New Worlds in Astroparticle Physics: Proceedings of the 4th International Workshop, Faro, Portugal 5 O 7 September 2002

NEW WORLDS IN >

H

Proceedings otthe Fourth International Workshop editors

Alexander Krasnitz Ana M. Mourao Mario Pimenta Robertus Potting

World Scientific

NEW WORLDS IN

n°

C J 1-3

n Proceedings of the Fourth International Workshop

This page is intentionally left blank

NEW WORLDS IN

)—I

n m Proceedings of the Fourth International Workshop

editors

Alexander Krasnitz CENTRA & Universidade do Algarve, Faro, Portugal

Ana M. Mourao CENTRA & Instituto Superior Tecnico, Lisbon, Portugal

Mario Pimenta UP & Instituto Superior Tecnico, Lisbon, Portugal

Robertus Potting CENTRA & Universidade do Algarve, Faro, Portugal

V f e World Scientific wB

New Jersey • London • Singapore Si. • Hong Kong

Published by World Scientific Publishing Co. Pte. Ltd. 5 Toh Tuck Link, Singapore 596224 USA office: Suite 202,1060 Main Street, River Edge, NJ 07661 UK office: 57 Shelton Street, Covent Garden, London WC2H 9HE

British Library Cataloguing-in-Publication Data A catalogue record for this book is available from the British Library.

Cover design is based on graphics material from the New Worlds in Astroparticle Physics—2002 poster by Dimensao 6.

NEW WORLDS IN ASTROPARTICLE PHYSICS Proceedings of the Fourth International Workshop Copyright © 2003 by World Scientific Publishing Co. Pte. Ltd. All rights reserved. This book, or parts thereof, may not be reproduced in any form or by any means, electronic or mechanical, including photocopying, recording or any information storage and retrieval system now known or to be invented, without written permission from the Publisher.

For photocopying of material in this volume, please pay a copying fee through the Copyright Clearance Center, Inc., 222 Rosewood Drive, Danvers, MA 01923, USA. In this case permission to photocopy is not required from the publisher.

ISBN 981-238-584-3

Printed in Singapore by World Scientific Printers (S) Pte Ltd

This page is intentionally left blank

PREFACE

The Fourth International Workshop on New Worlds in Astroparticle Physics took place from the 5th through the 7th of September 2002 on the Campus of Gambelas, at the University of the Algarve, near Faro. For three days we had talks and discussions, with presentation of recent results on distant supernovae and cosmological background radiation, of developments in new projects on gravitational wave searches and extremely high energy cosmic ray detection based on the International Space Station, and of theoretical insights into extremely dense matter and the possibility of recreating Big Bang at the CERN Large Hadron Collider. Astroparticle physics is growing and nourishing - even if we still do not know where the dark mass and the dark energy come from... The symposium was organised by the University of the Algarve, Instituto Superior Tecnico, CENTRA (Multidisciplinary Center for Astro­ physics) and CFIF (Center for Physics of Fundamental Interactions). Fi­ nancial support from FCT (Foundation for Science and Technology) under the Programa Operacional Ciencia, Tecnologia, Inovagao do Quadro Comunitario de Apoio II, from FLAD (Portuguese American Foundation) and from Gulbenkian Foundation is gratefully acknowledged.

Jorge Dias de Deus

VII

This page is intentionally left blank

CONTENTS Preface PART 1

vii OVERVIEWS IN ASTROPARTICLE PHYSICS

Dense Matter N. K. Glendenning

3

Extreme Energy Cosmic Rays and Fundamental Physics H. J. De Vega

20

Gravitational Waves: Probing the Extremes of Physics N. Andersson

28

Cosmological Parameters: Fashion and Facts A. Blanchard

50

PART 2

CONTRIBUTIONS

Astroparticle Physics Beyond the Standard Model Physical Implications of Quintessential Brane Cosmologies K. E. Kunze and M. A. Vdzquez-Mozo

75

Dark Radiation and Localization of Gravity on the Brane R. Neves and C. Vaz

82

Astrophysical Tests of Lorentz Symmetry in Electrodynamics M. Mewes

89

Apparent Lorentz Violation through Spacetime-Varying Couplings R. Lehnert

96

The CERN Axion Solar Telescope

103

M. D. Hasinoff for the CAST experiment Matter Under Extreme Conditions Properties of Dense and Cold QCD H. J. De Vega

113

IX

Stability of Quark Matter and Quark Stars M. Fiolhais. M. Malheiro and A. R. Taurines

123

Heavy Quarks or Compactified Extra Dimensions in the Core of Hybrid Stars G. G. Barnafoldi, P. Levai and B. Lukdcs

133

Ratios of Antibaryon/Baryon Yields in Heavy Ion Collisions

143

Yu. M. Shabelski Cosmic Rays EUSO: Basic Parameters M. C. Espirito Santo

151

The Radio Technique 40 Years Later: Where Do We Stand? E. Zas

161

GPS Synchronization in Cosmic Ray Experiments P. Assis

171

Results from the AMS01 1998 Shuttle Flight M. Steuer

178

Electric Charge Reconstruction with RICH Detector of the AMS Experiment L. Arruda. F. Barao, J. Borges, F. Carmo, P. Goncalves, A. Keating, M. Pimenta and I. Perez Velocity Reconstruction with the RICH Detector of the AMS Experiment L. Arruda, F. Barao, Joao P. Borges. M. Pimenta and I. Perez

191

202

Neutrino Physics and Astrophysics Results from the Sudbury Neutrino Observatory /. Maneira for the SNO Collaboration

217

Status Report on Borexino A. de Bellefon

227

Testing Neutrino Parameters at Future Accelerators J. C. Romdo

233

XI

Gravitational Waves and Tests of General Relativity Relativistic R-Modes in Slowly Rotating Neutron Stars S. Yoshida and U. Lee

