Past exam of the mathematics course of the University of Cambridge / 2019 / iii / Paper 310 / 2 / b / Solution

The reaction maintaining chemical equilibrium isFor $T<m_e$ and negligible chemical potentials, the electron and Positron are dilute and obey the nonrelativistic equilibrium number density. Including two spin states for each particle,The photon number density is $n_{\gamma }=2\zeta (3)T^3/{\pi }^2$, so the electron-positron equilibrium pair abundance below the electron mass isIf $n_e$ denotes only one of the charge-conjugate species, this expression is divided by two.