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

Revisiting the averaged annihilation rate of thermal relics at low temperature