The energy density $\epsilon$ and pressure $P$ of photons in the early universe is given by

$$\epsilon=\frac{4 \sigma}{c} T^{4}, \quad P=\frac{1}{3} \epsilon,$$

where $\sigma$ is the Stefan-Boltzmann constant. By using the first law of thermodynamics $d E=T d S-P d V+\mu d N$, deduce that the entropy differential $d S$ can be expressed in the form

$$d S=\frac{16 \sigma}{3 c} d\left(T^{3} V\right)$$

With the third law, show that the entropy density is given by $s=(16 \sigma / 3 c) T^{3}$.

While particle interaction rates $\Gamma$ remain much greater than the Hubble parameter $H$, justify why entropy will be conserved during the expansion of the universe. Hence, in the early universe (radiation domination) show that the temperature $T \propto a^{-1}$ where $a(t)$ is the scale factor of the universe, and show that the Hubble parameter $H \propto T^{2}$.