The formation and decay of metastable bound states can significantly decrease the thermal-relic dark matter density, particularly for dark matter masses around and above the TeV scale. Incorporating bound-state effects in the dark matter thermal decoupling requires in principle a set of coupled Boltzmann equations for the bound and unbound species. However, decaying bound states attain and remain in a quasi-steady state. Here we prove in generality that this reduces the coupled system into a single Boltzmann equation of the standard form, with an effective cross-section that describes the interplay among bound-state formation, ionisation, transitions and decays. We derive a closed-form expression for the effective cross-section for an arbitrary number of bound states, and show that bound-to-bound transitions can only increase it. Excited bound levels may thus decrease the dark matter density more significantly than otherwise estimated. Our results generalise the Saha ionisation equilibrium to metastable bound states, potentially with applications beyond the dark matter thermal decoupling.

亚稳态束缚态的形成和衰减可以显着降低热残留暗物质密度，特别是对于 TeV 尺度附近和以上的暗物质质量。在暗物质热解耦中结合束缚态效应原则上需要一组用于束缚和未束缚物种的耦合玻尔兹曼方程。然而，衰减束缚态达到并保持在准稳态。在这里，我们一般性地证明，这将耦合系统简化为标准形式的单个玻尔兹曼方程，其有效截面描述了束缚态形成、电离、跃迁和衰变之间的相互作用。我们推导出任意数量束缚态的有效横截面的封闭式表达式，并表明束缚到束缚的转换只能增加它。因此，激发的束缚能级可能比其他估计的更显着地降低暗物质密度。我们的结果将 Saha 电离平衡推广到亚稳态束缚态，可能会应用于暗物质热解耦之外的应用。