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.