Primordial Black Holes from Polynomial Potentials in Single Field Inflation

TL;DR

This paper presents a method to reconstruct inflaton potentials from power spectra, demonstrating how polynomial potentials can generate primordial black holes by violating slow-roll conditions and over-shooting local minima during inflation.

Contribution

It introduces a reverse engineering approach for inflaton potentials, specifically showing how polynomial potentials can produce PBHs by violating slow-roll conditions.

Findings

A polynomial potential can generate significant PBH formation.

Violating slow-roll conditions is necessary for large density perturbation spikes.

The method allows estimation of fine-tuning needed for PBH production in string landscape models.

Abstract

Within canonical single field inflation models, we provide a method to reverse engineer and reconstruct the inflaton potential from a given power spectrum. This is not only a useful tool to find a potential from observational constraints, but also gives insight into how to generate a large amplitude spike in density perturbations, especially those that may lead to primordial black holes (PBHs). In accord with other works, we find that the usual slow-roll conditions need to be violated in order to generate a significant spike in the spectrum. We find that a way to achieve a very large amplitude spike in single field models is for the classical roll of the inflaton to over-shoot a local minimum during inflation. We provide an example of a quintic polynomial potential that implements this idea and leads to the observed spectral index, observed amplitude of fluctuations on large scales, significant PBH formation on small scales, and is compatible with other observational constraints. We quantify how much fine-tuning is required to achieve this in a family of random polynomial potentials, which may be useful to estimate the probability of PBH formation in the string landscape.