Title: A slippery slope: how fast can the primordial power spectrum grow?

Abstract: In order to produce even just one primordial black hole, a very large boost in the primordial power spectrum is required. There have been various investigations into how sharp this boost can be, with the consensus settling around a spectral index of 4 or slightly larger. We show, however, that the precise slope of the power spectrum has very little effect on the PBH mass function. Furthermore, we show that sharp transitions in the inflationary parameters, which are often required to boost the power spectrum quickly, should be smoothed. This means that power spectra with slopes much greater than k^4 are difficult to manifest. We review the implications of this in the context of the hunt for primordial black holes, and the corresponding model of inflation that can produce them.