Steepest growth of the power spectrum and primordial black holes

Abstract

We derive analytic bounds on the shape of the primordial power spectrum in\nthe context of single-field inflation. In particular, the steepest possible\ngrowth has a spectral index of $n_s - 1 = 4$ once transients have died down.\nIts primary implication is that any constraint on the power spectrum at a\nparticular scale can be extrapolated to an upper bound over an extended range\nof scales. This is important for models which generate relics due to an\nenhanced amplitude of the primordial scalar perturbations, such as primordial\nblack holes. In order to generate them, the power spectrum needs to grow many\norders of magnitude larger than its observed value on CMB scales - typically\nachieved through a phase of ultra slow-roll inflation - and is thus subject to\nadditional constraints at small scales. We plot all relevant constraints\nincluding CMB spectral distortions and gravitational waves sourced by scalar\nperturbations at second order. We show how this limits the allowed mass of\nPBHs, especially for the large masses of interest following recent detections\nby LIGO and prospects for constraining them further with future observations.\nWe show that any transition from approximately constant $\\epsilon$ slow-roll\ninflation to a phase where the power spectrum rapidly rises necessarily implies\nan intervening dip in power. We also show how to reconstruct a potential that\ncan reproduce an arbitrary time-varying $\\epsilon$, offering a complementary\nperspective on how ultra slow-roll can be achieved.\n