The Last Crossing in Excursion-Set Theory of Cosmic Reionization

Abstract

I introduce the last crossing of the photon-counting barrier as a statistic for analytical models of cosmic reionization. Because ionizing galaxies are biased tracers of density and their photons propagate through gas of spatially varying opacity, the galaxy-IGM connection requires statistics beyond the global ionized fraction. The excursion-set model of Furlanetto et al. (2004) identifies the ionized-bubble scale around a point with the first crossing of a photon-counting barrier. I develop the last-crossing formalism and define the last-crossing scale, $R_\ell$, as the smallest scale on which the enclosed photon budget still ionizes the enclosed gas. I derive its distribution analytically. Using the empirical-barrier framework of Kaurov (2016) applied to CROC simulations, I measure both crossings in the simulated IGM. The last-crossing formalism separates externally ionized regions, whose photon-counting condition fails below $R_\ell$, from internally ionized regions whose trajectories remain above the barrier to the resolution scale. Resolved external regions are predominantly underdense, with larger $R_\ell$ selecting lower density and later reionization. At fixed local density, first-crossing scale, and reionization redshift, these regions show deficits in ionizing-luminosity and galaxy-number density at small radius relative to matched unresolved regions, and excesses in both at intermediate radius. Their combination of low gas density and a deficit of nearby sources suggests that Ly$α$ transmission spikes may preferentially arise in these regions: low density reduces the opacity, while sources on larger scales maintain the ionized state. The last-crossing picture connects the internal source geometry of ionized regions to a testable prediction for radial galaxy distributions around transmission-selected IGM locations.