Direct Estimate of the Post-Newtonian Parameter and Cosmic Curvature from Galaxy-scale Strong Gravitational Lensing

Einstein's theory of general relativity (GR) has been precisely tested on solar system scales, but extragalactic tests are still poorly performed. In this work, we use a newly compiled sample of galaxy-scale strong gravitational lenses to test the validity of GR on kiloparsec scales. In order to solve the circularity problem caused by the presumption of a specific cosmological model based on GR, we employ the distance sum rule in the Friedmann–Lemaître–Robertson–Walker metric to directly estimate the parameterized post-Newtonian (PPN) parameter γPPN and the cosmic curvature Ωk by combining observations of strong lensing and Type Ia supernovae. This is the first simultaneous measurement of γPPN and Ωk without any assumptions about the contents of the universe or the theory of gravity. Our results show that ${\gamma }_{\mathrm{PPN}}={1.11}_{-0.09}^{+0.11}$ and ${{\rm{\Omega }}}_{k}={0.48}_{-0.71}^{+1.09}$, indicating a strong degeneracy between the two quantities. The measured γPPN, which is consistent with the prediction of 1 from GR, provides a precise extragalactic test of GR with a fractional accuracy better than 9.0%. If a prior of the spatial flatness (i.e., Ωk = 0) is adopted, the PPN parameter constraint can be further improved to ${\gamma }_{\mathrm{PPN}}={1.07}_{-0.07}^{+0.07}$, representing a precision of 6.5%. On the other hand, in the framework of GR (i.e., γPPN = 1), our results are still marginally compatible with zero curvature (${{\rm{\Omega }}}_{k}=-{0.12}_{-0.36}^{+0.48}$), supporting no significant deviation from a flat universe.