Resonant decay of gravitational waves into dark energy

We study the decay of gravitational waves into dark energy fluctuations π, 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 m33 (cubic Horndeski) and ∼ m42 (beyond Horndeski) operators are present, the gravitational wave acts as a classical background for π and modifies its dynamics. In particular, π 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 π, its energy and the change of the gravitational-wave signal. The resonance is affected by π self-interactions in a way that we cannot describe analytically. This effect is very relevant for the operator m33 and it limits the instability. In the case of the ∼ m42 operator self-interactions can be neglected, at least in some regimes. The modification of the gravitational-wave signal is observable for 3 × 10-20 ≤ αH ≤ 10-17 with a LIGO/Virgo-like interferometer and for 10-16 ≤ αH ≤ 10-10 with a LISA-like one.