [44]. The potential along the MSSM %at direction is not completely %at because of supersymmetry breaking. In addition to the usual soft supersymmetry breaking, the non-zero energy density of the early Universe also breaks supersymmetry, in particular during in%ation when the Hubble expansion dominates over any low energy supersymmetry breaking scale [45,46]. Flatness can also be spoiled by higher-order non-renormalizable terms, and the details of the condensate dynamics depend on these. In most cases, the MSSM condensate along a %at direction is however not the state of lowest energy. The condensate typically has a negative pressure, which causes the in%ation-induced perturbations to grow. Because of this the condensate fragments, usually when the Hubble scale equals the supersymmetry breaking scale, into lumps of condensate matter which eventually settle down to


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non-topological solitons dubbed as Q-balls by Coleman [47]. Q-balls carry a global charge, which in the case of MSSM is either B or L. The properties of Q-balls depend on supersymmetry breaking. If transmitted to MSSM by gravity, the Q-balls turn out to be only semistable but nevertheless long-lived compared with the time scales of the very early Universe [48]. When they decay, they may provide not only the baryonic matter but also dark matter LSPs [49]. If supersymmetry breaking is transmitted from the hidden sector to MSSM by gauge interactions, the resulting Q-balls would be stable and could exist at present as a form of dark matter [50]. In this case one can make direct searches for their existence [51]. In both cases there is a prediction for the relation between the baryon and dark matter densities. Moreover, the condensate perturbations are inherited by the Q-balls, and can thus be a source of both isocurvature and adiabatic density perturbations [52–54]. This review is organized as follows. In Section 2, we recapitulate some basic cosmology, and in particular baryogenesis. We brie%y discuss various popular schemes of baryogenesis and describe the original A4eck–Dine baryogenesis. In Section 3, we present some background material for in%ation, mainly concentrating on supersymmetric models. Quantum %uctuations and reheating are also discussed. In Section 4, we present the MSSM %at directions and discuss their properties. Various contributions to the %at direction potential in the early Universe are listed. Low energy supersymmetry breaking schemes, such as gravity and gauge mediation, are also discussed. In Section 5, we discuss the dynamical properties of %at directions and the running of the %at direction potential due to gauge and Yukawa interactions. Leptogenesis along LHu %at direction, and the condensate evaporation in a thermal bath, is also described. We discuss fragmentation of the condensates for both gravity and gauge mediated supersymmetry breaking and present the relevant numerical studies. In Section 6, Q-ball properties are presented in detail. We describe various types of Q-balls, their interactions and their behavior at 1nite temperature. We discuss surface evaporation, diJusion, and dissociation of charge from Q-balls in a thermal bath. In Section 7, we focus on the cosmological consequences of Q-balls. We consider Q-ball baryogenesis and non-thermal dark matter generation through charge evaporation for diJerent types of Q-balls. We discuss Q-balls as self-interacting dark matter and present experimental and astrophysical constraints on stable Q-balls. In Section 8, we brie%y survey beyond-the-MSSM-condensates by considering in%atonic Q-balls and A4eck–Dine mechanism without MSSM %at directions. We also describe solitosynthesis, a process of accumulating large Q-balls in a charge asymmetric Universe. 2. Baryogenesis 2.1. Baryon asymmetric Universe There are only insigni1cant amounts of anti-particles within the solar system. Cosmic ray showers contain ∼ 10−4 anti-protons for each proton [55], but the anti-protons are by-products of the interaction of the primary beam with the interstellar dust medium. This strongly suggest that galaxies and intergalactic medium is made up of matter rather than anti-matter, and if there were any anti-matter, the abundance has to be smaller than one part in 104 . The absence of annihilation radiation from the Virgo cluster shows that little anti-matter is to be found within a 20 Mpc sphere, and the relic diJuse cosmic ray background virtually excludes domains of anti-matter in the visible Universe [17].

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The best present estimation for the baryon density comes from BBN [56] combined with the CMB experiments and it is given by [57] 0:010 6 b h2 6 0:022 ;

(1)

where b ≡ b =c de1nes the fractional baryon density b with respect to the critical energy density of the Universe: c = 1:88h2 × 10−29 g cm−3 . The observational uncertainties in the present value of the Hubble constant; H0 = 100h km s−1 Mpc−1 ≈ (h=3000)Mpc−1 are encoded in h. Various considerations such as Hubble Space Telescope observations and type Ia supernova data suggest that h = 0:70 [58]. However, from the age of the globular cluster which comes out to be 11 Gyr; h seems to take lower value of about 0.5 [59]. The present convention is to take 0:5 6 h 6 0:8. In terms of the baryon and photon number densities we may write ≡

nb − nbS = 2:68 × 10−8 b h2 ; n

(2)

where nb is the baryon number density and nbS is for anti-baryons. The photon number density is given by n ≡ (2(3)=2 )T 3 . Observations of the deuterium abundance in quasar absorption lines suggest [60] 4(3) × 10−10 6 6 7(10) × 10−10 :

(3)

The conservative bounds are in parentheses. Often in the literature the baryon asymmetry is given in relation to the entropy density s=1:8g∗ n , where g∗ measures the eJective number of relativistic species which itself a function of temperature. At the present time g∗ ≈ 3:36, while during BBN g∗ ≈ 10:11, rising up to 106:75 at T 100 GeV. In the presence of supersymmetry at T 100 GeV, the number of eJective relativistic species are doubled to 213.30. The baryon asymmetry B, de1ned as the diJerence of baryon and anti-baryon number densities relative to the entropy density, is bounded by 5:7(4:3) × 10−11 6 B ≡

nb − nbS 6 9:9(14) × 10−11 ; s

(4)

where the numbers in parenthesis are conservative bounds [60]. If at the beginning = 0, then the origin of this small number can not be understood in a CPT invariant Universe by a mere thermal decoupling of nucleons and anti-nucleons at T ∼ 20 MeV. The resulting asymmetry would be too small by at least nine orders of magnitude, see [22]. 2.2. Thermal history of the Universe 2.2.1. Expanding Universe The hot Big Bang cosmology assumes that the Universe is spatially homogeneous and isotropic and can be described by the Friedmann–Robertson–Walker (FRW) metric dr 2 2 " # 2 2 2 2 2 2 ds = g"# d x d x = dt − a (t) + r (d) + sin ) d* ) ; (5) 1 − Kr 2


107

where a(t) is the scale factor that determines the expansion or the contraction of the Universe; the constant K de1nes the spatial geometry. If K = 0, the Universe is %at and has Euclidean geometry, otherwise there is a spatial curvature corresponding either to a closed elliptic (K = +1) or an open hyperbolic (K =−1) geometries. The value of K cannot however 1x the global topology; for instance, an Euclidean topology can be %at and in1nite R3 , or a surface of a 3-torus T 3 . However, topology has other observable consequences, e.g., for the pattern of CMB temperature %uctuations [61]. There are two characteristic scales corresponding to the homogeneous and isotropic Universe: the curvature scale rcurv = a(t)|K|−1=2 , and the Hubble scale −1 a(t) ˙ −1 ; (6) H = a(t) where dot denotes derivative w.r.t. t. The Hubble time is denoted by f af dt : = ln tHub = −1 H ai i

(7)

The behavior of the scale factor depends on the energy momentum tensor of the Universe. For a perfect %uid T"# = −pg"# + ( + p)u" u# ;

(8)

where is the energy density and p is the pressure of a %uid and the four velocity u" ≡ d x" =ds. For the FRW metric and for the perfect %uid the equations of motion gives the Friedmann–Lemaitre equation H2 =

K − ; 2 a(t)2 3MP

(9)

also known as the Hubble equation. The acceleration equation is given by 1 a(t) U =− ( + 3p) ; a(t) 6MP2

(10)

and the conservation of the energy momentum tensor T;#"# = 0 gives d(a3 ) = −3pa2 : da

(11)

Note that a3 is constantly decreasing in an expanding Universe for a positive pressure. The early Universe is believed to have been radiation dominated with p = =3 and a(t) ˙ t 1=2 , followed by a matter dominated era with p = 0 and a(t) ˙ t 2=3 . The early Universe might also have had an era of acceleration, known as the in%ationary phase, which could have happened only if aU ¿ 0 ⇔ + 3p ¡ 0 :

(12)

A geometric way of de1ning in%ation is [9] d(H −1 =a(t)) ¡0 ; dt

(13)

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which states that the Hubble length as measured in comoving coordinates decreases during in%ation. We will use this particular de1nition of in%ation very often while discussing the number of e-foldings and density perturbations. The Hubble expansion rate is related to the temperature by 2 1=2 T H= = 1:66 × g ; (14) ∗ MP 3MP2 where g∗ is the total number of relativistic degrees of freedom and it is given by Ti 4 7 Ti 4 g∗ (T ) = gi + gi : T 8 T i=b

(15)

i=f

Here Ti denotes the eJective temperature of species i, which decouples at a temperature T = TD . During the radiation era when H = (1=2t), one 1nds 2 t −1=2 1 MeV ≈ 2:42g∗ : (16) 1s T 2.2.2. Entropy An ideal gas of particles respects the Fermi–Dirac or Bose–Einstein distributions fi (p; "; T ) = [exp((Ei − "i )=T ) ∓ 1]−1 ;

(17)

where Ei2 = |p|2 + m2 ; "i represents the chemical potential of the species i; −=+ corresponds to Bose/Fermi statistics. The value of " is equal and opposite for particles and anti-particles. Therefore in the early Universe a 1nite net chemical potential is proportional to the particle anti-particle asymmetry. The bound on charge asymmetry relative to the photon number density is severe, less than one part in 1043 at temperatures close to BBN [62], while baryon asymmetry is comparatively larger, but still small enough for "e ; "b ≈ 0 to be an excellent approximation. Neutrinos may however carry a net B −L charge which need not be vanishingly small at early times, although a large enough neutrino chemical potential can aJect nucleosynthesis, for example, see [28]. The number density n, energy density , and pressure p can be expressed in terms of temperature, and gi is the number of internal degrees of freedom [63] gi gi ni (T ) = fi (p; "; T ) d 3 p = 2 T 3 Ii11 (∓) ; 3 (2) 2 gi gi Ei (p)fi (p; "; T ) d 3 p = 2 T 4 Ii21 (∓) ; i (T ) = 3 (2) 2 |p|2 gi gi fi (p; "; T ) d 3 p = 2 T 4 Ii03 (∓) ; (18) pi (T ) = 3 (2) 3Ei (p) 6 where Iiab (∓) ≡

xi

∞

ya (y2 − xi2 )b=2 dy; (ey ∓ 1)

xi ≡

mi : T

(19)


109

For a relativistic case with xi 1, Ir11 (−) = 2(3);

Ir21 (−) = Ir03 (−) =

4 15

for bosons ;

3(3) 74 ; Ir21 (+) = Ir03 (+) = for fermions ; (20) 2 120 where denotes the Riemann Zeta function and (3) = 1:202. Thus the energy density of radiation reads Ir11 (+) =

2 g∗ T 4 : 30 For non-relativistic particles with xi 1, one obtains for both bosons and fermions mi T 3=2 −mi =T nr e ; pnr = 0 : nnr (T ) = = gi m 2 r =

(21)

(22)

If the chemical potential is non-zero, the exponential Eq. (22) also includes a factor e+"i =T . The entropy density is de1ned as i + p i S ; (23) s≡ = T T where d(sa3 ) = 0 is a thermodynamically conserved quantity. The decoupling temperature can be expressed as [64] TD g∗SA (TD ) g∗S −SA (T ) 1=3 ; (24) = T g∗SA (T ) g∗S −SA (TD ) where S is the total entropy and SA the entropy in the degrees of freedom that have decoupled at TD . 2.2.3. Nucleosynthesis According to BBN (for reviews see [3,57]) the light elements 2 H; 3 He, 4 He, and 7 Li have been synthesized during the 1rst few hundred seconds. The abundances depend on the baryon-to-photon ratio nB ≡ : (25) n All the relevant physical processes take place essentially in the range from a few MeV ∼ 0:1 s down to 60 –70 KeV ∼ 103 s. During this period only photons, e± pairs, and the three neutrino %avors contribute signi1cantly to the energy density. Any additional energy density may be parameterized in terms of the eJective number of light neutrino species N# , so that 7 (26) g∗ = 10:75 + (N# − 3) : 4 Nucleosynthesis starts oJ with a freezing out of the weak interaction between neutron and proton at TD ≈ 0:8 MeV. Free neutrons keep decaying until deuterium begins to form through n + p → d + . Deuterium synthesis is over by TD ≈ 0:086 MeV (assuming =5×10−10 ). At TD , neutron abundance has been depleted to Xn (tD ) ≡ n=(n+p) ≈ 0:122. All the surviving neutrons are now captured through

110


n + D → (3 H; 3 He), and subsequently by virtue of the process (3 H; 3 He) + n → 4 He, which has a binding energy of 28:3 MeV. The total mass fraction of primordial helium, which is denoted by YP (4 He), is given by YP (4 He) ≈ 2Xn (TD ) = 0:245 :

(27)

Adopting the experimentally allowed range of 0:22 ¡ YP ¡ 0:26, one can constraint that number of light neutrino species by [56] N# 6 4 :

(28)

The four LEP experiments combined give the best 1t as [65] N# = 2:994 ± 0:12 :

(29)

Nucleosynthesis also constrains many non-conventional ideas, for instance alternative theories of gravity such as scalar–tensor theories [66]. Besides 4 He, D and 3 He are produced at the level of 10−5 , and 7 Li at the level of 10−10 . The theoretical prediction has some slight problems in 1tting the observed 4 He and 7 Li abundances. Both seem to indicate 1:7×10−10 ¡ ¡ 4:7×10−10 , corresponding to 0:006 ¡ b h2 ¡ 0:017 with a central value b h2 = 0:009 [57]. The abundance ratio D/H is comparable with 4 He and 7 Li abundances at the 27 level in the range 4:7×10−10 ¡ ¡ 6:2×10−10 , which corresponds to 0:017 ¡ b h2 ¡ 0:023. The likelihood analysis which includes all the three elements (D, 4 He, and 7 Li) yields [57] 4:7 × 10−10 ¡ ¡ 6:2 × 10−10 ;

0:017 ¡ b h2 ¡ 0:023 :

(30)

Despite the uncertainties there appears to be a general concordance between theoretical BBN predictions and observations, which is now being bolstered by the CMB data from several diJerent experiments. The results from the ground based DASI experiment indicates b h2 = 0:022+0:004 −0:003 [12], while the results from the BOOMERANG balloon-borne experiment imply b h2 = 0:021+0:004 −0:003 [10]. MAXIMA, another balloon experiment, quotes a somewhat larger value b h2 = 0:0325 ± 0:006 [67]. 2.3. Requirements for baryogenesis As pointed out by Sakharov [23], baryogenesis requires three ingredients: (1) baryon number non-conservation, (2) C and CP violation, and (3) out-of-equilibrium condition. All these three conditions are believed to be met in the very early Universe. 2 2.3.1. Non-conservation of baryonic charge In the SM, baryon number B is violated by non-perturbative instanton processes [70,71]. At the quantum level both baryon number current JB" and the lepton number current JL" are not conserved because of chiral anomalies [72]. However the anomalous divergences of JB" and JL" come with an equal amplitude and an opposite sign. Therefore B − L remains conserved, while B + L may change 2

There have been attempts [68,69] for baryogenesis via a repulsive interaction between baryons and anti-baryons which would lead to their spatial separation before thermal decoupling of nucleons and anti-nucleons. However at such early times the causal horizon contained only a very small fraction of the solar mass so that the asymmetry could not be smooth at distances greater than the galactic size.


111

via processes which interpolate between the multiple non-Abelian vacua of SU (2). The probability for the B + L violating transition is however exponentially suppressed [70,71]. As was 1rst pointed out by Manton [73], at high temperatures the situation is diJerent, so that when T MW , baryon violating transitions are in fact copious (see Section 2.4.2). In addition to baryogenesis, B violation also leads to proton decay in GUTs. For instance, the dimension 6 operator (QQQL)=: generates observable proton decay unless : ¿ 1015 GeV. In the MSSM the bound is : ¿ 1026 GeV because the decay can take place via a dimension 5 operator. In the MSSM superpotential there are also terms which can lead to WL = 1 and WB = 1. Similarly there are other processes such as neutron–anti-neutron oscillations in SM and in supersymmetric theories which lead to WB = 2 and WB = 1 transitions [74]. These operators are constrained by the measurements of the proton lifetime, which yield the bound ;p ¿ 1033 years [65]. 2.3.2. C and CP violation Weak interactions ensures maximum C violation while neutral Kaon is an example of CP violation in the quark sector which has a relative strength ∼ 10−3 [65]. CP violation could also expected to be found in the neutrino sector. Beyond the SM there are many sources for CP violation. An example is the axion proposed for solving the strong CP problem in QCD [75]. Quantum %uctuations of light scalars in the early Universe, in particular during in%ation, can create diJerent domains of various C and CP phases. C and CP can also be spontaneously broken during a phase transition, so that domains of broken phases form with diJerent CP-charges [76]. 2.3.3. Departure from thermal equilibrium Departure from a thermal equilibrium cannot be achieved by mere particle physics considerations but is coupled to the dynamical evolution of the Universe. If B-violation processes are in thermal equilibrium, the inverse processes will wash out the pre-existing asymmetry (Wnb )0 [77]. This is a consequence of S-matrix unitarity and CPT -theorem [78]. However there are several ways of obtaining an out-of-equilibrium process in the early Universe. • Out-of-equilibrium decay or scattering: The Universe in a thermal equilibrium can not produce any asymmetry, rather it tries to equilibrate any pre-existing asymmetry. If the scattering rate < ¡ H , the process can take place out-of-equilibrium. Such a situation is appropriate for e.g. GUT baryogenesis [78,79]. • Phase transitions: Phase transitions are ubiquitous in the early Universe. The transition could be of ;rst, or of second (or of still higher) order. First order transitions proceed by barrier penetration and subsequent bubble nucleation resulting in a temporary departure from equilibrium. Second order phase transitions have no barrier between the symmetric and the broken phase. They are continuous and equilibrium is maintained throughout the transition. Prime examples of 1rst order phase transitions in the early Universe are the QCD and electroweak phase transitions. The nature and details of QCD phase transition is still very much an open debate [80,25,81], and although a mechanism for baryogenesis during QCD phase transition has been proposed [82], much more eJort has been devoted to the electroweak phase transition [24,26] (see Section 2.4.2).

112


• Non-adiabatic motion of a scalar ;eld: Any complex scalar 1eld carries C and CP, but the symmetries can be broken by terms in the scalar potential. This can lead to a non-trivial trajectory of a complex scalar 1eld in the phase space. If a coherent scalar 1eld is trapped in a local minimum of the potential and if the shape of the potential changes to become a maximum, then the 1eld may not have enough time to readjust with the potential and may experience completely non-adiabatic motion. This is similar to a second order phase transition but it is the non-adiabatic classical motion which prevails over the quantum %uctuations, and therefore, departure from equilibrium can be achieved. If the 1eld condensate carries a global charge such as the baryon number, the motion can charge up the condensate. This is the basis for the A4eck–Dine baryogenesis [44] (see Section 2.5). 2.3.4. Sphalerons In the SM B + L is very weakly violated in the vacuum [70]. At 1nite temperatures violation is large [73,83–86] by virtue of the sphaleron con1gurations, which mediate transitions between degenerate gauge vacua with diJerent Chern–Simons numbers related to the net change of B + L. Thermal scattering produces sphalerons which in eJect decay in B + L non-conserving ways below 1012 GeV [87], and thus can exponentially wash away B + L asymmetry. Sphalerons and associated electroweak baryogenesis has been reviewed in [88–91,24,26]. Let us here just give a brief summary of the main ingredients. • Chiral anomalies An anomaly means that a classical current conservation no longer holds at the quantum level; an example is the chiral anomaly [72]. In the SM there is classical conservation of the baryon and lepton number currents JB" and JL" , but because of chiral anomaly the currents are not conserved. Instead [70], 4 1 =2 =1 2 " "# ˜ F => F˜ => ; 9" JB = − Ng Wi W i"# + Ng + − 8 8 9 9 36 1 =2 =1 " "# ˜ Ng 1 − F => F˜ => ; 9" JL = − Ng Wi W i"# + (31) 8 8 2 where Ng is the number of generations, =2 and =1 (Wi"# and F"# ) are respectively the SU (2) and U (1) gauge couplings (1eld strengths), and the various numbers inside the brackets correspond to the squares of the hypercharges multiplied by the number of states. Note that while at the quantum level B + L is violated, B − L is still conserved. • Gauge theory vacua The vacuum structure of the gauge theories is very rich [71,92]. In case of SU (2), the vacua are classi1ed by their homotopy class {n (r)}, characterized by the winding number n which labels the so called )-vacua [71,92]. A gauge invariant quantity is the diJerence in the winding number (Chern–Simons number) =2 NCS ≡ n+ − n− = d 4 x Wa"# W˜ a"# : (32) 8


113

In the electroweak sector the 1eld density W W˜ is related to the divergence of B + L current. Therefore a change in B + L re%ects a change in the vacuum con1guration and is determined by the diJerence in the winding number =2 " 4 W(B + L) = d x 9" JB+L = − Ng d 4 x Wa"# W˜ a"# = −2Ng NCS : (33) 4 For three generations of SM leptons and quarks the minimal violation is W(B + L) = 6. Note that the proton decay p → e+ 0 requires W(B + L) = 2, so that despite B-violation, proton decay is completely forbidden in the SM. The probability amplitude for tunneling from an n vacuum at t → −∞ to an n + NCS vacuum at t → +∞ can be estimated by the WKB method [70] −4NCS ∼ 10−162NCS : (34) P(NCS )B+L ∼ exp =2 (MZ ) Therefore, as advertised, the baryon number violation rate is totally negligible in the SM at zero temperature, but as argued in a seminal paper by Kuzmin et al. [93], at 1nite temperatures the situation is completely diJerent. • Thermal tunneling The sphaleron is a 1eld con1guration sitting at the top of the potential barrier between two vacua with diJerent Chern–Simons numbers and can be reached simply because of thermal %uctuations [93]. Neglecting U (1)Y , the zero temperature sphaleron solution was 1rst found by Manton and Klinkhamer [73,83]. At 1nite temperature the energy obeys an approximate scaling law [85,86] Esph (T ) = Esph (0)@(T )=@(0): Esph (T ) =

2mW (T ) B (A=g2 ) ; =2

(35)

where mW (T ) = (1=2)g2 @(T ) is the mass of the W-boson and the function B has a weak dependence on A=g2 , where A is the quartic self-coupling of the Higgs. Below the critical temperature of the electroweak phase transition, the sphaleron rate is exponentially suppressed [94]: 4 E (T ) 7 sph 5 4 =2 < ∼ 2:8 × 10 BT e−Esph =T ; (36) 4 B(A2 =g) where B is the functional determinant which can take the values 10−4 6 B 6 10−1 [95]. Above the critical temperature the rate is however unsuppressed. Requiring that the Chern–Simons number changes at most by WNCS ∼ 1, one can estimate from Eq. (32) that WNCS ∼ g22 l2sph Wi2 ∼ 1 → Wi ∼ 1=g2 lsph . Therefore a typical energy of the sphaleron con1guration is given by Esph ∼ l3sph (9Wi )2 ∼

1 g22 lsph

:

(37)

At temperatures greater than the critical temperature there is no Boltzmann suppression, so that the thermal energy ˙ T ¿ Esph . This determines the size of the sphaleron as lsph ¿

1 g22 T

:

(38)

114


This is exactly the infrared cut-oJ generated by the magnetic mass of order ∼ g22 T . Therefore, based on this coherence length scale one can estimate the baryon number violation per volume ∼ l3sph , and per unit time ∼ lsph . On dimensional grounds the transition probability would then be given by 1 (39) <sph ∼ 3 ∼ B(=2 T )4 ; lsph t where B is a constant which incorporates various uncertainties. However, the process is inherently non-perturbative, and it has been argued that damping of the magnetic 1eld in a plasma suppresses the sphaleron rate by an extra power of =2 [96], with the consequence that <sph ∼ =25 T 4 . Lattice simulations with hard thermal loops also give <sph ∼ O(10)=25 T 4 [97]. 2.3.5. Washing out B + L In the early Universe the transitions WNCS = +1 and WNCS = −1 are equally probable. The ratio of rates for the two transitions is given by <sph+ = e−WF=T ; (40) <sph− where WF is the free energy diJerence between the two vacua. Because of a 1nite B + L density, there is a net chemical potential "B+L . Therefore n2 2 WF ∼ "B+L T 2 + O(T 4 ) ≡ B+L + O(T 4 ) : (41) T2 One then obtains [87] <sph dnB+L = <sph+ − <sph− ∼ Ng 3 nB+L : (42) dt T It then follows that an exponential depletion of nB+L due to sphaleron transitions remains active as long as <sph MP ¿ H ⇒ T 6 =24 1=2 ∼ 1012 GeV : (43) 3 T g∗ This result is important because it suggests that below T = 1012 GeV, the sphaleron transitions can wash out any B + L asymmetry being produced earlier in a time scale ; ∼ (T 3 =Ng <sph ). This seems to wreck GUT baryogenesis based on B − L conserving groups such as the minimal SU (5). 2.4. Alternatives for baryogenesis There are several scenarios for baryogenesis (for reviews on baryogenesis, see [78,90,91]), the main contenders being GUT baryogenesis, electroweak baryogenesis, leptogenesis, and baryogenesis through the decay of a 1eld condensate, or A4eck–Dine baryogenesis. Here we give a brief description of these various alternatives. 2.4.1. GUT-baryogenesis This was the 1rst concrete attempt of model building on baryogenesis which incorporates out-ofequilibrium decays of heavy GUT gauge bosons X; Y → qq, and X; Y → qSlS (see e.g., [78,79,88,98]).


115

The decay rate of the gauge boson goes as <X ∼ =X MX , where MX is the mass of the gauge boson and =X1=2 is the GUT gauge coupling. Assuming that the Universe was in thermal equilibrium at the GUT scale, the decay temperature is given by TD ≈ g∗−1=4 =X1=2 (MX MP )1=2 ;

(44)

which is smaller than the gauge boson mass. Thus, at T ≈ TD , one expects nX ≈ nXS ≈ n , and hence the net baryon density is proportional to the photon number density nB = WBn . However below TD the gauge boson abundances decrease and eventually they go out-of-equilibrium. The net entropy generated due to their decay heats up the Universe with a temperature which we denote here by Trh . Let us naively assume that the energy density of the Universe at TD is dominated by the X bosons with X ≈ MX nX , and their decay products lead to radiation with an energy density = (2 =30)g∗ Trh4 , where g∗ ∼ O(100) for T ¿ MGUT . Equating the expressions for the two energy densities one obtains nX ≈

T4 2 g∗ rh : 30 MX

(45)

Therefore the net baryon number comes out to be B≡

nB 3 Trh WBnX ≈ WB : ≈ s g∗ n 4 MX

Trh is determined from the relation <X2 ≈ H 2 (TD ) ∼ (2 =90)g∗ Trh4 =MP2 . Thus,

1=2 1=2

g∗−1=2 <X MP g∗−1=2 =X MP B≈ WB ≡ WB : MX MX2

(46)

(47)

Uncertainties in C and CP violation are now hidden in WB, but can be tuned to yield total B ∼ 10−10 in many models. Above we have tacitly assumed that the Universe is in thermal equilibrium when T ¿ MX . This might not be true, since for 2 ↔ 2 processes the scattering rate is given by < ∼ =2 T , which becomes smaller than H at suYciently high temperatures. Elastic 2 → 2 processes maintain thermal contact typically only up to a maximum temperature ∼ 1014 GeV, while chemical equilibrium is lost already at T ∼ 1012 GeV [99,100]. It has been argued that QCD gas, which becomes asymptotically free at high temperatures, never reaches a chemical equilibrium above ∼ 1014 GeV [101]. In supergravity the maximum temperature of the thermal bath should not exceed 1010 GeV [102] (see Section 3.6.3). 2.4.2. Electroweak baryogenesis A popular baryogenesis candidate is based on the electroweak phase transition, during which one can in principle meet all the Sakharov conditions. There is the sphaleron-induced baryon number violation above the critical temperature, various sources of CP violation, and an out-of-equilibrium environment if the phase transition is of the 1rst order. In that case bubbles of broken SU (2)×U (1)Y are nucleated into a symmetric background with a Higgs 1eld pro1le that changes through the bubble wall [93,103,24].

116


There are two possible mechanisms which work in a diJerent regime; local and non-local baryogenesis. In the local case both CP violation and baryon number violation takes place near the bubble wall. This requires the velocity of the bubble wall to be greater than the speed of the sound in the plasma [104,105], and the electroweak phase transition to be strongly 1rst order with thin bubble walls. The second alternative, where the bubble wall velocity speed is small compared to the sound speed in the plasma, appears to be more realistic. In this mechanism the fermions, mainly the top quark and the tau-lepton, undergo CP violating interactions with the bubble wall, which results in a diJerence in the re%ection and the transmission probabilities for the left and right chiral fermions. The net outcome is an overall chiral %ux into the unbroken phase from the broken phase. The %ux is then converted into baryons via sphaleron transitions inside the unbroken phase. The interactions are taking place in a thermal equilibrium except for the sphaleron transitions, the rate of which is slower than the rate at which the bubble sweeps the space. One great diYculty with the electroweak baryogenesis is the smallness of CP violation in the SM. It has been pointed out that an additional Higgs doublet [105–108] would provide an extra source for CP violation in the Higgs sector. However, the situation is much improved in the MSSM where there are two Higgs doublets Hu and Hd , and two important sources of CP violation [109]. The Higgses couple to the charginos and neutralinos at one loop level leading to a CP violating contribution. There is also a new source of CP violation in the mass matrix of the top squarks which can give rise to considerable CP violation [110]. Bubble nucleation depends on the thermal tunneling rate, and the expansion rate of the Universe. The tunneling rate has to overcome the expansion rate in order to have a successful phase transition via bubble nucleation at a given critical temperature Tc ¿ Tt ¿ T0 . The actual value of the baryon asymmetry produced at the electroweak baryogenesis is still an open debate [111–113,40,42], but in general it is hard to generate a large baryon asymmetry. For Tc ∼ 100 GeV, N = 3, =2 = 0:033, and B(A=g2 ) ∼ 1:87, one obtains the condition [114,87,115,26] Esph (Tc ) Esph (Tc ) + 9 log (10) + log (B) ; (48) ¿ 7 log Tc Tc which implies [115] Esph (Tc ) ¿ 45 Tc

for B = 10−1 ;

(49)

¿ 37

for B = 10−4 :

(50)

The standard bound is often taken to be that of Eq. (49). In terms of the Higgs 1eld value at Tc , one then obtains from Eq. (35) Esph (Tc ) g2 1 Esph (Tc ) @(Tc ) = ∼ 4B(A=g2 ) Tc 36 Tc Tc

(51)

for the above values of =2 ; B. Then the bounds in Eqs. (49), (50) translate to @(Tc ) ¿ 1:3 Tc

(1) ;

(52)


117

where the number in parentheses is for Eq. (50). Eq. (52) respectively, implies that the phase transition should be strongly 1rst order in order that sphalerons do not wash away all the produced baryon asymmetry. This result is the main constraint on electroweak baryogenesis. In order to save the SM electroweak baryogenesis some attempts such as matter induced effects have also been invoked; exciting the SM gauge degrees of freedom in a time varying Higgs background [116,117], or via dynamical scalar 1eld which couples to the SM 1elds [118]. Lattice studies suggest that in the SM the phase transition is strongly 1rst order only below Higgs mass mH ∼ 72 GeV [27,119–121]. Above this scale the transition is just a cross-over. Such a Higgs mass is clearly excluded by the LEP measurements [65], thus excluding electroweak baryogenesis within the SM. 2.4.3. Electroweak baryogenesis in MSSM In the MSSM the ratio @(Tc )=Tc can increase considerably. The MSSM Higgs sector at 1nite temperature has been considered in [122–125], for lattice studies see [126–128]. In the MSSM the right-handed stop tSR couples to the Higgs with a large Yukawa coupling. This leads to a strong 1rst order phase transition [123–125]. The LEP precision tests then indicate that the lightest left-handed stop should be heavy with mQ ¿ 500 GeV. This implies that for lightest right-handed stop mass

2 ˜ A m2t˜ ≈ m2U + 0:15MZ2 cos(2>) + m2t 1 − 2t ; (53) mQ where A˜ t = At − "=tan(>) is the stop mixing parameter, and " is the soft-SUSY breaking mass parameter for the right-handed stop. The coeYcient > of the cubic term >TH 3 in the eJective potential reads 3=2

A˜ 2t h3t sin3 (>) √ 1− 2 ; (54) >MSSM ≈ >SM + mQ 4 2 and can be at least one order of magnitude larger than >SM . In principle this modi1cation can give rise to a strong enough 1rst order phase transition. The sphaleron bound implies Higgs and stop masses in the range [41,42] 110 GeV 6 mH 6 115 GeV;

and

105 GeV 6 mt˜R 6 165 GeV :

(55)

The present LEP constraint on the Higgs mass is mH ¿ 115 GeV [129]. Hence, even an MSSM-based electroweak baryogenesis may be at the verge of being ruled out. The de1nitive test of the MSSM based electroweak baryogenesis will obviously come from the Higgs and the stop searches at the LHC and the Tevatron [40–42,130,111]. 2.4.4. Leptogenesis Even if B + L is completely erased by the sphaleron transitions, a net baryon asymmetry in the Universe can still be generated from a non-vanishing B − L [131], even if there were no baryon number violating interactions. Lepton number violation alone can produce baryon asymmetry B ∼ −L [132], a process which is known as leptogenesis (for a recent review [133], and references therein). For lepton number violation one however has to go beyond the SM.

