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.