STATISTICAL MECHANICS OF THE COSMOLOGICAL MANY-BODY PROBLEM

Farooq AhmadDepartment of Physics, University of Kashmir, Srinagar 1900006, Kashmir, India;and Inter-University Centre for Astronomy andAstrophysics, Pune 411007, India

William C. SaslawAstronomyDepartment, University of Virgina, Charlottesville, VA 22903; and Institute of Astronomy,Madingley Road,

Cambridge CB3 0HA, England; andNational Radio AstronomyObservatory,1 Charlottesville, VA 22903

and

Naseer Iqbal BhatDepartment of Physics, University of Kashmir, Srinagar 1900006, Kashmir, India;and Inter-University Centre for Astronomy andAstrophysics, Pune 411007, India

Received 2001 September 14; accepted 2002 February 7

ABSTRACT

We derive analytical expressions for the grand canonical partition functions of point masses, and ofextended masses (e.g., galaxies with halos), which cluster gravitationally in an expanding universe. From thepartition functions, we obtain the systems thermodynamic properties, distribution functions (includingvoids and counts in cells), and moments of distributions such as their skewness and kurtosis. This also pro-vides an analytical calculation of the evolution of the distribution. Our results apply to both linear and non-linear regimes of clustering. In the limit of point masses, these results reduce exactly to previous resultsderived from thermodynamics, thus providing a new, more fundamental foundation for the earlier results.

Subject headings: cosmology: theory galaxies: clusters: general gravitation large-scale structure of universe methods: analytical

1. INTRODUCTION

Galaxies cluster on very large scales under the inuenceof their mutual gravitation, and the characterization of thisclustering is a problem of current interest. Dierent techni-ques such as percolation (Zeldovich, Einesto, & Shandarin1982; Grimmet 1989), minimal spanning trees (Pearson &Coles 1995; Bhavsar & Splinter 1996), fractals (Mandelbrot1982), correlation functions (Totsuji &Kihara 1969; Peebles1980), and distribution functions (Saslaw &Hamilton 1984)have been introduced to understand the large-scale struc-ture in the universe. Of all the description so far, correla-tions and distribution functions have been related mostdirectly to physical theories of gravitational clustering.The consequences of these theories also agree well withobservations.Thus far, theories of the cosmological many-body galaxy

distribution function have been developed mainly from athermodynamic point of view. This starts from the rst twolaws of thermodynamics and, for quasi-equilibrium evolu-tion, derives gravitational many-body equations of state inthe context of the expanding universe. Application of thethermodynamic uctuation theory to these equations ofstate, considering the galaxies as point gravitating masses,then gives their distribution function. Comparisons of grav-itational thermodynamics to the cosmological many-bodyproblem have been discussed on the basis of N-body com-puter simulation results (e.g., Saslaw et al. 1990; Itoh, Ina-gaki, & Saslaw 1988, 1993). Comparisons with the observedgalaxy clustering (e.g., Sheth, Mo, & Saslaw 1994; Fang &Zou 1994; Raychaudhury & Saslaw 1996) along with other

theoretical arguments (e.g., Zhan 1989; Zhan & Dyer 1989)support it further.The applicability of thermodynamics to the cosmological

many-body problem suggests that statistical mechanicsshould also apply. This close relation occurs because statis-tical mechanics is the microscopic (and therefore perhapsmore fundamental) statistical description of particle (e.g.,galaxy) positions and motions whose ensemble averagesprovide the macroscopic thermodynamic description of thesystem. Hitherto it has not been possible to develop a statis-tical mechanical theory ofN-body galaxy clustering becausethe relevant partition function for the gravitational grandcanonical ensemble could not be solved.The general conditions under which statistical mechanics