245

Collision of Highly Relativistic Particles with Black Holes: The Gravitational Radiation Generated V. Cardoso and J. P. S. Lemos

252

Pair of Accelerated Black Holes in an Anti-de Sitter Background: The AdS C-Metric O. J. Dias and J. P. S. Lemos

260

Timing the PSR J2016+1947 Binary System: Testing the Fundamental Assumption of General Relativity P. C. Freire. J. A. Navarro and S. B. Anderson

270

Supernovae and Dark Matter Supernovae and Dark Energy A. Goobar Current Status of Type IA Supernovae Theory and Their Role in Cosmology S. Blinnikov

281

291

Intensive Supernovae Searches K. Schahmaneche

298

Probing the Dark Matter within the Solar Interior /. P. Lopes

306

List of participants

319

This page is intentionally left blank

PART 1

OVERVIEWS IN ASTROPARTICLE PHYSICS

This page is intentionally left blank

DENSE MATTER

NORMAN K. GLENDENNING Nuclear Science Division and Institute for Nuclear & Particle Astrophysics Lawrence Berkeley National Laboratory Berkeley, California 94720 USA Neutron stars are the densest objects in the universe today in which matter in adiabatic equilibrium can be found. Various high-density phases, both geometric and constitutional are spatially spread out by the pressure gradient in the star. Boundaries between phases slowly move, appear, or disappear as the density profile of the star is changed by the centrifugal force due to spindown caused by the magnetic torque of a pulsar, or the spinup of an x-ray neutron star due to the torque applied by mass accreted from a companion star. Phase transitions in turn produce their own imprint on the spin behavior through changes in the moment of inertia as one phase replaces another, in some cases on single stars, and in others on populations. These are the clues that we elucidate after first reviewing high-density phases.

1. A Brief History • 1054 Chinese astronomer "observed the apparition of a guest star ...its color an iridescent yellow". • 1933 Baade and Zwicky—binding energy of "closely packed neutrons" powers supernova. • 1939 Oppenheimer, Volkoff and Tolman—neutron fermi gas. • 1967 Pacini predicted magnetic dipole radiation. • 1967 Hewish & Bell's serendipitous discovery of neutron stars producing a radio pulse once every revolution from beamed radiation along the magnetic axis which is fixed in the star. They are believed to be the direct product of core collapse a mature massive star and its the subsequent supernova. • 1974 Hulse and Taylor binary neutron star pair in close orbit. • 1984 Bacher's discovery of first Millisecond pulsar. They are believed to be very old supernova products that have been spun up by mass

3

4

accretion from a low-mass companion star. • 1992 Wolszczan & Frail, discovery of 3 planets around a neutron star.

2. Gross Features of Neutron Stars • Surface gravity M/R of Black hole =0.5, Neutron star =0.2, Sun =10~ 6 • Gravitational binding / Nuclear binding ~ 10 • Radius = 10 - 12km, Mass> 1.44MQ • Spin periods from seconds to milliseconds • Neutron stars are degenerate objects (/i «T). • Stars are electrically neutral. (Z^et/A ~ (m/e) 2 < 10 - 3 6 ) • Baryon number and charge are conserved. • Strangeness not conserved (beyond 10~ 10 seconds). • Millisecond pulsars have remarkably stable pulses: P = 1.55780644887275 ± 0.00000000000003 ms (measured for PSR 1937+21 on 29 Nov 1982 at 1903 UT)

120< z> L_ CO CO ■D

100-

Q.

80-

n

r-lI

H—

o l_

i

U

CD

+

60-

CD -Q

E

fT 1

40-

CM

i

+


13>14 All details of our calculation can be found elsewhere. 15 ' 16 ' 17 7. Results The spin evolution of accreting neutron stars as determined by the changing moment of inertia and the evolution equation 15 is shown in Fig. 13. We assume that up to O.4M0 is accreted. Otherwise the maximum frequency

17

attained is less than shown. Three average accretion rates are assumed, M_io = 1, 10 and 100 (where M_ 1 0 is in units of l O _ l o M 0 / y ) .

1000

200 10°

10'

10

10"

time (years)

10'"

400 600 v (s-1)

800

1000

Figure 13. Evolution of spin frequencies of accreting neutron stars with (solid curves) and without (dashed curves) quark deconfinement if QAMQ is accreted. The spin plateau around 200 Hz signals the ongoing process of quark confinement in the stellar centers. Spin equilibrium is eventually reached. (From Ref. 1 5 .) Figure 14. Calculated spin distribution of the underlying population of x-ray neutron stars for one accretion rate (open histogram) is normalized to the number of observed objects (18) at the peak. Data on neutron stars in low-mass X-ray binaries (shaded histogram) is from Ref. 8 . The spike in the calculated distribution corresponds to the spinout of the quark matter phase. Otherwise the spike would be absent. (From Ref. 15

0

We compute a frequency distribution of x-ray stars in low-mass binaries (LMXBs) from Fig. 13, for one accretion rate, by assuming that neutron stars begin their accretion evolution at the average rate of one per million years. A different rate will only shift some neutron stars from one bin to an adjacent one. The donor masses in the binaries are believed to range between 0.1 and 0.4M© and we assume a uniform distribution in this range and repeat the calculation shown in Fig. 13 at intervals of 0.1M Q . The resulting frequency distribution of x-ray neutron stars is shown in Fig. 14; it is striking. A spike in the distribution signals spinout of the quark matter core as the neutron star spins up. This feature would be absent if there were no phase transition in our model of the neutron star.

