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.

Introduction

In this chapter we will introduce holonomies and some associated concepts which will be important in the description of gauge theories to be presented in the following chapters. We will describe the group of loops and its infinitesimal generators, which will turn out to be a fundamental tool in describing gauge theories in the loop language.

Connections and the associated concept of parallel transport play a key role in locally invariant field theories like Yang–Mills and general relativity. All the fundamental forces in nature that we know of may be described in terms of such fields. A connection allows us to compare points in neighboring fibers (vectors or group elements depending on the description of the particular theory) in an invariant form. If we know how to parallel transport an object along a curve, we can define the derivative of this object in the direction of the curve. On the other hand, given a notion of covariant derivative, one can immediately introduce a notion of parallel transport along any curve.

For an arbitrary closed curve, the result of a parallel transport in general depends on the choice of the curve. To each closed curve γ in the base manifold with origin at some point o the parallel transport will associate an element H of the Lie group G associated to the fiber bundle. The parallel transported element of the fiber is obtained from the original one by the action of the group element H.

Introductiong

Since the unification of the electromagnetic and weak interactions through the Glashow–Salam–Weinberg model [75], Yang–Mills theories [76] have been widely accepted as correctly describing elementary particle physics. This belief was reinforced when they proved to be renormalizable [77, 78]. Moreover, the discovery of color symmetry as the underlying gauge invariance associated with strong interactions raised the possibility that all interactions of nature could possibly be cast as Yang–Mills theories. This spawned interest in grand unified models and some partial successes were achieved in this direction.

A crucial ingredient in the description of elementary particle physics through gauge theories is the maintenance of the gauge invariance of physical results and the underlying theory and this is also crucial in order to be able to prove renormalizability.

The success of the electroweak model is yet to be achieved by the quark model of strong interactions. The reason is that perturbative techniques, which were adequate for the electroweak model, are only appropriate in the high energy regime of strong interactions. This motivated the interest in non-perturbative techniques, especially to prove the existence of a confining phase. A great effort took place in the late 1970s and suggestive arguments were put forward but a rigorous proof of quark confinement is still lacking.

In several of these attempts the use of loops played an important role. Loops were used in a variety of contexts and approaches including the one we are focusing on in this book, the loop representation.

Introduction

Continuing with the idea of describing gauge theories in terms of loops, we will now introduce a set of techniques that will aid us in the description of loops themselves. The idea is to represent loops with a set of objects that are more amenable to the development of analytical techniques. The advantages of this are many: whereas there is limited experience in dealing with functions of loops, there is a significant machinery to deal with analytic functions. They even present advantages for treatment with computer algebra.

Surprisingly, we will see that the end result goes quite beyond our expectations. The quantities we originally introduced to describe loops immediately reveal themselves as having great potential to replace loops altogether from the formulation and go beyond, allowing the development of a reformulation of gauge theories that is entirely new. This formulation introduces new perspectives with respect to the loop formulation that have not been fully developed yet, though we will see in later chapters some applications to gauge theories and gravitation.

The plan for the chapter is as follows: in section 2.2 we will start by introducing a set of tensorial objects that embody all the information that is needed from a loop to construct the holonomy and therefore to reconstruct any quantity of physical relevance for a gauge theory. In section 2.3 we will show how the group of loops is a subgroup of a Lie group with an associated Lie algebra, the extended loop group.

Introduction

Having cast general relativity as a Hamiltonian theory of a connection, we are now in a position to apply the same techniques we used to construct a loop representation of Yang–Mills theories to the gravitational case. We should recall that we are dealing with a complex SU(2) connection. However, we can use exactly the same formulae that we developed in chapter 5 since few of them depend on the reality of the connections. Whenever the presence of a complex connection introduces changes, we will discuss this explicitly.

As we have seen, we can introduce a loop representation either through a transform or through the quantization of a non-canonical algebra. The initial steps are exactly the same as those in the SU(2) Yang–Mills case. The differences arise when we want to write the constraint equations. In the Yang–Mills case the only constraint was the Gauss law and one had to represent the Hamiltonian in terms of loops. In the case of gravity one has to impose the diffeomorphism and Hamiltonian constraints in terms of loops. In order to do so one can either use the transform or write them as suitable limits of the operators in the T algebra. We will outline both derivations for the sake of comparison. As we argued in the Yang–Mills case both derivations are formal and in a sense equivalent, although the difficulties are highlighted in slightly different ways in the two derivations.