may describe the cosmological many-body problem areclosely related to those for the applicability of thermody-namics, described in detail previously (Saslaw & Hamilton1984; Saslaw & Fang 1996; Saslaw 2000), and we briey dis-cuss them in the present context. When the ensemble aver-aged thermodynamic quantities change more slowly thanlocal dynamical crossing or clustering timescales, then theform of the statistical distribution functions remains essen-tially the same, and only their macroscopic variables evolve.In this quasi-equilibrium evolution, equilibrium statisticalmechanics provides a good approximation to the distribu-tion of particles and velocities at any given time with the val-ues of the macroscopic variables at that time. Inequilibrium, all permissible microstates of the systems in theensemble have an equal a priori probability. This is the fun-damental postulate of statistical mechanics. It implies thatthe approximate probability of nding a specied macro-state in the system is proportional to the number of permis-sible microstates having the macrostates properties.Cosmological many-body systems generally satisfy the

timescale criterion of quasi-equilibrium statistical mechan-1 Operated by Associated Universities, Inc., under cooperative agree-

ment with the National Science Foundation.

The Astrophysical Journal, 571:576584, 2002 June 1

# 2002. The American Astronomical Society. All rights reserved. Printed in U.S.A.

576

ics since macroscopic global variables such as average tem-perature, density, and the ratio of gravitational correlationenergy to thermal energy change on timescales at least aslong as the Hubble time, whereas local dynamical timescalesin regions of clustering are shorter. The criterion of equal apriori probabilities for any microstate or conguration isless well understood, and its rigorous derivation remains animportant unsolved problem even in classical statisticalmechanics. It is closely related to statistical homogeneityand the absence of extensive very nonlinear structures overscales comparable with the system. N-body simulations inthe previous references show that relaxation to the observeddistributions (e.g., eq. [42] below) of quasi-equilibrium stat-istical mechanics occurs for initial power-law perturbationspectra with power-law indices between about 1 and +1.Systems with much stronger global initial correlations oranticorrelations relax only after many expansion timescales,or not at all. A more detailed analysis of the ranges of initialconditions that form the basin of attraction for equation(42) is an important future exploration.In the present paper, we investigate the problem of non-

linear gravitational galaxy clustering from the point of viewof statistical mechanics. The statistical mechanics ofN-bodysystems is based on the N-body Hamiltonian. From this thepartition function is formed as a function of an N-dimen-sional integral that incorporates the eects of interactionsamong all the particles. To evaluate such an integral analyti-cally is generally very complicated. However, we will showhow the partition function for the cosmological many-bodyproblem can be evaluated analytically. This serves as a basicresult for rigorously evaluating all the thermodynamicproperties of the system, starting with the free energy. It isparticularly interesting that the parameter b, which is theratio of the gravitational potential correlation energy totwice the kinetic energy of peculiar motion, emerges directlyin the partition function and in the equations of state. So wedo not need to make any assumptions in the derivation ofthe functional form of bnT3 as was done earlier (Saslaw& Hamilton 1984; Saslaw & Fang 1996). Once the many-body partition function is known, there is no diculty inevaluating the grand canonical partition function, whichrepresents the exchange both of particles and energy. Fromthe grand canonical partition function, the distributionfunction of galaxies follows directly.In addition to its generality and rigor, the main advantage

of this statistical mechanical approach is that it can easily beextended to nonpoint-mass systems. Until now, all thethermodynamic techniques developed for the study of gravi-tational clustering have been applicable only to point massgalaxies. Actual galaxies have extended structures and hal-oes, and we shall see how this aects previous results. Theintroduction of a softening parameter enables us to includeeects of large haloes of dark matter around galaxies. In x 2,we develop the analytical solution of the conguration inte-gral for the cosmological gravitational systems. This inte-gral may be applied to systems containing either point orextended masses. Then we analytically calculate the parti-tion function and the free energy for systems composed ofparticles assumed to be point masses. From the free energy,all the thermodynamic functions follow directly, in x 3,including the equations of state. The proper thermodynamicdependence of the parameter b emerges directly in the equa-tions of state. Then we calculate the distribution function,f N, simply and directly. All these results agree exactly

with earlier ones derived using thermodynamic arguments.These earlier results are then extended naturally to systemshaving nonpoint-mass particles. In x 4, the distributionfunction f N is calculated more generally, taking theextended nature of the particles into account. Section 5 givesthe moments, along with the skewness and kurtosis, for thedistribution functions of extended galaxies that are cluster-ing under their mutual gravitation. In x 6, we examine theevolution of b for extended galaxies. Finally, our results arediscussed in x 7.