18

8. Discussion The observed frequency clustering of x-ray neutron stars is about 100 Hz higher than what we calculate. This discrepancy should not be surprising in view of our ignorance of the equation of state above saturation density of nuclear matter and the fact that we employed a stellar model that was previously used, without change. Spyrou and Stergioulas are currently studying a selection of models to learn whether the position of the clustering of spin frequencies can be used to discriminate among them. What is clear is that however crude any model of hadronic matter may be, the physics underlying the effect of a phase transition on spin rate is robust, although not inevitable. We have cited an analogous phenomenon discovered in rotating nuclei. 18 ' 19 - 20 The data in Fig. 14 is gathered from Tables 2-4 of the review article of van der Klis concerning discoveries made with the Rossi X-ray Timing Explorer. 8

9. Conclusion The apparent clustering in rotation frequency of accreting x-ray neutron stars in low-mass binaries may be caused by the progressive conversion of quark matter in the core to confined hadronic matter, paced by the slow spinup due to mass accretion. When conversion is completed, normal accre­ tion driven spinup resumes. To distinguish this conjecture from others, one would have to discover the inverse phenomenon—a spin anomaly near the same frequency in an isolated ms pulsar. 6 If such a discovery were made, and the apparent clustering of x-ray accretors is confirmed, we would have some degree of confidence in the hypothesis that a dense matter phase, most plausibly quark matter, exists from birth in the cores of canonical neutron stars, is spun out if the star has a companion from which it ac­ cretes matter, and later, having consumed its companion, resumes life as a millisecond radio pulsar and spins down.

Acknowledgments This work was supported by the Director, Office of Energy Research, Office of High Energy and Nuclear Physics, Division of Nuclear Physics, of the U.S. Department of Energy under Contract DE-AC03-76SF00098.

19

References 1. N. K. Glendenning, Phys. Lett. 114B (1982) 392; Astrophys. J. 293 (1985) 470. 2. N. K. Glendenning, Compact Stars (Springer-Verlag New York, l'st ed. 1996, 2'nd ed. 2000). 3. N. K. Glendenning, Phys. Rev. D 46 (1992) 1274. 4. K. Rajagopal and F. Wilczek, Phys.Rev.Lett. 86 (2001) 3492. 5. M. Alford, K. Rajagopal, S. Reddy, and F. Wilczek, Phys.Rev. D 64 (2001) 074017. 6. N. K. Glendenning, S. Pei and F. Weber, Phys. Rev. Lett. 79 (1997) 1603. 7. N.K. Spyrou and N. Sterigoulas, Astron. & Astrophys., 395 (2002) 151. 8. M. van der Klis, Ann. Rev. Astron. Astrophys, 38 717 (2000). 9. N. K. Glendenning, Astrophys. J. 293 (1985) 470. 10. N. K. Glendenning and S. A. Moszkowski, Phys. Rev. Lett. 67 (1991) 2414. 11. E. Farhi and R. L. Jaffe, Phys. Rev. D 30 (1984) 2379. 12. R. F. Eisner and F. K. Lamb, Astrophys. J. 215 (1977) 897. 13. P. Ghosh, F. K. Lamb and C. J. Pethick, Astrophys. J. 217 (1977) 578. 14. V. M. Lupinov, Astrophysics of Neutron Stars, (Springer-Verlag, New York, 1992. 15. N. K. Glendenning and F. Weber, Astrophys. J. Lett 559 (2001) L119. 16. N. K. Glendenning and F. Weber, astro-ph/0010336 (2000). 17. N. K. Glendenning and F. Weber, Signal of quark deconfinement in millisec­ ond pulsars and reconGnement in accreting x-ray neutron stars, in Physics of Neutron Star Interiors, Ed. by Blaschke, Glendenning and Sedrakian (Springer-Verlag, Lecture Notes Series, 2001). 18. B. R. Mottelson and J. G. Valatin, Phys. Rev. Lett. 5 (1960) 511. 19. A. Johnson, H. Ryde and S. A. Hjorth, Nucl. Phys. A179 (1972) 753. 20. F. S. Stephens and R. S. Simon, Nucl. Phys. A183 (1972) 257.

E X T R E M E E N E R G Y COSMIC RAYS AND FUNDAMENTAL PHYSICS

H. J. DE VEGA LPTHE, Universite Pierre et Marie Curie (Paris VI) et Denis Diderot (Paris VII), UMR 7589 CNRS, Tour 16, ler. etage, 4, Place Jussieu, 75252 Paris, Cedex 05, France This is an overview on the physics behind cosmic rays and extreme energy cos­ mic rays. We discuss the acceleration mechanisms of cosmic rays, their possible astrophysical sources and the main open physical problems and difficulties. The top-down and bottom-up scenarii are contrasted.

Cosmic rays are one of the rare systems in the Universe which are not thermalized. Their energy spectrum follows approximately a power law over at least thirteen orders of magnitude [see fig. 1]. The understanding of the cosmic ray spectrum involves several branches of physics and astronomy. First, to explain how cosmic rays get energies up to 1020eV according to observations 1 ' 2 ' 3 and with such power spectrum. Then, to study the effect of galactic and extragalactic magnetic fields and explain the knee and the ankle effects. Furthermore, reconciliate the GZK effect 4 with the current observations of cosmic rays events beyond 1020eV. Last and not least to understand the interaction of the cosmic rays with the atmosphere, the extended air shower formation, especially at extreme energies and the fluorescence effects. At the same time, one has to identify the astronomical sources of cosmic rays and relate the observed events, composition and spectra with the properties and structure of the sources. The standard physical acceleration mechanism goes back to the ideas proposed by E. Fermi in the fifties. That is, charged particles can be effi­ ciently accelerated by electric fields in astrophysical shock waves6'7. This is the so-called diffusive shock acceleration mechanism yielding a power spectrum with n(E) ~ E~a ,

20

(1)

