Precision Joint Constraints on Cosmology and Gravity Using Strongly Lensed Gravitational Wave Populations

We present a unified Bayesian framework to jointly constrain the Hubble constant $H_0$ and the post-Newtonian parameter $\gamma$, a key probe of deviations from general relativity, using the population characteristics of strongly lensed gravitational wave (GW) events from binary black hole mergers. Unlike traditional methods that rely on electromagnetic counterparts or GW waveform modeling, our approach exploits the time-delay distribution and the total number of lensed events, achievable with third-generation detectors such as the Einstein Telescope. Assuming a flat $\Lambda$CDM cosmology, we demonstrate that this method can achieve precision levels of $0.4\% - 0.7\%$ for$ H_0$ and $0.5\% - 3.3\%$ for $\gamma$ at $68\%$ credibility, significantly outperforming existing joint constraints. These results underscore the power of lensed GW population statistics as a robust and efficient probe of both cosmic expansion and the nature of gravity.