2. THE COSMOLOGICAL MANY-BODYPARTITION FUNCTION

We start with the formalism of classical statisticalmechanics for an ensemble of comoving cells containinggravitating particles (galaxies) in an expanding universe. Ifthe size of the cells is smaller than the particle correlationlength, then each member of this ensemble is correlatedgravitationally with other cells. However, correlationswithin a cell are generally greater than correlations amongcells, so that extensivity is a good approximation (e.g.,Saslaw 2000; Sheth & Saslaw 2000). We consider a large sys-tem, which consists of an ensemble of cells, all of the samevolume V, or radius R (much smaller than the total volume)and average density nn. Both the number of galaxies and theirtotal energy will vary among these cells that, taken together,are represented by a grand canonical ensemble. In the sys-tem, galaxies have a gravitational pairwise interaction. Weassume that their distribution is statistically homogenousover large regions.Our starting point is the general partition function of a

system of N particles of mass m interacting gravitationallywith a potential energy , having momenta pi and averagetemperature T:

ZNT ; V 13NN!

Z

exp XNi1

p2i2m

r1; r2; . . . ; rN" #

T1( )

d3Np d3Nr : 1Here N! takes the distinguishability of classical particlesinto account, and normalizes the phase space volume cell.Integration over momentum space yields

ZNT ; V 1N!

2mT

2

3N=2QNT ; V 2

as usual, where

QNT ; V Z

Z

expr1; r2; . . . ; rNT1 d3Nr 3

is the conguration integral. In general, the gravitationalpotential energy r1; r2; . . . ; rN is a function of the rela-tive position vector rij jri rjj and is the sum of the poten-tial energies of all pairs. Evidently the evaluation ofQNT ; V is the main task and involves a very complicated3N-fold integration, which usually cannot be carried out forgeneral interparticle potentials. Often, as in the theory ofimperfect gases or liquids, this conguration integral isapproximated by a virial expansion in powers of the density,an approximation that becomes less accurate at higher den-sities. However, we show here that for the cosmological

MECHANICS OF COSMOLOGICAL MANY-BODY PROBLEM 577

gravitational many-body problem, it is not necessary toconstruct the usual virial expansion, and the partitionfunction has a relatively simple exact solution.In this gravitational system the potential energy

r1; r2; . . . ; rN is due to all pairs of particles (galaxies)composing the system, hence

r1; r2; . . . ; rN X

1i

tively small nite range of distances, this integration extendsonly over a limited region of space whose linear dimensionsare of the order where fij is appreciable. (Note that thismight not apply to some fractal density distributions.) Theintegration over the coordinates r1 of the particles gives astraightforward factor of V, while the second integral aftertransforming to spherical polar coordinates gives

Q2T ; V V 2 1 3Gm2=R1

2Tnn1=3" #

; 15

where

R1

1

R1

2s

R1

2ln

=R11

1 =R12

q : 16Here R1 is the radius of the cell, and nn is the average numberdensity per unit volume so that nn1=3 rr.Next consider the scale transformation for temperature T

and number density nn:

T!1T ;nn!3nn : 17

This leaves the factor 1=Tnn1=3 in the second term of equa-tion (15) invariant, i.e.,

1=Tnn1=3!nnT3 : 18But the ratio Gm2=Tnn1=3 is dimensionless, and thereforethe dimensionless scale invariance transformation forGm2=Tnn1=3 gives

Gm2=Tnn1=3!Gm23nnT3 : 19This is also clear from the scaling property of the partitionfunction derived by Landau & Lifshitz (1980) and veriedfor the scaling property of the gravitational partition func-tion (Saslaw & Fang 1996). Accordingly, the general solu-tion of equation (1) has the form

ZNT ; V T3=2N ; 20where nn and T enter as a function of just nnT3. Conse-quently, any modications of the thermodynamic functionsowing to the potential energy term in equation (1) willdepend on the intensive thermodynamic variables nn and Tonly in the combination nnT3. Therefore, equation (15) isinvariant under the transformation (17) and has the form

