Thermal relic abundances of particles with velocity-dependent interactions

We reexamine the evolution of thermal relic particle abundances for the case where the interaction rate depends on the particle velocities. For the case of Sommerfeld enhancement, we show that the standard analytic approximation, modified in a straightforward way, provides an estimate of the relic particle abundance that is accurate to within 10% (in comparison to <1%<1% error for the non-Sommerfeld-enhanced case). We examine the effect of kinetic decoupling on relic particle abundances when the interaction rate depends on the velocity. For the case of pure p-wave annihilation, the effect of kinetic decoupling is an increase in the relic abundance, but the effect is negligible when the kinetic decoupling temperature is much less than the chemical decoupling temperature. For the case of Sommerfeld-enhanced s-wave annihilations, after kinetic decoupling occurs, annihilations continue to change the particle abundance down to arbitrarily low temperatures, until either matter domination begins or the Sommerfeld effect cuts off. We derive analytic approximations to give the final relic particle abundances for both of these cases.