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

Primordial black holes and induced gravitational waves from inflation in the Horndeski theory of gravity