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/cm2 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 8h-1Mpc), consistent with recent estimates based on optical analysis of galaxy clusters (Girardi et al., 1998) and weak lensing measurements (Van Waerbeke et al., 2002).