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We reexamine the evolution of thermal relic particle abundances for the casewhere the interaction rate depends on the particle velocities. For the case ofSommerfeld enhancement, we show that the standard analytic approximation,modified in a straightforward way, provides an estimate of the relic particleabundance that is accurate to within 10% (in comparison to less than 1% errorfor the non-Sommerfeld-enhanced case). We examine the effect of kineticdecoupling on relic particle abundances when the interaction rate depends onthe velocity. For the case of pure p-wave annihilation, the effect of kineticdecoupling is an increase in the relic abundance, but the effect is negligiblewhen the kinetic decoupling temperature is much less than the chemicaldecoupling temperature. For the case of Sommerfeld-enhanced s-waveannihilations, after kinetic decoupling occurs, annihilations continue tochange the particle abundance down to arbitrarily low temperatures, untileither matter domination begins or the Sommerfeld effect cuts off. We deriveanalytic approximations to give the final relic particle abundances for both ofthese cases.

Thermal Relic Abundances of Particles with Velocity-Dependent Interactions