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).

In this chapter we give a brief overview of some aspects of the theory of dynamical systems. We assume that the reader is familiar with the theory of systems of linear differential equations, and with the elementary stability analysis of equilibrium points of systems of non–linear differential equations (e.g. Perko 1991). We emphasize instead the fundamental concept of the flow and various other geometrical concepts such as α– and ω–limit sets, attractors and stable/unstable manifolds, which have proved useful in applications in cosmology. In the interest of readability we have stated some of the definitions and theorems in a simplified form; full details may be found in the references cited. One important aspect of the theory that we do not discuss due to limitations of space is structural stability and bifurcations. We refer to Perko (1991, chapter 4) for an introduction to these matters. We also note that the discussion of chaotic dynamical systems is deferred until Chapter 11.

To date, applications of the theory of dynamical systems in cosmology have been confined to the finite dimensional case, corresponding to systems of ordinary differential equations, although in Chapter 13 we obtain a glimpse of the potential for using infinite dimensional dynamical systems. We restrict our discussion to the finite dimensional case, referring the interested reader to books such as Hale (1988), Temam (1988a,b) and Vishik (1992) for an introduction to the infinite dimensional case.

The cosmological models proposed by A. Einstein and W. de Sitter in 1917, based on Einstein's theory of general relativity, initiated the modern study of cosmology. The concept of an expanding universe was introduced by A. Friedmann and G. Lemaître in the 1920s, and gained credence in the 1930s because of Hubble's observations of galaxies showing a systematic increase of redshift with distance, together with Eddington's proof of the instability of the Einstein static model. Since the 1940s the implications of following an expanding universe back in time have been systematically investigated, with an emphasis on four distinct epochs in the history of the universe:

(1) The galactic epoch, which is the period of time extending from galaxy formation to the present. This is the epoch that is most accessible to observation. During this period, matter in a cosmological model is usually idealized as a pressure–free perfect fluid, with galaxy clusters or galaxies acting as the particles of the fluid. The cosmic background radiation has negligible dynamic effect in this period.

(2) The pre–galactic epoch, during which matter is idealized as a gas, with the particles being the gas molecules, atoms, nuclei, or elementary particles at different times. The epoch is divided into a post–decoupling period, when matter and radiation evolve essentially independently, and a pre–decoupling period, when matter is ionized and is strongly interacting with radiation through Thomson scattering. The observed cosmic microwave background radiation is interpreted as evidence for the existence of this pre–decoupling period.