4.3. Application to the determination of m
The abundance of X-ray clusters at z = 0.33 can be determined from the Henry sample (1997) containing 9 clusters. Despite the limited number of clusters and the limited range of redshift for which the above cosmological test can be applied, interesting answer can already be obtained, demonstrating the power of this test. Comparison of the local TDF and the high redshift TDF clearly show that there is a significant evolution in the abundance of X-ray clusters (see figure 1), such an evolution is unambiguously detected in our analysis. This evolution is consistent with the recent study of Donahue et al. (2000). We have performed a likelihood analysis to estimate the mean density of the universe from the detected evolution between z = 0.05 and z = 0.33. The likelihood function is written in term of all the parameters entering in the problem: the power spectrum index and the amplitude of the fluctuations. The best parameters are estimated as those which maximize the likelihood function. The results show that for the open and flat cases, one obtains high values for the preferred m with a rather low error bar :
(Blanchard et al., 2000). Interestingly, the best fitting model also reproduces the abundance of clusters (with T ~ 6 keV) at z = 0.55 as found by Donahue and Voit (2000).
Figure 5. Likelihoods from the measured abundance of EMSS clusters in the redshift range (0.3,0.4) based on Henry's sample (1997). The dashed line is for a flat universe while the continuous line is for an open cosmology.