Primordial black holes and induced gravitational waves from inflation in
  the Horndeski theory of gravity
  We investigate the production of primordial black holes (PBHs) and
scalar-induced gravitational waves (GWs) for cosmological models in the
Horndeski theory of gravity. The cosmological models of our interest
incorporate the derivative self-interaction of the scalar field and the kinetic
coupling between the scalar field and gravity. We show that the scalar power
spectrum of the primordial fluctuations can be enhanced on small scales due to
these additional interactions. Thus, the formation of PBHs and the production
of induced GWs are feasible for our model. Parameterizing the scalar power
spectrum with a local Gaussian peak, we first estimate the abundance of PBHs
and the energy spectrum of GWs produced in the radiation-dominated era. Then,
to explain the small-scale enhancement in the power spectrum, we reconstruct
the inflaton potential and self-coupling functions from the power spectrum and
their spectral tilt. Our results show that the small-scale enhancement in the
power spectrum can be explained by the local feature, either a peak or dip, in
the self-coupling function rather than the local feature in the inflaton
potential.
