Consider a homogenous and isotropic universe with mass density $\rho(t)$, pressure $P(t)$ and scale factor $a(t)$. As the universe expands its energy changes according to the relation $d E=-P d V$. Use this to derive the fluid equation

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

Use conservation of energy applied to a test particle at the boundary of a spherical fluid element to derive the Friedmann equation

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

where $k$ is a constant. State any assumption you have made. Briefly state the significance of $k$.