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We examine the counts-in-cells probability distribution functions that describe dark matter halos in the Dark Energy Universe Simulations (DEUS). We describe the measurements between redshifts $z=0$ to $z=4$ on both linear and non-linear scales. The best-fits of the gravitational quasi-equilibrium distribution (GQED), the negative binomial distribution (NBD), the Poisson-Lognormal distribution (PLN), and the Poisson-Lognormal distribution with a bias parameter (PLNB) are compared to simulations. The fits agree reasonably consistently over a range of redshifts and scales. To distinguish quintessence (RPCDM) and phantom ($w$CDM) dark energy from $\Lambda$ dark energy, we present a new method that compares the model parameters of the counts-in-cells probability distribution functions. We find that the mean and variance of the halo counts-in-cells on $2-25h^{-1}$Mpc scales between redshifts $0.65, Comment: 24 pages, 15 figures, 1 table; submitted to ApJ

We calculate the irreducible triplet contribution to galaxy clustering in the cosmological many-body problem. These triplets generally represent short-lived configurations in which three close objects interact with pairwise gravitational forces. From the resulting grand canonical partition function, we obtain higher order analytical expressions for thermodynamic quantities such as specific heats, isothermal compressibility, and thermal expansion in the system. Compared with previous analyses, which included reducible but not irreducible triplets, the additional terms are usually small, especially in the limit of large N. This confirms the thermodynamic results. We also derive the modified spatial distribution functions and show that they agree with recent observational results. The inclusion of triplet and higher order irreducible clusters does not significantly modify the results of the reducible terms in the partition function or of its thermodynamic consequences such as the distribution function.

We examine the peculiar velocity distribution function of galaxies in cosmological many-body gravitational clustering. Our statistical mechanical approach derives a previous basic assumption and generalizes earlier results to galaxies with haloes. Comparison with the observed peculiar velocity distributions indicates that individual massive galaxies are usually surrounded by their own haloes, rather than being embedded in common haloes. We then derive the density of energy states, giving the probability that a randomly chosen configuration of N galaxies in space is bound and virialized. Gravitational clustering is very efficient. The results agree well with the observed probabilities for finding nearby groups containing N galaxies. A consequence is that our local relatively low mass group is quite typical, and the observed small departures from the local Hubble flow beyond our group are highly probable., Paper in aastex 5.0 format and 9 figures. Replace a new version with figures and typos corrected

We explore the development of galaxy clustering after an extremely long time in a flat Einstein–de Sitter universe. The time-scale for growth of clustering in this universe is shorter than the global expansion time-scale. This eventually leads to a single cluster of arbitrary large size, which dominates each expanding particle horizon. Since the particle horizons expand faster than the metric, one cluster may ultimately dominate the universe. Assuming that the distribution of galaxies in such a static, bound cluster becomes approximately isothermal, we show how it can be represented by a general relativistic metric embedded in an expanding Einstein–de Sitter universe. The embedding is achieved by means of a ‘Schwarzschild membrane’. Pressure is important, unlike in previous Tolman–Bondi models of inhomogeneities. These new hybrid models generalize previous descriptions of growing clusters and show how a changing equation of state for the matter may alter the large-scale symmetry of space–time over growing regions in the universe.

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

We develop the properties of cosmological many-body clustering at high redshift and compare the resulting theoretical distributions with recent observations of groups of galaxies around z ≈ 3. Analyses of the observed variance of counts-in-cells and of the probability for forming strongly overdense regions suggest Ω0 ≈ 0.3 ± 0.2 and 0.1 Ω0 0.3, respectively, for Einstein-Friedmann universes. Since these models are essentially analytical, they allow us to examine statistical questions that have analogs in more complicated cold dark matter models based on small numbers of simulations. We show how future observations of the galaxy distribution function can test cosmological many-body clustering and determine Ω0 more precisely. For Ω0 0.3, these models predict that galaxies start clustering at redshifts z 10.

