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

**4.1. Local Cluster Number Density**

The determination of the local (*z*
0.3)
cluster abundance
plays a crucial role in assessing the evolution of the cluster
abundance at higher redshifts. The cluster XLF is commonly modeled
with a Schechter function:

(7) |

where is the faint-end
slope, *L*^{*}_{X} is the characteristic
luminosity, and ^{*} is directly
related to the space-density of clusters brighter than
*L*_{min}: *n*_{0} =
_{Lmin}
(*L*)
*dL*. The cluster XLF in the
literature is often written as: (*L*_{44}) = *K* exp(-
*L*_{X} / *L*^{*}_{X})
*L*_{44}^{-}, with
*L*_{44} = *L*_{X} / 10^{44} erg
s^{-1}. The normalization *K*, expressed in units of
10^{-7} Mpc^{-3}(10^{44} erg
s^{-1})^{-1}, is related to
^{*} by
^{*} = *K*
(*L*^{*}_{X} /
10^{44})^{1-}.

Using a flux-limited cluster sample with measured redshifts and
luminosities, a binned representation of the XLF can be
obtained by adding the contribution to the space density of each
cluster in a given luminosity bin
*L*_{X}:

(8) |

where *V*_{max} is the total search volume defined as

(9) |

Here *S*(*f*) is the survey sky coverage, which depends on the
flux *f* = *L* / (4
*d*_{L}^{2}), *d*_{L}(*z*) is the
luminosity distance, and *H*(*z*)
is the Hubble constant at *z* (e.g.
Peebles 1993,
pag. 312). We define *z*_{max}
as the maximum redshift out to which the object is included in the
survey. Equations 8 and 9 can be easily
generalized to compute the XLF in different redshift bins.

In Figure 6 we summarize the recent progress
made in computing
(*L*_{X}) using primarily low-redshift
*ROSAT* based surveys. This work improved the first determination
of the cluster XLF
(Piccinotti et al. 1982,
see Section 3.2). The BCS and REFLEX cover a
large *L*_{X} range and have good statistics at the bright
end, *L*_{X}
*L*^{*}_{X}
and near the knee of the XLF. Poor clusters and groups
(*L*_{X}
10^{43}
erg s^{-1}) are better studied using deeper surveys, such as the
RDCS. The very faint end of the XLF has been investigated using an
optically selected, volume-complete sample of galaxy groups detected
*a posteriori* in the RASS
(Burns et al. 1996).

From Figure 6, we note the very good agreement among
all these independent determinations. Best-fit parameters are
consistent with each other with typical values:
1.8
(with 15% variation), ^{*}
1 ×
10^{-7} *h*_{50}^{3} Mpc^{-3} (with
50% variation), and *L*^{*}_{X}
4 ×
10^{44} erg s^{-1} [0.5-2 keV]. Residual differences at
the faint end are
probably the result of cosmic variance effects, because the lowest
luminosity systems are detected at very low redshifts where the search
volume becomes small (see
Böhringer et
al. 2002b).
Such an overall
agreement is quite remarkable considering that all these surveys used
completely different selection techniques and independent
datasets. Evidently, systematic effects associated with different
selection functions are relatively small in current large cluster
surveys. This situation is in contrast with that for the galaxy
luminosity function in the nearby Universe, which is far from
well established
(Blanton et al. 2001).
The observational study of
cluster evolution has indeed several advantages respect to galaxy
evolution, despite its smaller number statistics. First, a robust
determination of the local XLF eases the task of measuring cluster
evolution. Second, X-ray spectra constitute a single parameter family
based on temperature and K-corrections are much easier to compute than
in the case of different galaxy types in the optical bands.