I'm slightly confused about the power spectrum of matter

[ergospherical]

How do you get these scalings for the matter power spectrum?$$P_{\Delta}(k) \sim \begin{cases} k & \quad k < k_{\mathrm{eq}} \\ k^{-3} & \quad k >k_{\mathrm{eq}} \end{cases}$$(N.B. $k_{\mathrm{eq}}$ is the scale of modes that enter the horizon $k \sim \mathcal{H}$ at matter-radiation equality. So the first scaling is for comfortably super horizon modes and the second is for comfortably sub horizon modes).

Take the sub horizon case, i.e. modes where $k \gg k_{\mathrm{eq}}$, as an example. From the Mészáros equation we know that for these modes we have $\Delta_m \sim \log a$ in radiation domination (early times) and $\Delta_m \sim a$ in matter domination (late times). How do I use this to deduce that $P_{\Delta}(k) \sim k^{-3}$?

 

[ergospherical]

I've gotten an answer back -- on super horizon scales we know that $\delta \sim \phi \sim \mathcal{R}$, and scale invariance of perturbations in $\mathcal{R}$ constrains $P_{\mathcal{R}} \sim k^{n_s - 4}$, so by the Poisson equation:
$$\Delta \sim (k/\mathcal{H})^2 \phi_k \implies P_{\Delta} \big{|}_{\tau = \tau_i} \sim k^4 \tau_i^4 P_{\Phi} \sim k^{n_s}$$at some initial time $\tau_i$. The perturbations evolve as $\sim \tau^2$ up until horizon crossing, so you have something like$$P_{\Delta} \sim (\tau/\tau_i)^4 P_{\Delta} \big{|}_{\tau = \tau_i} \sim k^{n_s}$$At horizon crossing, but still in radiation era, you have log growth, followed by $\tau^2$ growth again in the matter era. So the result is you pick up two more "transfer" factors:

$$P_{\Delta} = \left(\frac{\tau}{\tau_i}\right)^4 \left( 1 + \log{\frac{\tau_{eq}}{\tau_{cross}}} \right)^2 \left( \frac{\tau_{cross}}{\tau_i} \right)^4 P_{\Delta} \big{|}_{\tau = \tau_i} \sim (\log{k})^2 k^{n_s - 4}$$since at horizon crossing $k \sim \mathcal{H} \sim \tau^{-1}$ (so $\tau_{cross} \sim k^{-1}$)

So taking $n_s \approx 1$, you reproduce approximately the scalings $P_{\Delta} \sim k$ at early times and $P_{\Delta} \sim k^{-3}$ at late times.