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The X-ray luminosity function is an important statistic of the census ofgalaxy clusters and an important means to probe the cosmological model of ourUniverse. Based on our recently completed REFLEX II cluster sample we constructthe X-ray luminosity function of galaxy clusters for several redshift slicesfrom $z = 0$ to $z = 0.4$ and discuss its implications. We find no significantsignature of redshift evolution of the luminosity function in the redshiftinterval. We provide the results of fits of a parameterized Schechter functionand extensions of it which provide a reasonable characterization of the data.Using a model for structure formation and galaxy cluster evolution we comparethe observed X-ray luminosity function with predictions for differentcosmological models. For the most interesting constraints for the cosmologicalparameters $\Omega_m$ 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 andslope 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 ourresults with those of the SZ-cluster survey provided by the PLANCK mission andwe find very good agreement with the results using PLANCK clusters ascosmological probes, but we have some tension with PLANCK cosmological resultsfrom the microwave background anisotropies. We also make a comparison withother cluster surveys. We find good agreement with these previous results andshow that the REFLEX II survey provides a significant reduction in theuncertainties compared to earlier measurements.

astronomy and astrophysics

The extended ROSAT-ESO Flux Limited X-ray Galaxy Cluster Survey (REFLEX  II) IV. X-ray Luminosity Function and First Constraints on Cosmological  Parameters