The expansion scale factor, $a(t)$, for an isotropic and spatially homogeneous universe containing material with pressure $p$ and mass density $\rho$ obeys the equations

$$\begin{gathered} \dot{\rho}+3(\rho+p) \frac{\dot{a}}{a}=0, \\ \left(\frac{\dot{a}}{a}\right)^{2}=\frac{8 \pi G \rho}{3}-\frac{k}{a^{2}}+\frac{\Lambda}{3}, \end{gathered}$$

where the speed of light is set equal to unity, $G$ is Newton's constant, $k$ is a constant equal to 0 or $\pm 1$, and $\Lambda$ is the cosmological constant. Explain briefly the interpretation of these equations.

Show that these equations imply

$$\frac{\ddot{a}}{a}=-\frac{4 \pi G(\rho+3 p)}{3}+\frac{\Lambda}{3} .$$

Hence show that a static solution with constant $a=a_{\mathrm{s}}$ exists when $p=0$ if

$$\Lambda=4 \pi G \rho=\frac{k}{a_{\mathrm{s}}^{2}}$$

What must the value of $k$ be, if the density $\rho$ is non-zero?