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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.

Interpret $P(k)$ in the variance integral as the unsmoothed cosmological density power spectrum of $\delta$. A linear smoothing window gives $P_R(k)=P(k)|\tilde{W}(kR){|}^2$ for the smoothed field; the PDF's description of $P(k)$ as already belonging to ${\delta }_R$ would otherwise count the window twice.

For a scale-free matter power spectrum $P(k)=A{k}^{n_{\mathrm{eff}}}$, substitute $q=kR$ to obtainWhenever the dimensionless integral is finite and nonzero, this proves the scale-free smoothed density varianceThe printed condition $n_{\mathrm{eff}}<3$ is insufficient by itself. For a normalized window with $\tilde{W}(0)=1$, infrared convergence requires $n_{\mathrm{eff}}>-3$. Ultraviolet convergence depends on the filter: a spherical top-hat window function requires $n_{\mathrm{eff}}<1$, a Gaussian filter suppresses every ultraviolet power, and a sharp-k smoothing filter has finite Fourier support. In particular, the monotone relation $R\to \infty \Rightarrow S={\sigma }_R^2\to 0$ requires $n_{\mathrm{eff}}>-3$ in the scale-free model.

At a single variance $S_{\ast }$, the Gaussian random field has ${\delta }_{S_{\ast }}\sim N(0,S_{\ast })$. Thus its endpoint tail isHere $\mathrm{erfc}$ is the complementary error function.

For the crossing probability, the key assumption is independent, symmetric increments as the smoothing scale changes. A sharp-k smoothing filter, $\tilde{W}(kR)=1_{k<1/R}$, supplies this property: decreasing $R$ adds independent Fourier shells of the underlying Gaussian field. Parameterizing their accumulated variance by $S$ gives a Brownian motion with covariance $\min (S,S^{\prime })$. An arbitrary real-space window gives correlated increments and does not justify this argument.

Reflect a trajectory after its first hit of ${\delta }_c$. The Strong Markov property and symmetric future increments preserve its probability, and reflection maps an endpoint below the barrier to one above it. Every endpoint above the barrier has already crossed; the reflected paths give an equally probable set that crossed but ended below. The Brownian reflection principle therefore yields the excursion-set description of halo formation:Its derivative is the Brownian first-passage time densityThe factor of two resolves trajectories that crossed a larger-scale barrier but finish below it on the chosen smaller scale.

Define the halo peak height $\nu ={\delta }_c/{\sigma }_R$. The cumulative probability $\mathcal{P}(>M)$ is a dimensionless mass fraction. With $V_M=M/\bar{\rho }$, its conversion to halo number density givesConsequently the Press-Schechter halo mass function isIt is positive when ${\sigma }_R$ decreases with mass. A chosen mass-radius convention $M\propto R^3$ gives $d\log {\sigma }_R/d\log M=-(n_{\mathrm{eff}}+3)/6$. For a sharp Fourier filter that mass assignment needs a convention, since its real-space kernel is not a localized top-hat volume.

For warm dark matter, the cutoff introduces a new length and invalidates the pure power-law variance scaling near that length. With the same sharp-k filter and $n_{\mathrm{eff}}>-3$, the variance is exactlyThe Brownian walk runs only up to $S_{\max }$ and then stops. The cutoff does not destroy independence of the Fourier shells that are still present. Thus the first-crossing density and derivative mass-function formula remain valid on the invertible part of $S(M)$, using the actual cutoff variance. For $R<k_{\mathrm{WDM}}^{-1}$, $d{\sigma }_R/dM=0$ and this idealized sharp-filter model has no new crossings or new low-mass haloes. Its total collapsed fraction isThis is halo first crossing with a finite variance cutoff: some trajectories never cross, so one must not force the collapsed fraction to one. The uncut power-law abundance cannot be extrapolated to small masses. For other windows, a variance plateau still occurs but the Markov/reflection derivation is not exact; the same mass function is then a modelling approximation, rather than a result established by this calculation.

Set $\mathbf{q}={\mathbf{k}}_1$, $\mathbf{k}-\mathbf{q}={\mathbf{k}}_2$, $q=|\mathbf{q}|$, and $\mu =\hat{\mathbf{k}}\cdot \hat{\mathbf{q}}$. Substitution into $F_2=5{\alpha }_s/7+2\beta /7$ givesThe azimuthal integral in $d^{3}q=q^{2}dqd\mu d\varphi$ and the factor two in $P_{22}$ then give

For the scale-free matter power spectrum $P(k)=A{k}^n$, put $r=q/k$ andAll dimensional factors separate:where

As $r\to 0$, $K_{22}\sim r^n{\mu }^2/(8{\pi }^2)$, so the soft-$q$ integral converges exactly when $n>-1$. For $r\to \infty$, $K_{22}\sim r^{2n-2}$ times an angular function, requiring $n<1/2$. At fixed $\mu <1$, $r\to 1$ is regular. The corner $(r,\mu )=(1,1)$ makes $|\mathbf{k}-\mathbf{q}|$ soft; after resolving that corner in the local soft momentum, its radial behavior is again $p^{n}dp$, requiring $n>-1$. Thereforeis the infrared- and ultraviolet-convergence window for $P_{22}$.