Enhanced induced gravitational waves in Horndeski gravity

We study secondary gravitational wave production in Horndenski gravity, when the scalar field dominates the very early universe. We find that higher derivative interactions easily dominate the source term on subhorizon scales and significantly enhance the amplitude of induced GWs. We analytically derive, for the first time, the Horndeski induced GW spectrum for a general class of power-law solutions. The main effect of modifications of gravity are stronger resonances and a growth of tensor fluctuations on small scales. The maximum attainable amplitude of the induced GW spectrum is bounded by the possible backreaction of higher derivatives on curvature fluctuations, thereby shutting down the source term to induced GWs. We argue that the maximum attainable amplitude depends linearly on the primordial curvature spectrum ($\Omega_{\rm GW}\propto {\cal P}_\zeta$), as opposed to the standard case where it depends quadratically. Resonances may further enhance the maximum amplitude by a factor $(k/{\cal H}_t)^2$ or $(k/{\cal H}_t)$ respectively for sharp and broad peaks (including a scale invariant) primordial spectrum, where ${\cal H}_t$ is the comoving horizon at the time when standard gravity is recovered. Remarkably, in the scale invariant case, the Horndeski-induced GW spectrum grows as $k^3$. This opens up the interesting possibility that induced GWs might be observable despite no enhancement of the primordial curvature spectrum. Our formalism can be generalized to a wider class of solutions and to more general scalar-tensor theories, such as DHOST and spatially covariant gravity. In the appendices we provide discussions on the gauge issue and disformal transformations of induced GWs.