The number density $n=N / V$ for a photon gas in equilibrium is given by

$$n=\frac{8 \pi}{c^{3}} \int_{0}^{\infty} \frac{\nu^{2}}{e^{h \nu / k T}-1} d \nu$$

where $\nu$ is the photon frequency. By letting $x=h \nu / k T$, show that

$$n=\alpha T^{3}$$

where $\alpha$ is a constant which need not be evaluated.

The photon entropy density is given by

$$s=\beta T^{3}$$

where $\beta$ is a constant. By considering the entropy, explain why a photon gas cools as the universe expands.