Light propagation effects

The last-scattering surface is the most remote region which is observable using electromagnetic radiation. Since photons on their way pass through large-scale inhomogeneities such as voids, clusters and superclusters, it is important to know how the light propagation phenomena affect the CMB radiation. In the standard approach the CMB temperature fluctuations are analysed by solving the Boltzmann equation within linear perturbations around the homogeneous and isotropic FLRW model (Seljak and Zaldarriaga, 1996; Seljak et al., 2003). The use of the FLRW metric for the background model results in a remarkably good fit to the CMB data (Hinshaw et al., 2009). However, the assumption of homogeneity, which is also consistent with other types of cosmological observations, is not a direct consequence of them (Ellis, 2008). It is often said that such theorems as the Ehlers–Geren–Sachs (1968) theorem and the ‘almost EGS theorem’ (Stoeger et al., 1995), justify the application of the FLRW models. These theorems state that if anisotropies in the cosmic microwave background radiation are small for all fundamental observers, then locally the Universe is almost spatially homogeneous and isotropic. The founding assumption of these theorems, namely the local Copernican principle applied to the ‘U region’, i.e. ‘the region within and near our past light cone from decoupling to the present day’, is not mandatory and we have already stressed it needs still to be tested. Moreover, as shown by Nilsson et al. (1999), it is possible that the CMB temperature fluctuations are small but the Weyl curvature is large.

A central aim of this book is to show that there is much still to be learned about the evolution and effects of exact nonlinear inhomogeneities in Einstein's theory, and that this is highly relevant to the real Universe.

The Universe as we observe it is very inhomogeneous. Among its structures there are groups and clusters of galaxies, large cosmic voids and very large elongated structures such as filaments and walls. In cosmology, however, the homogeneous and isotropic models of the Robertson–Walker class have been used almost exclusively, and in these, structure formation is described by an approximate perturbation theory. This works well as long as the perturbations remain small, but cannot be applied once perturbations become large and evolution becomes nonlinear. This is where the methods of inhomogeneous cosmology must come into play. These methods can be employed not only to study the evolution of cosmic structures, but also to investigate the formation and evolution of black holes, as well as studying the geometry and dynamics of the Universe.

Whatever the successes of the Concordance model, based on an FLRW metric plus perturbation theory, structure evolution does sooner or later become nonlinear and non-Newtonian, and our understanding of present-day observations will be incomplete without the methods of inhomogeneous cosmology. The phenomena of fully relativistic inhomogeneous evolution must occur and cannot be ignored.

This book presents the application of inhomogeneous exact solutions of the Einstein equations in such areas as the evolution of galactic black holes and cosmic structures, and the impact of inhomogeneities on light propagation which allows us to solve the horizon and dark energy problems.

Horizon problem and inflation

In hot Big Bang models, the comoving region over which the CMB is observed to be homogeneous to better than one part in 105 at the last-scattering surface is much larger than the intersection of this surface with the future light cone from any point pB of the Big Bang. Since this light cone provides the maximal distance over which causal processes could have propagated from pB, the observed quasiisotropy of the CMB remains unexplained. As shown by Célérier (2000b), this ‘horizon problem’ develops sooner or later in any cosmological model exhibiting a spacelike singularity such as that occurring in the standard FLRW universes.

Even inflation, which was put forward in order to remove this drawback in the framework of standard homogeneous cosmology, only postpones the occurrence of the horizon problem, since it does not change the spacelike character of the singularity and is insufficient to solve it permanently. This is shown in Fig. 6.1, where thin lines represent light cones and where the CMB as seen by an observer O corresponds to the intersection of the observer's backward light cone with the last-scattering line. For a complete causal connection to occur between every pair of points in this intersection segment, all backward light signals issuing from points therein must reach the vertical axis before they reach the space like Big Bang curve. The event L is thus a limit beyond which any observer experiences the horizon problem. Adding an inflationary phase in the primordial history of the universe amounts to adding a slice of de Sitter spacetime, indicated here by the region between the dashed line and the Big Bang.

Cosmology, from its very birth, has been a science beset by insufficient data and ill-founded tenets. Faced with extreme difficulties in collecting observational data, it used to rely on simplifying working assumptions. Those assumptions had the tendency to evolve into dogmas or into elaborate theoretical constructs, immersed in which their practitioners all too easily forgot about these shaky foundations. It is enough to recall the original cosmological paper of Einstein from 1917, who, swayed by prevailing beliefs, was absolutely sure that the Universe must be spatially uniform and unchanging in time – so much so that he preferred to modify his freshly created theory rather than say that it contradicts the astronomical dogma. Another instructive example is the steady state theory. It had been very much in vogue for about 20 years, before it was proved wrong by a single discovery, that of the cosmic microwave background (CMB) radiation. It was glamorous and successful, in the eyes of its proponents, even though it relied on an assumption that is drastically at variance with laboratory physics (continuous creation of matter particles out of nothing).

Solutions are said to belong to Kundt's class if they admit a null geodesic congruence, generated by a vector field k, which is shear-free, twist-free and expansion-free. They include the pp-waves and plane wave space-times that have been described in the previous chapter. The whole class, initially investigated by Kundt (1961, 1962) in the case of vacuum or with an aligned pure radiation field, is however much wider. Because the null vector field k is not in general covariantly constant, the rays of the corresponding non-expanding waves are not necessarily parallel, as in the case of pp-waves, and the wave surfaces need not be planar. This greater freedom permits, for example, the presence of a cosmological constant Λ or aligned electromagnetic fields, both null and non-null.

For vacuum or some specific matter content, generalised Goldberg—Sachs theorems imply that the Kundt space-times must be algebraically special, that is of Petrov type II, D, III or N (or conformally flat), with k being a repeated principal null direction of the Weyl tensor. Any Einstein—Maxwell and pure radiation fields must also be aligned: that is, k is the common eigendirection of the Weyl and Ricci tensor.

A number of physically interesting space-times of this class are explicitly known. These will be briefly described in this chapter, with emphasis given to those that describe exact non-expanding gravitational waves of various kinds.

After Einstein first presented his theory of general relativity in 1915, a few exact solutions of his field equations were found very quickly. All of these assumed a high degree of symmetry. Some could be interpreted as representing physically significant situations such as the exterior field of a spherical star, or a homogeneous and isotropic universe, or plane or cylindrical gravitational waves. Yet it took a long time before some of the more subtle properties of these solutions were widely understood.

In their seminal review of “exact solutions of the gravitational field equations”, Ehlers and Kundt (1962) included the following statement. “At present the main problem concerning solutions, in our opinion, is not to construct more but rather to understand more completely the known solutions with respect to their local geometry, symmetries, singularities, sources, extensions, completeness, topology, and stability.” Since this was written, considerable progress has been made in the understanding of many exact solutions. However, this development has been very restricted compared to the enormous effort that has been put into the derivation of further “new” solutions. Although significant advance has been achieved in the interpretation of many solutions, it is a fact that some aspects of even the most frequently quoted exact solutions still remain poorly understood. The opinion of Ehlers and Kundt thus still indicates an even more urgent task.

In this work, the very traditional approach will be adopted that an exact solution of Einstein's equations is expressed in terms of a metric in particular coordinates. Specifically, it will be represented in the form of a 3+1-dimensional line element in which the coordinates have certain ranges.