(a) Consider the motion of three galaxies $O, A, B$ at positions $\mathbf{r}_{O}, \mathbf{r}_{A}, \mathbf{r}_{B}$ in an isotropic and homogeneous universe. Assuming non-relativistic velocities $\mathbf{v}(\mathbf{r})$, show that spatial homogeneity implies

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

that is, that the velocity field $\mathbf{v}$ is linearly related to $\mathbf{r}$ by

$$v_{i}=\sum_{j} H_{i j} r_{j},$$

where the matrix coefficients $H_{i j}$ are independent of $\mathbf{r}$. Further show that isotropy implies Hubble's law,

$$\mathbf{v}=H \mathbf{r} \text {, }$$

where the Hubble parameter $H$ is independent of $\mathbf{r}$. Presuming $H$ to be a function of time $t$, show that Hubble's law can be integrated to obtain the solution

$$\mathbf{r}(t)=a(t) \mathbf{x}$$

where $\mathbf{x}$ is a constant (comoving) position and the scalefactor $a(t)$ satisfies $H=\dot{a} / a$.

(b) Define the cosmological horizon $d_{H}(t)$. For models with $a(t)=t^{\alpha}$ where $0<\alpha<1$, show that the cosmological horizon $d_{H}(t)=c t /(1-\alpha)$ is finite. Briefly explain the horizon problem.