Extended Press-Schechter Formalism

Updated 3 December 2025

Extended Press-Schechter formalism is a statistical framework that generalizes the original model by integrating non-spherical collapse, environmental factors, and non-Gaussian fluctuations.
It uses an excursion-set approach with stochastic density contrasts to derive halo/PBH mass functions and conditional merger rates through first-crossing distributions.
Extensions like moving barriers and non-Markovian corrections enhance its predictive power, matching simulations for large-scale structure and precision cosmology.

The Extended Press-Schechter (EPS) formalism is a foundational statistical approach in cosmological structure formation theory, built to generalize the original Press-Schechter (PS) model by systematically incorporating the effects of environment, non-spherical collapse, and non-Gaussian fluctuations. The EPS formalism provides an excursion-set framework for deriving halo and object mass functions, conditional mass functions, merger rates, and related quantities, facilitating unified modeling in both dark matter halo and primordial black hole (PBH) scenarios (Prescod-Weinstein et al., 2010, Zheng et al., 2023, Tosone et al., 2020, Farooq et al., 27 Jan 2025, Benson et al., 2018, Lapi et al., 2021, Kushwaha et al., 30 Sep 2025, Simone et al., 2010).

1. Excursion-Set Theory and the Foundation of EPS

In the EPS approach, the smoothed linear density contrast $\delta(R)$ is considered as a stochastic variable, evolving as the smoothing scale (or equivalently, the variance $S(M) = \sigma^2(M)$ ) is varied. A region is defined to collapse and form a structure when its overdensity first crosses a critical barrier $B(S)$ —the so-called first-crossing problem. For a Gaussian, uncorrelated (sharp- $k$ filtered) field, $\delta(S)$ performs a Markovian random walk with

$\frac{d\delta}{dS} = \eta(S), \quad \langle\eta(S)\eta(S')\rangle = \delta_D(S-S'),$

where $D(z)$ is the linear growth factor. The fraction of mass in objects that first upcross $B(S)$ between $S$ and $S+dS$ is given by a first-passage distribution. For a constant barrier $B$ , the solution is

$f(S) = \frac{B}{\sqrt{2\pi} S^{3/2}} \exp\left(-\frac{B^2}{2S}\right).$

The comoving mass function follows by converting from $S$ to the mass $M$ :

$\frac{dn}{dM} = \frac{\bar\rho}{M} f(S) \left| \frac{dS}{dM}\right|.$

The Markov character enables analytic results for spherical collapse, while non-Markovian extensions (non-sharp filters, PBH horizon-tied smoothing) require explicit corrections (Zheng et al., 2023, Farooq et al., 27 Jan 2025, Kushwaha et al., 30 Sep 2025).

2. Environmental Dependence and Conditional Mass Functions

EPS formalism naturally extends to conditional predictions. For a region of mass $M_0$ with initial overdensity $\delta_0$ (and variance $S_0$ ), the conditional mass function for subregions of mass $M$ (variance $S > S_0$ ) is (Zheng et al., 2023)

$f_{\rm EPS}(S, \delta_1 | S_0, \delta_0) dS = \sqrt{\frac{2}{\pi}} \frac{\delta_1 - \delta_0}{(S - S_0)^{3/2}} \exp\left[-\frac{(\delta_1 - \delta_0)^2}{2(S - S_0)}\right] dS.$

Here, $\delta_1$ is the collapse barrier at the redshift of interest. This formulation predicts the suppression or enhancement of structure formation in over-/underdense regions and underpins the spatial bias of collapsed objects. Both mean (unconditional) and environment-conditioned mass functions arise as special cases by appropriate averaging over $\delta_0$ (Zheng et al., 2023, Benson et al., 2018).

3. Moving Barriers and Ellipsoidal Collapse

Real halo formation deviates from strict spherical symmetry. Sheth & Tormen and subsequent studies incorporated ellipsoidal collapse by adopting a moving barrier

$B(S) = \sqrt{a}\, \delta_c \left[1 + \beta \left(\frac{aS}{\delta_c^2}\right)^\gamma\right],$

with empirical parameters $(a, \beta, \gamma)$ tuned to $N$ -body simulations. This moving-barrier approach alters both the exponential cutoff and power-law scaling in the high- and low-mass ends of the mass function, improving the fit to numerical results (Zheng et al., 2023, Prescod-Weinstein et al., 2010, Simone et al., 2010). Expansions in path-integral or Fokker-Planck treatments yield analytic corrections for arbitrary barrier shapes, crucial for precision cosmology (Simone et al., 2010, Lapi et al., 2021).

