Precision joint constraints on cosmology and gravity using strongly lensed gravitational wave populations

We present a Bayesian framework to jointly constrain the Hubble constant ($H_0$) and the post-Newtonian parameter ($γ$), a key indicator of deviations from general relativity, using the population characteristics of strongly lensed gravitational wave (GW) events from binary black hole mergers. Our method extracts cosmological and gravitational information directly from the statistical properties of lensed GW populations, without relying on electromagnetic counterparts of the GW events, waveform modeling, and resolved stellar kinematics of the lens galaxy. This establishes lensed GW statistics as a clean and independent probe of cosmic expansion and gravitational physics. Assuming a flat $Λ$CDM cosmology and simulating a GW population observed by the third-generation detector Einstein Telescope, we demonstrate that this method can achieve precision levels of $0.60\%\sim0.99\%$ for $H_0$ and $0.53\%\sim3.3\%$ for $γ$ with various priors of matter density, significantly outperforming existing joint constraints, which typically achieve $2\%$ precision on $H_0$ and $20\%$ precision on $γ$. These results highlight the potential of lensed GW population statistics as a robust and efficient tool for probing both the expansion history of the Universe and the nature of gravity.