= Scale-free potential and bulk-flow variance
{title2=$\sigma_\Phi(R)\propto R^{(1-n)/2},\quad\sigma_v(R)\propto R^{-(n+1)/2}$}
For a <matter power spectrum> $P(k)=Ak^n$ at a fixed epoch, the <Poisson equation> gives potential Fourier amplitudes proportional to $\delta_k/k^2$, and the growing-mode <peculiar velocity> is proportional to $\delta_k\mathbf k/k^2$. With a <spherical top-hat window function>, their variances are proportional to $\int k^{n-2}W(kR)^2dk$ and $\int k^nW(kR)^2dk$, respectively. Rescaling $kR$ gives the displayed powers when the integrals converge: $1<n<5$ for the averaged potential and $-1<n<3$ for the bulk velocity. Absolute averaged potential has infrared sensitivity; band-limited fluctuations or potential differences can have different convergence requirements. The density variance converges only for $-3<n<1$, so these three absolute averages cannot all converge for one unbroken power law at every scale.
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