Cosmological many-body clustering agrees with the spatial and velocity distribution functions of galaxies at low redshifts, and it can be extended to high redshifts z ≈ 3 or more. The high redshift distribution functions are predicted to have a particular form. In the simplest case, there are no free parameters in this prediction, but the degree of clustering depends sensitively on Ω0. Current observations of small groups at high redshifts suggest that Ω0 = 0.3 ± 0.2 for Einstein-Friedmann cosmologies.

We analyze the three-dimensional spatial-velocity distribution of the counts-in-cells for 4501 galaxies in the Pisces-Perseus supercluster. This is done for the total sample, for a subset containing the most conspicuous galaxies (brightest members with large intrinsic linear diameters), and for a subset contain- ing the least conspicuous galaxies (faintest members with small intrinsic linear diameters). We also examine subsets limited by absolute magnitudes as well as volume limited samples. The counts-in-cells distribution functions for all these samples are described remarkably well by the gravitational quasi- equilibrium distribution which was previously found to -t the more statistically homogeneous Zwicky, CfA, ESO, and IRAS catalogs. For the complete Pisces-Perseus sample, the value of b, representing the ratio of gravitational correlation energy to the kinetic energy of peculiar motions is 0.8. The most con- spicuous galaxies are somewhat more uniformly distributed than the least conspicuous ones which tend to follow the overall distribution. A dendrogram analysis shows that groups of the most conspicuous galaxies tend to be located near groups of the least conspicuous ones. We also analyze the velocity distribution in di†erent parts of the supercluster. There is a dense core of about 750 galaxies which have an overall Gaussian velocity distribution with a dispersion of about 450 km s~1 around the mean of 5260 km s~1. However this region is not completely relaxed since it con- tains subgroups which may not have Gaussian distributions. Moreover, in most subgroups the crossing time exceeds the Hubble time, indicating incomplete relaxation. Irregular galaxies tend to concentrate in this region, perhaps indicating signi-cant tidal interactions. Spatial distribution functions measured for increasing numbers of galaxies around the dense core of Pisces-Perseus approach the form and have the value of b observed in larger statistically homogeneous catalogs. It appears that galaxies in this region constitute a "" fair sample II with a wide range of Z103 structures characteristic of gravitational clustering. Our results appear consistent with the supercluster forming gravitationally as an accumulation of smaller clusters. Accurate peculiar velocities of these galaxies would help decide whether this clustering was stimulated by a larger scale perturbation or occurred by chance. Subject headings: galaxies: clusters: individual (Pisces-Perseus) E gravitation

We study how merging affects the luminosity and the spatial distributions of galaxies. By using a master equation description and a distribution of physical merging cross sections, we find that the evolution of the galaxy luminosity distribution and the spatial counts-in-cells distribution are closely related. They place strong constraints on each other. The observed evolution of the galaxy luminosity function in the Autofib Redshift Survey (Ellis et al.) implies at least a 10% decrease in the galaxy comoving number density at z < 0.7. This significantly affects the evolution of galaxy spatial counts-in-cells distribution. The observed evolution of the galaxy luminosity function also requires a form of "differential dimming" for the less luminous galaxies even if galaxy merging drives the evolution. A counts-in-cells analysis for distant galaxies would help clarify the nature of the excess counts of faint-blue galaxies at high redshifts.

We derive the metric for a simple class of isothermal inhomogeneous universes in which the nonzero pressure balances gravity. These models are spherical and static. There exists a family of such models in which the isothermal universe is a special case. These models may represent the ultimate state of an Einstein-de Sitter universe that undergoes a phase transition caused by gravitational clustering.

