2. Multiplicity functions
We first review the method we use to obtain the mass functions of various astrophysical objects, specifically galaxies and Lyman- clouds. We consider objects of dark matter mass M to be defined by a density threshold . This constraint depends on the class of astrophysical objects one considers and it allows us for instance to distinguish clusters from galaxies which correspond to higher density contrasts (see VS II). Lyman- clouds are also formed by several populations of different objects which are not always defined by a constant density threshold (see Sect. 5). Note that such a goal is beyond the reach of the usual Press-Schechter prescription (Press & Schechter 1974) which only deals with "just-collapsed halos" while we wish to describe simultaneously a wide variety of objects. In any case, we attach to each halo a parameter x defined by:
is the average of the two-body correlation function over a spherical cell of radius R and provides the measure of the density fluctuations in such a cell. Then, we write the multiplicity function of these objects (defined by the constraint ) as (see VS I):
The scaling function depends only on the initial spectrum of the density fluctuations and must be obtained from numerical simulations. However, from theoretical arguments (see VS I and Balian & Schaeffer 1989) it is expected to follow the asymptotic behaviour:
with , , to 20 and by definition it must satisfy
The correlation function , that measures the non-linear fluctuations at scale R can be modelled in a way that accurately follows the numerical simulation. The mass functions obtained from (2) for various constraints were checked against the results of numerical simulations in Valageas et al. (1999b) in the case of a critical universe with an initial power-spectrum which is a power-law: with and -2. This study showed that this model provides a reasonable approximation to the mass functions obtained in the simulations and that it works quite well for the two cases we shall need in the present article: i) a constant density threshold and ii) a constant radius constraint (or ). Moreover, the results of Valageas et al. (1999b) showed that is close to a similar scaling function obtained from the counts-in-cells statistics, as expected from theoretical considerations (see for instance VS I).
It is clear that the model outlined above provides a unified description of various astrophysical objects which are obtained from the same non-linear density field. This is a great advantage of this approach since it ensures that we can model a wide variety of objects, from low density Lyman- clouds to high density bright galaxies, in a fully consistent way. Then, we can study the interplay between these various structures as they develop progressively.
© European Southern Observatory (ESO) 1999
Online publication: June 18, 1999