Warm dark matter at small scales: peculiar velocities and phase space
  density
  We study the scale and redshift dependence of the power spectra for density
perturbations and peculiar velocities, and the evolution of a coarse grained
phase space density for (WDM) particles that decoupled during the radiation
dominated stage. The (WDM) corrections are obtained in a perturbative expansion
valid in the range of redshifts at which N-body simulations set up initial
conditions, and for a wide range of scales. The redshift dependence is
determined by the kurtosis $\beta_2$ of the distribution function at
decoupling. At large redshift there is an enhancement of peculiar velocities
for $\beta_2 > 1$ that contributes to free streaming and leads to further
suppression of the matter power spectrum and an enhancement of the peculiar
velocity autocorrelation function at scales smaller than the free streaming
scale. Statistical fluctuations of peculiar velocities are also suppressed on
these scales by the same effect. In the linearized approximation, the coarse
grained phase space density features redshift dependent (WDM) corrections from
gravitational perturbations determined by the power spectrum of density
perturbations and $\beta_2$. For $\beta_2 > 25/21$ it \emph{grows
logarithmically} with the scale factor as a consequence of the suppression of
statistical fluctuations. Two specific models for WDM are studied in detail.
The (WDM) corrections relax the bounds on the mass.
