  Primordial black holes (PBHs) can be produced by a range of mechanisms in the
early universe. A particular formation channel that connects PBHs with
inflationary phenomenology invokes enhanced primordial curvature perturbations
at small scales. In this paper, we re-examine the impact of the growth of the
primordial power spectrum on PBH formation in terms of its implications for the
PBH mass function. We elaborate on how rapidly the background can transition
between different values of the parameters of the Hubble hierarchy, which must
ultimately derive from a consistent derivative expansion for the background
inflaton field. We discuss artefacts associated with matching calculations, and
highlight the robustness of the $k^4$ steepest growth previously found for
single-field inflation with conservatively smoothed transitions and limits on
how much the amplitude of the power spectrum can grow. We show that the mass
function is relatively insensitive to the steepness of the growth of the power
spectrum and subsequent decay, depending primarily on the peak amplitude and
the presence of any plateaus that last more than an e-fold. The shape of the
power spectrum can of course be constrained by other tracers, and so
understanding the physical limitations on its shape remains a pertinent
question.

Cosmology and Nongalactic Astrophysics