Temperature of a decoupled relativistic relic

If a relativistic species decouples when the coupled plasma has $g_{\ast s,\mathrm{dec}}$ entropy degrees of freedom, then after the plasma reaches $g_{\ast s}$ degrees of freedom,The ratio follows from separate redshifting of the decoupled species and cosmological entropy conservation in the coupled plasma.

An extra relativistic neutrino that decouples before ordinary neutrinos misses two stages of plasma heating. Until ordinary decoupling, entropy conservation gives a temperature ratio $(43/(4g_d){)}^{1/3}$. Electron-positron annihilation then heats photons but neither neutrino population, multiplying the ratio by $(4/11{)}^{1/3}$. Here $g_d$ counts the coupled bath at the extra species' decoupling, with no subsequent exotic entropy injection.