  We study the effects of modifying the expansions history of the Universe on
Dark Matter freezeout. We derived a modified Boltzmann equation for freeze-out
for an arbitrary energy density in the early Universe and provide an analytic
approach using some approximations. We then look at the required thermally
averaged cross sections needed to obtain the correct relic density for the
specific case where the energy density consists of radiation plus one extra
component which cools faster. We compare our analytic approximation to a
numerical solutions. We find that it gives reasonable results for most of the
parameter space explored, being at most a factor of order one away from the
measured value. We find that if the new contribution to the energy density is
comparable to the radiation density, then a much smaller cross section for Dark
Matter annihilation is required. This would lead to weak scale Dark Matter
being much more difficult to detect and opens up the possibility that much
heavier Dark Matter could undergo freezeout without violating perturbative
unitarity.
