Instantaneous photon heating by a decaying relic ($T_{\gamma }^{+}/T_{\gamma }^{-}=(1+{\rho }_{\chi }/{\rho }_{\gamma }{)}^{1/4}$)

If a nonrelativistic relic decays entirely into promptly thermalized photons, energy conservation at fixed scale factor gives ${\rho }_{\gamma }^{+}={\rho }_{\gamma }^{-}+{\rho }_{\chi }^{-}$. Since thermal photons have ${\rho }_{\gamma }\propto T_{\gamma }^4$, their temperature rises by the displayed factor while an already decoupled Cosmic neutrino background is not heated. Before decay the relic-to-photon density ratio grows as $a$.