Clusters of Galaxies (Space Telescope Science Institute Symposium Series)

Collected in this volume are the review papers from the Space Telescope Science Institute symposium on Clusters of Galaxies held in May 1989. Fifteen experts in the field have presented summaries of our current understanding of the formation and evolution of clusters and their constituent galaxies. Subjects covered include the existence and importance of subclustering, models of the evolution of clusters and the intracluster medium, the effect of the cluster environment on galaxies, observations of high redshift clusters, and the use of clusters as tracers of large-scale structure. This book provides a timely focus for future observational and theoretical work on clusters of galaxies.

SPACE TELESCOPE SCIENCE INSTITUTE SYMPOSIUM SERIES: 4 Series Editor S. Michael Fall, Space Telescope Science Institute

CLUSTERS OF GALAXIES

SPACE TELESCOPE SCIENCE INSTITUTE

Other titles in the Space Telescope Science Institute Symposium Series 1

Stellar Populations Edited by C.A. Norman, A Renzini and, M. Tosi 1987 0 521 33380 6 2 Quasar Absorption Lines Edited by C. Blades, C.A. Norman and, D.Tumshek 1988 0 521 34561 8 3 The Formation and Evolution of Planetary Systems Edited by H.A. Weaver and L.Danly 1989 36633 X 4 Clusters of Galaxies Edited by W.R. Oegerle, MJ. Fitchett and L.Danly 1990 0 521 38462 1

CLUSTERS OF GALAXIES Proceedings of the Clusters of Galaxies Meeting Baltimore 1989 May 15-17

Edited by WILLIAM R. OEGERLE Space Telescope Science Institute MICHAEL J. FITCHETT Space Telescope Science Institute LAURA DANLY Space Telescope Science Institute

Published for the Space Telescope Science Institute

The right of the University of Cambridge to print and tell alt manner of book} was granted by Henry VIII in 1534. The University hat printed and published continuously since IS84.

CAMBRIDGE UNIVERSITY PRESS Cambridge New York Melbourne

Port Chester Sydney

Published by the Press Syndicate of the University of Cambridge The Pitt Building, Trumpington Street, Cambridge CB2 1RP 40 West 20th Street, New York, NY 10011, USA 10 Stamford Road, Oakleigh, Melbourne 3166, Australia © Cambridge University Press 1990 First published 1990 Printed in Great Britain at the University Press, Cambridge British Library cataloguing in publication data Library of Congress cataloguing in publication data

ISBN 0 521 38462 1 hardback

CONTENTS Preface Participants

xi xiii

Chapter 1 Cosmology and Cluster Formation P. J. E. PEEBLES The Statistics of Clusters of Galaxies Biasing The Sequence of Creation Did Clusters Form From Gaussian Fluctuations? References Discussion — N. Bahcall, Chair

1 1 4 6 7 8 9

Chapter 2 Clusters of Galaxies: Structure, Infall, and Large-Scale Distribution M. J. GELLER Substructure: Does It Exist? Infall Patterns and Q The Large-Scale Distribution of Clusters (Groups) Conclusion References Discussion — N. Bahcall, Chair

25 25 31 34 36 38 40

Chapter 3 Cosmogony with Clusters of Galaxies A. CAVALIERE, S. COLAFRANCESCO Introduction Morphologies The Search for a Mass Function Local Luminosity Functions X-ray Clusters in Redshift Space Concluding Remarks References Discussion — A. Oemler, Chair

43 43 44 48 51 53 56 57 59

Chapter 4 Cosmogony and the Structure of Rich Clusters of Galaxies M. J. WEST Introduction Cluster Formation in Gravitational Instability Models Cluster Formation in the Explosion Scenario Cluster Formation with Dark Matter

65 65 68 90 95

viii

Contents

Summary References Discussion

—

A. Oemler, Chair

101 102 104

Chapter 5 The Dark Matter Distribution in Clusters M. J. FITCHETT Introduction Motivation Cluster Dynamics X-ray Constraints Substructure and the Mass Distribution Gravitational Lensing Conclusions References Discussion — A. Oemler, Chair

111 111 112 113 117 121 127 130 131 133

Chapter 6 T h e Effect of t h e Cluster Environment on Galaxies B. C. WHITMORE Introduction Possible Mechanisms The Morphology-Density Relation The Size of Galaxies in Clusters The Distribution of Mass for Galaxies in Clusters Summary References Discussion — 0 . Richter, Chair

139 139 140 144 151 156 161 164 167

Chapter 7 Evidence for Gas Deficiency in Cluster Galaxies M. P. HAYNES Introduction HI Deficiency in Clusters Observations of the Virgo Cluster Constraints on the Sweeping Mechanism Induced Star Formation Recent Results for Early Type Galaxies Summary and Conclusions References Discussion — O. Richter, Chair

177 177 179 181 183 187 188 190 192 194

Chapter 8 Properties of Galaxies in Groups and Clusters A. SANDAGE Introduction The Virgo Cluster Survey

201 201 202

Contents Survey of the Fornax Cluster and Loose Groups Field Survey for the Ratio of Dwarfs to Giants Variation of Effective Size and Surface Brightness with Absolute Magnitude for E and dE Galaxies Are There Transfigurations Along the Hubble Sequence? Galaxy Clusters Are Still Young References Discussion — W. Oegerle, Chair

ix 213 213 215 218 219 223 225

Chapter 9 Dynamical Evolution of Clusters of Galaxies D. RICHSTONE Introduction Physical Processes Evolution of Spherical Virialized Clusters Formation and Evolution of Subclusters Summary References Discussion — W. Oegerle, Chair

231 231 232 236 239 248 249 250

Chapter 10 Hot Gas in Clusters of Galaxies W. FORMAN, C. JONES Hot Gas In Galaxies, Groups, and Clusters Importance of Studies of the Hot Intracluster Medium Dynamical Classification of Clusters of Galaxies and the Role of the Central Galaxy The Origin of the Intracluster Medium Future Progress References Discussion — A. Meiksin, Chair

257 257 261 262 269 274 275 277

Chapter 11 Hydrodynamic Simulations of the Intracluster Medium A. E. EVRARD Introduction Theoretical Overview Numerical Details Not Another Coma Cluster The Hydrostatic Isothermal Model and Binding Mass Estimates Characteristics of the Ensemble Estimated Abundance Functions Summary and Discussion References Discussion — A. Meiksin, Chair

287 287 289 290 294 304 307 312 316 320 323

x

Contents

Chapter 12 Evolution of Clusters in the Hierarchical Scenario N. KAISER Introduction Self-Similar Clustering Allowed Range of Spectral Indices Application to Physically Plausible Spectra Optical Clusters X-ray Clusters References Discussion — M. Fall, Chair

327 327 328 330 332 332 333 336 336

Chapter IS Distant Clusters as Cosmological Laboratories J. E. GUNN Introduction Catalogs, Surveys and Outlook for the Future The Evolution of Cluster Galaxies The Implications of the Dynamics of Distant Clusters The Epoch of Galaxy Formation Parting Comments References Discussion — M. Fall, Chair

341 341 342 345 347 349 350 350 351

Chapter 14 Future Key Optical Observations of Galaxy Clusters J. P. HUCHRA Introduction Internal Properties of Clusters Connection to the Environment Clusters and Large-Scale Structure A Prescription for the Abell Blues Summary References Discussion — R. Burg, Chair

359 359 360 362 366 369 370 371 372

Chapter 15 Cluster Research with X-ray Observations R. GIACCONI, R. BURG Introduction X-ray Luminosity Function Interpretation of the Luminosity Function Future Surveys Summary References

377 377 379 384 385 394 394

PREFACE Clusters of galaxies are probably the largest gravitationally bound entities in the universe. They offer a laboratory for studying such diverse astrophysical problems as the form of the initial fluctuation spectrum, the evolution and formation of galaxies, environmental effects on galaxies, and the nature and quantity of dark matter in the universe, as well as providing tracers of the large-scale structure. The view that clusters are dynamically relaxed systems has been challenged by the demonstration of significant substructure in the galaxy and X-ray distribution within clusters (see the chapters herein by Geller, Cavaliere & Colafrancesco, Fitchett, Richstone, and Forman). There is, however, still some dissent on the reality of subclustering (see the discussion in West's chapter). New simulations of the formation and evolution of the dark matter and gas distributions in clusters are giving interesting results—their confrontation with observations may yield information on the nature of the initial density fluctuations required to form galaxies and enable us to solve some of the problems in this field (e.g., the so-called "^-discrepancy"). The simulations should also allow for better comparisons between theory and optical and X-ray observations (see the chapters by Cavaliere & Colafrancesco, Evrard and West). The abundance and velocity dispersions of rich clusters, and measurements of their clustering properties and peculiar motions may provide strong constraints on theories of galaxy formation (see the chapters by Kaiser, Peebles and West). The effect of the environment on galaxies in rich clusters and compact groups (eg. tidal and ram-pressure stripping of galaxy halos and the morphology—density relation) is discussed in the chapters by Whitmore, Haynes and Sandage. The discovery of 'luminous arcs' in several intermediate redshift clusters may lead to a better understanding of the dark matter distribution in clusters (see the chapter by Fitchett). The observations of high-redshift clusters are just beginning to provide clues to the evolution of clusters and their constituent galaxies (see Gunn's chapter). This book is a collection of review papers and discussions from the workshop entitled "Clusters of Galaxies" held at the Space Telescope Science Institute (ST Scl) during May 15-17, 1989. The workshop sought to bring together observers and theorists to discuss the observations, their interpretation, and the models of cluster formation and evolution. The program covered the dynamical state of clusters on the first day of the meeting, observations and theory of the influence of the cluster environment on galaxies during the second day, and the evolution of clusters on the third day. The meeting concluded with reviews of the key optical and X-ray observations that need to be obtained (reviews by Huchra and Giacconi & Burg, respectively). This book will appear in print just as the Hubble Space Telescope and the ROSAT X-ray telescope are launched. Several of the chapters herein discuss what should be investigated by these missions, and some predictions have been made. It would be very interesting indeed to hold this meeting again in five years time, and discuss the new results that we can only speculate on now. Many workers in the field contributed greatly to this meeting, either by the presentation of poster papers or the participation in the discussion sessions. The poster papers were were bound and distributed to workshop participants, and mailed to a number of Astronomy department libraries. One of the hallmarks of the ST Scl workshops is the large amount of time devoted to discussion after each talk. Discussion periods varied greatly in time, but averaged ~ 30 — 40 minutes each. We have painstakingly transcribed the discussions after each talk from audio tape, and they appear here at the end of each chapter. We hope that the discussions capture the true flavor of the

XII

meeting. The success of this workshop and the publication of it's proceedings are due to the efforts of many people at ST Scl. The local organizing committee, who set the scientific program, consisted of Neta Bahcall, Rich Burg, Holland Ford, Riccardo Giacconi, Colin Norman, Brad Whitmore, and the undersigned. The smooth running of the meeting was due to months of preparation by Barb Eller, who arranged for all the accomodations and food for more than 100 visitors, as well as taking care of endless details that we would never have dreamed of. Sarah Stevens-Rayburn and Rod Fansler took care of the finances (they kept us from deficit-spending). We thank the the Facilities department and the ST Scl Science Data Analysts for technical assistance in running the meeting. We thank Dave Paradise for making photographs of the figures. Finally, we thank Dorothy Whitman and Ron Meyers in our Publications department for editorial assistance, and especially Rob Miller for cheerfully and expertly making the numerous changes to the manuscripts to produce these proceedings.

Bill Oegerle Mike Fitchett Laura Danly

PARTICIPANTS Luis Aguilar James Annis Lee Armus John Bahcall Neta Bahcall Stephen Balbus Chantal Balkowski David Batuski Mark Bautz Timothy Beers Suketu Bhavsar Chris Blades Elihu Boldt Kirk Borne Gregory Bothun Richard Bower Richard Burg Jack Burns Chris Burrows Claude Canizares Alphonso Cavaliere Veronique Cayatte Stephane Chariot Dennis Cioffi Ray Cruddace Ruth Daly Laura Danly R. R. De Carvalho Herwig DeJonghe Van Dixon S. G. Djorgovski Megan Donahue Eli Dwek Joanne Eder A. C. Edge Jean Eilek Richard Elston August Evrard Michael Fall James Felten Harry Ferguson Michael Fitchett William Forman Bernard Fort Andrew Fruchter Margaret Geller Daniel Gerbal

Riccardo Giacconi Riccardo Giovanelli Daniel Golombek James Gunn Herbert Gursky Asao Habe Robert Hanisch Bill Harris Martha Haynes J. Patrick Henry Gary J. Hill John Hill Paul Hintzen John Huchra Walter Jaffe Fred Jaquin Robert Jedrzejewski Roman Juszkiewicz Nick Kaiser Neal Katz Stephen Kent Randy Kimball Anne Kinney Michael Kowalski Gerard Kriss Michael Kurtz Ofer Lahav Kenneth Lanzetta Tod Lauer Russel Lavery Ray Lucas Gerard Luppino Elliot Malumuth Eyal Maoz Bruno Marano A. Mazure Thomas McGlynn Brian McNamara Avery Meiksin Yannick Mellier Michael Merrifield Georges Meylan Richard Mushotzky Colin Norman William Oegerle Augustus Oemler R. P. Olowin

Frazer Owen Paolo Padovani James Peebles Vahe Petrosian Marc Postman Massimo Ramella George Rhee Douglas Richstone Otto Richter Rex Rivolo Hermann-Josef Roesser William Romanishin Vera Rubin Eduard Salvadore-Sole Allan Sandage Manuel Sanroma James Schombert Robert Schommer Ethan Schreier Patrick Seitzer William Snyder Noam Soker Mitchell Struble Alex Szalay Ed Smith Eric Smith Gustav Tammann Peter Teague Edgar Thomas Peter Thomas Chris Thompson Scott Tremaine Melville Ulmer C. Megan Urry Jacqueline VanGorkom Tiziana Ventura Duncan Walsh David Weinberg Michael West Raymond White Richard White Brad Whitmore Barbara Williams Michael Wise Gianni Zamorani Stephen Zepf Esther Zirbel

COSMOLOGY AND CLUSTER FORMATION

P. J. E. Peebles Joseph Henry Laboratories Princeton University Jadwin Hall Princeton NJ 08544

Abstract. I discuss some issues that arise in the attempt to understand what rich clusters of galaxies might teach us about cosmology. First, the mean mass per galaxy in a cluster, if applied to all bright galaxies, yields a mean mass density ~ 30 percent of the critical Einstein-de Sitter value. Is this because the mass per galaxy is biased low in clusters, or must we learn to live in a low density universe? Second, what is the sequence of creation? There are theories in which protoclusters form before galaxies, or after, or the two are more or less coeval. Third, can we imagine that clusters formed by gravitational instability out of Gaussian primeval density fluctuations? Or do the observations point to the non-Gaussian perturbations to be expected from cosmic strings, or explosions, or even some variants of inflation? These issues depend on a fourth: do we know the gross physical properties of clusters well enough to use them as constraints on cosmology? I argue that some are too well established to ignore. Their implications for the other issues are not so clear, but one can see signs of progress.

1. THE STATISTICS OF CLUSTERS OF GALAXIES To draw lessons for cosmology, we need not only the physical properties of individual clusters but also an understanding of how typical the numbers are. The issue here is whether the Abell catalog or any other now available is adequate for the purpose. There are known problems in the catalogs: they contain objects with suspiciously low velocity dispersions, and they miss systems whose X-ray properties might be consistent with massive clusters. Recently there has been considerable interest in the possible systematic errors this might introduce in estimates of cluster masses and spatial correlations (Sutherland 1988; Kaiser 1989; Dekel et al. 1989; Frenk et al. 1989). The points are well taken but I think the situation is not disastrous: if we take a balanced view, not attempting to push the data too hard, and taking care to look for supporting evidence from tests of reproducibility, we get some believable and useful measures. The cluster-galaxy cross correlation function, l+^cff(r), is the mean number density

2

P. J. E. Peebles

of galaxies as a function of distance r from a cluster, measured in units of the large-scale mean density. The fact that one finds consistent estimates of £Cg from different cluster distance classes (with reasonable choice of parameters in the luminosity function) is evidence that the typical richness of the cluster sample does not vary substantially with distance. The number of bright galaxies within the Abell radius r 0 = 1.5h~l Mpc (H = lOOh km sec" Mpc ) around a cluster is larger than expected for a homogeneous distribution by the factor

nVa

-4

The original estimate (Seldner and Peebles 1977) is N(< ra)/nVa = 360; the reanalysis by Lilje and Efstathiou (1988), which uses better cluster distances and galaxy luminosity function, is half that. I adopt the mean with twice the weight for the newer value:

(„

The scatter around the mean value of N(< ra) surely is large, even for a given nominal richness class, because richness estimates are compromised by groups and clusters seen in projection. The rms scatter in iV(< r) from cluster to cluster is measured by the cluster-galaxy-galaxy correlation function, £cgg (Fry and Peebles 1980). Estimates of £cgg should be reworked using the better current distance scales and luminosity function; the old result is 6N

((*(C . The central luminosity density is about 4.5 X 1 0 1 2 / J ~ 2 L 0 / M p c 3 (in the B band; see e.g., Dressier 1978). Thus the massto-light ratio for the cluster core is ~ 300/i MQ/LQ, again at B. This value for the mass-to-light ratio in the core of a typical rich cluster is a example of the 'missing mass' or dark matter problem originally discovered by Zwicky (1933) in his application of the virial theorem to the Coma cluster. This value of the mass-to-light ratio is within the range obtained in detailed studies of rich systems.

