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 \Ω\ B=0.0125 h⁻² (h is the Hubble constant in units of 100 km\⁻¹\⁻¹) and the {\ COBE} normalization of matter fluctuations, temperature and luminosity functions of X-ray clusters depend sensitively on the density parameter \Ω_0. Allowing for several uncertainties in observational data as well as in our simplified assumptions, we conclude that \Ω_0 \~ 0.2-0.5 and h\~ 0.7 CDM models with/without the cosmological constant reproduce simultaneously the observed temperature and luminosity functions of X-ray clusters at redshift z\~0.
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Kitayama et al. (1996) studied this question.