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

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The discussion focuses on deriving the scalings for the matter power spectrum, specifically the behavior of PΔ(k) in different regimes. For super horizon modes (k < keq), the scaling is PΔ(k) ∼ k, while for sub horizon modes (k > keq), it scales as PΔ(k) ∼ k^{-3}. The Mészáros equation indicates that in radiation domination, Δm grows logarithmically with scale factor a, while in matter domination, it grows linearly. The derivation involves applying the Poisson equation and considering the evolution of perturbations through horizon crossing, leading to a final expression that captures the transition between early and late-time behaviors. Ultimately, the scalings are confirmed as PΔ ∼ k at early times and PΔ ∼ k^{-3} at late times.
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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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