Evolution of the phase-space density and the Jeans scale for dark matter derivedfrom 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 = ρ/(σ1D2)3/2 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: λJ = (5π/G)1/2Q−1/3ρdm−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 × 10−6M⊙.