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At neutrino decoupling, neutrinos, electrons, positrons, and photons share one temperature. The neutrinos subsequently free stream, so $T_{\nu }\propto a^{-1}$. In the still-coupled electromagnetic plasma, the effective entropy degrees of freedom change during electron-positron annihilation in cosmology fromto $g_{\ast s,\mathrm{after}}=2$. Separate cosmological entropy conservation in that plasma gives $g_{\ast s}{T}_{\gamma }^3{a}^3=\mathrm{constant}$, while $T_{\nu }a$ remains constant. Consequently the Cosmic neutrino background temperature obeys

Before electron-positron annihilation in cosmology, photons and electron-positron pairs share a temperature. Their effective entropy degrees of freedom areAfter annihilation the electromagnetic plasma contains only the two photon polarizations, so $g_{\ast s}^{\mathrm{after}}=2$. The already decoupled neutrinos receive none of this entropy and simply cool as $T_{\nu }\propto a^{-1}$. Applying cosmological entropy conservation separately to the coupled electromagnetic plasma giveswhereas $T_{\nu }^3{a}^3=T_{\mathrm{before}}^3{a}_{mbefore}^3$. Division yields the Cosmic neutrino background temperature