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

In this paper, the criterion used to determine whether a density perturbation will collapse to form a primordial black hole (PBH) is re-examined, in respect of its use to determine the abundance of PBHs. There is particular focus on which parameter to use, the time at which the abundance should be calculated, and the use of different smoothing functions. It is concluded that, with the tools currently available, the smoothed density contrast should be used rather than the peak value, and should be calculated from the time-independent component of the density contrast in the super-horizon limit (long before perturbations enter the horizon) rather than at horizon crossing. For the first time the effect of the choice of smoothing function upon the formation criterion is calculated, and, for a given abundance of PBHs, it is found that the uncertainty in the amplitude of the power spectrum due to this is $\mathcal{O}(10\%)$, an order of magnitude smaller than previous calculations suggest. The relation between the formation criterion stated in terms of the density contrast and the curvature perturbation $\mathcal{R}$ is also discussed.