Monotonic photon cooling during pair annihilation ($d\ln T_{\gamma }/d\ln a=-3/[3+d\ln g_{\ast s}/d\ln T_{\gamma }]$)

Electromagnetic comoving entropy conservation gives $a^3{g}_{\ast s}(T_{\gamma })T_{\gamma }^3$ constant. Differentiation proves the displayed relation. During equilibrium electron-positron annihilation, $g_{\ast s}$ increases with temperature, so the absolute photon temperature still falls as space expands. The comoving temperature $a{T}_{\gamma }$ rises, and the asymptotic relativistic-to-pair-free entropy count heats photons relative to decoupled neutrinos by $(11/4{)}^{1/3}$. This relative heating must not be mistaken for an absolute temperature rise along the smooth expanding history.