Analytical warm dark matter power spectrum on small scales
  Using the Reduced Relativistic Gas (RRG) model, we analytically determine the
matter power spectrum for Warm Dark Matter (WDM) on small scales, $k>1\
h\text{/Mpc}$. 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, $m>5\ \text{keV}$,
the higher order multipoles, $\ell>2$, in the Einstein-Boltzmann system of
equations are negligible on scales $k<10\ h\text{/Mpc}$. 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
M\'esz\'aros 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-$\alpha$ 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.
