Number density of a relativistic neutrino relative to photons ($n_{\nu }/n_{\gamma }=\frac{3g_{\nu }}{8}(T_{\nu }/T_{\gamma }{)}^3$)

With negligible chemical potential, the Fermi-Dirac distribution has $3/4$ of the boson number density per degree of freedom at equal temperature. A species with $g_{\nu }$ populated states therefore has $n_{\nu }/n_{\gamma }=(3g_{\nu }/8)(T_{\nu }/T_{\gamma }{)}^3$, since photons have two states. This remains the number-density ratio after the neutrino becomes massive if its comoving particle number is conserved.