21

with 2.3 < a < 2.5 - 2.7. Such spectrum is well verified over more than ten orders of magnitude in energy. In short, electric fields accelerate charged particles in the wavefront of the shock. Then, magnetic fields deviate and diffuse the particles. Charged particles take energy from the wavefront of macroscopic (astronomic!) size. Particles trapped for long enough can acquire gigantic energies. This mechanism can accelerate particles till arbitrary high energies. The upper limit in energy is given by the time while the particle stays in the wavefront which depends on the size of the source and on the magnetic field strength. Particles gain energy by bouncing off hydromagnetic disturbances near the shock wave. Particles are both in the downstream and the upstream flow regions. If a particle in the upstream crosses the shock-front, reaches the downstream region and then crosses back to the upstream region This cross boosts its energy by a factor proportional to the Lorentz factor of the shock squared ~ T 2 6 . Such factor could be very large. Indeed, the larger is the acceleration energy the less probable is the process yielding a power spectrum of the type of eq.(l). Accelerated particles are focalized inside a cone of angle 9 < p. To rotate the particle momentump this angle takes a time At ~ rg 9 = Ze^ r where rg = zfB stand for the relativistic gyration radius of the particle in the magnetic field. It must be At < Rs where Rs is the radius of the spherical wave. Otherwise the particle is gone of the shock front. Therefore, E < Z e B± T. That is, one finds for the maximal available acceleration energy 6 ' 7 for a particle of charge Z e, Emax = ZeB±T

(2)

There are other estimates but all have the same structure. The maximal energy is proportional to the the magnetic field strength, to the particle charge, to Rs which is of the order of the size of the source and to a big numerical factor as T. Further important effects are the radiation losses from the accelerated charged particles and the back-reaction of the particles on the plasma. That is, the non-linear effects on the shock wave. Particle acceleration in shock-waves can be described at different levels. The simplest one is the test particle description where the propagation of charged particles in shock-waves is studied. A better description is obtained with transport equations. In addition, non-linear effects can be introduced in such a Fokker-Planck treatment 6 ' 7 . The distribution function for the particles in the plasma f(x,p, t) obeys

22

Figure 1. Cosmic ray spectrum[5].

the Fokker-Planck equation, ^

+ tf-V/ + | ^ d i v t f - d i v ( K V / ) = Q .

(3)

Here, u stands for the velocity field, the third term describes the adiabatic compression and it follows from the collision terms in the transport equation for small momentum transfer, K describes the spatial diffusion and Q is a injection or source term. The energy spectrum follows from this equation irrespective of the de­ tails of the diffusion. It must be recalled that the coefficients in this Fokker-

23

Planck equation are only sketchily known for relevant astrophysical plas­ mas. A microscopic derivation of the Fokker-Planck equation including reliable computation of its coefficients will be important to understand the acceleration of extreme energy cosmic rays (EECR). Let us consider stationary solutions of the Fokker-Planck equation (3) for a simple one dimensional geometry. Let us consider a step function as velocity field6. That is, v(x) = vi upstream, for x > 0 and v(x) = vi downstream, for x < 0 (4)

Eq.(3) then takes the form,

, , df d n(x,p) 9f] dx Yx = Yx

for x ^ 0 .

V{X)

Integrating upon x taking into account eq.(4) yields,

df v(x) f(x,p) = n(x,p) -^ + A(p) , where A(p) is an integration constant. Integrating again upon x gives the solution h(p) + 9i(p) e~Vl

" (I ' ,P> ,upstream, x > 0

f(x,P) = {

(5) f2(p) , downstream, x < 0 .

Matching the solutions at the shock-wave front at x = 0 yields, (r-l)p^=3r(/i-/

2

)

,

and the standard model Lagrangian, like the ones considered for example in Ref 6 , might result in an effective variation of fundamental constans while preserving the late time acceleration of the four-dimensional universe. This, and other issues, will be addressed else­ where. Acknowledgments M.A.V.-M. thanks the organizers of Astro 2002, and in particular Robertus Potting, for their kind invitation to present this work and their hospitality in Faro. References 1. S. Perlmutter et al., Astrophys. J. 517 (1999) 565; A. G. Riess et al, Astron. J. 116 (1998) 1009.

81

2. L. Dyson, M. Kleban and L. Susskind, J. High Energy Phys. 0210 (2002) Oil. 3. K.E. Kunze and M.A. Vazquez-Mozo, Phys. Rev. D 6 5 (2002) 044002. 4. E.W. Kolb, M.J. Perry and T.P. Walker, Phys. Rev. D 3 3 (1986) 869 F.S. Ascetta, L.M. Krauss and P. Romanelli, Phys. Lett/248 (1990) 146. 5. R. Bean, S.H. Hansen and A. Melchiorri, Phys. Rev. D 6 4 (2001) 103508. 6. P. Brax, C. van de Bruck, A.C. Davis and C.S. Rhodes, Varying constants in brane world scenarios, hep-ph/0210057.

D A R K RADIATION A N D LOCALIZATION OF GRAVITY ON T H E B R A N E

R U I N E V E S A N D C E N A L O VAZ Area Departamental de Fisica/CENTRA, FCT, Universidade Campus de Gambelas, 8000-117 Faro, Portugal E-mail: [email protected], [email protected]

do

Algarve

We discuss the dynamics of a spherically symmetric dark radiation vaccum in the Randall-Sundrum brane world scenario. Under certain natural assumptions we show that the Einstein equations on the brane form a closed system. For a de Sitter brane we determine exact dynamical and inhomogeneous solutions which depend on the brane cosmological constant, on the dark radiation tidal charge and on its initial configuration. We define the conditions leading to singular or globally regular solutions. We also analyse the localization of gravity near the brane and show that a phase transition to a regime where gravity propagates away from the brane may occur at short distances during the collapse of positive dark energy density.

