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Thermal photon number density is proportional to $T^3$. A sudden photon heating factor $(1+r{)}^{1/4}$ therefore raises photon number by $(1+r{)}^{3/4}$ and dilutes the baryon-to-photon ratio by its inverse when baryon number is conserved. Comparing Big Bang nucleosynthesis and Cosmic microwave background baryon-density determinations can test entropy injection between the two epochs, subject to changes in the expansion rate and nuclear abundances.

For $T\ll m_e$ and negligible chemical potentials, the combined equilibrium density of electrons and positrons divided by the photon number density isThe exponential suppression eventually makes pair equilibrium incompatible with the residual electrons required by charge neutrality.

The photon number density in a thermal zero-chemical-potential bath scales as $T_{\gamma }^3$. The heating factor from the previous calculation therefore increases photon number at fixed physical volume by $(1+r_d{)}^{3/4}$. The decay is nonbaryonic and preserves baryon number, giving baryon-to-photon dilution by entropy injection:If the decay occurs after the light-element abundances have frozen out in Big Bang nucleosynthesis but before the epoch probed by the Cosmic microwave background, abundance measurements infer the earlier ${\eta }_{\mathrm{BBN}}$, whereas the CMB measures the later ${\eta }_{\mathrm{CMB}}$. In this simple scenario,A significant discrepancy in this direction could indicate photon entropy injection by $\chi$. Several light-element abundances help distinguish a changed baryon density from a changed expansion history. If decay occurs during nucleosynthesis, the inference instead requires time-dependent $\eta$ and a modified expansion history; the before/after comparison cannot automatically be applied to every decay time allowed in the question.

The reaction maintaining chemical equilibrium isFor $T<m_e$ and negligible chemical potentials, the Electron and Positron are dilute and obey the nonrelativistic equilibrium number density. Including two spin states for each particle,The photon number density is $n_{\gamma }=2\zeta (3)T^3/{\pi }^2$, so the electron-positron equilibrium pair abundance below the electron mass isIf $n_e$ denotes only one of the charge-conjugate species, this expression is divided by two.