 Solution

In the nonrelativistic dilute regime, $E(p)=m{c}^2+p^2/(2m)+\cdots$ and the Fermi-Dirac distribution reduces to the Maxwell-Boltzmann distribution. HenceDifferentiating the Gaussian integral with respect to its coefficient gives ${\int }_0^{\infty }{p}^2{e}^{-a{p}^2}dp=\sqrt{\pi }/(4a^{3/2})$. ThereforeThe degeneracy factor here is the one in the supplied integral, namely one. Substituting $kT=m{c}^2/\alpha$, $\mu =0$ and dividing by the supplied photon density givesAt $\alpha =20$ this is about $4.8\times 10^{-8}$. The relic is already nonrelativistic at decoupling and becomes still colder as its momenta redshift, making it a possible cold dark matter component if it is stable and interacts sufficiently weakly with light.

For orientation, write ${\eta }_b=n_b/n_{\gamma }$. Ignoring subsequent entropy transfer, its density relative to baryons would be ${\rho }_X/{\rho }_b=(1/20)(n/n_{\gamma })/{\eta }_b$. With the cosmological baryon-to-photon ratio of order $6\times 10^{-10}$, this is of order four, demonstrating the right cosmological scale. The measured dark-to-baryonic density ratio is likewise of order five; an observational reference is arxiv.org/abs/1807.06209 . This is a candidate abundance argument, not an exact relic-density calculation: photon heating after decoupling changes $n/n_{\gamma }$, and stability and interaction properties were not specified. The supplied mass and freeze-out parameter alone therefore do not establish a precise fit.