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.

Introduction

There have been many different attempts to provide a quantum description of gravitational phenomena. Although there is at present no immediate experimental evidence of quantum effects of the gravitational field, it is expected on general grounds that at sufficiently high energies quantum effects may be relevant. The fact that quantum field theories in general involve virtual processes of arbitrarily high energies may suggest that an understanding of quantum gravity may be needed to provide a complete picture of quantum fields. Ultraviolet divergences arise as a consequence of an idealization in which one expects the field theory in question to be applicable up to arbitrarily high energies. It is generally accepted that for high energies gravitational corrections could play a role. On the other hand, classical general relativity predicts in very general settings the appearance of singularities in which energies, fields and densities become intense enough to suggest the need for quantum gravitational corrections.

In spite of the many efforts invested over the years in trying to apply the rules of quantum mechanics to the gravitational field, most attempts have remained largely incomplete due to conceptual and technical difficulties. There are good reasons why the merger of quantum mechanics and gravity as we understand them at present is a difficult enterprise. We now present a brief and incomplete list of the issues involved.

In this chapter we will study the quantization of the free Maxwell theory. Admittedly, this is a simple problem that certainly could be tackled with more economical techniques, and this was historically the case. However, it will prove to be a very convenient testing ground to gain intuitive feelings for results in the language of loops. It will also highlight the fact that the loop techniques actually produce the usual results of more familiar quantization techniques and guide us in the interpretation of the loop results.

We will perform the loop quantization in terms of real and Bargmann [70] coordinates. The reason for considering the complex Bargmann coordinatization is that it shares many features with the Ashtekar one for general relativity. It also provides a concrete realization of the introduction of an inner product purely as a consequence of reality conditions, a feature that is expected to be useful in the gravitational case.

The Maxwell field was first formulated in the language of loops by Gambini and Trias [62]. The vacuum and other properties are discussed in reference [63] and multiphoton states are discussed in referece [64]. The loop representation in terms of Bargmann coordinates was first discussed by Ashtekar and Rovelli [65].

The organization of this chapter is as follows: in section 4.1 we will first detail some convenient results of Abelian loop theory, which will simplify the discussion of Maxwell theory and will highlight the role that Abelian theories play in the language of loops.