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.
