Steepest growth re-examined: repercussions for primordial black hole
  formation
  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 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. 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 steepest growth
index previously found for single-field inflation with conservatively smoothed
transitions and limits on how much the power spectrum can grow.
