Derive the relation between the neutrino temperature $T_{\nu}$ and the photon temperature $T_{\gamma}$ at a time long after electrons and positrons have become non-relativistic.

[In this question you may work in units of the speed of light, so that $c=1$. You may also use without derivation the following formulae. The energy density $\epsilon_{a}$ and pressure $P_{a}$ for a single relativistic species a with a number $g_{a}$ of degenerate states at temperature $T$ are given by

$$\epsilon_{a}=\frac{4 \pi g_{a}}{h^{3}} \int \frac{p^{3} d p}{e^{p /\left(k_{B} T\right)} \mp 1}, \quad \quad P_{a}=\frac{4 \pi g_{a}}{3 h^{3}} \int \frac{p^{3} d p}{e^{p /\left(k_{B} T\right)} \mp 1},$$

where $k_{B}$ is Boltzmann's constant, $h$ is Planck's constant, and the minus or plus depends on whether the particle is a boson or a fermion respectively. For each species a, the entropy density $s_{a}$ at temperature $T_{a}$ is given by,

$$s_{a}=\frac{\epsilon_{a}+P_{a}}{k_{B} T_{a}} .$$

The effective total number $g_{*}$ of relativistic species is defined in terms of the numbers of bosonic and fermionic particles in the theory as,

$$g_{*}=\sum_{\text {bosons }} g_{\text {bosons }}+\frac{7}{8} \sum_{\text {fermions }} g_{\text {fermions }}$$

with the specific values $g_{\gamma}=g_{e^{+}}=g_{e^{-}}=2$ for photons, positrons and electrons.]