1. Introduction In the search for extra spatial dimensions the Randall and Sundrum (RS) brane world scenario is particularly interesting for its simplicity and depth 1 . In this model the Universe is a 3-brane boundary of a noncompact Z 2 symmetric 5-dimensional anti-de Sitter space. The matter fields live only on the brane but gravity inhabits the whole bulk and is localized near the brane by the warp of the infinite fifth dimension. Since its discovery many studies have been done within the RS scenario (see Ref. 2 for a recent review and notation). For a brane bound observer 3,4 s ' the interaction between the brane and the bulk introduces correction terms to the 4-dimensional Einstein equations, namely, a local high energy embedding term generated by the matter energy-momentum tensor and a non-local term induced by the bulk Weyl tensor. Such equations have an intrincate non-linear dynamics. For example, the exterior vaccum of collapsing matter on the brane is now filled with gravitational modes origi­ nated by the bulk Weyl curvature and can no longer be regarded as a static space 6 ' 7 .

82

83

Previous research on the RS scenario has been focused on static or ho­ mogeneous dynamical solutions. In this proceedings we report some new results on the dynamics of a spherically symmetric RS brane world vaccum. For a de Sitter brane we present exact dynamical and inhomogeneous so­ lutions, define the conditions to characterize them as singular or globally regular and discuss the localization of gravity to the vicinity of the brane (see Ref. 8 for more details). 2. Brane Vaccum Field Equations In the Gauss-Codazzi formulation of the RS model 3 ' 4 ' 5 , the Einstein vac­ cum field equations on the brane are given by G^v = — A#M„ — E^,

(1)

where A is the brane cosmological constant and the tensor S^u is the limit on the brane of the projected 5-dimensional Weyl tensor. It is a symmetric and traceless tensor constrained by the following conservation equations V„5£ = 0.

(2)

The projected Weyl tensor £M„ can be written in the following general form 5 K

I 4

U [ U^Uv + - h ^ ) + Vftv + Q^Uu +

QvUti

(3)

where uM such that u^u^ = - 1 is the 4-velocity field and /iM„ = (?M„ + wMu„ is the tensor projecting orthogonaly to uM. The forms U, V^v and Q p represent different aspects of the effects induced on the brane by the 5dimensional gravitational field. Thus, U is an energy density, Vp,v a stress tensor and QM an energy flux. Since the 5-dimensional metric is not known, in general £^v is not com­ pletely determined on the brane 3 ' 4 and so the effective 4-dimensional theory is not closed. To close it we need simplifying assumptions about the effects of the gravitational field on the brane. For instance we may consider a static and spherically symmetric brane vaccum with QM = 0, V^v ^ 0 and i / ^ 0 . This leads to the Reissner-Nordstrom black hole solution on the brane 9 . It is also possible to close the system of Einstein equations when consid­ ering a dynamical and spherically symmetric brane vaccum with QM = 0, U ^ O , and PM„ ^ 0. The general, spherically symmetric metric in comoving coordinates (t, r, 6, <j>) is given by ds2 = gllvdx>idxv = -eadt2

+ A2dr2 + R2dVl2,

(4)

84

where dtt2 = d62 + sm26d<j)2, a - a(t,r), A = A(t,r), R = R(t,r) and R is the physical spacetime radius. If the traceless stress tensor 7 V is isotropic then it will have the general form ? V = V [rfirv - -h^v

,

(5)

where V = V(t, r) and rM is the unit radial vector, given in the above metric by !•„ = (0,4,0,0). Then £1= ( - J diag (p, -pr, -Pr, ~PT) ,

(6)

where the energy density and pressures are, respectively, p — U, pr = (1/3) {U + IV) and pT = (1/3) (U - V). Consequently, the conservation Eq. (2) read 10

2j(p+pr)

= -2p-4-(p

+ pT),

JDl

a'(p+Pr)

= -2Pr'+4—(pT-pr),

(7)

where the dot and the prime denote, respectively, derivatives with respect to t and r. A synchronous solution is permitted with the equation of state p + pr = 0, equivalent to V = -111 where U has the dark radiation form

The constant Q is the dark radiation tidal charge. Hence, we get GV = ~ A 3 M " + -pi ( U M U ^

_ 2r r

n v + hnv),

(9)

an exactly solvable closed system for the unknown functions A(t, r) and R(t, r) which depends on the free parameters A and Q. Indeed, its solutions are of the LeMaitre-Tolman-Bondi type ds2 = -dt2 + ^—-dr2

+ R2dfl2,

(10)

where R satisfies (11) ^ = i * 2 ~ R^2 + fThe function / = f(r) > - 1 is interpreted as the energy inside a shell labelled by r in the dark radiation vaccum and is fixed by its initial config­ uration.

85

3. Localization of Gravity near the Brane As is clear in Eq. (9) the dark radiation dynamics depends on A and Q. It is important to point out that these parameters have a direct effect on the localization of gravity in the vicinity of the brane. Indeed, the tidal acceleration away from the brane 5 is given by 9 - lim kABGDnAuBncuD

= ^ A + %,

(12)

where UA is the extension off the brane of the 4-velocity field satisfying UAUA = 0 and UAUA = — 1. The gravitational field is only bound to the brane if the tidal acceleration points towards the brane. It must then be negative implying that ~AR4 < - %

(13)

As a consequence, gravity is only localized for all R if A < Ac with Ac = rc4A2/12 and Q < 0 or A = Ac and Q < 0. For A < Ac and Q > 0 the gravitational field will just remain localized if R > Rc where i? 4 = 3Q/(A C — A). On the other hand for A > Ac and Q < 0 localization is limitted to the epochs R < Rc. If A > Ac and Q > 0 then gravity is always free to propagate far away into the bulk. According to recent supernovae measurements (see e.g. Ref. n ) A ~ 10- 84 GeV 2 . On the other hand M p > 108GeV and M p ~ 1019GeV imply A > 108GeV4 12 because 6/t2 = AK 4 . Since Ac = K 2 A / 2 then Ac is bound from below, Ac > 10 _29 GeV 2 . Hence, observations demand A to be positive and smaller than the critical value A c , 0 < A < Ac. Note that is means an anti-de Sitter bulk, A < 0. The same conclusion is true if M p is in the TeV range because Ac increases when M p decreases. Since current observations do not yet constrain the sign of Q 13 we conclude that for 0 < A < Ac only for Q < 0 gravity is bound to the brane for all R. If Q > 0 then for R < Rc the tidal acceleration is positive and gravity is no longer localized near the brane. 4. Inhomogeneous Dynamics for a de Sitter Brane Assume from now on that 0 < A < A c . Non-static solutions correspond to / ^ 0. An example is

R

+

2A

= A//? cosh ±2\

-t + cosh~l '

3

I V ?

