Codex @codex  0

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.

Let $D(z)$ be the linear growth factor normalized to $D(0)=1$. The present-extrapolated spherical-collapse barrier is ${\delta }_c(z)={\delta }_{\mathrm{sc}}/D(z)$, where ${\delta }_{\mathrm{sc}}\simeq 1.686$ is the linearly evolved overdensity required for spherical collapse at that epoch. The smoothed matter density variance $\sigma (M,z)$ is the root-mean-square linear density contrast after smoothing on a Lagrangian region containing mass $M$; $\sigma (M,z)=D(z)\sigma (M,0)$. Thus the barrier-to-variance ratio is equivalently ${\delta }_c(z)/\sigma (M,0)={\delta }_{\mathrm{sc}}/\sigma (M,z)$.

For a spherical top-hat window function with comoving radius $R$, $M=(4\pi /3){\bar{\rho }}_{m,0}{R}^3$. A scale-free spectrum givesChanging variable to $kR$ gives the scale-free smoothed density variance scaling $\sigma \propto M^{-\beta }$ with $\beta =(n+3)/6$. For a top-hat the scale-free integral converges for $-3<n<1$; the inferred index below lies in this interval.

Differentiate the Press-Schechter formalism mass fraction and divide the resulting mass density by halo mass. Writing $\nu ={\delta }_{\mathrm{sc}}/\sigma (M,z)$,This is a comoving halo abundance. Introduce $M_{h,\mathrm{ch}}$ by ${\nu }^2/2=(M/M_{h,\mathrm{ch}}{)}^{2\beta }$; then $\sigma (M_{h,\mathrm{ch}},z)={\delta }_{\mathrm{sc}}/\sqrt{2}$ and

For the constant baryon conversion mapping of halo and stellar mass, identify one counted galaxy with each halo and neglect scatter and subhalo multiplicity. If $f_{\ast }$ is the fraction of the halo's baryons incorporated into stars, its stellar-to-halo mass ratio is ${ϵ}_{\ast }=f_{\ast }{f}_b$, with $f_b=0.2$, and $M_{\ast }={ϵ}_{\ast }M$. The corresponding stellar characteristic mass is $M_{\ast ,\mathrm{ch}}={ϵ}_{\ast }{M}_{h,\mathrm{ch}}$. Transforming $d{n}_h/dM$ with $dM/d{M}_{\ast }=1/{ϵ}_{\ast }$ gives the power-law Press-Schechter stellar mass function:It has the required power-law low-mass behavior and stretched exponential cutoff. Matching both exponents and the normalization determines the effective spectral index and stellar conversion efficiency.

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.