**4.2. The local temperature distribution function**

In order to estimate the amount of evolution in the number of clusters, one
obviously needs a reliable estimate of the number of clusters at
*z* ~ 0. This already is not so easy and is a serious limitation.
The estimation of the local temperature distribution function of X-ray
clusters
can be achieved from a sample of X-ray selected clusters for which the
selection function is known, and for which temperatures are
available. Until recently, the standard reference sample was the
Henry and Arnaud sample
(1991),
based on 25 clusters selected in the 2. - 10. keV band. The ROSAT
satellite has since provided better quality samples of X-ray clusters, like
the RASS and the BCS sample, containing several hundred of clusters.
Temperature information is still lacking for most of clusters in these
samples and therefore such clusters samples do not allow yet to improve
estimations of the TDF in practice. We have
therefore constructed a sample of X-ray clusters, by selecting all X-ray
clusters with a flux above
2.210^{-11} erg/s/cm^{2} with | *b*| > 20. Most
of the clusters come from the Abell XBACS sample, to which some non-Abell
clusters were added. The completeness was estimated by comparison with the
RASS and the BCS and found to be of the order of 85%. This sample
comprises 50
clusters, which makes it the largest one available for measuring the TDF
at the time it was published. The inferred TDF is in very good agreement
with the TDF derived
from the BCS luminosity function or from more recent comprehensive
survey
(Reiprich and
Böhringer, 2002)
(with ~ 65 clusters). The abundance of clusters is higher than
derived from the Henry and Arnaud sample as given by
Eke et al. (1998)
for instance. It is in good agreement with
Markevitch (1998)
for clusters with
*T* > 4 keV, but is slightly higher for clusters with *T* ~
3 keV. The power
spectrum of fluctuations can be normalized from the abundance of clusters,
leading to
_{8} =
_{c} = 0.6
(using PS formula) for
_{m} = 1 and
to _{c} = 0.7
for _{m} =
0.35 corresponding to
_{8} = 0.96 for
a *n* = - 1.5 power
spectrum index (contrary to a common mistake the cluster abundance does not
provide an unique normalization for
_{8} in low
density models, but on a scale ~ ^{-3}
(_{m})^{1/2} 8*h*^{-1}Mpc),
consistent with recent estimates based on optical analysis of galaxy
clusters
(Girardi et al., 1998)
and weak lensing measurements
(Van Waerbeke et al.,
2002).