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