(14)

86

where /? = (3/A)[3/ 2 /(4A) + Q] and + or - correspond respectively to ex­ pansion or collapse. If Q > 0 then / > — 1 but for Q < 0 the energy function / must satisfy in addition | / | > 2y/—QA/3. Since R is a non-factorizable function of t and r these solutions define new exact and inhomogeneous cosmologies for the spherically symmetric dark radiation de Sitter brane. 5. Singularities and Regular Bounces The dark radiation dynamics defined by Eq. (11) may produce shell focusing singularities at R — 0 or regular bouncing points at some R ^ 0. To see this consider R2R2 = V(R,r)

= ^R4 + fR2-Q.

(15)

If for all R > 0 the potential V is positive then a shell focusing singularity forms at R — Rs = 0. Alternatively, if there is an epoch R = R* ^ 0 for which V — 0 then a regular rebounce point appears at R = i?*. For the dark radiation vaccum at most two regular rebounce epochs can be found. Since A > 0 there is always a phase of continuous expansion to infinity with ever increasing rate. Depending on f(r) other phases may exist. To ilustrate take j3 > 0 and compare the settings Q < 0, / > —1, | / | > 2y / -0,/>-l. 0.1

0.08

0.06

V 0.04 0.02

0.2

0.4

R2

0.6

0.8

1

Figure 1. Plot of V for /3 > 0 and Q < 0. Non-zero values of / belong to the interval — 1 < / < —2^/—QA/3 and correspond to shells of constant r. If for Q < 0 we have / > 2^/-QA/3 then V > 0 for all R > 0. There are no rebounce points and the dark radiation shells may either expand continuously or collapse to a singularity at Rs = 0. However for — 1 < / < —2yJ— QA/3 (see Fig. 1) we find two rebounce epochs at R = i?*± with

87

Rl± = - 3 / / ( 2 A ) ± V/3- Since 1/(0, r) = -Q > 0 a singularity also forms at i? s = 0. Between the two rebounce points there is a forbidden zone where V is negative. The phase space of allowed dynamics is thus divided in two disconnected regions separated by the forbidden interval' i?*_ < R < R*+. For 0 < R < R*~ the dark radiation shells may expand to a maximum radius R = i?*-, rebounce and then fall to the singularity. If R > i?* + then there is a collapsing phase to the minimum radius R = R*+ followed by reversal and subsequent accelerated continuous expansion. The singularity at Rs = 0 does not form and so the solutions are globally regular. Since Q < 0 gravity is bound to the brane for all the values of R. 10

8 6

V 4 2

0.5

1

1.5

2

2.5

3

3.5

4

R2 Figure 2. Plot of V for (3 > 0 and Q > 0. Non-zero values of / belong to the interval / > — 1 and correspond to shells of constant r. The shaded region indicates where gravity is not localized near the brane. If Q > 0 (see Fig. 2) then we find globally regular solutions with a single rebounce epoch at R = R* where R% = - 3 / / ( 2 A ) + ^/0. This is the minimum possible radius for a collapsing dark radiation shell. It then reverses its motion and expands forever. The phase space of allowed dynamics denned by V and R is limitted to the region R> R*. Below i?» we find a forbidden region where V is negative. In particular, V(0,r) = —Q < 0 implying that the singularity at Rs = 0 does not form and so the solutions are globally regular. Note that if gravity is to be bound to the the brane for R > R* then i?» > Rc. If not then we find a phase transition epoch R = Rc such that for R < Rc the gravitational field is no longer localized near the brane. 6. Conclusions In this work we have reported some new results on the dynamics of a RS brane world dark radiation vaccum. Using an effective 4-dimensional

88

approach we have shown that some simplifying but natural assumptions lead to a closed and solvable system of Einstein field equations on the brane. We have presented a set of exact dynamical and inhomogeneous solutions for A > 0 showing they further depend on the dark radiation tidal charge Q and on the energy function f(r). We have also described the conditions under which a singularity or a regular rebounce point develop inside the dark radiation vaccum and discussed the localization of gravity near the brane. In particular, we have shown that a phase transition to a regime where gravity is not bound to the brane may occur at short distances during the collapse of positive dark energy density on a realistic de Sitter brane. Left for future research is for example an analysis of the dark radiation vaccum dynamics from a 5-dimensional perspective. Acknowledgements We thank the Fundacao para a Ciencia e a Tecnologia (FCT) for fi­ nancial support under the contracts POCTI/SFRH/BPD/7182/2001 and POCTI/32694/FIS/2000. We also would like to thank Louis Witten and T.P. Singh for kind and helpful comments. References 1. L. Randall and R. Sundrum, Phys. Rev. Lett. 83, 3370 (1999); Phys. Rev. Lett. 83, 4690 (1999). 2. R. Maartens, Geometry and Dynamics of the Brane World, arXiv:grqc/0101059. 3. T. Shiromizu, K. I. Maeda and M. Sasaki, Phys. Rev. D62, 024012 (2000). 4. M. Sasaki, T. Shiromizu and K. I. Maeda, Phys. Rev. D62, 024008 (2000). 5. R. Maartens, Phys. Rev. D62, 084023 (2000). 6. M. Bruni, C. Germani and R. Maartens, Phys. Rev. Lett. 87, 231302 (2001). 7. M. Govender and N. Dadhich, Phys. Lett. B538, 233 (2002). 8. R. Neves and C. Vaz, Dark Radiation Dynamics on the Brane, arXiv:hepth/0207173, to be published in Phys. Rev. D. 9. N. Dadhich, R. Maartens, P. Papadopoulos and V. Rezania, Phys. Lett. B487, 1 (2000). 10. T. P. Singh, Class. Quant. Grav. 16, 3307 (1999). 11. N. A. Bahcall, J. P. Ostriker, S. Perlmutter and P. J. Steinhardt, Science 284, 1481 (1999). 12. D. Langlois, R. Maartens, M. Sasaki and D. Wands, Phys. Rev. D63, 084009 (2001). 13. K. Ichiki, M. Yahiro, T. Kajino, M. Orito and G. J. Mathews, Phys.Rev. D66, 043521 (2002).