We identify low redshift clusters and groups in the Sloan Digital Sky Survey (SDSS) and estimate their kinetic and correlation potential energies. We compare the distribution of these energies to the predictions by Yang and Saslaw (2012) and in the process estimate a measure of an average 3-dimensional velocity and spatial anisotropy of a sample of clusters. We find that the inferred velocity anisotropy is correlated with the inferred spatial anisotropy. We also find that the general shape of the energy distribution agrees with theory over a wide range of scales from small groups to superclusters once the uncertainties and fluctuations in the estimated energies are included., Comment: 18 Pages, 8 figures. Accepted by ApJ. Latest version incorporates the referee's suggestions and comments

We develop a general theory for estimating the probability that a galaxy cluster of a given shape exists. The theory is based on the observed result that the distribution of galaxies is very close to quasi-equilibrium, in both its linear and nonlinear regimes. This places constraints on the spatial configuration of a cluster of galaxies in quasi-equilibrium. In particular, we show that that a cluster of galaxies may be described as a collection of nearly virialized subclusters of approximately the same mass. Clusters that contain more than 10 subclusters are very likely to be completely virialized. Using our theory, we develop a method for comparing probabilities of different spatial configurations of subclusters. As an illustrative example, we show that a cluster of galaxies arranged in a line is more likely to occur than a cluster of galaxies arranged in a ring., Comment: 18 pages, 6 figures. Corrected typos and incorporated referee suggestions. Accepted for publication in the ApJ

We examine the evolution of the spatial counts-in-cells distribution of galaxies and show that the form of the galaxy distribution function does not change significantly as galaxies merge and evolve. In particular, bound merging pairs follow a similar distribution to that of individual galaxies. From the adiabatic expansion of the universe we show how clustering, expansion and galaxy mergers affect the clustering parameter b. We also predict the evolution of b with respect to redshift., Comment: To appear in the Proceedings of the Conference in Honour of Murray Gell-Mann's 80th Birthday

This is the first exploration of the galaxy distribution function at redshifts greater than about 0.1. Redshifts are based on the North and South GOODS Catalogs. In each catalog we examine clustering in the two redshift bands 0.47 < z < 0.8 and 0.9 < z < 1.5. The mean redshifts of the samples in these bands are about 0.6 and 1.1. Our main result is that at these redshifts the galaxy spatial distribution function f_V(N) has the form predicted by gravitational quasi-equilibrium dynamics for cosmological many-body systems. This constrains related processes such as galaxy merging and the role of dark matter in the range of these redshifts., Accepted ApJ, 6 pages, 5 figures

The cosmological many‐body problem was stated over 300 years ago, but its solution is quite recent and still incomplete. Imagine an infinite expanding universe essentially containing a very large number of objects moving in response to their mutual gravitational forces. What will be the spatial and velocity distributions of these objects and how will they evolve? This question fascinates on many levels. Though inherently non‐linear, it turns out to be one of the few analytically solvable problems of statistical mechanics with long range forces. The partition function can be calculated. From this all the thermodynamic properties of the system can be obtained for the grand canonical ensemble. They confirm results derived independently directly from the first and second laws of thermodynamics. The behavior of infinite gravitating systems is quite different from their finite relations such as star clusters. Infinite gravitating systems have regimes of negative specific heat, an unusual type of phase transition, and a very close relation to the observed large‐scale structure of our universe. This last feature provides an additional astronomical motivation, especially since the statistical mechanics may be generalized to include effects of dark matter haloes around galaxies. Previously the cosmological many‐body problem has mostly been studied using the BBGKY hierarchy (not so suitable in the non‐linear regime) and by direct computer integrations of the objects' orbits. The statistical mechanics agrees with and substantially extends these earlier results. Most astrophysicists had previously thought that a statistical thermodynamic approach would not be applicable because: a) many‐body gravitational systems have no rigorous equilibrium state, b) the unshielded nature of the long‐range force would cause the partition function to diverge on large scales, and c) point masses would produce divergences on small scales. However, deeper considerations show that these are not significant difficulties. There remain many important questions for which we have only a very preliminary understanding. An example is to find the basin of attraction for initial states which are able to evolve into systems described by this gravitational partition function. It may well be that ideas and techniques developed for other types of statistical mechanical systems will help us answer these questions.

We show that the tenure lengths for managers of sport teams follow a power law distribution with an exponent between 2 and 3. We develop a simple theoretical model which replicates this result. The model demonstrates that the empirical phenomenon can be understood as the macroscopic outcome of pairwise interactions among managers in a league, threshold effects in managerial performance evaluation, competitive market forces, and luck at the microscopic level.

