Introduction

David Wands and Roy Maartens

Our Universe provides the grandest arena in which to test General Relativity as a theory of space, time and gravity. It becomes essential to consider both the causal propagation of matter and radiation through space-time and the dynamical evolution of space-time if we are to construct a consistent theoretical framework in which to interpret astronomical observations on the largest observable scales. Einstein himself originally tried to construct a static cosmology but it was soon appreciated that the field equations of General Relativity naturally accommodate dynamical and evolving space-times. Einstein's own static model of a 3-sphere, balancing the gravitational pull of matter against a positive spatial curvature and a cosmological constant was shown to be poised between expansion and collapse and hence unstable to infinitesimal disturbance.

Friedmann and Lemaître showed that Einstein's field equations admit expanding- universe solutions, which have become the basis for modern cosmology, despite Einstein's initial dismissal of the solutions. However persuasive the theoretical models, empirical observations are, of course, necessary to determine the actual dynamics of our observable Universe; the work of Slipher and Hubble in the 1920s [1] persuaded scientists that in fact our Universe is expanding. The logical consequence of this expansion is that either our Universe was hotter and denser in the past (coming ultimately from a Hot Big Bang) or, perhaps, that energy had to be continually created as the universe expanded (the Steady State model). The discovery of the Cosmic Microwave Background (CMB) radiation by Penzias and Wilson in 1965 [2] convinced most astronomers that the Universe did in fact begin at a Hot Big Bang, a finite time in the past. This hot dense plasma, in thermal equilibrium at early times, also provides a setting for the freezing out of the light atomic nuclei as the universe cools below several million degrees Kelvin [3], although heavier elements must be formed later in stars.

In approaching the issue of how the universe started, it is common cause that we have to face up to the unsolved problem of quantum gravity: the domain where Einstein's theory of gravity is expected to break down because quantum effects become so dominant that they affect the very nature of space and time. Comparing the gravitational constants of nature with those from quantum theory leads to the Planck length ℓP ≈ 10-33cm, which is taken to be the characteristic scale at which quantum gravity dominates. By contrast, most (but not all) variant classical gravitational theories modify GR at low energies (see Chapter 14).

Quantum gravity processes are presumed to have dominated the very earliest times, preceding inflation: the geometry and quantum state that provide the initial data for any inflationary epoch themselves are usually assumed to come from the as yet unknown quantum gravity theory. There are many theories of the quantum origin of the universe, but none has attained dominance. The problem is that we do not have a good theory of quantum gravity (Rovelli, 2004, Weltmann, Murugan and Ellis, 2010), so all these attempts are essentially different proposals for extrapolating known physics into the unknown. A key issue is whether quantum effects can remove the initial singularity and make possible universes without a beginning.

In addition, the weakness of the gravitational force implies that it will be very difficult, though perhaps not impossible, to observationally test theories of quantum gravity.

In the previous chapter, cosmological models which drop the isotropy assumption of FLRW models were considered; here we drop the homogeneity assumption. Of course, perturbed FLRW models also satisfy neither assumption, but they are treated only in perturbation theory. Here we aim to study models in the fully nonlinear theory. Inhomogeneous models have been applied both globally (as shown by the use of LTB models in Chapter 15) and to model localized inhomogeneities and, e.g. their fully nonlinear effects on observation via lensing (as shown by the use of Swiss cheese models in Chapter 16). In the global context, issues such as whether inflation could remove inhomogeneity, or whether hierarchical models could fit the data, can be examined: these are essential to judging the robustness of the assumptions of the standard model.

For example, the evidence cited as support for the standard model can be well fitted by nonstandard models, as we have seen in Chapter 15. Thus one can legitimately ask, what is the largest family of cosmological models that can fit the observations? One can then try to devise observational tests to eliminate as many of them as one can.

One may also wonder why we look for exact models of structure formation, when the perturbative theory is so successful? The inflationary paradigm coupled with the perturbation theory of FLRW models has offered the first viable explanation of the observed degree of inhomogeneity in the universe (see Chapter 10). However, the galaxies, clusters and voids we observe now have values of (e.g.) δρ/ρ outside the perturbative regime.

This book provides a survey of modern cosmology emphasizing the relativistic approach. It is shaped by a number of guiding principles.

• Adopt a geometric approach Cosmology is crucially based in spacetime geometry, because the dominant force shaping the universe is gravity; and the best classical theory of gravity we have is Einstein's general theory of relativity, which is at heart a geometric theory. One should therefore explore the spacetime geometry of cosmological models as a key feature of cosmology.

• Move from general to special One can best understand the rather special models most used in cosmology by understanding relationships which hold in general, in all spacetimes, rather than by only considering special high symmetry cases. The properties of these solutions are then seen as specialized cases of general relations.

