Amplification of Primordial Perturbations from the Rise or Fall of the
  Inflaton
  The next generation of cosmic microwave background, gravitational wave, and
large scale structure, experiments will provide an unprecedented opportunity to
probe the primordial power spectrum on small scales. An exciting possibility
for what lurks on small scales is a sharp rise in the primordial power
spectrum: This can lead to the formation of primordial black holes, providing a
dark matter candidate or the black holes observed by the LIGO-Virgo
collaboration. In this work we develop a mechanism for the amplification of the
small-scale primordial power spectrum, in the context of single-field inflation
with a step-like feature in the inflaton potential. Specifically, we consider
both the upward and the downward step in the potential. We also discuss the
possibility of the strong coupling between perturbations because the rapid
changes of the potential derivatives with the time-dependent field value,
caused by the step-like feature, could make the coupling stronger. As a result,
we find that the perturbations can remain weakly coupled yet sufficiently
enhanced if the step realizes the rapid changes of the potential derivatives in
some fraction of an e-fold, $\mathcal O(\mathcal P_{\mathcal R}^{1/2}) \lesssim
\Delta N < 1$, where $\mathcal P_\mathcal R$ is the power spectrum of the
curvature perturbation at that time. We also discuss the PBH formation rate
from the inflaton trapping at the local minimum, which can occur in the
potential with an upward step.
