= Two-stage entropy dilution of an early-decoupled neutrino
{title2=$T_\nu/T_\gamma=[43/(11g_d)]^{1/3}$}
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.
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