COSMOLOGY, CLUSTERING AND SUPERCLUSTERING WILLIAM C. SASLAW On large scales, the probability that galaxies occupy a given size volume of space is not random, like the toss of a coin, but is related to the presence of nearby galaxies. Positions of galaxies depend on one another. Details of this dependence may provide important clues to the origin and evolution of the universe. Modern systematic searches for these clues started in the 1930s with Edwin P. Hubble's galaxy counts, accelerated in the 1950s with the development of new statistical techniques, and surged in the 1970s and 1980s as computers became powerful enough to handle large amounts of data and calculate complex simulations of clustering physics. There are two main descriptions of galaxy clustering. The first is essentially pictorial, derived by searching the sky for filaments, voids, and overdense regions. Galaxies in projected high-density regions having similar redshift distances are likely to form a physically related cluster. If such a cluster has been bound gravitationally for a large fraction of the present Hubble time, it evolves fairly independently of its surrounding galaxies. Occasionally several contiguous large clusters form a supercluster. This may result partly from the initial positioning of clusters and partly from their subsequent motions. Most galaxies are not in large physically bound clusters. They are, however, clustered in a more general statistical sense. Rather than examining individual clusters, this second description looks for statistical departures from a Poisson distribution (which is the uniform random spatial distribution of objects whose positions are entirely independent of one another, i.e., totally uncorrelated). In a Poisson distribution there will be some regions where galaxies are strongly clustered just by chance, and other regions which are unusually empty, also by chance. The observed numbers and statistics of such regions may then be compared with those expected from various theories. STATISTICAL MEASURUES OF CLUSTERING The most useful and informative statistics can be measured objectively from observations and related analytically to physical processes of clustering and computer experiments. Low-order correlation functions are one example. The two-point correlation function *(*) in its simplest form for a homogeneous isotropic system is defined by ************************. Here ******is the average number of galaxies between radial distance * and *+dr from any given galaxy. The overall average number density of galaxies in the entire system, or over a very large volume of the universe, is *. For a random Poisson distribution, *(*)****, so **(*) is determined just by * and geometry. Therefore *(*) helps measure departures from the Poisson state. Higher-order correlation functions use the relative positions of three or more galaxies for a more refined description that, unfortunately, is harder to measure observationally. The first accurate observations of *(*) for galaxies, made by H. Totsuji and T. Kihara in 1969, gave a power law of the form *(*) ******* on scales 0.2 *******Mpc (for a Hubble constant of 50 km s***Mpc*). Large clusters of galaxies are observed to have a similar two-point correlation function if each cluster is represented by a single point, but this result is much more uncertain. On small scales *(*)***, so the observed clustering is highly nonlinear; that is, correlations dominate for ****Mpc. Another observed simple objective clustering statistic is the distribution function *(*). This is the probability for finding * galaxies in a volume of size V, or projected onto the sky in an area of size A. If the distribution is statistically homogeneous then it will not depend significantly on the shape of the volumes or areas, provided they are sufficiently large and numerous to give a fair average sample. Recent analyses of the area counts of galaxies show that they have a distribution of the form **************************. Where ****** is the average number in a volume V for an average number density *. The quantity b is a measure of clustering and is related to gravitational correlations. The observed value of b is */******* for galaxies whose separations are typically 1-10 Mpc. For large clusters with separations of ***********************. For a sample of faint radio sources with separations ****50Mpc,******, which is a random Poisson distribution. The *(*) distribution for **** gives the probability that a region is a void with no galaxies at all. Other statistics applied to galaxy clustering include minimal spanning trees (the shortest line connecting all the galaxies in a sample), topological patterns formed by contour maps of regions with the same density, and multifractal analyses (a single fractal dimension does not adequately describe galaxy clustering), which are related to how the average number density of galaxies around a given galaxy changes with distance from the galaxy. These other statistics also yield valuable information. Unlike *(*) and *(*), however, they have not yet been related generally to an underlying dynamical