The linearised equation for the growth of small inhomogeneous density perturbations $\delta_{\mathbf{k}}$ with comoving wavevector $\mathbf{k}$ in an isotropic and homogeneous universe is

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

where $\rho$ is the matter density, $c_{s}=(d P / d \rho)^{1 / 2}$ is the sound speed, $P$ is the pressure, $a(t)$ is the expansion scale factor of the unperturbed universe, and overdots denote differentiation with respect to time $t$.

Define the Jeans wavenumber and explain its physical meaning.

Assume the unperturbed Friedmann universe has zero curvature and cosmological constant and it contains only zero-pressure matter, so that $a(t)=a_{0} t^{2 / 3}$. Show that the solution for the growth of density perturbations is given by

$$\delta_{\mathbf{k}}=A(\mathbf{k}) t^{2 / 3}+B(\mathbf{k}) t^{-1}$$

Comment briefly on the cosmological significance of this result.