The number density of photons in thermal equilibrium at temperature $T$ takes the form

$$n=\frac{8 \pi}{c^{3}} \int \frac{\nu^{2} d \nu}{\exp (h \nu / k T)-1}$$

At time $t=t_{\mathrm{dec}}$ and temperature $T=T_{\mathrm{dec}}$, photons decouple from thermal equilibrium. By considering how the photon frequency redshifts as the universe expands, show that the form of the equilibrium frequency distribution is preserved, with the temperature for $t>t_{\mathrm{dec}}$ defined by

$$T \equiv \frac{a\left(t_{\mathrm{dec}}\right)}{a(t)} T_{\mathrm{dec}}$$

Show that the photon number density $n$ and energy density $\epsilon$ can be expressed in the form

$$n=\alpha T^{3}, \quad \epsilon=\xi T^{4},$$

where the constants $\alpha$ and $\xi$ need not be evaluated explicitly.