The extended ROSAT-ESO Flux Limited X-ray Galaxy Cluster Survey (REFLEX   II) IV. X-ray Luminosity Function and First Constraints on Cosmological   Parameters
  The X-ray luminosity function is an important statistic of the census of galaxy clusters and an important means to probe the cosmological model of our Universe. Based on our recently completed REFLEX II cluster sample we construct the X-ray luminosity function of galaxy clusters for several redshift slices from $z = 0$ to $z = 0.4$ and discuss its implications. We find no significant signature of redshift evolution of the luminosity function in the redshift interval. We provide the results of fits of a parameterized Schechter function and extensions of it which provide a reasonable characterization of the data. Using a model for structure formation and galaxy cluster evolution we compare the observed X-ray luminosity function with predictions for different cosmological models. For the most interesting constraints for the cosmological parameters $\Omega_m$ and $\sigma_8$ we obatain $\Omega_m \sim 0.27 \pm 0.03$ and $\sigma_8 \sim 0.80 \pm 0.03$ based on the statistical uncertainty alone. Marginalizing over the most important uncertainties, the normalisation and slope of the $L_X - M$ scaling relation, we find $\Omega_m \sim 0.29 \pm 0.04$ and $\sigma_8 \sim 0.77 \pm 0.07$ ($1\sigma$ confidence limits). We compare our results with those of the SZ-cluster survey provided by the PLANCK mission and we find very good agreement with the results using PLANCK clusters as cosmological probes, but we have some tension with PLANCK cosmological results from the microwave background anisotropies. We also make a comparison with other cluster surveys. We find good agreement with these previous results and show that the REFLEX II survey provides a significant reduction in the uncertainties compared to earlier measurements. 
H. B\"ohringer, G. Chon, C.A. Collins
astro-ph.CO