On the slope of curvature power spectrum in non-attractor inflation
  The possibility that primordial black holes constitute a fraction of dark
matter motivates a detailed study of possible mechanisms for their production.
Black holes can form by the collapse of primordial curvature fluctuations, if
the amplitude of their small scale spectrum gets amplified by several orders of
magnitude with respect to CMB scales. Such enhancement can for example occur in
single-field inflation that exhibit a transient non-attractor phase: in this
work, we make a detailed investigation of the shape of the curvature spectrum
in this scenario. We make use of an analytical approach based on a gradient
expansion of curvature perturbations, which allows us to follow the changes in
slope of the spectrum during its way from large to small scales. After
encountering a dip in its amplitude, the spectrum can acquire steep slopes with
a spectral index up to $n_s-1\,=\,8$, to then relax to a more gentle growth
with $n_s-1\,\lesssim\,3$ towards its peak, in agreement with the results found
in previous literature. For scales following the peak associated with the
presence of the non-attractor phase, the spectrum amplitude then mildly decays,
during a transitional stage from non-attractor back to attractor evolution. Our
analysis indicates that this gradient approach offers a transparent
understanding of the contributions controlling the slope of the curvature
spectrum. As an application of our findings, we characterise the slope in
frequency of a stochastic gravitational wave background generated at second
order from curvature fluctuations, using the more accurate information we
gained on the shape of curvature power spectrum.
