Describe the motion of light rays in an expanding universe with scale factor $a(t)$, and derive the redshift formula

$$1+z=\frac{a\left(t_{0}\right)}{a\left(t_{\mathrm{e}}\right)},$$

where the light is emitted at time $t_{\mathrm{e}}$ and observed at time $t_{0}$.

A galaxy at comoving position $\mathbf{x}$ is observed to have a redshift $z$. Given that the galaxy emits an amount of energy $L$ per unit time, show that the total energy per unit time crossing a sphere centred on the galaxy and intercepting the earth is $L /(1+z)^{2}$. Hence, show that the energy per unit time per unit area passing the earth is

$$\frac{L}{(1+z)^{2}} \frac{1}{4 \pi|\mathbf{x}|^{2} a^{2}\left(t_{0}\right)}$$