118


A popular example is SO(10) GUT model, which can either be broken into SU (5) and then subsequently to the SM, or into the SM gauge group directly. The most attractive aspect of SO(10) is that it is left–right symmetric (for details, see [134,135]), and has a natural foundation for the see-saw mechanism [32,33] as it incorporates a singlet right-handed neutrino NR with a mass MR . A lepton number violation appears when the Majorana right handed neutrino decays into the SM lepton doublet and Higgs doublet, and their CP conjugate state through NR → @ + l;

NR → @S + lS ;

(56)

There also exist WL = 0, and WL = 2 processes mediated by the right-handed neutrino through (l@)(l@) ; MR

ll@@ ; MR

(57)

which are dimension 5 operators [136,137]. (There are other processes involving t-quarks which may also be important [138,139].) CP asymmetry is generated through the interference between tree level and one-loop diagrams. The total baryon asymmetry and total lepton asymmetry can be found in terms of the chemical potentials as [140] B= (2"qi + "uR i + "dR i ); L = (2"li + "eR i ) ; (58) i

i

where i denotes three leptonic generations. The Yukawa interactions establish an equilibrium between the diJerent generations ("li = "l and "qi = "q , etc.), and one obtains expressions for B and L in terms of the number of colors N = 3, and the number of charged Higgs 1elds NH B=−

4N "l ; 3

L=

14N 2 + 9NNH "l ; 6N + 3NH

together with a relationship between B and B − L [140] 8N + 4NH B= (B − L) : 22N + 13NH

(59)

(60)

A similar expression has also been found in [141,88,133], although there seems to be small of order one diJerences. The baryon asymmetry based on the decays of right-handed neutrinos in a thermal bath has been computed in [138,142–145]. In a recent analysis [145], it was pointed out that the baryogenesis scale is tightly constrained together with the heavy right-handed neutrino mass √ 10 TB ∼ M1; R = O(10 ) GeV, with an upper bound on the light neutrino masses i mi ¡ 3 eV. The current bound on the right-handed neutrino mass is around MR ∼ O(1011 ) GeV for light neutrino masses m1# ≈ m2# ≈ m3# ∼ O(0:1) eV. High-scale leptogenesis is ruled out in a supersymmetric theory because of the gravitino problem (see Section 3.7.1). However, if the masses of the right-handed neutrinos are such that the mass splitting is comparable to their decay widths, it is possible to obtain an enhancement in the CP phase of order one [144], and possibly a low scale thermal leptogenesis [146]. Otherwise, one could resort to non-thermal leptogenesis [147,136,137,148], or, to the scattering process discussed in [149], or to sneutrino driven leptogenesis [150,137].


119

2.4.5. Baryogenesis through ;eld condensate decay Scalar condensates may have formed in the course of the evolution of the early Universe. In particular, during in%ation all scalar 1elds are subject to %uctuations driven by the non-zero in%aton energy density so that 1elds with very shallow potentials may easily take non-zero values. An example is the MSSM, where for the squark and slepton 1elds there are several directions in the 1eld space where the potential vanishes completely [46,43]. These directions are called (perturbatively) %at. Field %uctuations along such %at directions will soon be smoothed out by in%ation, which eJectively stretches out any gradients, and only the zero mode, or the scalar condensate, remains. This mechanism is quite general and applicable to any order parameter with %at enough potential. Baryogenesis can then be achieved by the decay of a condensate that carries baryonic charge, as was 1rst pointed out by A4eck and Dine [44]. As we will discuss, the %at direction condensate can get dynamically charged with a large B and/or L by virtue of CP-violating self-couplings. Baryogenesis from MSSM %at directions has the virtue that it only requires two already quite popular paradigms: supersymmetry and in%ation. In the old version [44] baryons were produced by a direct decay of the condensate, to be discussed in Section 2.5.2. It was however pointed out 1rst by Kusenko and Shaposhnikov [151] in the case of gauge mediated supersymmetry breaking, and then by Enqvist and McDonald in [48] in the case of gravity mediated supersymmetry breaking, that the MSSM %at direction condensate most often is not stable but fragments and eventually forms non-topological solitons called Q-balls [47]. These issues will be dealt in Sections 6 and 7. 2.5. Old AAeck–Dine baryogenesis 2.5.1. Classical motion of the order parameter In the original A4eck–Dine baryogenesis [44] it was assumed that the order parameter along the %at direction is displaced from the origin because of in%ationary %uctuations. Because of in%ation, only the long wave-length model of the order parameter will survive so that a spatially constant condensate 1eld is formed along the %at direction. This we shall sometimes call the A4eck–Dine (AD) 1eld. In an expanding Universe the coherent AD 1eld * obeys the usual equation of motion, 9V *U + 3H *˙ + =0 ; 9*

(61)

where H is the Hubble parameter. To follow the time evolution of the AD 1eld, let us consider a toy model with the potential |*|6 + ··· : (62) M2 Although this potential is unrealistic in that it does not take correctly into account of supersymmetry breaking induced by the non-zero cosmological constant of the in%ationary era, it nevertheless captures the main features of the initial cosmological evolution of the AD 1eld. The theory Eq. (62) has a partially conserved current j" = i*∗ 9" *, with V (*) = m2 |*|2 + A(*4 + *∗4 ) +

9" j " = 9" (i*∗ 9" * − i9" *∗ *) = iA(*∗4 − *4 ) :

(63)

The current is conserved for small *. The role of the higher order term |*|6 is just to stabilize the potential. In the toy model Eq. (62), we identify the baryon number density nB with j0 . The model

120


also has a CP invariance under which * ↔ *∗ but which is violated by the initial conditions, which are taken to be * = i*0 ;

*˙ = 0 ;

(64)

where *0 is real. Writing * = *R + i*I one 1nds the coupled equations of motion (see e.g., [152]) 4 3|*| 2 *I = 4A*3R ; *U I + 3H *˙ I + m + 12A*R *I + M2 3|*|4 2 U ˙ *R + 3H *R + m + *R = 4A(3*I *2R − *3I ) : (65) M2 In a matter dominated Universe H = 2=(3t), so that for large times tm−1 the motion is damped and Eq. (65) has then oscillatory solutions of the form Ak sin(mt + Ik ); k = I; R ; (66) mt where the amplitudes Ak and the phases Ik depend on the parameters m, A, M , and the initial conditions Eq. (64). For large times the baryon number is then found to be *k =

nB = 2(*I *˙ R − *R *˙ I ) =

2AI AR sin(Ii − IR ) : mt 2

(67)

If *20 mM , as was tacitly assumed by A4eck and Dine [44], one may disregard the higher-order terms. In that case one obtains [45] AI = *0 and AR = aR A*30 =m2 , where aR = 0:85 is determined numerically. Likewise, numerically one 1nds that II − IR = 1:54. Thus, 1:7A*40 ; m3 t 2 and the generated baryon number per particle is nB =

R=

mnB 1:7A*20 = : * m2

(68)

(69)

Eq. (69) is true for matter dominated Universe; for radiation dominated Universe one obtains a similar result, but the numerical prefactor 1.7 should be replaced by −1:3. If *20 ¿ mM , one 1nds AI =aI (mM )1=2 and AR =aR (M 3 =m)1=2 with aI =0:94; aR =−2:86; II =011, and IR = −0:41. It then follows from Eq. (67), that 2:7AM 2 : (70) mt 2 Thus, the baryon generation mechanism is remarkably robust. The initial conditions do not matter, nor the actual expansion rate of the Universe. The baryon number generated per *-particle is always large and, with A ∼ m2 =*2 , typically nB 1. Although these conclusions were derived in a toy model, similar results hold true also for the MSSM %at directions. Thus, to summarize, along a %at direction where squarks and sleptons have non-zero expectation values, evolution of the AD 1eld condensate, starting from a CP violating initial value, will dynamically generate large baryon number density and charge the condensate with B and/or L. nB = −


121

2.5.2. Condensate decay To provide the Universe with the observed baryon to entropy ratio, nB =s ∼ 10−10 , the AD condensate must eventually transform itself into ordinary quarks. Originally [44], it was thought that this could happen via the decay of the AD 1eld components (squarks and sleptons) to ordinary quarks and leptons. The AD condensate can be thought of as a coherent state of *-particles where *=*0 eimt and |*0 |m. When supersymmetry breaking is switched on, the AD 1eld starts to oscillate about the old vacuum *m. Writing * = * + * , one observes that all the 1elds to which the excitations * couple are heavy with masses O(*). The 1eld * itself has a mass of the order of supersymmetry breaking, O(m). Therefore, * can decay to light 1elds only through loop diagrams involving heavy 1elds, with an eJective coupling of the type (g2 =*)* 9 † , where is a light fermion and g some coupling constant. The decay rate is thus, [44] < ∼ g4

m3 : *2

(71)

Because of the oscillations of the AD 1eld, the Universe will eventually become dominated by the energy density in the oscillations, * m2 *2 , so that H ∼ *1=2 =MP . The AD 1eld will decay

when < H , or * (m2 MP )1=3 . This implies a reheating temperature Trh s1=3 *1=4 while the baryon number density is nB = R* =m, where R is given in Eq. (69). Therefore one 1nally obtains A*20 M 1=6 nB 2 : (72) s m m Depending on A, and the size of the initial %uctuation *0 of the AD condensate, nB =s can be either small or large. Therefore determining the initial value is of utmost importance [153]. This requires us to consider theories of in%ation in more detail, which will be done in the next section. Following that, we shall discuss the disappearance of the AD condensate by fragmentation into (quasi)stable lumps of condensate matter, whose state of lowest energy is a spherical non-topological soliton, a Q-ball [47]. 3. Field uctuations during ination Apart from explaining the initial condition for the hot Big Bang model, the %atness problem, and the horizon problem, cosmological in%ation [154,13,14] is one of the most favored candidate for the origin of structure in the Universe (for reviews on in%ation, see [15,155]). There are many models of in%ation, but by far the simplest is one in which in%ation is generated by the large energy density of a scalar 1eld. The scalar 1eld driven in%ation not only explains the homogeneity and the %atness problems but also the observed scale invariance of the density perturbations. In%ation based on a scalar 1eld theory is described by the following Lagrangian: MP2 1 R + 9" *9" * − V (*) ; 2 2 where R is the curvature scalar. The energy–momentum tensor reads 1 T"# = 9" *9# * − g"# 9 *9 * − g"# V (*) 2 L=

(73)

(74)

122


so that the energy density and the pressure are given by 1 1 ≡ T00 = *˙ 2 + 2 (∇*)2 + V (*) ; 2 2a (t) 1 1 Tii p ≡ 2 = *˙ 2 − 2 (∇*)2 − V (*) : a (t) 2 6a (t)

(75) (76)

One of the initial conditions for in%ation is that there must be a homogeneous patch of the Universe which is bigger than the size of the Hubble horizon [156] (also supported by numerical studies, see [157]). However such a stringent condition can be evaded in a chaotic in%ation beginning at the Planck scale [158,159,15,160]. More complicated situation can be obtained if there are several 1elds that participate in in%ation; the classic example is assisted in%ation [161,162]. 3.1. Fluctuation spectrum in de Sitter space The plane wave solution of a massive scalar 1eld *(x; t) in a spatially %at Robertson–Walker metric can be decomposed into Fourier modes by 1 *= d 3 k (*k (t)eik ·x + h:c:) : (77) (2)3=2 Solving the Klein–Gordon equation for the scalar 1eld in a conformal metric: ds2 = g"# d x" d x# = a2 (;; x)(d;2 − d x2 ), the mode function can be given by [163–166] 1=2 *k (;) = H |;|3=2 (c1 H#(1) (k;) + c2 H#(2) (k;)) ; 4 ; = −H −1 e−Ht ;

and

#2 =

m2 9 − 2 ; 4 H

(78)

where m is the mass of the scalar 1eld, H#(1) and H#(2) are the Hankel functions and c1 ; c2 are constants. The readers might be tempted to take the limit ;0, in order to match the above solution with the plane wave solution in a Minkowski background. However this leads to a quasi static de Sitter solution [167]. More technically, it has been shown that using a point splitting regularization scheme, it is possible to obtain a Bunch–Davies vacuum for a de Sitter background which actually corresponds to taking c1 = 0, and c2 = 1. A simple but intuitive way has been developed in [168], where it has been argued that during a de Sitter phase, the main contribution to the two point correlation function comes from the long wavelength modes; k|;|1 or kH exp(Ht). Therefore the two point function is de1ned by an infrared cutoJ which is determined by the Hubble expansion [168] H eHt 1 *2 ≈ d 3 k |*k |2 : (79) (2)3 H The result of the integration yields [165–168] an inde1nite increase in the variance with time H3 t : (80) 42 This result can also be obtained by considering the Brownian motion of the scalar 1eld [160]. *2 ≈


123

For a massive 1eld with mH , and # = 3=2, one does not obtain an inde1nite growth of the variance of the long wavelength %uctuations, but [165–168] 3H 4 2 2 (1 − e−(2m =3H )t ) : (81) 2 2 8 m In the limiting case when m → H , the variance goes as *2 ≈ H 2 . In the limit mH , the variance goes as *2 ≈ (H 3 =122 m) [167]. Only in a massless case *2 can be treated as a homogeneous background 1eld with a long wavelength mode. This result plays an important role for the rest of this review as it implies that in a de Sitter phase any scalar 1eld, including the AD condensate, are subject to quantum %uctuations. *2 =

3.2. Slow roll in9ation A completely %at potential can render in%ation eternal, provided the energy density stored in the %at direction dominates. The in%aton direction is however not completely %at but has a potential V (*) with some slope. An in%ationary phase is obtained while 1 H2 ≈ V (*) ; (82) 3MP2 3H *˙ ≈ −V (*) ; (83) where prime denotes derivative with respect to *. In the above the approximations are: *˙ 2 ¡ V (*), and *U ¡ V (*), which lead to the slow roll conditions (see e.g. [9]) MP2 V 2 1 ; (84) j(*) = 2 V V |(*)| = MP2 1 : (85) V Note that j is positive by de1nition. These conditions are necessary but not suYcient for in%ation. They only constrain the shape of ˙ Therefore a tacit assumption behind the success of the potential but not the velocity of the 1eld *. the slow roll conditions is that the in%aton 1eld should not have a large initial velocity. In%ation comes to an end when the slow roll conditions are violated, j ∼ 1, and ∼ 1. However, there are certain models where this need not be true, for instance in hybrid in%ation models [169], where in%ation comes to an end via a phase transition, or in oscillatory models of in%ation where slow roll conditions are satis1ed only on average [170]. One of the salient features of the slow roll in%ation is that there exists a late time attractor behavior. This means that during in%ation the evolution of a scalar 1eld at a given 1eld value has to be independent of the initial conditions. Therefore slow roll in%ation should provide an attractor behavior which at late times leads to an identical 1eld evolution in the phase space irrespective of the initial conditions [171]. In fact the slow roll solution does not give an exact attractor solution to the full equation of motion but is nevertheless a fairly good approximation [171]. A similar statement has been proven for multi-1eld exponential potentials without slow roll conditions (i.e., assisted in%ation) [161]. The attractor behavior of the in%aton leads to powerful predictions which can be distinguished from other candidates of galaxy formation [19].

124


The standard de1nition of the number of e-foldings is given by tend * V 1 a(tend ) = H dt ≈ 2 d* ; N ≡ ln a(t) MP *end V t

(86)

where *end is de1ned by j(*end ) ∼ 1, provided in%ation comes to an end via a violation of the slow roll conditions. The number of e-foldings can be related to the Hubble crossing mode k = ak Hk by comparing with the present Hubble length a0 H0 . The 1nal result is [9]

1=4 Vk1=4 1 Vend k 1016 GeV N (k) = 62 − ln − ln + ln 1=4 − ln 1=4 (87) a0 H 0 3 Vk1=4 Vend rh where the subscripts end (rh) refer to the end of in%ation (onset of reheating). The details of the thermal history of the Universe determine the precise number of e-foldings, but for most practical purposes it is suYcient to assume that N (k) ≈ 50, keeping all the uncertainties such as the scale of in%ation and the end of in%ation within a margin of 10 e-foldings. A signi1cant modi1cation can take place only if there is an epoch of late in%ation such as thermal in%ation [172], or in theories with a low quantum gravity scale [173,174].

3.3. Primordial density perturbations Initially, the theory of cosmological perturbations has been developed in the context of FRW cosmology [175], and for models of in%ation in [176–180]. For a complete review on this topic, see [16]. For a real single scalar 1eld there arise only adiabatic density perturbations. In case of several %uctuating 1elds there will in general also be isocurvature perturbations. We brie%y describe the two perturbations and their observational diJerences. 3.3.1. Adiabatic perturbations and the Sachs–Wolfe eBect Let us consider small inhomogeneities *(x; t) = *(t) + I*(x; t) such that I**. Perturbations in matter densities automatically induce perturbations in the background metric, but the separation between the background metric and a perturbed one is not unique. One needs to choose a gauge. A simple choice would be to 1x the observer to the unperturbed matter particles, where the observer will detect a velocity of matter 1eld falling under gravity; this is known as the Newtonian or the longitudinal gauge because the observer in the Newtonian gravity limit measures the gravitational potential well where matter is falling in and clumping. The induced metric can be written as ds2 = a2 (;)[(1 + 2@) d;2 − (1 − 2K)Iik d xi d xk ] ;

(88)

where @ has a complete analogue of Newtonian gravitational potential. In the case when the spatial part of the energy momentum tensor is diagonal, i.e. ITji = Iij , it follows that @ = K [16]. Right at the time of horizon crossing one 1nds a solution for I* as H (t∗ )2 |I*k |2 = ; (89) 2k 3 where t∗ denotes the instance of horizon crossing. Correspondingly, we can also de1ne a power spectrum 2 2

k3 H (t H ) ∗ P* (k) = 2 |I*k |2 = ≡ : (90)

2 2 2 k=aH


125

Note that the phase of I*k can be arbitrary, and therefore, in%ation has generated a Gaussian perturbation. In the limit k → 0, one can 1nd an exact solution for the long wavelength inhomogeneities kaH [181,16], which reads t · 1 H @k ≈ c1 ; (91) a dt + c2 a 0 a t I*k 1 c1 (92) = a dt − c2 ; a *˙ 0 where the dot denotes derivative with respect to physical time. The growing solutions are proportional to c1 , the decaying proportional to c2 . Concentrating upon the growing solution, it is possible to obtain a leading order term in an expansion with the help of the slow roll conditions: H˙ @k ≈ −c1 2 ; (93) H c1 I*k ≈ : (94) ˙ H * Note that at the end of in%ation, which is indicated by aU = 0, or equivalently by H˙ = −H 2 , one obtains a constant Newtonian potential @k ≈ c1 . This is perhaps the most signi1cant result for a single 1eld perturbation. In a long wavelength limit one obtains a constant of motion [180,182,16] de1ned as 2 H −1 @˙ k + @k p = + @k ; w = : (95) 3 1+w If the equation of state for matter remains constant there is a simple relationship which connects the metric perturbations at two diJerent times [180,182,16] @k (tf ) =

1 + 23 (1 + w(tf ))−1 @k (ti ) : 1 + 23 (1 + w(ti ))−1

(96)

The comoving curvature perturbation [183] reads in the longitudinal gauge [16] for the slow roll in%ation as Rk = @k −

H2 (H −1 @˙ k + @k ) : H˙

(97)

For CMB and structure formation we need to know the metric perturbation during the matter dominated era when the metric perturbation is @(tf ) ≈ (3=5)c1 . Substituting the value of c1 from Eq. (94), we obtain

3 I*k

@k (tf ) ≈ H : (98) 5 *˙ k=aH

In a similar way it is also possible to show that the comoving curvature perturbations is given by

H ; (99) I*

Rk ≈ *˙ k=aH

126


where I* denotes the 1eld perturbation on a spatially %at hypersurfaces, because on a comoving hypersurface I* = 0, by de1nition. Therefore, on %at hypersurfaces ˙ ; I*k = *It

(100)

where It is the time displacement going from %at to comoving hypersurfaces [19,9]. As a result Rk ≡ HIt :

(101)

Note that during matter dominated era the curvature perturbation and the metric perturbations are related to each other 3 @k = − Rk : 5

(102)

In the matter dominated era the photon sees this potential well created by the primordial %uctuation and the redshift in the emitted photon is given by WTk = −@k : T

(103)

At the same time, the proper time scale inside the %uctuation becomes slower by an amount It=t =@k . Therefore, for the scale factor a ˙ t 2=3 , decoupling occurs earlier with Ia 2 It 2 = @k : = a 3 t 3

(104)

By virtue of T ˙ a−1 this results in a temperature which is hotter by 2 @k WTk = −@k + @k = − : T 3 3

(105)

This is the celebrated Sachs–Wolfe eJect [184], which we shall revisit when discussing isocurvature %uctuations. 3.3.2. Spectrum of adiabatic perturbations Now, one can immediately calculate the spectrum of the metric perturbations. For a critical density Universe

k 2 I

2 =− @k ; (106) Ik ≡ k 3 aH where ∇2 → −k 2 , in the Fourier domain. Therefore, with the help of Eqs. (90) and (98), one obtains 4 4 9 H 2 H 2 2 ; (107) Ik ≡ P@ (k) = 9 9 25 *˙ 2 where the right-hand side can be evaluated at the time of horizon exit k = aH . In fact the above expression can also be expressed in terms of curvature perturbations [19,9] k 2 2 Ik = Rk ; (108) 5 aH


127

˙ 2 (H=2)2 , exactly the same expresand following Eq. (97), we obtain I2k = 4=25PR (k) = (4=25)(H= *) sion as in Eq. (107). With the help of the slow roll equation 3H *˙ = −V , and the critical density formula 3H 2 MP = V , one obtains I2k ≈

1 1 V3 V ; = 6 2 4 j 2 2 150 MP 75 MP V

(109)

where we have used the slow roll parameter j ≡ (MP2 =2)(V =V )2 . The COBE satellite measured the CMB anisotropy and 1xes the normalization of I@ (k) on a very large scale. For a critical density Universe, if we assume that the primordial spectrum can be approximated by a power law and ignoring gravitational waves: (n−1)=2 k −5 I@ (k) = 1:91 × 10 ; (110) kpivot where n is the spectral index and kpivot =7:5a0 H0 is the scale at which the normalization is independent of the spectral index. The spectral index n(k) is de1ned as d ln P@ n(k) − 1 ≡ : (111) d ln k This de1nition is equivalent to the power law behavior if n(k) is fairly a constant quantity over a range of k of interest. The power spectrum can then be written as P@ (k) ˙ k n−1 :

(112)

If n = 1, the spectrum is %at and known as Harrison–Zeldovich spectrum [185]. For n = 1, the spectrum is tilted and n ¿ 1 is known as blue spectrum. In terms of the slow roll parameters, one can write [171] dj = 2j − 4j2 ; d ln k

d = −2j + M2 d ln k

dM2 = −2jM2 + M2 + 73 ; d ln k

(113)

where V (d 3 V=d*3 ) ; V2 Thus one 1nds [186] M2 ≡ MP4

n − 1 = −6j + 2 :

73 ≡ MP6

V 2 (d 4 V=d*4 ) : V3

(114)

(115)

Slow roll in%ation requires that j1; ||1, and therefore naturally predicts small variation in the spectral index within W ln k ≈ 1. The recent Boomerang data suggest [10] |n − 1| 6 0:1 :

(116)

The rate of change in is also very small, and can be estimated in a similar way [187] dn = −16j + 24j2 + 2M2 : (117) d ln k It is possible to extend the calculation of metric perturbation beyond the slow roll approximation basing on a formalism similar to that developed in [188–191].

128


3.3.3. Gravitational waves Gravitational waves are linearized tensor perturbations of the metric and do not couple to the energy momentum tensor. Therefore, they do not give rise a gravitational instability, but carry the underlying geometric structure of the space–time. The 1rst calculation of the gravitational wave production was made in [192], and the topic has been considered by many authors [193]. For reviews on gravitational waves, see [16,194]. The gravitational wave perturbations are described by a line element ds2 + I ds2 , where ds2 = a2 (;)(d;2 − d xi d xi );

I ds2 = −a2 (;)hij d xi d xj :

(118) ij

The gauge invariant and conformally invariant 3-tensor hij is symmetric, traceless I hij = 0, and divergenceless ∇i hij = 0 (∇i is a covariant derivative). Massless spin 2 gravitons have two degrees of freedom and as a result are also transverse. This means that in a Fourier domain the gravitational wave has a form hij = h+ eij+ + h× eij× :

(119)

For the Einstein gravity, the gravitational wave equation of motion follows that of a massless Klein Gordon equation [16]. Especially, for a %at Universe 2 k i i hUj + 3H h˙ j + hij = 0 : (120) a2 As any massless 1eld, the gravitational waves also feel the quantum %uctuations in an expanding background. The spectrum mimics that of Eq. (90) 2

H 2 Pgrav (k) = 2 : (121)

MP 2 k=aH

Note that the spectrum has a Planck mass suppression, which suggests that the amplitude of the gravitational waves is smaller compared to that of the adiabatic perturbations. Therefore it is usually assumed that their contribution to the CMB anisotropy is small. The corresponding spectral index can be calculated as [186] d ln Pgrav (k) ngrav = = −2j : (122) d ln k Note that the spectral index is negative. 3.4. Multi-;eld perturbations In multi-1eld in%ation models contributions to the density perturbations come from all the 1elds. However unlike in a single scalar case, in the multi-1eld case there might not be a unique late time trajectory corresponding to all the 1elds. This is true in particular for those 1elds that are eJectively massless during in%ation, such as the MSSM %at direction 1elds. Therefore, in these cases scalar perturbations will depend on the 1eld trajectories and thus on the choice of initial conditions, with an ensuing loss of predictivity. In a very few cases it is possible to obtain a late time attractor behavior of all the 1elds; an example is assisted in%ation [161]. Let us here nevertheless assume that there is an underlying unique late time trajectory resulting in a simple expression for the amplitude of the density perturbations and the spectral index [195,155].


129

3.4.1. Adiabatic and isocurvature conditions There are only two kinds of perturbations that can be generated. The 1rst one is the adiabatic perturbation discussed previously; it is a perturbation along the late time classical trajectories of the scalar 1elds during in%ation. When the primordial perturbations enter our horizon they perturb the matter density with a generic adiabatic condition, which is satis1ed when the density contrast of the individual species is related to the total density contrast Ik 1 1 1 1 1 Ikb = Ikc = Ik# = Ik = Ik ; 3 3 4 4 4

(123)

where b stands for baryons, c for cold dark matter, for photons and # for neutrinos. The other type is the isocurvature perturbation. During in%ation this can be viewed as a perturbation orthogonal to the unique late time classical trajectory. Therefore, if there were N %uctuating scalar 1elds during in%ation, there would be N − 1 degrees of freedom which would contribute to the isocurvature perturbation. The isocurvature condition is known as I = 0: the sum total of all the energy contrasts must be zero. The most general density perturbations is then given by a linear combination of an adiabatic and an isocurvature density perturbations. 3.4.2. Adiabatic perturbations due to multi-;eld In a comoving gauge Eq. (97) with R = −HI*= *˙ holds good even for multi-1eld in%ation models, provided we identify each 1eld component of * along the slow roll direction. There also exists a relationship between the comoving curvature perturbations and the number of e-foldings N [181,196,195,155] R = IN =

9N I*a ; 9*a

(124)

where N is measured by a comoving observer while passing from %at hypersurface (which de1nes I*) to the comoving hypersurface (which determines R) [195,197]. The repeated indices are summed over and the subscript a denotes a component of the in%aton. A more intuitive derivation has been given in [155,9]. If again one assumes that the perturbations in I*a have random phases with an amplitude (H=2)2 , one obtains I2k =

V 9N 9N : 2 9* 9* 2 75 MP a a

(125)

For a single component 9N=9* ≡ (MP−2 V=V ), and then Eq. (125) reduces to Eq. (109). By using slow roll equations we can again de1ne the spectral index n−1=−

2 MP2 N; a N; b V; ab MP2 V; a V; a − + 2 ; V2 VN; c N; c MP2 N; a N; a

(126)

where V; a ≡ 9V=9*a , and similarly N; a ≡ 9N=9*a . For a single component we recover Eq. (115) from Eq. (126). These results prove useful in constraining the AD potential by cosmological density perturbations, as will be discussed in Section 5.3.

