Resonant decay of gravitational waves into dark energy

We study the decay of gravitational waves into dark energy fluctuations pi, taking into account the large occupation numbers. We describe dark energy using the effective field theory approach, in the context of generalized scalar-tensor theories. When the m(3)(3) (cubic Horndeski) and (m) over tilde (2)(4) (beyond Horndeski) operators are present, the gravitational wave acts as a classical background for pi and modifies its dynamics. In particular, pi fluctuations are described by a Mathieu equation and feature instability bands that grow exponentially. Focusing on the regime of small gravitational-wave amplitude, corresponding to narrow resonance, we calculate analytically the produced pi, its energy and the change of the gravitational-wave signal. The resonance is affected by pi self-interactions in a way that we cannot describe analytically. This effect is very relevant for the operator m(3)(3) and it limits the instability. In the case of the (m) over tilde (2)(4) operator self-interactions can be neglected, at least in some regimes. The modification of the gravitational-wave signal is observable for 3 x 10(-20) less than or similar to alpha(H) less than or similar to 10(-17) with a LIGO/Virgo-like interferometer and for 10(-16) less than or similar to alpha(H) less than or similar to 10(-10) with a LISA-like one.