A universe contains baryonic matter with background density $\rho_{B}(t)$ and density inhomogeneity $\delta_{B}(\mathbf{x}, t)$, together with non-baryonic dark matter with background density $\rho_{D}(t)$ and density inhomogeneity $\delta_{D}(\mathbf{x}, t)$. After the epoch of radiation-matter density equality, $t_{\mathrm{eq}}$, the background dynamics are governed by

$$H=\frac{2}{3 t} \quad \text { and } \quad \rho_{D}=\frac{1}{6 \pi G t^{2}}$$

where $H$ is the Hubble parameter.

The dark-matter density is much greater than the baryonic density $\left(\rho_{D} \gg \rho_{B}\right)$ and so the time-evolution of the coupled density perturbations, at any point $\mathbf{x}$, is described by the equations

$$\begin{aligned} &\ddot{\delta}_{B}+2 H \dot{\delta}_{B}=4 \pi G \rho_{D} \delta_{D} \\ &\ddot{\delta}_{D}+2 H \dot{\delta}_{D}=4 \pi G \rho_{D} \delta_{D} \end{aligned}$$

Show that

$$\delta_{D}=\frac{\alpha}{t}+\beta t^{2 / 3}$$

where $\alpha$ and $\beta$ are independent of time. Neglecting modes in $\delta_{D}$ and $\delta_{B}$ that decay with increasing time, show that the baryonic density inhomogeneity approaches

$$\delta_{B}=\beta t^{2 / 3}+\gamma$$

where $\gamma$ is independent of time.

Briefly comment on the significance of your calculation for the growth of baryonic density inhomogeneities in the early universe.