The detection of gravitational waves (GWs) propagating through cosmic structures can provide invaluable information on the geometry and content of our Universe as well as on the fundamental theory of gravity. In order to test possible departures from General Relativity, it is essential to analyze, in a modified gravity setting, how GWs propagate through a perturbed cosmological space-time. Working within the framework of geometrical optics, we develop tools to address this topic for a broad class of scalar-tensor theories, including scenarios with non-minimal, derivative couplings between scalar and tensor modes.

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.

We analyze the propagation of high-frequency gravitational waves (GW) in scalartensor theories of gravity, with the aim of examining properties of cosmological distances as inferred from GW measurements. By using symmetry principles, we first determine the most general structure of the GW linearized equations and of the GW energy momentum tensor, assuming that GW move with the speed of light. Modified gravity effects are encoded in a small number of parameters, and we study the conditions for ensuring graviton number conservation in our covariant set-up. We then apply our general findings to the case of GW propagating through a perturbed cosmological space-time, deriving the expressions for the GW luminosity distance d (GW) L and the GW angular distance d (GW) A. We prove for the first time the validity of Etherington reciprocity law dfor a perturbed universe within a scalar-tensor framework. We find that besides the GW luminosity distance, also the GW angular distance can be modified with respect to General Relativity. We discuss implications of this result for gravitational lensing, focussing on time-delays of lensed GW and lensed photons emitted simultaneously during a multimessenger event. We explicitly show how modified gravity effects compensate between different coefficients in the GW time-delay formula: lensed GW arrive at the same time as their lensed electromagnetic counterparts, in agreement with causality constraints.