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Choose the Fourier transform conventionStatistical homogeneity makes the Fourier covariance diagonal, and isotropy makes the matter power spectrum depend only on $k=|\mathbf{k}|$. The two-point correlation function $\xi (r)=⟨\delta (\mathbf{x})\delta (\mathbf{x}+\mathbf{r})⟩$ follows by inserting the transform and using this covariance:Integrating the polar angle relative to $\mathbf{r}$ gives $\int d{\Omega }_k{e}^{ikr\cos \theta }=4\pi \sin (kr)/(kr)$. HenceThe inverse isotropic transform is $P(k)=4\pi {\int }_0^{\infty }{r}^2\xi (r)\sin (kr)/(kr)dr$ whenever these transforms exist, with distributions or regularization otherwise.

For a spherical average of comoving radius $R$, the spherical top-hat window function isAt a fixed epoch let $P(k)=A{k}^n$. The scale-free smoothed density variance isThus the RMS fractional mass fluctuation scales as $R^{-(n+3)/2}$; the RMS mass fluctuation itself scales as $\bar{\rho }{R}^3{\sigma }_{\delta }\propto R^{(3-n)/2}$ at fixed epoch.

The comoving Poisson equation gives ${\Phi }_{\mathbf{k}}=-4\pi G{a}^2\bar{\rho }{\delta }_{\mathbf{k}}/k^2$. For the growing irrotational mode the linearized cosmological continuity equation gives ${\mathbf{v}}_{\mathbf{k}}=-iaHf\mathbf{k}{\delta }_{\mathbf{k}}/k^2$, where $f=d\log D_{+}/d\log a$. The averaged-potential and bulk-velocity variances are thereforeFor a single Cartesian component of the isotropic bulk velocity, divide its variance by three. Rescaling $kR$ givesThese are dimensional scaling laws, with an important convergence qualification. Since $W(y)\to 1$ at small $y$ and has large-$y$ envelope $O(y^{-2})$, the density average converges for $-3<n<1$, the absolute potential average for $1<n<5$, and the bulk velocity for $-1<n<3$. No one unbroken power law makes all three absolute averages finite on the entire range $0<k<\infty$. In particular, long-wavelength modes can dominate the potential. Physical infrared/ultraviolet cutoffs, potential differences, or fluctuation amplitudes in a band around $k\sim R^{-1}$ resolve the relevant divergences; with cutoffs the simple powers need not describe the full integrated variance. The scale-free potential and bulk-flow variance laws should be understood with these conditions, not as a claim that divergent RMS integrals are finite.