Relic cooling across muon annihilation ($T_H/T_{\gamma }=(43/57{)}^{1/3}$)

A relativistic relic that decouples between pion and muon annihilation keeps $a{T}_H$ constant. The coupled bath initially contains photons, electron-positron pairs, muon-antimuon pairs and three neutrino-antineutrino species, with entropy degrees of freedom $57/4$. After muon annihilation this is $43/4$, giving the displayed temperature ratio. After neutrino decoupling, electron-positron annihilation heats photons by $(11/4{)}^{1/3}$, so the final relic-to-photon ratio becomes $(172/627{)}^{1/3}$. This counting assumes the species are relativistic when counted and treats annihilations and decoupling as sharp equilibrium transitions.