(a) Consider a homogeneous and isotropic universe with a uniform distribution of galaxies. For three galaxies at positions $\mathbf{r}_{A}, \mathbf{r}_{B}, \mathbf{r}_{C}$, show that spatial homogeneity implies that their non-relativistic velocities $\mathbf{v}(\mathbf{r})$ must satisfy

$$\mathbf{v}\left(\mathbf{r}_{B}-\mathbf{r}_{A}\right)=\mathbf{v}\left(\mathbf{r}_{B}-\mathbf{r}_{C}\right)-\mathbf{v}\left(\mathbf{r}_{A}-\mathbf{r}_{C}\right)$$

and hence that the velocity field coordinates $v_{i}$ are linearly related to the position coordinates $r_{j}$ via

$$v_{i}=H_{i j} r_{j}$$

where the matrix coefficients $H_{i j}$ are independent of the position. Show why isotropy then implies Hubble's law

$$\mathbf{v}=H \mathbf{r}, \quad \text { with } H \text { independent of } \mathbf{r}$$

Explain how the velocity of a galaxy is determined by the scale factor $a$ and express the Hubble parameter $H_{0}$ today in terms of $a$.

(b) Define the cosmological horizon $d_{H}(t)$. For an Einstein-de Sitter universe with $a(t) \propto t^{2 / 3}$, calculate $d_{H}\left(t_{0}\right)$ at $t=t_{0}$ today in terms of $H_{0}$. Briefly describe the horizon problem of the standard cosmology.