The formalisms most widely used to describe structure formation and evolution are N-body simulations and the linear perturbation approach. However, in N-body simulations the interactions between particles are described by Newtonian mechanics where, at variance with GR, matter does not affect light propagation, there is instantaneous action at a distance, and there are no curvature effects. As regards the linear approach, we have discussed its drawbacks in Chapter 1.

To avoid such weaknesses, we now apply the methods of Chapter 2, for constructing Lemaître–Tolman, Lemaître and Szekeres models, to several explicit descriptions of structure formation, based on data from actual galaxies, clusters and voids in the observed Universe. Our approach requires certain data to be set at initial and final times, t1 and t2, and then calculates the model that evolves between them. Thus, we next consider the observational constraints at recombination, which we will use as an initial time. For constraints at t2, we will look to present-day observations of relatively nearby structures.

Initial conditions

Transforming scales in the background

It is a common-sense assumption that present-day cosmic structures evolved from small initial fluctuations whose traces can be observed in the CMB temperature fluctuations. We imagine that a condensed structure at t2 has accumulated its present mass by drawing matter in from the surroundings. Thus, a condensation (such as a galaxy cluster) will be enveloped in a region where the density is lower than the cosmic average, up to the distance Rc, called the compensation radius, at which the total mass within Rc is the same as it would be in a Friedmann (dust) model.

Since its discovery during the late 1990s (Riess et al., 1998; Perlmutter et al., 1999), the dimming of distant SN Ia has been mostly ascribed to the influence of a mysterious dark energy component. Formulated in a Friedmannian framework, based upon the ‘cosmological principle’, this interpretation has given rise to the ‘Concordance’ model. However, we have already stressed in Chapter 1 that a caveat of such a reasoning is that the ‘cosmological principle’ derives from the philosophical Copernican assumption and has never been tested.

Moreover, it is well known, since the work of Ellis and Stoeger (1987), that the inhomogeneities observed in our Universe can have an effect upon the values of the cosmological parameters derived for a smoothed-out or averaged Friedmannian model. A tentative estimate of such an effect was computed by Hellaby (1988). He found that the error obtained when using averaging procedures compared to the volume matching (i.e. the procedure proposed by Ellis and Stoeger, 1987) of Friedmann models to inhomogeneous L–T solutions with realistic density profiles implies that the mean density and pressure of the averaged Friedmann models are 10–30% underestimated as regards the volume-matched ones. Therefore, even if the cosmological constant is the only component in the Einstein equations beyond ordinary matter, the estimation of its actual value is less straightforward than the conventional wisdom has it.

As regards dark energy, i.e. a negative pressure fluid with an equation of state parameter ω ≠ −1, (ω = −1 being the signature of the pure geometric cosmological constant), it remains a phenomenon which cannot be explained in the framework of current physics.

For many years after its appearance, general relativity (GR) was regarded as an exotic extension of Newtonian gravity, that was only necessary for highprecision measurements in the Solar System and for describing the expansion of the Universe. However, the increasing precision of physical and astronomical measurement is transforming GR into an indispensable tool, and not merely a small correction to Newton's theory.

It is commonly stated that we have entered the era of precision cosmology, in which a number of important observations have reached a degree of precision, and a level of agreement with theory, that is comparable with many Earth-based physics experiments. One of the consequences of this advance is the need to examine at what point our usual, well-worn assumptions begin to compromise the accuracy of our models, and whether more general theoretical methods are needed to maintain calculational accuracy. Historically, each advance in astronomical measurement has produced many new discoveries, and revealed more of the structure of the cosmos, such as voids, walls, filaments, etc. As we map out the Universe around us – its mass distribution and flow patterns – in ever greater detail, the nonlinear behaviour of cosmic structures will become increasingly apparent, and the methods of inhomogeneous cosmology will come into their own. Inhomogeneous solutions of Einstein's field equations provide models of both small and large structures that are fully nonlinear.

It is widely assumed that the Universe, when viewed on a large enough scale, is homogeneous and can be described by an FLRW model. The successes of the Concordance model are built on using a spatially homogeneous and isotropic background metric combined with first-order perturbation theory.

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).