(i) A homogeneous and isotropic universe has mass density $\rho(t)$ and scale factor $a(t)$. Show how the conservation of total energy (kinetic plus gravitational potential) when applied to a test particle on the edge of a spherical region in this universe can be used to obtain the Friedmann equation

$$H^{2} \equiv\left(\frac{\dot{a}}{a}\right)^{2}=\frac{8 \pi G}{3} \rho-\frac{k c^{2}}{a^{2}},$$

where $k$ is a constant. State clearly any assumptions you have made.

(ii) Assume that the universe is flat $(k=0)$ and filled with two major components: pressure-free matter $\left(P_{\mathrm{M}}=0\right)$ and dark energy with equation of state $P_{\Lambda}=-\rho_{\Lambda} c^{2}$ where their mass densities today $\left(t=t_{0}\right)$ are given respectively by $\rho_{\mathrm{M} 0}$ and $\rho_{\Lambda 0}$. Assuming that each component independently satisfies the fluid conservation equation, $\dot{\rho}=-3 H\left(\rho+P / c^{2}\right)$, show that the total mass density can be expressed as

$$\rho(t)=\frac{\rho_{\mathrm{M} 0}}{a^{3}}+\rho_{\Lambda 0}$$

where we have set $a\left(t_{0}\right)=1$.

Hence, solve the Friedmann equation and show that the scale factor can be expressed in the form

$$a(t)=\alpha(\sinh \beta t)^{2 / 3},$$

where $\alpha$ and $\beta$ are constants which you should specify in terms of $\rho_{\mathrm{M} 0}, \rho_{\Lambda 0}$ and $t_{0}$.

[Hint: try the substitution $b=a^{3 / 2}$.]

Show that the scale factor $a(t)$ has the expected behaviour for a matter-dominated universe at early times $(t \rightarrow 0)$ and that the universe accelerates at late times $(t \rightarrow \infty)$.