Prospect of Measuring the Cosmic Dipole by Strongly Lensed Gravitational Waves Associated with Galaxy Surveys

The cosmic dipole observed in the cosmic microwave background (CMB) is traditionally interpreted as being caused by the observer's motion relative to the background. However, tensions with dipole measurements from radio galaxy counts motivate the need for independent probes. This work investigates the feasibility of using strongly lensed gravitational wave (GW) events to measure the cosmic dipole. Strongly lensed GWs produce multiple time-delayed images, which can be used to infer the distances to both the lens and the source. These distances, associated with the observed redshifts of the lens and the source from galaxy catalogues, encode information about the background cosmology and cosmic dipole effects. By reconstructing a statistical sample of doubly lensed GW events based on the singular isothermal sphere lens model, the cosmic dipole can be estimated jointly with background cosmological parameters. Using realistic simulations for Einstein Telescope and Cosmic Explorer, we forecast that a dipole magnitude $g$ consistent with both the CMB and number count measurement could be detected with 10 years of observation. Furthermore, constraints on $g$ are greatly improved by combining constraints from doubly lensed events with those from triply or quadruply lensed events. In the most optimistic scenario, where we measure the number count dipole magnitude with 10 years of observation, we obtain $g = (2.45^{+1.53}_{-1.28}) \times 10^{-3}$ from the combined constraint, provided that systematic uncertainties are mitigated. Although challenging, strongly lensed GWs offer a novel approach to measuring the cosmic dipole, providing an independent consistency test with different systematics from electromagnetic probes.