  We revisit the calculation of the abundance of primordial black holes (PBHs)
formed from primordial density perturbations, using a formation criterion
derived by Shibata and Sasaki which refers to a metric perturbation variable
rather than the usual density contrast. We implement a derivation of the PBH
abundance which uses peaks theory, and compare it to the standard calculation
based on a Press--Schechter-like approach. We find that the two are in
reasonable agreement if the Press--Schechter threshold is in the range
$\Delta_{{\rm th}} \simeq 0.3$ to 0.5, but advocate use of the peaks theory
expression which is based on a sounder theoretical footing.

Astrophysics