130


3.4.3. Isocurvature perturbations and CMB One may of course simply assume a purely isocurvature initial condition. For any species the entropy perturbation is de1ned by In Ini Si = − : (127) n ni Thus, if initially there is a radiation bath with a common radiation density contrast Ir , a baryon-density contrast Ib = 3Ir =4, and a CDM density contrast Ic , then 3 r Ic − (3=4)c Ir r + (3=4)c S = Ic − Ir = = Ic ≈ Ic ; (128) 4 r c r c where we have used the isocurvature condition Ir +Ic =0, and the last equality holds in a radiation dominated Universe. However a pure isocurvature perturbation gives 1ve times larger contribution to the Sachs–Wolfe eJect compared to the adiabatic case [198,19,9]. This result can be derived very easily in a matter dominated era with an isocurvature condition Ic = −Ir , which gives a contribution Rk = (1=3)Sk . Therefore from Eqs. (102) and (105), we obtain WTk =T =−Sk =15. There is an additional contribution from radiation because we are in a matter dominated era, see Eq. (128), S ≈ Ic ≡ −(3=4)Ir . The sum total isocurvature perturbation WTk =T = −S=15 − S=3 = −6S=15, where S is measured on the last scattering surface. The Sachs–Wolfe eJect for isocurvature perturbations 1xes the slope of the perturbations, rather than the amplitude [199,200]. Present CMB data rules out pure isocurvature perturbation spectrum [201,202], although a mixture of adiabatic and isocurvature perturbations remains a possibility [201–205]. In the latter case it has been argued that the adiabatic and isocurvature perturbations might naturally turn out to be correlated [206,207]. The most general power spectrum is not a single function but a 5 × 5 matrix, which contains all possible adiabatic and isocurvature perturbations together with their cross-correlations. As discussed in [207], resolving the perturbation spectrum in all its generality would be an observational challenge that probably would have to wait for the determination of the polarization spectrum by the Planck Surveyor Mission. It is sometimes useful to consider the ratio = of the total power spectra, de1ned as Ptot =Pad +Piso [208,204], where = is de1ned as [204]

i

I 16 Piso

(129) =

a

; ==

25 Pad I k=aH

Ii

is the perturbation in the photon energy density due to isocurvature perturbations and Ia where is the perturbation due to adiabatic perturbations. 3.5. In9ation models A detailed account on in%ation model building can be found in many reviews [15,98,155,9]. Here we brie%y recall some of the popular models with a particular emphasis on supersymmetric in%ation. First we recapitulate some aspects of non-supersymmetric models. 3.5.1. Non-supersymmetric in9ation The very 1rst attempt to build an in%ation model was made in [154], where one loop quantum correction to the energy momentum tensor due to the space–time curvature were taken into account,


131

resulting in terms of higher order in curvature invariants. Such corrections to the Einstein equation admit a de Sitter solution [209], which was presented in [154,210]. In%ation in Einstein gravity with an additional R2 term was considered in [211] (for a discussion of in%ation in pure R2 gravity, see [212]). Such a theory is conformally equivalent to a theory with a canonical gravity [213] with a scalar 1eld having a potential term. A similar situation arises in theories with a variable Planck mass, i.e., in scalar tensor theories [66]. In%ation in these models has been studied extensively [214]. The simplest single 1eld in%ation model is arguably chaotic in%ation [158,215] with a generic potential V=

A

MP=−4

*= ;

(130)

where = is a positive even integer. In chaotic in%ation slow roll takes place for *MP , and the two slow roll parameters are given by [9] j≡

=2 MP2 ; 2 *2

= =(= − 1)

MP2 : *2

(131)

In%ation ends when j ≡ 1, or, * ≈ =MP . The cosmological scales leave the horizon when * = √ 2N=MP , and the spectral indices for scalar and tensor perturbations turn out to be [9] 2+= 3:1= : (132) n=1− ; r= 2N N The amplitude of the density perturbations, if normalized at the COBE scale, yields the constraint A 4 × 10−14 . An exponential potential, such as might arise in string theories and theories with extra dimensions,

2 * V (*) = V0 exp − (133) p MP would give rise to a power law a(t) ˙ t p for the scale factor, so that in%ation occurs when p ¿ 1. Multiple exponentials with diJering slopes give rise to what has been dubbed as assisted in%ation [161]. 3.5.2. F-term in9ation In four dimensions the N = 1 supersymmetric potential receives two contributions: one from the F-term, which is related to the chiral supermultiplets, and the second from the D-term, which contains the gauge interactions. For a detailed discussion of supersymmetric in%ation we refer to the review by Lyth and Riotto [155]. Here we give a brief resume of the two types of in%ation. Historically, supersymmetric in%ation was 1rst introduced to cure some of the problems associated with the 1ne tuning of new in%ation [216], but since then utilizing supersymmetry as a tool for in%ation has gained in popularity (we describe supersymmetry in Section 4.2, and for supergravity, see Section 4.5.2). The F-term potential can be derived from the superpotential W 9W ∗ ∗ ∗i V (*; * ) = F Fi ; Fi = − ; (134) 9*i where for renormalizable interactions W has a mass dimension three.

132


Supersymmetry is broken whenever |F|2 = 0. A simple working example is to consider the superpotential W = AS(*2 − *20 )

(135)

which is invariant under a global R symmetry with the super1elds S and N carrying respectively the R charges 1 and 0. The scalar components of these super1elds can be written in the form 7 *1 + i*2 S = √ ; *= √ ; (136) 2 2 where we have used R-transformation in order to make S real. The potential follows from Eq. (134): A2 2 (* + *22 )2 + A2 72 (*21 + *22 ) : (137) 4 1 The supersymmetric vacuum is located at 7 = 0, *1 = *0 , and *2 = 0. Note that the potential has a %at direction along 7-axis when 7 ¿ 7inst = *0 . When 7 ¡ 7inst , the mass squared of *1 becomes negative and suggests a phase transition along the *1 direction. When this happens 7, *1 , and *2 begin to oscillate around their supersymmetry preserving vacua. If *1 = *2 = 0, the height of the potential is given by V = A2 *40 , and as a consequence one obtains a period of in%ation. This is the simplest example of a %at direction giving rise to an in%ation potential, and it is known as the hybrid model, 1rst described in a non-supersymmetric context in [215,160] and in a supersymmetric context in [217]. In order to have a graceful exit from in%ation one requires a slope for the %at direction such that 7 can roll down and approach 7inst . The %atness of the potential can be lifted in two ways: by radiative corrections [218], or by the low energy soft supersymmetry breaking eJects. Due to supersymmetry breaking the fermions obtain a mass of the order (92 W=9*2 ) = AS, while the two complex scalars receive a mass squared A2 S 2 ± A2 *20 . The one-loop radiative correction to the potential is given by [219] 1 Mi2 fi 4 IV = (−) M ln ; (138) i 642 i M2 V = A2 *40 − A2 *20 (*21 − *22 ) +

where fi denotes the number of fermions, Mi2 is the fermion mass squared, and M the cut-oJ or the renormalization scale. The summation should be taken over all helicity states i. In the present example the eJective potential along the %at direction is given by CA2 7 2 4 V = A *0 1 + 2 ln √ ; (139) 8 2M where C is a constant essentially counting the states running in the loops. If the loop correction dominates over the tree level potential, there is a period of in%ation which typically ends when CN 7=A MP : (140) 42 The COBE normalization sets the scale of in%ation to 1=4 50 1=4 C 1=4 A × 1015 GeV (141) V ∼ N


while the spectral index is given by 3CA2 1 1+ : n=1− N 162

133

(142)

Therefore, depending on the coupling constant A and the number of e-foldings N , it is possible to have a wide range of in%ation energy scales which all provide a spectral index n ∼ 0:96– 0.98. Soft supersymmetry breaking contributions induce m7 ∼ O(TeV). One could also imagine that the mass of 7 appears dynamically if 7 has couplings to bosons and fermions; these may induce a typical running mass ˙ 72 ln(7=M ) [220,155,221]. 3.5.3. D-term in9ation In the above discussion we have neglected the gauge contribution. The D-term Da = −ga (*∗ T a *) gives rise to a scalar potential (see [37,34]) 1 a D Da ; V (*; *∗ ) = (143) 2 where (T a )ji satisfy [T a ; T b ] = ifabc T c (fabc is the structure constant). The simplest realization of D-term in%ation reproduces the hybrid potential with three chiral super1elds, S, *+ , and *− with (non-anomalous) U (1) charges 0; +1; −1 [222]. The superpotential can be written as W = AS*+ *− :

(144)

The scalar potential then reads [222] g2 (|*+ |2 − |*− |2 + M2 )2 ; (145) 2 where g is the gauge coupling and M is the Fayet–Iliopoulos D-term. Note that the potential allows unique supersymmetry preserving vacua with a broken gauge symmetry S = *+ = 0, and *− = M. By virtue of the coupling, when |S| ¿ Sinst = gM=A, the 1elds *+ ; *− → 0, and therefore in%ation occurs because of the Fayet–Iliopoulos D-term V = g2 M4 =2. The slope along the in%aton direction S can be generated by the one-loop contribution and reads g2 g 2 M4 A2 |S|2 1+ V= : (146) ln 2 162 MP2 √ In%ation ends when slow roll condition breaks down for S ∼ (g=2 2)MP , and the predictions for the in%ationary parameters are similar to the previous discussion. D-term in%ation based on an anomalous U (1) symmetry (which could appear in string theory [223]) is no diJerent. Hybrid in%ation is successful but has also problems that are related to the initial conditions. In [224], it was pointed out that hybrid in%ation requires an extremely homogeneous 1eld con1guration for the 1elds orthogonal to the in%aton. In our example the orthogonal 1elds to the in%aton must be set to zero with a high accuracy over a region much larger than the initial size of the horizon. It is possible to solve this impasse by having a pre-in%ationary matter dominated phase when the 1eld orthogonal to the in%aton direction oscillates and decays into lighter degrees of freedom, gradually settling down to the bottom of its potential [224]. V = A2 |S|2 (|*+ |2 + |*− |2 ) + A2 |*+ *− |2 +

134


3.5.4. Supergravity corrections When the 1eld values are close to the Planck scale, supergravity (SUGRA) eJects become important and may ruin the %atness of the in%aton potential. The soft breaking mass of the scalar 1elds are typically [225,217,226,45,46] V m2soft ∼ ∼ O(1)H 2 : (147) 3MP2 Once the in%aton gains a mass ∼ H , the 1eld simply rolls down to the minimum of the potential and in%ation stops. Indeed, in SUGRA the slow roll parameter m2 V ∼ SUGRA || ≡ MP2 ∼ O(1) ; (148) V H2 where m2SUGRA ≈ m2SUSY + (VSUSY =3MP2 ) ∼ m2SUSY + O(1)H 2 . Note that the latter contribution dominates in an expanding Universe and violates the slow roll condition. For 1eld values smaller than Planck scale it is always possible to obtain j1, but in supergravity can never be made less than one for a single chiral 1eld with a minimal kinetic term. This is known as the problem in SUGRA models of in%ation [217]. When there are more than one chiral super1elds, it might be possible to cancel the dominant O(1)H correction to the in%aton mass by choosing an appropriate KUahler term [226,217] (see also discussion in [155].) In hybrid in%ation models derived from an F-term the dominant O(1)H correction in the mass term can be canceled if |N | = 0 exactly, which however seems to lead to an initial condition problem, as discussed above. The fact that the superpotential is linear in S in Eqs. (135) and (144) guarantees the cancellation of the dominant contribution in the mass term for a minimal KUahler term ∼ |S|2 . For non-minimal KUahler potential such as K = |S|2 + >|S|4 =MP2 + · · ·, one obtains (92 K=9S9S ∗ )−1 ∼ 1 − 4>|S|2 =MP2 . These contributions again lead to a problematic > × O(1)H contribution to the in%aton mass unless the value of the unknown constant > is suppressed. In [222], it was shown that the problem does not appear for D-term in%ation even for the non-minimal KUahler potential because the main contribution to the in%ation potential does not come from the vev of the in%aton 1eld alone, but from the Fayet–Iliopoulos term which belongs to the D-sector of the potential. Based on this fact many D-term in%ation models have been written down [227,228]. Therefore, hybrid in%ation, whether realized as an eJective potential coming from F-sector or from D-sector, appears to be among the most promising models for supersymmetric in%ation. 3.6. Reheating of the Universe 3.6.1. Perturbative in9aton decay Traditionally reheating has been assumed to be a consequence of the perturbative decay of the in%aton [229,198,22]. After the end of in%ation, when H 6 m* , the in%aton 1eld oscillates about the minimum of the potential. Averaging over one oscillation results in [230] pressureless equation of state where p = *˙ 2 =2 − V (*) vanishes, 3 so that the energy density redshifts as during matter domination with * = i (ai =a)3 (subscript i denotes the quantities right after the end of in%ation). If is not very large), the beta functions for (mass)2 of squarks of the 1rst and second generations and sleptons only receive signi1cant contributions from gauge/gaugino loops. A review of these eJects can be found in [314]. Here we only mention the main results for the case of universal boundary conditions, where at MGUT all the scalar masses are m20 and the gauginos have a common soft breaking mass m1=2 . For a low value of tan > = 1:65, 4 1 m2Hu − m20 − 2m21=2 2

(196)

at the weak scale, while m2u˜ 3 and m2Q˜ remain positive. The soft breaking (mass)2 of the 1rst and 3 second generations of squarks is m20 + (5 − 7)m21=2 , while for the right-handed and left-handed sleptons one gets m20 + 0:1m21=2 and m20 + 0:5m21=2 , respectively. The important point is that the sum m2Hu + m2L , which describes the mass along the Hu L %at direction, is driven to negative values at the weak scale only for m1=2 & m0 . This is intuitively understandable, since Eqs. (195) have a 1xed point solution [317] m2Hu + m2u˜ 3 + m2Q˜ = At = 0 when m1=2 = 0. 3

5.2. Hubble induced radiative corrections Here we describe radiative corrections in a cosmological set-up relevant for the AD mechanism [308]. When the Hubble induced supersymmetry breaking is dominant, i.e. for H ¿ O(TeV), the evolution of the soft terms is diJerent from the vacuum RG equations given in Eq. (195). For the low-energy supersymmetry breaking case, constraints from the weak scale (e.g. realization of electroweak symmetry breaking, and experimental limits on the sparticle masses) give information 4

This value corresponds to the case of maximal top Yukawa coupling, so called 1xed point scenario [316,317]. Such a low value of tan > is excluded by Higgs searches at LEP [318], unless one allows stop masses well above 1 TeV. We nevertheless include this scenario in our discussion since it represents an extreme case.


153

about the soft breaking parameters m20 and m1=2 . Together with 1ne tuning arguments, these constraints imply that m20 ¿ 0 and that m0 ; m1=2 are O(TeV). In the Hubble induced supersymmetry breaking case m20 and m1=2 are determined by the scale of in%ation (and the form of the KUahler potential). At low scales the Hubble induced terms are completely negligible because at temperature T ∼ MW , H ∼ O(1) eV; at present the Hubble parameter is tiny, H0 ∼ O(10−33 ) eV. There exists an even more fundamental diJerence between the Hubble induced and ordinary radiative corrections. In Minkowski space the loop contributions to beta functions freeze at a scale of the order of the mass of the particles in the loop. In an expanding Universe the horizon radius ˙ H −1 de1nes an additional natural infrared cut-oJ for the theory. The masses of particles coupled to the %at direction receive contributions from two sources. There is a supersymmetry preserving part proportional to the vev *, and the Hubble induced supersymmetry breaking part. The loop contributions to beta functions should thus be frozen at a scale given by the largest of |*| and H (recall that ht and gauge couplings are close to one). In particular, if the squared mass of the %at direction condensate is positive at very large scales but turns negative at some intermediate scale Qc , the origin of the %at direction potential will cease to be a minimum, provided the Hubble parameter is less than Qc . On the other hand, if m2* ¡ 0 at the GUT scale, its running should already be terminated at the scale |*| determined by Eq. (190). 5 In the following two subsections we discuss separately the cases of positive and negative GUT-scale (mass)2 for the %at direction condensate. 5.2.1. The case with CI ≈ −1 In this case all scalar 1elds roll towards the origin very rapidly and settle there during in%ation, provided radiative corrections condensate * is to their masses are negligible. A typical %at direction a linear combination * = Ni=1 ai ’i of the MSSM scalars ’i , implying that m2* = Ni=1 |ai |2 m2’ . In [308], it was noticed that with small values for the " parameter, the running of m2* crucially depends on m1=2 . Let us consider sample cases with gaugino masses m1=2 = (H ; 3H ; H=3), the A-term 6 At (MGUT ) = (±H ; ±3H ; ±H=3), top Yukawa ht (MGUT )=(2; 0:5) and couplings g1 (MGUT )=g2 (MGUT )=g3 (MGUT )= 0:71, and follow the running of scalar soft masses from MGUT down to 103 GeV, where low-energy supersymmetry breaking becomes dominant. The main result is that only the LHu %at direction can acquire a negative (mass)2 at low scales. In this case m2* = (m2Hu + m2L + "2 )=2, where the last term is from the Hubble induced " term. The results are summarized in Table 2, where it has been assumed that "(MGUT ) . H=4 so that the "-term contribution to m2* is negligible. In general m2* changes sign at a higher scale for ht (MGUT )=2. This is expected since a large Yukawa coupling naturally maximizes the running of m2Hu . Furthermore, the diJerence between At =m1=2 ¡ 0 and At =m1=2 ¿ 0 5

Here we note that the Hubble cut-oJ usually plays no role in loop corrections to the in%aton potential. In most in%ation models the masses of the 1elds which may run in the loop are larger than the Hubble expansion rate due to the presence of a 1nite coupling to the in%aton. This will happen if the (time varying) in%aton vev is large and the couplings are not very small. In those cases, which are somewhat similar to our case with CI ¿ 0, one can trust the usual loop calculation evaluated in a %at space time background [218]. 6 The RG equations (195) for At show that the relative sign between At and m1=2 matters, since it aJects the running of |At |, and subsequently, scalar soft masses. Without loss of generality we take the common gaugino mass m1=2 to be positive.

154


Table 2 The scale Qc (in GeV) where the squared mass of the Hu L %at direction changes sign, shown for CI = −1 and several values for the ratios At =H and m1=2 =H as well as the top Yukawa coupling ht , all taken at scale MGUT = 2 × 1016 GeV, from [308] At =H

m1=2 =H

Qc (ht = 2)

Qc (ht = 0:5)

+1=3 (−1=3) +1=3 (−1=3) +1=3 (−1=3)

1=3 1 3

× 106 –107 1011

× 103 106 –107

+1 (−1) +1 (−1) +1 (−1)

1=3 1 3

× 106 –107 1011

× 105 (×) 108 (106 )

+3 (−3) +3 (−3) +3 (−3)

1=3 1 3

× 1014 (107 ) 1015 (1011 )

107 109 (103 ) 1010 (106 )

becomes more apparent as |At =m1=2 | increases and ht decreases. The quasi 1xed-point value of At =m1=2 is positive [316,317]. Positive input values of At will thus lead to positive At at all scales, but a negative At (MGUT ) implies that At 0 for some range of scales, which diminishes its eJect in the RG equations, see Eq. (195). The sign of At (MGUT ) is more important for smaller ht , since then At =m1=2 will evolve less rapidly. It was noticed in [308] that the squared mass of the Hu L %at direction does not change sign when m1=2 = H=3, except for 7 At = ±3H and ht = 0:5. This can be explained by the fact that for small m1=2 and small or moderate |At | we are generally close to the 1xed point solution 1 1 (197) m2Hu − H 2 ; m2u˜3 0; m2Q˜ 3 H 2 : 2 2 Nevertheless, even for m1=2 H the squared mass of the LHu %at direction as well as m2u˜ 3 are ¡ 0:2H 2 above 1 TeV, because of the 1xed point behavior. This implies that the LHu %at direction can still be viable for baryogenesis, as pointed out by McDonald [324]. Flat directions built out of u˜ 3 will be marginal at best, since the decrease in m2u˜ 3 will be counteracted by other contributions to m2* ; e.g. for the uS 3 dS 1 dS 2 %at direction we 1nd m2* ¿ 2H 2 =3 at all scales. The AD mechanism for baryogenesis should always work if Qc ¿ HI , since in that case the global minimum of the potential during in%ation is located at |*| = 0. Note that in this case the vev |*| is usually determined by Qc rather than by Eq. (190). For scales close to Qc the mass term in the scalar potential Eq. (189) can be written as >* H 2 |*|2 log(|*|=Qc ), where the coeYcient >* can be obtained from the RG equations. If >* ¿ 0, which is true for the Hu L %at direction for CI ¡ 0, this term will reach a minimum at log(|*|=Qc ) = −1. If d−3 1=d−2 Qc ¡ (HI MGUT ) the non-renormalizable contributions to the scalar potential are negligible for |*| ∼ Qc , so that the minimum of the quadratic term essentially coincides with the minimum of 7

m2Hu

For this choice of parameters, At runs initially very slowly. It will therefore remain large for some time and helps to decrease quickly towards lower scales.


155

the complete potential given by Eq. (189). In models of high scale in%ation (e.g. chaotic in%ation models), the Hubble constant during in%ation HI can be as large as 1013 GeV. This implies that m2* for the Hu L %at direction can only become negative during in%ation if m21=2 H 2 , which includes the “no-scale” scenario studied in [305]. The region of the parameter space safely allowing AD leptogenesis is much larger in models of intermediate and low scale in%ation (e.g. some new in%ation models [14]) where HI is substantially smaller. In such models one can easily have HI ¡ Qc at least for the Hu L %at direction, unless m21=2 H 2 or "2 & m21=2 [308]. If Qc ¡ HI , the condensate * settles at the origin during in%ation and its post-in%ationary dynamics will depend on the process of thermalization. If the in%aton decay products thermalize very slowly, m2* is only subject to zero-temperature radiative corrections and * can move away from the origin once H . Qc ; a necessary condition for this scenario is that in%atons do not directly decay into 1elds that are charged under SU (3) × SU (2) × U (1)Y . If Qc 1 TeV, * will readily settle at the new minimum and AD leptogenesis can work. The situation will be completely diJerent if in%atons directly decay into some matter 1elds. In 2 such a case the plasma of in%aton decay products has a temperature T ∼ ( = 1 and A = 1). These are representative of all the other directions, too, except for Hu L. For M ∼ O(1), typical value for K is found to be about −0:05. Similar contours S 2 and (QLd) S 2 directions, see [329]. For all the squark directions can be obtained for the d = 6 (uSdS d) with no stop, as long as hb and hu can be neglected, K is always negative, and the contours of equal K do not depend on A. In the presence of stop mixing K ¡ 0 is no longer automatic even in the purely squark directions. The more there is stop, the larger value of M is required for K ¡ 0. Even for pure stop directions, positive K is typically obtained only for relatively light gaugino masses with M . 0:5. In contrast to the squark directions, K was found [329] to be always positive in the Hu L-direction. This is due to the fact Hu L does not involve strong interactions which in other directions are mainly

158

K. Enqvist, A. Mazumdar / Physics Reports 380 (2003) 99 – 234 2

2 d

d 1.5

1.5

c

1

ξ

ξ

c 1

b 0.5

0.5

b

a

(i) 0 -3

-2

-1

0

1

2

0 -3

3

(ii)

a -2

-1

A

2

3

2.5

2 d

d

1.5

c

c

ξ

ξ

1.5

1

A

2.5

2

0

1

1 b

b 0.5

0.5 a

a

(iv) 0 -3

-2

-1

0

A

1

2

(iii) 3

0 -3

-2

-1

0

1

2

3

A

Fig. 1. Contours of K for two d = 4 %at directions in the (A; M ≡ mg =m(t = 0))-plane: (a) K = 0 (b) K = −0:01; (c) K = −0:05; (d) K = −0:1. The directions are (i) Q3 Q3 QL; (ii) QQQL, no stop; (iii) uS 3 uSdSe; S (iv) uSuSdSeS with equal weight for all u-squarks, S from [329].

responsible for the decrease of the running scalar masses. Very roughly, instability is found when mg˜ & mt˜, although the exact condition should be checked case by case. In general, the sign of K could be deduced from the observation of SUSY parameters such as tan >, the gluino mass and the supersymmetry breaking parameter m* [329]. 5.3.2. Gauge mediated supersymmetry breaking A similar analysis can be made for the gauge mediated case, where supersymmetry breaking is transmitted to the observable sector below some relatively low messenger sector scale :S , above which the potential is completely %at (see Section 4.4.3). In the gravity mediated scenario the soft masses stay intact, modulo RG running, up to the Planck scale; in gauge mediation the masses simply disappear above :S . For a large condensate vev, one can integrate out the gauge and chiral 1elds coupled to the %at direction in order to obtain an eJective low energy theory. In such a case,


159

as was 1rst pointed out by Kusenko and Shaposhnikov [151], the potential along the %at direction obtains a logarithmic correction of the form [151,331]

A2 |@|2(d−1) |@|2 AA A@d 4 U (@) = m* log 1 + 2 + + + h:c: ; (204) m* dMPd−3 MP2(d−3) where m* ∼ 1–100 TeV. Because of the diJerences in the potential, the dynamical evolution of the condensate 1eld will be markedly diJerent from the gravity mediated case. Because the messenger sector is not constrained by experiments, one cannot provide a detailed description of the mass parameters. Here one should note that in order to have an AD condensate, the A-term is actually constrained. In the gauge mediated case |AA | 6 (10−4 –10−7 )m* , for d = 4; 6. Eqs. (200) and (204) are two book-keeping equations which are useful for the rest of this review. 5.4. Density perturbations from the 9at direction condensate The role of MSSM %at directions is not just limited to generating the lepton and/or baryon asymmetry in the Universe, but they also play an interesting role in the dynamics of density perturbations. 5.4.1. Energetics of 9at direction and the in9aton ;eld Once the %atness of the %at direction potential is lifted by non-renormalizable terms, for large 1eld values the condensate energy density can dominate over the in%aton potential. This could be disastrous: either in%ationary expansion would come to a halt, or the %at direction condensate %uctuations might ruin the successful predictions for the angular power spectrum [52–54,332]. In [53], the generation of adiabatic density perturbations was studied for both D- and F-term in%ation models. Note that in the former case there is no Hubble induced mass correction to the %at direction condensate. The scalar potential for F- and D-%at direction of dimension d is given by (see Eq. (191) in Section 4.6.2) A2 |@|2(d−1) ; (205) M 2(d−3) where only the dominant term from Eq. (191) corresponding to superpotential term of the form W = A@d =nM d−3 has been kept. Throughout this discussion R-parity is conserved and therefore we deal only with even dimensions d = 4; 6; 8. For illustrative purposes let us assume that the D-term in%ationary potential is given by (see Eq. (146) in Section 3.5.3) 2 g 4 M4 g 2 M4 S + V (S) = ; (206) ln 2 322 Q2 V (*) ≈

where S is the in%aton component; Q is here the renormalization scale. For a large initial vev for *; S ∼ O(MP ), the dynamics is 1rst dominated by V (*). For a suYciently large vev of * the eJective condensate mass squared V (*), becomes larger than H 2 . This occurs if * ¿ *H , where [53] g 1=(d−2) 2(d−1)=2(d−2) *H = M2=(d−2) MP(d−4)=(d−2) : (207) (6(2d − 2)(2d − 3))1=2(d−2) A

160


If initially *i ¿ *H , then * will 1rst oscillate in its potential with a decreasing amplitude: *(t) ˙ a−3=d (t) [230]. This period ends before the onset of in%aton domination. The system then enters a regime where both * and S are slowly rolling. The slow rolling dynamics of the scalar 1elds is given by the solution of 1=2 9V (Ka ) a V (Ka ) ˙ ; H= ; (208) 3H Ka = − 9Ka 3MP2 where Ka ≡ S; *. By taking the ratio of the equations for * and S, one obtains 9* 162 (d − 1)A2 *(2d−3) S ; = 9S 2d−2 g4 M4 MP2(d−3)

(209)

which has a general solution of the form −4 2 * = *i [1 + =d *2d (Si − S 2 )]−1=(2d−4) ; i

=d =

162 (d − 2)(d − 1)A2 2d−2 MP2(d−3) g4 M4

;

(210)

where *i and Si are the initial values at the onset of in%ation. There are two features about this solution. First, since Si is large compared with the value of S at N = 50 e-foldings before in%ation, we see that for suYciently large *i the value of * at late times is ;xed by Si , 1=(2d−4) 1 1 : (211) * ≡ *∗ ≈ 1=(d−2) =d Si This is true if *i ¿ *∗ , otherwise, * simply remains at *i . Second, we can relate Si to the total number of e-foldings during the V (S) dominated period of in%ation. In general, for suYciently large *i , we could have an initial period of V (*) dominated in%ation. During this period S does not signi1cantly change from Si . The potential is dominated by V (*) once * ¿ *S , where [53] √ (d−3)=(d−1) 2 4 1=2(d−1) 2MP gM : (212) *S = 1=(d − 1) A 2 *S is generally less than *H (see Eq. (207)), therefore * will be slow rolling during V (S) domination. From Eq. (210), we 1nd that the condition for S to change signi1cantly from Si at a given value of * is given by d−2 1 1 : (213) Si ¡ 1=2 * =d The condition for S to change signi1cantly during V (*) dominated in%ation is given by Eq. (213) with * = *S ; one 1nds (d−3)=(d−1)

2(d−2)=2(d−1) gd=(d−1) M2=(d−1) MP : (214) 4 A1=(d−1) Since Sic is small √ compared with MP , whereas the value of S required to generate 50 e-foldings of in%ation, S50 = g 50MP =(2) is close to MP , it follows that Si (¿ S50 ) will generally be larger than Sic , and so the in%aton will remain at Si until the Universe becomes in%aton dominated. Si ¡ Sic ≈


161

In this case the total number of e-foldings during in%aton domination is given by NS where Si = (g=2)NS1=2 MP . If *i ¿ *∗ , then * at N ≈ 50 e-foldings of in%ation will be given by [53] 1=(d−2) 1=(2d−4)

2 1 : (215) *∗ ≈ =d gMP NS1=2 Note that the dependence on NS is quite weak; for the case of d = 4 (d = 6) AD baryogenesis, *∗ ˙ NS−1=4 (NS−1=8 ). If there is no large number of in%ationary e-foldings one can essentially 1x the value of *∗ . In this case one can predict the magnitude of the baryonic isocurvature perturbation. Imposing a chaotic-type initial condition V (*i ) ≈ MP4 yields √ 2MP *i ≈ 1=(d−1) : (216) A By directly solving the slow roll equations for * and S, we obtain the total number of e-foldings of in%ation: N T = N* + N S ≈

*2i 1 42 Si2 + ; 4(d − 1) MP2 g2 MP2

(217)

where N* is the number of e-foldings during V (*) domination, provided *i ¿ *S . V (S) will dominate the total number of e-foldings only if NS &

1 : 2(d − 1)A2=(d−1)

(218)

Since NS ¿ 50, the above condition will be satis1ed so long as A is not very small (for example, if A ≈ 1=(d − 1)!). In this case the value of * at the time when the CMB perturbations are generated will be determined mainly by the total number of e-foldings of in%ation, i.e. NT ≈ NS . 5.4.2. Adiabatic perturbations during D-term in9ation The potential for the %at direction condensate is far from %at, and so if the magnitude of the %at direction condensate is large, it will cause a large deviation from scale-invariance to the adiabatic perturbation. This will impose an upper limit on the amplitude of the %at direction condensate at 50 e-foldings before the end of in%ation. If we assume that the %at direction follows a late time attractor trajectory together with the in%aton, then following the analyses in [53], and Kawasaki and Takahashi [54], the %at direction induced adiabatic density perturbation can be estimated from Eq. (125). For a potential of the form V = V (S) + V (*), one obtains (with the help of Eqs. (84), (85), (115) and (126)) [53] + V )(V V 2 + V V 2 ) 2(V MP2 S S S * * * =− (219) VS VS + V* V* − (VS + V* )V (VS2 + V*2 ) and M2 Q= P (VS + V* )V

(VS + V* )(VS2 + V*2 ) 2V

:

(220)

162


Particularly, for the case of D-term in%ation, if V* ¡ VS and V* ¡ VS , we obtain the conventional result, see Eq. (115). Here the isocurvature contribution to the spectral index has been neglected; we will discuss it in the next subsection. Since the main contribution to the scale-dependence of the perturbations comes from , let us estimate the deviation from scale-invariance due to the presence of the %at direction condensate. Note that when V* ¿ VS , with V* VS and V* VS still satis1ed, we can expand in order to obtain corrections to the conventional D-term in%ation model [53] ≈ MP2

V* V* VS − MP2 : VS VS VS

(221)

The condensate scalar induced deviation from the scale invariance in the spectral index is given by Wn* ≈ −

2V* V* MP2 VS VS

:

(222)

Requiring that |Wn* | ¡ K, the present CMB observations imply that n = 1:2 ± 0:3; adopting K ¡ 0:2 imposes an upper bound on *, K 1=(4d−7) 5=(4d−7) −4=(4d−7) 8=(4d−7) (4d−15)=(4d−7) g A M MP ; (223) * ¡ * c = kd √ N where

kd =

22(d−1) 128(d − 1)2 (2d − 3)

1=(4d−7) :

