In a homogeneous and isotropic universe, the scale factor $a(t)$ obeys the Friedmann equation

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

where $\rho$ is the matter density, which, together with the pressure $P$, satisfies

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

Here, $k$ is a constant curvature parameter. Use these equations to show that the rate of change of the Hubble parameter $H=\dot{a} / a$ satisfies

$$\dot{H}+H^{2}=-\frac{4 \pi G}{3}\left(\rho+3 P / c^{2}\right)$$

Suppose that an expanding Friedmann universe is filled with radiation (density $\rho_{R}$ and pressure $\left.P_{R}=\rho_{R} c^{2} / 3\right)$ as well as a "dark energy" component (density $\rho_{\Lambda}$ and pressure $\left.P_{\Lambda}=-\rho_{\Lambda} c^{2}\right)$. Given that the energy densities of these two components are measured today $\left(t=t_{0}\right)$ to be

$$\rho_{R 0}=\beta \frac{3 H_{0}^{2}}{8 \pi G} \quad \text { and } \quad \rho_{\Lambda 0}=\frac{3 H_{0}^{2}}{8 \pi G} \quad \text { with constant } \beta>0 \quad \text { and } \quad a\left(t_{0}\right)=1,$$

show that the curvature parameter must satisfy $k c^{2}=\beta H_{0}^{2}$. Hence derive the following relations for the Hubble parameter and its time derivative:

$$\begin{aligned} H^{2} &=\frac{H_{0}^{2}}{a^{4}}\left(\beta-\beta a^{2}+a^{4}\right) \\ \dot{H} &=-\beta \frac{H_{0}^{2}}{a^{4}}\left(2-a^{2}\right) \end{aligned}$$

Show qualitatively that universes with $\beta>4$ will recollapse to a Big Crunch in the future. [Hint: Sketch $a^{4} H^{2}$ and $a^{4} \dot{H}$ versus $a^{2}$ for representative values of $\beta$.]

For $\beta=4$, find an explicit solution for the scale factor $a(t)$ satisfying $a(0)=0$. Find the limiting behaviours of this solution for large and small $t$. Comment briefly on their significance.