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The patch like model of the hierarchical galaxy formation in the $\Lambda$CDMcosmological model with small damping scale is considered. In this modelgalaxies and clusters of galaxies are identified with rare high density peaks,what suppresses the action of random factors in the vicinity of peaks and makesthe process of halos formation more rapid and regular. High concentration ofirregular subhalos surrounding the central peaks and their subsequent mergingjust after formation allows to consider this medium as a mixture ofcollisionless dispersed dark matter (DM) particles and collisional subhalos.Merging of these subhalos with the central dominating halo is accompanied bytidal destruction of the central cusp, what progressively shallows the densityprofile and promotes formation of super massive central black holes. In theframework of this model we can reproduce the observed correlation of mass anddensity of virialized galaxies and clusters of galaxies known as the virialparadox [4,5]. These correlations are closely linked with the composition of DMand the shape of the power spectrum what allows to restrict them using alreadyavailable observations. These correlations put constraints on the HDM and WDMmodels and allow to test models of cosmological inflation. We confirm that themissing satellite problem is directly linked with the virial paradox andreheating of the Universe which increases temperature and entropy of thebaryons, prevents formation of first stars and divides halos into twopopulations: the first one includes galaxies formed before reheating which aremainly concentrated in the vicinity of the massive ones while the secondpopulation -- numerous dark halos formed after reheating -- accumulatesmajority of DM but does not contain stars. Their spatial distribution is morehomogeneous.

The patch like model of galaxies formation: the virial paradox, core-cusp and missing satellite problems