The Last Crossing in Excursion-Set Theory of Cosmic Reionization

This paper introduces the last crossing of the photon-counting barrier as a new statistic for analytical models of cosmic reionization, complementing the standard first-crossing formalism of the excursion-set model developed by Furlanetto, Zaldarriaga, and Hernquist (2004, hereafter FZH04).

Theoretical framework. The FZH04 model connects ionized regions to large-scale density fluctuations: overdense regions host more collapsed halos and therefore more ionizing sources. The model reduces the growth of ionized structure to a barrier-crossing calculation, where the barrier is defined as B(S, z) = δc(z) − √2 K(ζ) √(Smin − S), with δc the linear collapse threshold, ζ the ionizing efficiency, S the variance of the smoothed density field, and Smin the variance at the minimum source-halo scale. The first crossing identifies the largest self-ionizing region around the point—the excursion-set bubble scale that connects the biased galaxy distribution to the large H ii regions targeted by 21 cm measurements. The last crossing, introduced here, is defined as the smallest scale on which the enclosed photon budget still ionizes the enclosed gas: Sl = max S: δ(S) ≥ B(S, z), with Rl = R(Sl) the smallest smoothing scale on which the FZH04 condition is still met.

Mathematical derivation. The author derives a Volterra integral equation for the last-crossing distribution, adapting the approach of Hopkins (2012) to the FZH04 barrier. Two features require modification from the Hopkins setup: (1) the trajectory endpoint is at the finite, physically motivated scale Si ≡ Smin rather than a small-radius cutoff, and (2) the barrier at this endpoint is finite (Bi = δc(z)), so a nonzero fraction of trajectories sits above the barrier at Si and contributes a discrete endpoint probability pend = 1 − Φ(Bi/√Si). The resulting equation is a Volterra equation of the first kind: A(S) = ∫ dS′ fl(S′) K(S, S′), where A(S) is the joint probability that the trajectory is above the barrier at S and below it at the endpoint, and K(S, S′) is the conditional probability of being above the barrier at S given the trajectory touched the barrier at S′. The equation is solved numerically by binning from the endpoint outward using forward substitution, with the grid cast in u ≡ √(Si − S) to linearize the barrier near Smin.

Analytic results. The first-crossing distribution peaks at larger radii because it selects the largest self-ionizing region, while the last-crossing distribution peaks at smaller radii. Both distributions integrate to the theoretical ionized volume fraction ⟨xHII⟩th rather than to unity. As reionization proceeds, the first-crossing distribution moves to larger radii, mirroring the expansion of enclosing ionized regions predicted by FZH04, while "the last-crossing distribution evolves in an approximately self-similar manner: its large-scale extent moves outward together with the characteristic first-crossing scale, while its probability remains weighted toward smaller radii." Exact self-similarity is broken by the finite endpoint at Smin and the scale-dependent slope of the matter power spectrum.

Simulation comparison. The author constructs an empirical barrier from CROC radiative-transfer simulations following Kaurov (2016), defining the barrier as the trajectory value at which the prior-weighted neutral and ionized distributions are equal. The simulated barrier is fuzzy—locations with similar smoothed density can have different ionization states due to source environments, recombination histories, and the nonlocal radiation field. Comparing measured crossings with the analytic FZH04 prediction at matched ⟨xHII⟩ = 0.60, the CROC first-crossing distribution is broader than the analytic curve, and the last-crossing distribution shows an extended tail to larger Rl that lies above the analytic prediction. This difference reflects nonlocal photon transport: in CROC, photons travel across the simulation volume, and a region need not close its own photon budget at every aperture to be ionized.

Resolved versus unresolved last crossings. The crossing construction divides ionized regions into two classes. Regions with unresolved last crossings satisfy the empirical barrier all the way down to the minimum resolved smoothing scale, meaning their local photon budget closes at the smallest aperture the analysis can probe. Regions with resolved last crossings belong to a larger ionized bubble but fall below the empirical barrier on smaller scales, so their photon budget closes only across a larger aperture. Spatially, resolved-Rl regions form coherent structures that surround the remaining neutral pockets and trace the edges of ionized regions rather than their dense interiors. The unresolved-Rl population is concentrated at higher density, while the resolved-Rl population peaks well below mean density. Within the resolved population, the median density at fixed Rl decreases with increasing Rl, and the median reionization redshift shifts later. The top 10% largest-Rl regions occupy a distinct corner of the joint plane of density and reionization time, consistent with the inside-out picture where source-rich overdense regions ionize first and photons subsequently reach lower-density gas.

