Excursion-set for Primordial Black Holes I: white noise and moving barrier

TL;DR

This paper refines the excursion-set formalism for primordial black hole formation, addressing issues with correlated noise and moving barriers, and demonstrates its robustness for realistic scenarios.

Contribution

It introduces an efficient numerical framework for first-passage-time problems with moving barriers and clarifies the role of cloud-in-cloud effects in PBH formation.

Findings

Sampling on a synchronous surface removes noise correlation issues.

The new framework efficiently solves first-passage-time problems with moving barriers.

Press-Schechter estimates can fail without boundary evolution considerations.

Abstract

In the excursion-set formalism, the mass distribution of primordial black holes (PBHs) is derived from the first-passage time of a random walk describing the density contrast as the coarse-graining scale varies. We address two recent concerns that have been raised about this approach. First, it was argued that the random walks are subject to colored (i.e. correlated over time) noise, making the first-passage-time problem cumbersome. We show that this arises from an incorrect separation of drift and noise when sampling on the Hubble-crossing surface: if Fourier modes are uncorrelated, the noise is strictly white. Moreover, sampling along the Hubble-crossing surface precludes using the density dispersion as a time variable, explaining some pathologies. Sampling instead on a synchronous surface removes both issues. This requires solving a first-passage-time problem with a moving barrier, for which we provide an efficient numerical framework. Second, it was suggested that cloud-in-cloud (i.e. that large black holes may engulf smaller ones) is irrelevant for PBHs and that the excursion set is therefore not needed. While valid for widely separated scales, this statement fails for broad power spectra with enhanced continua of modes. We further show that Press-Schechter estimates neglecting boundary evolution can break down even without cloud-in-cloud effects. Our results establish the robustness and necessity of the excursion-set formalism in realistic PBH formation scenarios.