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Testing the Equivalence Principle with Strong Lensing Time Delay Variations

Strong lensing time delay measurements provide a valuable and almost model-independent tool for cosmological investigations. In this work we recognize that they also carry information on the strength of the gravitational coupling at the redshift of the lens, and thus could be in principle used to test the equivalence principle on extragalactic scales. For the case of an extended lens with a static mass distribution we explicitly derive an analytical relation between $\dot{G}/G$ and the relative variation of the time delay. For illustrative purpose, we apply our formula to the light curves of multiple images of the quasar DES J0408-5354 and simulated ones, which results in weak constraints on the variation of $\dot{G}/G$ of order $10^{-1}-10^{-2} \, yr^{-1}$ in the best scenario. Finally, we briefly discuss how those constraints can be improved in the next future.