The total energy of a gas can be expressed in terms of a momentum integral

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

where $p$ is the particle momentum, $\mathcal{E}(p)=c \sqrt{p^{2}+m^{2} c^{2}}$ is the particle energy and $\bar{n}(p) d p$ is the average number of particles in the momentum range $p \rightarrow p+d p$. Consider particles in a cubic box of side $L$ with $p \propto L^{-1}$. Explain why the momentum varies as

$$\frac{d p}{d V}=-\frac{p}{3 V}$$

Consider the overall change in energy $d E$ due to the volume change $d V$. Given that the volume varies slowly, use the thermodynamic result $d E=-P d V$ (at fixed particle number $N$ and entropy $S$ ) to find the pressure

$$P=\frac{1}{3 V} \int_{0}^{\infty} p \mathcal{E}^{\prime}(p) \bar{n}(p) d p .$$

Use this expression to derive the equation of state for an ultrarelativistic gas.

During the radiation-dominated era, photons remain in equilibrium with energy density $\epsilon_{\gamma} \propto T^{4}$ and number density $n_{\gamma} \propto T^{3}$. Briefly explain why the photon temperature falls inversely with the scale factor, $T \propto a^{-1}$. Discuss the implications for photon number and entropy conservation.