Peak statistics for the primordial black hole abundance

The primordial black hole (PBH) abundance evaluated by the conventional Press-Schechter (PS) probability distribution is shown to be equivalent to the high-peak limit of a special point-like peak statistics via unphysical dimensionality reduction of the Bardeen-Bond-Kaiser-Szalay (BBKS) theory. The fact that PBHs are formed at high peak values $\nu_c \gg 1$ leads to a systematic bias proportional to $\nu_c^3$ between the predictions of the PS method and the physical BBKS peak theory in a general three-dimensional spatial configuration. As long as realistic PBHs are collapsed from three-dimensional density peaks in space, the systematic bias led by $\nu_c^3$ implies a significant underestimation of the PBH abundance reported by the PS method. For the inflationary spectrum in the narrow-spike class, the underestimation in the extended mass functions is further enlarged by at least a factor of $10^{2.5}$ in all mass range, indicating a severer constraint to models in the favor of considering PBHs as all dark matter in a certain mass range.