• Explore geometric as well as matter degrees of freedom As well as exploring matter degrees of freedom in cosmology, one should examine the geometric degrees of freedom. This applies in particular in examining the possible explanations of the apparent acceleration of the expansion of the universe in recent times.

• Determine exact properties and solutions where possible Because of the nonlinearity of the Einstein field equations, approximate solutions may omit important aspects of what occurs in the full theory. Realistic solutions will necessarily involve approximation methods, but we aim where possible to develop exact relations that are true generically, on the one hand, and exact solutions of the field equations that are of cosmological interest, on the other.

The main part of scientific cosmology today deals with technical issues to do with modelling the origin and evolution of the universe, as discussed in the rest of this book. However cosmology also has wider connotations, as reflected in the broader use of the term in popular use. The link between these two aspects of cosmology resides in two interrelated issues: on the one hand, the relation between cosmology and local physics; and on the other, the foundational question of why the universe is as it is. Both can only be tackled properly by taking philosophical issues seriously. The underlying issue is fundamental: what is the nature of cosmology as a science? How does it relate to issues of testing and verification?

This book does not deal with these issues in depth, as to do so fully would take us too far from our main theme (and our competence), but it does not ignore them either, for to do so would exclude some of the most interesting issues in cosmology. This chapter considers, relatively briefly, how issues in relativistic cosmology relate to these two fundamental themes. It will emphasize two related key topics where these issues come to a head: namely the possible existence of a multiverse, and the question of whether the universe is probable or improbable.

We present theses that can be regarded as reasonable within the current framework of cosmology and physics.

The basic observational concepts and relations in cosmology were introduced in Chapter 7. Here we consider how this works out when applied to the standard models of cosmology: how do current observations support and refine the use of perturbed FLRW models, discussed in Chapters 9 to 12? This is crucial in determining how acceptable these models are as models of the real universe: observational testing is the core of scientific cosmology.

Extraordinary progress has been made in this regard in recent decades. Multi-wavelength observations (radio to γ-ray) of vast numbers of sources and various backgrounds have been made, gathering massive amounts of data. This has been based on developments in telescopes (ground-based, balloon-borne, space-based), detectors (photomultipliers, CCDs, fibre optics, adaptive optics, interferometric spectrometers, etc.), and high-performance computing power. We cannot go into those developments here, but simply refer to Lena, Lebrun and Mignard (2010) for a survey.

An important feature of the observational constraints is that they have two separate aspects. On the one hand, we can test the background model by observations that probe the expansion history, such as the magnitude–redshift diagram, and, on the other hand, we can test perturbations about the background by observations that probe the CMB anisotropies and the growth of structure.

For the background, we use galaxies, supernovae and the CMB as markers of the geometry of the background model; for this purpose we are only interested in their capacity to provide standard candles and standard rulers.

A central pillar of modern cosmology is the near-isotropy of the CMB, compatible with a perturbed FLRW model of the universe. The small deviations from isotropy in the CMB temperature contain a wealth of information. Temperature anisotropies due to inhomogeneities were predicted by Sachs and Wolfe (1967) soon after the discovery of the CMB in 1965 by Penzias and Wilson. Shortly afterwards, polarization was predicted in models with anisotropy in the expansion rate around the time of recombination (Rees, 1968). The detailed physics of CMB fluctuations in almost-FLRW models was essentially understood for models with only baryonic matter by 1970 (Silk, 1968, Peebles, 1968, Zel'dovich, Kurt and Sunyaev, 1968, Peebles and Yu, 1970, Sunyaev and Zel'dovich, 1970). By the early 1980s, CDM was included (Peebles, 1982, Bond and Efstathiou, 1984). Further milestones included the effect of spatial curvature (Wilson, 1983), polarization (Kaiser, 1983, Bond and Efstathiou, 1984) and gravitational waves (Dautcourt, 1969, Polnarev, 1985). All of this work used the standard metric-based approach to cosmological perturbation theory, but CMB physics has also been studied extensively in the 1+3-covariant approach (Ellis, Matravers and Treciokas, 1983b, Ellis, Treciokas and Matravers, 1983, Stoeger, Maartens and Ellis, 1995, Maartens, Ellis and Stoeger, 1995a,b, Dunsby, 1997, Uzan, 1998, Challinor and Lasenby, 1998, 1999, Maartens, Gebbie and Ellis, 1999, Challinor, 2000a,b, Lewis, Challinor and Lasenby, 2000, Gebbie and Ellis, 2000, Gebbie, Dunsby and Ellis, 2000, Lewis, 2004a,b, Pitrou, 2009).

A coherent view?

We conclude by returning to the question we started with: what is our best current picture of the physical universe, and what are its problems and uncertainties?

