Excursion-Set Framework in Cosmology

• Excursion-Set Framework is a stochastic approach modeling structure formation by tracking smoothed density fields and their first-crossing events over a threshold barrier.
• It uses both Markovian (sharp-k filter) and non-Markovian (Gaussian, Top-Hat) methods to derive halo abundances, bias, and merger histories with analytic and numerical tools.
• The framework extends to applications such as primordial black hole formation and reionization modeling, employing path-integrals, Monte Carlo simulations, and upcrossing approximations for accurate predictions.

Searching arXiv for recent and foundational papers on the excursion-set framework, especially the PBH-focused moving-barrier formulation and related non-Markovian/general excursion-set developments. The excursion-set framework is a stochastic description of thresholded structure formation in which a smoothed field, usually the density contrast, is followed as the smoothing scale varies, and the physically relevant event is the first crossing of a barrier by the resulting random walk. In cosmology, this construction is used to compute halo abundances, bias, conditional mass functions, merger histories, and primordial-black-hole mass functions; closely related excursion-set ideas also appear in reionization modeling, thresholded point-process analyses, and sensitivity analysis for random sets (Simone et al., 2011, Auclair et al., 4 Mar 2026, Park et al., 2021).

1. Random walks in smoothing scale

In its standard form, the framework starts from the smoothed overdensity

$$\delta(R)\equiv \int d^3x'\,W(|\mathbf{x}-\mathbf{x}'|,R)\,\delta(\mathbf{x}'),$$

or equivalently in Fourier space through a window $\tilde W(k,R)$. The variance

$$S(R)\equiv \sigma^2(R)=\int \frac{d^3k}{(2\pi)^3}\,P(k)\,\tilde W^2(k,R)$$

plays the role of a pseudo-time variable, so that decreasing smoothing radius $R$ corresponds to increasing $S$. For hierarchical spectra, $S$ decreases monotonically with $R$, and one studies a stochastic trajectory $\delta(S)$ at fixed spatial position (Ma et al., 2010, Simone et al., 2011).

With a sharp-$k$ filter, $\delta(S)$ becomes Markovian. In that limit the walk obeys

$\tilde W(k,R)$0

and the corresponding probability density satisfies a Fokker–Planck equation with an absorbing barrier. Real-space top-hat and Gaussian filters, by contrast, correlate successive increments because the same Fourier modes contribute across nearby smoothing scales; the walk then becomes non-Markovian and the first-crossing problem ceases to be exactly local in $\tilde W(k,R)$1 (Ma et al., 2010, Musso et al., 2012).

This random-walk picture is the core abstraction of the framework. It translates the physics of collapse or activation into a stochastic first-passage problem while keeping the dependence on the input power spectrum and on the chosen coarse-graining procedure explicit.

2. Barriers, first crossing, and physical observables

A collapse model is encoded in a barrier $\tilde W(k,R)$2. In the simplest spherical-collapse case the barrier is constant, $\tilde W(k,R)$3, whereas more realistic descriptions admit moving or stochastic barriers. The first-crossing distribution $\tilde W(k,R)$4 is the probability density that a trajectory crosses $\tilde W(k,R)$5 for the first time at pseudo-time $\tilde W(k,R)$6. It is the central object from which mass functions and related quantities follow (Simone et al., 2011).

For dark-matter halos, the mapping to the mass function is

$\tilde W(k,R)$7

In the constant-barrier, Markovian limit this yields the Press–Schechter result, while generic moving barriers require either derivative expansions, path-integral methods, or integral-equation solvers. De Simone, Maggiore, and Riotto developed a path-integral treatment for generic moving barriers and used it to derive conditional probabilities, two-barrier crossing rates, formation-time distributions, and the first two scale-independent halo-bias parameters (Simone et al., 2011).

The same formalism naturally supports conditional statistics. Conditioning on a long-wavelength perturbation leads to peak-background-split bias, while conditioning on an earlier barrier crossing yields progenitor distributions and merger rates. In non-Markovian settings, these conditional probabilities retain memory of the trajectory history, so environment and assembly enter explicitly rather than only through halo mass. This is the basis of analytic assembly-bias calculations in non-Markovian excursion-set theory (Zhang et al., 2013).

A useful alternative representation is the upcrossing approximation. For correlated steps, Musso–Sheth-type formulas replace the full no-earlier-crossing constraint by a local upcrossing condition involving the walk height and slope. This approximation is accurate when crossings are rare and became one of the main analytic tools for non-Markovian excursion-set calculations (Musso et al., 2012).