Source environments. The author matches each resolved-Rl region to an ionized region without a distinct resolved last crossing, holding fixed local density, enclosing first-crossing scale Rf, and reionization redshift. Results show that "in the resolved-Rl sample, the innermost shell, R < 0.25Rl, has an ionizing-luminosity density lower by approximately a factor of two, and a galaxy number density lower by roughly one third. Both ratios rise above unity outside the central shell and peak near R ≃ 0.5Rl, where the luminosity and number densities are enhanced by roughly 75% and 65%, respectively. This demonstrates that Rl retains information beyond local density, enclosing first-crossing scale, and reionization redshift—the classification identifies a source geometry with a locally source-poor center embedded within a richer surrounding environment."

Observational implications. The author proposes that resolved-Rl regions may preferentially host Lyα transmission spikes: low gas density is the primary condition producing spikes in CROC, and resolved-Rl regions combine low density with a deficit of nearby sources and an excess at larger radius. The physical picture is that low density reduces the Lyα opacity, while sources distributed over larger scales maintain the ionized state. This predicts a central galaxy deficit and an excess at finite separation around transmission-selected IGM locations, which can be tested with transmission-selected galaxy–IGM measurements. The author notes that the galaxy underdensities and tentative finite-separation transmission signal reported by Zhu et al. (2026) bear on this prediction.

Summary of contributions. The paper develops the last-crossing formalism for the FZH04 barrier, derives its distribution analytically with a finite-endpoint correction, validates the solver against Monte Carlo realizations, measures both crossings in CROC simulations using an empirical barrier, characterizes the physical environments selected by resolved versus unresolved last crossings, and connects the classification to testable predictions for the galaxy–IGM connection during reionization.

Improvements to AI systems:

1. Analytical solver for Volterra integral equations with finite endpoints. The paper derives a Volterra equation of the first kind with a discrete endpoint probability and solves it via forward substitution on a transformed grid. An AI system can be improved to automatically detect and handle such finite-endpoint boundary conditions in integral-equation solvers, including the linearization of the kernel near the endpoint, enabling accurate numerical solutions without manual tuning.

2. Self-similarity detection and symmetry-breaking analysis. The paper identifies approximate self-similarity in the last-crossing distribution and pinpoints its breaking due to finite endpoints and scale-dependent power spectra. An AI system can be trained to automatically test for self-similarity in evolving distributions, quantify deviations, and attribute them to specific physical or numerical causes (e.g., boundary effects, spectral slope changes), improving model validation and physical interpretation.

3. Empirical barrier construction from noisy simulation data. The author builds a fuzzy empirical barrier by equating prior-weighted neutral and ionized distributions from radiative-transfer simulations. An AI system can be improved to construct probabilistic or fuzzy barriers from discrete simulation outputs, including uncertainty quantification, and to compare such empirical barriers against analytic predictions using matched global quantities (e.g., ionized fraction).

4. Crossing-statistics classification for spatial structure. The paper classifies ionized regions into resolved versus unresolved last crossings, revealing distinct spatial morphologies (e.g., resolved regions trace edges, unresolved regions trace dense interiors). An AI system can be enhanced to automatically classify regions based on multi-scale crossing statistics and to extract spatial correlations (e.g., density–radius–reionization-time joint distributions) from large simulation datasets.

5. Causal source–environment matching. The author matches resolved-Rl regions to control regions with identical density, first-crossing scale, and reionization redshift, isolating the effect of source geometry. An AI system can be improved to perform causal matching in high-dimensional parameter spaces, identifying which physical variables carry independent information beyond known covariates, and to quantify effect sizes (e.g., luminosity density ratios) with confidence intervals.

6. Predictive linking of geometric statistics to observable signatures. The paper predicts that resolved-Rl regions preferentially host Lyα transmission spikes, linking a geometric classification to a specific observable. An AI system can be improved to generate testable predictions from abstract geometric classifications, including expected galaxy–IGM cross-correlations (e.g., central deficit, finite-separation excess), and to propose specific observational tests (e.g., transmission-selected galaxy surveys) with quantitative thresholds.

7. Multi-scale photon-budget accounting in nonlocal transport. The paper highlights that in simulations, photons travel nonlocally, so a region need not close its own photon budget at every aperture. An AI system can be improved to model nonlocal radiation transport effects in reionization, using crossing statistics to distinguish locally self-ionized regions from those ionized by distant sources, thereby refining predictions for 21 cm and Lyα observations.

8. Automated extraction of physical environments from crossing classifications. The paper characterizes resolved-Rl regions as source-poor centers embedded in richer surroundings. An AI system can be improved to automatically extract such environmental profiles (e.g., radial luminosity density, galaxy number density) from simulation outputs, and to identify which combinations of local and large-scale properties best predict observable signatures like transmission spikes.