The conventional paradigm for the formation of structures in the universe is based on the growth of small perturbations due to gravitational instabilities. In this picture, some mechanism is invoked to generate small perturbations in the energy density in the very early phase of the universe. These perturbations grow due to gravitational instability and eventually form the different structures which we see today. Such a scenario can be constrained most severely by observations of cosmic microwave background radiation (CMBR) at z 103. Since the perturbations in CMBR are observed to be small (10-5 - 10-4 depending on angular scales), it follows that the energy density perturbations were small compared to unity at the redshift of z 1000.
The central quantity one uses to describe the growth of structures during 0 < z < 103 is the density contrast defined as (t, x) = [(t, x) - bg(t)] / bg(t) which characterizes the fractional change in the energy density compared to the background. (Here bg(t) is the mean background density of the smooth universe.) Since one is often interested in the statistical behaviour of structures in the universe, it is conventional to assume that and other related quantities are elements of an ensemble. Many popular models of structure formation suggest that the initial density perturbations in the early universe can be represented as a Gaussian random variable with zero mean (that is, < > = 0) and a given initial power spectrum. The latter quantity is defined through the relation P(t, k) = < |k(t)|2 > where k is the Fourier transform of (t, x) and < ... > indicates averaging over the ensemble. It is also conventional to define the two-point correlation function (t, x) as the Fourier transform of P(t, k) over k. Though gravitational clustering will make the density contrast non Gaussian at late times, the power spectrum and the correlation function continue to be of primary importance in the study of structure formation.
When the density contrast is small, its evolution can be studied by linear perturbation theory and each of the spatial Fourier modes k(t) will grow independently. It follows that (t, x) will have the form (t, x) = D(t)f (x) in the linear regime where D(t) is the growth factor and f (x) depends on the initial configuration. When 1, linear perturbation theory breaks down and one needs to either use some analytical approximation or numerical simulations to study the non linear growth. A simple but effective approximation is based on spherical symmetry in which one studies the dynamics of a spherical region in the universe which has a constant over-density compared to the background. As the universe expands, the over-dense region will expand more slowly compared to the background, will reach a maximum radius, contract and virialize to form a bound nonlinear system. If the proper coordinates of the particles in a background Friedmann universe is given by r = a(t)x we can take the proper coordinates of the particles in the over-dense region to be r = R(t)x where R(t) is the expansion rate of the over-dense region. The relative acceleration of two geodesics in the over-dense region will be g = x = ( / R)r. Using (8) and . r = 3, we get
where the subscript `non-dust' refers to all components of matter other than the one with equation of state P = 0; the dust component is taken into account by the first term on the right hand side with M = (4/3) NR R3. The density contrast is related to R by (1 + ) = ( / bg) = (a/R)3. Given the equation (64) satisfied by R and (20), it is easy to determine the equation satisfied by the density contrast. We get (see p. 404 of ):
This is a fully nonlinear equation satisfied by the density contrast in a spherically symmetric over-dense region in the universe.