We study the cosmological propagation of gravitational waves (GWs) beyond
general relativity (GR) across homogeneous and isotropic backgrounds. We
consider scenarios in which GWs interact with an additional tensor field and
use a parametrized phenomenological approach that generically describes their
coupled equations of motion. We analyze four distinct classes of derivative and
non-derivative interactions: mass, friction, velocity, and chiral. We apply the
WKB formalism to account for the cosmological evolution and obtain analytical
solutions to these equations. We corroborate these results by analyzing
numerically the propagation of a toy GW signal. We then proceed to use the
analytical results to study the modified propagation of realistic GWs from
merging compact binaries, assuming that the GW signal emitted is the same as in
GR. We generically find that tensor interactions lead to copies of the
originally emitted GW signal, each one with its own possibly modified
dispersion relation. These copies can travel coherently and interfere with each
other leading to a scrambled GW signal, or propagate decoherently and lead to
echoes arriving at different times at the observer that could be misidentified
as independent GW events. Depending on the type of tensor interaction, the
detected GW signal may exhibit amplitude and phase distortions with respect to
a GW waveform in GR, as well as birefringence effects. We discuss observational
probes of these tensor interactions with both individual GW events, as well as
population studies for both ground- and space-based detectors.
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