Small density perturbations $\delta_{\mathbf{k}}(t)$ in pressureless matter inside the cosmological horizon obey the following Fourier evolution equation

$$\ddot{\delta}_{\mathbf{k}}+2 \frac{\dot{a}}{a} \dot{\delta}_{\mathbf{k}}-4 \pi G \bar{\rho}_{\mathrm{c}} \delta_{\mathbf{k}}=0$$

where $\bar{\rho}_{\mathrm{c}}$ is the average background density of the pressureless gravitating matter and $\mathbf{k}$ is the comoving wavevector.

(i) Seek power law solutions $\delta_{\mathbf{k}} \propto t^{\beta}(\beta$ constant) during the matter-dominated epoch $\left(t_{\mathrm{eq}}<t<t_{0}\right)$ to find the approximate solution

$$\delta_{\mathbf{k}}(t)=A(\mathbf{k})\left(\frac{t}{t_{\mathrm{eq}}}\right)^{2 / 3}+B(\mathbf{k})\left(\frac{t}{t_{\mathrm{eq}}}\right)^{-1}, \quad t \gg t_{\mathrm{eq}}$$

where $A, B$ are functions of $\mathbf{k}$ only and $t_{\mathrm{eq}}$ is the time of equal matter-radiation.

By considering the behaviour of the scalefactor $a$ and the relative density $\bar{\rho}_{\mathrm{c}} / \bar{\rho}_{\text {total }}$, show that early in the radiation era $\left(t \ll t_{\mathrm{eq}}\right)$ there is effectively no significant perturbation growth in $\delta_{\mathbf{k}}$ on sub-horizon scales.

(ii) For a given wavenumber $k=|\mathbf{k}|$, show that the time $t_{\mathrm{H}}$ at which this mode crosses inside the horizon, i.e., $c t_{\mathrm{H}} \approx 2 \pi a\left(t_{\mathrm{H}}\right) / k$, is given by

$$\frac{t_{\mathrm{H}}}{t_{0}} \approx \begin{cases}\left(\frac{k_{0}}{k}\right)^{3}, & t_{\mathrm{H}} \gg t_{\mathrm{eq}} \\ \left(1+z_{\mathrm{eq}}\right)^{-1 / 2}\left(\frac{k_{0}}{k}\right)^{2}, & t_{\mathrm{H}} \ll t_{\mathrm{eq}}\end{cases}$$

where $k_{0} \equiv 2 \pi /\left(c t_{0}\right)$, and the equal matter-radiation redshift is given by $1+z_{\mathrm{eq}}=$ $\left(t_{0} / t_{\mathrm{eq}}\right)^{2 / 3}$.

Assume that primordial perturbations from inflation are scale-invariant with a constant amplitude as they cross the Hubble radius given by $\left\langle\left|\delta_{\mathbf{k}}\left(t_{\mathrm{H}}\right)\right|^{2}\right\rangle \approx V^{-1} A / k^{3}$, where $A$ is a constant and $V$ is a large volume. Use the results of (i) to project these perturbations forward to $t_{0}$, and show that the power spectrum for perturbations today will be given approximately by

$$P(k) \equiv V\left\langle\left|\delta_{\mathbf{k}}\left(t_{0}\right)\right|^{2}\right\rangle \approx \frac{A}{k_{0}^{4}} \times \begin{cases}k, & k<k_{\mathrm{eq}} \\ k_{\mathrm{eq}}\left(\frac{k_{\mathrm{eq}}}{k}\right)^{3}, & k>k_{\mathrm{eq}}\end{cases}$$