On primordial black holes from an inflection point
  Recently, it has been claimed that inflationary models with an inflection
point in the scalar potential can produce a large resonance in the power
spectrum of curvature perturbation. In this paper however we show that the
previous analyses are incorrect. The reason is twofold: firstly, the inflaton
is over-shot from a stage of standard inflation and so deviates from the
slow-roll attractor before reaching the inflection. Secondly, on the (or close
to) the inflection point, the ultra-slow-roll trajectory supersede the
slow-roll one and thus, the slow-roll approximations used in the literature
cannot be used. We then reconsider the model and provide a recipe for how to
produce nevertheless a large peak in the matter power spectrum via fine-tuning
of parameters.
