  We apply the in-in formalism to address the question of whether the size of
the one-loop spectrum of curvature fluctuations in ultra-slow-roll inflation
models designed for producing a large population of primordial black holes
implies a breakdown of perturbation theory. We consider a simplified piece-wise
description of inflation, in which the ultra-slow-roll phase is preceded and
followed by slow-roll phases linked by transitional periods. We work in the
$\delta\phi$-gauge, including all relevant cubic and quartic interactions and
the necessary counterterms to renormalize the ultraviolet divergences,
regularized by a cutoff. The ratio of the one-loop to the tree-level
contributions to the spectrum of curvature perturbations is controlled by the
duration of the ultra-slow-roll phase and of the transitions. Our results
indicate that perturbation theory does not necessarily break in well-known
models proposed to account for all the dark matter in the form of primordial
black holes.
