Detecting dark energy fluctuations with gravitational waves

Luminosity distance estimates from electromagnetic and gravitational wave sources are generally different in models of dynamical dark energy gravity beyond the standard cosmological scenario. We show that this leaves a unique imprint on angular power-spectrum fluctuations luminosity gravitational-wave observations which tracks inhomogeneities field. Exploiting synergy supernovae measurements, we build joint estimator directly probes fluctuations, providing conclusive evidence for their existence case detection. Moreover, such measurement would also allow to probe running Planck mass. discuss experimental requirements detect these signals.