  The ultra-slow-roll (USR) inflationary models predict large-amplitude scalar
perturbations at small scales which can lead to the primordial black hole
production and scalar-induced gravitational waves. In general scalar
perturbations in the USR models can only be obtained using numerical method
because the usual slow-roll approximation breaks. In this work, we propose an
analytical approach to estimate the scalar spectrum which is consistent with
the numerical result. We find that the USR inflationary models predict a peak
with power-law slopes in the scalar spectrum and energy spectrum of
gravitational waves, and we derive the expression of the spectral indexes in
terms of the inflationary potential. In turn, the inflationary potential near
the USR regime can be reconstructed from the negative spectral index of the
gravitational wave energy spectrum.
