Revisiting the averaged annihilation rate of thermal relics at low temperature

We derive a low-temperature expansion of the formula to compute the average annihilation rate $\langle \sigma v \rangle$ for dark matter in $\mathbb{Z}_2$-symmetric models, both in the absence and the presence of mass degeneracy in the spectrum near the dark matter candidate. We show that the result obtained in the absence of mass degeneracy is compatible with the analytic formulae in the literature, and that it has a better numerical behaviour for low temperatures. We also provide as ancillary files two Wolfram Mathematica notebooks which perform the two expansions at any order.