On primordial black holes from an inflection point

Abstract

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.

Quantum diffusion during inflation and primordial black holes

“…[80][81][82]. This, for instance, happens when the inflationary potential has a flat inflection point [83], around which slow roll is transiently violated and one can even enter an ultra slow-roll phase.…”

Constant roll and primordial black holes

“…where the first term in the parenthesis was determined in (11). The friction term is removed by using the MS variable v k ≡ √ 2M Pl z ζ k , instead of directly using the curvature perturbation ζ k in the comoving gauge.…”

Inflation without gauge redundancy

“…The latter, in turn, could generate a peak in the scalar power spectrum at small scales. In such cases, it is well-known that the slow-roll approximation does not capture all the relevant physics, and one is forced to solve the inflationary dynamics exactly [6][7][8]. Naïvely, assuming the slow-roll approximation to be applicable, one is tempted to carry on the same analysis discussed before by simply replacing the tree-level potential in eq.…”

“…(3.43) only enters by means of the anomalous dimension of the scalar field φ. 7 In this respect, the discussion 7 It means that we can write…”

“…As mentioned in the introduction, there are well-motivated cases in which eq. (2.70) is not enough to fully describe the power spectrum at small scales in the presence of an ultra slow-roll phase [5][6][7][8]. In such cases, one has to solve eq.…”