A S T R O P H Y S I C A L TESTS OF LORENTZ S Y M M E T R Y I N ELECTRODYNAMICS

MATTHEW MEWES Physics Department, Indiana University, Bloomington, IN 47405, E-mail: [email protected]

U.S.A.

In this talk presented at the Fourth International Workshop on New Worlds in Astroparticle Physics, I discuss recent constraints on Lorentz violation in electro­ dynamics. The observed absence of birefringence of light that has propagated over cosmological distances bounds some coefficients for Lorentz violation to 2 x 10 - 3 2 .

1. Introduction The exact character of physics beyond the standard model is an open ques­ tion. The standard model is commonly believed to be the low-energy limit of Planck-scale physics which unifies all known forces. Due to the energy scales involved, an experimental search for this new physics would seem pointless given the current technology. However, some high-energy theories may lead to violations in symmetries which hold exactly in the standard model. 1 ' 2 In particular, spontaneous symmetry breaking in the fundamental theory might result in apparent violations in the Lorentz and CPT symme­ tries. Furthermore, Lorentz and CPT violations can be tested to extremely high precision using today's technology.3 A general Lorentz-violating extension to the standard model has been constructed. 3 ' 4 It consists of the minimal standard model plus small Lorentz- and CPT-violating terms. The standard-model extension has pro­ vided a theoretical framework for many searches for Lorentz and CPT vio­ lations. To date, experiments involving hadrons, 5 ' 6 protons and neutrons, 7 electrons, 8 ' 9 photons, 10 ' 11 and muons 12 have been performed. In practice, one often works with a particular limiting theory extracted from the standard-model extension. For example, the photon sector of the standard-model extension yields a Lorentz-violating modified electro­ dynamics. The theory predicts several unconventional features that lead to sensitive tests of Lorentz symmetry. For example, in the presence of

89

90 certain forms of Lorentz violation, light propagating through the vacuum will experience birefringence. The absence of birefringence in light emitted from distant sources leads to tight bounds on some of the coefficients for Lorentz violation. 10 ' 11 In this work, I review some of these bounds. This research was done in collaboration with Alan Kostelecky. A detailed discussion can be found in the literature. 11 2. Lorentz-Violating Electrodynamics The modified electrodynamics maintains the usual gauge invariance and is covariant under observer Lorentz transformations. It includes both CPTeven and -odd terms. The CPT-odd terms have been the subject of nu­ merous experimental and theoretical investigations. 4 ' 10 ' 13,14 For example, some of these terms have been bounded to extremely high precision using polarization measurements of distant radio galaxies. 10 In contrast, until re­ cently, the CPT-even terms have received little attention. Here, I review a recent study of these terms. 11 The CPT-even lagrangian for the modified electrodynamics is 4 £ = -\F^F"V

- \{kF)KXliVFKXF^

,

(1)

where F^v is the field strength, FM„ = d^Av — duA^. The first term is the usual Maxwell lagrangian. The second is an unconventional Lorentzviolating term. The coefficient for Lorentz violation, {kp)^^, is real and comprised of 19 independent components. The absence of observed Lorentz violation implies {kp)^^ is small. The equations of motion for this la­ a grangian are daF^ + (kp)^a/}JdaFf}'Y = 0. These constitute modified source-free inhomogeneous Maxwell equations. The homogeneous Maxwell equations remain unchanged. A particularly useful decomposition of the 19 independent components can be made. 11 The lagrangian in terms of this decomposition is C = |[(1 + ktv)E2 - (1 - ktr)B2} + \E ■ (ke+ + ke_) ■ E -\B

■ (ke+ - « e _) -B + E- (k0+ +

K0_)

•B ,

(2)

where E and B are the usual electric and magnetic fields. The 3 x 3 matrices Ke+, Ke_, k0+ and K 0 _ are real and traceless. The matrix k0+ is antisym­ metric, while the remaining three are symmetric. The real coefficient ktr corresponds to the only rotationally invariant component of (/CF)/KI/37From the form of Eq. (2), we see that the component ktr can be thought of as a shift in the effective permittivity e and effective permeability \x by

