Small scale aspects of warm dark matter: Power spectra and acoustic oscillations

We provide a semianalytic derivation of approximate evolution equations for density perturbations of warm dark matter candidates that decoupled while relativistic with arbitrary distribution functions, their solutions at small scales, and a simple numerical implementation that yields their transfer functions and power spectra. Density perturbations evolve through three stages: radiation domination when the particle is relativistic and nonrelativistic and matter domination. An early integrated Sachs-Wolfe effect during the first stage leads to an enhancement of density perturbations and a plateau in the transfer function for kkfs, the free-streaming wave vector. An effective fluid description emerges at small scales which includes the effects of free streaming in initial conditions and inhomogeneities. The transfer function features warm dark matter acoustic oscillations at scales k2kfs. A simple analytic interpolation of the power spectra between large and small scales and a numerical implementation valid for arbitrary distribution functions is provided. As an application we study the power spectra for two models of sterile neutrinos with m~keV produced nonresonantly and compare our results to those obtained from Boltzmann codes.