Integrated Sachs-Wolfe effect for gravitational radiation

Gravitational waves (GWs) are messengers carrying valuable information about their sources. For sources at cosmological distances, the waves will also contain the imprint left by the intervening matter. The situation is in close analogy with cosmic microwave photons, for which the large-scale structures the photons traverse contribute to the observed temperature anisotropies, in a process known as the integrated Sachs-Wolfe effect. We derive the GW counterpart of this effect for waves propagating on a Friedman-Robertson-Walker background with scalar perturbations. We find that the phase, frequency, and amplitude of the GWs experience Sachs-Wolfe-type integrated effects, in addition to the magnification effects on the amplitude from gravitational lensing. We show that for supermassive black hole binaries, the integrated effects could account for measurable changes on the frequency, chirp mass, and luminosity distance of the binary, thus unveiling the presence of inhomogeneities, and potentially dark energy, in the universe.