Cosmological constraints from the local X-ray luminosity function of the   most X-ray luminous galaxy clusters
  We present precise constraints on the normalization of the power spectrum of mass fluctuations in the nearby universe, sigma_8, as a function of the mean local matter density, Omega_m. Using the observed local X-ray luminosity function of galaxy clusters from the extended BCS and REFLEX studies, a mass-luminosity relation determined from Chandra and ROSAT X-ray data and weak gravitational lensing observations, and the mass function predicted by the Hubble Volume simulations of Evrard et al., we obtain sigma_8 = (0.508\pm0.019) Omega_m^-(0.253\pm0.024), with Omega_m < 0.34 at 68 per cent confidence. The degeneracy between sigma_8 and Omega_m can be broken using Chandra measurements of the X-ray gas mass fractions in dynamically relaxed clusters. Using this information and including Gaussian priors on the mean baryon density of the universe and the Hubble constant, we obtain sigma_8=0.695\pm0.042 and Omega_m=0.287\pm0.036, for an assumed flat LambdaCDM cosmology (marginalized 68 per cent confidence limits). Our results are in good agreement with some recent studies based on the local X-ray temperature function of clusters, the redshift evolution of the X-ray luminosity and temperature functions of clusters, early results from the Sloan Digitized Sky Survey, the most recent results from studies of cosmic shear, and combined analyses of the 2dF galaxy redshift survey and cosmic microwave background anisotropies. 
S.W. Allen, R.W. Schmidt, A.C. Fabian (IoA, Cambridge) and H. Ebeling   (IfA, Hawaii)