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