Structure, Infall, and Large-Scale Distribution I

i

i

i

i

I

|

i

t

— r

i

I

r

1

20 —

V

i

|

i

i

i

1

27 1

1

n = 65 °r. mm = 744 km s"'

18

n (cD) = 25 (7,. r i (cD) = 773 km s"'

1

J

16

14 — to

0

| E

10

—

1 1 1 1 1

2

8 6

,

1

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czmin, there are three corresponding values of r. The caustics thus represent the boundary of the triplevalued region. Outside the caustics, the solutions are single-valued; inside they are triple-valued. For an optimal comparison of the data with an infall model, we need a complete redshift survey which extends to sufficiently large angular radius from the cluster center. Few (if any) such samples exist. However, there are at least four systems with sufficient data to make a preliminary and demonstrative comparison: A539, A1656 (Coma), A1367, and A2670. Figure 7 shows the azimuthally averaged data (Ostriker et al. 1988; Huchra et al. 1990; Sharpies et al. 1988) for these systems (see Regos and Geller (1989) for a complete description of the construction of these plots). The redshift samples shown are not magnitude limited; they include all the available data. In each cluster we use the observed angular distribution for a magnitude limited sample of galaxies to obtain an estimate of the spatial distribution. We assume spherical symmetry. With an estimate of the mean galaxy density in the field (de Lapparent, Geller, and Huchra 1989), we can take the observed galaxy number density enhancement as the matter density enhancement A(r). In so doing, we tacitly assume that the galaxies trace the matter distribution. Given A(r) we can calculate the caustics as a

Structure, In/all, and Large-Scale Distribution

S3

TURN (H0R0 cos 9)

9

9 'mm

a)

^intersect

b)

Figure 5. a). The geometry of a cluster of galaxies where 0, C, and G are the observer, the cluster center and a galaxy, respectively. Ro is the distance to the cluster center, b). A cluster in redshift space. The curves A and B are the caustics and the arrows denote the direction of increasing phase space density. function of il from Equation (6). Note that for a particular A(r) the amplitude of the caustics is a function of Si only. The caustics in Figure 7 are marked with the relevant value of fi. For a given Si, variation in the form of A(r) changes the form of the caustics. Thus, in principle, by fitting the caustics to the data, we could test the assumption that the galaxies trace the matter distribution. For all four clusters in Figure 7, the density of points (galaxies) in redshift space drops substantially outside the caustics for Si ~ 0.2 — 0.5. Given the assumptions, the data appear to be consistent with low values of Si. In the data, the caustics are not apparent near the turnaround radius (where the predicted caustics meet). In this region the infall velocities are small and the density contrast associated with the caustics is not observable. These preliminary comparisons indicate that it is probably worthwhile to carry out magnitude limited redshift surveys (deep enough to include £ 100 galaxies in the infall region) which cover the infall region in a judiciously chosen set of clusters. The estimates of Si obtained by fitting the caustics are independent of those derived from the cluster core and apply to a larger spatial scale (~ 1 — bh Mpc). Fitting the caustics to sufficiently dense data could also provide constraints on the relative distribution of dark and light-emitting matter in the region. The technique is limited by some of the

34

M. J. Geller

N O

Figure 6. Observed velocity, cz, as a function ofr for fixed 0 in the range 0< 0 < 90°. problems which plague analysis of the core region. Substructure (groups) in the infall region are a problem as is asymmetry of the system. These problems might be amenable to treatment with N-body simulations.

3. THE LARGE-SCALE DISTRIBUTION OF CLUSTERS (GROUPS) On the largest scale, clusters of galaxies could be convenient markers of large-scale structure in the distribution of galaxies. However, the results of statistical analyses of existing catalogs remain poorly understood. So far several groups have used the language of correlation functions (Peebles 1980) to describe the cluster distribution. Bahcall and Soneira (1983) calculated the two-point correlation function for the Hoessel, Gunn, and Thuan (1980) sample of 104 nearby Abell clusters. Postman, Geller, and Huchra (1986) calculated the cluster correlation function for a variety of other samples drawn from both the Abell and Zwicky catalogs, and Shectman (1985) analyzed a sample of clusters drawn from the Shane-Wirtanen counts (1967). More recently Huchra et al. (1990) observed and analyzed a sample of 145 Abell (richness R> 0) clusters at high galactic latitude and with distance class D1 Abell clusters with velocities cz < 15,000 km s""1, at least two are superpositions of groups (and/or foreground galaxies) along the line-of-sight. Neither of these systems has a close neighbor which is also a cluster in the catalog (Zabludoff, Geller, and Huchra 1990). Erroneous mean redshifts also arise from the superposition problem with somewhat surprising frequency; in fact, there appears to be a bias in the Abell catalog toward superpositions and toward identifying concentrations of galaxies on the sky which appear to be associated with an apparently bright galaxy (which may well be foreground). It is dangerous to base cluster redshifts on a single redshift measurement and the danger, of course increases with redshift! It seems that the only reliable approach to these problems is well-defined sampling of the redshift distribution in the direction of each cluster. The flip side of the superposition problem is the failure to identify systems which show up as fingers in complete redshift surveys (Ramella, Geller, and Huchral1989). In the first two slices of the CfA redshift survey there are 2 R = 1 clusters; there are 4 groups (including the 2 Abell clusters) which have physical properties indistinguishable from those of Abell R = 1 systems. Although the rich clusters selected by Abell are apparently biased tracers of the distribution of individual galaxies, groups of galaxies selected from complete redshift surveys do appear to trace the structure in the galaxy distribution (Ramella, Geller, and Huchra 1990). Figure 8a shows the distribution of galaxies in the first two slices of the CfA redshift survey extension. These slices cover the declination range 26.5° < 6 < 38.5°. Figure 8b shows the distribution of group centers in the two slices. Note that the group centers trace the large-scale features visible in the galaxy distribution of Figure 8a. Perhaps not surprisingly the correlation function for the 128 groups is consistent with the correlation function for the galaxy distribution (Ramella, Geller, and Huchra 1990). It would be valuable to have model predictions of the correlation function for groups selected from N-body simulations in the same way that they are selected from the data.

4. CONCLUSION In principle clusters of galaxies offer probes of the development of large-scale structure on scales from a fraction of a Megaparsec to hundreds of Megaparsecs. However, on each scale there are profound, but clearly defined issues which could be at least partially resolved by a combination of well-designed observations and models. On scales ^ lh Mpc, the internal dynamics of individual clusters can provide insight into their history. There are a number of systems which clearly have substructure. Controversy centers Figure 8. (opposite page) a) A cone diagram for galaxies in the declination range 26.5° < 8 < 38.5° and with cz < 12,000 km s~l. b). A cone diagram showing the distribution of the centers of 128 groups in the same declination slice. The crosses denote the Abell clusters in the region, all of which are detected by the group-finding algorithm.

Structure, Infall, and Large-Scak Distribution

right ascension

15

16

10000

5000

10.0 < m § 26.5 § 6
Af~2 at the low end, albeit close to their resolution limit, is interesting as it puts considerable strain on theories of hierarchical clustering that suggest flatter slopes. Models of hierarchical clustering that attempt to include non-linear collapses end up in mass distribution functions (MFs) of Schechter-like form (3) N(M,z)dM oc p(z)M~1(z) f(m)dm, f(m) = m " r c" 1 " 6 with the mass m normalized to a unit proportional to Mc. This general structure is that expected on dimensional grounds, and the form of f(rn) quantifies the notion of a wide distribution with considerable weight (modulated by F < 2) at the high-M end before a cutoff (modulated by 0 ^ 1). In fact, Press and Schechter (1974) obtained a MF of the above form with F, 0 specified in terms of initial conditions, based on a two-step derivation. A golden rule N(M, z)MdM = —dF is assumed to relate the mass density of clumps just in the range M — M + dM with the differential of the fractional mass in all objects gone non-linear by the redshift z. An ansatz for F suggests adding up the independent volumes of size Re, and mass Ms oc pRg, wherein the mean overdensity exceeds a threshold of non-linearity 6C ^ 1: F(Ms,z)

oc p f°°dv e-" 2 /2 .

(4)

JVc

The mass appears explicitly only in the limit vc = 6c/-1.8.

-5

-10

-2

-1 0 Log M/M o

Figure 3. The mass functions for cores collapsed at density peaks (dashed line), and for maximal halos (continuous line), are compared with that from the full Press-Schechter theory (dotted line). Here and in the following: CDM spectrum, threshold Se = 1.33, Qo = 1. Here we adopt b/B = 1.5.

Cosmogony with Clusters of Galaxies

51

The number of low mass objects may be increased by additional cores arising from constructive superpositions of smaller sub-threshold volumes: an upper bound may be estimated granting priority to the peak collapses, but adding a maximal collapse chance Fps/2 to overdense random spheres in all the truly residual volume. This means using the golden rule F = Fp(M) + (1 — Fp)Fps(M)/2. A large infall may dominate the statistics of added collapses in the MF. To fix the realistic size of the halos Cavaliere and Colafrancesco (1989, in preparation) first re-interpret in terms of energy-like conditions the geometrical S/N £ 1 definition of a gravitational range. Thus the asymptote is seen to be set by any breakdown of the homogeneity, isotropy and high density conditions in the environment of the accreting objects; such breakdowns are implied by the transient cellular structure delineated by Doroshkevich (1970) and by Shandarin and Zeldovich (1989) on the basis of quasilinear deformations rather than linear overdensities. By the time taken for the potential associated with a peak to draw mass inflows from large radii, other potentials on comparable scales (albeit in their quasilinear regime) exert a long-A forcing (% oc 6%k~*) of the flows into filaments or sheets (with intervening voids), setting to infall an effective dimensionality D < 3 on large scales. The upper limit is decreased to Daa < 1 if on halo scales n > —OAD holds. A limit Daa £> 1/2 holds in subcritical conditions. The shape of the leading edge of the MF, on the other hand, is very sensitive to the selection of these objects. To demonstrate this sensitivity, consider that linear bias in the simple form a oc b~ < 1 competes with non-linear delays introduced by small-scale substructures: to lowest order, this may be represented with a dispersion increased by a factor B i£ 1, to yield a cutoff oc exp[—(6/B)2m2a/2]. In addition, given the average collapse threshold 6C, soft clipping of the kind envisaged by Szalay 1988 and by Bonometto and Borgani 1989 may further soften the cutoff. 4. LOCAL LUMINOSITY FUNCTIONS Over and above the complexities in determining the parameters F (slope) and 0 (cutoff) in the basic form of Equation (3), one general point that emerges is that the local MF will contain some time-integrated information relevant to cosmogony, especially when effects of finite time are important. The next question is, how these features are reflected in the luminosity functions (LFs). In the optical band, the mass-to-luminosity ratios show an apparent increasing trend M/Lo ~ Me with a slope e ~ 0.3, cf. Hoffman et al. 1982; such a trend is consistent with an overall density at the critical value. This implies some steepening of the optical LF relative to the MF, namely

N(L0) oc 4-r+0/d-0.

(9)

The result for the core LF falls short of existing data for groups (see Figure 4), but only marginally considering that the actual uncertainties may be larger than the formal errors (Figure 4). In X-rays, instead, a flattening is expected because Equation (1) yields M/Lx oc g~ M~ ' . For the LF of cores localized around peaks the flattening is considerable: N{LX) = N(M)dLx/dM -+ L"?, (10) with 7 = 0.75(F + 0.3) that takes on the value ~ 1.4 for CDM. Compared with the observations by Johnson et al. 1983, and Kowalski et al. 1984, this is too flat - see

52

A. Cavaliere and S. Colafrancesco

-5

-

2

.3

L V /1O 1 3 L G Figure 4. 7Vie optical luminosity functions for cores collapsed at peaks with M/L = 200 (dashed line) is compared with that for halos (Daa = 1, continuous line), assuming a considerable mass fraction to be visible with M/L oc M"'3 (see discussion in the text). Data points from Bahcall 1979, with the rich cluster luminosities shifted as shown by the triangles to convert to luminosities at given contrast. Figure 5. Note that the X-ray emission tends to enhance the inner regions, so that the mass sampled is not necessarily the same as in the optical. We are examining the "visibility" of the halos, and the prospects are as follows. In the optical, halos to be relevant must be flatter than p oc r~^ + e (which is likely, cf. Ryden 1988 and references therein), and old, so as to stand out against the background. In X-rays, where the background is essentially given, the brightness distribution may remain marginal in spite of a ICP disposition flatter than that of the galaxies (the /? models again); in any case, low brightness halos in the local objects call for wide

Cosmogony with Clusters of Galaxies

53

apertures. On the other hand, accreting halos may cause a secular increase of the central h (t); similar increasing trends are seen in N-body experiments, but they are sensitive to physical, and also to numerical, dissipation. Note that a further flattening of the X-ray LF may be associated with a varying ICP fraction. Indications that MjM = g ^ constant holds, come from comparing galaxies accessible to detailed X-ray mappings that contain or retain only a small ICP mass compared with the mass in stars M/M* & 10 (c/. Fabbiano 1988), with rich clusters where the ratio is quite large, up to values ~ 5 (Blumenthal et al. 1984). Clearly at some intermediate scale or formation epoch more diffuse baryons (relative to those bound into stars) are to be differentially produced, retained or engulfed in the deepening and enlarging potential wells. Because stars, once formed under a reasonable IMF, return on average £> 1/2 of their mass in diffuse baryons, engulfing or infall is bound to dominate eventually, and is likely to concern "failed" galaxies. Considerable dilution by material of primeval composition is required anyway to maintain the definitely subsolar composition observed in the ICP, starting from the high yields of leading stellar evolutionary models (cf. reviews by Matteucci 1989, Giannone and Angeletti 1989). Quantitative indications for M in the range from groups to clusters are obtained by David et al. 1989 (and concurringly by Oemler, private communication), who find the ratio within a fixed radius to increase by a factor ~ 4. Now, there may be a size dependence g(M) or an epoch dependence g(t). A limit to a pure size dependence q(M) ex MJJ is given by n ~ 0.3. In fact, the total X-ray luminosity follows Lx oc M ' + ** oc JV^ ' (where JV^ = number of galaxies within the Abell radius, see Kaiser 1986); comparing with the data collations by Bahcall 1979b, Mushotzky 1984, the resulting upper bound is n £> 0.4. A size dependence g oc M * would further flatten the slope of the X-ray LF by an additional 0.2.