4. Conditional Merger Trees and Mass Assembly

EPS formalism produces conditional probabilities for sequential upcrossings, establishing analytic merger tree algorithms for halo formation histories. The conditional mass function

$f(S_1 | S_0) = \frac{\delta_c}{\sqrt{2\pi} (S_1 - S_0)^{3/2}} \exp\left[ - \frac{ \delta_c^2 }{2 (S_1 - S_0) } \right]$

characterizes progenitor mass distributions at earlier epochs for halos of mass $M_0$ today (Lapi et al., 2021, Benson et al., 2018). Markovian algorithms (e.g., Cole, Parkinson et al.) stochastically implement these transitions for merger trees, enabling direct calculation of formation times, mass accretion histories, and subsequent halo properties such as concentration (Benson et al., 2018).

5. Extensions to Primordial Black Holes and Non-Gaussianity

EPS theory underlies calculations of PBH mass functions by reinterpreting "collapse" as the first upcrossing of a barrier tied to the horizon threshold during radiation domination. For PBHs, non-Markovian stochasticity is intrinsic, invalidating the usual PS "fudge factor 2" and necessitating careful evaluation of both direct and prior upcrossing components for a positive-definite mass spectrum (Kushwaha et al., 30 Sep 2025, 2207.13689, Farooq et al., 27 Jan 2025, Sureda et al., 2020). The formalism incorporates power spectra with spikes or non-Gaussian initial perturbations, yielding complex, often multimodal, PBH mass distributions that can be matched against observational constraints.

A table summarizes key features in different contexts:

Context	Barrier Form	Markovianity
Halos (ΛCDM)	Constant/moving	Markovian
PBHs (horizon entry)	Moving (mass dep.)	Non-Markovian
Halo merger history	Moving	Markovian
Non-Gaussian initial	Arbitrary	Non-Markovian

Non-Gaussianity and scale-dependent statistics can be included using path-integral or Edgeworth (moment) expansions, leading to analytic corrections to the first-crossing rates and altered mass functions (Simone et al., 2010).

6. Applications: Large-Scale Structure, PBH Constraints, and Semi-Analytic Models

EPS underlies analytic/semianalytic predictions of halo mass functions, their spatial bias, and merger histories, forming the backbone of many semi-analytic galaxy, black hole, and cosmic dawn models (Ellis et al., 2024, Bohr et al., 2021, Tosone et al., 2020). As shown in simulations and data, EPS reproduces conditional mass functions and bias relations to percent-level accuracy over decades in mass and redshift, generalizing the Press-Schechter principle to extended environments (Zheng et al., 2023, Benson et al., 2018). In the PBH context, EPS allows tailored modeling of extended, peaked, or hybrid mass functions and encompasses environmental and filter-dependent effects (Farooq et al., 27 Jan 2025, 2207.13689, Kushwaha et al., 30 Sep 2025). Observational constraints (microlensing, CMB, dynamical bounds) are typically implemented by integrating the predicted mass function against empirical limit curves (2207.13689, Sureda et al., 2020).

7. Limitations, Calibration, and Numerical Validation

Despite its analytic power, standard EPS theory omits assembly bias and nonlocal effects, predicting median mass assembly and concentration relations but systematically underestimating the scatter at fixed mass—an effect remedied only by introducing explicit environmental dependence in barrier shape or merger rates (Benson et al., 2018). Its Markovian foundation is formally exact only for sharp- $k$ filtering; realistic window functions induce correlated steps, requiring non-Markovian corrections. Numerical simulation suites such as VVV (Zheng et al., 2023) demonstrate that EPS, with calibrated barrier and filter choices, remains accurate for both unconditional and conditional mass functions across wide ranges of redshift and environment. For PBH formation, marked deviations due to nonlocality and non-Gaussianity necessitate further extensions of the basic formalism (Kushwaha et al., 30 Sep 2025, Simone et al., 2010).

For in-depth theoretical descriptions, derivations, and calibration to $N$ -body or cosmological simulation data, the following constitute core references: (Prescod-Weinstein et al., 2010, Zheng et al., 2023, 2207.13689, Tosone et al., 2020, Benson et al., 2018, Kushwaha et al., 30 Sep 2025, Lapi et al., 2021, Simone et al., 2010, Farooq et al., 27 Jan 2025).