Q2T ; V V 2 1 nnT3 R1

; 21

where 3=2 Gm23 is a positive constant. Proceeding inthe same way, we nd Q3T ; V, Q4T ; V, andQNT ; V:

Q3T ; V Z Z Z

1 f121 f23 d3r1d3r2d3r3

V3 1 nnT3 R1

2; 22

Q4T ; VZ Z Z Z

1 f141 f241 f34 d3r1d3r2d3r3d3r4

V4 1 nnT3 R1

3; 23

and, in general,

QNT ; V VN 1 nnT3 R1

N1: 24

Finally, substituting equation (24) into equation (2) givesthe partition function for gravitational clustering of galaxiesincorporating their extended structures, such as sphericalhalos:

ZNT ; V 1N!

2mT

2

3N=2

VN 1 nnT3 R1

N125

with 3=2 G3m6 and =R1 1 =R121=2=R12 lnf=R1=1 1 =R121=2g. The eect of=R1 on the behavior of =R1 is shown in Figure 1. For=R1 0, corresponding to point masses, =R1 1. Itdecreases monotonically as =R1 increases up to its maxi-mum realistic value less than 1.

3. THERMODYNAMIC FUNCTIONS FOR A SYSTEMOF EXTENDED GALAXIES

Once the partition function is known, it is straightfor-ward to calculate the thermodynamic functions and the

Fig. 1.Variation of =R1 as a function of =R1

No. 2, 2002 MECHANICS OF COSMOLOGICAL MANY-BODY PROBLEM 579

equations of state. For example, the free energy is given by

F T lnZNT ; V

NT ln NVT3=2

NT ln 1 nnT3

R1

NT 32NT ln

2m

2

: 26

In this expression we have approximated N 1 by N forlarge N. Note that nn appears in the factor multiplying since this is an average over the system. From this freeenergy the other thermodynamic quantities such as entropyS, internal energy,U, pressure, P, and chemical potential, l,follow directly:

S @F@T

V ;N

NT ln NVT3=2

N ln 1 nnT3

R1

3Nb 52N 3

2N ln

2m

2

; 27

U F TS 32NT1 2b ; 28

P @F@V

T ;N

NTV

1 2b ; 29

and

l

T

1T

@F

@V

T ;V

ln NVT3=2

ln1 b b 3

2ln

2m

2

: 30

In these results, the quantity b appears naturally and isgiven by

b nnT3=R1

1 nnT3=R1 : 31

Equation (31) for b reduces to the usual result for pointmasses for 0, implying 1, from equation (16), andb is therefore related to b (point masses) by

b b=R11 b=R1 1 : 32

Thus, for point masses 0, 1), we get the sameexpressions originally derived by assuming the point massform of equation (31) for b (Saslaw & Hamilton 1984) andlater conrmed by various authors. More complicated rela-tions for bnnT3 have been postulated from time to timebut one can show (Saslaw & Fang 1996; F. Ahmad &S. Masood 2002, in preparation) that they lead to undesir-able physical consequences such as distribution functionshaving negative probabilities, or violating the rst or secondlaws of thermodynamics. However, in the present deriva-tion, the functional form of b emerges directly from the par-tition function and its equations of state. This indicates adeep connection between the statistical mechanics and ther-modynamics of gravitational galaxy clustering.

The eect of =R1 on b can be understood from the dier-ence b b, as b changes for various values of =R1 as shownin Figure 2. It can easily be seen that b is not much alteredfor either small or reasonably large values of =R1. How-ever, this eect is maximum at intermediate values of b irre-spective of =R1. This is because at small values of b, theparticle behavior is close to the ideal gas situation, and thesoftening parameter has very little eect on gravitationalgalaxy clustering. On the other hand, clustering is close tothe completely virialized state for b close to one. From equa-tion (32) we notice that b depends strongly on , which inturn is a function of =R1. The softening parameter intro-duces correlations that lower the correlation energy, andconsequently b decreases relative to b 0.We conclude from the above equations of state that all

the thermodynamic quantities are aected by the introduc-tion of a softening parameter that may correspond to agalactic halo. This means that the softening parameter andcell size may have nonnegligible eects on the thermody-namics of gravitating interacting masses in an expandinguniverse.

