Primordial Black Hole Formation in Starobinsky's Linear Potential Model

Abstract

We study the power spectrum of the comoving curvature perturbation $\cal R$ in the model that glues two linear potentials of different slopes, originally proposed by Starobinsky. We find that the enhanced power spectrum reaches its maximum at the wavenumber which is $\pi$ times the junction scale. The peak is $\sim2.61$ times larger than the ultraviolet plateau. We also show that its near-peak behavior can be well approximated by a constant-roll model, once we define the effective ultra-slow-roll $e$-folding number appropriately by considering the contribution from non-single-clock phase only. Such an abrupt transition to non-attractor phase can leave some interesting characteristic features in the energy spectrum of the scalar-induced gravitational waves, which are detectable in the space-borne interferometers if the primordial black holes generated at such a high peak are all the dark matter.