5. X-RAY CLUSTERS IN REDSHIFT SPACE The ICP content may have a primary epoch dependence, g(t). The issue is best discussed in comparison with a reference baseline, provided by non-evolutionary cluster sources distributed homogeneously in look-back time. To represent empirically the X-ray local data one can use a Schechter-like function, N(Lx)dLx

oc A f - ^ - T e " * it

(11)

with Lx normalized to a unit proportional to Lc oc pc Mc ; the slope consistent with the data by Johnson et al. 1983, Kowalski et al. 1984 is 7 ~ 1.7 - see Figure 5. If one assumes the LFs at higher z to be invariant with constant normalizations Mc and Lc, the expected number counts and z- distributions are as given in Figures 6 and 7. Note how - given the evolution, no evolution in this particular case - both these integrals over the distant LFs depend on their shape because the objects observed in a flux-limited survey "slide" down the LF with increasing redshift. Can we expect the invariant case to be realistic? Actually, any HCS for the formation of cosmic structure implies strong changes outside the local environment; specifically, it implies strong changes of the density-evolution type along with some luminosity antievolution (Kaiser 1986). This is because the HCS holds the same amount of mass to be reshuffled into larger and larger units along the cosmic arrow of time. As we look back, we expect to see more numerous, smaller units, that are also denser and cooler, see Equation (2). Thus the opposite behaviours of p(z), M(z) tend to cancel each

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A. Cavaliere and S. Colafrancesco

o

-8

-10

Figure 5. The local X-ray luminosity function fitted with the empirical Schechter-like function given in eq. 11: 0 = 1 and 7 = 1.7 (dotted curve). We also show the LF for cores collapsed around peaks (dashed curve). Here and in the following figures. b/B = 1. Data points from Kowalski et al. 1984- Luminosity unit: 10 erg/s in the range 2 — 6 keV; vertical unit: # Mpc~ 3 /10 44 erg s~l. other out of the X-ray emission, and if a = constant is assumed, one expects a weakly decreasing Lc(z) <x (1 + z)( 5 + 7n )/ 2 ("+ ; 0. Correspondingly, the comoving LF behaves scale-invariantly as given by N(Lx,z) K

oc M-\z)L-\z)

/ ( / ) oc (1 + zf

= [7 + 5 7 + 7n( 7 - l)]/2(n + 3)

(12)

(Cavaliere and Colafrancesco 1988). Figures 6 and 7 include representations of number

Cosmogony with Clusters of Galaxies

55

Eh

A

-2

-1

Log F/F_ 12

Figure 6. Integral counts of X-ray clusters for representative evolutions of the empirical Schechter-like local LF of previous figure: A) homogeneous distribution in redshift; B) scale-invariant evolution of the luminosity function, corresponding to the approximate Equation 12; C) epoch-dependent ICP content g(z) ~ (1 -f z ) ~ 1 2 . Curve D) shows instead the scale-invariant evolution of the LF for peaks. Preliminary counts from the EMSS of Gioia et al. (1988) are also shown. Flux unit: 10 erg/s cm in the range 0.3 - 3.5 keV; counts: # sr'1. counts and ^-distributions under these conditions. To what extent does the scale-invariant limit hold? Actually, we have already mentioned indications that the ICP content may vary or change. Within the limits derived in the previous Section for a pure size dependence, the alteration to the scaleinvariant evolution is small. Epoch dependence can be effectively tested at z ^ 0.5. Cavaliere and Colafrancesco 1988 considered an average dependence g(z) oc (1+z)" 3 ^/ 2 . In the limit of constant time-rate £ = 1, it follows that g (z) oc p~ (z) cancels the dependence p(z) from the emission and gives way to the strong decrease of Mc(z). The result is the addition to the exponent K in the previous equation of a negative AK = —3£(7 — 1), which implies counts considerably flatter, and z-distributions cut off faster than by the mere sliding down the LF. Figures 6 and 7 illustrate the difference between these alternatives, computed numerically to include relevant details. Cavaliere and Colafrancesco 1988 discuss the robustness of these behaviours of the counts and zdistributions: in particular, it is plain that a flatter shape of the LF may well flatten the counts at medium and low fluxes, but cannot sharpen the decline in the z-distribution. Note that an epoch dependence g(z) converts to an apparent size dependence, since groups and clusters of richness 0 are on average smaller and older as well: on the basis of Mc(z) from Equation (2), the equivalence is given by rj = 3a£/2 ~ 0.3. To wit, typical groups formed at z ~ 2.5 would have an ICP content smaller by (1 + z)~ 1 °/( n + 3 ) compared with rich clusters, in accord with the result of David et al. 1989.

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A. Cavaliere and S. Colafrancesco 1

1

1

1

! ^ a)

,

_r

c -1 i...

"1

->—I

—1_

-

b)

D ....... r-

—L_J

I

j 1

.5

1

1

11

B i

i

i

1.5

Figure 7. The redshift distributions corresponding to Figure 6, to the flux level of 10" 1 3 erg/s cm2. Refinements of the selection procedure for cluster sources from the EMMS (Gioia et al. 1987), or the new survey planned from ROSAT, will discriminate between these evolutionary patterns.

6. CONCLUDING REMARKS Epoch dependence of the ICP content may be expected in a biased HCS. This is because stellar time scales may well start and govern the first build up of the ICP, in regions of high total overdensity where galaxies can condense and star formation is efficient. But with the progress of the clustering, the deeper and larger potential wells accrete or engulf regions originally underdense on large scales; there galaxy formation is delayed and star generations are hindered by sub-threshold conditions, so that the content in diffuse baryons of primeval composition remains high. After such failed galaxies are engulfed into the hot ICP, star formation activity is permitted to resume only in localized cooling flows. A dearth of early ICP will maintain the cooling time comparable to the age, and so preclude early pervasive cooling flows with a large soft X-ray emission. It also implies

Cosmogony with Clusters of Galaxies

57

statistical Sunyaev-Zel'dovich effects somewhat less than the canonical estimate, cf. Cole and Kaiser 1988, because the dearth would be already effective at z ~ 0.5. The other implication is a metal abundance diluted toward us, in particular a decreasing equivalent width of the Fe lines. To sum up, we have stressed that linear perturbation theory may be well defined (e.g., the CDM spectrum depends specifically on the normalization o~o[8Mpc] = b ) but non-linear dynamics introduces many complexities. Some of these are welcome, like the increased variance that within the HCS may yield in priciple a range of morphologies as wide as observed. But specific models with little power on small scales, like CDM, may have a hard time in achieving this. This is an area still open to calibrated N-body experiments. As for the luminosity functions, these complexities require more parameters to enter the theories, to reflect such important features as sites and actual masses of successful collapses (including secondary infall), or the feedback of internal substructure and ambient geometry (including the effective dimensionality of the collapses). At the bright end of the luminosity function, where today rich clusters lie, the effects of non-linear variance or soft clipping may counteract those of linear bias: the relative abundances of these rare objects are particularly sensitive to the balance. Here stands another crucial knot for ensemble N-body experiments to disentangle. At the faint end, for poor clusters and groups, we see requirements for steeper luminosity functions especially in X-rays, relative to the predictions by a simple peak theory. A concurring indication for the mass function comes from the N-body experiments. Added collapses (and persistence), or added age-dependent mass, constitute possible solutions. We have explored effects of infall, which is inevitable with gravitational instability, and is now directly observable around rich clusters (see Regos and Geller 1989). Infall is conceivably enhanced by spectra flat towards small masses in a flat universe. The associated steepening effect looks interesting for the MF, possibly also for the optical LF, but not necessarily so in X-rays, unless numerical experiments prove flat brightness distributions or a stimulated increase of the ICP central densities. We end by noting that delays associated with non-linear dynamics (and bias) may further enhance the appreciable amount of groups still forming today in the timeresolved picture we propose. In these terms one could understand the finding by Hickson et al. 1988 of a population of compact, but dynamically unevolved groups. We are grateful for many discussions to S. Matarrese and N. Vittorio, and especially to R. Scaramella for fruitful exchanges.

REFERENCES Arnaud, K.A., Johnstone, R.M., Fabian, A.C., Crawford, C.S., Nulsen, P.E.J., Shafer, R.A., and Mushotzky, R.F. 1987, M.N.R.A.S., 227, 241. Bahcall, N.A. 1979, Ap. J., 232, 689. Bahcall, N.A. 1979b, Ap. J. Lett., 232, L83. Bardeen, J.M., Bond, J.R., Kaiser, N., and Szalay, A.S. 1986, Ap. J., 304, 15. Bertschinger, E. 1987, Ap. J. Lett, 323, L103. Binggeli, B., Tammann, G.A., and Sandage, A. 1987, A.J., 94, 251. Blumenthal, G.R., Faber, S.M., Primack, J.R., and Rees, M.J. 1984, Nature, 311, 517. Bond, J.R. 1989, in Large Scale Motions in the Universe, Rubin V.C. and Coyne G. eds., Princeton: Princeton University Press, p. 465.

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Bonometto, S., and Borgani, S., 1989, preprint. Cavaliere, A., Gursky, H., and Tucker, W.H 1971, Nature, 231, 437. Cavaliere, A., Santangelo, P., Tarquini, G., and Vittorio, N. 1986, Ap. J., 305, 651. Cavaliere, A. 1980, in X-ray Astronomy, R. Giacconi and G. Setti eds., Dordrecht: Reidel, p.217. Cavaliere, A., and Colafrancesco, S. 1988, Ap. J., 331, 660. — 1989, in Large Scale Structure and Motions in the Universe, M. Mezzetti et al. eds., Dordrecht: Reidel, p.73. Cole, S., and Kaiser, N. 1988, M.N.R.A.S., 233, 637. David, L.P., Arnaud, K.A., Forman, W., and Jones, C. 1989, preprint. Davis, M., and Peebles, P.J.E. 1983, Ap. J., 267, 465. Dekel, A. 1989, in Large Scale Motions in the Universe, Rubin V.C. and Coyne G. eds., Princeton: Princeton University Press, p. 465. Doroshkevich, A.G., 1970, Astrophysica, 6, 320. Dressier, A., and Shectman, S.A. 1988, A.J., 95, 985. Efstathiou, G., Frenk, C.S., White, S.D.M., and Davis, M. 1988, M.N.R.A.S., 235, 715. Evrard, A.E. 1987, Ap. J., 316, 36. Fabian, A.C. 1988, in Hot Thin Plasmas in Astrophysics, R. Pallavicini ed., Dordrecht: Kluwer, p.293. Fabbiano, G., 1988, preprint. Fitchett, M.J. 1988, in Proc. of the Minnesota Astrophysics Lecture Series Large Scale Structure and Its Relation to Clusters of Galaxies. Fitchett, M.J., and Webster, R. 1987, Ap. J., 317, 653. Forman, W., and Jones, C. 1982, Ann. Rev. Astr. Ap., 20, 547. Geller, M.J. 1988, in Saas Fee Lectures, Large-Scale Structure in the Universe, Martinet L. and Major M. eds., Sauverny: Geneva Observatory, p.69. Geller, M.J., and Beers, T.C. 1982, P.A.S.P., 94, 421. Giannone, P., and, Angeletti, L. 1989, preprint. Gioia, I.M., Maccacaro, T., Morris, S.L., Schild, R.E., Stocke, J.T., and Wolter, A. 1988, in High Redshift and Primeval Galaxies, J. Bergeron et al. eds., Paris: E. Frontiers, p.231. Gunn, J.E., and Gott, J.R. 1972, Ap. J. 176, 1. Gott, J.R., and Turner, E.L. 1977, Ap. J., 216, 357. Hickson, P., Kindl, E., and Huchra, J.P. 1988, Ap. J., 331, 64. Hoffman, Y., Shaham, J., and Shaviv, G., 1982, Ap. J.. 262, 413. Johnson, M.W., Cruddace, R.G., Ulmer, M.P., Kowalski, M.P., and Wood, K.S. 1983, Ap. J., 266, 425. Kaiser, N. 1986, M.N.R.A.S. 222, 323. Kowalski, M.P., Ulmer, M.P., Cruddace, R.G., and Wood, K.S. 1984, Ap. J. Suppl., 56, 403. Lucchin, F., 1988, Morphological Cosmology, P. Flin ed., Lecture Notes in Physics, 332, p. 284. Matteucci, F. 1989, preprint. Mellier, Y., Mathez, G., Mazure, A., Chauvineau, B., and Proust, D. 1988, A. A. 199, 67. Mushotzky, R.F. 1984, Physica Scripta, T7, 157. Oegerle, W.R., Fitchett, M.J., and Hoessel, J.G. 1989, A.J., 97, 627.

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Peebles, P.J. 1974, Ap. J. Lett. 189, L51. Peebles, P.J. 1980, The Large Scale Structure of the Universe, Princeton: Princeton University Press. Press, W.H., and Schechter, P. 1974, Ap. J., 187, 425. Regos, E., and Geller, M. 1989, A. J., 98, 755. Ryden, B.S., 1988, Ap. J., 333, 78. Sarazin, C.L. 1988, X-ray Emission from Clusters of Galaxies, Cambridge: Cambridge University Press. Scaramella, R. 1988, Ph.D. Thesis at S.I.S.S.A. Schaeffer, R., and Silk, J., 1988, Ap. J., 332, 1. Shandarin, S.F., and Zel'dovich, Ya. B., 1989, Rev. Mod. Phys., 61, 185. Szalay, A., 1988, Ap. J., 333, 21. Triimper, J. 1988, in Hot Thin Plasmas in Astrophysics, R. Pallavicini ed., Dordrecht: Kluwer, p.355. West, M.J., Oemler, A., and Dekel, A. 1988, Ap. J., 327, 1. West, M.J., and Richstone, D.O. 1988, Ap. J., 335, 532. White, S.D.M. 1982, in Morphology and Dynamics of Galaxies, Martinet L. and Mayor M. eds., Geneva: Geneva Observatory, p. 289.

DISCUSSION Peebles: You showed us the results of N-body simulations in which you reproduced remarkably well the bimodal structures seen now in some clusters, both in the X-ray and galaxy counts. How would such an object look at a redshift of 1? Cavaliere: I would say, more fragmented and more sparse. Peebles: What is the effect of that as an X-ray source? Cavaliere: That depends on the brightness selection or brightness bias. Giacconi: The answer is certainly yes. The subclumps are denser and of higher surface brightness. Cavaliere: Not always, there is a point here which I would like to stress. These are bimodal configurations at earlier times. Felten: Would you say what the time sequence is in these plots? Cavaliere: Yes. We are seeing here three different projections at two different times. These are at three time units, and twenty-three time units, like ten Gyr from the beginning. The other question was, how that would be observable in X-rays? And this comes back to the figure that I show you (see Figure 2). So it is true, Riccardo, that

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a typical condensation would have a large surface-brightness, but the point that I am trying to make is that there is a large variance, so along with some typical condensations there are also many marginal ones. Those are subjected to what I call the brightness bias, which is strong because it is nonlinear, proportional to the density squared. So one is almost certainly bound to underestimate observationally the number of objects at earlier times in X-rays. Not to mention the other point that the amount of gas relative to the dynamical mass may have been less at those times than it is now: that is a separate issue. Fitchett: Many people have claimed that the bimodal structures that we have seen in simulation are well described by a linear orbit model. Is this actually accurate or do your structures persist for so long at maximum expansion that this doesn't fit? Cavaliere: We produced a number of plots which illustrate the run of the relative velocities of the two components. We found that in a typical bimodal run these are not badly off the data for A98, for example. We never published these results. Fitchett: I wanted to make the point that what is interesting is that the relative velocities observed for known bimodal clusters are small, and if you are observing these systems at a random orientation and time you would expect to see some large velocities. Cavaliere: So you mean that an interesting point is the velocity vs. time. That's right, I'll try to dig out those results for the Proceedings. Sandage: In both Virgo and in Coma we have evidence for substructure and it is very strong optical evidence. What is the X-ray distribution? Is it spherically symmetric about a center, or do you have two lobes? Cavaliere: Do you mean the observed or simulated X-ray distribution? Sandage: The observed. Cavaliere: There are people in the audience that know this more directly than me, but I just mention that there is a slight hint of an ellipticity in the Coma distribution of X-ray luminosity. As for the simulations, I stress the following point. In some numerical runs, it happens that even when you don't have a very strong bimodal configuration and you have only a hint of dumpiness at very early times, later on when the thing appears at a first look as a sphere, if you go to examine the simulated X-ray maps you will still find a trace of inner asymmetry. It is usually weak, ellipses with axial ratios 1.5 to 1 or something like that. Sandage: Why is it in Virgo where we have very strong evidence for two clusters the X-ray emission is spherically symmetric only about one. Cavaliere: Can some X-ray observer comment on Virgo? Mushotzky: There is X-ray emission centered around 4472. The surface brightness is low enough that it can't be that far away from the galaxy. The main emission is centered around M87 and the surface brightness is high enough to be well mapped out to a large angular scale.

