Past exam of the general relativity and cosmology course of the University of Oxford / 2018 / 4 / b / Solution

Let $V$ be proper volume, let $g_i$ count the populated internal states, and take ${\mu }_i=0$ and $ϵ=pc$. Spherical integration in momentum space givesUsing the fermion-to-boson thermal integral ratio, the minus-denominator boson integral is $2\zeta (3)$ and the plus-denominator fermion integral is $3\zeta (3)/2$. HenceThe energy integral has one extra factor $pc=k_{\mathrm{B}}Tx$:The supplied distribution identity gives the fermion integral as $(1-2^{-3}){\pi }^4/15=7{\pi }^4/120$. ThereforeFor an isotropic ultrarelativistic gas, momentum flux gives $P_i=\frac{1}{3}\int pv{f}_i{d}^{3}p$. Since $v=c$ and $pc=ϵ$,Photons have $g_{\gamma }=2$, electrons and Positrons each have $g=2$, and each neutrino or antineutrino species has one populated helicity in the standard thermal bath.