Press-Schechter formalism

The Press-Schechter formalism estimates the halo abundance from the Gaussian probability that a smoothed linear overdensity crosses the spherical-collapse threshold, with a factor of two enforcing total mass conservation.

A halo is associated with the first upcrossing of the linear spherical-collapse threshold by the smoothed density contrast as its variance increases. With a sharp-k smoothing filter, the Brownian reflection principle makes the crossing probability twice the one-scale Gaussian upper tail. This handles trajectories that crossed earlier but finish below the barrier and leads to the Press-Schechter halo mass function.

A small-scale spectral cutoff bounds the smoothing variance by $S_{\max }$. A sharp-k excursion remains Brownian up to this terminal variance, with total crossed fraction $\mathrm{erfc}({\delta }_c/\sqrt{2S_{\max }})<1$. The ordinary first-crossing density remains valid before the cutoff. Below the sharp-filter cutoff radius the variance is constant, so there are no new crossings and the idealized differential halo abundance vanishes. The missing fraction need not be artificially renormalized to one.

A sharp-k filter includes every Fourier mode below a cutoff with unit weight and excludes all modes above it. For a Gaussian random field, increasing the cutoff adds independent Fourier shells, so the smoothed density contrast is a Brownian motion when parameterized by its variance. The real-space window is not a localized spherical top-hat, so its halo mass assignment needs a specified convention.

The Press-Schechter differential abundance is

The virial theorem and hydrostatic equilibrium relate a cluster's gas temperature to $GM/R$, while $M=(4\pi /3){\Delta }_v{\rho }_c{R}^3$. A calibrated mass-temperature relation converts the Press-Schechter halo mass function into $dn/dT=(dn/dM)|dM/dT|$. Its exponential rare-cluster tail is sensitive to the density contrast variance on cluster scales, hence to the power-spectrum amplitude, often parameterized by ${\sigma }_8$. Mass calibration, temperature scatter, nonthermal pressure, sample selection and the matter density must be included rather than treating temperature counts as an independent measurement of amplitude alone.

If the linearly extrapolated rms amplitude scales as $\sigma (M)\propto M^{-s}$, a fixed peak-height cutoff in the Press-Schechter halo mass function has $M_{\ast }\propto D^{1/s}$, where $D$ is the linear growth factor. The differential prefactor scales as $M^{s-2}$ and the exponential argument as $M^{2s}$. The cutoff convention must be stated when fixing its absolute normalization: $\nu =1$ and ${\nu }^2/2=1$ select different masses, although their redshift scaling is the same.

For comoving differential number density $n(M)=dn/dM$, the abundance in an order-unity logarithmic mass bin is approximately $Mn(M)$. A characteristic mean spacing is therefore $[Mn(M){]}^{-1/3}$. A cumulative mass fraction cannot replace this number directly; cumulative halo counts require integrating $n(M)$ over mass.

For a fixed narrow mass bin with time-independent smoothing slope, the Press-Schechter halo mass function has time dependence $n\propto \nu e^{-{\nu }^2/2}$, with halo peak height $\nu \propto D^{-1}$. Differentiating gives the displayed rate. It is positive only for $\nu >1$; the bin can lose objects to higher masses later. Cumulative counts require integrating the differential abundance, and a moving mass boundary adds a selection term.

At fixed mass, the Press-Schechter halo mass function has abundance per logarithmic interval proportional to $\nu e^{-{\nu }^2/2}$. This increases as $\nu$ falls from infinity to one and then decreases as $\nu$ falls toward zero, reflecting incorporation of small dark-matter haloes into larger ones. A target below the maximum therefore has two mathematical crossings if the linear growth factor can increase without limit. The large-$\nu$ crossing describes first appearance; the small-$\nu$ crossing is late depletion. Late vacuum domination limits growth, so a formal future crossing need not be reached physically.

Defining characteristic mass by an exponential factor $\exp [-(M/M_{h,\mathrm{ch}}{)}^{2\beta }]$ fixes its peak height to $\nu =\sqrt{2}$, rather than one. Hence $\sigma (M_{h,\mathrm{ch}},z)={\delta }_{\mathrm{sc}}/\sqrt{2}$ and the variance at other masses follows the same power-law scaling. The precise characteristic-mass convention determines this factor.

For $\sigma (M,0)=S{M}^{-q}$ with $q=(n+3)/6>0$, matching the Press-Schechter halo mass function to $A{M}^{q-2}\exp [-(M/M_{\ast }{)}^{2q}]$ requires $\sigma (M_{\ast },0)={\delta }_c(z)/\sqrt{2}$. The comoving normalization is $A=2q{\bar{\rho }}_{m,0}{M}_{\ast }^{-q}/\sqrt{\pi }$. Consequently both $A$ and $M_{\ast }$ evolve with the linear growth factor; $M_{\ast }\propto D^{1/q}$.

For a power-law matter spectrum, smoothed variance scales as $\sigma \propto M^{-\beta }$ with $\beta =(n+3)/6$. Differentiating the Press-Schechter collapsed fraction gives a halo mass function proportional to $M^{\beta -2}\exp [-(M/M_{h,\mathrm{ch}}{)}^{2\beta }]$. A constant stellar-to-halo mass ratio preserves both exponents and rescales characteristic mass and normalization.