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Astron. Astrophys. 317, 1-13 (1997)

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Appendix

In this appendix, we give a simple derivation in the case of a low density universe, of the amplitude of the initial fluctuations that collapse at some redshift, as well as the density contrast of the virialized objects. The spherical collapse in an expanding universe was first studied by Lemaître (1933). Gunn & Gott (1972) did cluster formation analysis through this model and the modern version of the spherical collapse can be found in Peebles (1980). Here, we derive in an elegant way the quantities that are used in this paper. Although these expressions were used by Oukbir & Blanchard, they were not given. The expression for the threshold density contrast of the non-linear collapse, though derived in a different way, has also been given by Lacey & Cole (1993) (see their appendix).

1. The density threshold

At high redshifts, the amplitude of the density fluctuations are small and the perturbations grow according to linear theory: [FORMULA]. In the case of a low density open model, the growth factor D(t) is given by (Weinberg 1972):

[EQUATION]

In this equation, [FORMULA] is the development angle defined as,

[EQUATION]

with a(t being the expansion factor of the universe. When the amplitude of [FORMULA] approaches unity, the overdense region decouples from the Hubble expansion, turns around, collapses and virializes. A simple analytic model allows one to calculate the critical overdensity corresponding to the redshift [FORMULA] at which a region virializes. A useful quantity is the critical linear overdensity extrapolated to the present day: [FORMULA].

One considers a spherical overdensity within a homogeneous universe. In the case where there is no shell crossing, Birkhoff's theorem allows one to treat the evolution of the overdensity as if it were an unperturbed region of the universe with average density [FORMULA]. Since [FORMULA], the parameteric equations of motion of the overdense shell are:

[EQUATION]

[EQUATION]

[EQUATION]

with [FORMULA] and [FORMULA] being the density parameter and the Hubble constant relative to the overdense shell. The overdense shell reaches maximum expansion for [FORMULA], with the mean density of the shell at maximum expansion being:

[EQUATION]

Virialization occurs at [FORMULA] with a radius [FORMULA]. The mean density of the overdense sphere at virialization is then:

[EQUATION]

Eq. ( A.5) can be written as [FORMULA]. Since we want to express [FORMULA] as a function of [FORMULA], we use Eq. ( A.6) to obtain:

[EQUATION]

The parametric equation of the mean density of the universe is:

[EQUATION]

[FORMULA] and [FORMULA] are the density parameter and the Hubble constant of the universe. The ratio of the perturbation density over the mean background density of the universe is then:

[EQUATION]

In this equation we want to express ( [FORMULA] ) and ( [FORMULA] ) as a function of time. We use the fact that for [FORMULA] (or equivalently, [FORMULA] ), the parametric equation of the time,

[EQUATION]

becomes:

[EQUATION]

We invert this equation to obtain [FORMULA] as a function of time:

[EQUATION]

to obtain:

[EQUATION]

Similarly, using the fact that for [FORMULA] ( [FORMULA] ) Eq. ( A.4) can be written

[EQUATION]

we obtain:

[EQUATION]

Now, we can use the developments of [FORMULA] and [FORMULA] as a function of time, to eliminate [FORMULA] et [FORMULA] from Eq. ( A.9):

[EQUATION]

In this equation, we want to substitute the variable t for the redshift z defined as:

[EQUATION]

From Eq. ( A.2), and for [FORMULA]:

[EQUATION]

Using the development of Eq. ( A.10) until the third order we obtain:

[EQUATION]

We use this expression of [FORMULA] in Eq. ( A.11) to finally obtain:

[EQUATION]

The initial overdensity, [FORMULA], of a perturbation which collapses at [FORMULA] is:

[EQUATION]

Linearly extrapolated to z=0 , this overdensity is

[EQUATION]

with D being the linear growth factor given by Eq. ( A.1). For [FORMULA], [FORMULA] could be approximated by [FORMULA]. Using Eq. ( A.12), [FORMULA] becomes: [FORMULA]. Finally:

[EQUATION]

2. The density contrast

The density contrast over the mean density of the background universe of an object which virializes at redshift [FORMULA], is the ratio of Eq. ( A.6) to Eq. ( A.8):

[EQUATION]

which can be expressed analytically by mean of Eqs. ( A.2) and ( A.9),

[EQUATION]

similar to the expression derived by Maoz (1990). In this equation, X is defined as,

[EQUATION]

In the case [FORMULA], we recover the usual constant value [FORMULA]. In the case [FORMULA], [FORMULA] is higher and grows with decreasing the redshift.

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