4. DISTRIBUTION FUNCTIONS

The overall clustering of galaxies is partially character-ized by the full probability distribution function f N. Itcontains the void distribution f0V as well as the objectiveclustering statistic fV N of counts of the number of galaxiesin cells of a given size distributed throughout the system.We next use our result for the partition function to give anew much simpler derivation of the distribution functionfor the cosmological many-body system.For a system of many particles (galaxies), the thermody-

namic quantities obtained in x 3 from the canonical parti-tion function for the canonical ensemble are the same as inthe grand canonical ensemble, since they are average rela-tions for the entire system (e.g., Huang 1987). However, to

Fig. 2.Eect of =R1 on b. This illustrates the variation of b-b as afunction of b for dierent values of =R1.

580 AHMAD, SASLAW, & BHAT Vol. 571

obtain the probability distribution function, we need to con-sider the departures from these averages, since the distribu-tion functions describe the probabilities of ndinguctuations of all amplitudes. For the system of galaxies,where galaxies as well as energy can cross cell boundaries,we must use the grand canonical ensemble. The rst step isthus to obtain the grand canonical partition function ZG,which is a weighted sum of all the canonical partition func-tions dened by

ZGT ; V ; z X1N0

eNl=TZNT ; V : 33

The probability of nding N-particles in a cell of the grandcanonical ensemble is the sum over all of the energy states:

f N PN

i0 eNl=TeUi=T

ZGT ; V ; z ;

eNl=TZNT ; VZGT ; V ; z : 34

This is the basic equation for evaluating the distributionfunction. Recall that the weighting factor z el=T , theactivity, is an average value over NN, while the canonical par-tition function ZNT ; V in equation (1) is a sum over allN-particles. For a system of point masses, eNl=T andZNT ; V are given by equations (30) and (25) with 1and therefore 0:

eNl=T NN

VT3=2

N1 bNeNb 2m

2

3N=235

and

ZNT ; V 1N!

2m

2

3N=2VT3=2

N

1 NbNN1 b N1

: 36

Also, the grand canonical partition function ZG is generallyrelated to the thermodynamic potential and thermody-namic variables by

lnZG PVT

NN1 b : 37

Substituting equations (35)(37) into equation (34) gives thedistribution function for point mass galaxies directly:

f N NN1 b

N!NN1 b Nb N1e NN1bNb : 38

This is the same distribution function for a system of pointmass particles derived earlier by Saslaw & Hamilton (1984)and Saslaw & Fang (1996), although here we have derived itdierently and quite simply, directly from statisticalmechanics.We proceed in a similar way to derive the distribution

function for a cosmological system of galaxies or particlesrepresented by nonpoint masses. For nonpoint masses,

eNl=T and ZNT ; V are obtained from equations (30),(25), and (31):

eNl=T NN

VT3=2

N1 bNeNb 2m

2

3N=239

and

ZnT ; V 1N!

2m

2

3N=2

VT3=2

N

1 NbNN1 b N1

: 40

Also, the grand partition function ZG for a nonpoint-masssystem is

lnZG PVT

NN1 b : 41

Thus, substituting equations (39)(41) into equation (34)gives the distribution function for nonpoint-massparticles:

f N; NN1 b

N! NN1 b NbN1e NN1bNb :

42As ! 0 in f N; , then this distribution tends to the distri-bution function of the point mass distribution.We next compare the results of the distribution function

for extended particles obtained in equation (42) with theearlier point mass distribution function of equation (38).Figure 3 shows two examples of the eects of nite galaxysize on the counts-in-cells distribution fV N; for cells con-taining an average of NN nnV 1 galaxy (Fig. 3a) andNN 10 galaxies (Fig. 3b). The solid lines plot the distribu-tions for =R1 0 for comparison, showing how the distri-butions are modied for b 0:1, 0.6, and 0.9 if =R1 0:5.These indicate that quite large values of =R1 are necessaryto produce signicant departures of the distributions fromthose of point masses.

