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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.

A relativistic species decoupling before ordinary neutrino decoupling cools as $T_X\propto a^{-1}$. Cosmological entropy conservation in the remaining bath gives $T_X/T_{\nu }=[10.75/g_{\ast s,D}{]}^{1/3}$, where $g_{\ast s,D}$ counts that bath just after the early species has left it. Subsequent electron-positron annihilation in cosmology gives $T_{\nu }/T_{\gamma }=(4/11{)}^{1/3}$. For the ideal high-temperature relativistic entropy content of the Standard Model, the resulting ratio is about $0.332$.

In natural units with unreduced Planck mass $M_P=G^{-1/2}$, the radiation-era Hubble parameter scales as $\sqrt{g_{\ast }}{T}^2/M_P$, while a low-energy weak interaction rate scales as $G_F^2{T}^5$. Their equality gives a decoupling scale of order one MeV. Distinct weak processes have distinct rate coefficients: neutrino decoupling and cosmological weak freeze-out of neutron-proton conversion are nearby events, not an identical precise temperature inferred from the power laws alone.