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 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.
