Primordial black holes from the perturbations in the inflaton potential
  in peak theory
  The primordial black hole (PBH) is an effective candidate for dark matter. In
this work, the PBH abundance $f$ is calculated in peak theory, with one or two
perturbations in the inflaton potential. We construct an antisymmetric
perturbation that can create a perfect plateau in the inflaton potential,
leading inflation to the ultra-slow-roll stage. During this stage, the power
spectrum of primordial curvature perturbation is remarkably enhanced on small
scales, generating abundant PBHs. The PBH abundance $f\sim 0.1$ can be achieved
in one or two typical mass windows at $10^{-17}M_\odot$, $10^{-13}M_\odot$, and
$30M_\odot$, without spoiling the nearly scale-invariant power spectrum on
large scales. For comparison, $f$ is calculated in two approximate methods of
peak theory (with different spectral moments) and also in the Press--Schechter
theory. It is found that the Press--Schechter theory systematically
underestimates $f$ by two or three orders of magnitude compared with peak
theory.
