(i) Consider a one-dimensional model universe with "stars" distributed at random on the $x$-axis, and choose the origin to coincide with one of the stars; call this star the "homestar." Home-star astronomers have discovered that all other stars are receding from them with a velocity $v(x)$, that depends on the position $x$ of the star. Assuming non-relativistic addition of velocities, show how the assumption of homogeneity implies that $v(x)=H_{0} x$ for some constant $H_{0}$.

In attempting to understand the history of their one-dimensional universe, homestar astronomers seek to determine the velocity $v(t)$ at time $t$ of a star at position $x(t)$. Assuming homogeneity, show how $x(t)$ is determined in terms of a scale factor $a(t)$ and hence deduce that $v(t)=H(t) x(t)$ for some function $H(t)$. What is the relation between $H(t)$ and $H_{0}$ ?

(ii) Consider a three-dimensional homogeneous and isotropic universe with mass density $\rho(t)$, pressure $p(t)$ and scale factor $a(t)$. Given that $E(t)$ is the energy in volume $V(t)$, show how the relation $d E=-p d V$ yields the "fluid" equation

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

where $H=\dot{a} / a$.

Show how conservation of energy applied to a test particle at the boundary of a spherical fluid element yields the Friedmann equation

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

for constant $k$. Hence obtain an equation for the acceleration $\ddot{a}$ in terms of $\rho, p$ and $a$.

A model universe has mass density and pressure

$$\rho=\frac{\rho_{0}}{a^{3}}+\rho_{1}, \quad p=-\rho_{1} c^{2},$$

where $\rho_{0}$ is constant. What does the fluid equation imply about $\rho_{1}$ ? Show that the acceleration $\ddot{a}$ vanishes if

$$a=\left(\frac{\rho_{0}}{2 \rho_{1}}\right)^{\frac{1}{3}}$$

Hence show that this universe is static and determine the sign of the constant $k$.