 Potential fluctuations per logarithmic wavenumber

For $P_{\delta }(k)=A{k}^n$, the cosmological Poisson equation gives $P_{\phi }(k)=(4\pi G{a}^2\bar{\rho }{)}^2{P}_{\delta }(k)/k^4$. The contribution to variance per logarithmic wavenumber is $k^3{P}_{\phi }(k)/(2{\pi }^2)$, giving the displayed expression and rms scaling ${\phi }_{\mathrm{rms}}(R)\propto R^{(1-n)/2}$ for a band around $k\sim R^{-1}$. The Harrison-Zeldovich spectrum $n=1$ has equal potential variance per logarithmic interval. An uncut power law over all scales has divergent total potential variance for every $n$, so the scale-local amplitude must not be confused with a finite global rms.