5. MOMENTS FOR EXTENDED STRUCTURES

Sometimes it may not be necessary or practicable to knowthe exact shape of the distribution function. Then its generalproperties can be characterized by a sequence of momentsof the distribution. Fluctuations may be related to varia-tions of the numbers of particles, energy, etc., throughmoments. Higher order uctuations are also informativeand can be calculated straightforwardly as discussed byCallen (1985) and Saslaw (1969). For nonpoint masses, theuctuation moments for the number of particles are

hDN2i NN

1 b2; 43

hDN3i NN

1 b41 2b ; 44

and

hDN4i NN

1 b61 8b 6b2 3hDN2i2 : 45

No. 2, 2002 MECHANICS OF COSMOLOGICAL MANY-BODY PROBLEM 581

In calculating these moments, we have used the rst-, sec-ond-, and third-order derivatives of equation (39) directly.In a similar way, we can evaluate uctuations in energy andcorrelated energy-number uctuations as

hDU2i 3NNT2

41 b25 20b 34b2 163 ; 46

hDNDUi 3NNT

21 b21 4b 2b2 ; 47

and

hDN2DUi 3NNT

21 b41 6b 2b2 : 48

Another aspect of these moments is their relation to cor-relation functions (e.g., Saslaw 2000). In addition, from thedensity moments we can measure the skewness and kurtosisof the distributions of extended galaxies clustering undergravitation. The skewness is given by

S NN hDN3i

hDN2i2 1 2b ; 49

and the kurtosis is given by

K NN2 hDN4i 3hDN2i2hDN2i3

1 8b 6b2 : 50These relations in equations (49) and (50) are especiallyimportant as they can easily be related to observed catalogsand to simulations.

6. EVOLUTION OF b t As the universe expands, clustering generates entropy,

which is reected in the increase of b in equation (27). For astatistically homogeneous universe, the local expansion incomoving coordinates would be adiabatic and satisfy therst law of thermodynamics dQ dU PdV with no netheat ow into or out of a volume, which implies

dQ 0 dU PdV : 51Using the cosmic energy equation to calculate the adiabaticevolution for values of b (for a point-mass system) overlarge scales as a function of expansion scale, a, gives (Saslaw1992)

a

a b1=8

1 b7=8; 52

where a is a constant that determines the value of b at someducial expansion state.For extended masses, the time evolution of b as a func-

tion of the expansion scale can be obtained from equation(51) by regarding U and b as functions of P and T, sincethen

3N

21 2bdTP @U

@b

T

PdV 0: 53

To determine b as a function of P and T, we use equations(29) and (31) to nd

PT4 b : 54Equation (54) can be reexpressed after dierentiation as

dTP T4b

db : 55

Fig. 3a Fig. 3b

Fig. 3.(a) Eects of nite galaxy size for =R1 0:5 and NN 1 on the galaxy counts-in-cells distribution function fV N; . Solid lines: The b-valuesrepresent point particles with =R1 0. Dotted and dashed lines: The b-values are modied by =R1 0:5, illustrating distribution functions for galaxies withhaloes. (b) Same as (a), but for NN 10.

582 AHMAD, SASLAW, & BHAT Vol. 571

Substituting dTP from equation (55) and @U=@bT fromequation (28) into equation (53), and making use ofdV=V 3da=a , leads after integration to

a

a b1=8

1 b7=8 b

1=8

1 b7=8

R1

1=8

1 b R1

1

3=4: 56

Therefore, the evolution of b for extended galaxies hasthe same form as for point galaxies but with b substitutingfor b.

