Past exam of the mathematics course of the University of Cambridge / 2024 / iii / Paper 310 / 2 / b / Solution

Photons are bosons with two polarizations and temperature $T$, so ${\rho }_{\gamma }=({\pi }^2/30)2T^4$. Each relativistic neutrino species includes a neutrino and antineutrino helicity state, is fermionic, and has temperature $T_{\nu }=(4/11{)}^{1/3}T$. ConsequentlyAdding the two contributions gives the defining expression for the effective number of neutrino species: