Small scale aspects of warm dark matter : power spectra and acoustic
  oscillations
  We provide a semi-analytic study of the small scale aspects of the power
spectra of warm dark matter (WDM) candidates that decoupled while relativistic
with arbitrary distribution functions. These are characterized by two widely
different scales $k_{eq} \sim 0.01\,(\mathrm{Mpc})^{-1}$ and $k_{fs}=
\sqrt{3}\,k_{eq}/2\,< V^2_{eq} >^{1/2} $ with $< V^2_{eq} >^{1/2} \ll 1 $ the
velocity dispersion at matter radiation equality. Density perturbations evolve
through three stages: radiation domination when the particle is relativistic
and non-relativistic and matter domination. An early ISW effect during the
first stage leads to an enhancement of density perturbations and a plateau in
the transfer function for $k \lesssim k_{fs}$. An effective fluid description
emerges at small scales which includes the effects of free streaming in initial
conditions and inhomogeneities. The transfer function features
\emph{WDM-acoustic oscillations} at scales $k \gtrsim 2 \,k_{fs}$. We study the
power spectra for two models of sterile neutrinos with $m \sim \,\mathrm{keV}$
produced non-resonantly, at the QCD and EW scales respectively. The latter case
yields acoustic oscillations on mass scales $\sim 10^{8}\,M_{\odot}$. Our
results reveal a \emph{quasi-degeneracy} between the mass, distribution
function and decoupling temperature suggesting caveats on the constraints on
the mass of a sterile neutrino from current WDM N-body simulations and
Lyman-$\alpha$ forest data. A simple analytic interpolation of the power
spectra between large and small scales and its numerical implementation is
given.