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Sandage: How about in Coma? Mushotzky: In Coma, as Cavaliere said, it's elliptical and I don't remember exactly, but it is roughly about 0.6 elliptical. And then again, the angular scale is such that the Einstein data is not mapped as far out as the galaxies. Rhee: I have a question about the theoretical prediction of the frequency of occurrence of substructure in clusters, because from what I heard this morning, it is about 40% or less. And it seems that West using a rather rigorous approach starting with Gaussian fluctuations and then fixing the normalization using the amplitude and slope of the correlation function doesn't find evidence for the substructure today and yet you do, in your simulations, using rather more simple initial conditions. And I was wondering how simple your results are. There are things you didn't mention such as how you start your simulation at l/6th of the turn-around radius with 500 particles, and what I meant about the initial conditions is how sensitive are your results to those two parameters? Cavaliere: When speaking about substructure, I think that we must divide the subject into two parts: large scale substructure, with separations of the order of megaparsecs or more, and probably here everybody agrees that such things do exist in the simulations and do exist in the real world! And the small scales, very sensitive to the initial conditions, as I said. Now, here there are a number of contrasting facts. First of all, observationally there have been contrasting claims: Margaret Geller this morning has reviewed this matter and I will say no more on that. As for the simulations, there are discrepancies also there. Now, when you take n ~ 0 initial condition, simple random extraction from a uniform sphere, you overplay the small scale non-uniformities, that is, you put much power on small scales. So the small scale dumpiness in these simulations is emphasized. On the other hand, looking at the simulations by Efstathiou et al. or by West et al. , even they have some small scale substructure. Now, the point is semantic to some extent. That is, what do you define as substructure, what level of Ap//>, or mass of the largest subclumps compared to the total mass, do you call substructure? Even the substructures appearing in white-noise simulations, apart from bimodals, often consist of clumps with mass much smaller than the total, of order 10 % or so. So, this involves a threshold in a continuous distribution depending on n—some small scale substructure arises in all simulations and the thing that really varies is the contrast. The average mass fraction in the second largest clumps or so is enhanced in simulations with n ~ 0. My summary of the numerical experiments is that the merging mode of evolution is emphasized by initial white noise, while accretion of small subclumps onto a dominant one prevails for spectra flatter toward low masses. Oemler: Wouldn't it be fair to say that you have found more of the sort of binary structures for separations of the Mpc scale or so, than we found in our simulations? Cavaliere: Do you have a statistics? How many of your simulations do show bimodals? Oemler: Not many — 5 or 10% . Cavaliere: Whereas we have on the order of 20-30%. Another point is, how do you gauge the present time in your simulations. Oemler: It's normalized by the correlation function.

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Cavaliere: Do you notice in your simulation that the proportion of bimodals is strongly decreasing with time? Oemler: What's the answer to that, Mike? West: I'm going to address this in my talk. Oemler: Maybe we should wait until after Mike talks to answer that. Peebles: Could I ask how you handle the dark mass? Cavaliere: As a matter of fact, the dark matter is not included in these particular simulations. Peebles: Would it not worry you, for example, that the number of particles in those very small clumps is not large, perhaps 10. And, therefore, two body relaxation might be excessively over emphasized in the simulations as well as in the real world. Cavaliere: The two body relaxation should be coped with anyway by the appropriate choice of the softening length. This is generally on the order of 30 kpc for our simulations. Peebles: If you doubled the number of particles, would you reproduce the structure or would it look different? Cavaliere: We tried that, with the E4 simulation for example, and the structure persisted. There might be another point to consider—the settling of the galaxies relative to the dark matter caused by dynamical friction or energy absorbed into internal degrees already mentioned here. That might be a possible way of making more compact cores and hence a more visible X-ray emission. However, the amount of baryons relative to dark matter then is a very crucial parameter, the effect in the general potential would be very different if that fraction changed. Huchra: One thing that I want to say sort of in response to Allan's question but also in reference to people who like to make X-ray maps of clusters. If you look at the Virgo cluster there are indeed two very dense clumps of galaxies—one centered on M87, one centered on 4472. They have roughly, within a factor of 2, the same total luminosity in galaxies. But one is a relatively strong extended X-ray source and the other isn't. The one that isn't has a velocity dispersion of only about 450 or 500 kilometers per second, the one around 4472, whereas the one around M87 is 300 or 400 kilometers per second higher, about 800 kilometers per second. And the X-ray emission is going as a relatively strong power of the temperature, and the velocity dispersion is a measure of the temperature, which is what a lot of people are saying. I don't think it's crazy to say that this could mean that low velocity dispersion clumps like you might see in your simulations don't show up in X-rays because the temperature for the velocity dispersion is just too low. Cavaliere: Yes. Let me point out on my maps, what is mapped here is the total emission, but what you say is obviously very true once we are in an observational window. For instance it would be interesting to compare ROSAT with Einstein and see

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what the difference is. Sandage: May I ask you if it is fair to say that any cluster that you see clumps or subclusters, that cluster is in the epoch of its formation. Cavaliere: Well, that is a very interesting point. That is related to the problem of how we normalize a luminosity function in the presence of increased dispersion. I mean that if the collapse times have a large dispersion, then what you get is a rather soft cutoff at high masses. Now, what you call a critical mass or mass being non-linear now, that is observationally influenced very much by the dispersion of your actual luminosity function. So, I would respond that indeed if an object has a very clumpy appearance, that particular object should be near the time of its formation or on the way to recollapse down for its first time, but still far from the virialization or the violent relaxation for the object as a whole. Sandage: And also, don't parts of a given cluster collapse at different rates? So, those parts that have the lowest density initially will take a lot longer to collapse, so it may be that you have an old core that was high density to begin with and then later you are having all these things raining down, which are now the spirals, which will then give you the morphology density relation. Cavaliere: As a matter of fact, if you take seriously the peak approach, you have the central peak collapsing, and then a lot of mass infalling onto it in a halo, especially with flat power on small scales in a critical universe. That infall is gradual, in fact it might take up to a few Hubble times for those objects collapsing now. Sandage: And is still going on? Cavaliere: Still going on, probably in many clusters. Some direct evidence has been mentioned Geller. Considering the distribution of halo masses, that is even more influenced by the point that I was mentioning before: the spread, or variance, is even longer for the halos than for the cores, so you are quite right. Giacconi: I wanted to go back to something that you mentioned, namely the evidence for lack of clusters at z > 0.5. Cavaliere: Please do not quote me as accepting that as final. I consider it as an intriguing but preliminary result. Giacconi: But you know those data are incorrect because they have not been corrected for the exposure time. So that in fact real data could be very different. In addition it was very difficult to detect any clusters with z > 0.5 on the visual optical material that was used to identify clusters. So, I believe the discussion about the evolution of mass of the gas is very interesting, but I think the data on which you base your comparison is nonexistent (laughter). Cavaliere: Riccardo, I fully appreciate your view. I consider those counts as a normalization, if you allow me. (laughter) But it is well understood that on ROSAT with a carefully planned survey can solve the issue and say whether there is any real dearth at z > 0.5 or not. So those results provide a motivation, if you like, for looking

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the corner and predicting what will be observed eventually. Henry: Can I make a comment on that? We are analysing the extended medium survey again, and I guess the point that Riccardo is making is that the survey detects objects at fixed angular size bins so you get different variable fractions of the emission at different redshifts, so the number count is going to be very hard to interpret and for that reason we are looking at just the luminosity function in a narrow redshift range, which will at least eliminate this problem. And, in fact, the first results we got (I didn't get a chance to discuss them yet), if anything, are a lot lower than the low redshift luminosity function. Djorgovski: What's the redshift? Henry:

0.3-0.4

COSMOGONY AND THE STRUCTURE OF RICH CLUSTERS OF GALAXIES

Michael J. West Department of Astronomy University of Michigan Ann Arbor, MI 48109

Abstract. N-body simulations of the formation of clusters of galaxies allow a detailed, quantitative comparison of theory with observations, from which one can begin to address two fundamental and related questions: Can the observed properties of rich clusters of galaxies tell us something about the cosmological initial conditions? Can we use N-body simulations of clusters to test/constrain theories for the formation of the large-scale structure of the universe?

1. INTRODUCTION A wide range of theories have been proposed to explain the origin of galaxies, clusters of galaxies, and the large-scale structure of the universe. Broadly speaking, these can be divided into two classes. Most currently popular models for the formation of structure in the universe are based on the idea of gravitational instability in an expanding universe, in which it is assumed that structure has grown gravitationally from small-amplitude, Gaussian primordial density fluctuations. A second class of cosmogonic scenarios, which will be referred to here as non-Gaussian models, appeal to other processes besides simple gravitational clustering as the driving force behind the genesis of structure. Within the basic framework of the gravitational instability picture, there are several rival theoretical scenarios that are viable at present. Depending on the the details of the cosmological initial conditions and dominant mass component of the universe, the sequence of formation of structure may have proceeded in quite different ways. If, for instance, the universe is dominated by weakly interacting, non-baryonic particles (i.e., cold dark matter, hereafter CDM) then the formation of structure is expected to proceed hierarchically from small to large scales, with galaxy and cluster formation preceding the collapse of superclusters. A similar hierarchical scenario would arise in a baryondominated universe if the primordial fluctuations were isothermal (e.g., Peebles 1980). If, on the other hand, the universe is baryon-dominated and the initial perturbations were adiabatic, or if the mass density of the universe is dominated by massive neutrinos,

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then small-scale fluctuations would have been erased prior to recombination by photon diffusion or by free streaming of the neutrinos. In this case the sequence of formation of structure would proceed from large to small scales, with the collapse of superclustersized pancakes occurring first, followed by their fragmentation to produce galaxies and clusters {e.g., Zel'dovich, Einasto, and Shandarin 1982). Hybrids of these different scenarios are also possible {e.g., Dekel 1983; Dekel and Aarseth 1984). Among non-Gaussian models, the two most popular at present are the explosion scenario {e.g., Ostriker and Cowie 1981; Ikeuchi 1981) and the cosmic string model {e.g., Zel'dovich 1980; Vilenkin 1985; Albrecht and Turok 1985). In the explosion scenario, it is assumed that there was an early generation of some unknown sort of "seed" objects which exploded at high redshifts. The ensuing shock waves would have swept up surrounding material into thin, dense, expanding shells which might have subsequently cooled and fragmented to produce a new generation of seed objects that also exploded, resulting in an amplification process that could conceivably lead to the formation of very large scale structures, with the most likely sites for the formation of rich clusters being the points where shells intersect. In cosmic string scenarios, the primordial perturbations are assumed to have been correlated, rather than uncorrelated as in gravitational instability models. Cosmic strings which might have arisen in the early universe could accrete surrounding matter to produce clusters and the largescale structure. Other non-Gaussian scenarios have also been proposed, such as the generation of structure by primordial turbulence or by radiation pressure. Discriminating between the various scenarios that have been proposed for the origin of the large-scale structure is one of the major goals of modern cosmology. Attempts to confront these theories with observations have generally tended to concentrate on largescale objects such as superclusters and voids, since these are believed to be relatively unevolved at present and hence still likely to retain some information about the initial conditions from whence they arose. However, as this article will attempt to show, rich clusters of galaxies can also provide a powerful means of testing theories for the origin of the large-scale structure. Rich clusters offer several advantages which are summarized below:

1.1 Advantages • Rich clusters have been studied for decades and so are a fairly well-observed class of objects for which a large body of observational data exists. Although it would seem that, ideally, the best way to learn about the origin of the large-scale structure of the universe would be to map the large-scale galaxy distribution using extensive redshift surveys, gathering the many redshifts needed for such an approach is very difficult and time-consuming. Furthermore, with existing samples such as the CfA and Southern Sky Redshift surveys, it is not clear that we even have a fair sample yet of what the large-scale structure is really like, since features can be seen in the galaxy distribution having sizes which span the surveyed volumes. • Clusters are believed to be dynamically young systems. This is based on such observations as their relatively low mean densities {6p/p ?« 200 within an Abell radius compared, say, to the typical mean densities of individual galaxies, which are generally two or more orders of magnitude greater) and the frequency with which substructure may appear within clusters. Because they are likely to be dynamically young and because dissipation has probably not played an important role in their

Cosmogony and the Structure of Rich Clusters of Galaxies

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formation, there is a chance that the observed properties of rich clusters might still reflect some traces of the initial conditions at the time of their formation and could therefore provide a useful probe of conditions at, say, the recombination epoch. • Since rich clusters can be identified to redshifts as great as z ~ 1, it should be possible to examine the ways in which clusters and clustering evolve with time, which could provide further important constraints on cosmogonic scenarios. There are, however, also some disadvantages: 1.2 Disadvantages • Although dynamically young, clusters of galaxies are nevertheless non-linear systems today, having already collapsed and probably virialized. Consequently there is no guarantee that any traces of the initial conditions which might have been present at the time of cluster formation have not already been erased by subsequent dynamical evolution. • Because clusters are highly non-linear systems today, studies of cluster formation are generally not amenable to analytic methods such as the simple linear theory that has been developed to describe the growth of structure in an expanding universe (e.g., Peebles 1980). Consequently, one must resort to N-body simulations, with all their inherent benefits and limitations (see Efstathiou et al. 1985 for a discussion), in order to extrapolate the evolution of structure from some assumed set of primordial conditions in the linear regime into the non-linear phase of clustering. A fairly large number of numerical simulations of clusters of galaxies have been performed to date. These can be roughly divided into three types: 1) Simulations of isolated clusters. These simulations focus on the expansion, turn around, and collapse phases of cluster formation, while essentially ignoring any surrounding cosmology. They usually assume very simple initial conditions, such as a spherical "top hat" sort of perturbation. Owing largely to their simplicity these were the earliest sorts of cluster simulations that were performed, with Aarseth pioneering much of this work (e.g., Aarseth 1963, 1966, 1969). Other well-known studies have also been done by Peebles (1970), White (1976), Cavaliere et al. (1986), and others. 2) Cosmological simulations. Large-scale cosmological simulations have become very popular in the last few years as a means of comparing the predictions of various theories with observations. Simulations have now been performed for most of the cosmological scenarios discussed earlier (e.^., Aarseth, Gott, and Turner 1979; Efstathiou and Eastwood 1981; Centrella and Melott 1983; Frenk, White, and Davis 1983; Klypin and Shandarin 1983; Dekel and Aarseth 1984; Albrecht and Turok 1985; Davis et al. 1985; Saarinen, Dekel, and Carr 1987; Bennet and Bouchet 1988; Weinberg, Dekel, and Ostriker 1989, and others). Such simulations allowed White, Davis, and Frenk (1984), for example, to argue that a neutrino- dominated universe could be ruled out because it would produce clusters which are much larger than those observed. Unfortunately, resolution on the scale of clusters in most large-scale cosmological simulations is usually too poor to permit a detailed study of the formation and systematic properties of clusters. This is because the dynamical range

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of such large-scale simulations is quite limited; one is attempting to simulate the mass distribution in a large volume of space using a limited number of particles and consequently each individual particle must be rather massive. 3) Large-scale cluster distribution. Such simulations are not intended for examining the properties of individual clusters per se, rather, these are studies of the largescale clustering properties of clusters, as reflected by the cluster-cluster correlation function and the morphology of superclusters (e.g., Barnes et al. 1985; Batuski, Melott, and Burns 1987; White et al. 1987; Weinberg, Ostriker, and Dekel 1989). Such studies have suggested, for example, that most simple gravitational instability scenarios may have difficulty accounting for the very large-scale structure indicated by recent observations. The above list is by no means meant to be complete, but rather is intended simply to give some idea of the sorts of cluster simulations that have been done to date. As a means of illustrating in more detail just what can be learned from comparing N-body simulations of clusters with observations, the following sections discuss several numerical studies of cluster formation that have been done in various collaborations between Dekel, Oemler, Richstone, Weinberg, and West. In what follows, many of the results will be presented in a rather qualitative way, however, more detailed, quantitative discussions can be found in the original papers cited below.

