Constraints on neutrino masses from the study of the nearby large-scale structure and galaxy cluster counts

The high precision measurements of the cosmic microwave background by the Planck survey yielded tight constraints on cosmological parameters and the statistics of the density fluctuations at the time of recombination. This provides the means for a critical study of structure formation in the Universe by comparing the microwave background results with present epoch measurements of the cosmic large-scale structure. It can reveal subtle effects such as how different forms of Dark Matter may modify structure growth. Currently most interesting is the damping effect of structure growth by massive neutrinos. Different observations of low redshift matter density fluctuations provided evidence for a signature of massive neutrinos. Here we discuss the study of the cosmic large-scale structure with a complete sample of nearby, X-ray luminous clusters from our REFLEX cluster survey. From the observed X-ray luminosity function and its reproduction for different cosmological models, we obtain tight constraints on the cosmological parameters describing the matter density, Omega_m, and the density fluctuation amplitude, sigma_8. A comparison of these constraints with the Planck results shows a discrepancy in the framework of a pure LambdaCDM model, but the results can be reconciled, if we allow for a neutrino mass in the range of 0.17 to 0.7 eV. Also some others, but not all of the observations of the nearby large-scale structure provide evidence or trends for signatures of massive neutrinos. With further improvement in the systematics and future survey projects, these indications will develop into a definitive measurement of neutrino masses.