Probing modified gravitational-wave propagation with extreme mass-ratio inspirals

Extreme mass-ratio inspirals (EMRIs), namely binary systems composed of a massive black hole and a compact stellar-mass object, are anticipated to be among the gravitational wave (GW) sources detected by the Laser Interferometer Space Antenna (LISA). Similarly to compact binary mergers detected by current GW detectors, EMRIs can be used as cosmic rulers to probe the expansion of the Universe. Motivated by tensions in current cosmological observations as well as by alternative models of dark energy, modified gravity theories can affect the propagation of GWs across cosmological distances, with modifications commonly parametrised in terms of two phenomenological parameters, $\Xi_0$ and $n$. In this work we adopt a Bayesian approach to constrain for the first time parametrised deviations from General Relativity using the loudest simulated EMRIs detected by LISA as dark sirens with a simulated galaxy catalog. Assuming all the cosmological parameters except $\Xi_0$ are already tightly constrained, our forecasts show that $\Xi_0$ can be constrained to a few percent level (90% C.I.) with 4 years of LISA observations, unless EMRI detection rates turn out to be closer to current pessimistic expectations. These results quickly degrade if additional cosmological parameters are inferred simultaneously, but become more robust with an extended LISA observation period of 10 years. Overall, we find that EMRIs with LISA are better at constraining modified GW propagation than current second-generation ground-based GW detectors, but they will only be comparable to third-generation detectors in the most optimistic scenarios.