I I'm slightly confused about the power spectrum of matter

ergospherical
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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}##?
 
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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.
 
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