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Cosmological evolution of warm dark matter fluctuations II: Solution from small to large scales and keV sterile neutrinos

We solve the cosmological evolution of warm dark matter (WDM) density fluctuations with the Volterra integral equations of paper I. In the absence of neutrinos, the anisotropic stress vanishes and the Volterra equations reduce to a single integral equation. We solve numerically this equation both for DM fermions decoupling at equilibrium and DM sterile neutrinos decoupling out of equilibrium. We give the exact analytic solution for the density fluctuations and gravitational potential at zero wavenumber. We compute the density contrast as a function of the scale factor a for a wide range of wavenumbers k. At fixed a, the density contrast grows with k for k < k_c while it decreases for k > k_c, (k_c ~ 1.6/Mpc). The density contrast depends on k and a mainly through the product k a exhibiting a self-similar behavior. Our numerical density contrast for small k gently approaches our analytic solution for k = 0. For fixed k < 1/(60 kpc), the density contrast generically grows with a while for k > 1/(60 kpc) it exhibits oscillations since the RD era which become stronger as k grows. We compute the transfer function of the density contrast for thermal fermions and for sterile neutrinos in: a) the Dodelson-Widrow (DW) model and b) in a model with sterile neutrinos produced by a scalar particle decay. The transfer function grows with k for small k and then decreases after reaching a maximum at k = k_c reflecting the time evolution of the density contrast. The integral kernels in the Volterra equations are nonlocal in time and their falloff determine the memory of the past evolution since decoupling. This falloff is faster when DM decouples at equilibrium than when it decouples out of equilibrium. Although neutrinos and photons can be neglected in the MD era, they contribute in the MD era through their memory from the RD era.