The past is our future frontier. Distribution functions at high redshift have not yet been observed. Therefore this chapter will be very short.

Here and there we have glimpses of how galaxy clustering may have developed. These are from observations of two-particle correlations, Lyman alpha clouds, merging protogalaxies, and rich clusters, all at great distances. Eventually, when the halfdozen or so high-redshift catalogs now being started accumulate complete and well-chosen samples, they will yield up the distribution functions of the past. Insofar as these agree with the GQED, their evolution is represented by the changing of b. Thus there is time for genuine predictions, such as those of (30.12) and (30.19)–(30.20) shown in Figure 30.5. These catalogs will also test the importance of merging, which would alter N's conservation (see Chapter 36).

At high redshifts the value of b for gravitational quasi-equilibrium evolution depends quite strongly on Ω0. igures 30.5, 31.12, and 31.13 indicate that as we look further into the past, b will decrease more slowly for lower Ω0. This is essentially because the clustering pattern is “frozen” at higher redshifts for lower Ω0. Equivalently, for higher Ω0 most of the evolution occurs more recently. Zhan (1989) and Saslaw and Edgar (1999) give useful diagrams to show how this can help determine Ω0 and the redshift at which galaxies start clustering.

The total mass of the globular cluster system of our Galaxy makes only ∼10−3 the mass of the Galactic disk. It contains, however, ∼20% of all known low-mass binary X-ray sources, about one half of all binary pulsars, and more than a half of the millisecond pulsars in the Galaxy. Close binary systems containing neutron stars should thus form much more easily in the dense stellar environment of globular clusters than elsewhere in the Galaxy. In these lectures we first review the formation mechanism of neutron stars. Then, we present the evolutionary scenarios leading to the formation of binary X-ray sources and binary and millisecond pulsars in the Galactic disk and the Galactic bulge. We later discuss the specific mechanisms to form neutron star binaries in globular clusters. We end by discussing the open issues concerning the origin and evolution of X–ray sources and millisecond pulsars in globular clusters, and their relationship with the structure, dynamics and evolution of the clusters themselves.

Low–mass binary X–ray sources and millisecond pulsars

An early, unexpected result of X–ray astronomy was the discovery of several bright X–ray sources in globular clusters. Later on, searches for radio pulsars have produced many detections, especially of short period pulsars. We begin these lectures with a very schematic presentation of those two kinds of objects.

X–ray binaries

Luminous Galactic binary X–ray sources provided the first evidence of neutron star binaries, that is binary star systems containing neutron stars (Giacconi et al. 1971; Lewin et al. 1971; Schreier et al. 1972; Tananbaum et al. 1972).

Current theoretical predictions concerning the evolution of old, metal-poor stars in galactic globulars are revisited in the light of recent improvements in the input physics. After a short introduction, the role of fundamental physics in constraining stellar structure all along the various evolutionary phases is shortly recalled, together with the additional role played by the evaluation of some macroscopic mechanisms, like convection and diffusion (Sect. 2). Theoretical predictions concerning the CM-diagram location of the best-known evolutionary phases are discussed in some detail, with particular regard to the existing uncertainties in modeling stellar structure as well as in handling observational data (Sect. 3). This discussion is thus extended to the faint stars recently revealed by HST, either at the faint end of the MS or along the WD cooling sequence (Sect. 4). Additional theoretical constraints given by the pulsational properties of RR Lyrae pulsators are recalled (Sect. 5) and the case of extragalactic globulars in the Local Group are briefly discussed (Sect. 6). Some general and methodological considerations close the paper.

The CM diagram: an introduction

The birth of modern physics dates back to the time when Galileo Galilei stated that any attempt to understand the world around us must—first of all—save the phenomena (“salvare i fenomeni”). In modern words, we say that physics is studying relations between observable quantities, so that the identification of suitable “observables” is a first-priority step in any physical investigation.

Hundreds of examples of two-point correlations have been published for simulated and observed systems. Here we will stick to the basics. First we see how point masses, each representing a galaxy, can start with a Poisson distribution and then correlate gravitationally for different initial velocity dispersions in universes with different critical densities Ω0. Then I summarize some effects of incorporating a range of galaxy masses, and of other initial distributions, and of intergalactic dark matter.

Originally, astronomers hoped that these sorts of results could be compared directly with observations to read the value of Ω0 off the sky. But it was not to be. Too many ambiguities and combinations of conditions gave similar results for ξ2(r). Current observations are not yet sensitive enough to distinguish among all views of the past. Here I describe just some modern examples of the observed form of ξ2 and consider how they may differ for different types of galaxies.

Simulations

Starting a simulation from a Poisson distribution has the attractive feature of starting with minimum structural information. The initial power spectrum (14.35) has n = 0 and thus there is equal power on all scales: a democratic beginning. Then we can watch how pure gravitational interaction builds up different structures. This approach also helps isolate the effects of more complex initial conditions and processes.

First, we consider a related set of simulations with N = 4,000, values of Ω0 = 1, 0.1, and 0.01, and a Poisson initial spatial distribution.

To the denizens of Iolanthe, we can only say with wonder that their motions gravitational are very much more rational, and easier to understand. Not individually, but statistically. To test the special function (29.4) for many-body motions in the context of cosmology we return to simulations in Sections 31.1–31.4.

Figures 15.5 and 15.6 in Section 15.3 illustrated some results of these simulations, which we now examine in more systematic detail. As in Section 31, we start with the simplest Ω0 = 1 case and identical masses. Then we explore the effects of smaller Ω0 and of components with different masses. Unlike the spatial distributions, velocity distribution functions are just beginning to be computed for experiments with dark matter and non-Poisson initial conditions. This is partly because the definition of which particles constitute a galaxy is still unsettled for such cases and partly because observations of f(v) for representative samples are just starting to be analyzed. Both these situations should improve. Then velocity distribution functions will become very valuable because they are more sensitive than the spatial distributions to some of the basic cosmological parameters.

Figure 32.1 shows the velocity distribution functions at four different expansion factors for the 4,000-body, Ω0 = 1, initially cold Poisson simulations with particles of identical masses.

Quick and capacious computers, increasing realization of the importance of largescale structure, and the first glimpses into related many-body physics all combined to change our understanding of galaxy clustering in the early 1970s. So with our historical perspective concluded (though never complete) we now change our approach and describe selected ways to characterize the galaxy distribution. With the large, automatically analyzed catalogs now available, there is no lack of positional data. Successful new observational techniques are also providing many galaxy redshifts, which are being refined into peculiar velocities relative to the general expansion. Nor is there any lack of statistical techniques for analyzing the data. Dozens of quantitative descriptions, many based on analogies in subjects ranging from archaeology to zoology, have been proposed. The main problem is to select those which give most insight into the physical causes of the structure we see. In the next chapters, I sketch several examples, their strengths and weaknesses, and some of their accomplishments. It helps provide a perspective for the two descriptions that will dominate subsequent chapters: correlation functions and distribution functions.

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.