 Solution

Interpret the requested scale-dependent rms as the contribution from a logarithmic band of wavelengths, or equivalently a band centered at $k\sim R^{-1}$. A global uncut rms is not finite for this scale-free spectrum. The dimensionless density power per logarithmic wavenumber isThe cosmological Poisson equation gives ${\phi }_{\mathbf{k}}=-(4\pi G{a}^2\bar{\rho }){\delta }_{\mathbf{k}}/k^2$. ThereforeThis is the potential fluctuations per logarithmic wavenumber scaling. A simple dimensional argument gives the same answer: the density amplitude on scale $R$ is proportional to $R^{-(n+3)/2}$, and its potential amplitude is of order $G\bar{\rho }(aR{)}^2{\delta }_R$, supplying two additional powers of $R$.

For $n=1$, the Harrison-Zeldovich spectrum has scale-independent potential amplitude per logarithmic interval. If the density-spectrum amplitude grows as $A(t)=b^2(t)A_{\mathrm{in}}$, then $4\pi G{a}^2\bar{\rho }=C/a$, so the potential amplitude evolves as $b/a$. It is constant in the pure growing matter mode of an Einstein-de Sitter universe, where $b=a$.

A literal total variance would be proportional to ${\int }_0^{\infty }{k}^{n-2}dk$. Infrared convergence requires $n>1$ and ultraviolet convergence requires $n<1$, so no index makes both ends finite. At $n=1$ both divergences are logarithmic. Thus cutoffs or a band specification are essential for a numerical rms; the boxed statement describes the intended scale-local fluctuations rather than claiming a finite unsmoothed potential variance.