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

TL;DR

This paper generalizes the Saha equilibrium to metastable bound states, deriving an effective cross-section that simplifies the calculation of dark matter freeze-out, highlighting the importance of bound-to-bound transitions in reducing dark matter density.

Contribution

It provides a unified framework to incorporate bound-state effects into dark matter freeze-out calculations using a single Boltzmann equation with an effective cross-section.

Findings

Bound-to-bound transitions increase the effective cross-section.

Excited bound states can significantly reduce dark matter density.

The Saha equilibrium is generalized for metastable bound states.

Abstract

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.