2. CLUSTER FORMATION IN GRAVITATIONAL INSTABILITY MODELS In a series of papers, Avishai Dekel, Gus Oemler, and I have performed simulations of the formation of clusters of galaxies in a wide range of cosmogonic scenarios within the framework of the gravitational instability picture (West, Dekel, and Oemler 1987, 1989; West, Oemler, and Dekel 1988, 1989). Our goal was to examine the systematic properties of these simulated clusters, with the hope that some differences could be found between clusters formed from different initial conditions. If such differences exist, then comparing the properties of the simulated clusters with those of observed rich clusters might allow one to place constraints on cosmogonic models or perhaps even rule out one or more of the competing scenarios. We assume that in the early universe there was a power-law spectrum of density fluctuations,



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Cosmogony and the Structure of Rich Clusters of Galaxies

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of the density profiles of rich clusters might depend on the form of the initial spectrum of density fluctuations (e.g., Fillmore and Goldreich 1984; Hoffman and Shaham 1985), and thus cluster density profiles might provide an interesting test of the cosmological initial conditions. Figure 5 shows the projected surface mass density profiles obtained for all clusters in various gravitational instability scenarios. It is striking that, although the clusters have formed in quite different ways, their density profiles appear nonetheless to be quite similar. Quantitative measurements of the shapes of these profiles confirm their similarity. Hence, it seems that the density profiles are insensitive to the initial conditions. Only profiles of clusters formed in an open universe (not shown here) differ, appearing significantly steeper in their inner regions. We interpret the similarity of profiles of clusters formed in quite different scenarios as a consequence of violent relaxation during cluster collapse, which is an efficient means of erasing traces of the initial conditions. Similar results have been obtained by van Albada (1982) and Villumsen (1984) for stellar systems. However, it is important to note that quite different results have been obtained by Quinn, Salmon, and Zurek (1986) and Efstathiou et al. (1988), who found that the density profiles of bound objects in their simulations beginning from different initial fluctuation spectra did in fact show a rather strong dependence on the initial conditions. Why these various studies produce such discrepant results is not clear, although there are several possible explanations. One possibility is that West, Dekel, and Oemler (1987) looked at systems of much lower mean overdensities (appropriate for rich clusters) than those of Quinn, Salmon, and Zurek (1986) and Efstathiou et al. (1988), who focused on higher density systems comparable to galactic halos. Second, according to standard lore, structure should evolve in a self-similar way from a scale-free initial fluctuation spectrum in a flat universe, and therefore one would expect galactic halos and galaxy clusters to be indistinguishable from one another except for a change of scale. However, the fact is that individual objects do not grow in an entirely self-similar manner but rather pass through several well-defined stages of evolution. Hence, it is entirely possible that systems of different dynamical ages will not necessarily exhibit similar properties. Specifically, the density profile which results from cluster collapse and violent relaxation may be altered by later secondary infall of outlying material, so that the density profile at later times may differ from that at earlier epochs. In hierarchical scenarios, the higher densities of galaxy halos imply that they must have collapsed at correspondingly earlier epochs than clusters and thus have had longer to accrete a significant fraction of their total mass by infall. Clusters of galaxies, on the other hand, because they are dynamically young may still reflect the universal density profile that results from violent relaxation. And, of course, a third possibility is that the different numerical results may simply be a consequence of the different ways in which the initial conditions were generated and the simulations performed by each group. Resolving this discrepancy would be an important step towards a better understanding of the relationship between density profiles and the cosmological initial conditions, as would a detailed numerical study of the competing effects of violent relaxation and secondary infall. Does the universal density profile of the simulated clusters agree with the observed density profiles of rich clusters? To answer this question we determined profiles for 27 Abell clusters for which reliable data are available. These are shown in Figure 5. In general, the observed profiles exhibit the same general shape as the profiles of the simulated clusters. This may be telling us something very interesting about the distributions of luminous and non-luminous matter in clusters of galaxies. If one takes

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2.4 Velocity Dispersion Profiles Another property worth examining is the run of velocity dispersion with radius in both the simulated and observed clusters. Figure 6 shows a comparison of the mean line-of-sight velocity dispersion profiles of the simulated clusters in different scenarios. Once again the cluster profiles are all remarkably similar, with the exception of those clusters formed in an open universe. To compare the simulation results with observations, Figure 6 also shows the composite velocity dispersion profile for a sample of 15 clusters taken from studies by West, Dekel, and Oemler (1987) and Dressier and Shectman (1988). Within the rather considerable scatter, the theoretical velocity dispersion profiles certainly appear to be consistent (or perhaps more fairly stated, at least not inconsistent with the observa-

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2.5 Subclustering Substructure in clusters of galaxies might provide a very useful means of probing the initial fluctuation spectrum. The basic idea behind this approach is quite simple. In the gravitational instability picture, the amount of small-scale clustering (i.e., binaries, small groups of galaxies, etc.) that develops will naturally depend on the amount of small-scale power present in the initial fluctuation spectrum. Thus, in pancake scenarios one would expect to find negligible small-scale structure since all small-scale perturbations were erased from the initial fluctuation spectrum prior to recombination, whereas in hierarchical scenarios one would expect to find a good deal of small-scale clustering. One way in which this small-scale clustering may manifest itself is in the form

Cosmogony and the Structure of Rich Clusters of Galaxies

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of substructure within clusters. West, Oemler, and Dekel (1988) attempted to quantify the amount of subclustering which occurs in clusters and their surroundings to see if this could indeed provide a useful diagnostic of the cosmological initial conditions. While this approach sounds straightforward in principle, in practice developing tests to objectively search for substructure and assessing the statistical significance of the results are quite difficult tasks. Numerous studies have claimed to detect substructure within rich clusters (e.g., Geller and Beers 1982; Dressier and Shectman 1988; Fitchett and Webster 1988; Mellier et al. 1988, and many others). However, based on three different tests that we developed, West, Oemler, and Dekel (1988) claimed to find little evidence of significant substructure within the inner regions of most of their simulated clusters. We then applied these same statistical tests to the observational data published by Dressier (1980) and also found little significant substructure in the inner regions of most of these clusters within, say, ~ 1 — 2 h~ Mpc of their centers. Similar results have also been obtained recently by West and Bothun (1989), who applied a different set of statistical tests using both the projected galaxy distribution and available velocity information. It is important to emphasize that clumps can often be seen in the projected galaxy distribution but in many cases these are consistent with what would be expected from simple Poisson noise and hence do not represent genuine dynamical entities. West, Oemler, and Dekel (1988) argued that the lack of significant substructure in the inner regions of most clusters is once again a consequence of violent relaxation, which acts to quickly obliterate any trace of substructure that might otherwise have been present. A similar conclusion has been reached by Efstathiou et al. (1988). We would argue that the lack of significant substructure in the inner regions of most Abell clusters implies that they are most likely dynamically relaxed systems at present. Although we found little substructure in the inner regions of most rich clusters, we did find that the amount of small-scale structure present in the regions immediately surrounding clusters can provide a sensitive test of cosmogony, with those scenarios originating from initial fluctuation spectra with more small-scale power showing the greatest amount of small-scale clustering in the cluster environs. I would like to digress here for a moment to discuss an important point regarding the interpretation of substructure. It is often argued that the (supposed) frequent occurrence of substructure implies that many rich clusters have formed only recently or are perhaps still forming today. However, it is important to remember that other equally plausible interpretations of apparent substructure are also possible. As emphasized by West and Bothun (1989), when discussing substructure in the context of cluster formation, it is essential to make a distinction between different types of "substructure" that might be observed: 1) those subclusters which are the surviving vestiges of smaller systems of galaxies that may have recently merged to produce a rich cluster and, thus, represent a genuine signal of recent cluster formation, 2) those subclusters which presently reside within an otherwise relaxed cluster, for example secondary infall of some bound group presently undergoing tidal disruption within the cluster (see the article by Fitchett in this volume for further discussion of this possibility), 3) those groups of galaxies which are bound to the cluster but still outside the cluster confines, destined to fall in at some later time, and 4) apparent subclustering in the form of groups of galaxies which are not dynamically bound to the cluster, but rather are expanding with the general Hubble flow and appear as substructure simply because of projection along the line of

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sight (e.g., the Cancer cluster, see Bothun et al. 1983). While all four types of apparent substructure are important, they have quite different implications for theories of cluster formation and the present dynamical state of clusters. Even with complete radial velocity information, it is still a very difficult task to unambiguously determine whether some apparent clump of galaxies represents a genuine statistically significant region of substructure (rather than simply Poisson noise) and whether the apparent subcluster is actually physically associated with the rich cluster itself. For instance, given a typical cluster line-of-sight velocity dispersion of ~ 1000 kms , galaxies in some group lying as much as 10 h~^ Mpc in front of or behind a rich cluster could easily be erroneously classified as cluster members even with complete redshift information. X-ray observations of clusters also suffer from these same ambiguities. Perhaps the only way to truly distinguish between these possibilities is with independent distance information with which one could, in principle, obtain the peculiar velocity field around clusters. Without precise distance and peculiar velocity information, it is difficult, if not impossible, to distinguish between the different forms which subclustering may take. Thus, it seems worth stressing that it may be premature to assume that the prevalence of observed substructure necessarily implies that many rich clusters are in an unrelaxed state at the present epoch. Another point that has been emphasized by both Geller and Beers (1982) and West and Bothun (1989) is that it is dangerous to make sweeping statements about the frequency with which substructure occurs in rich clusters based on the very inhomogeneous observational samples that are available at present. For instance, Dressier (1980) surveyed regions extending anywhere from ~ 0.5 — 5 h~l Mpc from the centers of 55 clusters, whereas Colless and Hewett (1988) surveyed only the innermost regions (out to radii of ~ 0.5 — 1 h~l Mpc) of a sample of 14 clusters. Consequently, there is a clear bias for substructure to be found more often in those clusters which have been observed over a greater area. Lastly, Alfonso Cavaliere and I have been asked by the organizers of this meeting to address possible reasons why our simulations seem to predict quite different dynamical states for clusters of galaxies today. My answer is that I think that our numerical results do not really differ at all, rather, it is our interpretations of the results that differ. Consider Figure 7, which shows a typical simulation of hierarchical clustering by West, Dekel, and Oemler (1987). This simulation shows the same clumpy appearance seen in many of the simulations of Cavaliere et al. (1986) and others. However, when we assign a physical scale to these simulations, which, as discussed earlier, we can do in an unambiguous manner using the two-point correlation function, then the simulation shown in Figure 7 extends over 20 h Mpc on a side and the individual clumps are separated by distances of ~ 5 — 10 h Mpc. In such a case we would call these clumps three distinct clusters, whereas Cavaliere et al. (1986) would label this a cluster with strong subclustering. Perhaps these clumps will merge someday, perhaps not. Thus, it seems to me that in essence the question of substructure boils down to a semantic point - just how does one choose to define substructure, and how does one assign some statistical significance to the observations? Further thoughts on substructure can be found in the articles by Cavaliere, Fitchett, Forman, and Geller in this volume.

2.6 Cluster Alignments The possibility that clusters of galaxies may show some tendency to be aligned with one another and/or with the surrounding galaxy distribution has profound implications

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for cosmogonic scenarios. Observations indicate that: • the major axes of rich clusters tend to point toward neighboring clusters over scales of ~ 15 - 30 h'1 Mpc, perhaps more (Binggeli 1982; Flin 1987; Rhee and Katgert 1987; West 1989), and • in the regions surrounding clusters, galaxy counts are preferentially higher along the direction defined by cluster major axes, with this effect also extending to ~ 15 - 30 h'1 Mpc (Argyres, Groth, Peebles, and Struble 1987; Lambas, Groth, and Peebles 1988). Such observations provide strong evidence that the galaxy distribution is characterized by a filamentary topology, and suggest a connection between the formation of clusters and the large-scale structure. It should be emphasized here that the two types of observed alignments are not simply redundant measures of the same effect; one could, for instance, have neighboring clusters being aligned with one another due to their mutual tidal forces without necessarily finding a corresponding excess number of galaxies in filaments between clusters. Binggeli (1982) was the first to show the tendency for neighboring rich clusters to point towards one another. While other studies by Struble and Peebles (1985) and Ulmer, McMillan, and Kowalski (1989) were unable to find any significant alignment tendency, I think that it is fairly safe to consider this effect established. As discussed by West (1989), a likely cause of the discrepant results from different studies is the large uncertainties in position angle determinations of cluster major axes. Binggeli's (1982) original results are illustrated in Figure 8, where 9 is the angle between the major axis of an Abell cluster and the line connecting its center to that of its nearest neighboring cluster and D is the spatial separation between the clusters. In the absence of any alignments, the distribution of 9 should be uniform with a mean (9) = 45°. There is a clear tendency for small values of 9 when neighboring clusters are separated by distances D < 15 — 30 h Mpc which indicates a general propensity for neighboring clusters to be aligned with one another. For D < 30 h Mpc, (9) w 30° ± 4 when only nearest neighbor clusters are considered, and (9) R* 36° ± 5 when all neighbors are included. Dekel, West, and Aarseth (1984) searched for similar cluster alignments in different gravitational instability scenarios. Their results are shown in Figure 9 for pancake, hierarchical clustering (n = 0), and hybrid simulations. A strong tendency for alignments can be seen for the pancake scenario for cluster separations less than ~ 30 ft Mpc ((#) w 25° ± 3). A weaker, though still significant, tendency can be seen for the hybrid model ({9) fa 36° ±2). No alignments are found for clusters formed in the n = 0 hierarchical clustering simulations ((9) « 44° ± 2 ) . It is reassuring to note that these results appear to be quite robust; Figure 10 shows the alignment tendency for clusters in pancake and n = 0 hierarchical clustering simulations taken from Frenk, White, and Davis (1983) (in this case (9) ta 30° ± 2 for the pancake scenario, and (9) « 44° ± 1 for hierarchical clustering). The fact that quite different alignment tendencies are found for different cosmogonic scenarios is encouraging, as it means that this simple test may provide a very powerful means of distinguishing between different models. A complementary test for filamentary structure is that of Argyres et al. (1986), who looked for correlations between the orientations of rich clusters and the distribution of galaxies in the regions surrounding them. Given the direction defined by the major axis of a cluster, one can ask whether galaxies in the cluster environs are uniformly distributed with respect to this axis or whether they show some sort of non-uniform distribution. If clusters reside within filaments, for example, the density of galaxies

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Figure 7. Projected galaxy distribution in typical n = 0 hierarchical clustering simulation. The box is ~ 20 h Mpc on a side. in the cluster surroundings should show a systematic tendency to be higher in some preferred direction relative to the cluster major axis. Plotted in Figure 11 are the results obtained by Argyres et al. (1986) when they compared the orientations of a large number of rich clusters with the Shane-Wirtanen galaxy counts. Shown is the surface density of galaxies in excess of the expected mean density if the galaxies were uniformly distributed, for bins of different angular separation from the cluster center. At the typical redshifts in this sample, the bins 0.25 < 6 < 0.5, 0.5 < 6 < 1.0, 1.0 < 0 < 2.0, and 2.0 < 0 < 4.0 should correspond to distances of roughly 2, 4, 8, and 16 A" 1 Mpc from the cluster center. If galaxies are uniformly distributed around clusters, there should be no density excess seen in Figure 11. Instead, there is a clear tendency to find more galaxies in the direction defined by the major axes of the clusters. Similar results have been obtained by Lambas, Groth, and Peebles (1988). Thus, this test provides strong evidence of a filamentary pattern for the large-scale galaxy distribution, with rich clusters residing within these filaments and tending to be oriented such that their major axes are parallel to the filaments. This same analysis was applied by West, Dekel, and Oemler (1989) to the various gravitational instability simulations. These results are shown in Figures 12a-d. A strong signal of alignments is clearly seen in the pancake and the n = —2 hierarchical clustering simulations and at a weaker level in the hybrid model. No evidence of signif-

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Cosmogony and the Structure of Rich Clusters of Galaxies

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in the luminous matter that were not also present in the underlying mass distribution. I would especially encourage a detailed study of alignment tendencies in CDM, as it still remains to be seen whether CDM can reproduce the large-scale filaments and alignments of neighboring clusters that are observed. Results presented by Dekel (1984), as well as those discussed here, suggest that CDM may have problems in this area.

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5. SUMMARY The goal here has not been to argue for or against the merits of any particular theoretical scenario. Rather, what has hopefully been accomplished is simply to convince the reader of just how powerful this approach of using N-body simulations of cluster formation in comparison with observations can be in attempting to better understand the origin of rich clusters and the large-scale structure of the universe. Numerical simulations allow one to make a more detailed, quantitative comparison of theory with observations than would otherwise be possible with purely analytic methods. With the new generation of N-body codes that are being developed, such as tree codes (e.g., Barnes and Hut 1986; Hernquist 1987; Bouchet and Hernquist 1988), very high-resolution particle-mesh codes (e.^., Villumsen 1989; Melott and Shandarin 1989), and codes incorporating gas dynamics (see the article by Evrard in this volume), it will be possible to perform better and more sophisticated simulations than those described here. Similarly, with the launch of the Hubble Space Telescope, the amount and quality of observational data for clusters will increase tremendously in the near future. The next few years will no doubt prove to be a very exciting time for both theorists and observers interested in the origin of clusters of galaxies. It's a pleasure to thank Avishai Dekel, Gus Oemler, Doug Richstone, and David

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Weinberg for making our collaborations together very enjoyable and rewarding. In addition to those already mentioned, I would also like to thank Greg Bothun, Mike Fitchett, and Jim Schombert for many stimulating conversations on clusters and cluster formation. All the simulations discussed here were performed using the N-body codes developed by Sverre Aarseth, to whom I am grateful for sharing them with me as well as for providing helpful advice. I thank Cheryl Samsel for a careful reading of this manuscript and for helpful suggestions. Finally, I would like to thank the organizers of this meeting, Bill Oegerle and Mike Fitchett, for having invited me to give a talk on this subject.

