Abstract
Using the Reduced Relativistic Gas (RRG) model, we analytically determine
the matter power spectrum for Warm Dark Matter (WDM) on small scales,
k > 1 h/Mpc. The RRG is a simplified model for the ideal relativistic
gas, but very accurate in the cosmological context. In another work,
we have shown that, for typical allowed masses for dark matter particles,
m>5 keV, the higher order multipoles, ℓ ≥ 2, in the Einstein-Boltzmann
system of equations are negligible on scales k < 10 h/Mpc.
Hence, we can follow the perturbations of WDM using the ideal fluid
framework, with equation of state and sound speed of perturbations
given by the RRG model. We derive a Mészáros-like equation for WDM
and solve it analytically in radiation, matter and dark energy dominated
eras. Joining these solutions, we get an expression that determines
the value of WDM perturbations as a function of redshift and wavenumber.
Then we construct the matter power spectrum and transfer function
of WDM on small scales and compare it to some results coming from
Lyman-α forest observations. Besides being a clear and pedagogical
analytical development to understand the evolution of WDM perturbations,
our power spectrum results are consistent with the observations considered
and the other determinations of the degree of warmness of dark matter particles.