The number density of photons in equilibrium at temperature $T$ is given by

$$n=\frac{8 \pi}{(h c)^{3}} \int_{0}^{\infty} \frac{\nu^{2} d \nu}{e^{\beta h \nu}-1}$$

where $\beta=1 /\left(k_{B} T\right)\left(k_{B}\right.$ is Boltzmann's constant). Show that $n \propto T^{3}$. Show further that $\epsilon \propto T^{4}$, where $\epsilon$ is the photon energy density.

Write down the Friedmann equation for the scale factor $a(t)$ of a flat homogeneous and isotropic universe. State the relation between $a$ and the mass density $\rho$ for a radiation-dominated universe and hence deduce the time-dependence of $a$. How does the temperature $T$ depend on time?