X-Ray galaxy clusters: constraints on models of galaxy formation
  We present a self-consistent approach to the modeling of X-ray clusters of galaxies in a flat universe. Employing the Press and Schechter (1974) formalism to derive the mass function and relating the observable properties of clusters to their virial mass allows us to study the cluster X-ray distribution functions and their evolution with redshift. This approach differs from some results based on the X-ray luminosity function, which we argue is subject to modeling uncertainties. We obtain stringent constraints on the power spectrum of the initial density perturbations assumed to seed galaxy formation by comparing the theoretical temperature function with available data: the amplitude $\sigma_8$ is found to be $0.57 \pm 0.05$, while the local spectral index $n$ is falling in the range $-2.4 \leq n \leq -1.5$ on scales between 5 and 15 $h^{-1}$ Mpc: the standard CDM model is clearly ruled out, independently of the normalization. We further examine the evolutionary properties of X-ray clusters. Our approach greatly clarifies the situation: contrary to some previous claims, we find that flat models are in reasonable agreement with the observed number of clusters at high redshif ts. We also examine the contribution of clusters to the X-ray counts and to the soft X-ray background and compare the expected values in the case of the $\Omega_0=1$ and $\Omega_0=0.2$ universes. The number counts are in agreement with the observations, further confirming the relevance of our modeling. We conclude that although clusters are not the primary source of the soft X-ray background, their contribution is nevertheless non-negligible. This is particularly important for they could escape the constraints imposed on possible point sources contributing to the background. Finally, we briefly 
J. Oukbir, J.G. Bartlett, A. Blanchard (Observatoire astronomique de   Strasbourg)