## 2. The galaxy mass functionMass condensations may be characterized by two parameters, such as
for instance velocity dispersion and radius, or equivalently mass and
radius. As will be explained in Sect. 3.1, we define galaxies as halos
of mass ## 2.1. Press-Schechter approach (PS)We shall first devise a "Press-Schechter approach" (PS), in the
sense that we wish to recognize in the early universe, when
overdensities still grow according to linear theory, the density
fluctuations which will eventually become halos of mass Let us first consider the case .
According to the spherical model, an overdensity characterized by the
density contrast given by the linear
theory, , will slowly separate from
the general expansion, its radius will reach a maximum
at time
and collapse at time
when
. We shall assume that in fact the
halo virializes at this collapse time at the radius
, as implied by energy conservation
and virial equilibrium if kinetic energy is negligible at the
turn-around. Hence at time the
density contrast of the halo is .
Then we assume that the density of this object does not evolve
significantly so that . Thus, a given
halo of mass as a function of mass and redshift. Then, as long as , we have Then we consider that each halo is characterized by its parameter so that the mass fraction in objects between and is: where we assumed gaussian initial fluctuations and we corrected by a factor 2 as in the traditional Press-Schechter prescription. Then the comoving multiplicity function of dark matter halos is: where is the mean density of the present universe. When the density contrast is constant and equal to the formulation (4) is exactly the usual Press-Schechter multiplicity function. In the case , we have where is the growing mode of the linear theory and the density contrast of a halo which virializes at redshift is . Hence we now get: where is defined by: The multiplicity function of dark matter halos is still given by (4). Note that we have the normalization condition: ## 2.2. Non-linear hierarchical scaling modelWe shall now devise a second method which deals directly with the
non-linear regime. We define for each halo the parameter where is the average of the two-body correlation function
over a spherical cell of radius
while the mass fraction in halos of mass between The function is a universal
function that depends only on the initial spectrum of fluctuations and
that has to be taken from numerical simulations although its
qualitative behaviour is well-known:
for small It has been shown in VS that this function
is quite close to a similar scaling
function that is obtained from the
counts in cells. Bounds to estimate the difference between these two
functions are given in VS while numerical checks are presented in
Valageas et al. (1999a). Hereafter we use the function
in place of
. The correlation function
that measures the non-linear
fluctuations at scale ## 2.3. Comparison of the PS model with the non-linear hierarchical scaling approachThese two formulations look very different, especially for
evolution effects. The power at which redshift enters the expressions
(4) and (9) is different even in the exponential. It happens, however,
that the parameter of the linear
theory can be expressed in terms of the parameter where is a parameter close to
unity. Using this relation one can obtain for a power-law initial
spectrum of index which shows (VS) that the expression (4) deduced from linear theory using the Press-Schechter ideas can be brought into a form quite similar to the non-linear scaling one (9) with: This holds in the highly non-linear regime, that is for
and
. The definition of galaxies we shall
use, which considers that galaxies are objects which have already
virialized (and have since undergone a significant cooling) ensures
that we always have for all galaxies
at any time. The condition implies
that (14) will hold until the redshift
such that
. This similar scaling as a function
of the parameter © European Southern Observatory (ESO) 1999 Online publication: April 19, 1999 |