Dependence of freeze-out spectra on Γ/H during radiation-dominated expansion

Investigate the dependence of thermalization and the resulting freeze-out momentum spectrum on the ratio Γ/H, where Γ ≡ n⟨σv⟩ is the interaction rate and H is the Hubble rate, in a radiation-dominated cosmological expansion; determine how different Γ/H regimes shape the momentum-resolved distribution governed by the Boltzmann equation and identify the conditions under which freeze-out occurs.

Background

In cosmological settings, thermalization proceeds only if the interaction rate Γ ≡ n⟨σv⟩ exceeds the Hubble expansion rate H; otherwise, expansion redshifts momenta more rapidly than collisions can redistribute them, leading to freeze-out.

The authors note that their framework supports radiation-dominated expansion and can explicitly track the competition between Γ and H, but they explicitly defer a systematic study of how Γ/H controls the freeze-out spectrum, identifying this as future work.