For the case of minimal d = 4 AD baryogenesis, one obtains [53] K 1=9 5 −4 8 (g A M MP )1=9 ∼ 1016 GeV ; *c = 0:53 √ N while for d = 6 AD baryogenesis scenario, one gets K 1=17 5 −4 8 9 1=17 (g A M MP ) ∼ 1017 GeV : *c = 0:77 √ N

(224)

(225)

(226)

5.4.3. Adiabatic perturbations during F-term in9ation During F-term in%ation, the dominant part of the %at direction potential is given by (see Eq. (189), Section 4.6.1) CI H 2 * 2 + V (*) ; (227) 2 where V (*) is the usual part from the non-renormalizable superpotential term. Here we assume that CI ≈ −O(1). In such a case the local minimum of the %at direction condensate is given by Eq. (190), denoted here by *m . Note that if * is close to *m (|I*| ≡ |* − *m | . *m ), then in%ation will damp I* to be close to zero. The equation of motion for perturbations around this local minimum is given by Vtotal (*) ≈

I*U + 3HI*˙ = −kH 2 I*;

k = (2d − 4)CI & 1 ;

(228)


163

which has a solution of the form: √ 1 I* = I*o e=Ht ; = = (−3 + 9 − 4k) : (229) 2 As long as Ht1, i.e. there are a signi1cant number of e-foldings, the amplitude of the %at direction condensate will be damped to be exponentially close to the minimum of its local minimum. In general, it is likely that the initial value of * will not be close to *m . It has been shown that the deviation of the adiabatic perturbation from scale-invariance implies that the value of the potential at N ≈ 50 cannot be very much larger than *m [53]. The deviation from scale-invariance due to the %at direction condensate is then Wn* = −

3V (*) 9* 2 dM =− : M dN V (*) + V (S) 9N

(230)

For **m , the * 1eld will be rapidly oscillating in its potential and the change in the amplitude of * over an e-folding due to damping by expansion will be 9*=9N ∼ −*. Requiring that |Wn* | ¡ K imposes an upper bound on * 1=2(d−1) √ 1=(d−1) (d−2)=(d−1) Kd 2H M : (231) *. 2 6(d − 1)A For d = 4, one 1nds [53] 0:8 AMP 1=6 * . 1=4 ; *m H CI while for d = 6 0:9 * . 1=8 *m CI

AMP H

1=20

(232)

;

(233)

where we have used K =0:2. For typical values of HI , the scale-invariance of the density perturbations implies that * at N ≈ 50 e-foldings cannot be much more than an order of magnitude greater than *m . Since there is no reason for * to be close to this upper limit when N ≈ 50, it is most likely that * will be close to *m when the primordial perturbations responsible for large scale structure formation have left the horizon during in%ation. 5.4.4. Isocurvature 9uctuations in D-term in9ation The isocurvature perturbation in the baryon number arises from the AD scalar if the angular direction is eJectively massless, i.e. mass is small compared with H during and after in%ation [52–54]. The resulting perturbations will be unsuppressed until the baryon number of the Universe is generated. This in turn requires that there are no order H corrections to the supersymmetry-breaking A-terms. The baryon number from AD baryogenesis is generated at H ≈ msusy ∼ 100 GeV when the A-term can introduce B and CP violation into the coherently oscillating AD scalar. If the phase of the AD scalar relative to the real direction (de1ned by the A-term) is ), then the baryon number density given by (see Eq. (253), in Section 5.5) nB ≈ msusy *2o sin2) ;

(234)

164


where *o is the amplitude of the coherent oscillations at H ≈ msusy . One can then obtain %uctuation in the baryon number or an isocurvature perturbation as InB 2I) H = = ; (235) nB tan(2)) * tan(2)) where I) ≈ (H=2*) is generated by quantum %uctuations of the AD 1eld at the time when the perturbations cross the horizon. The magnitude of the CMB isocurvature perturbation relative to the adiabatic perturbation can be written as [53,203,208], (see Eq. (129), in Section 3.4.3)

i

I ! 2M 2 V (S)

; (236) = = a = I 3 V (S)tan(2))* where V (S) is the in%aton potential (see Section 5.4.1). For purely baryonic isocurvature perturbations ! = B =m , where B is the baryon density and m is the total matter density. For the case of D-term in%ation one obtains [53] g!M 1 ; (237) == 1=2 6 *N tan(2)) where N ≈ 50. Requiring that the deviations from the spectral index due to the AD scalar are acceptably small, for d = 4, one 1nds = ¿ =c =

3:3!(gA)4=9 ; K 1=9 tan(2))

(238)

0:18!(g3 A)4=17 : K 1=17 tan(2))

(239)

and for d = 6 = ¿ =c =

The range of B allowed by nucleosynthesis is 0:006 . B . 0:036 (for 0:6 . h . 0:87) [3]. For m = 0:4, K = 0:2, and for d = 4, we obtain =c = (0:06 − 0:36)

(gA)4=9 ; tan(2))

(240)

and for d = 6 =c = (3:0 × 10−3 − 0:018)

(g3 A)4=17 : tan(2))

(241)

(The lower limits should be multiplied by 0.4 for the case m = 1.) If, for example, g ∼ A ∼ 0:1 and tan(2)) . 1, one would obtain a lower bound = & 10−2 for d = 4 and = & 10−3 for d = 6. Such small isocurvature contamination could be detectable in future CMB experiments. Present CMB and large-scale structure observations require that = . 0:1 [203,208]. COBE normalization combined with the value of 78 (the rms of the density 1eld on a scale of 8 Mpc) as obtained from X-ray observations of the local cluster together with the shape parameter < ≈ m h=0:25±0:05 [333] from galaxy surveys, which is also consistent with the recent observations of high-redshift supernovae [20,21]) yields the limit = . 0:07. The limit may however rely too much on COBE normalization, which is just one experimental result among many.


165

Future CMB observations by MAP will be able to probe down to = ≈ 0:1, while PLANCK (with CMB polarization measurements) should be able to see isocurvature perturbations as small as 0:04 [204] (see also [205]). For the case of minimal (d = 4) AD baryogenesis, there is a good chance that PLANCK will be able to observe isocurvature perturbations at least if in%ation is driven by D-term. For higher dimension AD baryogenesis (d ¿ 6) the situation is less certain. All this assumes that * can take any value. This is true if *i ¡ *∗ , in which case * remains at its initial value *i . We have seen that the dynamics of the %at direction during D-term in%ation implies that if *i ¿ *∗ then * will equal *∗ at N ≈ 50. In this case we can 1x the magnitude of the isocurvature perturbation. For d = 4, N ≈ 50 and m = 0:4 it is given by [53] 1=4 NS (gA)1=2 : (242) = = =∗ ≈ (0:17–1:03) 50 tan(2)) (For m = 1 this should be multiplied by 0.4.) For d = 6 and m = 0:4, 1=8 3=4 1=4 NS g A = = =∗ ≈ (4:4 × 10−3 –2:6 × 10−2 ) : (243) 50 tan(2)) If g; A & 0:1 then for the d = 4 case one expects =∗ ≈ 0:01– 0.1. For the d = 6 case the isocurvature perturbation might just be at the observable level. It is important that one can 1x the isocurvature perturbation to be not much larger than the lower bound coming from adiabatic perturbations. This is because there is typically a very small range of values of * over which the isocurvature perturbation is less than the present observational limit, = . 0:1, but larger than the adiabatic perturbation lower bound, = & 0:01 for d = 4. 5.4.5. Isocurvature 9uctuations in F-term in9ation If the %at direction condensate is stuck in a local minimum * ≈ *m given by Eq. (190), the isocurvature perturbation is given by [52,53] H 2! =≈ ; (244) 3 tan(2))I *m where I = 3IT=T ≈ 3 × 10−5 is the density perturbation. Given H , d, and the value of *m , the magnitude of the isocurvature perturbation is essentially 1xed. For d=4 and m =0:4, the isocurvature perturbation has been found to be HI 1=2 A1=2 2 = = (3:1–18:6) × 10 1=4 (245) CI tan(2)) MP while for d = 6 = = (2:9–17:4) × 10

2

A1=4 CI1=8 tan(2))

HI MP

3=4

:

(246)

If we require that = . 0:1 we 1nd the upper bounds HI =MP . 10−7 =A (for d = 4) and HI =MP . 10−5 =A1=3 (for d = 6). For typical values of H the isocurvature perturbation in the F-term in%ation can be close to the present observational limits. In [54], however, it was pointed out that there would be negligible isocurvature perturbations produced from the MSSM %at directions, and the present observations would not be able place any independent constraint upon the initial amplitude of the %at directions.

166


5.4.6. MSSM 9at directions as a source for curvature perturbations In a very recent development, a new paradigm has been laid, where adiabatic density perturbations were generated from the decay of the pure isocurvature perturbations. Though it was 1rst suggested in [319], but it was implemented recently in a pre-Big-Bang scenario [320], where the axion 1eld which generates isocurvature perturbations decays late in the Universe. In order to create pure adiabatic density perturbations it is important that the 1eld, known as curvaton [321], is subdominant during in%ation, but becomes dominant during the late phase of the Universe especially when it is decaying. The curvaton 1eld 7 generates isocurvature perturbations during in%ation, assuming that the perturbations generated from the in%aton 1eld can be negligible, 2 and the curvaton mass m27 ≈ V77 Hinf . In this limiting case the power spectrum for the curvaton will be equivalent to a massless scalar 1eld during in%ation; P7 = H∗2 =42 , where ∗ denotes the epoch when perturbations are crossing the horizon k = a∗ H∗ . The curvaton 1eld follows it trajectory after the end of in%ation, and when H ∼ m7 , the curvaton oscillates and eventually decays through its coupling to the SM relativistic degrees of freedom. While oscillating it produces density contrast I7 = 2I7=7, assuming that H∗ ¡ 7∗ [321], and the perturbation spectrum is given by H∗ P71=2 = = 2 : (247) PI1=2 7 7 7∗ When the curvaton decays it converts all its isocurvature perturbations to the adiabatic ones by following that, before decaying the relativistic degrees of freedom due to the in%aton decay products gives rise to density perturbations in the radiation as r = (1=4)Ir , and 7 = (1=3)I7 . With these results the curvature perturbations is given by [321] 1 4r r + 37 7 ≈ I7 ; (248) = 4r + 47 3 supposing that before decaying r is negligible. If the curvaton decay products do not dominate the Universe, then there will adiabatic and isocurvature perturbations both [321,322]. In [323], the authors have pointed out that the MSSM %at directions which are lifted by the non-renormalizable operators such as d = 7; 9 are the best candidate for the curvaton. The %at S and d = 9; QuQ directions d = 7; LLdS dS dS (lifted by Hu LLLdS dS d), S uQ S uSeS (lifted by QuQ S uQ S uH S d eSe), S d− 1 which are lifted by superpotential: W ∼ 7 =M d−3 . Note that is the super1eld other than the curvaton, does not produce any A-term in the potential, since = 0, and therefore does not give rise to any U (1) violating terms in the %at direction potential. It was shown in [323], that these %at directions can dominate the energy density of the Universe and while decaying the squarks and sleptons can directly decay in the MSSM relativistic degrees of freedom. Therefore the virtue of this scenario is that the MSSM %at direction is solely responsible for reheating the Universe, barring any need for speculation from the in%aton coupling to the SM 1elds. In%ation was supposed to happen in the hidden sector of the theory, which does not necessarily couple to the SM sector. 5.5. Baryon number asymmetry In both D- and F-term in%ation the in%aton and other scalar 1elds begin to oscillate coherently about the minimum of their respective potential after the end of in%ation, and the post-in%ationary evolution of the %at direction condensate is no exception. If CI and C are positive in Eqs. (189) and


167

(192), the corresponding vevs are either given by Eqs. (190) or (193). In fact, in D-term in%ation models for |C| less than about 0.5, it is possible to have a positive H 2 correction and still generate the observed baryon asymmetry as shown by McDonald [324]. In addition, there has been attempts for AD baryogenesis in F-term in%ation, basing on the low energy eJective action of the heterotic string theory, where in%ation is driven by T -moduli [334]. Flat directions beyond MSSM involving a triplet Higgs has also been considered [335]. Here we will mainly concentrate on the negative H 2 correction only. After in%ation, * initially continues to track the instantaneous local minimum of the scalar potential, which can be derived by replacing HI with H (t) in Eq. (190). Once H m0 , the low-energy soft terms take over. The condensate mass squared turns positive, and since the phase of * diJers from the phase of A, * starts to change non-adiabatically. In an absence of H corrections to the A-terms, the initial phase ) of the AD 1eld relative to the real direction is random and so typically ≈ 1. As a result * starts a spiral motion in the complex plane (see Figs. 2, and forthcoming discussion on the condensate trajectory), which leads to a generation of a net baryon and/or lepton asymmetry [45,46]. 8 The baryon number density is related to the AD 1eld as ˙ ; nB; L = >i(*˙ † * − *† *)

(249)

where > is corresponding baryon and/or lepton charge of the AD 1eld. The equations of motion for the AD 1eld are given by 9V (*) *U + 3H *˙ + =0 : 9*∗ The above two equations lead to 9V (*) n˙B; L + 3HnB; L = 2> Im * ; 9*∗

(250)

=2>

m* Im(a*d ) dM d−3

By integrating Eq. (251), we obtain the baryon and/or lepton number as t m* a3 (t)nB; L (t) = 2>|a| d−3 a3 (t )|*(t )|d sin()) dt ; M

(251)

(252)

Note that a introduces an extra CP phase which we may parameterize as sin(I). After a few expansion times, the amplitude of the oscillations will become damped by the expansion of the Universe and the A-term, which is proportional to a large power of *, will become gradually negligible. The net baryon and/or lepton asymmetry is then given by [46] nB; L (tosc ) = > ≈>

2(d − 2) m* *20 sin 2) sin I ; 3(d − 3) 2(d − 2) m* (m* M d−3 )2=(d−2) sin 2) sin I ; 3(d − 3)

where sin I ∼ sin 2) ≈ O(1). 8

There have been attempts for AD baryogenesis in local domains, see [336].

(253)

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When the in%aton decay products have completely thermalized with a reheat temperature Trh , the baryon and/or lepton asymmetry is given by [337] Trh nB; L 1 = nB; L (tosc ) ; 2 s 4 MP H (tosc )2 =

Trh d−2 > 2 (m* M d−3 )2=(d−2) sin 2) sin I ; 6(d − 3) MP m*

(254)

where we have used H (tosc ) ≈ m* , and s is the entropy density of the Universe at the time of reheating. For d = 4, the baryon-to-entropy ratio turns out to be [337] m3=2 nB; L M Trh −10 ; (255) ×> ≈ 1 × 10 s m* MP 109 GeV and for d = 6 nB; L ≈ 5 × 10−10 × > s

m3=2 m*

1 TeV m*

1=2

M MP

3=2

Trh 100 GeV

;

(256)

where we have taken the net CP phase to be ∼ O(1). The asymmetry remains frozen unless there is additional entropy production afterwards. Note that for d = 4, the required reheat temperature of the Universe is below the gravitino overproduction bound (see Section 3.6.2). For higher dimensional non-renormalizable operators, a low reheat temperature is favorable, which is indeed a good news. In this regard low scale in%ation, which guarantees a low reheat temperature, has been given some consideration [338] (see also [339] where AD baryogenesis after a brief period of thermal in%ation, required to solving the cosmological moduli problem, has been discussed). Although, in [340], it was pointed out that in gauge mediated supersymmetry breaking it is hard to reconcile AD baryogenesis with a moduli problem. Among the host of MSSM %at directions which are lifted by non-renormalizable operator and listed in Table 1, the LHu %at direction carrying the lepton number is the candidate for producing lepton asymmetry in the Universe (there has been some earlier attempts of direct baryogenesis via uSdS dS directions, see e.g. [341]). The lepton asymmetry calculated above in Eqs. (255) and (256) can be transformed into baryon number asymmetry via sphalerons nB =s = (8=23)nL =s. AD leptogenesis has important implications in neutrino physics also, because in the MSSM, the LHu direction is lifted by the d = 4 non-renormalizable operator which also gives rise to neutrino masses [46]: W=

1 m#i (Li Hu )2 = (Li Hu )2 ; 2Mi 2Hu 2

(257)

where we have assumed the see-saw relation m#i = Hu 2 =Mi with diagonal entries for the neutrinos #i , i = 1; 2; 3. The 1nal nB =s can be related to the lightest neutrino mass since the %at direction moves furthest along the eigenvector of Li Lj which corresponds to the smallest eigenvalue of the neutrino mass matrix [46]. −6 m3=2 10 eV nL Trh −10 ≈ 1 × 10 ; (258) ×> s m* 108 GeV m#l where m#l denotes the lightest neutrino.


169

5.6. Thermal eBects In our discussion on the baryon/lepton asymmetry we have tacitly assumed that the asymmetry has been generated before the Universe has thermalized and reheated. This might not be the case if there were light degrees of freedom which have thermalized with an instantaneous plasma temperature Tinst 6 (H2 R2 ; = dt

(363)

where xst = >R is de1ned by the Gaussian thick wall pro1le ’(r) = ’(0)exp(−r 2 =R2 ), and therefore > = log(g’(0)=3T ). In order to evade complete dissociation WEWm* Q, where Wm* ≈ |K|m* . The dissociation will not be completed provided the temperature of the thermal bath is given by [49] 1=4 |K|2 T6 m* Q1=4 ; (364) 4g∗ (T )kr T >2 where the dynamical time scale is assumed to be ∼ kr =m* , with kr ¿ 1. In a realistic case dissociation alone cannot erode the Q-ball completely. The Q-ball will rather come into thermal equilibrium with the ambient plasma. In an expanding Universe the minimum energy is con1gured in such a way that the Q-charges are always present in the Universe along


197

with the Q-balls. In this case the energy of the Q-balls decreases as the temperature of the Universe decreases. 6.6.2. DiBusion DiJusion takes place only through the soft edge of the Q-ball at a distance over which ’ does not change much. There are two factors which determine the diJusion rate; 1rstly, how eYciently the edge of the Q-ball diJuses, and secondly how fast the core of the Q-ball readjusts itself in order to compensate for the loss of charge. At large temperatures the diJusion rate is large as the Q-ball tries to relax into chemical equilibrium with a thermal plasma. The net diJusion rate is given by [398,349,351] Q Q2=3 GeV h2 10 where M ≡ (g∗ (TD ) − g∗ (T ))1=3 , the temperature TD signi1es the decoupling temperature of a Q-ball with a rest of the plasma, and >Q = (3m3* =4!’20 )1=3 . One should require that TF ¡ Ts , which is satis1ed for a very low -mass. In order to have some feeling for the numbers involved, we note that for the smallest charged Q-ball with Q = 2, the freeze-out temperature is smaller than Ts only when m 6 O(1) GeV. During the matter dominated epoch, due to the gravitational clustering, Q-ball synthesis is favorable but for ambient temperature T 6 O(1) eV, m 6 O(1) MeV.

220


8.3.2. Phase transition via solitogenesis An interesting observation has been made in [442,443], where it was pointed out that solitosynthesis may also lead to a 1rst order phase transition. The gradual absorption of charge may lead to a critical charge Q = Qc for which the false vacuum inside the Q-ball becomes unstable and expands as the whole space is 1lled with the true vacuum. The value of the critical charge is determined by minimizing the energy of the Q-ball with respect to the radius when dE=dR = 0 and d 2 E=dR2 = 0 are satis1ed simultaneously. In a thin wall limit when Qc 1, the critical charge is given by [443] √ 100 10 ’0 S13 Qc = ; (434) 81 U (’0 )5=2 ’ where S1 = 0 0 d’ 2(U (’) − (!2 =2)’2 ). One can also enquire how large the critical charge should be in order to facilitate the destabilization of the otherwise cosmologically stable false vacuum. It has been found that the decay of a metastable false vacuum at zero temperature requires a small charge QC ∼ 28(’0 =U (’0 )1=4 ) for A ∼ (100 GeV)4 [443]. Similar considerations at 1nite temperature result in a larger critical charge Q ¿ Qc ∼ 146’0 T=U (’0 ; T )1=2 for the phase transition to proceed. Solitosynthesis-catalyzed phase transitions in supersymmetric models would require a local violation of lepton or baryon number. As an example (see [443]), one could consider a lepton number violating local minimum along Hd L˜ L L˜ R = 0. This false vacuum can decay into the standard true vacuum but the sphaleron induced transition rate is almost negligible compared to the cosmological time scale because SU (2) × U (1) is broken. If the typical mass scale of squarks and sleptons is considerably heavier than 1 TeV, L-balls will accumulate when Ts ¿ TF ≈ m* =40, and can catalyze a phase transition within one Hubble time at temperatures Ts [442,443]. Acknowledgements The authors are thankful to Rouzbeh Allahverdi, Mar Bastero-Gil, Zurab Berezhiani, Ed Copeland, Masaaki Fujii, Katrin Heitmann, Asko Jokinen, Shinta Kasuya, Alex Kusenko, Mikko Laine, U Andrew Liddle, John McDonald, Tuomas MultamUaki, Altug Ozpineci, Abdel Pèrez-Lorenzana and Iiro Vilja for useful discussions. A.M. was partially supported by The Early Universe Network: HPRN-CT-2000-00152. A.M. is also thankful to the Helsinki Institute of Physics where part of the project has been carried out. K.E. acknowledges the Academy of Finland grant no. 51433. References [1] S. Weinberg, The Quantum Theory of Fields, vol. 2: Modern applications, Cambridge University Press, Cambridge, 1995. [2] S. Sarkar, Rep. Prog. Phys. 59 (1996) 1493 [arXiv:hep-ph/9602260]. [3] K.A. Olive, G. Steigman, T.P. Walker, Phys. Rep. 333 (2000) 389 [arXiv:astro-ph/9905320]. [4] http://map.gsfc.nasa.gov/ [5] http://aether.lbl.gov/www/projects/cosa/ [6] P.J.E. Peebles, Principles of Physical Cosmology, Princeton University Press, Princeton, 1993. [7] Cosmic Background Explorer homepage: http://space.gsfc.nasa.gov/astro/cobe/


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Cosmological constant—the weight of the vacuum T. Padmanabhan IUCAA, Pune University Campus, Ganeshkhind, Pune 411 007, India Accepted 1 March 2003 editor: M.P. Kamionkowski

Abstract Recent cosmological observations suggest the existence of a positive cosmological constant with the magnitude (G˝=c3 ) ≈ 10−123 . This review discusses several aspects of the cosmological constant both from the cosmological (Sections 1– 6) and .eld theoretical (Sections 7–11) perspectives. After a brief introduction to the key issues related to cosmological constant and a historical overview, a summary of the kinematics and dynamics of the standard Friedmann model of the universe is provided. The observational evidence for cosmological constant, especially from the supernova results, and the constraints from the age of the universe, structure formation, Cosmic Microwave Background Radiation (CMBR) anisotropies and a few others are described in detail, followed by a discussion of the theoretical models (quintessence, tachyonic scalar .eld, : : :) from di4erent perspectives. The latter part of the review (Sections 7–11) concentrates on more conceptual and fundamental aspects of the cosmological constant like some alternative interpretations of the cosmological constant, relaxation mechanisms to reduce the cosmological constant to the currently observed value, the geometrical structure of the de Sitter spacetime, thermodynamics of the de Sitter universe and the role of string theory in the cosmological constant problem. c 2003 Elsevier Science B.V. All rights reserved. PACS: 98.80.−k; 98.80.Es; 98.80.Cq; 98.80.Qc; 04.60.−m Keywords: Cosmological constant; Dark energy; Cosmology; CMBR; Quintessence; de Sitter spacetime; Horizon; Tachyon; String theory

Contents 1. Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 236 1.1. The many faces of the cosmological constant . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 237 1.2. A brief history of cosmological constant . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 240 E-mail address: [email protected] (T. Padmanabhan). c 2003 Elsevier Science B.V. All rights reserved. 0370-1573/03/$ - see front matter doi:10.1016/S0370-1573(03)00120-0

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T. Padmanabhan / Physics Reports 380 (2003) 235 – 320

2. Framework of standard cosmology . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.1. Kinematics of the Friedmann model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.2. Dynamics of the Friedmann model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.3. Composition of the universe . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.4. Geometrical features of a universe with a cosmological constant . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3. Evidence for a nonzero cosmological constant . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.1. Observational evidence for accelerating universe . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.2. Age of the universe and cosmological constant . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.3. Gravitational lensing and the cosmological constant . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.4. Other geometrical tests . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4. Models with evolving cosmological “constant” . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.1. Parametrized equation of state and cosmological observations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.2. Theoretical models with time dependent dark energy: cosmic degeneracy . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5. Structure formation in the universe . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.1. Linear evolution of perturbations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.2. Nonlinear growth of perturbations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.3. Structure formation and constraints on dark energy . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6. CMBR anisotropies . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7. Reinterpreting the cosmological constant . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7.1. Cosmological constant as a Lagrange multiplier . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7.2. Cosmological constant as a constant of integration . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7.3. Cosmological constant as a stochastic variable . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7.4. Anthropic interpretation of the cosmological constant . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7.5. Probabilistic interpretation of the cosmological constant . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8. Relaxation mechanisms for the cosmological constant . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9. Geometrical structure of the de Sitter spacetime . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10. Horizons, temperature and entropy . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10.1. The connection between thermodynamics and spacetime geometry . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10.2. Temperature of horizons . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10.3. Entropy and energy of de Sitter spacetime . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10.4. Conceptual issues in de Sitter thermodynamics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11. Cosmological constant and the string theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Acknowledgements . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

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1. Introduction This review discusses several aspects of the cosmological constant both from the cosmological and .eld theoretical perspectives with the emphasis on conceptual and fundamental issues rather than on observational details. The plan of the review is as follows: This section introduces the key issues related to cosmological constant and provides a brief historical overview. (For previous reviews of this subject, from cosmological point of view, see [1–3,139].) Section 2 summarizes the kinematics and dynamics of the standard Friedmann model of the universe paying special attention to features involving the cosmological constant. Section 3 reviews the observational evidence for cosmological constant, especially the supernova results, constraints from the age of the universe and a few others. We next study models with evolving cosmological ‘constant’ from di4erent perspectives.


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(In this review, we shall use the term cosmological constant in a generalized sense including the scenarios in which cosmological “constant” is actually varying in time.) A phenomenological parameterization is introduced in Section 4.1 to compare theory with observation and is followed up with explicit models involving scalar .elds in Section 4.2. The emphasis is on quintessence and tachyonic scalar .eld models and the cosmic degeneracies introduced by them. Section 5 discusses cosmological constant and dark energy in the context of models for structure formation and Section 6 describes the constraints arising from CMBR anisotropies. The latter part of the review concentrates on more conceptual and fundamental aspects of the cosmological constant. (For previous reviews of this subject, from a theoretical physics perspective, see [4–6].) Section 7 provides some alternative interpretations of the cosmological constant which might have a bearing on the possible solution to the problem. Several relaxation mechanisms have been suggested in the literature to reduce the cosmological constant to the currently observed value and some of these attempts are described in Section 8. Section 9 gives a brief description of the geometrical structure of the de Sitter spacetime and the thermodynamics of the de Sitter universe is taken up in Section 10. The relation between horizons, temperature and entropy are presented at one go in this section and the last section deals with the role of string theory in the cosmological constant problem. 1.1. The many faces of the cosmological constant Einstein’s equations, which determine the dynamics of the spacetime, can be derived from the action (see, e.g. [7]): √ √ 1 4 A= R −g d x + Lmatter ( ; 9 ) −g d 4 x ; (1) 16G where Lmatter is the Lagrangian for matter depending on some dynamical variables generically denoted as . (We are using units with c = 1.) The variation of this action with respect to will lead to the equation of motion for matter (Lmatter = ) = 0, in a given background geometry, while the variation of the action with respect to the metric tensor gik leads to the Einstein’s equation Rik −

1 Lmatter gik R = 16G ≡ 8GTik ; 2 gik

(2)

where the last equation de.nes the energy momentum tensor of matter to be Tik ≡ 2(Lmatter =gik ). Let us now consider a new matter action Lmatter = Lmatter − (=8G) where is a real constant. Equation of motion for the matter (Lmatter = ) = 0, does not change under this transformation since is a constant; but the action now picks up an extra term proportional to √ √ 1 4 R −g d x + Lmatter − −g d 4 x A= 16G 8G √ √ 1 4 (R − 2) −g d x + Lmatter −g d 4 x (3) = 16G and Eq. (2) gets modi.ed. This innocuous looking addition of a constant to the matter Lagrangian leads to one of the most fundamental and fascinating problems of theoretical physics. The nature of

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this problem and its theoretical backdrop acquires di4erent shades of meaning depending which of the two forms of equations in (3) is used. The .rst interpretation, based on the .rst line of Eq. (3), treats as the shift in the matter Lagrangian which, in turn, will lead to a shift in the matter Hamiltonian. This could be thought of as a shift in the zero point energy of the matter system. Such a constant shift in the energy does not a4ect the dynamics of matter while gravity—which couples to the total energy of the system—picks up an extra contribution in the form of a new term Qik in the energy-momentum tensor, leading to: 1 i ≡ ik : (4) Rik − ik R = 8G(Tki + Qki ); Qki ≡ 2 8G k The second line in Eq. (3) can be interpreted as gravitational .eld, described by the Lagrangian of the form Lgrav ˙ (1=G)(R − 2), interacting with matter described by the Lagrangian Lmatter . In this interpretation, gravity is described by two constants, the Newton’s constant G and the cosmological constant . It is then natural to modify the left hand side of Einstein’s equations and write (4) as: Rik − 12 ik R − ik = 8GTki :

(5)

In this interpretation, the spacetime is treated as curved even in the absence of matter (Tik = 0) since the equation Rik − (1=2)gik R − gik = 0 does not admit Pat spacetime as a solution. (This situation is rather unusual and is related to the fact that symmetries of the theory with and without a cosmological constant are drastically di4erent; the original symmetry of general covariance cannot be naturally broken in such a way as to preserve the sub group of spacetime translations.) In fact, it is possible to consider a situation in which both e4ects can occur. If the gravitational interaction is actually described by the Lagrangian of the form (R − 2), then there is an intrinsic cosmological constant in nature just as there is a Newtonian gravitational constant in nature. If the matter Lagrangian contains energy densities which change due to dynamics, then Lmatter can pick up constant shifts during dynamical evolution. For example, consider a scalar .eld with the Lagrangian Lmatter = (1=2)9i 9i − V ( ) which has the energy momentum tensor Tba = 9a 9b − ab ( 12 9i 9i − V ( )) :

(6)

For .eld con.gurations which are constant [occurring, for example, at the minima of the potential V ( )], this contributes an energy momentum tensor Tba =ab V ( min ) which has exactly the same form as a cosmological constant. As far as gravity is concerned, it is the combination of these two e4ects— e4 of very di2erent nature—which is relevant and the source will be Tab = [V ( min ) + (=8G)]gab , corresponding to an e4ective gravitational constant e4 = + 8GV ( min ) :

(7)

If min and hence V ( min ) changes during dynamical evolution, the value of e4 can also change in course of time. More generally, any .eld con.guration which is varying slowly in time will lead to a slowly varying e4 . The extra term Qik in Einstein’s equation behaves in a manner which is very peculiar compared to the energy momentum tensor of normal matter. The term Qki = ik is in the form of the energy momentum tensor of an ideal Puid with energy density and pressure P = − ; obviously, either the pressure or the energy density of this “Puid” must be negative, which is unlike conventional laboratory systems. (See, however, Ref. [8].)


