Beyond the standard $\Lambda$CDM cosmology: the observed structure of DM
  halos and the shape of the power spectrum
  Recent advances in observational astronomy allow to study various groups of
Dark Matter (DM) dominated objects from the dwarf spheroidal (dSph) galaxies to
clusters of galaxies that span the mass range from $10^{6} M_{\odot}$ to
$10^{15} M_{\odot}$. To analyze data of this divers collection of objects we
used a simple toy model of spherical DM halo formation that was initially
proposed by Peebles. This model introduced the concept of the epoch or redshift
of halo formation. Using this concept we analyzed selected sample of DM
dominated objects and we have found empirical correlations between the virial
mass, $M_{vir}$, of halos and basic parameters of their cores, namely, the mean
DM density, pressure and entropy. These correlations are a natural result of
similar evolution of all such objects. It is driven mainly by gravitational
interactions what implies a high degree of self similarity of both the process
of halos formation and their internal structure.
  We confirmed the CDM--like shape of both the small and large scale power
spectrum. However, our reconstruction of the evolutionary history of observed
objects differs from expectations of the standard $\Lambda$CDM cosmology and
requires either a multicomponent composition of DM or a more complex primordial
power spectrum of density perturbations with significant excess of power at
scales of clusters of galaxies and larger. We demonstrated that a model with
suitable combination of the heavy DM particles (CDM) and DM particles with
large damping scale (HDM) could provide a successful description of the
observational data in a wide range of masses.
