Gravitational Wave Decay into Dark Energy
  We study the decay of gravitational waves into dark energy fluctuations
$\pi$, through the processes $\gamma \to \pi\pi$ 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 $\frac12 \tilde m_4^2(t) \, \delta g^{00}\, \left(
{}^{(3)}\! R + \delta K_\mu^\nu \delta K^\mu_\nu -\delta K^2 \right)$, 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.
