  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 examine how rapidly the background can
transition between different values of the parameters of the Hubble hierarchy
in the context of single-clock inflation, which must ultimately derive from a
consistent derivative expansion for the background inflaton field. We discuss
artefacts associated with instant or very rapid transitions, and consider the
impact on the steepest power spectrum growth and the formation of PBHs. In
particular, we 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, and
demonstrate that the PBH mass distribution is sensitive to the artefacts, which
go away when the transitions are smoothed. We also show that the mass
distribution is relatively insensitive to the steepness of the growth and
subsequent decay of the power spectrum, 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.
