The ROSAT Deep Cluster Survey (RDCS) has provided a new large deep sample of X-ray selected galaxy clusters. Observables such as the flux number counts n(S), the redshift distribution n(z) and the X-ray luminosity function (XLF) over a large redshift baseline (zlesssim 0.8) are used here in order to constrain cosmological models. Our analysis is based on the Press-Schechter approach, whose reliability is tested against N-body simulations. Following a phenomenological approach, no assumption is made a priori on the relation between cluster masses and observed X-ray luminosities. As a first step, we use the local XLF from RDCS, along with the high-luminosity extension provided by the XLF from the BCS, in order to constrain the amplitude of the power spectrum, sigma_8, and the shape of the local luminosity-temperature relation. We obtain sigma_8=0.58 +/- 0.06 for Omega_0=1 for open models at 90% confidence level, almost independent of the L-T shape. The density parameter Omega_0 and the evolution of the L-T relation are constrained by the RDCS XLF at z>0 and the EMSS XLF at z=0.33, and by the RDCS n(S) and n(z) distributions. By modelling the evolution for the amplitude of the L-T relation as (1+z)^A, an Omega_0=1 model can be accommodated for the evolution of the XLF with 1<A<3 at 90% confidence level, while Omega_0=0.4^{+0.3}_{-0.2} and Omega_0<0.6 are implied by a non--evolving L-T for open and flat models, respectively.
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