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Such an equation of state, = −P also has another important implication in general relativity. The spatial part g of the geodesic acceleration (which measures the relative acceleration of two geodesics in the spacetime) satis.es the following exact equation in general relativity (see e.g., p. 332 of [9]): ∇ · g = −4G( + 3P) ;

(8)

showing that the source of geodesic acceleration is ( + 3P) and not . As long as ( + 3P) ¿ 0, gravity remains attractive while ( + 3P) ¡ 0 can lead to repulsive gravitational e4ects. Since the cosmological constant has ( + 3P ) = −2 , a positive cosmological constant (with ¿ 0) can lead to repulsive gravity. For example, if the energy density of normal, nonrelativistic matter with zero pressure is NR , then Eq. (8) shows that the geodesics will accelerate away from each other due to the repulsion of cosmological constant when NR ¡ 2 . A related feature, which makes the above conclusion practically relevant is the fact that, in an expanding universe, remains constant while NR decreases. (More formally, the equation of motion, d( V )=−P dV for the cosmological constant, treated as an ideal Puid, is identically satis.ed with constant ; P .) Therefore, will eventually dominate over NR if the universe expands suQciently. Since ||1=2 has the dimensions of inverse length, it will set the scale for the universe when cosmological constant dominates. It follows that the most stringent bounds on will arise from cosmology when the expansion of the universe has diluted the matter energy density suQciently. The rate of expansion of the universe today is usually expressed in terms of the Hubble constant: H0 = 100h km s−1 Mpc−1 where 1 Mpc ≈ 3 × 1024 cm and h is a dimensionless parameter in the range 0:62 . h . 0:82 (see Section 3.2). From H0 we can form the time scale tuniv ≡ H0−1 ≈ 1010 h−1 yr and the length scale cH0−1 ≈ 3000h−1 Mpc; tuniv characterizes the evolutionary time scale of the universe and H0−1 is of the order of the largest length scales currently accessible in cosmological observations. From the observation that the universe is at least of the size H0−1 , we can set a bound on to be || ¡ 10−56 cm−2 . This stringent bound leads to several issues which have been debated for decades without satisfactory solutions. • In classical general relativity, based on the constants G; c and , it is not possible to construct any dimensionless combination from these constants. Nevertheless, it is clear that is extremely tiny compared to any other physical scale in the universe, suggesting that is probably zero. We, however, do not know of any symmetry mechanism or invariance principle which requires to vanish. Supersymmetry does require the vanishing of the ground state energy; however, supersymmetry is so badly broken in nature that this is not of any practical use [10,11]. • We mentioned above that observations actually constrain e4 in Eq. (7), rather than . This requires and V ( min ) to be .ne tuned to an enormous accuracy for the bound, |e4 | ¡ 10−56 cm−2 , to be satis.ed. This becomes more mysterious when we realize that V ( min ) itself could change by several orders of magnitude during the evolution of the universe. • When quantum .elds in a given curved spacetime are considered (even without invoking any quantum gravitational e4ects) one introduces the Planck constant, ˝, in the description of the physical system. It is then possible to form the dimensionless combination (G˝=c3 ) ≡ L2P . (This equation also de.nes the quantity L2P ; throughout the review we use the symbol ‘≡’ to de.ne variables.) The bound on translates into the condition L2P . 10−123 . As has been mentioned several times in literature, this will require enormous .ne tuning.

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• All the above questions could have been satisfactorily answered if we take e4 to be zero and assume that the correct theory of quantum gravity will provide an explanation for the vanishing of cosmological constant. Such a view was held by several people (including the author) until very recently. Current cosmological observations however suggests that e4 is actually nonzero and e4 L2P is indeed of order O(10−123 ). In some sense, this is the cosmologist’s worst nightmare come true. If the observations are correct, then e4 is nonzero, very tiny and its value is extremely 5ne tuned for no good reason. This is a concrete statement of the .rst of the two ‘cosmological constant problems’. • The bound on L2P arises from the expansion rate of the universe or—equivalently—from the energy density which is present in the universe today. The observations require the energy density of normal, nonrelativistic matter to be of the same order of magnitude as the energy density contributed by the cosmological constant. But in the past, when the universe was smaller, the energy density of normal matter would have been higher while the energy density of cosmological constant does not change. Hence we need to adjust the energy densities of normal matter and cosmological constant in the early epoch very carefully so that & NR around the current epoch. If this had happened very early in the evolution of the universe, then the repulsive nature of a positive cosmological constant would have initiated a rapid expansion of the universe, preventing the formation of galaxies, stars, etc. If the epoch of ≈ NR occurs much later in the future, then the current observations would not have revealed the presence of nonzero cosmological constant. This raises the second of the two cosmological constant problems: Why is it that ( =NR ) = O(1) at the current phase of the universe? • The sign of determines the nature of solutions to Einstein’s equations as well as the sign of ( + 3P ). Hence the spacetime geometry with L2P = 10−123 is very di4erent from the one with L2P = −10−123 . Any theoretical principle which explains the near zero value of L2P must also explain why the observed value of is positive. At present we have no clue as to what the above questions mean and how they need to be addressed. This review summarizes di4erent attempts to understand the above questions from various perspectives. 1.2. A brief history of cosmological constant Originally, Einstein introduced the cosmological constant in the .eld equation for gravity (as in Eq. (5)) with the motivation that it allows for a .nite, closed, static universe in which the energy density of matter determines the geometry. The spatial sections of such a universe are closed 3-spheres with radius l = (8GNR )−1=2 = −1=2 where NR is the energy density of pressureless matter (see Section 2.4) Einstein had hoped that normal matter is needed to curve the geometry; a demand, which—to him—was closely related to the Mach’s principle. This hope, however, was soon shattered when de Sitter produced a solution to Einstein’s equations with cosmological constant containing no matter [12]. However, in spite of two fundamental papers by Friedmann and one by Lemaitre [13,14], most workers did not catch on with the idea of an expanding universe. In fact, Einstein originally thought Friedmann’s work was in error but later published a retraction of his comment; similarly, in the Solvay meeting in 1927, Einstein was arguing against the solutions describing expanding universe. Nevertheless, the Einstein archives do contain a postcard from Einstein


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to Weyl in 1923 in which he says: “If there is no quasi-static world, then away with the cosmological term”. The early history following de Sitter’s discovery (see, for example, [15]) is clearly somewhat confused, to say the least. It appears that the community accepted the concept of an expanding universe largely due to the work of Lemaitre. By 1931, Einstein himself had rejected the cosmological term as superPous and unjusti.ed (see Ref. [16], which is a single authored paper; this paper has been mis-cited in literature often, eventually converting part of the journal name “preuss” to a co-author “Preuss, S. B”!; see [17]). There is no direct record that Einstein ever called cosmological constant his biggest blunder. It is possible that this often repeated “quote” arises from Gamow’s recollection [18]: “When I was discussing cosmological problems with Einstein, he remarked that the introduction of the cosmological term was the biggest blunder he ever made in his life.” By 1950s the view was decidedly against and the authors of several classic texts (like Landau and Liftshitz [7], Pauli [19] and Einstein [20]) argued against the cosmological constant. In later years, cosmological constant had a chequered history and was often accepted or rejected for wrong or insuQcient reasons. For example, the original value of the Hubble constant was nearly an order of magnitude higher [21] than the currently accepted value thereby reducing the age of the universe by a similar factor. At this stage, as well as on several later occasions (e.g., [22,23]), cosmologists have invoked cosmological constant to reconcile the age of the universe with observations (see Section 3.2). Similar attempts have been made in the past when it was felt that counts of quasars peak at a given phase in the expansion of the universe [24–26]. These reasons, for the introduction of something as fundamental as cosmological constant, seem inadequate at present. However, these attempts clearly showed that sensible cosmology can only be obtained if the energy density contributed by cosmological constant is comparable to the energy density of matter at the present epoch. This remarkable property was probably noticed .rst by Bondi [27] and has been discussed by McCrea [28]. It is mentioned in [1] that such coincidences were discussed in Dicke’s gravity research group in the sixties; it is almost certain that this must have been noticed by several other workers in the subject. The .rst cosmological model to make central use of the cosmological constant was the steady state model [29–31]. It made use of the fact that a universe with a cosmological constant has a time translational invariance in a particular coordinate system. The model also used a scalar .eld with negative energy .eld to continuously create matter while maintaining energy conservation. While modern approaches to cosmology invokes negative energies or pressure without hesitation, steady state cosmology was discarded by most workers after the discovery of CMBR. The discussion so far has been purely classical. The introduction of quantum theory adds a new dimension to this problem. Much of the early work [32,33] as well as the de.nitive work by Pauli [34,35] involved evaluating the sum of the zero point energies of a quantum .eld (with some cut-o4) in order to estimate the vacuum contribution to the cosmological constant. Such an argument, however, is hopelessly naive (inspite of the fact that it is often repeated even today). In fact, Pauli himself was aware of the fact that one must exclude the zero point contribution from such a calculation. The .rst paper to stress this clearly and carry out a second order calculation was probably the one by Zeldovich [36] though the connection between vacuum energy density and cosmological constant had been noted earlier by Gliner [37] and even by Lemaitre [38]. Zeldovich assumed that the lowest order zero point energy should be subtracted out in quantum .eld theory and went on to compute the gravitational force between particles in the vacuum Puctuations. If E is an energy scale of a virtual process

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corresponding to a length scale l = ˝c=E, then l−3 = (E=˝c)3 particles per unit volume of energy E will lead to the gravitational self energy density of the order of G(E=c2 )2 −3 GE 6 l = 8 4 : ≈ (9) l c ˝ This will correspond to L2P ≈ (E=EP )6 where EP = (˝c5 =G)1=2 ≈ 1019 GeV is the Planck energy. Zeldovich took E ≈ 1 GeV (without any clear reason) and obtained a which contradicted the observational bound “only” by nine orders of magnitude. The .rst serious symmetry principle which had implications for cosmological constant was supersymmetry and it was realized early on [10,11] that the contributions to vacuum energy from fermions and bosons will cancel in a supersymmetric theory. This, however, is not of much help since supersymmetry is badly broken in nature at suQciently high energies (at ESS ¿ 102 Gev). In general, one would expect the vacuum energy density to be comparable to the that corresponding to the supersymmetry braking scale, ESS . This will, again, lead to an unacceptably large value for . In fact the situation is more complex and one has to take into account the coupling of matter sector and gravitation—which invariably leads to a supergravity theory. The description of cosmological constant in such models is more complex, though none of the attempts have provided a clear direction of attack (see e.g., [4] for a review of early attempts). The situation becomes more complicated when the quantum .eld theory admits more than one ground state or even more than one local minima for the potentials. For example, the spontaneous symmetry breaking in the electro-weak theory arises from a potential of the form V = V0 − 2 2 + g 4

(2 ; g ¿ 0) :

(10) 4

At the minimum, this leads to an energy density Vmin = V0 − ( =4g). If we take V0 = 0 then (Vmin =g) ≈ −(300 GeV)4 . For an estimate, we will assume that the gauge coupling constant g is comparable to the electromagnetic coupling constant: g = O( 2 ), where ≡ (e2 =˝c) is the .ne structure constant. Then, we get |Vmin | ∼ 106 GeV4 which misses the bound on by a factor of 1053 . It is really of no help to set Vmin = 0 by hand. At early epochs of the universe, the temperature dependent e4ective potential [39,40] will change minimum to = 0 with V ( ) = V0 . In other words, the ground state energy changes by several orders of magnitude during the electro-weak and other phase transitions. Another facet is added to the discussion by the currently popular models of quantum gravity based on string theory [41,42]. The currently accepted paradigm of string theory encompasses several ground states of the same underlying theory (in a manner which is as yet unknown). This could lead to the possibility that the .nal theory of quantum gravity might allow di4erent ground states for nature and we may need an extra prescription to choose the actual state in which we live in. The di4erent ground states can also have di4erent values for cosmological constant and we need to invoke a separate (again, as yet unknown) principle to choose the ground state in which L2P ≈ 10−123 (see Section 11). 2. Framework of standard cosmology All the well developed models of standard cosmology start with two basic assumptions: (i) The distribution of matter in the universe is homogeneous and isotropic at suQciently large scales.


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(ii) The large scale structure of the universe is essentially determined by gravitational interactions and hence can be described by Einstein’s theory of gravity. The geometry of the universe can then be determined via Einstein’s equations with the stress tensor of matter Tki (t; x) acting as the source. (For a review of cosmology, see e.g. [43–47]). The .rst assumption determines the kinematics of the universe while the second one determines the dynamics. We shall discuss the consequences of these two assumptions in the next two subsections. 2.1. Kinematics of the Friedmann model The assumption of isotropy and homogeneity implies that the large scale geometry can be described by a metric of the form dr 2 2 2 2 2 2 2 2 2 2 2 ds = dt − a (t) dx = dt − a (t) + r (d$ + sin $ d ) (11) 1 − kr 2 in a suitable set of coordinates called comoving coordinates. Here a(t) is an arbitrary function of time (called expansion factor) and k =0; ±1. De.ning a new coordinate % through % =(r; sin−1 r; sinh−1 r) for k = (0; +1; −1) this line element becomes ds2 ≡ dt 2 − a2 dx2 ≡ dt 2 − a2 (t)[d%2 + Sk2 (%)(d$ 2 + sin2 $ d 2 )] ;

(12)

where Sk (%) = (%; sin %; sinh %) for k = (0; +1; −1). In any range of time during which a(t) is a monotonic function of t, one can use a itself as a time coordinate. It is also convenient to de.ne a quantity z, called the redshift, through the relation a(t) = a0 [1 + z(t)]−1 where a0 is the current value of the expansion factor. The line element in terms of [a; %; $; ] or [z; %; $; ] is 2 da 1 2 −2 ds = H (a) − a2 dx2 = [H −2 (z) d z 2 − d x2 ] ; (13) a (1 + z)2 where H (a) = (a=a), ˙ called the Hubble parameter, measures the rate of expansion of the universe. This equation allows us to draw an important conclusion: The only nontrivial metric function in a Friedmann universe is the function H (a) (and the numerical value of k which is encoded in the spatial part of the line element.) Hence, any kind of observation based on geometry of the spacetime, however complex it may be, will not allow us to determine anything other than this single function H (a). As we shall see, this alone is inadequate to describe the material content of the universe and any attempt to do so will require additional inputs. Since the geometrical observations often rely on photons received from distant sources, let us consider a photon traveling a distance rem (z) from the time of emission (corresponding to the redshift z) till today. Using the fact that ds = 0 for a light ray and the second equality in Eq. (13) we .nd that the distance traveled by light rays is related to the redshift by d x = H −1 (z) d z. Integrating this relation, we get 1 z −1 rem (z) = Sk ( ); ≡ H (z) d z : (14) a0 0 All other geometrical distances can be expressed in terms of rem (z) (see e.g., [44]). For example, the Pux of radiation F received from a source of luminosity L can be expressed in the form F =L=(4d2L )

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where dL (z) = a0 rem (z)(1 + z) = a0 (1 + z)Sk ( )

(15)

is called the luminosity distance. Similarly, if D is the physical size of an object which subtends an angle to the observer, then—for small —we can de.ne an angular diameter distance dA through the relation = D=dA . The angular diameter distance is given by dA (z) = a0 rem (z)(1 + z)−1

(16)

with dL = (1 + z)2 dA . If we can identify some objects (or physical phenomena) at a redshift of z having a characteristic transverse size D, then measuring the angle subtended by this object we can determine dA (z). Similarly, if we have a series of luminous sources at di4erent redshifts having known luminosity L, then by observing the Pux from these sources L, one can determine the luminosity distance dL (z). Determining any of these functions will allow us to use relations (15) [or (16)] and (14) to obtain H −1 (z). For example, H −1 (z) is related to dL (z) through −1=2 d dL (z) d dL (z) kd2L (z) −1 → ; (17) H (z) = 1 − 2 dz 1 + z dz 1 + z a0 (1 + z)2 where second equality holds if the spatial sections of the universe are Pat, corresponding to k = 0; then dL (z), dA (z); rem (z) and H −1 (z) all contain the (same) maximal amount of information about the geometry. The function rem (z) also determines the proper volume of the universe between the redshifts z and z + d z subtending a solid angle d+ in the sky. The comoving volume element can be expressed in the form d3L dV (1 + z)dL 2 dr ˙ −1 ; (18) ˙ rem d z d+ dz (1 + z)4 dL where the prime denotes derivative with respect to z. Based on this, there has been a suggestion [48] that future observations of the number of dark matter halos as a function of redshift and circular velocities can be used to determine the comoving volume element to within a few percent accuracy. If this becomes possible, then it will provide an additional handle on constraining the cosmological parameters. The above discussion illustrates how cosmological observations can be used to determine the metric of the spacetime, encoded by the single function H −1 (z). This issue is trivial in principle, though enormously complicated in practice because of observational uncertainties in the determination of dL (z); dA (z), etc. We shall occasion to discuss these features in detail later on. 2.2. Dynamics of the Friedmann model Let us now turn to the second assumption which determines the dynamics of the universe. When several noninteracting sources are present in the universe, the total energy momentum tensor which appear on the right hand side of the Einstein’s equation will be the sum of the energy momentum tensor for each of these sources. Spatial homogeneity and isotropy imply that each Tba is diagonal and has the form Tba = dia[i (t); −Pi (t); −Pi (t); −Pi (t)] where the index i = 1; 2; : : : ; N denotes N


245

di4erent kinds of sources (like radiation, matter, cosmological constant etc.). Since the sources do a not interact with each other, each energy momentum tensor must satisfy the relation Tb;a = 0 which 3 3 translates to the condition d(i a ) = −Pi da . It follows that the evolution of the energy densities of each component is essentially dependent on the parameter wi ≡ (Pi =i ) which, in general, could be a function of time. Integrating d(i a3 ) = −wi i da3 , we get a a 3 d aW 0 wi (a) i = i (a0 ) exp −3 W (19) a a0 aW which determines the evolution of the energy density of each of the species in terms of the functions wi (a). This description determines (a) for di4erent sources but not a(t). To determine the latter we can use one of the Einstein’s equations: a˙2 8G k H 2 (a) = 2 = i (a) − 2 : (20) a 3 a i This equation shows that, once the evolution of the individual components of energy density i (a) is known, the function H (a) and thus the line element in Eq. (13) is known. (Evaluating this equation at the present epoch one can determine the value of k; hence it is not necessary to provide this information separately.) Given H0 , the current value of the Hubble parameter, one can construct a critical density, by the de.nition: c =

3H02 = 1:88h2 × 10−29 gm cm−3 = 2:8 × 1011 h2 M Mpc−3 8G

= 1:1 × 104 h2 eV cm−3 = 1:1 × 10−5 h2 protons cm−3

(21)

and parameterize the energy density, i (a0 ), of di4erent components at the present epoch in terms of the critical density by i (a0 ) ≡ +i c . [Observations [49,50] give h = 0:72 ± 0:03 (statistical) ±0:07 (systematic).] It is obvious from Eq. (20) that k = 0 corresponds to +tot = i +i = 1 while +tot ¿ 1 and +tot ¡ 1 correspond to k = ±1. When +tot = 1, Eq. (20), evaluated at the current epoch, gives (k=a20 ) = H02 (+tot − 1), thereby .xing the value of (k=a20 ); when, +tot = 1, it is conventional to take a0 = 1 since its value can be rescaled. 2.3. Composition of the universe It is important to stress that absolutely no progress in cosmology can be made until a relationship between and P is provided, say, in the form of the functions wi (a)s. This fact, in turn, brings to focus two issues which are not often adequately emphasized: (i) If we assume that the source is made of normal laboratory matter, then the relationship between and P depends on our knowledge of how the equation of state for matter behaves at di4erent energy scales. This information needs to be provided by atomic physics, nuclear physics and particle physics. Cosmological models can at best be only as accurate as the input physics about Tki is; any de.nitive assertion about the state of the universe is misplaced, if the knowledge about Tki which it is based on is itself speculative or nonexistent at the relevant energy scales. At present we have laboratory results testing the behavior of matter up to about 100 GeV and hence we can, in principle, determine the equation of state for

246


matter up to 100 GeV. By and large, the equation of state for normal matter in this domain can be taken to be that of an ideal Puid with giving the energy density and P giving the pressure; the relation between the two is of the form P = w with w = 0 for nonrelativistic matter and w = (1=3) for relativistic matter and radiation. (ii) The situation becomes more complicated when we realize that it is entirely possible for the large scale universe to be dominated by matter whose presence is undetectable at laboratory scales. For example, large scale scalar .elds dominated either by kinetic energy or nearly constant potential energy could exist in the universe and will not be easily detectable at laboratory scales. We see from (6) that such systems can have an equation of state of the form P = w with w = 1 (for kinetic energy dominated scalar .eld) or w = −1 (for potential energy dominated scalar .eld). While the conservative procedure for doing cosmology would be to use only known forms of Tki on the right hand side of Einstein’s equations, this has the drawback of preventing progress in our understanding of nature, since cosmology could possibly be the only testing ground for the existence of forms of Tki which are diQcult to detect at laboratory scales. One of the key issues in modern cosmology has to do with the conPict in principle between (i) and (ii) above. Suppose a model based on conventional equation of state, adequately tested in the laboratory, fails to account for a cosmological observation. Should one treat this as a failure of the cosmological model or as a signal from nature for the existence of a source Tki not seen at laboratory scales? There is no easy answer to this question and we will focus on many facets of this issue in the coming sections. Fig. 1 provides an inventory of the density contributed by di4erent forms of matter in the universe. The x-axis is actually a combination +hn of + and the Hubble parameter h since di4erent components are measured by di4erent techniques. (Usually n = 1 or 2; numerical values are for h = 0:7.) The density parameter contributed today by visible, nonrelativistic, baryonic matter in the universe is about +B ≈ (0:01– 0.2) (marked by triangles in the .gure; di4erent estimates are from di4erent sources; see for a sample of Refs. [51–60]). The density parameter due to radiation is about +R ≈ 2 × 10−5 (marked by squares in the .gure). Unfortunately, models for the universe with just these two constituents for the energy density are in violent disagreement with observations. It appears to be necessary to postulate the existence of: • Pressure-less (w = 0) nonbaryonic dark matter which does not couple with radiation and having a density of about +DM ≈ 0:3. Since it does not emit light, it is called dark matter (and marked by a cross in the .gure). Several independent techniques like cluster mass-to-light ratios [61] baryon densities in clusters [62,63] weak lensing of clusters [64,65] and the existence of massive clusters at high redshift [66] have been used to obtain a handle on +DM . These observations are all consistent with +NR = (+DM + +B ) ≈ +DM ≈ (0:2– 0.4). • An exotic form of matter (cosmological constant or something similar) with an equation of state p ≈ − (that is, w ≈ −1) having a density parameter of about + ≈ 0:7 (marked by a .lled circle in the .gure). The evidence for + will be discussed in Section 3. So in addition to H0 , at least four more free parameters are required to describe the background universe at low energies (say, below 50 GeV). These are +B ; +R ; +DM and + describing the fraction of the critical density contributed by baryonic matter, radiation (including relativistic particles like e.g., massive neutrinos; marked by a cross in the .gure), dark matter and cosmological constant


247

Fig. 1. Cosmic inventory of energy densities. See text for description (.gure adapted from [46]).

respectively. The .rst two certainly exist; the existence of last two is probably suggested by observations and is de.nitely not contradicted by any observations. Of these, only +R is well constrained and other quantities are plagued by both statistical and systematic errors in their measurements. The top two positions in the contribution to + are from cosmological constant and nonbaryonic dark matter. It is unfortunate that we do not have laboratory evidence for the existence of the .rst two dominant contributions to the energy density in the universe. (This feature alone could make most of the cosmological paradigm described in this review irrelevant at a future date!) The simplest model for the universe is based on the assumption that each of the sources which populate the universe has a constant wi ; then Eq. (20) becomes a0 3(1+wi ) k a˙2 2 = H +i − 2 ; (22) 0 2 a a a i where each of these species is identi.ed by density parameter +i and the equation of state characterized by wi . The most familiar form of energy densities are those due to pressure-less matter with wi = 0 (that is, nonrelativistic matter with rest mass energy density c2 dominating over the kinetic energy density, v2 =2) and radiation with wi =(1=3). Whenever any one component of energy density dominates over others, P w and it follows from Eq. (22) (taking k = 0, for simplicity) that ˙ a−3(1+w) ;

a ˙ t 2=[3(1+w)] :

(23)

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For example, ˙ a−4 ; a ˙ t 1=2 if the source is relativistic and ˙ a−3 ; a ˙ t 2=3 if the source is nonrelativistic. This result shows that the past evolution of the universe is characterized by two important epochs (see e.g. [43,44]): (i) The .rst is the radiation dominated epoch which occurs at redshifts greater than zeq ≈ (+DM =+R ) ≈ 104 . For z & zeq the energy density is dominated by hot relativistic matter and the universe is very well approximated as a k = 0 model with a(t) ˙ t 1=2 . (ii) The second phase occurs for zzeq in which the universe is dominated by nonrelativistic matter and—in some cases—the cosmological constant. The form of a(t) in this phase depends on the relative values of +DM and + . In the simplest case, with +DM ≈ 1, + =0, +B +DM the expansion is a power law with a(t) ˙ t 2=3 . (When cosmological constant dominates over matter, a(t) grows exponentially.) During all the epochs, the temperature of the radiation varies as T ˙ a−1 . When the temperature falls below T ≈ 103 K, neutral atomic systems form in the universe and photons decouple from matter. In this scenario, a relic background of such photons with Planckian spectrum at some nonzero temperature will exist in the present day universe. The present theory is, however, unable to predict the value of T at t = t0 ; it is therefore a free parameter related +R ˙ T04 . 2.4. Geometrical features of a universe with a cosmological constant The evolution of the universe has di4erent characteristic features if there exists sources in the universe for which (1 + 3w) ¡ 0. This is obvious from equation (8) which shows that if ( + 3P) = (1 + 3w) becomes negative, then the gravitational force of such a source (with ¿ 0) will be repulsive. The simplest example of this kind of a source is the cosmological constant with w = −1. To see the e4ect of a cosmological constant let us consider a universe with matter, radiation and a cosmological constant. Introducing a dimensionless time coordinate 2 = H0 t and writing a = a0 q(2) Eq. (20) can be cast in a more suggestive form describing the one dimensional motion of a particle in a potential 1 dq 2 + V (q) = E ; (24) 2 d2 where V (q) = −

+NR 1 +R 2 + + ; + q 2 q2 q

1 E = (1 − +) : 2

(25)

This equation has the structure of the .rst integral for motion of a particle with energy

E in a potential V (q). For models with + =+R ++NR ++ =1, we can take E =0 so that (dq=d2)= V (q). Fig. 2 is the phase portrait of the universe showing the velocity (dq=d2) as a function of the position q =(1+z)−1 for such models. At high redshift (small q) the universe is radiation dominated and q˙ is independent of the other cosmological parameters; hence all the curves asymptotically approach each other at the left end of the .gure. At low redshifts, the presence of cosmological constant makes a di4erence and—in fact—the velocity q˙ changes from being a decreasing function to an increasing function. In other words, the presence of a cosmological constant leads to an accelerating universe at low redshifts. For a universe with nonrelativistic matter and cosmological constant, the potential in (25) has a simple form, varying as (−a−1 ) for small a and (−a2 ) for large a with a maximum in between at q = qmax = (+NR =2+ )1=3 . This system has been analyzed in detail in literature for both constant


249

Fig. 2. The phase portrait of the universe, with the “velocity” of the universe (dq=d2) plotted against the “position” q = (1 + z)−1 for di4erent models with +R = 2:56 × 10−5 h−2 ; h = 0:5; +NR + + + +R = 1. Curves are parameterized by the value of +NR = 0:1; 0:2; 0:3; 0:5; 0:8; 1:0 going from bottom to top as indicated. (Figure adapted from [46].)

cosmological constant [67] and for a time dependent cosmological constant [68]. A wide variety of explicit solutions for a(t) can be provided for these equations. We briePy summarize a few of them. • If the “particle” is situated at the top of the potential, one can obtain a static solution with aX = a˙ = 0 by adjusting the cosmological constant and the dust energy density and taking k = 1. This solution, crit = 4GNR =

1 ; a20

(26)

was the one which originally prompted Einstein to introduce the cosmological constant (see Section 1.2). • The above solution is, obviously, unstable and any small deviation from the equilibrium position will cause a → 0 or a → ∞. By .ne tuning the values, it is possible to obtain a model for the universe which “loiters” around a = amax for a large period of time [69–71,24–26]. • A subset of models corresponds to those without matter and driven entirely by cosmological constant . These models have k = (−1; 0; +1) and the corresponding expansion factors being proportional to [sinh(Ht); exp(Ht); cosh(Ht)] with 2 = 3H 2 . These line elements represent three di4erent characterizations of the de Sitter spacetime. The manifold is a four dimensional hyperboloid embedded in a Pat, .ve dimensional space with signature (+ − −−). We shall discuss this in greater detail in Section 9. • It is also possible to obtain solutions in which the particle starts from a = 0 with an energy which is insuQcient for it to overcome the potential barrier. These models lead to a universe which collapses back to a singularity. By arranging the parameters suitably, it is possible to make a(t) move away or towards the peak of the potential (which corresponds to the static Einstein universe) asymptotically [67].

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T. Padmanabhan / Physics Reports 380 (2003) 235 – 320 0.5

6 (0.3, 0.7) (0.3, 0)

0.3

(0.3, 0)

4

(1, 0)

0.2

(0.3, 0.7)

5 d LH0

d AH0

0.4

(1, 0)

3 2

0.1

1 0.5

1

1.5 z

2

2.5

3

0.5

1

1.5 z

2

2.5

3

Fig. 3. The left panel gives the angular diameter distance in units of cH0−1 as a function of redshift. The right panel gives the luminosity distance in units of cH0−1 as a function of redshift. Each curve is labelled by (+NR ; + ).

