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 ultraslow-roll stage. During this stage, the power spectrum of primordial curvature perturbation is remarkably enhanced on small scales, generating abundant PBHs. The PBH abundance $f\ensuremath{\sim}0.1$ can be achieved in one or two typical mass windows at ${10}^{\ensuremath{-}17}\text{ }\text{ }{M}_{\ensuremath{\bigodot}}$, ${10}^{\ensuremath{-}13}\text{ }\text{ }{M}_{\ensuremath{\bigodot}}$, and $30\text{ }\text{ }{M}_{\ensuremath{\bigodot}}$, 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 2 or 3 orders of magnitude compared with peak theory.
Read full abstract