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5.2. Nonlinear growth of perturbations

In a purely matter dominated universe, equation (64) reduces to R ddot = - GM / R2. Solving this equation one can obtain the non linear density contrast delta as a function of the redshift z:

Equation 82 (82)
Equation 83 (83)

Here, delta0 is the density contrast at present if the initial density contrast was evolved by linear approximation. In general, the linear density contrast deltaL is given by

Equation 84 (84)

When theta = (2pi/3), deltaL = 0.568 and delta = 1.01 appeq 1. If we interpret delta = 1 as the transition point to nonlinearity, then such a transition occurs at theta = (2pi/3), deltaL appeq 0.57. From (82), we see that this occurs at the redshift (1 + znl) = (delta0 / 0.57). The spherical region reaches the maximum radius of expansion at theta = pi. This corresponds to a density contrast of deltam approx 4.6 which is definitely in the nonlinear regime. The linear evolution gives deltaL = 1.063 at theta = pi. After the spherical over dense region turns around it will continue to contract. Equation (83) suggests that at theta = 2pi all the mass will collapse to a point. A more detailed analysis of the spherical model [199], however, shows that the virialized systems formed at any given time have a mean density which is typically 200 times the background density of the universe at that time in a OmegaNR = 1. This occurs at a redshift of about (1 + zcoll) = (delta0 / 1.686). The density of the virialized structure will be approximately rhocoll appeq 170rho0(1 + zcoll)3 where rho0 is the present cosmological density. The evolution is described schematically in figure 16.

Figure 16

Figure 16. Evolution of an over dense region in spherical top-hat approximation.

In the presence of dark energy, one cannot ignore the second term in equation (64). In the case of a cosmological constant, w = - 1 and rho = constant and this extra term is independent of time. This allows one to obtain the first integral to the equation (64) and reduce the problem to quadrature (see, for example [200, 201, 202]). For a more general case of constant w with w neq - 1, the factor (rho + 3P) = rho(1 + 3w) will be time dependent because rho will be time dependent even for a constant w if w neq - 1. In this case, one cannot obtain an energy integral for the equation (64) and the dynamics has to be determined by actual numerical integration. Such an analysis leads to the following results [189], [203], [204]:

(i) In the case of matter dominated universe, it was found that the linear theory critical threshold for collapse, deltac, was about 1.69. This changes very little in the presence of dark energy and an accurate fitting function is given by

Equation 85 (85)

(ii) The over density of a virialized structure as a function of the redshift of virialization, however, depends more sensitively on the dark energy component. For -1 leq w leq - 0.3, this can be fitted by the function

Equation 86 (86)


Equation 87 (87)

and Theta(z) = 1 / OmegaNR(z) - 1 = (1 / Omega0 - 1)(1 + z)3w.

The importance of deltac and Deltavir arises from the fact that these quantities can be used to study the abundance of non linear bound structures in the universe. The basic idea behind this calculation [205] is as follows: Let us consider a density field deltaR(x) smoothed by a window function WR of scale radius R. As a first approximation, we may assume that the region with delta(R, t) > deltac (when smoothed on the scale R at time t) will form a gravitationally bound object with mass M propto rhobar R3 by the time t. The precise form of the M - R relation depends on the window function used; for a step function M = (4pi / 3) rhobar R3, while for a Gaussian M = (2pi)3/2 rhobar R3. Here deltac is a critical value for the density contrast given by (85). Since delta(t) = D(t) for the growing mode, the probability for the region to form a bound structure at t is the same as the probability delta > deltac[D(ti) / D(t)] at some early epoch ti. This probability can be easily estimated since at sufficiently early ti, the system is described by a Gaussian random field. This fact can be used to calculate the number density of bound objects leading to the result

Equation 88 (88)

The quantity sigma here refers to the linearly extrapolated density contrast. We shall now describe the constraints on dark energy arising from structure formation.

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