*Astron. Astrophys. 347, 1-20 (1999)*
## 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:
where
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):
where is the mean density of the
universe at redshift *z*, while the mass fraction in halos of
mass between *M* and is:
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
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