Globular cluster systems represent only a small fraction of the total stellar mass of galaxy halos, but provide unique tracers which can be used to address models of galaxy formation. Several “case studies” of individually important galaxies are presented, in which we look at the characteristics of their globular clusters including the metallicity distributions, specific frequencies, luminosity (mass) distributions, and kinematics. Among these galaxies are the Milky Way, the nearby giant elliptical NGC 5128, the Virgo ellipticals NGC 4472 and M87, and the supergiant cD galaxies at the centers of rich clusters. In each case the possible roles of mergers, small-satellite accretions, and in situ formation in the growth of the galaxy are discussed. We also briefly touch on the connection between the globular clusters and the much more numerous field-halo stars. We conclude that in all formation scenarios, the presence or absence of gas at any stage of the galaxy's evolution plays a crucial role in determining the total cluster population, the number of distinguishable subpopulations, and the metallicity distribution of the clusters.

The analysis of globular cluster systems (GCSs) in other galaxies is starting to fulfil its long-held promise of informing us about galaxy formation in ways that are unique. The more we learn about GCSs, the more we realize that their role in galaxy formation is an intricate and varied process – yet with common themes that apply particularly to the old-halo population that is found in every type of galaxy.

In the past, there has been a widely held mythology that thermodynamics and gravity are incompatible. The main arguments for that view were threefold. First, thermodynamics applies to equilibrium systems. But self-gravitating systems continually evolve toward more singular states, so they are never in equilibrium. Second, to obtain a thermodynamic or statistical mechanical description it must be possible to calculate the partition function for the relevant ensemble, as in (23.60). But self-gravitating systems contain states where two objects can move arbitrarily close and contribute infinite negative gravitational energy, making the partition function diverge. Third, the fundamental parameters of thermodynamics must be extensive quantities. But self-gravitating systems cannot localize their potential energy in an isolated cell; it belongs to the whole system.

All three of these arguments have a common basis in the long-range nature of the gravitational force and the fact that it does not saturate. By contrast, in the electrostatic case of a plasma, although the Coulomb forces are also long range, the positive and negative charges effectively cancel on scales larger than a Debye sphere where the plasma is essentially neutral, and its net interaction energy is zero. So one can describe plasmas thermodynamically (e.g., Landau & Lifshitz, 1969).

Visual impressions of filamentary structure in the distribution of galaxies are easy to find, as Figure 8.1 and Chapter 8 with its caveats showed. Percolation statistics provide an objective basis for their existence. Minimal spanning trees characterize filaments in an even more refined but less physical way.

A set of points, each of which may represent a galaxy, can be connected by line segments in various ways. Each figure of this sort is called a graph, and the points are its vertices. The connecting segments, which may be straight or curved, are called edges. If every vertex is connected to every other vertex by some sequence of edges (i.e., a route) the graph is said to be “connected.” It may contain circuits, which are closed routes that return to their initial vertex. But if it does not have any circuits, it is a tree. A collection of trees, which need not be connected, forms a forest.

A spanning tree connects all the vertices in the set being considered. Each edge of the spanning tree can be characterized by a property such as its length, or its relative or absolute angle, or the ratio of its length to the length of its neighboring edges, or some weighted combination of properties.

Spatial distribution functions, for all their usefulness, describe only half the phase space. Velocities are the other half. In the cosmological many-body problem, with galaxies interacting just through their mutual gravity, these velocities must be consistent with the galaxies' positions. On nonlinear scales where dissipation prevails, only weak memories of initial conditions or spatial coherence remain.

One measure of galaxy velocities, much studied in the 1980s (see Peebles, 1993 for a detailed summary), is the relative velocity dispersion of galaxy pairs as a function of their projected separation on the sky. For small separations this has an approximately exponential distribution. Simple dynamical models, including projection effects, can relate this pairwise distribution to the galaxy two-point correlation function in redshift space. Today, with increasing ability to measure redshifts and better secondary distance indicators, more accurate radial peculiar velocities are becoming available. So, with an eye to the future, this chapter considers the three-dimensional and radial peculiar velocity distributions predicted for the cosmological many-body problem. In Chapters 32 and 34, we will see that these predictions agree well with relevant N-body experiments and with the first observational determinations.

The peculiar velocity distribution function f(v) dv is just the probability for finding a galaxy in the system with a peculiar velocity between v and v + dv (relative to the Hubble expansion).

Of all the processes that might produce correlations among galaxies in our Universe, we know only one that definitely exists. It is gravity. Inevitably and inexorably, as Newton told Bentley, gravitational instability causes the galaxies to cluster.

