I came across this equation while studying about Dark Matter relic density but I can't seem to prove it.

In the early universe, the number density of dark matter in the thermal bath is approximately given by $$ n(T) = \frac{g}{(2 \pi)^3} \int d^3p e^{-\frac{E}{T}} = \frac{g}{(2 \pi)^3} \int d^3p e^{-\frac{\sqrt{m^2 + p^2}}{T}} $$ where $E$ is energy, $T$ is temperature, $g$ is the degrees of freedom.

The solution is $$ n(T)= \frac{1}{2\pi^2}gm^2TK_2(\frac{m}{T})$$ where $K_n(x)$ is the modified Bessel function of the second kind.

All the resources I have referred state the solution like it is trivial but I wasn't able to solve it. Can someone give me a more detailed solution for it? Thanks!



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