Are primordial black holes produced by entropy perturbations in single
  field inflationary models?
  We show that in single field inflationary models the super-horizon evolution
of curvature perturbations on comoving slices $\mathcal{R}$, which can cause
the production of primordial black holes (PBH), is not due to entropy
perturbations, but to the background evolution effect on the conversion between
entropy and curvature perturbations. We derive a general relation between the
time derivative of comoving curvature perturbations and entropy perturbations,
in terms of a conversion factor depending on the background evolution. Contrary
to previous results derived in the uniform density gauge assuming the gradient
term can be neglected on super-horizon scales, the relation is valid on any
scale for any minimally coupled single scalar field model, also on sub-horizon
scales where gradient terms are large.
  We apply it to the case of quasi-inflection inflation, showing that while
entropy perturbations are decreasing, $\mathcal{R}$ can grow on super-horizon
scales, due to a large increase of the conversion factor. This happens in the
time interval during which a sufficiently fast decrease of the equation of
state $w$ transforms into a growing mode that in slow-roll models would be a
decaying mode. The same mechanism also explains the super-horizon evolution of
$\mathcal{R}$ in globally adiabatic systems, for which entropy perturbations
vanish on any scale, such as ultra slow-roll inflation and its generalizations.