REFERENCES Aarseth, S.J. 1963, M.N.R.A.S., 126, 223. Aarseth, S.J. 1966, M.N.R.A.S., 132, 35. Aarseth, S.J. 1969, M.N.R.A.S., 144, 537. Aarseth, S.J., Gott, J.R., and Turner, E.L. 1979, Ap. J., 236, 43. Albrecht, A., and Turok, N. 1985, Phys. Rev. Lett, 54, 1868. Argyres, P.C., Groth, E.J., Peebles, P.J.E., and Struble, M.F. 1986, A. J., 91, 471. Barnes, J. 1984, M.N.R.A.S., 208, 873. Barnes, J., and Hut, P. 1986, Nature, 324, 446. Barnes, J., Dekel, A., Efstathiou, G., and Frenk, C.S. 1985, Ap. J., 295, 368. Batuski, D.J., Melott, A.L., and Burns, J.O. 1987, Ap. J., 322, 48. Bennet, D., and Bouchet, F. 1988, Phys. Rev. Lett, 60, 257. Binggeli, B. 1982, Astron. Astrophys., 107, 338. Bothun, G.D., Geller, M.J., Beers, T.C., and Huchra, J.P. 1983, Ap. J., 268, 47. Bouchet, F.R., and Hernquist, L. 1988, Ap. J. Suppl, 68, 521. Cavaliere, A., Santangelo, P., Tarquini, G., and Vittorio, N. 1986, Ap. J., 305, 651. Centrella, J., and Melott, A.L. 1983, Nature, 305, 196. Colless, M., and Hewett, P. 1987, M.N.R.A.S., 224, 453. Davis, M., Efstathiou, G., Frenk, C.S., and White, S.D.M. 1985, Ap. J., 292, 371. Dekel, A. 1983, Ap. J., 264, 373. Dekel, A. 1984, in the Eighth Johns Hopkins Workshop on Current Problems in Particle Theory, eds. G. Domokos and S. Koveski-Domokos, (Singapore: World Scientific), p. 191. Dekel, A., and Aarseth, S.J. 1984, Ap. J., 283, 1. Dekel, A., West, M.J., and Aarseth, S.J. 1984, Ap. J., 279, 1. Dressier, A. 1980, Ap. J. Suppl., 42, 565. Dressier, A., and Shectman, S. 1988, A. J., 95, 985. de Lapparent, V., Geller, M.J., and Huchra, J.P. 1986, Ap. J. (Letters), 302, LI. Efstathiou, G., and Eastwood, J.W. 1981, M.N.R.A.S., 194, 503. Efstathiou, G., Frenk, C.S., White, S.D.M., and Davis, M. 1988, M.N.R.A.S., 235, 715. Evrard, A.E. 1987, Ap. J., 316, 36. Fillmore, J.A., and Goldreich, P., 1984, Ap. J., 281, 1. Fitchett, M., and Merritt, D. 1988. Ap. J., 335, 18. Fitchett, M.J., and Webster, R.L. 1987, Ap. J., 317, 653. Flin, P., 1987, M.N.R.A.S., 228, 941. Frenk, C.S., White, S.D.M., and Davis, M. 1983, Ap. J., 271, 417.

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Geller, M.J., and Beers, T.C. 1982, Pub.A.S.P., 94, 421. Hernquist, L. 1987, Ap. J. Suppl., 64, 715. Hoffman, Y., and Shaham, J. 1985, Ap. J., 297, 16. Ikeuchi, S. 1981, Pub. Astr. Soc. Japan, 33, 211. Klypin, A.A., and Shandarin, S.F. 1983, M.N.R.A.S., 204, 891. Lambas, D.G., Groth, E.J., and Peebles, P.J.E. 1988, A. J., 95, 996. Mellier, Y., Mathez, G., Mazure, A., Chavineau, B., and Proust, D. 1988, Astron. Astrophys., 199, 67. Melott, A.L., and Shandarin, S.F. 1989, Ap. J., 343, 26. Ostriker, J.P., and Cowie, L.L. 1981, Ap. J. (Letters), 243, L127. Peebles, P.J.E. 1970, Astron. Astrophys., 75, 13. Peebles, P.J.E. 1980, The Large-Scale Structure of the Universe, (Princeton: Princeton University Press). Quinn, P.J., Salmon, J.K., and Zurek, W.H. 1986, Nature, 322, 392. Rhee, G.F.R.N., and Katgert, P. 1987, Astron. Astrophys., 183, 217. Roos, N., and Aarseth, S.J. 1982, Astron. Astrophys., 114, 41. Saarinen, S., Dekel, A., and Carr, B. 1987, Nature, 325, 598. Struble, M.F., and Peebles, P.J.E. 1985, A. J., 90, 582. Ulmer, M.P., McMillan, S.L.W., and Kowalski, M.P. 1989, Ap. J., 338, 711. van Albada, T.S. 1982, M.N.R.A.S., 201, 939. Vilenkin, A. 1985, Phys. Rep., 121, 264. Villumsen, J.V. 1984, Ap. J., 284, 75. Villumsen, J.V. 1989, Ap. J. Suppl., in press. Weinberg, D.H., Dekel, A., and Ostriker, J.P., 1989, in preparation. Weinberg, D.H., Ostriker, J.P., and Dekel, A. 1989, Ap. J., 336, 9. West, M.J. 1989, Ap. J., in press. West, M.J., and Bothun, G.D. 1989, Ap.J., submitted. West, M.J., Dekel, A., and Oemler, A. 1987, Ap. J., 316, 1. West, M.J., Dekel, A., and Oemler, A. 1989, Ap. J., 336, 46. West, M.J., Oemler, A., and Dekel, A. 1988, Ap. J., 327, 1. West, M.J., Oemler, A., and Dekel, A. 1989, Ap. J., in press. West, M.J., and Richstone, D.O. 1988, Ap. J., 335, 532. West, M.J., Weinberg, D.H., and Dekel, A. 1989, Ap.J., submitted. White, S.D.M. 1976, M.N.R.A.S., 177, 717. White, S.D.M., Davis, M., and Frenk, C.S. 1984, M.N.R.A.S., 209, 27P. White, S.D.M., Frenk, C.S., Davis, M., and Efstathiou, G. 1987, Ap. J., 313, 505. White, S.D.M., and Ostriker, J.P. 1988, preprint. Zel'dovich, Ya. B. 1970, Astron. Astrophys., 5, 84. Zel'dovich, Ya. B. 1980, M.N.R.A.S., 192, 663. Zel'dovich, Ya. B., Einasto, J., and Shandarin, S.F. 1982, Nature, 300, 407.

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DISCUSSION Peebles: Isn't one of the problems that there really aren't very many galaxies here and that the segregation effects that you get are typically two small body processes? West: Yes, I think that's true. But the important point is that within the subclumps that's always going to be the case to some extent. We've run simulations with a thousand particles where we had only ten galaxies and simulations where we had ten thousand particles and a hundred galaxies, and we get indistinguishable results. The same rapid segregation effect occurs in all cases. The important point is that in any hierarchical clustering scenario there will always be the formation of these small dense clumps, and thus it will always be possible to develop some segregation within them. I think this is a very general effect which should happen in any hierarchical clustering case. Neta Bahcall: Mike, back to the mass-radius relation. Another thing you can do is rather than comparing with the light-radius relation, is to use velocity dispersions to get the dynamical mass directly. West: That's right, Kashlinsky did that. He examined the relationship between the observed radius and velocity dispersion for a number of rich clusters and found a result which was consistent with constant density systems, although there was a lot of uncertainty. We could do the same thing with the simulations, but we haven't done that. Djorgovski:

But would you get the same answer?

West: Yes, given the correlation between cluster velocity dispersion and luminosity, we should get the same answer. Weinberg: I was quite struck that in your first set of simulations you are assuming that H = 1, light traces mass, you scale with the correlation function and you get velocity dispersions that are consistent with the observed clusters, whereas Jim Peebles told us this morning that when you look at observed clusters, you get il = 0.4. What is the resolution of this puzzle? Huchra: That one's easy. His clusters have dispersions of about 1000 kilometers per second instead of 750. The mass goes as the square. Peebles: Could I ask you about the density profile? In many cases, you assumed density profiles such that the log density is a function of log radius is convex away from the axis, which allows you to define a half-mass or half-count radius. The statistical analog of the density run is the cluster galaxy cross correlation function? That average density run is convex toward the axis if it is convex at all, which means that a half count radius is ill-defined. Can you reconcile these two, results? West: The half-mass radius is well defined when you consider something like a de Vaucouleurs. law.

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Peebles: I agree but then for a de Vaucouleurs law it is well defined because you have convexity in the right direction. West: That's right, and that is true of any density profile provided it is falling rapidly. Peebles: So long as it is falling rapidly enough. How about the observations? West: That's what the observed profiles showed. The cluster luminosity profile showed a steep fall-off at large radii. Peebles: I wonder if that is not a result of problems with the mean. You always have to subtract the background. Oemler: We worried very hard about that. That's an obvious problem, but I don't think that's true. It might be that if you at least operationally define where you go out to, several Abell radii, and normalize things there you would consistently treat the models and the observations. Peebles: If you consistently normalize at several Abell radii, you are subtracting the cluster. Right? The cluster galaxy cross-correlation function is appreciably large at several Abell radii. Oemler: No, I don't think so. That doesn't matter for comparing theory with observations. Petrosian: I just want to make a comment about mass-to-light ratio and its dependance on environment. You showed a lot of results. I don't know which side of the coin you're arguing for—you showed that light traces mass and also that it doesn't. West: I have no bias! (laughter) Petrosian: There is some evidence that I must point out, on luminous arcs. There is A 370, where we can definitely show by modeling of the arc with a gravitational lens model that light does not trace mass. The dark matter which we need for producing the gravitational lens effect is quite differently distributed to the light. Fitchett: My reading of the literature suggested that you have the same M/L within the arc as exterior to it. (laughter) Sellwood: Your amazing conclusion that no matter what initial spectrum you start with, you end up with the same cluster, why should we believe that? Why is that not an artifact of your N-body code? West: I can give you a number of reasons for that. One check is that we compare to other N-body results (laughter). While that's certainly no guarantee, at least it says that there is nothing peculiar about the code that we are using as compared to other N-body codes. Also in simulations of the dissipationless collapse of stellar systems, van Albada and Villumsen have found very similar results where, beginning from a wide range of initial conditions, they get a final universal density profile which looks like a de Vaucouleurs law. In terms of two-body relaxation effects within our simulations, some

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hand-waving calculations suggest that we've used a sufficiently large softening length that we should have been able to survive several tens of crossing times of these clusters before two-body relaxation effects become important. Another interesting thing to note is that in the pancake model, cluster formation cannot have occurred before the pancakes collapse, which occurs at an expansion factor of 4. We examined the clusters at an expansion factor of 6, so this universal density profile that we get must have developed very rapidly, before two-body effects within the clusters could have become significant. Yet we also find the same profile when we look at the hierarchical clustering simulations at much later times. So I think that is consistent with two-body relaxation not having become a big concern in these simulations. In terms of the explosion scenario simulations we've done with David Weinberg, we actually have in some simulations heavy "dark matter" particles, "gas" particles, and lighter "galaxy" particles, so we can look directly for two-body relaxation effects in the form of segregation between these different mass particles, and we don't find any. And these simulations also produce the same final density profile. Sellwood: Softening is a two-edged sword, it can help stop two-body relaxation but it also stops very dense concentration from building up. You loose your substructure that way. West: Yes, that's true. But for the density profiles I showed, the softening length is always much smaller than the scales we measured the density profile on. So I don't think we have poor resolution in the core of clusters, for example. Our softening length was chosen to be comparable to the size of a large galaxy, and thus is also smaller than the size of any interesting substructure which might develop. Tremaine: In your simulations that showed equipartition, of course the particles don't have the appropriate masses for galaxies in clusters, but you've argued that the same process should happen anyway. Of course, you can carry that to an extreme. For example, you could say that if you had an initial soup made out of oranges and grapefruits, now if you made a cluster out of them, the grapefruits would all be in the center and the oranges would all be on the outside. So, what's the dimensionless parameter, the difference between the two cases? What guarantees that the galaxies are going to come to the middle whereas, for example, grapefruits won't? West: This has mostly to do with the ratio of the mass of the galaxies to the dark particles. Provided that ratio is very large, we should expect to always get the same relaxation effect. Tremaine: So if it was grapefruits and neutrinos, the grapefruits would all still be in the center? West: Yes, that's my impression. Jaffe: I think there's two numbers. One you have to have not too many grapefruits. And the difference between the mass has to be large. I think that if you had in these simulations a million grapefruits and one bunch of raisins, we would not see this. West: I agree, for this mechanism to operate requires both that the individual galaxies

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be much more massive than the individual dark particles and that the total dark mass exceed the total luminous mass. Sandage: I have a similar question. We know for sure observationally that there is a factor of ten to the fourth in the mass of galaxies we can see in the Virgo cluster. What would be your prediction if you say that all Abell clusters are virialized for the distribution of the faint dwarfs compared to the giants? West: That's a question that Jim Peebles had asked Doug Richstone and I in an earlier discussion. The answer isn't clear. Again, these are very simple simulations with a limited dynamical range, but I think, as Doug is going to talk about tomorrow, it is possible to make some arguments which suggest that there is an energy transfer or heat transfer between different mass particles species, which in the extreme case where the galaxies are much larger than the dark particles leads to segregation, but may not necessarily do the same for different mass galaxies. Although dwarf galaxies are much smaller than the giant galaxies, it's not obvious that significant mass segregation should occur between them if, for instance, there is a large population of even smaller (dark) objects which could act as a sort of heat sink. But that is a possibility which still needs to be explored further. We tried looking at mass segregation among galaxies in some of our simulations by running a few cases where we had different mass galaxies. We didn't find any mass segregation as a general rule of thumb, but the galaxies only spanned a factor of two in mass, and we are also clobbered by small number statistics in these cases. So, I think it is still an open question both observationally and theoretically. Sandage: It is not an open question observationally because we know what the answer from observations is, and it is that the faint dwarf elliptical galaxies are distributed like the massive ones with a mass ratio of practically a thousand. So, you've got to produce that somehow. I'll show the evidence tomorrow. West: That may or may not be a problem for this mechanism. More work is needed. Kaiser: I would like to follow that line of questioning of the last two speakers. What worries me about what you have done is that you've put 10% of the mass into the galaxies which seems reasonable to me if we want to believe an 17 = 1 universe for this reason, fi = 1 means a total mass to light ratio of a couple of thousand. You are giving galaxies mass-to-light ratios of a couple of hundreds. It certainly implies very extended halos, and I'm not sure if that really renders your results unrealistic. The question seems to be would galaxies retain those halos? Even if the galaxies have such mass associated with them in the process of merging to form groups, would they retain their individual halos out to such large radii. The reason you are getting your effect is that on the group scale you've got about 50% or more of the mass in a couple of galaxies. West: I don't think that this segregation mechanism is terribly sensitive to the specific fraction of mass we actually attribute to each individual galaxy, provided again that this mass is much larger than the dark particle mass. Whether or not galaxies possess large extended halos, at early stages in the clustering hierarchy each galaxy will contribute a significant fraction of the total mass within the subcluster in which it resides, and consequently will rapidly settle to the bottom of the subcluster potential well. So in a sense, the galaxy will still be surrounded by an even larger dark halo. As subclusters merge, the galaxies will end up at the bottom of the potential well of the resulting

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larger subclusters. This has been demonstrated numerically in several studies in which people have done mergers of galaxies, etc., and found that in subsequent mergers on progressively larger scales the most tightly bound particles still end up at the center of the merger remnant. So I think that because we always reach the condition in the early stages of the simulations where the segregation can develop, and because this segregation is likely to be preserved through later stages as subclusters merge, the segregation of luminous and dark matter should exist at present, regardless of whether or not individual galaxies can be considered to retain whatever dark halos they may have had initially. Kaiser: But the baryons will always be a much smaller fraction if the mass-to-light ratio of the baryonic material is a few or ten. Could you do the same kind of experiment but reducing the mass of the galaxy by an order of magnitude or more? West: We didn't change the ratio of the galaxy to dark matter mass. We changed the faction of galaxies we put in, and found that provided you had more than 50% of the mass in dark matter, you always got the same segregation effect, so it seemed to be insensitive to our assumptions of 10% vs. 90% of the matter in the form of galaxies and dark matter. It seems to be a pretty general phenomenon. Djorgovski: It seems to me that you start with unrealistic conditions in that the dark matter is perfectly smoothly distributed, in which you embed these little raisins. Suppose you took seriously a picture in which most of the mass was, say, within a very bounded heavy halo say, 100 kpc? Would you get anything like this? West: Well, Gus Evrard has already explored that case. He started his simulations with dark halos around the galaxies, and found . . . Evrard: Can I speak for my simulations? There is significant agreement with what you have done in the sense of mass-light ratio being low relative to the global value, but there are some subtle discrepancies. One is that the density profile I found is steeper for the galaxies than for the total mass. The mass-light ratio should rise roughly as R^- in clusters of galaxies. Another thing I found in doing the cosmological simulations is that the two-point correlation function had a cusp at small separations because of this effect. This could probably be ruled out by the Turner-Gott evaluation of the correlation function at small scales. West: But how reliable are the density profiles for your small groups, don't you suffer from small numbers and two-body effects? How many galaxies...? Evrard: Yes. They were not rich clusters at all. We are talking about groups of 20 or 30 galaxies. And finally, the last point I would like to make is that I did the same simulations using test particles instead of galaxies, that is zero mass particles, and found no segregation in those particles, as you would probably expect. So, it is a mass dependant effect. So, it is sensitive to the amount of mass that you assume is embedded firmly in galaxies. West: Yes, but again, provided that the galaxy mass is appreciably greater than the mass of the individual dark particles, I would argue that the same effect should always occur in any hierarchical dark matter dominated cosmogony.

