The detection of fluctuations in the sky temperature of the cosmic microwave background (CMB) by the COBE team (Smoot et al. 1992) was an important milestone in the development of cosmology. Aside from the discovery of the CMB itself, it was probably the most important event in this field since Hubble's discovery of the expansion of the universe in the 1920s (Hubble 1929). The importance of the COBE detection lies in the way these fluctuations are supposed to have been generated, and their relation to the present matter distribution. As we shall explain shortly, the variations in temperature are thought to be associated with density perturbations existing at the epoch trec, when matter and radiation decoupled. If this is the correct interpretation, then we can actually look back directly at the power spectrum of density fluctuations at early times, before it was modified by non-linear evolution and without having to worry about the possible bias of galaxy power spectra.

The search for anisotropies in the CMB has been going on for around 25 years. As the experiments got better and better, and the upper limits placed on the possible anisotropy got lower and lower, theorists concentrated upon constructing models which predicted the smallest possible temperature fluctuations. The baryononly models of the 1970s were discarded primarily because they could not be modified to produce low enough CMB fluctuations. The introduction of dark matter allowed such a reduction and the culmination of this process was the introduction of bias, which reduces the expected temperature fluctuation still further.

One of the great successes of the hot-big-bang model is the agreement between the observed abundances of light elements and the predictions of nucleosynthesis calculations in the primordial fireball. However, this agreement can only be made quantitative for certain values of physical parameters, particularly the number of light neutrino types, neutron half-life and cosmological entropyper-baryon. Since the temperature of the cosmic microwave background radiation is now so strongly constrained, the latter dependence translates fairly directly into a dependence of the relative abundances of light nuclei upon the contribution of baryonic material to Ω0. It is this constraint, the way it arises and its implications that we shall discuss in this chapter. For more extensive discussions of both theoretical and observational aspects of this subject, see the technical review articles of Boesgaard & Steigman (1985), Bernstein et al. (1988), Walker et al. (1991) and Smith et al. (1993).

Theory of nucleosynthesis

Prelude

We begin a brief description of the standard theory of cosmological nucleosynthesis in the framework of the big-bang model with some definitions and orders of magnitude. The abundance by mass of a certain type of nucleus is the ratio of the mass contained in such nuclei to the total mass of baryonic matter contained in a suitably large volume. As we shall explain, the abundance of 4He, usually indicated with the symbol Y, has a value Y ≃ 0.25, or about 6% of all nuclei, as determined by various observations (of diverse phenomena such as stellar spectra, cosmic rays, globular clusters and solar prominences).

This book had its origins in a coffee-time discussion in the QMW common room in 1993, in the course of which we discussed many issues pertaining to the material contained here. At the time we had this discussion, there was a prevailing view, particularly among cosmologists working on inflationary models, that the issue of the density of matter in the Universe was more-or-less settled in favour of a result very near the critical density required for closure. Neither of us found the arguments made in this direction to be especially convincing, so we determined at that time to compile a dossier of the arguments – theoretical and observational, for and against – in order to come to a more balanced view. This resulted in a somewhat polemical preprint, ‘The Case for an Open Universe’, which contained many of the arguments we now present at greater length in this book, and which was published in a much abridged form as a review article in Nature (Coles & Ellis 1994).

The format of a review article did not, however, permit us to expand on the technical aspects of some of the arguments, nor did it permit us to foray into the philosophical and methodological issues that inevitably arise when one addresses questions such as the origin and ultimate fate of the Universe, and which form an important foundation for the conclusions attained. The need for such a treatment of this question was our primary motivation for writing this book.

In this chapter we shall discuss the dark matter inferred from astrophysical measurements. We divide these astrophysical arguments into three broad categories: galaxies, rich clusters of galaxies and the intergalactic medium. Because astrophysical processes (with the exception of gravitational effects) generally involve baryonic material only, the constraints we discuss frequently, though not exclusively, relate only to the baryonic contribution to the total density. We shall discuss constraints from large-scale structure in the matter distribution (i.e. clustering on scales greater than the scale of individual rich clusters) in the next chapter. For an extensive and influential survey of much of the astrophysical evidence see Peebles (1971), which serves as a standard reference for much that we discuss in this chapter; see also Faber & Gallagher (1979).

Galaxies

It was suggested as early as the 1930s that the total amount of matter in our own galaxy, the Milky Way, is greater than can be accounted for by the visible matter within it (e.g. Oort 1932). We shall not, however, go into any detail here concerning the evidence from stellar dynamics that there is dark matter in the disk of the Milky Way; see, for example, Bahcall (1984). This is still an open question. What we are interested in is the evidence for massive haloes of dark matter surrounding our own and other galaxies.

The mass-to-light ratio

Before discussing the evidence for dark matter in galaxies and clusters, we need to introduce some notation.

The issue

The issue we plan to address in this book, that of the average density of matter in the universe, has been a central question in cosmology since the development of the first mathematical cosmological models. As cosmology has developed into a quantitative science, the importance of this issue has not dimininished and it is still one of the central questions in modern cosmology.

