Press–Schechter Formalism

The Press–Schechter formalism is an analytical statistical framework introduced by American astrophysicists William H. Press and Paul L. Schechter in 1974 to predict the comoving number density of gravitationally bound, virialized dark matter halos as a function of mass M and cosmological redshift z. By linking the linear growth of primordial Gaussian density perturbations in the early universe to the non-linear spherical collapse of overdense regions, the formalism derives a closed-form halo mass function dn(M, z)/dM without requiring computationally expensive N-body integrations. Although simplified in its spherical geometry, the Press–Schechter model and its excursion-set extensions—most notably the Bond–Cole–Efstathiou–Kaiser (1991) random-walk formulation and the Sheth–Tormen (1999) ellipsoidal collapse modification—remain foundational to modern physical cosmology, galaxy formation modeling, and cluster abundance constraints on the dark energy equation of state.

In the standard ΛCDM cosmological model, primordial density fluctuations seeded during cosmic inflation are described by the dimensionless fractional overdensity field δ(x, t) = [ρ(x, t) − ρ̄(t)] / ρ̄(t), where ρ̄(t) is the mean background matter density of the universe. While perturbations remain small (|δ| ≪ 1), each Fourier mode grows self-similarly with cosmic time according to the linear growth factor D(z), which obeys δ(x, z) = D(z) δ₀(x). To associate a spatial scale R with a halo mass M, the density field is smoothed with a spherical top-hat window function W(r; R) enclosing a comoving mass M = (4π/3) ρ̄_m,0 R³, where ρ̄_m,0 is the present-day mean matter density.

The variance of the smoothed linear density field on mass scale M at redshift z is determined by integrating the linear matter power spectrum P_lin(k, z)—the same field probed in redshift space via the Kaiser effect—against the Fourier transform of the spherical top-hat filter W̃(kR) = 3[sin(kR) − kR cos(kR)] / (kR)³:

σ²(M, z) = D²(z) ∫₀^∞ (k² dk / 2π²) P_lin(k, 0) |W̃(kR)|²

Under the spherical collapse model for an Einstein–de Sitter (Ω_m = 1) matter-dominated universe, an isolated, spherically symmetric overdense shell expands to a maximum turnaround radius before decoupling from the Hubble flow and collapsing toward a virialized equilibrium state—ultimately Relaxing into a density profile described by the Navarro–Frenk–White profile or Einasto profile. At the exact epoch when the non-linear spherical shell reaches virial equilibrium (at half its turnaround radius, corresponding to a non-linear overdensity Δ_vir = 18π² ≈ 178), the linearly extrapolated overdensity of that same perturbation reaches a universal threshold constant:

δ_c = (3/20) (12π)^2/3 ≈ 1.686

When evaluating halo collapse at redshift z using the present-day (z = 0) linear variance σ(M), the effective collapse barrier scales inversely with the linear growth factor as δ_c(z) = δ_c,0 / D(z) = 1.686 / D(z).

Because the smoothed primordial density field δ_M is a zero-mean Gaussian random variable with variance σ²(M), the probability that a randomly selected point in space has a smoothed linear overdensity exceeding δ_c(z) is given by the Gaussian tail integral:

p(δ_M > δ_c(z)) = ∫_δc(z)^∞ [1 / √(2π σ²(M))] exp[−δ² / (2σ²(M))] dδ = (1/2) erfc[δ_c(z) / (√2 σ(M))]

Press and Schechter postulated that the fraction of cosmic matter locked inside virialized halos of mass greater than M at redshift z is proportional to this exceedance probability. However, in the limit M → 0 (where the variance σ(M) → ∞), the Gaussian integral evaluates to 1/2 rather than 1, implying that only half of all matter in the universe would ever accrete into collapsed objects because initially underdense regions (δ_M < 0) were neglected. To ensure that 100% of the background mass is accounted for across all halo scales, Press and Schechter multiplied the cumulative mass fraction F(>M) by a heuristic factor of 2, setting F(>M) = erfc[ν / √2], where ν ≡ δ_c(z) / σ(M) is the dimensionless peak height.

Differentiating F(>M) with respect to M and dividing by the individual halo mass M / ρ̄_m,0 yields the canonical Press–Schechter differential mass function:

dn(M, z) / dM = √(2/π) (ρ̄_m,0 / M²) [δ_c(z) / σ(M)] |d ln σ / d ln M| exp[−δ_c²(z) / (2σ²(M))]

Expressed in terms of the dimensionless multiplicity function f(ν) = (M / ρ̄_m,0) (dn / d ln ν), the Press–Schechter law takes the universal, cosmology-independent form f_PS(ν) = √(2/π) ν exp(−ν²/2). At low masses (ν ≪ 1), the halo abundance follows a power law driven by |d ln σ / d ln M|, whereas at high masses (ν ≫ 1, above the characteristic nonlinear collapse mass M* where σ(M*) = δ_c), the abundance follows an exp(−ν²/2) exponential cutoff—the dark-matter progenitor of the empirical Schechter luminosity function for galaxies.

