A solution to the Boltzmann equation governing the thermal relic abundance of cold dark matter is constructed by matched asymptotic approximations. The approximation of the relic density is an asymptotic series valid when the abundance does not deviate significantly from its equilibrium value until small temperatures. Resonance and threshold effects are taken into account at leading order and found to be negligible unless the annihilation cross section is negligible at threshold. Comparisons are made to previously attempted constructions and to the freeze out approximation commonly employed in the literature. Extensions to higher order matching is outlined, and implications for solving related systems are discussed. We compare our results to a numerical determination of the relic abundance using a benchmark model and find a fantastic agreement. The method developed also serves as a solution to a wide class of problems containing an infinite order turning point.
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