• In the case of +NR + + = 1, the explicit solution for a(t) is given by 2=3 3 a(t) ˙ sinh Ht ; k = 0; 3H 2 = : 2

(27)

This solution smoothly interpolates between a matter dominated universe a(t) ˙ t 2=3 at early stages and a cosmological constant dominated phase a(t) ˙ exp(Ht) at late stages. The transition from deceleration to acceleration occurs at zacc = (2+ =+NR )1=3 − 1, while the energy densities of the cosmological constant and the matter are equal at zm = (+ =+NR )1=3 − 1. The presence of a cosmological constant also a4ects other geometrical parameters in the universe. Fig. 3 gives the plot of dA (z) and dL (z); (note that angular diameter distance is not a monotonic function of z). Asymptotically, for large z, these have the limiting forms, dA (z) ∼ = 2(H0 +NR )−1 z −1 ;

dL (z) ∼ = 2(H0 +NR )−1 z :

(28)

The geometry of the spacetime also determines the proper volume of the universe between the redshifts z and z + d z which subtends a solid angle d+ in the sky. If the number density of sources of a particular kind (say, galaxies, quasars, : : :) is given by n(z), then the number count of sources per unit solid angle per redshift interval should vary as 2 n(z)a20 rem dN (z)H −1 (z) dV = : = n(z) d+ d z d+ d z (1 + z)3

(29)

Fig. 4 shows (dN=d+ d z); it is assumed that n(z) = n0 (1 + z)3 . The y-axis is in units of n0 H0−3 . 3. Evidence for a nonzero cosmological constant There are several cosmological observations which suggests the existence of a nonzero cosmological constant with di4erent levels of reliability. Most of these determine either the value of +NR or some combination of +NR and + . When combined with the strong evidence from the CMBR observations that the +tot = 1 (see Section 6) or some other independent estimate of +NR , one is led

T. Padmanabhan / Physics Reports 380 (2003) 235 – 320 (0.3, 0.7)

0.5 dN/dΩ dz

251

(0.3, 0)

0.4 0.3 0.2

(1, 0)

0.1 0.5

1

1.5 z

2

2.5

3

Fig. 4. The .gure shows (dN=d+ d z): it is assumed that n(z) = n0 (1 + z)3 . The y-axis is in units of n0 H0−3 . Each curve is labelled by (+NR ; + ).

to a nonzero value for + . The most reliable ones seem to be those based on high redshift supernova [72–74] and structure formation models [75–77]. We shall now discuss some of these evidence. 3.1. Observational evidence for accelerating universe Fig. 2 shows that the evolution of a universe with + = 0 changes from a decelerating phase to an accelerating phase at late times. If H (a) can be observationally determined, then one can check whether the real universe had undergone an accelerating phase in the past. This, in turn, can be done if dL (z), say, can be observationally determined for a class of sources. Such a study of several high redshift supernova has led to the data which is shown in Figs. 5, 9. Bright supernova explosions are brief explosive stellar events which are broadly classi.ed as two types. Type-Ia supernova occurs when a degenerate dwarf star containing CNO enters a stage of rapid nuclear burning cooking iron group elements (see e.g., Chapter 7 of [78]). These are the brightest and most homogeneous class of supernova with hydrogen poor spectra. An empirical correlation has been observed between the sharply rising light curve in the initial phase of the supernova and its peak luminosity so that they can serve as standard candles. These events typically last about a month and occurs approximately once in 300 years in our galaxy. (Type II supernova, which occur at the end of stellar evolution, are not useful as standard candles.) For any supernova, one can directly observe the apparent magnitude m [which is essentially the logarithm of the Pux F observed] and its redshift. The absolute magnitude M of the supernova is again related to the actual luminosity L of the supernova in a logarithmic fashion. Hence the relation F = (L=4d2L ) can be written as dL + 25 : (30) m − M = 5 log10 Mpc The numerical factors arise from the astronomical conventions used in the de.nition of m and M . Very often, one will use the dimensionless combination (H0 dL (z)=c) rather than dL (z) and the above

252


Fig. 5. The luminosity distance of a set of type Ia supernova at di4erent redshifts and three theoretical models with +NR + + = 1. The best .t curve has +NR = 0:32; + = 0:68.

equation will change to m(z) = M + 5 log10 (H0 dL (z)=c) with the quantity M being related to M by cH0−1 = M − 5 log10 h + 42:38 : M = M + 25 + 5 log10 (31) 1 Mpc If the absolute magnitude of a class of Type I supernova can be determined from the study of its light curve, then one can obtain the dL for these supernova at di4erent redshifts. (In fact, we only need the low-z apparent magnitude at a given z which is equivalent to knowing M.) Knowing dL , one can determine the geometry of the universe. To understand this e4ect in a simple context, let us compare the luminosity distance for a matter dominated model (+NR = 1; + = 0) dL = 2H0−1 [(1 + z) − (1 + z)1=2 ] ;

(32)

with that for a model driven purely by a cosmological constant (+NR = 0; + = 1) dL = H0−1 z(1 + z) :

(33)

It is clear that at a given z, the dL is larger for the cosmological constant model. Hence, a given object, located at a .xed redshift, will appear brighter in the matter dominated model compared to the cosmological constant dominated model. Though this sounds easy in principle, the statistical analysis turns out to be quite complicated. The supernova cosmology project (SCP) has discovered [74] 42 supernova in the range (0.18– 0.83). The high-z supernova search team (HSST) discovered 14 supernova in the redshift range (0.16 – 0.62) and another 34 nearby supernova [73] and used two separate methods for data .tting. (They also included two previously published results from SCP.) Assuming +NR + + = 1, the analysis of this data gives +NR = 0:28 ± 0:085 (stat) ±0:05 (syst).


253

Fig. 6. Con.dence contours corresponding to 68%, 90% and 99% based on SN data in the +NR − M plane for the Pat models with +NR + + = 1. Frame (a) on the left uses all data while frame (b) in the middle uses low redshift data and the frame (c) in the right uses high redshift data. While neither the low-z or high-z data alone can exclude the +NR = 1; + = 0 model, the full data excludes it to a high level of signi.cance.

Fig. 5 shows the dL (z) obtained from the supernova data and three theoretical curves all of which are for k = 0 models containing nonrelativistic matter and cosmological constant. The data used here is based on the redshift magnitude relation of 54 supernova (excluding 6 outliers from a full sample of 60) and that of SN 19974 at z = 1:755; the magnitude used for SN 19974 has been corrected for lensing e4ects [79]. The best .t curve has +NR ≈ 0:32; + ≈ 0:68. In this analysis, one had treated +NR and the absolute magnitude M as free parameters (with +NR + + = 1) and has done a simple best .t for both. The corresponding best .t value for M is M = 23:92 ± 0:05. Frame (a) of Fig. 6 shows the con.dence interval (for 68%, 90% and 99%) in the +NR − M for the Pat models. It is obvious that most of the probability is concentrated around the best .t value. We shall discuss frame (b) and frame (c) later on. (The discussion here is based on [80].) The con.dence intervals in the + −+NR plane are shown in Fig. 7 for the full data. The con.dence regions in the top left frame are obtained after marginalizing over M. (The best .t value with 16 error is indicated in each panel and the con.dence contours correspond to 68%, 90% and 99%.) The other three frames show the corresponding result with a constant value for M rather than by marginalizing over this parameter. The three frames correspond to the mean value and two values in the wings of 16 from the mean. The dashed line connecting the points (0,1) and (1,0) correspond to a universe with +NR + + = 1. From the .gure we can conclude that: (i) The results do not change signi.cantly whether we marginalize over M or whether we use the best .t value. This is a direct consequence of the result in frame (a) of Fig. 6 which shows that the probability is sharply peaked. (ii) The results exclude the +NR = 1; + = 0 model at a high level of signi.cance in spite of the uncertainty in M. The slanted shape of the probability ellipses shows that a particular linear combination of +NR and + is selected out by these observations [81]. This feature, of course, has nothing to do with supernova and arises purely because the luminosity distance dL depends strongly on a particular linear combination of + and +NR , as illustrated in Fig. 8. In this .gure, +NR ; + are treated as free parameters [without the k = 0 constraint] but a particular linear combination q ≡ (0:8+NR − 0:6+ ) is held .xed. The dL is not very sensitive to individual values of +NR ; + at low redshifts when (0:8+NR − 0:6+ ) is in the range of (−0:3; −0:1). Though some of the models have unacceptable parameter

254


Fig. 7. Con.dence contours corresponding to 68%, 90% and 99% based on SN data in the +NR − + plane. The top left frame is obtained after marginalizing over M while the other three frames uses .xed values for M. The values are chosen to be the best-.t value for M and two others in the wings of 16 limit. The dashed line corresponds to the Pat model. The unbroken slanted line corresponds to H0 dL (z = 0:63) = constant. It is clear that: (i) The data excludes the +NR = 1; + = 0 model at a high signi.cance level irrespective of whether we marginalize over M or use an accepted 16 range of values for M. (ii) The shape of the con.dence contours are essentially determined by the value of the luminosity distance at z ≈ 0:6.

values (for other reasons), supernova measurements alone cannot rule them out. Essentially the data at z ¡ 1 is degenerate on the linear combination of parameters used to construct the variable q. The supernova data shows that most likely region is bounded by −0:3 . (0:8+NR − 0:6+ ) . −0:1. In Fig. 7 we have also over plotted the line corresponding to H0 dL (z = 0:63)= constant. The coincidence of this line (which roughly corresponds to dL at a redshift in the middle of the data) with the probability ellipses indicates that it is this quantity which essentially determines the nature of the result. We saw earlier that the presence of cosmological constant leads to an accelerating phase in the universe which—however—is not obvious from the above .gures. To see this explicitly one needs to display the data in the a˙ vs a plane, which is done in Fig. 9. Direct observations of supernova is converted into dL (z) keeping M a free parameter. The dL is converted into dH (z) assuming k = 0 and using (17). A best .t analysis, keeping (M; +NR ) as free parameters now lead to the results


255

Fig. 8. The luminosity distance for a class of models with widely varying +NR ; + but with a constant value for q ≡ (0:8+NR − 0:6+ ) are shown in the .gure. It is clear that when q is .xed, low redshift observations cannot distinguish between the di4erent models even if +NR and + vary signi.cantly.

Fig. 9. Observations of SN are converted into the ‘velocity’ a˙ of the universe using a .tting function. The curves which are over-plotted corresponds to a cosmological model with +NR + + = 1. The best .t curve has +NR = 0:32; + = 0:68.

256


shown in Fig. 9, which con.rms the accelerating phase in the evolution of the universe. The curves which are over-plotted correspond to a cosmological model with +NR + + = 1. The best .t curve has +NR = 0:32; + = 0:68. In the presence of the cosmological constant, the universe accelerates at low redshifts while decelerating at high redshift. Hence, in principle, low redshift supernova data should indicate the evidence for acceleration. In practice, however, it is impossible to rule out any of the cosmological models using low redshift (z . 0:2) data as is evident from Fig. 9. On the other hand, high redshift supernova data alone cannot be used to establish the existence of a cosmological constant. The data for (z & 0:2) in Fig. 9 can be moved vertically up and made consistent with the decelerating + = 1 universe by choosing the absolute magnitude M suitably. It is the interplay between both the high redshift and low redshift supernova which leads to the result quoted above. This important result can be brought out more quantitatively along the following lines. The data displayed in Fig. 9 divides the supernova into two natural classes: low redshift ones in the range 0 ¡ z . 0:25 (corresponding to the accelerating phase of the universe) and the high redshift ones in the range 0:25 . z . 2 (in the decelerating phase of the universe). One can repeat all the statistical analysis for the full set as well as for the two individual low redshift and high redshift data sets. Frame (b) and (c) of Fig. 6 shows the con.dence interval based on low redshift data and high redshift data separately. It is obvious that the +NR = 1 model cannot be ruled out with either of the two data sets! But, when the data sets are combined—because of the angular slant of the ellipses—they isolate a best .t region around +NR ≈ 0:3. This is also seen in Fig. 10 which plots the con.dence intervals using just the high-z and low-z data separately. The right most frame in the bottom row is based on the low-z data alone (with marginalization over M) and this data cannot be used to discriminate between cosmological models e4ectively. This is because the dL at low-z is only very weakly dependent on the cosmological parameters. So, even though the acceleration of the universe is a low-z phenomenon, we cannot reliably determine it using low-z data alone. The top left frame has the corresponding result with high-z data. As we stressed before, the +NR = 1 model cannot be excluded on the basis of the high-z data alone either. This is essentially because of the nature of probability contours seen earlier in frame (c) of Fig. 6. The remaining 3 frames (top right, bottom left and bottom middle) show the corresponding results in which .xed values of M—rather than by marginalizing over M. Comparing these three .gures with the corresponding three frames in 7 in which all data was used, one can draw the following conclusions: (i) The best .t value for M is now M = 24:05 ± 0:38; the 16 error has now gone up by nearly eight times compared to the result (0.05) obtained using all data. Because of this spread, the results are sensitive to the value of M that one uses, unlike the situation in which all data was used. (ii) Our conclusions will now depend on M. For the mean value and lower end of M, the data can exclude the +NR = 1; + = 0 model [see the two middle frames of Fig. 10]. But, for the high-end of allowed 16 range of M, we cannot exclude the +NR = 1; + = 0 model [see the bottom left frame of Fig. 10]. While these observations have enjoyed signi.cant popularity, certain key points which underly these analysis need to be stressed. (For a sample of views which goes against the main stream, see [82,83].) • The basic approach uses the supernova type I light curve as a standard candle. While this is generally accepted, it must be remembered that we do not have a sound theoretical understanding of the underlying emission process.


257

Fig. 10. Con.dence contours corresponding to 68%, 90% and 99% based on SN data in the +NR − + plane using either the low-z data (bottom right frame) or high-z data (the remaining four frames). The bottom right and the top left frames are obtained by marginalizing over M while the remaining three uses .xed values for M. The values are chosen to be the best-.t value for M and two others in the wings of 16 limit. The dashed line corresponds to the Pat model. The unbroken slanted line corresponds to H0 dL (z = 0:63)= constant. It is clear that: (i) The 16 error in top left frame (0.38) has gone up by nearly eight times compared to the value (0.05) obtained using all data (see Fig. 7) and the results are sensitive to the value of M. (ii) The data can exclude the +NR = 1; + = 0 model if the mean or low-end value of M is used [see the two middle frames]. But, for the high-end of allowed 16 range of M, we cannot exclude the +NR = 1; + = 0 model [see the bottom left frame]. (iii) The low-z data [bottom right] cannot exclude any of the models.

• The supernova data and .ts are dominated by the region in the parameter space around (+NR ; + ) ≈ (0:8; 1:5) which is strongly disfavoured by several other observations. If this disparity is due to some other unknown systematic e4ect, then this will have an e4ect on the estimate given above. • The statistical issues involved in the analysis of this data to obtain best .t parameters are nontrivial. As an example of how the claims varied over time, let us note that the analysis of the .rst 7 high redshift SNe Ia gave a value for +NR which is consistent with unity: +NR = (0:94+0:34 −0:28 ). However, adding a single z = 0:83 supernova for which good HST data was available, lowered the value to +NR = (0:6 ± 0:2). More recently, the analysis of a larger data set of 42 high redshift SNe Ia gives the results quoted above. 3.2. Age of the universe and cosmological constant From Eq. (24) we can also determine the current age of the universe by the integral 1 dq

H 0 t0 = : 2(E − V ) 0

(34)

258


1 1.08

0.94

0.9

0.85

.82

.8

0.8

ΩΛ

0.6

0.4 0.7 0.2 0.67 0

0

0.2

0.4

0.6 Ω NR

0.8

1

Fig. 11. Lines of constant H0 t0 in the +NR −+ plane. The eight lines are for H0 t0 =(1:08; 0:94; 0:9; 0:85; 0:82; 0:8; 0:7; 0:67) as shown. The diagonal line is the contour for models with +NR + + = 1.

Since most of the contribution to this integral comes from late times, we can ignore the radiation term and set +R ≈ 0. When both +NR and + are present and are arbitrary, the age of the universe is determined by the integral ∞ dz 2

H 0 t0 = ≈ (0:7+NR − 0:3+ + 0:3)−0:3 : (35) 3 3 (1 + z) +NR (1 + z) + + 0 The integral, which cannot be expressed in terms of elementary functions, is well approximated by the numerical .t given in the second line. Contours of constant H0 t0 based on the (exact) integral are shown in Fig. 11. It is obvious that, for a given +NR , the age is higher for models with + = 0. Observationally, there is a consensus [49,50] that h ≈ 0:72 ± 0:07 and t0 ≈ 13:5 ± 1:5 Gyr [84]. This will give H0 t0 = 0:94 ± 0:14. Comparing this result with the .t in (35), one can immediately draw several conclusions: • If +NR ¿ 0:1, then + is nonzero if H0 t0 ¿ 0:9. A more reasonable assumption of +NR ¿ 0:3 we will require nonzero + if H0 t0 ¿ 0:82. • If we take +NR = 1; + = 0 and demand t0 ¿ 12 Gyr (which is a conservative lower bound from stellar ages) will require h ¡ 0:54. Thus a purely matter dominated + = 1 universe would require low Hubble constant which is contradicted by most of the observations.


259

• An open model with +NR ≈ 0:2; + = 0 will require H0 t0 ≈ 0:85. This still requires ages on the lower side but values like h ≈ 0:6; t0 ≈ 13:5 Gyr are acceptable within error bars. • A straightforward interpretation of observations suggests maximum likelihood for H0 t0 =0:94. This can be consistent with a + = 1 model only if +NR ≈ 0:3; + ≈ 0:7. If the universe is populated by dust-like matter (with w =0) and another component with an equation of state parameter wX , then the age of the universe will again be given by an integral similar to the one in Eq. (35) with + replaced by +X (1 + z)3(1+wX ) . This will give H 0 t0 =

∞

dz

(1 + z) +NR (1 + z)3 + +X (1 + z)3(1+wX ) 1=2 1 q = dq : +NR + +X q−3wX 0 0

(36)

The integrand varies from 0 to (+NR ++X )−1=2 in the range of integration for w ¡ 0 with the rapidity of variation decided by w. As a result, H0 t0 increases rapidly as w changes from 0 to −3 or so and then saturates to a plateau. Even an absurdly negative value for w like w = −100 gives H0 t0 of only about 1.48. This shows that even if some exotic dark energy is present in the universe with a constant, negative w, it cannot increase the age of the universe beyond about H0 t0 ≈ 1:48. The comments made above pertain to the current age of the universe. It is also possible to obtain an expression similar to (34) for the age of the universe at any given redshift z H0 t(z) =

z

∞

d z ; (1 + z )h(z )

h(z) =

H (z) H0

(37)

and use it to constrain + . For example, the existence of high redshift galaxies with evolved stellar population, high redshift radio galaxies and age dating of high redshift QSOs can all be used in conjunction with this equation to put constrains on + [85–90]. Most of these observations require either + = 0 or +tot ¡ 1 if they have to be consistent with h & 0:6. Unfortunately, the interpretation of these observations at present requires fairly complex modeling and hence the results are not water tight.

3.3. Gravitational lensing and the cosmological constant Consider a distant source at redshift z which is lensed by some intervening object. The lensing is most e4ective if the lens is located midway between the source and the observer (see, e.g., p. 196 of [46]). This distance will be (rem =2) if the distance to the source is rem . (To be rigorous, one should be using angular diameter distances rather than rem for this purpose but the essential conclusion does not change.) To see how this result depends on cosmology, let us consider a source at redshift z = 2, and a lens, located optimally, in: (a) + = 1 matter dominated universe, (b) a very low density matter dominated universe in the limit of + → 0, (c) vacuum dominated universe with

260


+ = +tot . In case (a), dH ≡ H −1 (z) ˙ (1 + z)−3=2 , so that z 1 rem (z) ˙ : dH (z) d z ˙ 1 − √ 1+z 0 The lens redshift is determined by the equation 1 1 1 1− √ 1− √ = : 2 1 + zL 1+z

(38)

(39)

For z = 2, this gives zL = 0:608. For case (b), a ˙ t giving dH ˙ (1 + z)−1 and rem (z) ˙ ln(1 + z). The equation to be solved is (1 + zL ) = (1 + z)1=2 which gives zL = 0:732 for z = 2. Finally, in the case of (c), dH is a constant giving rem (z) ˙ z and zL = (1=2)z. Clearly, the lens redshift is larger for vacuum dominated universe compared to the matter dominated universe of any +. When one considers a distribution of lenses, this will a4ect the probability for lensing in a manner which depends on + . From the observed statistics of lensing, one can put a bound on + . More formally, one can compute the probability for a source at redshift zs being lensed in a + + +NR = 1 universe (relative to the corresponding probability in a +NR = 1; + = 0 model). This relative probability is nearly .ve times larger at zs =1 and about 13 times larger for zs =2 in a purely cosmological constant dominated universe [91–95,2,3]. This e4ect quanti.es the fact that the physical volume associated with unit redshift interval is larger in models with cosmological constant and hence the probability that a light ray will encounter a galaxy is larger in these cases. Current analysis of lensing data gives somewhat di4ering constraints based on the assumptions which are made [96–99]; but typically all these constraints are about + . 0:7. The result [100] from +0:12 cosmic lens all sky survey (CLASS), for example, gives +NR = 0:31+0:27 −0:14 (68%) −0:10 (systematic) for a k = 0 universe. 3.4. Other geometrical tests The existence of a maximum for dA (z) is a generic feature of cosmological models with +NR ¿ 0. For a k =0; +NR =1 model, the maximum occurs at zmax ≈ 1:25 and zmax increases as + is increased. To use this as a cosmological test, we require a class of objects with known transverse dimension and redshift. The most reliable quantity used so far corresponds to the physical wavelength acoustic vibrations in the baryon–photon gas at z ≈ 103 . This length scale is imprinted in the temperature anisotropies of the CMBR and the angular size of these anisotropies will depend on dA and hence on the cosmological parameters; this is discussed in Section 6. In principle, one could also use angular sizes of galaxies, clusters of galaxies, or radio galaxies [101–103]. Unfortunately, understanding of di4erent physical e4ects and the redshift evolution of these sources make this a diQcult test in practice. There is another geometrical feature of the universe in which angular diameter distance plays an interesting role. In a closed Friedmann model with k = +1, there is possibility that an observer at % = 0 will be able to receive the light from the antipodal point % = . In a purely matter dominated universe, it is easy to see that the light ray from the antipodal point % = reaches % = 0 exactly at the time of maximum expansion; therefore, in a closed, matter dominated universe, in the expanding phase, no observer can receive light from the antipodal point. The situation, however, is di4erent in


the presence of cosmological constant. In this case, dA (z) ˙ (1 + z)−1 sin where z H (z) d z 1=2 ; h(z) = = |+tot − 1| : ) h(z H0 0

261

(40)

It follows that dA → 0 when → . Therefore, the angular size of an object near the antipodal point can diverge making the object extremely bright in such a universe. Assuming that this phenomena does not occur up to, say z = 6, will imply that the redshift of the antipodal point za (+ ; +NR ) is larger than 6. This result can be used to constrain the cosmological parameters [104,105,68] though the limits obtained are not as tight as some of the other tests. Another test which can be used to obtain a handle on the geometry of the universe is usually called Alcock–Paczynski curvature test [106]. The basic idea is to use the fact that when any spherically symmetric system at high redshift is observed, the cosmological parameters enter di4erently in the characterization of radial and transverse dimensions. Hence any system which can be approximated a priori to be intrinsically spherical will provide a way of determining cosmological parameters. The correlation function of SDSS luminous red galaxies seems to be promising in terms of both depth and density for applying this test (see for a detailed discussion, [107,108]). The main sources of error arises from nonlinear clustering and the bias of the red galaxies, either of which can be a source of systematic error. A variant of this method was proposed using observations of Lymanforest and compare the correlation function along the line of sight and transverse to the line of sight. In addition to the modeling uncertainties, successful application of this test will also require about 30 close quasar pairs [109,110]. 4. Models with evolving cosmological “constant” The observations which suggest the existence of nonzero cosmological constant—discussed in the last section—raises serious theoretical problems which we mentioned in Section 1.1. These diQculties have led people to consider the possibility that the dark energy in the universe is not just a cosmological constant but is of more complicated nature, evolving with time. Its value today can then be more naturally set by the current expansion rate rather than predetermined earlier on— thereby providing a solution to the cosmological constant problems. Though a host of models have been constructed based on this hope, none of them provides a satisfactory solution to the problems of .ne-tuning. Moreover, all of them involve an evolving equation of state parameter wX (a) for the unknown (“X ”) dark energy component, thereby taking away all predictive power from cosmology [166]. Ultimately, however, this issue needs to settled observationally by checking whether wX (a) is a constant [equal to −1, for the cosmological constant] at all epochs or whether it is indeed varying with a. We shall now discuss several observational and theoretical issues connected with this theme. While the complete knowledge of the Tba [that is, the knowledge of the right hand side of (20)] uniquely determines H (a), the converse is not true. If we know only the function H (a), it is not possible to determine the nature of the energy density which is present in the universe. We have already seen that geometrical measurements can only provide, at best, the functional form of H (a). It follows that purely geometrical measurements of the Friedmann universe will never allow us to determine the material content of the universe. [The only exception to this rule is when we assume that each of the components in the universe has constant wi . This is fairly strong assumption

262


and, in fact, will allow us to determine the components of the universe from the knowledge of the function H (a). To see this, we .rst note that the term (k=a2 ) in equation (22) can be thought of as contributed by a hypothetical species of matter with w = −(1=3). Hence Eq. (22) can be written in the form a0 3(1+wi ) a˙2 2 = H +i (41) 0 a2 a i with a of the i , the can be

term having wi = −(1=3) added to the sum. Let ≡ 3(1 + w) and +( ) denote the fraction critical density contributed by matter with w = ( =3) − 1. (For discrete values of wi and function +( ) will be a sum of Dirac delta functions.) In the continuum limit, Eq. (41) rewritten as ∞ 2 2 H = H0 d +( )e− q ; (42) −∞

where (a=a0 )=exp(q). The function +( ) is assumed to have .nite support (or decrease fast enough) for the expression on the right hand side to converge. If the observations determine the function H (a), then the left hand side can be expressed as a function of q. An inverse Laplace transform of this equation will then determine the form of +( ) thereby determining the composition of the universe, as long as all matter can be described by an equation of state of the form pi = wi i with wi = constant for all i = 1; : : : ; N .] More realistically one is interested in models which has a complicated form of wX (a) for which the above analysis is not applicable. Let us divide the source energy density into two components: k (a), which is known from independent observations and a component X (a) which is not known. From (20), it follows that 8G X (a) = H 2 (a)(1 − Q(a)); 3

Q(a) ≡

8Gk (a) : 3H 2 (a)

(43)

Taking a derivative of ln X (a) and using (19), it is easy to obtain the relation 1 d ln[(1 − Q(a))H 2 (a)a3 ] : (44) 3 d ln a If geometrical observations of the universe give us H (a) and other observations give us k (a) then one can determine Q and thus wX (a). While this is possible, in principle the uncertainties in measuring both H and Q makes this a nearly impossible route to follow in practice. In particular, one is interested in knowing whether w evolves with time or a constant and this turns out to be a very diQcult task observationally. We shall now briePy discuss some of the issues. wX (a) = −

4.1. Parametrized equation of state and cosmological observations One simple, phenomenological, procedure for comparing observations with theory is to parameterize the function w(a) in some suitable form and determine a .nite set of parameters in this function using the observations. Theoretical models can then be reduced to a .nite set of parameters which can be determined by this procedure. To illustrate this approach, and the diQculties in determining the equation of state of dark energy from the observations, we shall assume that w(a) is given by the simple form: w(a) = w0 + w1 (1 − a); in the k = 0 model (which we shall assume for simplicity),


263

Fig. 12. Con.dence interval contours in the w0 − w1 plane arising from the full supernova data, for Pat models with +NR + + = 1. The three frames are for +NR = (0:2; 0:3; 0:4). The data cannot rule out cosmological constant with w0 = −1; w1 = 0. The slanted line again corresponds to H0 dL (z = 0:63)=constant and shows that the shape of the probability ellipses arises essentially from this feature.

w0 measures the current value of the parameter and −w1 gives its rate of change at the present epoch. In addition to simplicity, this parameterization has the advantage of giving .nite w in the entire range 0 ¡ a ¡ 1. Fig. 12 shows con.dence interval contours in the w0 − w1 plane arising from the full supernova data, obtained by assuming that +NR + + = 1. The three frames are for +NR = (0:2; 0:3; 0:4). The following features are obvious from the .gure: (i) The cosmological constant corresponding to w0 = −1; w1 = 0 is a viable candidate and cannot be excluded. (In fact, di4erent analysis of many observational results lead to this conclusion consistently; in other words, at present there is no observational motivation to assume w1 = 0.) (ii) The result is sensitive to the value of +NR which is assumed. This is understandable from Eq. (44) which shows that wX (a) depends on both Q ˙ +NR and H (a). (We shall discuss this dependence of the results on +NR in greater detail below). (iii) Note that the axes are not in equal units in Fig. 12. The observations can determine w0 with far greater accuracy than w1 . (iv) The slanted line again corresponds to H0 dL (z = 0:63) = constant and shows that the shape of the probability ellipses arises essentially from this feature. In summary, the current data de.nitely supports a negative pressure component with w0 ¡ − (1=3) but is completely consistent with w1 = 0. If this is the case, then the cosmological constant is the simplest candidate for this negative pressure component and there is very little observational motivation to study other models with varying w(a). On the other hand, the cosmological constant has well known theoretical problems which could possibly be alleviated in more sophisticated models with varying w(a). With this motivation, there has been extensive amount of work in the last few years investigating whether improvement in the observational scenario will allow us to determine whether w1 is nonzero or not. (For a sample of references, see [111–126].) In the context of supernova based determination of dL , it is possible to analyze the situation along the following lines [80]. Since the supernova observations essentially measure dL (a), accuracy in the determination of w0 and w1 from (both the present and planned future [127]) supernova observations will crucially depend on how sensitive dL is to the changes in w0 and w1 . A good measure of the sensitivity is provided

264


Fig. 13. Sensitivity of dL to the parameters w0 ; w1 . The curves correspond to constant values for the percentage of change in dL H0 for unit change in w0 (top frames), and w1 (bottom frames). Comparison of the top and bottom frames shows that dL H0 varies by few tens of percent when w0 is varied but changes by much lesser amount when w1 is varied.

by the two parameters d A(z; w0 ; w1 ) ≡ ln(dL (z; w0 ; w1 )H0 ) ; dw0 d ln(dL (z; w0 ; w1 )H0 ) : (45) dw1 Since dL (z; w0 ; w1 ) can be obtained from theory, the parameters A and B can be computed form theory in a straight forward manner. At any given redshift z, we can plot contours of constant A and B in the w0 − w1 plane. Fig. 13 shows the result of such an analysis [80]. The two frames on the left are at z = 1 and the two frames on the right are at z = 3. The top frames give contours of constant A and bottom frame give contours of constant B. From the de.nition in Eq. (45) it is clear that A and B can be interpreted as the fractional change in dL for unit change in w0 ; w1 . For example, along the line marked A = 0:2 (in the top left frame) dL will change by 20 percent for unit change in w0 . It is clear from the two top frames that for most of the interesting region in the w0 − w1 plane, changing w0 by unity changes dL by about 10 percent or more. Comparison of z = 1 and z = 3 (the two top frames) shows that the sensitivity is higher at high redshift, as to be expected. The shaded band across the picture corresponds to the region in which −1 6 w(a) 6 0 which is of primary interest in constraining dark energy with negative pressure. One concludes that determining w0 from dL fairly accurately will not be too daunting a task. The situation, however, is quite di4erent as regards w1 as illustrated in the bottom two frames. For the same region of the w0 − w1 plane, dL changes only by a few percent when w1 changes by unity. That is, dL is much less sensitive to w1 than to w0 . It is going to be signi.cantly more B(z; w0 ; w1 ) ≡


265

Fig. 14. Contours of constant H0 dL in the +NR − w0 and +NR − w1 planes at two redshifts z = 1 and 3. The two top frames shows that a small variation of +NR in the allowed range of, say, (0:2; 0:4) corresponds to fairly large variation in w1 along the curve of constant dL .

diQcult to determine a value for w1 from observations of dL in the near future. Comparison of z = 1 and z = 3 again shows that the sensitivity is somewhat better at high redshifts but only marginally. The situation is made worse by the fact that dL also depends on the parameter +NR . If varying +NR mimics the variation of w1 or w0 , then, one also needs to determine the sensitivity of dL to +NR . Fig. 14 shows contours of constant H0 dL in the +NR − w0 and +NR − w1 planes at two redshifts z = 1 and 3. The two top frames shows that if one varies the value of +NR in the allowed range of, say, (0:2; 0:4) one can move along the curve of constant dL and induce fairly large variation in w1 . In other words, large changes in w1 can be easily compensated by small changes in +NR while maintaining the same value for dL at a given redshift. This shows that the uncertainty in +NR introduces further diQculties in determining w1 accurately from measurements of dL . The two lower frames show that the situation is better as regards w0 . The curves are much less steep and hence varying +NR does not induce large variations in w0 . We are once again led to the conclusion that unambiguous determination of w1 from data will be quite diQcult. This is somewhat disturbing since w1 = 0 is a clear indication of a dark energy component which is evolving. It appears that observations may not be of great help in ruling out cosmological constant as the major dark energy component. (The results given above are based on [80]; also see [128] and references cited therein.) 4.2. Theoretical models with time dependent dark energy: cosmic degeneracy The approach in the last section was purely phenomenological and one might like to construct some physical model which leads to varying w(a). It turns out that this is fairly easy, and—in fact—it is possible to construct models which will accommodate virtually any form of evolution.

