Dissecting the growth of the power spectrum for primordial black holes
  We consider the steepest rate at which the power spectrum from single field
inflation can grow, with the aim of providing a simple explanation for the
$k^4$ growth found recently. With this explanation in hand we show that a
slightly steeper $k^5 (\log k )^2$ growth is in fact possible. Moreover, we
argue that the power spectrum after a steep growth cannot immediately decay,
but must remain large for the $k$ modes which exit during a $\sim2$ e-fold
period. We also briefly consider how a strong growth can affect the spectral
index of longer wavelengths preceding the growth, and show that even the
conversion of isocurvature modes likely cannot lead to a stronger growth. These
results have implications for the formation of primordial black holes, and
other phenomena which require a large amplitude of power spectrum at short
scales.
