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.

The first goal of theoretical cosmology is to find a model of the universe, the simplest model, that is in agreement with observational data. The second goal is to explore the range of models that are compatible with observational data, in order to understand whether the simplest model is highly probable, and to understand the full range of cosmological possibilities in epochs that are not constrained by observations. This book describes results and techniques of analysis that pertain to the second goal.

The FL models are widely accepted as meeting the first goal (e.g. Peebles et al. 1991), although some uncertainties remain. First, insufficient evidence is available from redshift and peculiar velocity surveys to convincingly establish the averaging scale over which the universe can be regarded as isotropic and homogeneous. Second, a fully satisfactory theory of the formation of structure (i.e. of galaxies and their distribution in space) in a FL model has not yet been found. Third, the fact that the FL models (with Λ = 0), in particular the flat model, are unstable makes it implausible that the real universe can be approximated by a FL model over its entire evolution up to the present and into the future. Fourth, inflation is motivated by the desire to make a flat FL universe in the present epoch inevitable, or at least highly probable. In attempting to reach this goal one has to work with models more general than FL in the pre–inflation epoch.

It is well known that solutions of non–linear differential equations in three and higher dimension can display apparently random behaviour referred to as deterministic chaos, or simply, chaos. The associated dynamical system is then referred to as being chaotic. It was recognized some years ago that the oscillatory approach to the past or future singularity of Bianchi IX vacuum models displays random features (e.g. Belinskii et al. 1970, which we shall refer to as BKL, and Barrow 1982b), and hence is a potential source of chaos. This oscillatory behaviour is also believed to occur in other classes of models, provided that the Bianchi type and/or source terms are sufficiently general (see Sections 8.1 and 8.4). The goal in this chapter is to address the question of whether the dynamical systems which describe the evolution of Bianchi models are chaotic.

Historically, both Poincaré and Birkhoff in the late nineteenth and early twentieth centuries were aware that non–linear DEs could admit complicated aperiodic or quasi–periodic solutions. The modern development of a theory of chaotic dynamical systems was stimulated in a large part by two papers, namely Lorenz (1963), a numerical simulation of a three–dimensional DE, and Smale (1967), a theoretical analysis of discrete dynamical systems. The field developed rapidly once computer simulations of dynamical systems became widely available. Despite a lengthy history, complete agreement on a definition of chaotic dynamical system has not been reached.

The FL universes, based on the RW metric, are the standard models of current cosmology. In this chapter we discuss cosmological observations with a view to assessing the evidence for these models.

There are two stages in this process of assessment:

1. to discuss to what extent observations require the universe to be close to FL during the different epochs in its evolution,

2. assuming the universe is close to FL, to discuss the observational constraints on the parameters that characterize an FL universe.

We group the observations that pertain to the first stage under three headings, namely, discrete sources (Section 3.1), the cosmic microwave background radiation (Section 3.2) and the light–element abundances arising from nucleosynthesis in the early universe (Section 3.3). In Section 3.4 we assess the extent to which these observations require the universe to be close to FL in different epochs. We do not discuss events at earlier epochs (e.g. baryogenesis; see Kolb & Turner 1990, Chapter 6) since we regard our current knowledge of the physics concerned as too tentative to lead to reliable constraints. Finally, in Section 3.5 we discuss the ‘best–fit’ FL parameters and the ‘age problem’.

Observations of discrete sources

Observations of discrete sources (primarily galaxies, radio sources, infrared sources and quasars) provide information about the structure of the universe in the galactic epoch (say z ≲ 5).

Over the past four decades cosmological perturbation theory has played an important role in our attempts to understand the formation of large–scale structures in the universe. So far, most of the work done in this field has been concerned with linear perturbations of the FL cosmologies, the underlying assumption being that on a sufficiently large scale the universe can be described by a homogeneous and isotropic model. A number of approaches to this problem have been presented in the literature since the pioneering work of Lifshitz, notably the gauge–invariant formulation of Bardeen (1980). Although this approach has been widely used to describe both the origin and evolution of small perturbations from the quantum era through to the time when the linear approximation breaks down, it has three shortcomings. First, the variables are non–local, depending on unobservable boundary conditions at infinity. Second, many of the key variables have a clear physical meaning only in a particular gauge. Finally, the approach is inherently limited to linear perturbations of FL models.

Recently, Ellis & Bruni (1989), building on Hawking (1966), developed a geometrical method for studying cosmological density perturbations. This approach, which is based on the spatial gradients of the energy density μ and Hubble scalar H, is both coordinate–independent and gauge–invariant, and the variables have an unambiguous physical interpretation. In addition their approach is of a general nature, because it starts from exact non–linear equations that can in principle be linearized about any FL or non–tilted Bianchi model.