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In this paper, the criterion used to determine whether a density perturbationwill collapse to form a primordial black hole (PBH) is re-examined, in respectof its use to determine the abundance of PBHs. There is particular focus onwhich parameter to use, the time at which the abundance should be calculated,and the use of different smoothing functions. It is concluded that, with thetools currently available, the smoothed density contrast should be used ratherthan the peak value, and should be calculated from the time-independentcomponent of the density contrast in the super-horizon limit (long beforeperturbations enter the horizon) rather than at horizon crossing. For the firsttime the effect of the choice of smoothing function upon the formationcriterion is calculated, and, for a given abundance of PBHs, it is found thatthe uncertainty in the amplitude of the power spectrum due to this is$\mathcal{O}(10\%)$, an order of magnitude smaller than previous calculationssuggest. The relation between the formation criterion stated in terms of thedensity contrast and the curvature perturbation $\mathcal{R}$ is alsodiscussed.

The primordial black hole formation criterion re-examined: parameterisation, timing, and the choice of window function