(i) Explain briefly how the relative motion of galaxies in a homogeneous and isotropic universe is described in terms of the scale factor $a(t)$ (where $t$ is time). In particular, show that the relative velocity $\mathbf{v}(t)$ of two galaxies is given in terms of their relative displacement $\mathbf{r}(t)$ by the formula $\mathbf{v}(t)=H(t) \mathbf{r}(t)$, where $H(t)$ is a function that you should determine in terms of $a(t)$. Given that $a(0)=0$, obtain a formula for the distance $R(t)$ to the cosmological horizon at time $t$. Given further that $a(t)=\left(t / t_{0}\right)^{\alpha}$, for $0<\alpha<1$ and constant $t_{0}$, compute $R(t)$. Hence show that $R(t) / a(t) \rightarrow 0$ as $t \rightarrow 0$.

(ii) A homogeneous and isotropic model universe has energy density $\rho(t) c^{2}$ and pressure $P(t)$, where $c$ is the speed of light. The evolution of its scale factor $a(t)$ is governed by the Friedmann equation

$$\dot{a}^{2}=\frac{8 \pi G}{3} \rho a^{2}-k c^{2}$$

where the overdot indicates differentiation with respect to $t$. Use the "Fluid" equation

$$\dot{\rho}=-3\left(\rho+\frac{P}{c^{2}}\right)\left(\frac{\dot{a}}{a}\right)$$

to obtain an equation for the acceleration $\ddot{a}(t)$. Assuming $\rho>0$ and $P \geqslant 0$, show that $\rho a^{3}$ cannot increase with time as long as $\dot{a}>0$, nor decrease if $\dot{a}<0$. Hence determine the late time behaviour of $a(t)$ for $k<0$. For $k>0$ show that an initially expanding universe must collapse to a "big crunch" at which $a \rightarrow 0$. How does $\dot{a}$ behave as $a \rightarrow 0$ ? Given that $P=0$, determine the form of $a(t)$ near the big crunch. Discuss the qualitative late time behaviour for $k=0$.

Cosmological models are often assumed to have an equation of state of the form $P=\sigma \rho c^{2}$ for constant $\sigma$. What physical principle requires $\sigma \leqslant 1$ ? Matter with $P=\rho c^{2}$ $(\sigma=1)$ is called "stiff matter" by cosmologists. Given that $k=0$, determine $a(t)$ for a universe that contains only stiff matter. In our Universe, why would you expect stiff matter to be negligible now even if it were significant in the early Universe?