Probing the Dark Universe with Gravitational Waves

Gravitational waves (GWs) are expected to interact with dark energy and dark matter, affecting their propagation on cosmological scales. To model this interaction, we derive a gauge-invariant effective equation and action valid for all GW polarizations. This is achieved by encoding the effects of GW interactions at different orders of perturbation into a polarization-, frequency-, and time-dependent effective speed. The invariance of perturbations under time-dependent conformal transformations and the gauge invariance of GWs allow us to derive the unitary gauge effective action in any conformally related frame, thereby clarifying the relationship between the Einstein and Jordan frames. Tests of the polarization and frequency dependencies in the propagation time and luminosity distance of different GW polarizations allow us to probe the dark Universe, which acts as an effective medium, modeled by the GW effective speed.