Gravitational Wave Decay: Implications for cosmological scalar-tensor theories

The recent discovery that gravitational waves and light travel with the same speed, with an error below $10^{-15}$, has greatly constrained the parameter space of infrared modifications of gravity. In this thesis we study the phenomenology of gravitational-wave propagation in modifications of gravity relevant for dark energy with an additional scalar degree of freedom. Of particular interest are Horndeski and Beyond Horndeski models surviving after the event GW170817. Here the dark energy field is responsible for the spontaneous breaking of Lorentz invariance on cosmological scales. This implies that gravitons $gamma$ can experience new dispersion phenomena and in particular they can decay into dark energy fluctuations $pi$. First, we study the perturbative decay channels $gamma ightarrow pipi$ and $gamma ightarrowgammapi$ in Beyond Horndeski models. The first process is found to be large and thus incompatible with recent gravitational-wave observations. This provides a very stringent constraint for the particular coefficient $ m{4}{2}$ of the Effective Field Theory of Dark Energy or, in the covariant language, on quartic Beyond Horndeski operators. We then study how the same coupling affects at loop level the propagation of gravitons. It is found that the new contribution modifies the dispersion relation in a way that is incompatible with current observations, giving bounds of the same magnitude as the decay. Next, we improve our analysis of the decay by taking into account the large occupation number of gravitons and dark energy fluctuations in realistic situations. When the operators $m_3^3$ (cubic Horndeski) and $ m{4}{2}$ are present, we show that the gravitational wave acts as a classical background for $pi$ and affects its dynamics, with $pi$ growing exponentially. In the regime of small gravitational-wave amplitude, we compute analytically the produced $pi$ and the change in the gravitational wave. For the operator $m_3^3$, $pi$ self-interactions are of the same order as the resonance and affect the growth in a way that cannot be described analytically. For the operator $ m{4}{2}$, in some regimes self-interactions remain under control and our analysis improves the bounds from the perturbative decay, ruling out quartic Beyond Horndeski operators from having any relevance for cosmological applications. Finally, we show that in the regime of large amplitude for the gravitational wave $pi$ becomes unstable. If $m_3^3$ takes values relevant for cosmological applications, we conclude that dark energy fluctuations feature ghost and gradient instabilities in presence of gravitational waves of typical binary systems. Taking into account the populations of binary systems, we find that the instability is triggered in the whole Universe. The fate of the instability and the subsequent time-evolution of the system depends on the UV completion, so that the theory may end up in a state very different from the original one. In conclusion, the only dark-energy theories with sizeable cosmological effects that avoid these problems are $k$-essence models, with a possible conformal coupling with matter.