Peak statistics for the primordial black hole abundance
Résumé
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 νc≫1 leads to a systematic bias proportional to νc3 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 νc3 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 102.5 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.