Primordial black holes from the perturbations in the inflaton potential in peak theory

Abstract: 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.