Semi-analytic predictions for statistical properties of X-ray clusters of galaxies in cold dark matter universes

Temperature and luminosity functions of X-ray clusters are computed semi-analytically, combining a simple model for the cluster gas properties with the distribution functions of halo formation epochs proposed by Lacey & Cole (1993) and Kitayama & Suto (1996). In contrast to several previous approaches which apply the Press--Schechter mass function in a straightforward manner, our method can explicitly take into account the temperature and luminosity evolution of clusters. In order to make quantitative predictions in a specific cosmological context, we adopt cold dark matter (CDM) universes. Assuming the baryon density parameter $\Omega_{\rm B}=0.0125 h^{-2}$ ($h$ is the Hubble constant in units of 100 km$\cdot$sec$^{-1}\cdot$Mpc$^{-1}$) and the {\it COBE} normalization of matter fluctuations, temperature and luminosity functions of X-ray clusters depend sensitively on the density parameter $\Omega_0$. Allowing for several uncertainties in observational data as well as in our simplified assumptions, we conclude that $\Omega_0 \sim 0.2-0.5$ and $h\sim 0.7$ CDM models with/without the cosmological constant reproduce simultaneously the observed temperature and luminosity functions of X-ray clusters at redshift $z\sim0$.