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.
