Baryon-to-photon dilution by entropy injection (${\eta }^{+}/{\eta }^{-}=(1+{\rho }_{\chi }/{\rho }_{\gamma }{)}^{-3/4}$)

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.