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 less than 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.
