Astron. Astrophys. 363, 851-862 (2000)

Appendix: dark matter halos in any cosmology

We suppose that perturbations of the matter density field when the universe becomes matter dominated are completely characterised by their power spectrum :

where is the transfer function (see fit given in Appendix G of Bardeen et al. 1986). We further assume a post-inflation Harrison-Zel'dovich power spectrum () for these perturbations, and take the shape parameter used in the computation of to be (Sugiyama 1995):

where is the current matter density (in critical density units), is the baryon density, and is the reduced Hubble constant.

In the linear regime, the equation of motion is solved for the expansion factor, a, and the solutions for the growth of the density contrast (see e.g. Peebles 1980) are derived. There are two such solutions (Heath 1977), which form a complete set (Zel'dovich 1965) and read:

where the subscripts d and g respectively stand for the decaying and growing modes, and is the reduced cosmological constant. If one further assumes that the initial peculiar velocity of the perturbation is zero (i.e. that the perturbation simply moves along with the expanding universe), the density contrast in the linear regime grows as:

where the subscript i stands for the initial quantities.

In the non-linear regime, one considers an isolated spherical perturbation of radius , at time , which has a uniform overdensity with respect to the background (), and encloses a mass . We assume that the perturbation is bound, and that its peculiar velocity is nil. The equation of motion is integrated to obtain the time at which the perturbation reaches its maximum expansion radius :

where

and is the first real root of the cubic equation (c.f. Richstone et al. 1992):

After it has reached this maximum radius at time , the perturbation, by symmetry, collapses on a time scale . Thus, with the previous equations, the critical density contrast linearly extrapolated till today (Eq. A.5) is explicitly related to the collapse redshift of the perturbation for any cosmology, just by computing the redshift to which the collapse time of a perturbation with overdensity corresponds in the unperturbed universe (provided is smaller than the age of the universe):

If the resulting virialized perturbation can be approximated by a singular isothermal sphere truncated at virial radius , then is the solution of the following cubic equation (see Devriendt 1999):

Finally, the velocity of a test particle moving on a circular orbit in the isothermal sphere reads:

Implementing these results into the peaks formalism described in GHBM enables one to derive formation rates for dark matter halos as a function of redshift. Fig. A.1 illustrates this halo formation rate for three typical cosmologies gathered in Table 1.

Fig. A.1. Evolution of the formation rate of dark matter halos (masses indicated on the figure) for three different cosmologies: SCDM (solid line), OCDM (dashes), and CDM (dots). The parameters used for these models are given in Table 1.

© European Southern Observatory (ESO) 2000

Online publication: December 5, 2000
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