Past exam of the mathematics course of the University of Cambridge / 2020 / ii / Paper 3 / 9D / Solution

For equal internal degeneracy, the number density of an isotropic ultrarelativistic species is proportional toWith $x=\beta pc$, compare the Fermi-Dirac distribution and Bose-Einstein distribution usingFor $s>0$, a change of variables in the second term gives the fermion-to-boson thermal integral ratioParticle number has $s=2$, so $n_f=(1-2^{-2})n_b=3n_b/4$. The energy density contains one further factor $E=pc$, so it has $s=3$ and