Primordial Black Holes and Slow-Roll Violation
  For primordial black holes (PBH) to be the dark matter in single-field
inflation, the slow-roll approximation must be violated by at least ${\cal
O}(1)$ in order to enhance the curvature power spectrum within the required
number of efolds between CMB scales and PBH mass scales. Power spectrum
predictions which rely on the inflaton remaining on the slow-roll attractor can
fail dramatically leading to qualitatively incorrect conclusions in models like
an inflection potential and misestimate the mass scale in a running mass model.
We show that an optimized temporal evaluation of the Hubble slow-roll
parameters to second order remains a good description for a wide range of PBH
formation models where up to a $10^7$ amplification of power occurs in $10$
efolds or more.
