Light of wavelength $\lambda_{e}$ emitted by a distant object is observed by us to have wavelength $\lambda_{0}$. The redshift $z$ of the object is defined by

$$1+z=\frac{\lambda_{0}}{\lambda_{e}}$$

Assuming that the object is at a fixed comoving distance from us in a homogeneous and isotropic universe with scale factor $a(t)$, show that

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

where $t_{e}$ is the time of emission and $t_{0}$ the time of observation (i.e. today).

[You may assume the non-relativistic Doppler shift formula $\Delta \lambda / \lambda=(v / c) \cos \theta$ for the shift $\Delta \lambda$ in the wavelength of light emitted by a nearby object travelling with velocity $v$ at angle $\theta$ to the line of sight.]

Given that the object radiates energy $L$ per unit time, explain why the rate at which energy passes through a sphere centred on the object and intersecting the Earth is $L /(1+z)^{2}$.