There is an agreed basic view of the universe, the standard model of cosmology, in which the universe expands from a hot big bang early phase to a late-time cool, accelerating phase driven by a cosmological constant, with structure formation taking place through gravitational instability around the ‘scaffolding’ provided by dominant CDM, acting on seed perturbations generated by inflation. This seems to provide a statistically good fit to all the data up to now, with the same set of parameters.

Given the major physical uncertainties concerning this model, we need to continue subjecting it to further observational tests with ever-improving data – and we need to query its uniqueness, and probe the alternative possibilities. A key feature is that uncertainty about both geometry and physics increases with time in the past, and with distance from our world line. Our cosmological claims should make this feature clear. Use of the FLRW models as a starting point of analysis tends to hide this fact. The perturbative inhomogeneity imposed in these models changes the physics equations from PDEs to ODEs and so hides the nature of causality and associated causal domains.

The visible universe

The visible universe is the part of spacetime within our past light-cone since decoupling. The CMB emitted from where our past light-cone intersects the last scattering surface marks the boundary of this domain – the matter emitting this light represents our visual horizon, the furthest away matter that we can detect by electromagnetic radiation of whatever wavelength.

As discussed in the previous chapter, the explanation of dark energy is a central preoccupation of present-day cosmology. Its presence is indicated by the apparent recent speeding up of the expansion of the universe indicated by SNIa observations, which is usually taken to be caused by quintessence or a cosmological constant, and is consistent with other observations such as those of anisotropies and large-scale structure studies. Like dark matter, its existence was discovered, not predicted. The astronomical observations are being refined in many sophisticated ways and used to confirm the acceleration data and test the equation of state of the hypothetical dark energy. Whether its density is constant or varying, its existence is a major problem for theoretical physics. It is therefore crucial to pursue the possibility of other theoretical explanations.

The deduction of the existence of dark energy assumes that the universe has a RW geometry on large scales. However, the interpretation of the observations is ambiguous. They can at least in principle be accounted for without the presence of dark energy, if we allow inhomogeneity. This can contribute in one or both of two ways: locally via backreaction and associated observational effects, as discussed in the next chapter, and globally via large-scale inhomogeneity, considered in this chapter.

Here we consider the possibility that what appears to be the acceleration of an FLRW universe due to dark energy, is in fact rather a manifestation of Hubble-scale inhomogeneity in a universe such as that described by the Lemaître–Tolman–Bondi (LTB) models discussed in the next section, where we are near the centre of a void.

Physics usually begins with some concept of a space of events, or of positions at which objects or fields can be present. While this could be a discrete space (e.g. in lattice models) or a topological space without extra structure, it is usually assumed to be a continuum on which one can carry out the operations of calculus, i.e. a differential manifold.

Most of modern theoretical physics can be written in the language of differential geometry and topology, though it has only become common to do so since gauge theories assumed their present prominent role. Many advanced notions in these areas find a place in physics (see e.g. Nakahara (1990)). We shall be careful below to distinguish between concepts dependent on and independent of the presence of a metric, since gauge theories usually do not assume a metric.

While it is not true that every geometric object is of physical significance, it will be true of the geometric quantities we discuss in subsequent chapters. So when we consider geometric questions, it is important to recognize that these can also be understood as physical. Indeed, one of our aims is to show how powerful geometric methods can be in discussing cosmological questions, once one has mastered the necessary tools.

Dirac in his classic book (Dirac, 1981) emphasizes that quantum mechanics rests on a number of principles. The first two are that observables are operators, which can be expressed in different bases, and that the core of the dynamics lies in non-zero commutators for these operators.

The fundamental problem considered in this chapter is, how do we relate the FLRW model to the non-uniform real world? The point we emphasize here is that there is a hidden averaging scale in all our descriptions of the universe. There is a hierarchy of different scales of description we can use, with effective equations occurring at each scale. The relation between the descriptions, dynamics, and observations at each of the scales is a key issue. The FLRW model only applies at the largest scales; how does it relate to the inhomogeneities at smaller scales?

A fundamental feature is the non-commutativity of averaging in relation to both dynamics and observations. Through this effect, inhomogeneities in the universe, such as vast walls, filaments, clusters and voids in the distribution of galaxies, can affect both the dynamics and observational properties of the universe. Hence they have the potential to explain at least part, if not all, of the apparent acceleration of the universe indicated by the SNIa data. Note that these are smaller-scale inhomogeneities compared with those considered in the previous chapter, which are of the order of the Hubble scale.

Overall, the question is how to describe the real universe by an (almost) FLRW model, when it is nothing like that on small cosmological scales. What is the meaning of the FLRW metric in relation to the real lumpy universe?

Different scale descriptions

Any mathematical description of a physical system depends on an implicit averaging scale characterizing the nature of the envisaged model (Section 1.4.1), and its tests also have specific scales.