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 that is closely related to the cluster mass function is an important statistic of the census of galaxy clusters in our Universe. It is also an important means to probe the cosmological model of our Universe. Based on our recently completed REFLEX II cluster sample comprising 910 galaxy clusters with redshifts we construct the X-ray luminosity function of galaxy clusters for the nearby Universe and discuss its implications. We derived the X-ray luminosity function of the REFLEX II clusters on the basis of a precisely constructed selection function for the full sample and for several redshift slices from z = 0 to z = 0.4. In this redshift interval we find no significant signature of redshift evolution of the luminosity function. We provide the results of fits of a parameterized Schechter function and extensions of it which provide a reasonable characterization of the data. We also use a model for structure formation and galaxy cluster evolution to compare the observed X-ray luminosity function with the theoretical predictions for different cosmological models. The most interesting constraints can be derived for the cosmological parameters Omega(m) and sigma(8). We explore the influence of several model assumptions on which our analysis is based. We find that the scaling relation of X-ray luminosity and mass introduces the largest systematic uncertainty. From the statistical uncertainty alone we can constrain the matter density parameter, Omega(m) similar to 0.27 +/- 0.03 and the amplitude parameter of the matter density fluctuations, sigma(8) similar to 0.80 +/- 0.03. Marginalizing over the most important uncertainties, the normalisation and slope of the L-X - M scaling relation, we have larger error bars and a result of Omega(m) similar to 0.29 +/- 0.04 and sigma(8) similar to 0.77 +/- 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 there is some tension with Planck cosmological results from the microwave background anisotropies, which we discuss in the paper. We also make a comparison with results from the SDSS cluster survey, several cosmological X-ray cluster surveys, and recent Sunyaev-Zel'dovich effect 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.