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One other point, I think it's fair to say that our simulations, at some early stage after the beginning, are probably somewhat similar to your simulations, Gus, in that each galaxy particle in our simulations has by then effectively accreted its surrounding dark matter to form a halo, so I think our simulations end up being similar to yours. Struble: With regard to the Argyres et al. alignments, could you comment on the strength of the alignments you find? West: Yes, in the pancake models, we find very strong alignments - stronger than you find from the observations Mitch. But observationally it is a difficult problem to determine the major axis of clusters, what with contamination problems, projection effects, and things like that. So because of that, Argyres et al. may be detecting a reduced signal from what is really an intrinsically stronger effect. In a perfect world, free of contamination and projection effects, you might have observed an even stronger signal of alignments. So I think it might be partly that. But, also, these pancake simulations are a pretty gross oversimplification of what is really expected. So I'd take the actual numbers with a grain of salt, take the effect qualitatively. Felten: From your answer to Nick Kaiser's question, I infer that you may not know the answer to my question, but let me ask it anyway. That the factor 5 to 10 mass discrepancy which was on your last view-graph, do you have any idea whether that will just scale inversely proportional to the 10% you started out with. In other words, if you take the fraction 10% up to 50% or down to 1%, do you have any idea whether that factor 5 to 10 just goes inversely? West: No, we did that and we got the same basic results. Felten: Same meaning what? West: That it was a factor 5 to 10 different. Richstone: Isn't it fair to say that we did that going up to 50% but not going down, that's harder. Felten: You might want to go down to 1% . Richstone: Exactly, but that's harder because of the number of particles. Felten: When you went up to 50% what happened ? Richstone: It stayed the same. Felten: You underestimated the mass by a factor of 5? West: Yes, it was consistent with our other results.

THE DARK MATTER DISTRIBUTION IN CLUSTERS

M. J. Fitchett Space Telescope Science Institute 3700 San Martin Drive Baltimore, MD 21218

Abstract. This article describes the current status of various methods for determining the dark matter distribution in clusters. Despite a great deal of progress recently we still do not have good mass constraints for even one cluster. The reasons for this are discussed. New observational tools and methods of analysis should however lead to some results in the near future.

1. INTRODUCTION One of the many interesting aspects of clusters of galaxies is that they appear to contain large amounts of missing mass. The evidence for this has largely been based on the application of the standard virial theorem. More sophisticated approaches which utilize cluster velocity dispersion profiles came to similar conclusions but assumed that the mass distribution in clusters was the same as that of the light (galaxy) distribution. While this may be true it is definitely at present an assumption. Much recent theoretical work has argued for different distributions for the dark and luminous components of the universe. One of the consequences of this is that we should not assume that the mass distribution in clusters parallels the light distribution. Without this assumption it is very difficult to constrain the mass distribution in clusters, and consequently total cluster masses are not as yet well determined (Bailey 1982, The & White 1986, Merritt 1987). The cluster mass distribution is an important 'parameter' in that it directly influences many of the physical processes that occur in clusters. It is therefore essential that one tries to determine the cluster mass distribution in an as assumption free way as possible. This review will concentrate on ways to do this. A definite problem is that many Abell clusters have probably not yet virialized (see reviews by Geller in this volume and Fitchett 1988). The methods discussed here are mainly only applicable to virialized systems and so the subset of clusters to which they can be applied is small. Even for the simplest spherical clusters, with lots of data, it is still currently extremely difficult to derive reliable mass estimates and constraints on the mass distribution. I hope that this review, despite its pessimism, will stimulate new ideas and at least make us all wary of simple assumptions concerning clusters, since even the simplest question

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(what is the mass distribution in clusters?) is still far from answered. This article will be split into several sections. The first will be motivational and will briefly discuss problems associated with the virial theorem, what cosmological theories predict for the dark matter distribution in clusters, and describe how physical processes in clusters depend on the dark matter distribution. The following sections will describe the current status of the methods for determining the dark matter distribution in clusters observationally. There are various options; we can use the dynamics of tracer particles (galaxies), moving in the cluster potential; we can use the observed properties of the X-ray emitting gas which sits in the cluster potential; we might even use substructure as a probe of the matter distribution (related to this I will discuss the mass distribution in clusters with small scale substructure); finally gravitational lensing by clusters may provide new limits on the dark matter distribution and its morphology at small scales in clusters. Another potential method for determining cluster mass distributions is the use of Infall patterns around rich clusters. I will not discuss that method here as Geller will review it in her article. The Coma cluster will be used to illustrate some of these methods, not because it is typical of clusters, but because it is well studied. There is a large amount of redshift and X-ray data on this cluster, and using one cluster allows comparisons to be made between the various methods. For similar review articles see Mushotzky (1987) and Merritt (1988).

2. MOTIVATION So far every rich cluster analyzed appears to have a high mass to light ratio. Clusters therefore present strong evidence for the presence of large amounts of dark matter in the universe. In this article I will try to be consistent and quote mass to light ratios using blue luminosities, Lg, and scale to a Hubble constant of 50 km s Mpc . In these units the M/Lg required to close the universe is ~ 12OO/i5o. Additionally I will try to state the scale on which various M/Lg were determined since this might in general vary throughout the cluster. Average cluster M/Lg ratios scatter around ~ 3OO/15O (e.g., The & White 1986, Merritt 1987, Colless 1987, Sharpies et al. 1988). It is important to note that many of these papers quote large ranges of allowed M/Lg. These ranges do not reflect cases of statistically poorly determined M/Lg . Rather they are cases where the authors have relaxed some of the standard assumptions. I will return to this point later. Notice that if cluster mass to light ratios are typical of the M/Lg for the luminous components of the universe then fi ~ 0.2 — 0.3 (we do not know that this is the case however). Early estimates of cluster masses were based on the standard virial theorem or one of its variants (see Heisler et al. 1985). Bailey (1982), The & White (1986) and Merritt (1987) have shown that these estimates can be grossly wrong once one drops the assumption that mass traces light. Following Merritt (1987) the virial theorem for a spherical system can be written in the form

where the brackets indicate spatial averages and F(r) is the fraction of the total mass of the system within radius r (this function is unknown). Under the assumption that mass traces light one can fit a function (e.g., a deVaucouleurs profile) to the luminosity

The Dark Matter Distribution in Ousters

US

profile and from that evaluate the denominator. With an estimate of (v) 2 one can derive the standard virial mass estimate which I shall denote Mgfj. Once one allows for physically reasonable, but arbitrary, F(r) there is a great deal of leeway in the allowed values of M y j - For instance Myj will be minimized if F(r) = 1 at all r, corresponding to a point mass at the cluster centre. The ratio of Mmin/M8t(j depends on the assumed F(r). For a deVaucouleur profile fitted to the Coma data it is ~ 0.2 (Merritt 1987). Merritt also showed that the virial theorem constrains the minimum density of the system to be />min = 3C?~1(t;2)/47r(r2), and that cluster masses might be very high if the dark matter is fairly uniformly distributed. For Coma Merritt showed that Mmax ~ 80A/ g ^. Once one allows for different distributions of dark and luminous matter virial mass estimates may be incorrect. One should therefore be wary of using the standard virial mass estimator unless one has other reasons to suppose that mass traces light for the system. Better methods which take into account the detailed spatial-dynamical structure of the cluster are described later. What do cosmological theories predict for the dark matter distribution in clusters? West and collaborators (see article by West in this volume) have concluded that most initial fluctuation spectra form clusters with very similar mass profiles. West and Richstone (see articles in this volume by each author) have also recently described a mechanism whereby the galaxies in rich clusters may become more centrally concentrated than the cluster dark matter. This mechanism has important consequences for our determinations of il from clusters, since the additional cluster dark matter "halo" might contribute significantly to the mass density of the universe. Additionally X-ray data has important consequences for theories which postulate that the dark matter is baryonic. Hughes (1989) has shown that the X-ray emission in the Coma cluster is not consistent with a dark matter distribution which parallels the distribution of the hot X-ray emitting gas. Aside from the global cosmological importance of the cluster mass distribution the interior properties of clusters and their constituent galaxies depend on the dark matter distribution in various ways: • Dynamical friction: all galaxies in clusters experience a drag force. This is due to the overdensity generated by the gravitational focussing of the less massive material behind galaxies moving in the cluster. Chandrasekhar (1942) derived the drag force for a point mass in a constant density sea of less massive particles which have an isotropic Maxwellian velocity distribution. He found that the drag force is linear in the background particle density or, in other words, the dark matter density. • Tidal stripping of cluster galaxies by the cluster depends on the dark matter distribution through its second derivative, ie the tidal field. • Ram pressure stripping of gas from galaxies (see the article by Haynes in this volume) depends on the orbits of the galaxies and their velocities - both of these quantities are intimately related to the cluster mass distribution. • Substructure: the interaction between galaxies within clusters depends on the amount of substructure present in the cluster. cD galaxies for example are believed to form through galaxy mergers. This process should be more efficient in subclusters before the final relaxation of the cluster.

3. CLUSTER DYNAMICS As discussed in Section 2, simple mass estimates of clusters based on the mass traces light assumption might be in error. This section describes recent work which attempts

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to assess the importance of deviations from the mass traces light hypothesis. The first papers on this topic were due to The k White (1987) and Merritt (1987). Each derived constraints on the mass distribution in the Coma cluster without making assumptions about the dark matter distribution. Bailey (1982) had previously suggested that the Coma data was consistent with a small dark mass located at the cluster center. These more recent analyses explored a larger range of mass distributions. Analyses of the type presented in these papers were probably motivated by similar work on the dynamics of individual galaxies (e.g., Binney & Mamon 1982). For some galaxies one has radial luminosity and velocity dispersion profiles. Binney and Mamon described a scheme for analyzing data of this form to yield measurements of the galactic mass to light ratio. With the advent of automated fiber systems and the consequent ability to obtain large numbers of redshifts in clusters (Colless and Hewett, 1987, Dressier and Shectman, 1988) it became possible to apply the same techniques to clusters. The projected velocity dispersion profile, <rp(R), where R is the projected distance of a point form the cluster center, is calculated for the cluster by breaking the cluster into disjoint annuli and for each evaluating the dispersion in the line of sight galaxy velocities

where Nann is the number of galaxies in the chosen annulus. This function is plotted for the Coma data in Figure 1. In order to determine (rp(R) at as many projected radii R as possible Nann is chosen to be small, typically 10. This leads to relatively large error bars on this profile. Also such estimates can be strongly affected by a few "outliers", galaxies which are not cluster members but are hard to exclude on the basis of their velocities. Beers et al. (1989) have emphasized the need for, and described several, statistical measures of dispersion (or, technically 'scale') which have small errors even for small samples with outliers. They also made the interesting point that no matter how much data is acquired on clusters, the methodology at present is to break it down into as many subsamples as possible, and so the need for these more resistant scale estimators will continue. One typically needs ~ 100 measured galaxy redshifts in a cluster in order to construct a reasonable velocity dispersion profile. As a simple illustration of the power of the velocity dispersion profile consider the case of a centrally concentrated dark matter distribution (consistent with the virial theorem). Such a mass concentration would cause the velocity dispersion profile to fall as R I at large radii. This is inconsistent with the observed velocity dispersion profile and so the extremely centrally concentrated model can be ruled out. I will now describe the method used by Merritt and The & White. The projected velocity dispersion profile, crp(R), and galaxy surface density Sfla/(i?) are derived observationally. These are related to the actual 3-dimensional properties of the cluster via the usual Abell integral

and by the following equation

crP(R)%al(R) = 2J™ nj$_ ^?r(r? - ^ M r ) 2 - ot(r)2)]

(4)

where r is the three dimensional distance from the cluster center, R is the projected (observed) distance from the cluster center, and 0>(r) and 680/i^"0 kpc in the Coma cluster for each of the three models described above. Each of these models is consistent with the velocity dispersion data. Figure 3 shows these results and the actual data. Clearly none of the theoretical curves is a particularly good fit to the observations because the velocity histogram is skew ( at a significance level ~ 3 % ). This may indicate that either Coma has not yet equilibrated or that there are galaxies projected onto the cluster that are not members. Merritt was not able to strongly select one model in preference to the other two based on this test. See Merritt (1988) for further discussion. Although the shape test was not useful for galaxies at large radii in Coma one might wonder if it might discriminate between different models for galaxies at small radii. Unfortunately the Coma cluster appears to have substructure on small scales (Fitchett & Webster 1987, Mellier et al. 1988). One would be hard pressed to argue that the galaxy velocities in the core are representative of those expected in a smooth potential well. Therefore it is very difficult to reduce the range inherent in Merritt's extreme models. Of course one could for this, or any other cluster with sufficient data, determine the optimal place at which to apply this test (i.e., the radius /?* beyond which the theoretical curves shown in Figure 3 would be most different from each other). Given the skewness of the Coma data it is unlikely that this approach will help in this case. However for an isolated spherical cluster with a large number of measured redshifts (probably

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. ID

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4. X-RAY CONSTRAINTS Mass distributions derived from observations of the hot X-ray emitting gas in clusters will undoubtedly, in the future, prove far superior to those derived optically. Cur-

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1 a re probably due to clusters where the velocity dispersion has been overestimated due to subclustering. Interestingly Perseus, which has the largest /? discrepancy appears to have spatial-velocity substructure at the ~ 2% level (Fitchett and Smail 1989) which cannot be seen in the velocity data alone (a very similar situation to the Cancer cluster - see Bothun et al. 1983). Thus it is likely that the /? discepancy can be reduced in this cluster. In any case it is still not clear from these analyses that the cluster gas is isothermal. Indeed Henriksen and Mushotzky (1986) presented evidence that the spectrum of the Coma cluster is not consistent with Thermal Bremsstrahlung radiation from an isothermal plasma. In addition spatially resolved spectral data in Coma shows different temperatures at different distances from the cluster centre (see later). The most recent approach is due to Hughes (1989). He argues that since, for the Coma cluster, the IPC data yield an accurate determination of pgas(r), this quantity should be assumed well known and parametrized. Assuming functional forms for the dark matter distribution like those used by Merritt in the optical study (see Section 3) and the equation of Hydrostatic equilibrium one can then derive a temperature profile for each mass profile which can be compared to all available data. Figure 4 (taken from Hughes 1989) shows the results of Hughes' calculation for a model in which the dark matter distribution is given by PDM(r) = P°(^ + ( r / r c) ) > corresponding to Merritt's N = 4 model (see Section 3). Each box corresponds to a different value of rc (given in arcmins, recall l' corresponds to 40/I^Q kpc at Coma) and the different curves represent different po. Lower curves have higher p0. The simplest constraint from the Coma X-ray data is that the X-ray emission must extend to at least 40' we see X-ray emission at this radius. Thus solutions which have T(r) —> 0 at r < 40' can be excluded, in fact they are not plotted in this Figure. These correspond to high central density models and so even this simple constraint can set an interesting upper limit on the central dark matter density in Coma (for this mass distribution). Secondly one knows that the central temperature of Coma is > 4 keV, while the average temperature over a 3° field is 7.5 ± 0.2 keV. This eliminates solutions which have too rapid a rise in temperature - these would yield low central temperatures for a given

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average temperature. Such models have also been excluded from this figure. There are additional constraints which can be applied. Hughes used the following; the Tenma spectrum of a 3° region of the cluster; EXOSAT temperature and flux information at the cluster center and 45' off-center (EXOSAT has a field of view of 45'). These measurements gave JCTQ = 8.5 ±0.3 keV and a ratio of center to off-center temperatures of 1.15^QQy (Hughes et al. 1988). These constraints rule out the solutions shown as dotted lines. Clearly dark matter distributions of the form PDM{r) — Po(l + ( r / r c ) )~ 2 with rc > 25.5' are ruled out by the X-ray data. This represents a significant improvement over the optical constraint where for the same mass distribution one could only rule out r c > 80'. This is an important improvement because the large core radius models give large total masses. The smaller allowed rc reduces the allowed upper limit on Mcoma, and hence the range in M/Lg. In fact the X-ray analysis of Hughes allows M/LQ to lie in the range 90 — 25O/i5o as opposed to a range of 70 — 525/i5o from the optical data. Despite these improvements there is still a considerable amount of uncertainty - and this is a well studied cluster!! To conclude, the X-ray approach certainly will prove to be the most useful method for determining the mass distribution in clusters in the future. For the Coma cluster the X-ray data has improved the limits on M{r) and M/Lg. One should perhaps be a little cautious too in that the X-ray observations do not extend as far as the optical data in Coma (40' versus 3°) - see Merritt (1988) for more discussion. For more distant clusters it is unlikely that the current X-ray data will be a strong contender with the optical methods. In the near future the X-ray constraints on the mass distribution in Coma should be improved by a measurement of the central temperature from the coded mask experiment flown on Spacelab 2 (Skinner et al. 1987). The detection of X-ray emitting gas beyond 40' in this cluster would further constrain the mass distribution.

