The energy density of a particle species is defined by

$$\epsilon=\int_{0}^{\infty} E(p) n(p) d p$$

where $E(p)=c \sqrt{p^{2}+m^{2} c^{2}}$ is the energy, and $n(p)$ the distribution function, of a particle with momentum $p$. Here $c$ is the speed of light and $m$ is the rest mass of the particle. If the particle species is in thermal equilibrium then the distribution function takes the form

$$n(p)=\frac{4 \pi}{h^{3}} g \frac{p^{2}}{\exp ((E(p)-\mu) / k T) \mp 1}$$

where $g$ is the number of degrees of freedom of the particle, $T$ is the temperature, $h$ and $k$ are constants and $-$ is for bosons and $+$ is for fermions.

(a) Stating any assumptions you require, show that in the very early universe the energy density of a given particle species $i$ is

$$\epsilon_{i}=\frac{4 \pi g_{i}}{(h c)^{3}}(k T)^{4} \int_{0}^{\infty} \frac{y^{3}}{e^{y} \mp 1} d y$$

(b) Show that the total energy density in the very early universe is

$$\epsilon=\frac{4 \pi^{5}}{15(h c)^{3}} g^{*}(k T)^{4}$$

where $g^{*}$ is defined by

$$g^{*} \equiv \sum_{\text {Bosons }} g_{i}+\frac{7}{8} \sum_{\text {Fermions }} g_{i}$$

[Hint: You may use the fact that $\left.\int_{0}^{\infty} y^{3}\left(e^{y}-1\right)^{-1} d y=\pi^{4} / 15 .\right]$