For seventeen years after 1974, the ad-hoc factor of 2 in the Press–Schechter derivation remained a theoretical puzzle known as the cloud-in-cloud problem: a point whose density δ_M falls below δ_c when smoothed on a small mass scale M₁ may nevertheless reside inside a larger region of mass M₂ > M₁ whose smoothed density exceeds δ_c. In 1991, J. Richard Bond, Shaun Cole, George Efstathiou, and Nick Kaiser resolved this problem rigorously using excursion-set theory (also termed the extended Press–Schechter or EPS formalism).

In the excursion-set framework, one tracks the smoothed overdensity δ(S) at a fixed Lagrangian coordinate as a function of the variance S ≡ σ²(M), starting from S = 0 (M → ∞, where δ = 0) and increasing S toward smaller mass scales. When the density field is smoothed with a sharp k-space filter, increments Δδ between successive resolution steps ΔS are statistically independent Gaussian steps, so the trajectory δ(S) executes a Markovian Brownian random walk in the (S, δ) plane. A fluid element belongs to a halo of mass ≥ M if and only if its random walk crosses the absorbing collapse barrier δ = δ_c at some variance ≤ S. By Chandrasekhar's reflection principle for Brownian motion against a flat absorbing barrier, every trajectory that exceeds δ_c at variance S has an equal probability of ending above or below δ_c at S if the barrier is removed, yielding a first-passage probability distribution that recovers the factor of 2 from first principles and enables conditional mass functions and halo merger trees.

High-resolution cosmological N-body simulations demonstrated that the standard spherical Press–Schechter formula overpredicts the abundance of galaxy-sized halos below M* while underpredicting the abundance of high-mass galaxy clusters above M*. In 1999 and 2001, Ravi K. Sheth, H. J. Mo, and Giuseppe Tormen showed that this discrepancy arises because primordial density peaks are triaxial rather than spherical: tidal shear from the surrounding cosmic web deforms collapsing proto-halos into ellipsoids. Because tidal shear resists gravitational collapse most strongly in low-mass systems (large S), the effective collapse barrier increases with variance as a moving barrier B(S, z) = √a δ_c(z) [1 + β (S / (a δ_c²(z)))^γ], where a ≈ 0.707, p ≈ 0.3, and normalization A ≈ 0.3222 bring analytical mass functions into ~10% agreement with simulations across five orders of magnitude in halo mass.

Why did Press and Schechter multiply their mass fraction by a factor of 2?

In the original 1974 derivation, integrating the Gaussian probability distribution above δ_c at a single smoothing scale accounted for only 50% of the total mass in the universe as M → 0, because points that are locally underdense on small scales were excluded. The excursion-set formalism of Bond et al. (1991) proved that this missing 50% consists of locally underdense regions embedded inside larger overdense structures ("cloud-in-cloud"), which cross the collapse barrier δ_c at smaller variance S along a Brownian random walk.

What is the characteristic nonlinear mass M* in the Press–Schechter formalism?

The characteristic mass M*(z) is defined as the mass scale at which the root-mean-square linear density fluctuation equals the collapse threshold, σ(M*, z) = δ_c ≈ 1.686 (corresponding to peak height ν = 1). At the present epoch (z = 0), M* is approximately 10¹² to 10¹³ M_☉, marking the transition between power-law scaling for galaxy-scale halos and the exponential cutoff for rich galaxy clusters.

How is the Press–Schechter formalism used to build halo merger trees?

Through the extended Press–Schechter (EPS) conditional probability distribution, the random walk from a parent halo at (S₁, δ₁) to progenitor halos at an earlier redshift (S₂ > S₁, δ₂ > δ₁) follows the same barrier-crossing statistics with shifted origin (ΔS = S₂ − S₁, Δδ = δ₂ − δ₁). Semi-analytic galaxy formation models sample these conditional distributions backward in time to construct Monte Carlo merger trees of dark matter halos and the central supermassive black holes obeying the M–sigma relation.

1. Press, W. H., & Schechter, P. (1974). "Formation of galaxies and clusters of galaxies by self-similar gravitational condensation." The Astrophysical Journal, 187, 425–438. https://doi.org/10.1086/152650
2. Bond, J. R., Cole, S., Efstathiou, G., & Kaiser, N. (1991). "Excursion set mass functions for hierarchical Gaussian fluctuations." The Astrophysical Journal, 379, 440–460. https://doi.org/10.1086/170520
3. Lacey, C., & Cole, S. (1993). "Merger rates in hierarchical models of galaxy formation." Monthly Notices of the Royal Astronomical Society, 262(3), 627–649. https://doi.org/10.1093/mnras/262.3.627
4. Sheth, R. K., & Tormen, G. (1999). "Large-scale bias and the peak background split." Monthly Notices of the Royal Astronomical Society, 308(1), 119–126. https://doi.org/10.1046/j.1365-8711.1999.02692.x
5. Tinker, J., Kravtsov, A. V., Klypin, A., Abazajian, K., Warren, M., Yepes, G., Gottlöber, S., & Holz, D. E. (2008). "Toward a halo mass function for precision cosmology: The limits of universality." The Astrophysical Journal, 688(2), 709–728. https://doi.org/10.1086/591439