A Note on the Abundance of Primordial Black Holes: Use and Misuse of the Metric Curvature Perturbation

TL;DR

This paper clarifies that accurately calculating primordial black hole abundance requires detailed joint probability distributions of curvature perturbations, highlighting limitations of commonly used metrics.

Contribution

It demonstrates through a path-integral approach that the standard curvature perturbation metric is insufficient for precise PBH formation probability calculations.

Findings

Standard curvature perturbation metrics are inadequate for accurate PBH abundance estimates.

Exact PBH formation probability depends on multivariate joint probabilities and connected correlators.

Using the proper statistical framework improves the reliability of PBH abundance predictions.

Abstract

The formation of Primordial Black Holes (PBHs) through the collapse of large fluctuations in the early universe is a rare event. This manifests itself, for instance, through the non-Gaussian tail of the formation probability. To compute such probability and the abundance of PBHs, the curvature perturbation is frequently adopted. In this note we emphasize that its use does not provide the correct PBH formation probability. Through a path-integral approach we show that the exact calculation of the PBH abundance demands the knowledge of multivariate joint probabilities of the curvature perturbation or, equivalently, of all the corresponding connected correlators.