Annu. Rev. Astron. Astrophys. 2002. 40:
539-577
Copyright © 2002 by Annual Reviews. All rights reserved |

**5.1. The cosmological mass function**

The mass distribution of dark matter halos undergoing spherical
collapse in the framework of hierarchical clustering is described by
the Press-Schechter distribution (PS,
Press &Schechter
1974).
The number of such halos in the mass range [*M*, *M* +
*dM*] can be written as

(10) |

where
is the cosmic mean density. The function *f* depends
only on the variable =
_{c}(*z*)
/ _{M}, and is
normalized so that
*f* ()
*d* = 1.
_{c}(*z*)
is the linear-theory
overdensity extrapolated to the present time for a uniform spherical
fluctuation collapsing at redshift *z*. This quantity conveys
information about the dynamics of fluctuation evolution in a generic
Friedmann background. It is convenient to express it as
_{c}(*z*) =
_{0}(*z*)
[*D*(0) / *D*(*z*)], where *D*(*z*)
is the linear fluctuation growth factor, which depends on the density
parameters contributed by matter,
_{m} and by
cosmological constant,
_{} (e.g.
Peebles 1993).
The quantity
_{0}(*z*)
has a weak dependence on
_{m} and
_{} (e.g.
Kitayama & Suto
1997).
For a critical-density Universe it is
_{0} = 1.686,
independent of *z*.

The r.m.s. density fluctuation at the mass scale *M*,
_{M}, is
connected to the fluctuation power spectrum, *P*(*k*), by the
relation

(11) |

The dependence of the power spectrum on the wavenumber *k* is usually
written as
*P*(*k*)
*k*^{npr} *T*^{2}(*k*), where
*T*(*k*) is the
transfer function, which depends both on the cosmological parameters
of the Friedmann background and on the cosmic matter constituents
(e.g. fraction of cold, hot and baryonic matter, number of
relativistic species; see
Kolb & Turner 1989).
For a pure cold dark matter (CDM) model, *T*(*k*) depends to a
good approximation only on the shape parameter
=
_{m}
*h* (e.g.
Bardeen et al. 1986),
while a correction to this dependence needs to be introduced to
account for the presence of the baryonic component (e.g.
Eisenstein & Hu 1999).
The Harrison-Zel'dovich spectrum is generally assumed with the
primordial index, *n*_{pr} = 1, consistent with the most
recent analyses of the CMB anisotropies
(de Bernardis et al. 2001
and references therein). The amplitude of *P*(*k*) is usually
expressed in terms of
_{8}, the
r.m.s. density fluctuation within a top-hat sphere of
8 *h*^{-1} Mpc radius. Finally, in Equation
11 *W*(*x*) is the Fourier representation of the window
function, which describes the shape of the volume from which the
collapsing object is accreting matter. The comoving fluctuation size
*R* is connected to the mass scale *M* as
*R* = (3*M*/4
)^{1/3} for the top-hat window, i.e.
*W*(*x*) = 3(sin *x* - *x* cos *x*) /
*x*^{3}.

In their original derivation of the cosmological mass function,
Press & Schechter
(1974)
obtained the expression
*f* () =
(2)^{-1/2}
exp(- ^{2}/2) for
Gaussian density fluctuations. Despite its subtle simplicity (e.g.,
Monaco 1998),
the PS mass function has served for more than a decade as a guide to
constrain cosmological parameters from the mass function of galaxy
clusters. Only with the advent of the last generation of N-body
simulations, which are able to span a very large dynamical range,
significant deviations of the PS expression from the exact
numerical description of gravitational clustering have been noticed (e.g.
Gross et al. 1998,
Governato et al. 1999,
Jenkins et al. 2001,
Evrard et al. 2002).
Such deviations are interpreted in terms of
corrections to the PS approach. For example, incorporating the effects
of non-spherical collapse
(Sheth et al. 2001)
generalizes the above PS expression for *f*
() to

(12) |

where *a* = 0.707, *C* = 0.3222 and *q* = 0.3
(Sheth & Tormen
1999).
The above equation reduces to the PS expression for *a* = 1,
*C* = 1/2 and *q* = 0. Fitting formulae for *f*
(), which reproduce N-body
results to an accuracy of about 10% (e.g.
Evrard et al. 2002)
are now currently used to derive cosmological constraints from the
evolution of the cluster population.

In practical applications, the observational mass function of clusters
is usually determined over about one decade in mass. Therefore, it
probes the power spectrum over a relatively narrow dynamical range,
and does not provide strong constraints on the shape
of the power
spectrum. Using only
the number density of nearby clusters of a given mass *M*, one can
constrain the amplitude of the density perturbation at the physical
scale *R*
(*M* /
_{m}
_{crit})^{1/3} containing this
mass. Since such a scale depends both on *M* and on
_{m}, the
mass function of nearby (*z*
0.1) clusters is
only able to constrain a relation between
_{8} and
_{m}. In the
left panel of Figure 12 we show that, for a
fixed value of
the observed cluster mass function, the implied value of
_{8}
from Equation 12 increases as the density parameter decreases.

Determinations of the cluster mass function in the local Universe
using a variety of samples and methods indicate that
_{m}^{} = 0.4 - 0.6,
where
0.4 - 0.6, almost
independent of the presence of a
cosmological constant term providing spatial flatness (e.g.
Bahcall & Cen 1993,
Eke et al. 1996,
Girardi et al. 1998,
Viana & Liddle 1999,
Blanchard et al. 2000,
Pierpaoli et al. 2001,
Reiprich &
Böhringer 2002,
Seljak 2002,
Viana et al. 2002).
It is worth pointing out that formal statistical uncertainties in the
determination of
_{8} from the
different analyses are always far
smaller, 5%, than
the above range of values. This suggests that current discrepancies on
_{8} are likely
to be ascribed to
systematic effects, such as sample selection and different methods
used to infer cluster masses. We comment more on such
differences in the following section.
Completely independent constraints on a similar combination of
_{8} and
_{m} can be
obtained with measurements of the cosmic gravitational lensing shear (e.g.
Mellier 1999).
The most recent results give
_{m}^{0.6} = 0.45 ± 0.05
(van Waerbecke et
al. 2001,
and references therein).

The growth rate of the density perturbations depends primarily on
_{m} and, to
a lesser extent, on
_{}, at
least out
to *z* ~ 1, where the evolution of the cluster population is
currently studied. Therefore, following the evolution of the cluster
space density over a large redshift baseline, one can break the
degeneracy between
_{8} and
_{m}. This
is shown in a pictorial way in Figure 1
and quantified in the right panel of
Figure 12: models with different values of
_{m}, which
are normalized to yield the same number density of
nearby clusters, predict cumulative mass functions that progressively
differ by up to orders of magnitude at increasing redshifts.