7. DISCUSSION

It has long been an outstanding problem to relate thedynamical, thermodynamic, and statistical descriptions ofgravitational many-body systems in various contexts. Forcanonical ensembles of nite systems, such as individualglobular clusters or virialized clusters, which can exchangeenergy but not particles, these relations have been workedout in considerable detail (e.g., Saslaw 1987; de Vega &Sanchez 2001a, 2001b and references therein). For the cos-mological many-body system, where linear and nonlinearclustering of galaxies and their haloes are present over manyscales simultaneously, grand canonical ensembles areneeded since cells can exchange galaxies as well as energy.This system has been described dynamically by N-bodysimulations and thermodynamically by quasi-equilibriumequations of state and uctuations around them. But it hasnot hitherto had a statistical mechanical description.There have been several perceived diculties in providing

a statistical mechanical description of the cosmologicalmany-body system. One is the long-range nature of thegravitational eld leading to divergences in integrals ofpotential. Another is the expansion of the universe. How-ever, it has been realized (e.g., Saslaw & Fang 1996) that themean eld and the expansion essentially cancel each other inthe Friedmann-Robertson-Walker universes, showing thatthe dynamical, thermodynamic, and statistical descriptionsall depend only on the local departures from the mean eld.This is the opposite situation from that prevailing in othermany-body systems where the mean eld dominates.A third perceived diculty with gravitational statistical

mechanics are the divergences in the Hamiltonian, whichresult from two or more point particles approaching oneanother arbitrarily closely. Here we have resolved this prob-lem by using a softened potential to eliminate the diver-gence, and also to represent galaxy haloes. We found thatthe statistical description continues to be valid even in thelimit when the softening scale length vanishes. The reasonfor this, as in the thermodynamic description (Saslaw 2000),is that the probability of such point collisions in the expand-ing universe (although not always in nite globular clusters)is very small, and eliminating them does not signicantlyinuence the thermodynamic state of the system, inagreement also with dynamical N-body simulations ofpoint galaxy clustering.

Therefore, it has become reasonable to expect that a stat-istical mechanical description of the cosmological many-body system is possible, and we have found it by calculatingthe partition function analytically under these circumstan-ces. The basic assumption necessary for this is that the uni-verse is undervirialized, i.e., that it does not have a virializedhierarchy of structures on all scales, so b W=2T < 1(with W the average gravitational correlation energy in acell). This is observed to be the case in our universe forwhich b 0:75.From the closed form of the resulting gravitational

grand canonical partition function, we readily derive allthe thermodynamic quantities and their equations ofstate. These give exactly the same previous results founddirectly from thermodynamic arguments in the limit ofthe zero-softening parameter and generalize them for anite softening parameter . Only for softening parame-ters that are a signicant fraction, say e0:3, of the cellsize in physical coordinates, are the thermodynamicsmodied signicantly. We calculate the modied galaxydistribution functions f N; and their moments, as wellas their evolution.Analyses of earlier galaxy surveys, mentioned in x 1 and

having approximately 104105 galaxies, have determinedvalues of b to within an uncertainty of about 0.05. FromFigure 3, this is only sucient to provide an upper limit ofabout 0.1 to =R1. Much of the uncertainty comes from cat-alog incompleteness and resulting statistical uctuations inthe values of f N. Present ongoing catalogs, such as the2dF survey, with about 105106 galaxies can decrease theuncertainty in b as long as their selection criteria are homo-geneous and their samples contiguous. Future catalogs with106107 galaxies (e.g., the Sloan Survey) may reduce theuncertainty in b by a factor of about 10 and provide a directdetermination of =R1.Another method for uncovering the eects of =R1

may be to examine b over a range of redshifts and com-pare the results with equation (56). Both and R1 willvary with time as galaxy haloes form and dissipate andas the scale length of galaxy clustering increases. Follow-ing these developments naturally requires detailed mod-els, and their eects on f N may be detectable andseparable from other evolutionary properties in futureextensive catalogs that observe out to z 3. At these dif-ferent redshifts, it may also be possible to determine howevolving haloes aect the value of b as a function ofscale, i.e., cell size.In addition to its observational implications, the avail-

ability of the partition function in the form of equation (25)can provide new insights into the theory of cosmologicalmany-body gravitational clustering, for example, into thenature of associated phase transitions.

F. A. is particularly grateful to the Institute of Astron-omy, Cambridge University, for providing facilities to carryout this work, and to Jesus College, Cambridge, forproviding accommodation during his visit. We thank MarkWilkinson for helpful discussions.

No. 2, 2002 MECHANICS OF COSMOLOGICAL MANY-BODY PROBLEM 583

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584 AHMAD, SASLAW, & BHAT