theory. Some specialized computer experiments have examined their behavior. THEORIES OF CLUSTERING To understand the observed statistics we need to know the initial conditions for clustering as well as a physical theory for its subsequent evolution. Initial conditions may indicate properties of the early universe before galaxies formed and perhaps even close to the Big Bang. Some possibilities are that galaxy clustering started from a random Poisson distribution, or from a state with local clustering or from large-scale coherent structures. No clear observational evidence for any particular initial state has been found. On small scales the clustering processes themselves tend to destroy this evidence, while on large scales it is difficult to detect. Different types and distributions of dark matter may also be important for forming galaxies and clusters. For example, massive neutrinos, other weakly interacting massive particles, cosmic strings, quark nuggets, or other currently speculative objects of various high-energy theories may influence galaxy clustering if they exist in sufficient quantity. All known forms of matter gravitate and gravitation promotes clustering. Therefore, astronomers have developed analytical theories and examined many computer simulations to describe the gravitational clustering of galaxies. Results for different models are then compared with *(*) and *(*). The models generally differ in their initial conditions, the role of dark matter, and the time available for clustering. Computer simulations calculate the gravitational orbits of many thousands of particles, each one representing a galaxy, in the background of the expanding universe and any dark matter present. The orbits are found either by integrating the thousands of equations of motion-each with thousands of terms-directly, or by averaging the gravitational forces in different ways to simplify the problem. Averaging sacrifices detailed information in order to include a larger number of galaxies. Computer models which start with strong structure on scales of tens of megaparsecs frequently do not agree with the observed correlation and distribution functions. Those that do, often agree only for a short span of their evolution. On the other hand, models with fairly homogeneous initial distributions, such as an uncorrelated Poisson state, evolve gravitationally to agree reasonably well with the observations and remain in agreement as they continue to evolve. In other words, they relax into the observed state and remain there rather than just pass through it. This may make them more aesthetically pleasing, although it does not guarantee they are correct. For example, some models in which galaxies have formed and clustered very recently may conflict with the uniformity of the cosmic microwave background. Gravitational theory predicted the observed form of *(*) given earlier for relaxed statistically homogeneous clustering of point masses in a slowly expanding universe. The value of b is essentially the ratio of gravitational correlation energy (representing departures from a uniform Poisson distribution) to the kinetic energy of galaxies' peculiar velocities (representing departures from the Hubble flow). Computer experiments such as the example in Fig.1 show that this relaxed state is a very good description for universes which start with no or little large-scale correlation, have ***=1, and have expanded by more than several times their initial radius. As differences with these conditions become greater, agreement with the observed *(*) decreases. These conditions also lead to two-Point correlation functions in reasonable agreement with the observations provided, for example, that clustering started at redshift *** in ** models and ***** in ***, =0.1 models. Therefore, gravitational clustering starting from fairly simple initial conditions seems likely to account for the objective statistical evidence now available. These include large underdense regions and filamentary structures, some of which could have formed just by chance concentration of independently clustering regions or their boundaries. When more subtle statistics are developed further and related to dynamical evolution, perhaps they will reveal clear evidence for other processes such as primordial explosions, or large-scale initial structures. Additional Reading Hut, P. and MacMillan, S., eds.(1986). The Use of Supercomputers in Stellar Dynamics. Springer-Verlag, Berlin. Itoh, M., Inagaki, S., and Saslaw, W.C.(1988). Gravitational clustering of galaxies: Comparison between thermodynamic theory and N-body simulations. Ap.J.331 45. Oort, J.H.(1983). Superclusters. Ann. Rev. Astron. Ap.21 373. Peebles, P.J.E.(1980). The Large-Scale Structure of the Universe. Princeton University Press, Princeton, NJ. Rubin, V.C. and Coyne, G.V., eds.(1988). Large-Scale Motions in the Universe. Princeton University Press, Princeton, NJ. Saslaw, W.C.(1987). Gravitational Physics of Stellar and Galactic Systems. Cambridge University Press, Cambridge. See also Clusters of Galaxies; Galaxies, Formation; Superclusters, Dynamics and Models.