266


We shall now discuss some examples. A simple form of the source with variable w are scalar .elds with Lagrangians of di4erent forms, of which we will discuss two possibilities: Lquin = 12 9a 9a − V ( );

Ltach = −V ( )[1 − 9a 9a ]1=2 :

(46)

Both these Lagrangians involve one arbitrary function V ( ). The .rst one, Lquin , which is a natural generalisation of the Lagrangian for a nonrelativistic particle, L = (1=2)q˙2 − V (q), is usually called quintessence (for a sample of models, see [129–139]). When it acts as a source in Friedman universe, it is characterized by a time dependent w(t) with 1 1 1 − (2V= ˙ 2 ) : (47) q (t) = ˙ 2 + V ; Pq (t) = ˙ 2 − V ; wq = 2 2 1 + (2V= ˙ 2 ) The structure of the second scalar .eld can be understood by a simple analogy from special relativity. A relativistic particle with (one dimensional) by the Lagrangian

position q(t) and mass m is described

L=−m 1 − q˙2 . It has the energy E =m= 1 − q˙2 and momentum p=mq= ˙ 1 − q˙2 which are related by E 2 =p2 +m2 . As is well known, this allows the possibility of having massless particles with .nite energy for which E 2

= p2 . This is achieved by taking the limit of m → 0 and q˙ → 1, while keeping the ratio in E = m= 1 − q˙2 .nite. The momentum acquires a life of its own, unconnected with the velocity q, ˙ and the energy is expressed in terms of the momentum (rather than in terms of q) ˙ in the Hamiltonian formulation. We can now construct a .eld theory by upgrading q(t) to a .eld . Relativistic invariance now requires to depend on both space and time [ = (t; x)] and q˙2 to be replaced by 9i 9i . It is also possible now to treat the mass parameter m as a function of ,

say, V ( ) thereby obtaining a .eld theoretic Lagrangian L = −V ( ) 1 − 9i 9i . The Hamiltonian structure of this theory is algebraically very similar to the special relativistic example we started with. In particular, the theory allows solutions in which V → 0, 9i 9i → 1 simultaneously, keeping the energy (density) .nite. Such solutions will have .nite momentum density (analogous to a massless particle with .nite momentum p) and energy density. Since the solutions can now depend on both space and time (unlike the special relativistic example in which q depended only on time), the momentum density can be an arbitrary function of the spatial coordinate. This provides a rich gamut of possibilities in the context of cosmology [140–166]. This form of scalar .eld arises in string theories [167] and—for technical reasons—is called a tachyonic scalar .eld. (The structure of this Lagrangian is similar to those analyzed in a wide class of models called K-essence; see for example, [160]. We will not discuss K-essence models in this review.) The stress tensor for the tachyonic scalar .eld can be written in a perfect Puid form Tki = ( + p)ui uk − pik

(48)

with 9k

uk =

9i 9i

;

=

V ( ) 1 − 9i 9i

;

p = −V ( )

1 − 9i 9i :

(49)

The remarkable feature of this stress tensor is that it could be considered as the sum of a pressure less dust component and a cosmological constant [165]. To show this explicitly, we break up the density and the pressure p and write them in a more suggestive form as = + DM ;

p = pV + pDM ;

(50)


267

where V ( )9i 9i ; DM =

1 − 9i 9i

pDM = 0;

= V ( )

1 − 9i 9i ;

pV = − :

(51)

This means that the stress tensor can be thought of as made up of two components—one behaving like a pressure-less Puid, while the other having a negative pressure. In the cosmological context, the tachyonic .eld is described by t (t) = V [1 − ˙ 2 ]−1=2 ;

Pt = −V [1 − ˙ 2 ]1=2 ;

wt = ˙ 2 − 1 :

(52)

When ˙ is small (compared to V in the case of quintessence or compared to unity in the case of ˙ tachyonic .eld), both these sources have w → −1 and mimic a cosmological constant. When V , 6 the quintessence has w ≈ 1 leading to q ˙ (1 + z) ; the tachyonic .eld, on the other hand, has w ≈ 0 for ˙ → 1 and behaves like nonrelativistic matter. In both the cases, −1 ¡ w ¡ 1, though it is possible to construct more complicated scalar .eld Lagrangians with even w ¡ − 1. (See for example, [168]; for some other alternatives to scalar .eld models, see for example, [169].) Since the quintessence .eld (or the tachyonic .eld) has an undetermined free function V ( ), it is possible to choose this function in order to produce a given H (a). To see this explicitly, let us assume that the universe has two forms of energy density with (a) = known (a) + (a) where known (a) arises from any known forms of source (matter, radiation, : : :) and (a) is due to a scalar .eld. When w(a) is given, one can determine the V ( ) using either (47) or (52). For quintessence, (47) along with (43) gives 3H 2 (a) ˙ 2 (a) = (1 + w) = (1 − Q)(1 + w) ; 8G 2V (a) = (1 − w) =

3H 2 (a) (1 − Q)(1 − w) : 8G

(53)

For tachyonic scalar .eld, (52) along with (43) gives ˙ 2 (a) = (1 + w);

V (a) = (−w)1=2 =

3H 2 (a) (1 − Q)(−w)1=2 : 8G

(54)

Given Q(a), w(a) these equations implicitly determine V ( ). We have already seen that, for any cosmological evolution speci.ed by the functions H (a) and k (a), one can determine w(a); see Eq. (44). Combining (44) with either (53) or (54), one can completely solve the problem. Let us .rst consider quintessence. Here, using (44) to express w in terms of H and Q, the potential is given implicitly by the form [170,166] 1 aHQ ; (55) V (a) = H (1 − Q) 6H + 2aH − 16G 1−Q

1 (a) = 8G

1=2

1=2 da d ln H 2 aQ − (1 − Q) ; a d ln a

(56)

268


where Q(a) ≡ [8Gknown (a)=3H 2 (a)]. We shall now discuss some examples of this result: • Consider a universe in which observations suggest that H 2 (a) = H02 a−3 . Such a universe could be populated by nonrelativistic matter with density parameter +NR = + = 1. On the other hand, such a universe could be populated entirely by a scalar .eld with a potential V ( ) = V0 exp[ − (16G=3)1=2 ]. One can also have a linear combination of nonrelativistic matter and scalar .eld with the potential having a generic form V ( ) = A exp[ − B ]. • Power law expansion of the universe can be generated by a quintessence model with V ( ) = − . In this case, the energy density of the scalar .eld varies as ˙ t −2 =(2+ ) ; if the background density bg varies as bg ˙ t −2 , the ratio of the two energy densities changes as ( =bg =t 4=(2+ ) ). Obviously, the scalar .eld density can dominate over the background at late times for ¿ 0. • A di4erent class of models arise if the potential is taken to be exponential with, say, V ( ) ˙ exp(−: =MPl ). When k = 0, both and bg scale in the same manner leading to 3(1 + wbg ) = ; bg + :2

(57)

where wbg refers to the background parameter value. In this case, the dark energy density is said to “track” the background energy density. While this could be a model for dark matter, there are strong constraints on the total energy density of the universe at the epoch of nucleosynthesis. This requires + . 0:2 requiring dark energy to be subdominant at all epochs. • Many other forms of H (a) can be reproduced by a combination of nonrelativistic matter and a suitable form of scalar .eld with a potential V ( ). As a .nal example [68], suppose H 2 (a) = H02 [+NR a−3 +(1−+NR )a−n ]. This can arise, if the universe is populated with nonrelativistic matter with density parameter +NR and a scalar .eld with the potential, determined using Eqs. (55), (56). We get V ( ) = V0 sinh2n=(n−3) [ ( − )] ;

(58)

where 1=(n−3) n +NR (6 − n)H02 V0 = ; 16G (1 − +NR )3 and

= (3 − n)(2G=n)1=2

(59)

is a constant.

Similar results exists for the tachyonic scalar .eld as well [166]. For example, given any H (t), one can construct a tachyonic potential V ( ) so that the scalar .eld is the source for the cosmology. The equations determining V ( ) are now given by: aQ 2 aH 1=2 da − ; (60) (a) = aH 3(1 − Q) 3 H 1=2 2 aH aQ 3H 2 (1 − Q) 1 + − V= : 8G 3 H 3(1 − Q)

(61)


269

Eqs. (60) and (61) completely solve the problem. Given any H (t), these equations determine V (t) and (t) and thus the potential V ( ). As an example, consider a universe with power law expansion a = t n . If it is populated only by a tachyonic scalar .eld, then Q = 0; further, (aH =H ) in Eq. (60) is a constant making ˙ a constant. The complete solution is given by 1=2 2 1=2 1 2 3n2 1− (t) = t + 0 ; V (t) = ; (62) 3n 8G 3n t2 where n ¿ (2=3). Combining the two, we .nd the potential to be 2 1=2 n 1− ( − 0 )−2 : V ( ) = 4G 3n

(63)

For such a potential, it is possible to have arbitrarily rapid expansion with large n. (For the cosmological model, based on this potential, see [159].) A wide variety of phenomenological models with time dependent cosmological constant have been considered in the literature. They involve power law decay of cosmological constant like ˙ t − [171–176,68] or ˙ a− , [177–192], exponential decay ˙ exp(− a) [193] and more complicated models (for a summary, see [68]). Virtually all these models can be reverse engineered and mapped to a scalar .eld model with a suitable V ( ). Unfortunately, all these models lack predictive power or clear particle physics motivation. This discussion also illustrates that even when w(a) is known, it is not possible to proceed further and determine the nature of the source. The explicit examples given above shows that there are at least two di4erent forms of scalar .eld Lagrangians (corresponding to the quintessence or the tachyonic .eld) which could lead to the same w(a). A theoretical physicist, who would like to know which of these two scalar .elds exist in the universe, may have to be content with knowing w(a). The accuracy of the determination of w(a) depends on the prior assumptions made in determining Q, as well as on the observational accuracy with which the quantities H (a) can be measured. Direct observations usually give the luminosity distance dL or angular diameter distance dA . To obtain H (a) from either of these, one needs to calculate a derivative [see, for example, (17)] which further limits the accuracy signi.cantly. As we saw in the last section, this is not easy.

5. Structure formation in the universe The conventional paradigm for the formation of structures in the universe is based on the growth of small perturbations due to gravitational instabilities. In this picture, some mechanism is invoked to generate small perturbations in the energy density in the very early phase of the universe. These perturbations grow due to gravitational instability and eventually form the di4erent structures which we see today. Such a scenario can be constrained most severely by observations of cosmic microwave background radiation (CMBR) at z ≈ 103 . Since the perturbations in CMBR are observed to be small (10−5 –10−4 depending on angular scales), it follows that the energy density perturbations were small compared to unity at the redshift of z ≈ 1000. The central quantity one uses to describe the growth of structures during 0 ¡ z ¡ 103 is the density contrast de.ned as (t; x) = [(t; x) − bg (t)]=bg (t) which characterizes the fractional change in the energy density compared to the background.

270


(Here bg (t) is the mean background density of the smooth universe.) Since one is often interested in the statistical behavior of structures in the universe, it is conventional to assume that and other related quantities are elements of an ensemble. Many popular models of structure formation suggest that the initial density perturbations in the early universe can be represented as a Gaussian random variable with zero mean (that is, = 0) and a given initial power spectrum. The latter quantity is de.ned through the relation P(t; k) = |k (t)|2 where k is the Fourier transform of (t; x) and · · · indicates averaging over the ensemble. It is also conventional to de.ne the two-point correlation function 2 ˙ k 3 P(k) ˙ k (n+3) . Further, when the wavelength of the mode is larger than the Hubble radius, dH (t) = (a=a) ˙ −1 , during the radiation 2 2 4 (n+3) dominated phase, the perturbation grows as a making > ˙ a k . We need to determine how > scales with k when the mode enters the Hubble radius dH (t). The epoch aenter at which this occurs is determined by the relation 2aenter =k = dH . Using dH ˙ t ˙ a2 in the radiation dominated phase, we get aenter ˙ k −1 so that >2 (k; aenter ) ˙ a4enter k (n+3) ˙ k (n−1) :

(74)

It follows that the amplitude of Puctuations is independent of scale k at the time of entering the Hubble radius, only if n = 1. This is the essence of Harrison-Zeldovich and which is independent of the inPationary paradigm. It follows that veri.cation of n = 1 by any observation is not a veri.cation of inPation. At best it veri.es a far deeper principle of scale invariance. We also note that the power spectrum of gravitational potential P scales as P ˙ P=k 4 ˙ k (n−4) . Hence the Puctuation in the gravitational potential (per decade in k) >2 ˙ k 3 P is proportional to >2 ˙ k (n−1) . This Puctuation in the gravitational potential is also independent of k for n = 1 clearly showing the special nature of

274


this choice. [It is not possible to take n strictly equal to unity without specifying a detailed model; the reason has to do with the fact that scale invariance is always broken at some level and this will lead to a small di4erence between n and unity]. Given the above description, the basic model of cosmology is based on seven parameters. Of these 5 parameters (H0 ; +B ; +DM ; + ; +R ) determine the background universe and the two parameters (A; n) specify the initial Puctuation spectrum. It is possible to provide simple analytic .tting functions for the transfer function, incorporating all the above e4ects. For models with a cosmological constant, the transfer function is well .tted by [195] ln2 (1 + 2:34p) [1 + 3:89p + (16:1p)2 + (5:46p)3 + (6:71p)4 ]−1=2 ; (75) (2:34p)2 √ where p = k=(?h Mpc−1 ) and ? = +NR h exp[ − +B (1 + 2h=+NR )] is called the ‘shape factor’. The presence of dark energy, with a constant w, will also a4ect the transfer function and hence the .nal power spectrum. An approximate .tting formula can be given along the following lines [196]. Let the power spectrum be written in the form agQ 2 P(k; a) = AQ k n TQ2 (k) ; (76) gQ; 0 T2 (p) =

where AQ is a normalization, TQ is the modi.ed transfer function and gQ =(D=a) is the ratio between linear growth factor in the presence of dark energy compared to that in + = 1 model. Writing TQ as the product TQ T where T is given by (75), numerical work shows that TQ (k; a) ≡

TQ + q2 = ; T 1 + q2

where k is in Mpc−1 , and by = (−w)s with

q=

k ; ?Q h

(77)

is a scale-independent but time-dependent coeQcient well approximated

s = (0:012 − 0:036w − 0:017=w)[1 − +NR (a)] + (0:098 + 0:029w − 0:085=w)ln +NR (a)

(78)

where the matter density parameter is +NR (a) = +NR =[+NR + (1 − +NR )a−3w ]. Similarly, the relative growth factor can be expressed in the form gQ ≡ (gQ =g ) = (−w)t with t = −(0:255 + 0:305w + 0:0027=w)[1 − +NR (a)] − (0:366 + 0:266w − 0:07=w)ln +NR (a) : (79) Finally the amplitude AQ can be expressed in the form AQ = 2H (c=H0 )n+3 =(4), where H = 2 × 10−5 and

0

−1

0

(+NR )c1 +c2 ln +NR exp[c3 (n − 1) + c4 (n − 1)2 ]

(80)

= (a = 1) of Eq. (78), and

c1 = −0:789|w|0:0754−0:211 ln |w| ; c3 = −1:037;

c4 = −0:138 :

This .t is valid for −1 . w . −0:2.

c2 = −0:118 − 0:0727w ; (81)


275

5.2. Nonlinear growth of perturbations In a purely matter dominated universe, Eq. (64) reduces to RX = −GM=R2 . Solving this equation one can obtain the nonlinear density contrast as a function of the redshift z: 2=3 2=3 4 5 4 i (1 + zi ) 0 = ; (1 + z) = 2=3 3 ($ − sin $) 3 3 ($ − sin $)2=3 =

9 ($ − sin $)2 −1 : 2 (1 − cos $)3

(82)

(83)

Here, i ¿ 0 is the initial density contrast at the redshift zi and 0 is the density contrast at present if the initial density contrast was evolved by linear approximation. In general, the linear density contrast L is given by WL 3 −1= L = b 5

2=3 3 ($ − sin $)2=3 : 4

(84)

When $ = (2=3); L = 0:568 and = 1:01 1. If we interpret = 1 as the transition point to nonlinearity, then such a transition occurs at $=(2=3), L 0:57. From (82), we see that this occurs at the redshift (1+znl )=(0 =0:57). The spherical region reaches the maximum radius of expansion at $ = . This corresponds to a density contrast of m ≈ 4:6 which is de.nitely in the nonlinear regime. The linear evolution gives L = 1:063 at $ = . After the spherical over dense region turns around it will continue to contract. Eq. (83) suggests that at $ = 2 all the mass will collapse to a point. A more detailed analysis of the spherical model [200], however, shows that the virialized systems formed at any given time have a mean density which is typically 200 times the background density of the universe at that time in a +NR = 1. This occurs at a redshift of about (1 + zcoll ) = (0 =1:686). The density of the virialized structure will be approximately coll 1700 (1 + zcoll )3 where 0 is the present cosmological density. The evolution is described schematically in Fig. 16. In the presence of dark energy, one cannot ignore the second term in Eq. (64). In the case of a cosmological constant, w=−1 and =constant and this extra term is independent of time. This allows one to obtain the .rst integral to Eq. (64) and reduce the problem to quadrature (see, for example [201–203]). For a more general case of constant w with w = −1, the factor ( + 3P) = (1 + 3w) will be time dependent because will be time dependent even for a constant w if w = −1. In this case, one cannot obtain an energy integral for Eq. (64) and the dynamics has to be determined by actual numerical integration. Such an analysis leads to the following results [190,204,205]: (i) In the case of matter dominated universe, it was found that the linear theory critical threshold for collapse, c , was about 1.69. This changes very little in the presence of dark energy and an accurate .tting function is given by c =

3(12)2=3 [1 + log10 +NR ] ; 20

= 0:353w4 + 1:044w3 + 1:128w2 + 0:555w + 0:131 :

(85)

276

T. Padmanabhan / Physics Reports 380 (2003) 235 – 320 θ = (2 π/3)

θ=π

θ=2π

Linear

Non-Linear

density of the perturbed region Bound Structures

Density

Turn Around

8

2

5.5 background density 1.87 a-3

1.6 a nl

a max

a coll

Expansion Factor

Fig. 16. Evolution of an over dense region in spherical top-hat approximation.

(ii) The over density of a virialized structure as a function of the redshift of virialization, however, depends more sensitively on the dark energy component. For −1 6 w 6 − 0:3, this can be .tted by the function >vir (z) = 182 [1 + a@b (z)] ;

(86)

where a = 0:399 − 1:309(|w|0:426 − 1);

b = 0:941 − 0:205(|w|0:938 − 1) ;

(87)

and @(z) = 1=+NR (z) − 1 = (1=+0 − 1)(1 + z)3w . The importance of c and >vir arises from the fact that these quantities can be used to study the abundance of nonlinear bound structures in the universe. The basic idea behind this calculation [206] is as follows: Let us consider a density .eld R (x) smoothed by a window function WR of scale radius R. As a .rst approximation, we may assume that the region with (R; t) ¿ c (when smoothed on the scale R at time t) will form a gravitationally bound object with mass M ˙ R W 3 by the time t. The precise form of the M − R relation depends on the window function used; for a step function M = (4=3)R W 3 , while for a Gaussian M = (2)3=2 R W 3 . Here c is a critical value for the density contrast given by (85). Since (t) = D(t) for the growing mode, the probability for the region to form a bound structure at t is the same as the probability ¿ c [D(ti )=D(t)] at some early epoch ti . This probability can be easily estimated since at su=ciently early ti , the system is described by a Gaussian random .eld. This fact


can be used to calculate the number density of bound objects leading to the result 2c W 2 1=2 c 96 exp − dM : N (M ) dM = − M 62 9M 262

277

(88)

The quantity 6 here refers to the linearly extrapolated density contrast. We shall now describe the constraints on dark energy arising from structure formation. 5.3. Structure formation and constraints on dark energy Combining the initial power spectrum, P(k) = Ak n , n ≈ 1, with the transfer function in (75) we .nd that the .nal spectrum has the approximate form −3 2 Ak ln k (kkeq ) (89) P(k) ˙ Ak (kkeq ) −1 ≈ d (z ) ≈ 13(+ h2 )−1 Mpc = 13(?h)−1 h−1 Mpc [see Eq. (68)] where ? ≡ + h is with 2keq H eq NR NR the shape parameter (see Eq. (75); we have assumed +B ≈ 0 for simplicity.) From Eq. (89), it is clear that P(k) changes from an increasing function to a decreasing function at keq , the numerical value of which is decided by the shape parameter ?. Smaller values of +NR and ? will lead to more power at longer wavelengths. One of the earliest investigations which used power spectrum to determine + was based on the APM galaxy survey [207]. This work showed that the existence of large scale power requires a nonzero cosmological constant. This result was con.rmed when the COBE observations .xed the amplitude of the power spectrum unequivocally (see Section 6). It was pointed out in [208,209] that the COBE normalization led to a wrong shape for the power spectrum if we take +NR = 1; + = 0, with more power at small scales than observed. This problem could be solved by reducing +NR and changing the shape of the power spectrum. Current observations favour ? ≈ 0:25. In fact, an analysis of a host of observational data, including those mentioned above suggested [210] that + = 0 even before the SN data came up. Another useful constraint on the models for structure formation can be obtained from the abundance of rich clusters of galaxies with masses M ≈ 1015 M . This mass scale corresponds to a length scale of about 8h−1 Mpc and hence the abundance of rich clusters is sensitive to the root-mean-square Puctuation in the density contrast at 8h−1 Mpc. It is conventional to denote this quantity (=)2 1=2 , evaluated at 8h−1 Mpc, by 68 . To be consistent with the observed abundance of rich clusters, −1=2 Eq. (88) requires 68 ≈ 0:5+NR . This is consistent with COBE normalization for +NR ≈ 0:3; + ≈ 0:7. [Unfortunately, there is still some uncertainty about the 68 − +NR relation. There is a −0:6 claim [211] that recent analysis of SDSS data gives 68 ≈ 0:33 ± 0:03+NR .] The e4ect of dark energy component on the growth of linear perturbations changes the value of 68 . The results of Section 5.1 translate into the .tting function [190]

68 = (0:50 − 0:1@)+−B(+; @) ;

(90)

where @ = (n − 1) + (h − 0:65) and B(+; @) = 0:21 − 0:22w + 0:33+ + 0:25@. For constant w models with w =−1; −2=3 and −1=3, this gives 68 =0:96; 0:80 and 0.46, respectively. Because of this

278


e4ect, the abundance of clusters can be used to put stronger constraints on cosmology when the data for high redshift clusters improves. As mentioned before, linear perturbations grow more slowly in a universe with cosmological constant compared to the +NR = 1 universe. This means that clusters will be comparatively rare at high redshifts in a +NR = 1 universe compared to models with cosmological constant. Only less than 10 percent of massive clusters form at z ¿ 0:5 in a +NR = 1 universe whereas almost all massive clusters would have formed by z ≈ 0:5 in a universe with cosmological constant [212–215,75]. (A simple way of understanding this e4ect is by noting that if the clusters are not in place by z ≈ 0:5, say, they could not have formed by today in models with cosmological constant since there is very little growth of Puctuation between these two epochs.) Hence the evolution of cluster population as a function of redshift can be used to discriminate between these models. An indirect way of measuring this abundance is through the lensing e4ect of a cluster of galaxy on extended background sources. Typically, the foreground clusters shears the light distribution of the background object and leads to giant arcs. Numerical simulations suggest [216] that a model with +NR = 0:3; + = 0:7 will produce about 280 arcs which is nearly an order of magnitude larger than the number of arcs produced in a +NR = 1; + = 0 model. (In fact, an open model with +NR = 0:3; + = 0 will produce about 2400 arcs.) To use this e4ect, one needs a well de.ned data base of arcs and a controlled sample. At present it is not clear which model is preferred though this is one test which seems to prefer open model rather than a -CDM model. Given the solution to (64) in the presence of dark energy, we can repeat the above analysis and obtain the abundance of di4erent kinds of structures in the universe in the presence of dark energy. In particular this formalism can be used to study the abundance of weak gravitational lenses and virialized X-ray clusters which could act as gravitational lenses. The calculations again show [205] that the result is highly degenerate in w and +NR . If +NR is known, then the number count of weak lenses will be about a factor 2 smaller for w = −2=3 compared to the CDM model with a cosmological constant. However, if +NR and w are allowed to vary in such a way that the matter power spectrum matches with both COBE results and abundance of X-ray clusters, then the predicted abundance of lenses is less than 25 percent for −1 6 w 6 − 0:4. It may be possible to constrain the dark energy better by comparing relative abundance of virialized lensing clusters with the abundance of X-ray under luminous lensing halos. For example, a survey covering about 50 square degrees of sky may be able to di4erentiate a CDM model from w = −0:6 model at a 36 level. The value of 68 and cluster abundance can also be constrained via the Sunyaev–Zeldovich (S–Z) e4ect which is becoming a powerful probe of cosmological parameters [217]. The S–Z angular power spectrum scales as 687 (+B h)2 and is almost independent of other cosmological parameters. Recently the power spectrum of CMBR determined by CBI and BIMA experiments (see Section 6) showed an excess at small scales which could be interpreted as due to S–Z e4ect. If this interpretation is correct, then 68 (+B h=0:035)0:29 = 1:04 ± 0:12 at 95 percent con.dence level. This 68 is on the higher side and only future observations can decide whether the interpretation is correct or not. The WMAP data, for example, leads to a more conventional value of 68 = 0:84 ± 0:04. Constraints on cosmological models can also arise from the modeling of damped Lyman- systems [75,109,110,218–220] when the observational situation improves. At present these observations are consistent with +NR = 0:3; + = 0:7 model but do not exclude other models at a high signi.cance level.


279

Finally, we comment on a direct relation between (a) and H (a). Expressing Eq. (65) in terms of H (a) will lead to the form a2 H 2 + (3H 2 + aHH )a =

3 H02 +NR 4 a2 H 2 2 (1 + ) + : 2 a3 3 (1 + )

(91)

This can be used to determine H 2 (a) from (a) since this equation is linear and .rst order in Q(a) ≡ H 2 (a) (though it is second order in ). Rewriting it in the form A(a)Q + B(a)Q = C(a) ; where

A=

1 2 a ; 2

(92)

4 a 2 B = 3a + a − 3 1+

2

We can integrate it to give the solution a (1 + )8=3 2 2 da H (a) = 3H0 +NR : (1 + )5=3 a6 2

;

C=

3 H02 +NR (1 + ) : 2 a3

(93)

(94)

This shows that, given the nonlinear growth of perturbations (a) as a function of redshift and the approximate validity of spherical model, one can determine H (a) and thus w(a) even during the nonlinear phases of the evolution. [A similar analysis with the linear equation (66) was done in [221], leading to the result which can be obtained by expanding (94) to linear order in .] Unfortunately, this is an impractical method from observational point of view at present. 6. CMBR anisotropies In the standard Friedmann model of the universe, neutral atomic systems form at a redshift of about z ≈ 103 and the photons decouple from the matter at this redshift. These photons, propagating freely in spacetime since then, constitute the CMBR observed around us today. In an ideal Friedmann universe, for a comoving observer, this radiation will appear to be isotropic. But if physical process has led to inhomogeneities in the z = 103 spatial surface, then these inhomogeneities will appear as angular anisotropies in the CMBR in the sky today. A physical process operating at a proper length scale L on the z = 103 surface will lead to an e4ect at an angle $ = L=dA (z). Numerically, + Lz :0 ∼ : (95) = 34:4 (+h) $(L) = 2 1 Mpc H0−1 To relate the theoretical predictions to observations, it is usual to expand the temperature anisotropies in the sky in terms of the spherical harmonics. The temperature anisotropy in the sky will provide > = [T=T as a function of two angles $ and . If we expand the temperature anisotropy distribution on the sky in spherical harmonics: ∞

[T >($; ) ≡ alm Ylm ($; ) : ($; ) = T l; m

(96)

280


all the information is now contained in the angular coeQcients alm . If n and m are two directions in the sky with an angle between them, the two-point correlation function of the temperature Puctuations in the sky can be de.ned as C( ) = S(n)S(m) = alm a∗l m Ylm (n)Yl∗ m (m) : (97) Since the sources of temperature Puctuations are related linearly to the density inhomogeneities, the coeQcients alm will be random .elds with some power spectrum. In that case alm a∗l m will be nonzero only if l = l and m = m . Writing alm a∗l m = Cl ll mm

(98)

and using the addition theorem of spherical harmonics, we .nd that C( ) =

(2l + 1) l

4

Cl Pl (cos )

(99)

with Cl = |alm |2 . In this approach, the pattern of anisotropy is contained in the variation of Cl with l. Roughly speaking, l ˙ $ −1 and we can think of the ($; l) pair as analogue of (x; k) variables in 3-D. The Cl is similar to the power spectrum P(k). In the simplest scenario, the primary anisotropies of the CMBR arise from three di4erent sources. (i) The .rst is the gravitational potential Puctuations at the last scattering surface (LSS) which will contribute an anisotropy ([T=T )2 ˙ k 3 P (k) where P (k) ˙ P(k)=k 4 is the power spectrum of gravitational potential . This anisotropy arises because photons climbing out of deeper gravitational wells lose more energy on the average. (ii) The second source is the Doppler shift of the frequency of the photons when they are last scattered by moving electrons on the LSS. This is proportional to ([T=T )2D ˙ k 3 Pv where Pv (k) ˙ P=k 2 is the power spectrum of the velocity .eld. (iii) Finally, we also need to take into account the intrinsic Puctuations of the radiation .eld on the LSS. In the case of adiabatic Puctuations, these will be proportional to the density Puctuations of matter on the LSS and hence will vary as ([T=T )2int ˙ k 3 P(k). Of these, the velocity .eld and the density .eld (leading to the Doppler anisotropy and intrinsic anisotropy described in (ii) and (iii) above) will oscillate at scales smaller than the Hubble radius at the time of decoupling since pressure support will be e4ective at these scales. At large scales, if P(k) ˙ k, then [T 2 [T 2 [T 2 2 −2 ˙ constant; ˙k ˙$ ; ˙ k 4 ˙ $ −4 ; (100) T T D T int where $ ˙ : ˙ k −1 is the angular scale over which the anisotropy is measured. Obviously, the Puctuations due to gravitational potential dominate at large scales while the sum of intrinsic and Doppler anisotropies will dominate at small scales. Since the latter two are oscillatory, we sill expect an oscillatory behavior in the temperature anisotropies at small angular scales. There is, however, one more feature which we need to take into account. The above analysis is valid if recombination was instantaneous; but in reality the thickness of the recombination epoch is about [z 80 ([222,44, Chapter 3]). This implies that the anisotropies will be damped at scales smaller than the length scale corresponding to a redshift interval of [z = 80. The typical value for the peaks of the oscillation are


281

at about 0:3◦ to 0:5◦ depending on the details of the model. At angular scales smaller than about 0:1◦ , the anisotropies are heavily damped by the thickness of the LSS. The fact that several di4erent processes contribute to the structure of angular anisotropies make CMBR a valuable tool for extracting cosmological information. To begin with, the anisotropy at very large scales directly probe modes which are bigger than the Hubble radius at the time of decoupling and allows us to directly determine the primordial spectrum. Thus, in general, if the angular dependence of the spectrum at very large scales is known, one can work backwards and determine the initial power spectrum. If the initial power spectrum is assumed to be P(k) = Ak, then the observations of large angle anisotropy allows us to .x the amplitude A of the power spectrum [208,209]. Based on the results of COBE satellite [223], one .nds that the amount of initial power per logarithmic band in k space is given by 4 k k 3 |k |2 Ak 4 ∼ 2 > (k) = = 2 = (101) 22 2 0:07hMpc−1 (This corresponds to A (29h−1 Mpc)4 . Since the actual ([T=T ) is one realization of a Gaussian random process, the observed small-l results are subject to unavoidable Puctuations called the ‘cosmic variance’.) This result is powerful enough to rule out matter dominated, +=1 models when combined with the data on the abundance of large clusters which determines the amplitude of the power spectrum at R ≈ 8h−1 Mpc. For example the parameter values h = 0:5; +0 ≈ +DM = 1; + = 0, are ruled out by this observation when combined with COBE observations [208,209]. As we move to smaller scales we are probing the behavior of baryonic gas coupled to the photons. The pressure support of the gas leads to modulated acoustic oscillations with a characteristic wavelength at the z = 103 surface. Regions of high and low baryonic density contrast will lead to anisotropies in the temperature with the same characteristic wavelength. The physics of these oscillations has been studied in several papers in detail [224–232]. The angle subtended by the wavelength of these acoustic oscillations will lead to a series of peaks in the temperature anisotropy which has been detected [233,234]. The structure of acoustic peaks at small scales provides a reliable procedure for estimating the cosmological parameters. To illustrate this point let us consider the location of the .rst acoustic peak. Since all the Fourier components of the growing density perturbation start with zero amplitude at high redshift, the condition for a mode with a given wave vector k to reach an extremum amplitude at t = tdec is given by tdec n kcs dt ; (102) a 2 0 √ where cs = (9P=9)1=2 ≈ (1= 3) is the speed of sound in the baryon–photon Puid. At high redshifts, −1=2 t(z) ˙ +NR (1 + z)−3=2 and the proper wavelength of the .rst acoustic peak scales as :peak ∼ tdec ˙ −1=2 . The angle subtended by this scale in the sky depends on dA . If +NR + + = 1 then the h−1 +NR −0:4 −1 angular diameter distance varies as +NR while if + =0, it varies as +NR . It follows that the angular size of the acoustic peak varies with the matter density as 1=2 +NR (if + = 0) ; zdec :peak ˙ $peak ∼ (103) a0 r +−0:1 (if + + +NR = 1) : NR

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Fig. 17. The variation of the anisotropy pattern in universes with +NR =(0:25; 0:45; 1:0; 1:15); + =0 with the .rst acoustic peak moving from right to left. The y-axis is essentially a measure of ([T=T )2 while the x-axis is a measure of 1=$. (.gure courtesy: S. Sethi).