Why is this so? As our discussion unfolds, the reason for this importance should become clear, but we can outline three essential reasons right at the beginning. First, the density of matter in the universe determines the geometry of space, through Einstein's equations of general relativity. More specifically, it determines the curvature of the spatial sections: flat, elliptic or hyperbolic. The geometrical properties of space sections are a fundamental aspect of the structure of the universe, but also have profound implications for the space-time curvature and hence for the interpretation of observations of distant astronomical objects. Second, the amount of matter in the universe determines the rate at which the expansion of the universe is decelerated by the gravitational attraction of its contents, and thus its future state: whether it will expand forever or collapse into a future hot big crunch. Both the present rate of expansion and the effect of deceleration also need to be taken into account when estimating the age of the universe.

As we mentioned in Chapter 1, the main reasons for a predisposition towards a critical density universe are theoretical. We will address these issues carefully, but please be aware at the outset of our view that, ultimately, the question of Ω0 is an observational question and our theoretical prejudices must bow to empirical evidence.

Simplicity

In the period from the 1930s to the 1970s, there was a tendency to prefer the Einstein–de Sitter (critical density) model simply because – consequent on its vanishing spatial curvature – it is the simplest expanding universe model, with the simplest theoretical relationships applying in it. It is thus the easiest to use in studying the nature of cosmological evolution. It is known that, on the cosmological scale, spatial curvature is hard to detect (indeed we do not even know its sign), so the real value must be relatively close to zero. Moreover, many important properties of the universe are, to a good approximation, independent of the value of Ω. The pragmatic astrophysicist thus uses the simplest (critical density) model as the basis of his or her calculations – the results are good enough for many purposes (e.g. Rees 1995).

There are, in addition to this argument from simplicity, a number of deeper theoretical issues concerning the Friedman models which have led many cosmologists to adopt a stronger theoretical prejudice towards the Einstein–de Sitter cosmology than is motivated by pragmatism alone.

We now turn our attention to the evidence from observations of galaxy clustering and peculiar motions on very large scales. In recent years this field has generated a large number of estimates of Ω0 many of which are consistent with unity. Since these studies probe larger scales than the dynamical measurements discussed in Chapter 5, one might be tempted to take the large-scale structure as providing truer indications of the cosmological density of matter. On the other hand, it is at large scales that accurate data are hardest to obtain. Moreover, very large scale structures are not fully evolved dynamically, so one cannot safely employ equilibrium arguments in this case. The result is that one is generally forced to employ simplified dynamical arguments (based on perturbation theory), introduce various modelling assumptions into the analysis, and in many cases adopt a statistical approach. The global value of Ω0 is just one of several parameters upon which the development of galaxy clustering depends, so results are likely to be less direct than obtained by other approaches. Moreover, it may turn out that the gravitational instability paradigm, which forms the basis of the discussion in this chapter, is not the right way to talk about structure formation. Perhaps some additional factor, such as a primordial magnetic field (Coles 1992) plays the dominant role. Nevertheless, there is a persuasive simplicity about the standard picture and it seems to accommodate many diverse aspects of clustering evolution, so we shall accept it for the sake of this argument.

This book had its origins in a workshop held in Cape Town from June 27 to 2 July 1994, with participants from South Africa, USA, Canada, UK, Sweden, Germany, and India. The meeting considered in depth recent progress in analyzing the evolution and structure of cosmological models from a dynamical systems viewpoint, and the relation of this work to various other approaches (particularly Hamiltonian methods). This book is however not a conference report. It was written by some of the conference participants, based on what they presented at the workshop but altered and extended after reflection on what was learned there, and then extensively edited so as to form a coherent whole. This process has been very useful: a considerable increase in understanding has resulted, particularly through the emphasis on relating the results of the qualitative analysis to possible observational tests. Apart from describing the development of the subject and what is presently known, the book serves to delineate many areas where the answers are still unknown. The intended readers are graduate students or research workers from either discipline (cosmological modeling or dynamical systems theory) who wish to engage in research in the area, tackling some of these unsolved problems.

The role of the two editors has been somewhat different.

In Section 5.1 we give an overview of the use of qualitative methods in analyzing Bianchi cosmologies, expanding on the brief remarks in the Introduction to the book. Section 5.2 provides an introduction to the use of expansion–normalized variables in conjunction with the orthonormal frame formalism, thereby laying the foundation for the detailed analysis of the Bianchi models with non–tilted perfect fluid source in Chapters 6 and 7. In Section 5.3 we discuss, from a general perspective, the use of dynamical systems methods in analyzing the evolution of Bianchi cosmologies, referring to the background material in Chapter 4.

Overview

As explained in Section 1.4.2 there are two main approaches to formulating the field equations for Bianchi cosmologies:

1. the metric approach,

2. the orthonormal frame approach.

In the metric approach the basic variables are the metric components gαβ(t) relative to a group–invariant, time–independent frame (see (1.89)). This approach was initiated by Taub (1951) in a major paper. After a number of years researchers became aware that the Bianchi models admitted additional structure, namely the automorphism group, which plays an important role in identifying the physically significant variables (also referred to as gauge–invariant variables, or the true degrees of freedom). This group is defined to be the set of time–dependent linear transformations (1.87) of the spatial frame vectors that preserve the structure equations (1.88).