5. SUBSTRUCTURE AND THE MASS DISTRIBUTION Many Abell clusters exhibit a clumpy galaxy distribution, consistent with the early phase of hierarchical clustering seen in N-body experiments (e.g., Cavaliere et al. 1986). For some of these clusters, for example the bimodal ones such as A98 (Beers et al. 1982), there is good correspondence between the morphology of the X-ray emission in the cluster and the optical galaxy distribution. Recently however several fairly smooth, apparently relaxed, clusters have been found to contain subclumps of galaxies (Coma - Fitchett and Webster 1987, Mellier et al. 1988; Virgo - Binggeli et al. 1987; Hydra I - Fitchett and Merritt 1988). For these clusters the correspondence between the optical and X-ray morphologies is less clear. This is perhaps most striking in the case of the Coma cluster where the central substructure in the galaxy distribution is not reflected in the X-ray emission from this region (see the map in Helfand et al. 1980). This has often been used as an argument against the existence of substructure in this cluster. Another interesting related observation is that X-ray determined cluster centroids differ in the mean by ~ 240/i^g1 kpc from galaxy determined cluster centroids (Beers and Tonry 1986). These observations lead one to question how the mass is distributed in these clusters - is it traced by the galaxies, and therefore clumpy, or is it more smoothly distributed? For a similar discussion of these questions see Ulmer (1988). This section describes a simple model which attempts to explain the substructure observed in the Coma cluster core as the consequence of the accretion of a poor cluster

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Figure 4. Model temperature distributions for the N = 4 mass model discussed in the text. The dashed curves satisfy the simple X-ray constraints, while the solid curves satsify all X-ray spectral data on Coma (taken from Hughes 1989). or group. Observationally this is a reasonable hypothesis as clusters do live in the presence of large scale structure (I thank John Huchra for pointing this out to me) and sometimes have neighbouring groups of galaxies (for example Virgo - Binggeli et al.

The Dark Matter Distribution in Clusters

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1987). Also the densest rich clusters forming in N-body simulations appear to grow at late stages by the accretion of groups (coincidentally the model of the Coma cluster presented by Evrard in his chapter ingested a group very recently in its history). This simple model lends some insight into the issues discussed above, especially the question of whether the dark matter is clumpy or not. Substructure in fairly relaxed clusters might also help constrain the dark matter distribution - I will discuss that idea briefly. This work was carried out in collaboration with Steve Zepf (ST Scl), Tom McGlynn (ST Scl), and Peter Quinn (Stromlo). We have run several N-body simulations of a group infalling into a cluster using a hierarchical N-body code (Barnes and Hut 1986, Hernquist 1988). We have varied both the group and cluster parameters. Cluster parameters were chosen to reflect the three models for the Coma cluster described in Section 3. The results shown in this section are for a dense group (/>Jw*P ~ 35/>£^™a) of mass Mgroup = 1.5 X l O 1 4 / ^ M© falling radially from rest at 4A^g Mpc into the "mass traces light" Coma potential. Less dense groups exhibit qualitatively similar behaviour (see later). Under the mass traces light assumption the total mass of Coma is 3.7/»^Q x 10 MQ. Thus the ratio Mgroup/Mcoma ~ 0.04 and the group is only a small fraction of the mass of the whole system. The initial mass distribution of the infalling group and the Coma cluster were modelled by King models in the results shown here. For all simulations we find basically the same behaviour. The group falls through the cluster center, and for a period of time gives rise to a bimodal structure consisting of the cluster center and the core of the group. The group leaves the cluster center and is either disrupted or returns to reproduce the bimodal structure on subsequent passages. The principal differences between the simulations are the timescales, tf,, over which bimodality is apparent, and the number of passages the group survives through the cluster center Np. For the high density case we estimate that for the first encounter the double structure appears to be detectable for a total period of t^ ~ 3 — 6/i£~g x 10 years, where the uncertainty reflects the viewing angle and precisely where the line is drawn between a double structure in the core, and a group outside the cluster core. This group has Np = 3. For a low density group (with / C ~ p^™) NP = l but a bimodal structure is still formed in the cluster center on this single passage. This structure persists for approximately half of the time for the densest groups. The timescale estimates are based on inspection of the spatial distribution of our simulations. Since one would also expect a signature of the substructure in velocity space these timescales are underestimates of the true timescales. To compare this model with the Coma cluster it is necessary to assume a viewing angle for the simulation. Since the relative velocity of the subclumps in the center of the Coma cluster is small (~ 600 km s —see Fitchett and Webster 1987) compared to the velocity of the infalling group as it passes through the cluster center (typically ~ 3000 km s ), we assume that Coma corresponds to our looking at a group falling in perpendicular to the line of sight. Figure 5 compares the matter distribution as observed from this angle in the central region of the simulation with the galaxy distribution in the center of Coma (the scale is in arcminutes—recall l' corresponds to ~ 40A^Q kpc at Coma). The group fell in from the right and we show the simulation just after the group has passed through the cluster center. The group itself gives one density maximum (the left density peak) and the cluster center forms the second density peak. Although the group was started with a much greater central density than the cluster, the two density maxima observed appear to be roughly equal. Clearly the spatial distributions agree fairly well. We have also analyzed the velocity histograms of the central region of the

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M. J. Fitchett

-10

-10

20

Figure 5. Density contours of the central region of the Coma cluster (upper plot) and the corresponding region for the simulation. The data for the Coma cluster is taken from Godwin and Peach (1977) and galaxies down to V25 = 15.5 are plotted. The contours used in the cluster and simulation represent the same density contrasts.

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simulation and the 'core' of the Coma cluster, and find consistency. Figure 6 shows an X-ray map generated under the assumption of hydrostatic equilibrium from the simulation as observed in Figure 5, and viewed from the same direction (for method see Cavaliere et al. 1986). The contours are smooth and and reasonably fit with ellipses. The center of the X-ray distribution is not coincident with either of the density maxima of the simulation but appears to lie between them. The contours become rounder as one gets further from the center of the cluster. This can be compared to the X-ray map shown in Helfand et al. (1980). There is good qualitative agreement with the simulated X-ray map. The observed X-ray center of Coma lies between the two concentrations of galaxies in the cluster shown in Figure 5.

L-20

-10

x (arcmins)

Figure 6. X-ray emission for the simulation shown in Figure 5. This was calculated using the Hydrostatic assumption with 7 = 5/3. The contours are logarithmically spaced and are in arbitrary units. If the galaxies trace the mass distribution then this simple model is consistent with both the observed spatial subclustering in the Coma core and the almost elliptical X-ray emission from this region. Many authors have argued in the past that substructure in the mass distribution should cause substructure in X-ray maps. Indeed since the X-ray emissivity scales as pjias it has also been argued that subclumps should be even more obvious in X-rays. However pgas{r) is determined by the cluster potential DM{T) which, even for a clumpy mass distribution, might have only one minima (being an integral over the mass distribution this is not too surprising). Clearly the number of minima in the potential depends on the details of the mass distribution - subclumps

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within smooth clusters probably perturb the potential little and it remains close to elliptical or circular, whereas large scale substructure (in the form of well separated subclumps, such as A98) will give rise to a potential which has more than one local minima. The main point though is that one should be wary of assuming that smooth X-ray emission automatically guarantees a smooth matter distribution. All that can be safely said is that the potential is smooth. Very high signal to noise ratio X-ray observations would be needed to find the actual mass distribution for clusters with substructure (i.e., to solve PQM = -V2<J>DM/4ITG). I wish to stress that this is not to say that X-ray determined masses will be in large error in these cases. Indeed since the mass of the group remnant is small the X-ray determined mass will be very close to correct. Rather the point is that the X-ray maps might not show the detailed matter distribution in these systems. It is important to stress that by varying the cluster and group parameters, as well as the viewing angle, our scenario might be able to explain substructure seen in other clusters. For example the velocity substructure observed in the Hydra I cluster (Fitchett & Merritt 1988) is probably due to our observing a group infalling along the line of sight to the cluster. Spatially resolved subclumps with large relative velocities most likely correspond to the viewing angle being somewhat intermediate between the Coma and Hydra I cases. The substructure seen in the simulations is most pronounced for the more massive, centrally concentrated groups. Less massive and less dense groups do give rise to similar observable consequences, but their easier disruption leads to a shorter time over which their effects are observable. Observationally the effect of the infall of a larger number of these smaller groups is similar to that of one very massive and dense group. It would be useful to know the frequency of this effect. Large scale cosmological simulations could in principle determine this. Here a simple estimate of the frequency of central substructure will be made. For the high density group simulations spatial bimodality on cluster 'core' scales is observed approximately 10 per cent of the time (assuming an angle between the line-of-sight and the axis of infall of 60 degrees). Most of the remainder of the time, the group is too far from the core to be detected as central substructure, and a small percentage of the time is spent in a configuration in which the cluster and the groups cannot be separated spatially. Encounters with dense groups may be unlikely but our simulations have shown that the less dense groups can give rise to the same feature. This appears to last ~ 1.5 — 3/I^Q X 10 years, and the group disrupts after first passage. It seems likely that clusters will typically accrete at least a few low density groups during the age of the universe. If only three low density groups or one high density group is accreted in a Hubble time then there is a ~ 10 per cent chance of seeing a cluster with central substructure. Indeed this estimate should be viewed as a lower limit since it may be that we are actually living in the epoch of cluster formation. Also it is likely that the densest clusters may accrete at the largest rate. There is also some evidence that the high density groups we used might not be so unusual; some of Hickson's compact groups are found to have spatial densities approximately an order of magnitude larger than that of Coma (Hickson et al. 1988) and some poor clusters containing cDs have central densities at least as high as that in Coma (inferred from the data of Beers et al. 1984). Alternatively we might argue that this phenomenon must be very common as the infall of groups of galaxies into clusters is both observed (e.g., Mellier et al. 1988, Bingelli et al. 1987) and seen in simulations (e.g., Evrard 1989). Furthermore some well-studied clusters do show evidence for central substructure. Careful analysis of a large cluster sample is necessary in order to quantify

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the frequency of this. To summarize, this very simple model gives rise to substructure which appears to be very similar to that observed in the Coma cluster, while producing an almost elliptical distribution of X-ray emission. Clearly in our model, the X-ray centroid will not always coincide with the (ill-determined) galaxy centroid. This reminds us that the X-ray emission from clusters reflects the gravitational potential of the cluster, which is invariably much smoother than the cluster mass distribution. Current X-ray images may give reasonable mass estimates for regions of clusters, but do not have sufficient signal to noise to describe the detailed cluster mass distribution - ironically the galaxies may do a better job. If the accreted group is a poor cD cluster, this scenario gives a natural explanation of the phenomena of 'speeding' cDs (Hill et al. 1988, Sharpies et al. 1988). The cD galaxy belongs not to the cluster but to the group being accreted. This might also be the appropriate mechanism in Coma itself, since the central D galaxies in Coma resemble the central galaxies of poor cD clusters which often lack the extended envelopes of cDs in larger clusters. It was our hope at the start of this project that the observed substructure in the Coma cluster could be used to set limits on the cluster mass distribution. For example if the dark matter were very centrally concentrated infalling groups would be tidally disrupted as they fell into the cluster and not reach the core intact. Simulations of a high density group falling into a centrally concentrated model (corresponding to Merritt's most extreme, but allowed, centrally concentrated model) do show that the group is disrupted after first passage through the cluster center. However there is a time at which the model looks similar to the observations and so this cannot strictly rule out the centrally concentrated model. Allowing the infalling groups to have some angular momentum might lead to more optimistic results in that then the group would explore a larger region of the cluster tidal field and be disrupted before reaching the cluster center. The probability of observing subclumps close to the cluster center might also be higher in this case than in the simple case of radial infall. Constraining the cluster mass distribution by this method is unfortunately further complicated by the need to make some assumptions about the properties of the infalling groups. At this stage it is safest to conclude that this model shows that the dark matter in Coma could be distributed just like the galaxies (i.e., clumpy in the center of the cluster) and yet still give rise to X-ray emission which is close to elliptical in shape.

6. GRAVITATIONAL LENSING Recently giant luminous arcs have been discovered in some high redshift clusters (CL2244-02 Lynds & Petrosian 1986, A370 Soucail et al. 1987). These arcs are typically ~ 20" in length (corresponding to ~ 150/I^Q kpc for A370), and lie ~ 25" from the cluster center with their centers of curvature close to the cluster center. Several mechanisms have been proposed to explain how the arcs were formed - star formation in cooling flows, galaxy-galaxy collisions, explosions and gravitational lensing. The lensing hypothesis is that the arcs are the highly distorted images of background galaxies (Paczynski 1987). This hypothesis can be tested by measuring the redshift of the arc. Since the arcs are intrinsically faint this is a difficult procedure but Soucail et al. (1988), using a curved slit and 6 hours of integration on the ESO 3.6m telescope, found the arc in A370 to be at a redshift zarc — 0.724. The cluster has zc\ = 0.37, suggesting that the arc is not physically associated with the cluster. Coupling this observation with the beautiful symmetry of these arcs, and their location and orientation strongly

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suggests that they arise as a result of gravitational lensing. Under this hypothesis the giant luminous arcs provide a new and independent method for determining cluster masses interior to the arc, and so probe the cluster mass distribution on fairly small scales. As will be discussed later the numerous smaller arcs (arclets) which should accompany their more spectacular cousins provide additional probes of the cluster mass distribution and its morphology. With the launch of The Hubble Space Telescope and its ability to image high redshift clusters at high resolution, especially in the UV (where arcs are most visible) this method will become even more useful. Several detailed calculations have been carried out to determine the mass distribution in clusters using the giant luminous arcs (e.g., Grossman & Narayan 1989, Hammer & Rigaut 1989, Bergmann et al. 1989). These approaches are quite complex and so to get the idea of the basic physics across I will examine an idealized case. Suppose an observed luminous arc is large in angular extent, and can be assumed to represent one of the images which arises if the perfect Einstein ring configuration (source, center of lens and observer all aligned) is perturbed slightly. Assume also that the cluster mass distribution is axially symmetric about the line joining its center to the observer. Then the geometry of the situation and the bending angle formula for the cluster show that for the Einstein ring (and thereby for a small perturbation to it)

M(