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 O (1) in order to enhance the curvature power spectrum within the required number of e-folds between cosmic microwave background 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 e-folds or more.