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 = \rho / ( \sigma _ { 1 D } ^ { 2 } ) ^ { 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 : \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 \times 10 ^ { -6 } ~ { } M _ { \odot } .