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 $\tilde m_4^2$ (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 $\tilde m_4^2$ operator self-interactions can be neglected, at least in some regimes. The modification of the gravitational-wave signal is observable for $3 \times 10^{-20} \lesssim \alpha_{\rm H} \lesssim 10^{-17}$ with a LIGO/Virgo-like interferometer and for $10^{-16} \lesssim \alpha_{\rm H} \lesssim 10^{-10}$ with a LISA-like one.