Primordial Black Holes from Polynomial Potentials in Single Field
  Inflation
  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.