Therefore, the angle subtended by acoustic peak is quite sensitive to +NR if = 0 but not if +NR + + = 1. More detailed computations show that the multipole index corresponding to the −1=2 acoustic peak scales as lp ≈ 220+NR if = 0 and lp ≈ 220 if +NR + + = 1 and 0:1 . +NR . 1. This is illustrated in Fig. 17 which shows the variation in the structure of acoustic peaks when + is changed keeping + = 0. The four curves are for + = +NR = 0:25; 0:45; 1:0; 1:15 with the .rst acoustic peak moving from right to left. The data points on the .gures are from the .rst results of MAXIMA and BOOMERANG experiments and are included to give a feel for the error bars in current observations. It is obvious that the overall geometry of the universe can be easily .xed by the study of CMBR anisotropy. The heights of acoustic peaks also contain important information. In particular, the height of the .rst acoustic peak relative to the second one depends sensitively on +B . However, not all cosmological parameters can be measured independently using CMBR data alone. For example, di4erent models with the same values for (+DM + + ) and +B h2 will give anisotropies which are fairly indistinguishable. The structure of the peaks will be almost identical in these models. This shows that while CMBR anisotropies can, for example, determine the total energy density (+DM ++ ), we will need some other independent cosmological observations to determine individual components. At present there exists several observations of the small scale anisotropies in the CMBR from the balloon Pights, BOOMERANG [233], MAXIMA [234], and from radio telescopes CBI [235], VSA [236], DASI [237,238] and—most recently—from WMAP [235]. These CMBR data have been extensively analyzed [239,59,60,223,235,236,240–245] in isolation as well as in combination with


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other results. (The information about structure formation arises mainly from galaxy surveys like SSRS2, CfA2 [246], LCRS [247], Abell-ACO cluster survey [248], IRAS-PSC z [243], 2-D survey [249,242] and the Sloan survey [250].) While there is some amount of variations in the results, by and large, they support the following conclusions. 0:03 from the • The data strongly supports a k = 0 model of the universe [245] with +tot = 1:00±0:02 pre-MAP data and +tot = 1:02 ± 0:02 from the WMAP data. • The CMBR data before WMAP, when combined with large scale structure data, suggest +NR = 0:29±0:05±0:04 [59,60,245,251]. The WMAP result [239] is consistent with this giving 0:27±0:04. The initial power spectrum is consistent with being scale invariant and the pre-MAP value is n = 1:02 ± 0:06 ± 0:05 [59,60,245]. The WMAP gives the spectral index at k = 0:05 Mpc−1 to be 0:93±0:03. In fact, combining 2dF survey results with CMBR suggest [252] + ≈ 0:7 independent of the supernova results. • A similar analysis based on BOOMERANG data leads to +tot =1:02±0:06 (see for example, [241]). Combining this result with the HST constraint [49] on the Hubble constant h = 0:72 ± 0:08, galaxy 0:09 0:10 clustering data as well SN observations one gets + = 0:620:10 −0:18 ; + = 0:55−0:09 and + = 0:73−0:07 respectively [253]. The WMAP data gives h = 0:71+0:04 −0:03 . • The analysis also gives an independent handle on baryonic density in the universe which is consistent with the BBN value: The pre-MAP result was +B h2 = 0:022 ± 0:003 [59,60]. (This is gratifying since the initial data had an error and gave too high a value [254].) The WMAP data gives +B h2 = 0:0224 ± 0:0009.

There has been some amount of work on the e4ect of dark energy on the CMBR anisotropy [255–263]. The shape of the CMB spectrum is relatively insensitive to the dark energy and the main e4ect is to alter the angular diameter distance to the last scattering surface and thus the position of the .rst acoustic peak. Several studies have attempted to put a bound on w using the CMB observations. Depending on the assumptions which were invoked, they all lead to a bound broadly in the range of w . −0:6. (The preliminary analysis of WMAP data in combination with other astronomical data sets suggest w ¡ − 0:78 at 95 per cent con.dence limit.) At present it is not clear whether CMBR anisotropies can be of signi.cant help in discriminating between di4erent dark energy models.

7. Reinterpreting the cosmological constant It is possible to attack the cosmological constant problem from various other directions in which the mathematical structure of Eq. (3) is reinterpreted di4erently. Though none of these ideas have been developed into a successful formal theory, they might contain ingredients which may eventually provide a solution to this problem. Based on this hope, we shall provide a brief description of some of these ideas. (In addition to these ideas, there is extensive literature on several di4erent paradigms for attacking the cosmological constant problem based on: (i) Quantum .eld theory in curved spacetime [264–266], (ii) quantum cosmological considerations [267], (iii) models of inPation [268], (iv) string theory inspired ideas [269], and (v) e4ect of phase transitions [270].)

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7.1. Cosmological constant as a Lagrange multiplier The action principle for gravity in the presence of a cosmological constant √ 1 (R − 2) −g d 4 x A= 16G √ √ 1 4 R −g d x − = −g d 4 x (104) 16G 8G can be thought of as a variational principle extremizing the integral over R, subject to the condition that the 4-volume of the universe remains constant. To implement the constraint that the 4-volume is a constant, one will add a Lagrange multiplier term which is identical in structure to the second term in the above equation. Hence, mathematically, one can think of the cosmological constant as a Lagrange multiplier ensuring the constancy of the 4-volume of the universe when the metric is varied. If we take this interpretation seriously, then it is necessary to specify the 4-volume of the universe before the variation is performed and determine the cosmological constant so that the 4-volume has this speci.ed volume. A Friedmann model with positive cosmological constant in Minkowski space will lead to a .nite 3-volume proportional to −3=2 on spatial integration. (To achieve this, we should use the coordinates in which the spatial sections are closed 3-spheres.) The time integration, however, has an arbitrary range and one needs to restrict the integration to part of this range by invoking some physical principle. If we take this to be typically the age of the universe, then we will obtain a time dependent cosmological constant (t) with (t)H (t)−2 remaining of order unity. While this appears to be a conceptually attractive idea, it is not easy to implement it in a theoretical model. In particular, it is diQcult to obtain this as a part of a generally covariant theory incorporating gravity. 7.2. Cosmological constant as a constant of integration Several people have suggested modifying the basic structure of general relativity so that the cosmological constant will appear as a constant of integration. This does not solve the problem in the sense that it still leaves its value undetermined. But this changes the perspective and allows one to think of the cosmological constant as a nondynamical entity [271,272]. One simple way of achieving this is to assume that the determinant g of gab is not dynamical and admit only those variations which obey the condition gab gab = 0 in the action principle. This is equivalent to eliminating the trace part of Einstein’s equations. Instead of the standard result, we will now be led to the equation 1 i 1 i i i Rk − k R = 8G Tk − k T ; (105) 4 4 which is just the traceless part of Einstein’s equation. The general covariance of the action, however, implies that T;bab =0 and the Bianchi identities (Rik − 12 ik R);i =0 continue to hold. These two conditions imply that 9i R = −8G9i T requiring R + 8GT to be a constant. Calling this constant (−4) and combining with equation (105), we get Rik − 12 ik R − ik = 8GTki

(106)


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which is precisely Einstein’s equation in the presence of cosmological constant. In this approach, the cosmological constant has nothing to do with any term in the action or vacuum Puctuations and is merely an integration constant. Like any other integration constant its value can be .xed by invoking suitable boundary conditions for the solutions. There are two key diQculties in this approach. The .rst, of course, is that it still does not give us any handle on the value of the cosmological constant and all the diQculties mentioned earlier still exists. This problem would have been somewhat less serious if the cosmological constant was strictly zero; the presence of a small positive cosmological constant makes the choice of integration constant fairly arbitrary. The second problem is in interpreting the condition that g must remain constant when the variation is performed. It is not easy to incorporate this into the logical structure of the theory. (For some attempts in this direction, see [273].) 7.3. Cosmological constant as a stochastic variable Current cosmological observations can be interpreted as showing that the e2ective value of (which will pick up contributions from all vacuum energy densities of matter .elds) has been 2 −2 2 reduced from the natural value of L− P to LP (LP H0 ) where H0 is the current value of the Hubble constant. One possible way of thinking about this issue is the following [274]: Let us assume that the quantum micro structure of spacetime at Planck scale is capable of readjusting itself, soaking up any vacuum energy density which is introduced—like a sponge soaking up water. If this process is fully deterministic and exact, all vacuum energy densities will cease to have macroscopic gravitational e4ects. However, since this process is inherently quantum gravitational, it is subject to quantum Puctuations at Planck scales. Hence, a tiny part of the vacuum energy will survive the process and will lead to observable e4ects. One may conjecture that the cosmological constant we measure corresponds to this small residual Puctuation which will depend on the volume of the spacetime 2 2 2 region that is probed. It is small, in the sense that it has been reduced from L− to L− P P (LP H0 ) , which indicates the fact that Puctuations—when measured over a large volume—is small compared to the bulk value. It is the wetness of the sponge we notice, not the water content inside. This is particularly relevant in the context of standard discussions of the contribution of zero-point energies to cosmological constant. The correct theory is likely to regularize the divergences and make 4 the zero point energy .nite and about L− P . This contribution is most likely to modify the microscopic structure of spacetime (e.g. if the spacetime is naively thought of as due to stacking of Planck scale volumes, this will modify the stacking or shapes of the volume elements) and will not a4ect the bulk gravitational .eld when measured at scales coarse grained over sizes much bigger than the Planck scales. Given a large 4-volume V of the spacetime, we will divide it into M cubes of size ([x)4 and label the cubes by n = 1; 2; : : : ; M . The contribution to the path integral amplitude A, describing long wavelength limit of conventional Einstein gravity, can be expressed in the form

√ ic1 2 i([x)4 =L4P A= d 4 x −g(RL2P ) ; [exp(c1 (RLP ) + · · ·)] → exp 4 (107) LP n where we have indicated the standard continuum limit. (In conventional units c1 = (16)−1 .) Let us now ask how one could modify this result to describe the ability of spacetime micro structure to readjust itself and absorb vacuum energy densities. This would require some additional dynamical

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degree of freedom that will appear in the path integral amplitude and survive in the classical limit. It can be shown that [274] the simplest implementation of this feature is by modifying the standard path integral amplitude [exp(c1 (RL2P ) + · · ·)] by a factor [ (x n )= 0 ] where (x) is a scalar degree of freedom and 0 is a pure number introduced to keep this factor dimensionless. In other words, we modify the path integral amplitude to the form (i([x)4 =L4P )

(x n ) [c1 RL2P +···] e : (108) Amodify = 0 n In the long wavelength limit, the extra factor in (108) will lead to a term of the form

i([x)4 =L4P i([x)4 = exp ln 4 0 0 LP n n i 4 √ d x −g ln : → exp 4 0 LP

(109)

4 Thus, the net e4ect of our assumption is to introduce a ‘scalar .eld potential’ V ( ) = −L− P ln( = 0 ) in the semi classical limit. It is obvious that the rescaling of such a scalar .eld by → q 4 is equivalent to adding a cosmological constant with vacuum energy −L− P ln q. Alternatively, any vacuum energy can be reabsorbed by such a rescaling. The fact that the scalar degree of freedom occurs as a potential in (109) without a corresponding kinetic energy term shows that its dynamics is unconventional and nonclassical. The above description in terms of macroscopic scalar degree of freedom can, of course, be only approximate. Treated as a vestige of a quantum gravitational degrees of freedom, the cancellation cannot be precise because of Puctuations in the elementary spacetime volumes. These Puctuations will reappear as a “small” cosmological constant because of two key ingredients: (i) discrete spacetime structure at Planck length and (ii) quantum gravitational uncertainty principle. To show this, we use the fact noted earlier in Section 7.1 that the net cosmological constant can be thought of as a Lagrange multiplier for proper volume of spacetime in the action functional for gravity. In any quantum cosmological models which leads to large volumes for the universe, phase of the wave function will pick up a factor of the form e4 V E ˙ exp(−iA0 ) ˙ exp −i (110) 8L2P

from (104), where V is the four volume. Treating (e4 =8L2P ; V) as conjugate variables (q; p), we can invoke the standard uncertainty principle to predict [ ≈ 8L2P =[V. Now we use the earlier assumption regarding the microscopic structure of the spacetime: Assume that there is a zero point length of the order of LP so that the volume of the universe is made of a large number (N ) of cells, each of volume ( √ LP )4 where is a numerical constant. Then V = N ( LP )4 , implying a Poisson Puctuation [V ≈ V( LP )2 and leading to 8 1 8 8L2P √ ≈ 2 H02 : = (111) [ = 2 [V V


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√ This will give + = (8=3 2 ) which will—for example—lead to + = (2=3) if = 2 . Thus Planck length cuto4 (UV limit) and volume of the universe (IR limit) combine to give the correct [. (A similar result was obtained earlier in [273] based on a di4erent model.) The key idea, in this approach, is that is a stochastic variable with a zero mean and Puctuations. It is the rms Puctuation which is being observed in the cosmological context. This has three implications: First, FRW equations now need to be solved with a stochastic term on the right hand side and one should check whether the observations can still be explained. The second feature is that stochastic properties of need to be described by a quantum cosmological model. If the quantum state of the universe is expanded in terms of the eigenstates of some suitable operator (which does not commute the total four volume operator), then one should be able to characterize the Puctuations in each of these states. Third, and most important, the idea of a cosmological constant arising as a >uctuation makes sense only if the bulk value is rescaled away. The nontriviality of this result becomes clear when we compare it with few other alternative ways of estimating the Puctuations—none of which gives the correct result. The .rst alternative approach is based on the assumption that one can associate an entropy S = (AH =4L2P ) with compact space time horizons of area AH (We will discuss this idea in detail in Section 10). A popular interpretation of this result is that horizon areas are quantized in units of L2P so that S is proportional to the number of bits of information contained in the horizon area. In this approach, horizon areas can be expressed in 2 the form AH = AP N where AP ˙ √ LP is√a quantum of area and N−1is an integer. Then the >uctuations in the area will be [AH = AP N = AP AH . Taking AH ˙ for the de Sitter horizon, we .nd that [ ˙ H 2 (HLP ) which is a lot smaller than what one needs. Further, taking AH ˙ rH2 , we .nd that [rH ˙ LP ; that is, this result essentially arises from the idea that the radius of the horizon is uncertain within one Planck length. This is quite true, of course, but does not lead to large enough Puctuations. A more sophisticated way of getting this (wrong) result is to relate the Puctuations in the cosmological constant to that of the volume of the universe is by using a canonical ensemble description for universes of proper Euclidean 4-volume [275]. Writing V ≡ V=8L2P and treating V and as the relevant variables, one can write a partition function for the 4-volume as ∞ Z(V ) = g()e−V d : (112) 0

Taking the analogy with standard statistical mechanics (with the correspondence V → G and → E), we can evaluate the Puctuations in the cosmological constant in exactly the same way as energy Puctuations in canonical ensemble. (This is done in several standard text books; see, for example, [276, p. 194].) This will give ([)2 =

C ; V2

C=

9 9 = −V 2 ; 9(1=V ) 9V

(113)

where C is the analogue of the speci.c heat. Taking the 4-volume of the universe to be V=bH −4 = 1 −1=2 9b−2 where b is a numerical factor and using V = (V=8L2P ) we get ˙ L− . It follows P V from (113) that ([)2 =

C 12 (LP H 3 )2 : = 2 V b

(114)

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In other words [ ˙ H 2 (HLP ), which is the same result from area quantization and is a lot smaller than the cosmologically signi.cant value. Interestingly enough, one could do slightly better by assuming that the horizon radius is√quantized in units of Planck length, so that rH =H −1 =NLP . This will lead to the Puctuations [rH = rH LP and using rH =H −1 ˙ −1=2 , we get [ ˙ H 2 (HLP )1=2 — larger than (114) but still inadequate. In summary, the existence of two length scales H −1 and LP allows di4erent √ √ results for [ √ depending on how exactly the Puctuations are characterized ([V ˙ N ; [A ˙ N or [rH ˙ N ). Hence the result obtained above in (111) is nontrivial. These conclusions stress, among other things, the di4erence between >uctuations and the mean values. For, if one assumes that every patch of the universe with size LP contained an energy EP , then a universe with characteristic size H −1 will contain the energy E = (EP =LP )H −1 . The corresponding energy density will be = (E=H −3 ) = (H=LP )2 which leads to the correct result. But, of course, we do not know why every length scale LP should contain an energy EP and—more importantly—contribute coherently to give the total energy. 7.4. Anthropic interpretation of the cosmological constant The anthropic principle [277,278] is an interpretational paradigm which argues that, while discussing the origin of physical phenomena and the values of constants of nature, we must recognize the fact that only certain combination and range of values will lead to the existence of intelligence observers in the universe who could ask questions related to these issues. This paradigm has no predictive power in the sense that none of the values of the cosmological parameters were ever predicted by this method. 1 In fact some cosmologists have advocated the model with +NR = 1; + = 0 strongly and later—when observations indicated + = 1—have advocated the anthropic interpretation of cosmological constant with equal Puency. This is defended by the argument that not all guiding principles in science (Darwinian evolution, Plate tectonics, etc.) need to be predictive in order to be useful. In this view point, anthropic principle is a back drop for discussing admittedly complicated conceptual issues. Within this paradigm there have been many attempts to explain (after the fact) the values of several fundamental constants with varying degree of success. In the context of cosmological constant, the anthropic interpretation works as follows. It is assumed that widely disparate values for the constants of nature can occur in an ensemble of universes (or possibly in di4erent regions of the universe causally unconnected with each other). Some of these values for constants of nature—and in particular for the cosmological constant—will lead broadly to the kind of universe we seem to live in. This is usually characterized by formation of: (i) structures by gravitational instability, (ii) stars which act as gravitationally bound nuclear reactors that synthesize the elements and distribute them and (iii) reasonably complex molecular structures which could form the basis for some kind of life form. Showing that such a scenario can exist only for a particular range of values for the cosmological constant is considered an explanation for the value of cosmological constant by the advocates of anthropic principle. (More sophisticated versions of this principle exist; see, for example [279], and references cited therein.) 1

Some advocates of the anthropic principle cite Fred Hoyle predicting the existence of excited state of carbon nucleus, thereby leading to eQcient triple alpha reaction in stellar nucleosynthesis, as an example of a prediction from anthropic principle; it is very doubtful whether Hoyle applied anthropic considerations in arriving at this conclusion.


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The simplest constraint on the cosmological constant is that it should not be so high as to cause rapid expansion of the universe early on preventing the formation of galaxies [280]. If the energy density of the cosmological constant has to be less than that of energy density of matter at the redshift zgal (≈ 4) at which galaxy formation takes place, then we must have + . (1 + zgal )3 ≈ 125 : +NR

(115)

This gives a bound on + which is “only” a couple of orders of magnitude larger than what is observed. More formally, one could ask: What is the most probable value of + if it is interpreted as the value that would have been observed by the largest number of observers [281,282]? Since a universe with + ≈ +NR will have more galaxies than one with a universe with + ≈ 102 +NR , one could argue that most observers will measure a value + ≈ +NR . The actual probability dP for measuring a particular value for + in the range (+ ; + +d+ ) is the product (dP=d+ )=Q(+ )P(+ ) where P is the a priori probability measure for a speci.c value of + in a member of an ensemble of universes (or in a region of the universe) and Q(+ ) is the average number of galaxies which form in a universe with a given value of + . There has been several attempts to estimate these quantities (see, for example, [283,284]) but all of them are necessarily speculative. The .rst—and the most serious—diQculty with this approach is the fact that we simply do not have any reliable way of estimating P; in fact, if we really had a way of calculating it from a fundamental theory, such a theory probably would have provided a deeper insight into the cosmological constant problem itself. The second issue has to do with the dependence of the results on other parameters which describe the cosmological structure formation (like for example, the spectrum of initial perturbations). To estimate Q one needs to work in a multiparameter space and marginalize over other parameters— which would involve more assumptions regarding the priors. And .nally, anthropic paradigm itself is suspect in any scienti.c discussion, for reasons mentioned earlier. 7.5. Probabilistic interpretation of the cosmological constant It is also possible to produce more complex scenarios which could justify the small or zero value of cosmological constant. One such idea, which enjoyed popularity for a few years [285–288], is based on the conjecture that quantum wormholes can change the e4ective value of the observed constants of nature. The wave function of the universe, obtained by a path integral over all possible spacetime metrics with wormholes, will receive dominant contributions from those con.gurations for which the e4ective values of the physical constants extremize the action. Under some assumptions related to Euclidean quantum gravity, one could argue that the con.gurations with zero cosmological constant will occur at late times. It is, however, unlikely that the assumptions of Euclidean quantum gravity has any real validity and hence this idea must be considered as lacking in concrete justi.cation. 8. Relaxation mechanisms for the cosmological constant One possible way of obtaining a small, nonzero, cosmological constant at the present epoch of the universe is to make the cosmological constant evolve in time due to some physical process.

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At a phenomenological level this can be done either by just postulating such a variation and explore its consequences or—in a slightly more respectable way—by postulating a scalar .eld potential as described in Section 4. These models, however, cannot explain why a bare cosmological constant [the .rst term on the right hand side of (7)] is zero. To tackle this issue, one can invoke some .eld [usually a scalar .eld] which directly couples to the cosmological constant and decreases its “e4ective value”. We shall now examine two such models. The key idea is to introduce a .eld which couples to the trace T = Taa of the energy momentum tensor. If T depends on and vanishes at some value = 0 , then will evolve towards = 0 at which T = 0. This equilibrium solution will have zero cosmological constant [289–292]. While this idea sounds attractive, there are general arguments as to why it does not work in the simplest context [4]. A related attempt was made by several authors, [289,293–295], who coupled the scalar .eld directly to R which, of course, is proportional to T because of Einstein’s equations. Generically, these models have the Lagrangian 1 1 9 9 + (R − 2) − U ( )R : (116) L= 2 16G The .eld equations of this model has Pat spacetime solutions at = 0 provided U ( 0 ) = ∞. Unfortunately, the e4ective gravitational constant in this model evolves as Ge4 =

G 1 + 16GU ( 0 )

(117)

and vanishes as U → ∞. Hence these models are not viable. The diQculty in these models arise because they do not explicitly couple the trace of the Tab of the scalar .eld itself. Handling this consistently [296] leads to a somewhat di4erent model which we will briePy describe because of its conceptual interest. Consider a system consisting of the gravitational .elds gab , radiation .elds, and a scalar .eld which couples to the trace of the energy-momentum tensor of all .elds, including its own. The zeroth order action for this system is given by (0) A(0) = Agrav + A(0) + Aint + Aradn ;

where −1

Agrav = (16G) A(0)

1 = 2

i

√

√

4

(118)

R −g d x − 4

i −g d x;

A(0) int

√ −g d 4 x ;

=I

√ Tf( = 0 ) −g d 4 x :

(119) (120)

Here, we have explicitly included the cosmological constant term and I is a dimensionless number which ‘switches on’ the interaction. In the zeroth order action, T represents the trace of all .elds other than . Since the radiation .eld is traceless, the only zeroth-order contribution to T comes from the term, so that we have T = 4. The coupling to the trace is through a function f of the scalar .eld, and one can consider various possibilities for this function. The constant 0 converts to a dimensionless variable, and is introduced for dimensional convenience.


291

To take into account the back-reaction of the scalar .eld on itself, we must add to T the contribution T = − l l of the scalar .eld. If we now add T to T in the interaction term A(0) int further modi.es T ik . This again changes T . Thus to arrive at the correct action an in.nite iteration will have to be performed and the complete action can be obtained by summing up all the terms. (For a demonstration of this iteration procedure, see [297,298].) The full action can be found more simply by a consistency argument. Since the e4ect of the iteration is to modify the expression for A and A , we consider the following ansatz for the full action: √ √ √ 1 1 R −g d 4 x − G( ) i i −g d 4 x + Arad : A= ( ) −g d 4 x + (121) 16G 2 Here ( ) and G( ) are functions of to be determined by the consistency requirement that they represent the e4ect of the iteration of the interaction term. (Since radiation makes no contribution to T , we expect Arad to remain unchanged.) The energy–momentum tensor for and is now given by 1 T ik = ( )gik + G( ) i k − gik (122) 2 so that the total trace is Ttot = 4 ( ) − G( ) i i . The functions ( ) and G( ) can now be determined by the consistency requirement √ √ 1 G( ) i i −g d 4 x − ( ) −g d 4 x + 2 √ √ 1 4 i √ 4 i −g d x + I Ttot f( = 0 ) −g d 4 x : (123) = − −g d x + 2 Using Ttot and comparing terms in the above equation we .nd that ( ) = [1 + 4If]−1 ;

G( ) = [1 + 2If]−1 :

Thus the complete action can be written as √ i i √ 1 1 √ 4 4 R −g d x − A= −g d x + −g d 4 x + Arad : 16G 1 + 4nf 2 1 + 2nf

(124)

(125)

(The same action would have been obtained if one uses the iteration procedure.) The action in (125) leads to the following .eld equations: 1 1 ik i k traceless Rik − gik R = −8G G( ) − g + ( )gik + Tik ; (126) 2 2 8G +

1 G ( ) i ( ) i + =0 : 2 G( ) 8G G( )

(127)

Here, stands for a covariant d’Lambertian, Tiktraceless is the stress tensor of all .elds with traceless stress tensor and a prime denotes di4erentiation with respect to .

292


In the cosmological context, this reduces to 3a˙ f f (1 + 2If) X + ˙ = I ˙ 2 ; +I a 1 + 2If 2G (1 + 4If)2 a˙2 + k 8G = 2 a 3

1 ˙ 2 1 0 + + 4 2 1 + 2If 8G (1 + 4If) a

(128) :

(129)

It is obvious that the e4ective cosmological constant can decrease if f increases in an expanding universe. The result can be easily generalized for a scalar .eld with a potential by replacing by V ( ). This model is conceptually attractive since it correctly accounts for the coupling of the scalar .eld with the trace of the stress tensor. The trouble with this model is two fold: (a) If one uses natural initial conditions and do not .ne tune the parameters, then one does not get a viable model. (b) Since the scalar .eld couples to the trace of all sources, it also couples to dust-like matter and “kills” it, making the universe radiation dominated at present. This reduces the age of the universe and could also create diQculties for structure formation. These problems can be circumvented by invoking a suitable potential V ( ) within this model [299]. However, such an approach takes away the naturalness of the model to certain extent.

9. Geometrical structure of the de Sitter spacetime The most symmetric vacuum solution to Einstein’s equation, of course, is the Pat spacetime. If we now add the cosmological constant as the only source of curvature in Einstein’s equation, the resulting spacetime is also highly symmetric and has an interesting geometrical structure. In the case of a positive cosmological constant, this is the de Sitter manifold and in the case of negative cosmological constant, it is known as anti-de Sitter manifold. We shall now discuss some features of the former, corresponding to the positive cosmological constant. (For a nice, detailed, review of the classical geometry of de Sitter spacetime, see [300].) To understand the geometrical structure of the de Sitter spacetime, let us begin by noting that a spacetime with the source Tba = ab must have three-dimensional section which are homogeneous and isotropic. This will lead us to the Einstein’s equations for a FRW universe with cosmological constant as source a˙2 k 8G ≡ H 2 : + 2= 2 a a 3

(130)

This equation can be solved with any of the following three forms of (k; a(t)) pair. The .rst pair is the spatially Pat universe with (k = 0; a = eHt ). The second corresponds to spatially open universe with (k = −1; a = H −1 sinh Ht) and the third will be (k = +1; a = H −1 cosh Ht). Of these, the last pair gives a coordinate system which covers the full de Sitter manifold. In fact, this is the metric on a four-dimensional hyperboloid, embedded in a .ve-dimensional Minkowski space with the metric ds2 = dt 2 − d x2 − dy2 − d z 2 − dv2 :

(131)


293

The equation of the hyperboloid in 5-D space is t 2 − x2 − y2 − z 2 − v2 = −H −2 :

(132)

We can introduce a parametric representation of the hyperbola with the four variables (2; %; $; ) where x = H −1 cosh(H2)sin % sin $ cos ;

y = H −1 cosh(H2)sin % sin $ sin ;

z = H −1 cosh(H2)sin % cos $ ; v = H −1 cosh(H2)cos % ; t = H −1 sinh(H2) :

(133)

This set, of course, satis.es (132). Using (131), we can compute the metric induced on the hyperboloid which—when expressed in terms of the four coordinates (2; %; $; )—is given by ds2 = d22 − H −2 cosh2 (H2)[d%2 + sin2 %(d$ 2 + sin2 $ d 2 )] :

(134)

This is precisely the de Sitter manifold with closed spatial sections. All the three forms of FRW universes with k =0; ±1 arise by taking di4erent cuts in this four-dimensional hyperboloid embedded in the .ve-dimensional spacetime. Since two of these dimensions (corresponding to the polar angles $ and ) merely go for a ride, it is more convenient (for visualization) to work with a 3-dimensional spacetime having the metric ds2 = dt 2 − d x2 − dv2 :

(135)

instead of the .ve-dimensional metric (131). Every point in this three-dimensional space corresponds to a 2-sphere whose coordinates $ and are suppressed for simplicity. The (1+1) de Sitter spacetime is the two-dimensional hyperboloid [instead of the four-dimensional hyperboloid of (132)] with the equation t 2 − x2 − v2 = −H −2

(136)

embedded in the three-dimensional space with metric (135). The three di4erent coordinate systems which are natural on this hyperboloid are the following: • Closed spatial sections: This is obtained by introducing the coordinates t = H −1 sinh(H2); x = H −1 cosh(H2)sin %; v = H −1 cosh(H2)cos % on the hyperboloid, in terms of which the induced metric on the hyperboloid has the form ds2 = d22 − H −2 cosh2 (H2) d%2 :

(137)

This is the two-dimensional de Sitter space which is analogous to the four-dimensional case described by (134). • Open spatial sections: These are obtained by using the coordinates t = H −1 sinh(H2)cosh

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