3. Filters, non-Markovianity, peaks, and stochastic barriers

The choice of filter controls the correlation structure of the walk. In the non-Markovian extension of halo excursion-set theory, the covariance is written as

$\tilde W(k,R)$8

with $\tilde W(k,R)$9 parameterized by a filter-dependent coefficient $$S(R)\equiv \sigma^2(R)=\int \frac{d^3k}{(2\pi)^3}\,P(k)\,\tilde W^2(k,R)$$0. Representative values quoted for Gaussian initial conditions are $$S(R)\equiv \sigma^2(R)=\int \frac{d^3k}{(2\pi)^3}\,P(k)\,\tilde W^2(k,R)$$1 for sharp-$$S(R)\equiv \sigma^2(R)=\int \frac{d^3k}{(2\pi)^3}\,P(k)\,\tilde W^2(k,R)$$2, $$S(R)\equiv \sigma^2(R)=\int \frac{d^3k}{(2\pi)^3}\,P(k)\,\tilde W^2(k,R)$$3 for a Gaussian filter, and $$S(R)\equiv \sigma^2(R)=\int \frac{d^3k}{(2\pi)^3}\,P(k)\,\tilde W^2(k,R)$$4 for a real-space top-hat. A stochastic barrier can be incorporated through a diffusion parameter $$S(R)\equiv \sigma^2(R)=\int \frac{d^3k}{(2\pi)^3}\,P(k)\,\tilde W^2(k,R)$$5, equivalently $$S(R)\equiv \sigma^2(R)=\int \frac{d^3k}{(2\pi)^3}\,P(k)\,\tilde W^2(k,R)$$6, so that barrier scatter and non-Markovian memory enter the mass function and bias in a controlled perturbative way (Ma et al., 2010).

A second major refinement is the peak constraint. Excursion-set peaks combines first crossing with the requirement that halos form around maxima of the initial density field. This introduces peak-curvature weighting, mixed filter moments, and a mass-dependent collapse threshold motivated by triaxial collapse. With a TopHat mass definition, a square-root barrier $$S(R)\equiv \sigma^2(R)=\int \frac{d^3k}{(2\pi)^3}\,P(k)\,\tilde W^2(k,R)$$7, and scatter in $$S(R)\equiv \sigma^2(R)=\int \frac{d^3k}{(2\pi)^3}\,P(k)\,\tilde W^2(k,R)$$8, the excursion-set-peak framework matches halo abundances at the $$S(R)\equiv \sigma^2(R)=\int \frac{d^3k}{(2\pi)^3}\,P(k)\,\tilde W^2(k,R)$$9 level and gives linear bias in excellent agreement with simulations, especially at high mass where traditional peak-background-split fits underpredict the measured bias by $R$0 (Paranjape et al., 2012). Related analytical approximations in the excursion-set-peak framework also produce halo mass functions and bias at the $R$1–$R$2 level for CDM spectra with realistic filters and moving barriers (Lapi et al., 2014).

The self-consistency problem—whether excursion sets should average over all Lagrangian positions or only over special positions such as protohalo centers—has been examined directly. One analysis found that a drifting diffusing barrier calibrated on halo abundances predicts the distribution of first-crossing overdensities measured around random protohalo particles, while the overdensities around protohalo centers of mass are larger and more sharply distributed, consistent with ellipsoidal-collapse expectations. This resolves the apparent tension by separating the all-positions ensemble from the special-positions ensemble (Achitouv et al., 2012).

The same logic extends beyond $R$3CDM gravity. In chameleon modified gravity, the collapse threshold becomes environment dependent, and correlated steps materially alter both unconditional and conditional mass functions and the resulting halo bias. In that setting, abundance and clustering in underdense and overdense environments become jointly constraining observables (Lam et al., 2012).

4. Primordial-black-hole excursion sets

For primordial black holes, the framework is adapted to horizon reentry in radiation domination. The smoothed density contrast is constructed from the primordial curvature spectrum, and PBH formation is identified with the first crossing of a collapse threshold. A common mapping is

$R$4

with the mass linked to the horizon scale at reentry (Auclair et al., 4 Mar 2026).

A central recent issue is the status of the noise. One PBH analysis argued that the apparent colored noise found when sampling on the Hubble-crossing surface is a bookkeeping artifact caused by mixing deterministic drift with stochastic increments. With sharp-$R$5 filtering and uncorrelated Fourier modes, the genuine noise is strictly white; on a synchronous hypersurface, $R$6, the drift vanishes, $R$7 is monotonic, and the canonical stochastic equation

$R$8

is recovered. In this formulation the PBH threshold specified at horizon reentry becomes a moving barrier on the synchronous slice, and the first-passage problem is solved through a stable Volterra integral equation (Auclair et al., 4 Mar 2026).

This moving-barrier formulation is directly relevant to two controversies. First, it shows that the excursion-set treatment remains necessary for broad spectra: cloud-in-cloud is suppressed only for widely separated scales, whereas a continuum of enhanced modes produces repeated crossings that reshape the mass function by suppressing low masses and enhancing the high-mass tail. Second, Press–Schechter-like flux estimates can fail even without cloud-in-cloud, because moving barriers can make the naive flux negative; the full excursion-set solution avoids these pathologies and yields positive-definite first-crossing distributions (Auclair et al., 4 Mar 2026).

