(a) The equilibrium distribution for the energy density of a massless neutrino takes the form

$$\epsilon=\frac{4 \pi c}{h^{3}} \int_{0}^{\infty} \frac{p^{3} d p}{\exp (p c / k T)+1} .$$

Show that this can be expressed in the form $\epsilon=\alpha T^{4}$, where the constant $\alpha$ need not be evaluated explicitly.

(b) In the early universe, the entropy density $s$ at a temperature $T$ is $s=$ $(8 \sigma / 3 c) \mathcal{N}_{S} T^{3}$ where $\mathcal{N}_{S}$ is the total effective spin degrees of freedom. Briefly explain why $\mathcal{N}_{S}=\mathcal{N}_{*}+\mathcal{N}_{S D}$, each term of which consists of two separate components as follows: the contribution from each massless species in equilibrium $\left(T_{i}=T\right)$ is

$$\mathcal{N}_{*}=\sum_{\text {bosons }} g_{i}+\frac{7}{8} \sum_{\text {fermions }} g_{i}$$

and a similar sum for massless species which have decoupled,

$$\mathcal{N}_{S D}=\sum_{\text {bosons }} g_{i}\left(\frac{T_{i}}{T}\right)^{3}+\frac{7}{8} \sum_{\text {fermions }} g_{i}\left(\frac{T_{i}}{T}\right)^{3}$$

where in each case $g_{i}$ is the degeneracy and $T_{i}$ is the temperature of the species $i$.

The three species of neutrinos and antineutrinos decouple from equilibrium at a temperature $T \approx 1 \mathrm{MeV}$, after which positrons and electrons annihilate at $T \approx 0.5 \mathrm{MeV}$, leaving photons in equilibrium with a small excess population of electrons. Using entropy considerations, explain why the ratio of the neutrino and photon temperatures today is given by

$$\frac{T_{\nu}}{T_{\gamma}}=\left(\frac{4}{11}\right)^{1 / 3}$$