Effects of cold dark matter decoupling and pair annihilation on cosmological perturbations

Weakly interacting massive particles (WIMPs) are part of the lepton-photon plasma in the early universe until kinetic decoupling, after which time the particles behave like a collisionless gas with nonzero temperature. The Boltzmann equation for WIMP-lepton collisions is reduced to a Fokker-Planck equation for the evolution of the WIMP distribution including scalar density perturbations. This equation and the Einstein and fluid equations for the plasma are solved numerically including the acoustic oscillations of the plasma before and during kinetic decoupling, the frictional damping occurring during kinetic decoupling, and the free-streaming damping occurring afterwards and throughout the radiation-dominated era. An excellent approximation reduces the solution to quadratures for the cold dark matter density and velocity perturbations. The subsequent evolution is followed through electron pair annihilation and the radiation-matter transition; analytic solutions are provided for both large and small scales. For a 100 GeV WIMP with bino-type interactions, kinetic decoupling occurs at a temperature $T_d=23\mathrm{MeV}$. The transfer function in the matter-dominated era leads to an abundance of small cold dark matter halos; with a smooth window function the Press-Schechter mass distribution is $dn/d\ln M\propto M^{-1/3}$ for $M<{10}^{-4}(T_d/10\mathrm{MeV}{)}^{-3}{\mathrm{M}}_{\odot }$.