The mass density perturbation equation for non-relativistic matter $\left(P \ll \rho c^{2}\right)$ with wave number $k$ in the late universe $\left(t>t_{\mathrm{eq}}\right)$ is

$$\ddot{\delta}+2 \frac{\dot{a}}{a} \dot{\delta}-\left(4 \pi G \rho-\frac{c_{s}^{2} k^{2}}{a^{2}}\right) \delta=0 .$$

Suppose that a non-relativistic fluid with the equation of state $P \propto \rho^{4 / 3}$ dominates the universe when $a(t)=t^{2 / 3}$, and the curvature and the cosmological constant can be neglected. Show that the sound speed can be written in the form $c_{s}^{2}(t) \equiv d P / d \rho=$ $\bar{c}_{s}^{2} t^{-2 / 3}$ where $\bar{c}_{s}$ is a constant.

Find power-law solutions to $(*)$ of the form $\delta \propto t^{\beta}$ and hence show that the general solution is

$$\delta=A_{k} t^{n_{+}}+B_{k} t^{n_{-}}$$

where

$$n_{\pm}=-\frac{1}{6} \pm\left[\left(\frac{5}{6}\right)^{2}-\bar{c}_{s}^{2} k^{2}\right]^{1 / 2}$$

Interpret your solutions in the two regimes $k \ll k_{J}$ and $k \gg k_{J}$ where $k_{J}=\frac{5}{6 \bar{c}_{s}}$.