Other PBH formulations emphasize different coarse-graining choices. With a Gaussian window, the noise can become fully correlated across scales, and the mass-function peak is then governed predominantly by the scale-by-scale exceedance probability rather than by the Markovian first-crossing logic; this was highlighted explicitly for smooth coarse graining at horizon crossing (Saito et al., 26 Dec 2025). A broader study of window functions found that colored noise generically modifies the low-mass tail relative to Carr’s formula, although Carr’s estimate remains a practical approximation near the characteristic mass when a smooth Fourier-space window is used (Iizuka et al., 21 May 2026).

The PBH literature has also developed problem-specific variants of the crossing rule. A “first touch” prescription was introduced to account for the fact that smaller scales reenter the horizon earlier; it assigns the PBH mass to the earliest reentering crossing rather than to the largest-scale crossing, and was used to study broad blue-tilted spectra and abundance bounds on $R$9 (Erfani et al., 2021). Small-scale PBH clustering has likewise been formulated as a two-trajectory excursion-set problem with shared history up to a clustering scale, yielding joint formation probabilities for PBH pairs and a one-to-one relation between blue tilt and the mass ranges where PBHs form and cluster (Kameli et al., 3 Aug 2025).

5. Computational methods

Because correlated steps and moving barriers make exact first-passage solutions difficult, the excursion-set framework has accumulated a substantial numerical toolkit. Path-integral methods expand the trajectory measure in connected correlators and remain the standard analytic route for generic moving barriers and non-Gaussian initial conditions (Simone et al., 2011). For correlated Gaussian walks with smooth filters, the Stratonovich approximation resums the exact crossing series into a positive, normalized approximation that is substantially more accurate than naive truncations, and Cholesky decomposition provides a fast method for generating Monte Carlo trajectories with the required covariance (Nikakhtar et al., 2018).

For PBHs with a moving barrier on a synchronous slice, a Volterra equation of the second kind gives an efficient solver. Discretizing $S$0 turns the first-passage problem into a lower-triangular linear system,

$S$1

which can be solved by forward substitution in $S$2. The method converges rapidly on modest grids and is far less noisy than brute-force Monte Carlo when crossings are rare (Auclair et al., 4 Mar 2026).

Monte Carlo constructions remain important when the walk itself is the primary object. Correlated random walks can be generated either directly from a prescribed kernel $S$3 or from the covariance matrix implied by the filter and power spectrum. Such constructions have been used to model real-space top-hat walks for halos, where correlated-step Monte Carlo improves agreement with simulation-based mass functions relative to earlier analytic approximations, especially at $S$4 (Hiotelis et al., 2018).

These methods underscore a general point: once the covariance structure and barrier are specified, the excursion-set problem is algorithmically well posed, even when the walk is non-Markovian. The challenge is not the existence of a formulation but the choice of approximation appropriate to the filter, barrier, and observable of interest.

6. Broader uses and open issues

Although the framework is most developed in structure formation, the same thresholded-set logic appears in several other settings. In excursion-set reionization, one smooths the density field and tests an ionization criterion such as $S$5, or, in photon-counting form, whether the ionizing photon density exceeds the local number of atoms plus recombinations. The resulting semi-numerical models reproduce the large-scale topology of reionization efficiently, but they do not exactly conserve photons once ionized regions overlap; an on-the-fly $S$6 calibration approximately restores photon conservation while preserving the excursion-set topology (Park et al., 2021).

In auroral radar observations, coherent backscatter points can be treated as a threshold sample of the electric field because the Farley–Buneman instability requires the local $S$7 drift to exceed the ion-acoustic speed. The resulting point process is an excursion set of the underlying field, and the reduced structure factor satisfies, to leading order,

$S$8

In co-moving frames this yields spectra with indices near $S$9, consistent with in-situ measurements (Ivarsen et al., 25 Jun 2026).

A more abstract generalization treats excursion sets as random set-valued outputs. In that context, the symmetric-difference kernel

$S$0

is characteristic, allowing HSIC-based sensitivity analysis and ANOVA-style screening and ranking for set-valued simulators (Fellmann et al., 2023). This suggests that the excursion-set framework is not merely a cosmological technique but a general thresholded-random-field formalism.

Several open issues persist. Non-Markovian corrections for realistic filters remain technically involved; barrier models are often effective descriptions rather than first-principles collapse laws; non-Gaussian initial conditions complicate both path integrals and first-crossing statistics; and the relation between all-position excursion sets and special-position peak formalisms remains subtle outside regimes where it has been tested explicitly. In PBH applications, the status of window functions, collapse criteria, and critical-collapse mass mappings remains consequential. Nevertheless, across halos, PBHs, reionization, and related thresholded systems, the excursion-set framework remains the standard stochastic language for translating field statistics into first-passage observables (Ma et al., 2010, Achitouv et al., 2012, Auclair et al., 4 Mar 2026).