Past exam of the mathematics course of the University of Cambridge / 2019 / ii / Paper 4 / 9B (Cosmology)

For a relativistic species the supplied relation $P=ϵ/3$ makes its entropy density proportional to $T^3$. The fermion-to-boson thermal integral ratio gives the factor $7/8$ for each fermionic degree of freedom, so the electromagnetic plasma immediately before electron-positron annihilation in cosmology hasAfter annihilation only the two photon polarizations remain, so $g_{\ast s,\mathrm{after}}=2$.

The neutrinos have already undergone neutrino decoupling. They therefore receive none of the electron-positron entropy and cool as $T_{\nu }\propto a^{-1}$. In the still-coupled electromagnetic plasma, cosmological entropy conservation gives $g_{\ast s}{T}_{\gamma }^3{a}^3=\mathrm{constant}$. If $T_d$ and $a_d$ denote the common temperature and scale factor just before annihilation, then at a later scale factor $a$,Dividing gives the Cosmic neutrino background temperature ratio