## 2. Theoretical mass functionIn the gravitational instability scenario, virialized objects like galaxy clusters form from initially small density fluctuations that grow under the influence of gravity. The density field is specified by its power spectrum and the statistical nature of the fluctuations. It is generally assumed that the field is Gaussian, although some consequences of non-Gaussianity have been examined (Weinberg and Cole 1992), a question to which we return in Sect. 7. For the power spectrum, we adopt a simple power law, over the mass range corresponding to galaxy clusters. In
hierarchical models, such as CDM, the variance of the density field
diverges on small scales, and so one must work with a smoothed version
(see, for instance, Bardeen et al. 1986). The variance of the density
field smoothed with some window function where is the Fourier transform of the window
function. Davis & Peebles (1983) found that the variance of galaxy
counts within spheres of radius , quoted as
, is close to one. If the galaxy distribution
follows the mass distribution, then the variance of the density
perturbations on the same scale would also be equal to unity. However,
if the galaxy distribution is biased relative to the mass
distribution, then the variance of mass fluctuations in a sphere of
radius Accordingly, in the following, the value of Since in one dimension the early stage of the non-linear collapse
is entirely determined by the amplitude of the local mean density, the
exact solution for the collapse of an overdensity is calculable until
the first orbit crossing occurs. In three dimensions, the solution is
also known for the case of a spherical matter distribution
(Lemaître 1933, Gunn & Gott 1972, Peebles 1980). This
so-called spherical model can be used to model the non-linear collapse
of a cluster, which one then finds is driven by the value of the
density field smoothed with a top-hat window of size comparable to
that of the cluster. It is well known that when the linear density
field reaches the value 1.68, the density becomes singular for a
purely spherical collapse. In reality, the spherical symmetry is
broken by the development of substructures which instead leads to the
formation of a stationary state, the "virial" equilibrium. The final
radius of the collapsed object is expected to be half of its maximum
expansion radius, corresponding to a density contrast of the order of
200. In the absence of significant fragmentation during the collapse,
initial density fluctuations of the field smoothed on the scale
In practice one deals with the initial density field linearly
extrapolated to the present epoch . For
instance, the rms fluctuation on some scale In this relation Despite the fact that the spherical top-hat model permits a
considerable simplification of the actual development of
non-linearities, the precise calculation of the number density of
collapsed objects of mass Using the spherical model, Press & Schechter (1974) proposed a
derivation of the mass function of virialized objects. They argued
that the fraction of matter in the form of non-linear objects with
mass greater than where is the threshold for non-linear collapse, which in the spherical model is given by For an Einstein-de Sitter universe, . The mass distribution function is then easily derived: Unfortunately, this result predicts that only half of the mass of the universe ends up in virialized objects, whereas one expects that at each epoch in a hierarchical scenario all of the mass should be bound in virialized objects. Press & Schechter (1974) side-stepped this problem by arbitrarily multiplying this mass function by a factor of 2. Several authors have recently re-examined the issue and proposed solutions (Peacock & Heavens 1990, Cole 1991, Blanchard et al. 1992a). The problem has also been considered by Bond et al. (1991). Instead
of smoothing the density contrast with a filter
, they adopted a low-pass, top-hat filter in
Fourier space. At a given point On the other hand, Blanchard et al. (1992a) point out that with the usually adopted simplifications (i.e. the collapse being driven essentially by the mean local density and the neglect of fragmentation), it is always possible to write the mass function in an exact way as: where represents the fraction of volume
covered by non-linear spheres of radius greater than We justify our use of the PS formalism by its surprisingly good fit
to the mass functions found in numerical simulations. This was first
emphasized by Efstathiou et al. (1988), who examined the cluster
multiplicity function for different values of © European Southern Observatory (ESO) 1997 Online publication: June 30, 1998 |