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

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 probability of the curvature perturbation, which is frequently adopted to calculate the abundance of PBHs. 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.