Abundance and evolution of galaxy clusters in cosmological models with massive neutrino

The time evolution of the number density of galaxy clusters and their mass and temperature functions are used to constrain cosmological parameters in the spatially at dark matter models containing hot particles (massive neutrino) as well as cold and baryonic matter. We test the modified MDM (Lambda = 0) models with cosmic gravitational waves and show that they neither pass the cluster evolution test nor reproduce the observed height of the first acoustic peak in DeltaT/T spectrum, and therefore should be ruled out. The models with a non-zero cosmological constant are in better agreement with observations. We estimate the free cosmological parameters in DeltaMDM with a negligible abundance of gravitational waves, and find that within the parameter ranges h is an element of (0.6; 0.7), n is an element of (0.9; 1.1), f(nu) = Omega(nu)/Omega(m) is an element of (0, 0.2), (i) the value of Omega(Lambda) is strongly affected by a small fraction of hot dark matter: 0.45 < Omega(Lambda) < 0.7 (1sigma CL), and (ii) the redshift evolution of galaxy clusters alone reveals the following explicit relation between Omega(Lambda) and f(nu): Omega(Lambda) + 0.5 f(nu) = 0.65 +/- 0.1. This degeneracy is also expected in LSS tests (with a smaller error). The present accuracy of observational data allows to bound the fraction of hot matter, f(nu) is an element of (0, 0,2); the number of massive neutrino species remains undelimited, N-nu = 1, 2, 3.