91

(e — 1) = —(A*"1 — 1) = ktI. The result of this shift is a shift in the speed of light. Normally, this may be viewed as a distortion of the metric. In fact, this result generalizes to the nine independent coefficients in « t r , ke- and k0+. To leading order, these may be viewed as a distortion of the spacetime metric of the form rfv —> ri^" + h1*", where ¥lv is small, real and symmetric. Small distortions of this type are unphysical, since they can be elimi­ nated through coordinate transformations and field redefinitions. However, each sector of the full standard-model extension contains similar terms. Eliminating these terms from one sector will alter the other sectors. There­ fore, the effects of such terms can not be removed completely from the theory. As a consequence, in experiments where the properties of light are compared to the properties of other sectors, these terms are relevant. How­ ever, in experiments where only the properties of light are relevant, the nine coefficients in ktT, ke- and k0+ are not expected to appear. The tests discussed here rely on measurements of birefringence. This involves com­ paring the properties of light with different polarizations. Therefore, these tests compare light with light and are only sensitive to the ten independent components of Ke+ and k0-. Constraints on birefringence have been expressed in terms of a tendimensional vector ka containing the ten independent components of ke+ and Ho-.11 The relationship between ke+, «„_ and ka is given by k

(ke+y

f-(k3+k4) =A;5 V k6

(2k2 -k9 9 («„_)''* = -k -2k1

V ks

k5 k6\ k3 k7 , k7k*J ka k10

\ .

(3)

kw 2{kl - k 2 ) J

Bounds on birefringence appear as bounds on \ka\ = Vkaka, the magnitude of the vector ka. 3. Birefringence In order to understand the effects of Lorentz violation on the propagation of light, we begin by considering plane-wave solutions. Adopting the ansatz F^u{x) — Ftlve~WaX'" and solving the modified Maxwell equations yields the dispersion relation p°± = (l +

p±a)\p\.

(4)

92 In a frame where the phase velocity is along the z-axis, the electric field takes the form E± o c ( s i n f , ± l - c o s f , 0 ) + O ( f c f ) .

(5)

To leading order, the quantities p, a sin £ and a cos £ are linear combinations of {kp)K\]xv and depend on v, the direction of propagation. A prediction of these solutions is the birefringence of light in the vacuum. Birefringence is commonly found in conventional electrodynamics in the presence of anisotropic media. In the present context, the general vacuum solution is a linear combination of the E+ and £?_. For nonzero a, these solutions obey different dispersion relations. As a result, they propagate at slightly different velocities. At leading order, the difference in the velocities is given by Aw = v+ — V- = 2(7 .

(6)

For light propagating over astrophysical distances, this tiny difference may become apparent. As can be seen from the above solutions, birefringence depends on the linear combination a sin £ and a cos £. As expected, these only contain the ten independent coefficients which appear in ke+ and re0_. Expressions for a sin £ and a cos £ in terms of these ten independent coefficients and the direction of propagation can be found in the literature. 11 Next I discuss two observable effects of birefringence. The first effect is the spread of unpolarized pulses of light. The second is the change in the polarization angles of polarized light.

3.1. Pulse-Dispersion

Constraints

The narrow pulses of radiation from distant sources such as pulsars and gamma-ray bursts are well suited for searches for birefringence. In most cases, the pulses are relatively unpolarized. Therefore, the components E± associated with each mode will be comparable. The difference in velocity will induce a difference in the observed arrival time of the two modes given by At ~ AvL, where L is the distance to the source. Sources which produce radiation with rapidly changing time structure may be used to search for this difference in arrival time. For example, the sources mentioned above produce pulses of radiation. The pulse can be regarded as the superposition of two independent pulses associated with each mode. As they propagate, the difference in velocity will cause the two

93

pulses to separate. A signal for Lorentz violation would then be a mea­ surement of two sequential pulses of similar time structure. The two pulses would be linearly polarized at mutually orthogonal polarization angles. The above signal for birefringence has not yet been observed. However, existing pulse-width measurements place constraints on Lorentz violation. To see this, suppose a source produces a pulse with a characteristic time width w$. As the pulse propagates, the two modes spread apart and the width of the pulse will increase. The observed width can be estimated as w0 ~ ws + At. Therefore, observations of w0 place conservative bounds on At ~ AvL ~ 2aL. The resulting bound on a constrains the tendimensional parameter space of ke+ and /c 0 _. Since a single source con­ strains only one degree of freedom, at least ten sources located at different positions on the sky are required to fully constrain the ten coefficients. Using published pulse-width measurements for a small sample of fifteen pulsars and gamma-ray bursts, we found bounds on a for fifteen different propagation directions v. Combining these bounds constrained the tendimensional parameter space. At the 90% confidence level, we obtained a bound of \ka\ < 3 x 10~ 16 on the coefficients for Lorentz violation. 11 3.2. Polarimetry

Constraints

The difference in the velocities of the two modes results in changes in the polarization of polarized light. Decomposing a general electric field into its birefringent components, we write E(x) = (E+e~ip+t + E_e~ip-t)etp''s. Each component propagates with a different phase velocity. Consequently, the relative phase between modes changes as the light propagates. The shift in relative phase is given by A = (p0+-p°_)t~4iraL/\,

(7)

where L is the distance to the source and A is the wavelength of the light. This phase change results in a change in the polarization. The L/X dependence suggests the effect is larger for more distant sources and shorter wavelengths. Recent spectropolarimetry of distant galaxies at wavelengths ranging from infrared to ultraviolet has made it possible to achieve values of L/X greater than 10 31 . Given that measured polarization parameters are typically of order 1, we find an experimental sensitivity of 10~ 31 or better to components of ( & F ) K A ^ In general, plane waves are elliptically polarized. The polarization el­ lipse can be parameterized with angles tp, which characterizes the orienta­ tion of the ellipse, and x — ±arctan " ' * ] " !"ds' w m c n describes the shape

94 of the ellipse and helicity of the wave. The phase change, Acf>, results in a change in both tp and x- However, measurements of \ are not commonly found in the literature. Focusing our attention on if), we seek an expression for Sip = tjj — ip0, the difference between ip at two wavelengths, A and AoWe find11 5ib = - tan" 1

sin

£ c o s C o + cos|sinCocos((ty - = iira(L/X — L/\Q), £ = £ — 1i\)§ a n d -0.5

-5.