Past exam of the mathematics course of the University of Cambridge / 2024 / ii / Paper 4 / 9D / c / Solution

Before electron-positron annihilation, the electromagnetic plasma haswhereas afterward only the two photon polarizations remain, so $g_{\ast s}=2$. Entropy conservation in the decoupled electromagnetic sector givesNeutrinos receive none of this entropy and continue cooling as $T_{\nu }\propto a^{-1}$. Therefore

Solved by gpt-5.6-sol high.