  We combine observed properties of galaxies as the core density and radius
with the theoretical linear evolution of density fluctuations computed from
first principles since the end of inflation till today. The halo radius r_0 is
computed in terms of cosmological parameters. The theoretical density profiles
rho(r)/rho(0) have an universal shape as a function of r/r_0 which reproduces
the observations. We show that the linear approximation to the Boltzmann-Vlasov
equation is valid for very large galaxies and correctly provides universal
quantities which are common to all galaxies, as the surface density and density
profile. By matching the theoretically computed surface density to its observed
value we obtain (i) the decreasing of the phase-space density during the MD era
(ii) the mass of the dark matter particle which turns to be between 1 and 2 keV
and the decoupling temperature T_d which turns to be above 100 GeV (iii) the
core vs. cusp discrimination: keV dark matter particles produce cored density
profiles while wimps (m \sim 100 GeV, T_d \sim 5 GeV) produce cusped profiles
at scales about 0.003 pc. These results are independent of the particle model
and vary very little with the statistics of the dark matter particle.
Non-universal galaxy quantities (which need to include non-linear effects as
mergers and baryons) are reproduced in the linear approximation up to a factor
of order one for the halo radius r_0, galaxy mass M_{gal}, halo central density
rho_{0} and velocity dispersion sqrt{{\bar {v^2}}_{halo}} in the limiting case
of large galaxies (both r_0 and M_{gal} large). This shows the power of the
linear approximation scheme: although it cannot capture the whole content of
the structure formation, it correctly provides universal quantities which as
well as the main non-universal galaxy properties.
