The linearised equation for the growth of density perturbations, $\delta_{\mathbf{k}}$, in an isotropic and homogenous universe is

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

where $\rho$ is the density of matter, $c_{s}$ the sound speed, $c_{s}^{2}=d P / d \rho$, and $\mathbf{k}$ is the comoving wavevector and $a(t)$ is the scale factor of the universe.

What is the Jean's length? Discuss its significance for the growth of perturbations.

Consider a universe filled with pressure-free matter with $a(t)=\left(t / t_{0}\right)^{2 / 3}$. Compute the resulting equation for the growth of density perturbations. Show that your equation has growing and decaying modes and comment briefly on the significance of this fact.