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