Past exam of the mathematics course of the University of Cambridge / 2018 / iii / Paper 310 / 3 / c

Before decay, the nonrelativistic $\chi$ density redshifts as $a^{-3}$, while the photon density redshifts as $a^{-4}$. Therefore their ratio grows as $a$. Let a minus sign denote the instant just before decay and defineUsing the assumed radiation domination law $a\propto t^{1/2}$ gives $r_d=(t_d/t_{\mathrm{ref}}{)}^{1/2}{\rho }_{\chi }(t_{\mathrm{ref}})/{\rho }_{\gamma }(t_{\mathrm{ref}})$.

Across an instantaneous decay the scale factor does not change. Energy conservation and complete thermalization into a zero-chemical-potential photon bath giveSince blackbody radiation has ${\rho }_{\gamma }\propto T_{\gamma }^4$, instantaneous photon heating by a decaying relic givesThe already decoupled Cosmic neutrino background is not heated. Dividing its unchanged temperature by the new photon temperature therefore yieldsThe relation written with $a_d/a_{\mathrm{ref}}$ is independent of the expansion approximation. Replacing it by the time ratio uses the radiation-dominated background assumed in the question; an appreciable matter contribution would require solving for that background instead. This photon entropy injection is not adiabatic, so the pre-decay photon entropy need not be conserved across the decay.