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For statistical homogeneity and statistical isotropy, with the paper's finite-volume Fourier normalization,The angular integral is $4\pi \sin (kr)/(kr)$. In particular $\xi (r)=\int {\Delta }^2(k)\sin (kr)/(kr)d\ln k$, confirming the stated dimensionless power spectrum normalization.

For the scale-free matter power spectrum $P(k)=C{k}^n$, set $u=kr$ to obtainAt zero the integrand behaves as $u^{n+2}$, requiring $n>-3$; at infinity its oscillatory integral converges for $n<-1$. Thus in the ordinary convergent range $-3<n<-1$, the scale-free density correlation transform hasThe Gamma function expression follows by introducing $e^{-ϵu}$, evaluating the complex Laplace integral where it converges absolutely at zero, and continuing within the convergent oscillatory range; its removable singularity at $n=-2$ is handled by the limit. Absolute convergence of the original correlation integral requires the smaller range $-3<n<-2$.

The exponent relation is a useful scaling statement beyond that interval, but a pure primordial $n\simeq 1$ spectrum over all wavenumbers has no ordinary unsmoothed Fourier correlation integral. Physical transfer functions, cutoffs or smoothing must be specified, and a distributional extension need not yield a positive $r_0$ power law. Even within the convergent interval, the point variance is ultraviolet divergent without smoothing; the finite result here is at nonzero separation.

For a representative present-day ordinary galaxy sample over roughly $0.1$–$10h^{-1}\mathrm{Mpc}$, the familiar approximate fit iswhere $H_0=100h\mathrm{km}{\mathrm{s}}^{-1}\mathrm{Mp}{\mathrm{c}}^{-1}$. These are empirical galaxy clustering parameters, not universal matter-spectrum constants; luminosity, color, selection and scale change the fit, as shown in the galaxy correlation measurements. A formal power law of that slope corresponds to $n\simeq -1.2$ on those scales, not to the nearly scale-invariant primordial spectral index.