Evolution of the phase-space density and the Jeans scale for dark matter
  derived from the Vlasov-Einstein equation
  We discuss solutions of Vlasov-Einstein equation for collisionless dark
matter particles in the context of a flat Friedmann universe. We show that,
after decoupling from the primordial plasma, the dark matter phase-space
density indicator Q remains constant during the expansion of the universe,
prior to structure formation. This well known result is valid for
non-relativistic particles and is not "observer dependent" as in solutions
derived from the Vlasov-Poisson system. In the linear regime, the inclusion of
velocity dispersion effects permits to define a physical Jeans length for
collisionless matter as function of the primordial phase-space density
indicator: \lambda_J = (5\pi/G)^(1/2)Q^(-1/3)\rho_dm^(-1/6). The comoving Jeans
wavenumber at matter-radiation equality is smaller by a factor of 2-3 than the
comoving wavenumber due to free-streaming, contributing to the cut-off of the
density fluctuation power spectrum at the lowest scales. We discuss the
physical differences between these two scales. For dark matter particles of
mass equal to 200 GeV, the derived Jeans mass is 4.3 x 10^(-6) solar masses.
