Ultra slow-roll with a black hole

We investigate ultra slow-roll inflation with a seed black hole in a de Sitter background. By numerically tracking transitions from slow-roll to ultra slow-roll inflation, we find that quasi-normal mode solutions of the scalar field are excited following the decay of the slow-roll attractor, depending on the mass of the black hole. For small black holes, the picture is similar to standard inflation with the usual damping of the scalar field; with a large black hole, we find that the ringing modes dominate. It is believed that the transition to ultra slow-roll in the pure inflationary case enhances the peak of the primordial power spectrum, thereby increasing the likelihood of primordial black hole formation. We comment on how the novel ringing behaviour due to the seed black hole might impact on cosmological perturbations.