In an expanding spacetime, the density contrast $\delta(\mathbf{x}, t)$ satisfies the linearised equation

$$\ddot{\delta}+2 H \dot{\delta}-c_{s}^{2}\left(\frac{1}{a^{2}} \nabla^{2}+k_{J}^{2}\right) \delta=0,$$

where $a$ is the scale factor, $H$ is the Hubble parameter, $c_{s}$ is a constant, and $k_{J}$ is the Jeans wavenumber, defined by

$$c_{s}^{2} k_{J}^{2}=\frac{4 \pi G}{c^{2}} \bar{\rho}(t)$$

with $\bar{\rho}(t)$ the background, homogeneous energy density.

(i) Solve for $\delta(\mathbf{x}, t)$ in a static universe, with $a=1$ and $H=0$ and $\bar{\rho}$ constant. Identify two regimes: one in which sound waves propagate, and one in which there is an instability.

(ii) In a matter-dominated universe with $\bar{\rho} \sim 1 / a^{3}$, use the Friedmann equation $H^{2}=8 \pi G \bar{\rho} / 3 c^{2}$ to find the growing and decaying long-wavelength modes of $\delta$ as a function of $a$.

(iii) Assuming $c_{s}^{2} \approx c_{s}^{2} k_{J}^{2} \approx 0$ in equation $(*)$, find the growth of matter perturbations in a radiation-dominated universe and find the growth of matter perturbations in a curvature-dominated universe.