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The formation and decay of metastable bound states can significantly decreasethe thermal-relic dark matter density, particularly for dark matter massesaround and above the TeV scale. Incorporating bound-state effects in the darkmatter thermal decoupling requires in principle a set of coupled Boltzmannequations for the bound and unbound species. However, decaying bound statesattain and remain in a quasi-steady state. Here we prove in generality thatthis reduces the coupled system into a single Boltzmann equation of thestandard form, with an effective cross-section that describes the interplayamong bound-state formation, ionisation, transitions and decays. We derive aclosed-form expression for the effective cross-section for an arbitrary numberof bound states, and show that bound-to-bound transitions can only increase it.Excited bound levels may thus decrease the dark matter density moresignificantly than otherwise estimated. Our results generalise the Sahaionisation equilibrium to metastable bound states, potentially withapplications beyond the dark matter thermal decoupling.

Saha equilibrium for metastable bound states and dark matter freeze-out