We determine the spatial distribution function of galaxies from a wide range of samples in the 2MASS survey. The results agree very well with the form of the distribution predicted by the theory of cosmological gravitational many-body galaxy clustering. On large scales we find a value of the clustering parameter b = 0.867 +/- 0.026, in agreement with b = 0.83 +/- 0.05 found previously for the Pisces-Perseus supercluster. We measure b(theta) as a function of scale, since this is a powerful test of the applicability of computer simulations. The results suggest that when galaxies clustered they were usually surrounded by individual, rather than by communal haloes., Astrophysical Journal, accepted: 14 pages with 23 embedded reduced resolution Postscript figures & 2 tables

Even before astronomers knew that galaxies existed, they found that they are clustered. Though this statement may seem contradictory, it is not quite as contradictory as it seems. In compiling his famous catalog of nebulae, Messier (1781) noticed that about a dozen of his hundred or so objects were clumped together in the constellation of Virgo. A few years later, William Herschel (1784, 1811) sweeping the heavens with his telescope in Slough, recognized great inhomogeneities in the distribution of his much larger collection of thousands of nebulae. Herschel speculated, as Hubble and others would show definitively in the 1920s, that many of these nebulae were vast star systems, like our own Milky Way, held together and perhaps even caused to cluster by the forces of gravity. Most of Messier’s objects in Virgo were, in fact, such galaxies.

The clustering of galaxies is sometimes said to resemble a phase transition from a uniform distribution to an inhomogeneous collection of clusters. We examine how close this resemblance actually is using the cosmological many-body model for galaxy clustering. This model agrees well with the observed galaxy distribution and correlation functions. It also has exact analytic equations of state which can be used to explore its phase transition properties simply and directly. While there are superficial resemblances to a second-order phase transition, this description is modified by the infinite dynamical timescale needed for a hierarchy of clustering to develop at all levels throughout the universe. In an Einstein-deSitter cosmology, a first-order phase transition may occur asymptotically in the infinite future.

The expansion of the flat Einstein-deSitter model universe constantly slows, and ultimately ceases. This facilitates the eventual condensation of galaxies into a small number of great clusters. Consequently the model may evolve asymptotically into a cosmological isothermal sphere. The symmetries and metric of an isothermal model differ from those of its Robertson-Walker progenitor, suggesting that the evolution leads to a phase transition in the infinite future.

Some basic physical properties of the cosmological many-body problem are discussed and the successes and limitations of its applicability to galaxy clustering are reviewed, especially regarding recent observations of high redshift clustering.

IF black holes exist in the contracting phase of a closed universe, they will give rise to a pressure and entropy catastrophe. First, the black holes absorb all the radiation; then their apparent horizons merge, and coalesce with the cosmological apparent horizon. All external observers become internal observers. It is possible that the internal metric of some of the merging black holes will be contracting, and others expanding. I suggest here that the resulting violent inhomogeneities can lead to a reexpansion in a significant portion of the universe. Global reexpansion, prompted by the merging of black holes, may thus begin in a semi-classical rather than fully quantum gravitational era, at densities greater than those at which nucleosynthesis occurs. Surviving black holes and inhomogeneities could initiate the formation of structures such as galaxies in the 'new' universe. The behaviour of such an oscillating universe would differ in detail from cycle to cycle.

This topical volume examines one of the leading problems in astronomy - how galaxies cluster in our Universe. This book, first published in 2000, describes gravitational theory, computer simulations and observations related to galaxy distribution functions. It embeds distribution functions in a broader astronomical context, including other exciting contemporary topics such as correlation functions, fractals, bound clusters, topology, percolation and minimal spanning trees. Key results are derived and the necessary gravitational physics provided to ensure the book is self-contained. Throughout the book, theory, computer simulation and observation are carefully interwoven and critically compared. The book also shows how future observations can test the theoretical models for the evolution of galaxy clustering at early times in our Universe. This clear and authoritative volume is written at a level suitable for graduate students, and will be of key interest to astronomers, cosmologists, physicists and applied statisticians.