In a homogeneous and isotropic universe, describe the relative displacement $\mathbf{r}(t)$ of two galaxies in terms of a scale factor $a(t)$. Show how the relative velocity $\mathbf{v}(t)$ of these galaxies is given by the relation $\mathbf{v}(t)=H(t) \mathbf{r}(t)$, where you should specify $H(t)$ in terms of $a(t)$.

From special relativity, the Doppler shift of light emitted by a particle moving away radially with speed $v$ can be approximated by

$$\frac{\lambda_{0}}{\lambda_{\mathrm{e}}}=\sqrt{\frac{1+v / c}{1-v / c}}=1+\frac{v}{c}+\mathcal{O}\left(\frac{v^{2}}{c^{2}}\right)$$

where $\lambda_{e}$ is the wavelength of emitted light and $\lambda_{0}$ is the observed wavelength. For the observed light from distant galaxies in a homogeneous and isotropic expanding universe, show that the redshift defined by $1+z \equiv \lambda_{0} / \lambda_{\mathrm{e}}$ is given by

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

where $t_{\mathrm{e}}$ is the time of emission and $t_{0}$ is the observation time.