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.