Of all the descriptions of galaxy clustering, the correlation functions are connected most closely to the underlying gravitational dynamics. The next several chapters develop this connection. It resembles a great fugue, starting with simple themes and variations, then combining, developing, and recombining them to obtain a grand theoretical structure whose main insights have formed over the last three decades and which still continues to expand.

How should the irregular distribution of galaxies be described statistically? Are clusters the basic unit of structure among the galaxies, or is this unit an individual galaxy itself? Two new themes and the start of a theory emerged during the 1950s and early 1960s to answer these questions. One theme built rigorous multiparameter statistical models of clusters to compare with the catalogs. The other approach looked at basic measures of clustering, mainly the two-particle correlation function, without presupposing the existence of any specific cluster form. The theory successfully began Lemaître's program to calculate kinetic gravitational clustering in an infinite system of discrete objects – the problem whose root, we have seen, goes back to Bentley and Newton. All these developments were being stimulated by the new Lick Catalog of galaxy counts. More than a million galaxies were having their positions and magnitudes measured. Although this would supercede the catalogs of the Herschels, Dreyer, Hubble, and Shapley, its refined statistics would reveal new problems.

Neyman and Scott (1952, 1959) gambled on the idea that clusters dominate the distribution. Their model generally supposed all galaxies to be in clusters, which could, however, overlap. The centers of these clusters were distributed quasi-uniformly at random throughout space. This means that any two nonoverlapping volume elements of a given size have an equal chance of containing N cluster centers, regardless of where the volumes are.

Despite appearances, it is not the Epilogue, but the Prologue that is often left for last. Only after seeing what is done, can one acknowledge and apologize. My main acknowledgments are to many students and collaborators, for they have taught me much. My apologies are to those colleagues who may not find enough of their own results in the pages still ahead. For them I can only echo Longfellow that “Art is long and Time is fleeting.” The subject of large-scale structure in the universe, of which the distribution of the galaxies represents only a part, has burgeoned beyond all previous bounds as the new millennium approaches. Driven as much by the scope and depth of its questions as by new streams of data from the depths of time, there is an increasing excitement that fundamental answers are almost in reach. And there will be no stopping until they are found.

On the timescales of the physical processes we are about to consider, millennia count for very little. But on the timescale of our own understanding, years, decades, and certainly centuries have changed the whole conceptual structure surrounding our views. This may happen again when the role of dark matter becomes more transparent.

Meanwhile, this monograph is really no more than an extended essay on aspects of galaxy clustering that I've found especially interesting.

Percolation describes the shape and connectivity of clustering in a quantitative way. It is related to topological and fractal patterns of a distribution. Although these descriptions have not yet been derived from fundamental dynamical theories, they are useful for characterizing the evolution of N-body experiments and for discriminating between different distributions. They are also related to basic properties of phase transitions.

Among the many applications of percolation descriptions are the spread of forest fires, the flow of liquids (particularly oil) through cracks and pores, the shapes and linkage of polymer molecules, atomic spin networks and magnetic domains, and galaxy clustering. Most of these applications occur on a lattice where the distance between interacting neighbors is fixed. Lattice models simplify the analysis greatly, while often retaining some of the essential properties of the system. Of course galaxies are not confined to a lattice, and so we will need a more general approach. However, a simple lattice defines the basic ideas and can be linked to a more continuous distribution.

Starting in one dimension, imagine a straight line divided into segments of equal length and suppose that a galaxy can be found in any segment with probability p. In the simplest case p depends neither on position along the line nor on the presence of neighboring galaxies.

Definitions

Although galaxy distribution functions were known to Herschel, measured by Hubble, and analyzed statistically by Neyman and Scott, a new chapter in their understanding has opened in recent years. This relates distribution functions to the gravitational clustering of point masses in an expanding Universe. Calculations of the resulting cosmological many-body problem provide new insights into observed galaxy clustering as well as into the results of computer simulations.

Why gravity? When a reporter asked Willie Sutton, a well-known American bank robber, why he robbed banks, he supposedly replied “Because that's where the money is.” As money is the most obvious motivating force of banks, gravity is the most obvious motivating force of galaxy clustering. Unlike the economic parallel, studies of gravitational clustering have the advantage that the rules do not change as the system evolves and is understood better.

Still, the mutual gravitation of galaxies may not be the only significant influence on their clustering. Initial positions and velocities of galaxies when they first form as dynamical entities will help determine their subsequent distribution. This is particularly true on large scales where the distribution has not had time to relax from its initial state. On smaller relaxed scales, the nonlinear interactions of orbits will have dissipated most of the memory of the initial conditions. The nature of this relaxed state will be one of our main themes in subsequent sections and following chapters.

