Enhanced Localization of Dark Lensed Gravitational Wave Events Enables Host Galaxy Identification and Precise Cosmological Inference

Lensed gravitational wave (GW) events are expected to be powerful new probes of cosmology, contingent on redshift measurement by electromagnetic observations. Host galaxy identification is thus crucial but challenging due to poor localization by GW signal alone. In this paper, we show that the third-generation ground-based GW detectors will detect a population of lensed events with three or more detectable images (including the central one), each arriving at distinct times and Earth locations in the space, forming an effective network that reduces the typical localization area to $\sim0.01$ deg$^2$. For at least $90\%$ (or $50\%$) of these events, the localization improves by more than a factor of $10$ (or $30$) comparing with unlensed cases. Such precise localization and multiple-image detections enable robust host-galaxy identification and, through lens modelling, further yield sub-arcsecond position. As ``dark lensed sirens", these events become powerful probes of cosmological parameters. Using simulated lensed compact-binary mergers, we show that two-year or longer observations with third-generation GW detectors can measure the Hubble constant to $\lesssim1$\% precision via ``dark lensed sirens" (even when relying solely on lensed stellar-mass binary black hole events), while simultaneously constraining other cosmological parameters. This approach will provide an independent, complementary avenue for measuring cosmological parameters.