For a homogeneous and isotropic universe filled with pressure-free matter $(P=0)$, the Friedmann and Raychaudhuri equations are, respectively,

$$\left(\frac{\dot{a}}{a}\right)^{2}+\frac{k c^{2}}{a^{2}}=\frac{8 \pi G}{3} \rho \quad \text { and } \quad \frac{\ddot{a}}{a}=-\frac{4 \pi G}{3} \rho,$$

with mass density $\rho$, curvature $k$, and where $\dot{a} \equiv d a / d t$. Using conformal time $\tau$ with $d \tau=d t / a$, show that the relative density parameter can be expressed as

$$\Omega(t) \equiv \frac{\rho}{\rho_{\text {crit }}}=\frac{8 \pi G \rho a^{2}}{3 \mathcal{H}^{2}}$$

where $\mathcal{H}=\frac{1}{a} \frac{d a}{d \tau}$ and $\rho_{\text {crit }}$ is the critical density of a flat $k=0$ universe (Einstein-de Sitter). Use conformal time $\tau$ again to show that the Friedmann and Raychaudhuri equations can be re-expressed as

$$\frac{k c^{2}}{\mathcal{H}^{2}}=\Omega-1 \quad \text { and } \quad 2 \frac{d \mathcal{H}}{d \tau}+\mathcal{H}^{2}+k c^{2}=0$$

From these derive the evolution equation for the density parameter $\Omega$ :

$$\frac{d \Omega}{d \tau}=\mathcal{H} \Omega(\Omega-1)$$

Plot the qualitative behaviour of $\Omega$ as a function of time relative to the expanding Einsteinde Sitter model with $\Omega=1$ (i.e., include curves initially with $\Omega>1$ and $\Omega<1$ ).