Gravitational Wave Decay into Dark Energy

We study the decay of gravitational waves into dark energy fluctuations $pi$, through the processes $gamma to pipi$ and $gamma to gamma pi$, made possible by the spontaneous breaking of Lorentz invariance. Within the EFT of Dark Energy (or Horndeski/beyond Horndeski theories) the first process is large for the operator $frac12 tilde m_4^2(t) , delta g^{00}, left( {}^{(3)}! R + delta K_mu^ u delta K^mu_ u -delta K^2 right)$, so that the recent observations force $tilde m_4 =0$ (or equivalently $alpha_{rm H}=0$). This constraint, together with the requirement that gravitational waves travel at the speed of light, rules out all quartic and quintic GLPV theories. Additionally, we study how the same couplings affect the propagation of gravitons at loop order. The operator proportional to $tilde m_4^2$ generates a calculable, non-Lorentz invariant higher-derivative correction to the graviton propagation. The modification of the dispersion relation provides a bound on $tilde m_4^2$ comparable to the one of the decay. Conversely, operators up to cubic Horndeski do not generate sizeable higher-derivative corrections.