Why is it so difficult for current observations to determine the initial state of galaxy clustering and even earlier of galaxy formation? The answer, in aword, is dissipation. Much energy changed as its entropy gained, first as galaxies formed and then as they clustered.

To see the magnitude of this transformation, imagine a cube now a hundred megaparsecs across in a universe with Ω0 = 1 as an example. If the matter in these million cubic megaparsecs had not condensed at all as the universe expanded, if its temperature had decayed adiabatically α R–2 since decoupling so that now T ≈ 3 × 10–3 K, then the total random kinetic energy in this volume would be about 3 × 1056 erg. Gravitational condensation produces dissipation. In gaseous condensation, much of the energy exits as radiation, and some leaves as hot particles. If the dissipation is mostly particulate, as in many-body clustering, escaping orbits carry energy away. The remaining part of the system condenses into a deepening gravitational well and acquires the increased random kinetic energy it needs for quasi-stability. The magnitude of this kinetic energy, K ≈ |W|/2 ≈ –Etotal, provides an estimate of dissipation.

Velocities are the current frontier, as spectrographs with fiber optic cameras scramble to produce larger catalogs. Some galaxies will have good secondary distance indicators and be useful for peculiar velocities. Catalogs whose peculiar velocities are homogeneously sampled can provide direct comparisons with theoretical predictions.

So far, most catalogs of peculiar velocities are for restricted samples such as spirals in clusters, or for isolated field galaxies. This again involves finding a suitable definition of a cluster or field galaxy. Relating convenient morphological definitions of clusters to the underlying physics of their formation is often quite difficult. It is simpler, and perhaps less model dependent, to consider the combined peculiar velocity distribution for all clustering scales and degrees from isolated field galaxies and small groups to the richest dense clusters. The corresponding predictions of gravitational quasi-equilibrium clustering are given for f(v) by (29.4) and for the observed radial velocity distribution f(vr) by (29.16). Chapter 32 describes their agreement with cosmological many-body simulations.

The GQED predictions are for local peculiar velocities averaged over a large and varied system. They do not include the motion of the system itself, which corresponds to regional bulk flow produced by distant, rare, and massive attractors. There is general agreement that such bulk flows exist, though their detailed nature is subtle and still controversial.

Despite difficulties in solving the BBGKY kinetic equations directly, judicious approximations give valuable insights into their physical properties. As soon as the system leaves the linear regime, these approximations replace a more rigorous analysis. Computer simulations, although they require further approximations, can tell if these insights are genuine and provide a useful description. In this chapter we discuss several physical approximations; following chapters summarize simulations and observations.

Scaling

Scaling has acquired many meanings in different contexts of large-scale structure and thereby caused some confusion. One form of scaling is the simple geometric relation between angular correlation functions W(θ) in catalogs of various limiting magnitudes, given by (14.38). This is closely related to the manufacture of catalogs rather than to the underlying dynamics of clustering. We mention it further in Chapter 20. A more physical form of scaling results from noticing that the Ω0 = 1 Einstein–de Sitter cosmology contains no scale length. Nor is there any scale length in the Newtonian gravitational force between galaxies. Therefore, the argument goes, correlation functions should not contain any scale lengths either and should be power laws.

Unfortunately this argument neglects the actual dynamical development of correlations. Initially, if the positions of newly formed galaxies are not correlated in a gravitationally self-consistent way on all scales, gravitational interactions of galaxies will tend to produce self-consistent correlations.

Stubbornness, stamina, boldness, and luck enabled William Herschel to connect our Universe with Newton's and Kant's speculations. Leaving Hanover in 1757 after the French occupation, he settled in England as an itinerant teacher, copier, and composer of music, becoming organist of the Octagon Chapel at Bath in 1766. But his real interest from childhood was astronomy. He privately built a succession of larger and larger reflecting telescopes and systematically swept the heavens. His sister, Caroline, emigrating in 1772, helped with these nightly observations, to the eventual destruction of her own singing career. In 1781, Herschel had the great luck to find Uranus, the first planet discovered since the days of the ancients, although he originally thought it was just a comet. Fame followed quickly, and fortune soon after when George III granted him a pension for life. He supplemented this by building small telescopes for sale (until his wealthy marriage in 1788) and became a full-time astronomer. Career paths, like the subject itself, have changed considerably since then.

For twenty years, starting in 1783, Herschel searched for nebulae with his 20-foot telescope and its 18 7/10 inch speculum mirror. Messier's catalog, available in 1781, had inspired him first to try to resolve known nebulae with his superior telescope, and then to discover more.