Robust test of general relativity at the galactic scales by combining strong lensing systems and gravitational wave standard sirens

The measurement of the parametrized post-Newtonian parameter γ_PPN is a robust test of general relativity (GR). In some modified theories of gravity, γ_PPN may evolve with the redshift and deviate from one at high redshifts. This means that precise constraints on γ_PPN acquired in the solar system experiments could not be sufficient to test such theories and it is necessary to constrain γ_PPN with high precision at high redshifts. However, in many approaches aimed at extragalactic tests of GR, the results might be biased due to entanglement of various factors, such as cosmic curvature, cosmic opacity, and the Hubble constant. Strong lensing systems naturally provide a laboratory to test γ_PPN at galactic scales and high redshifts, but there is degeneracy between measured strength of gravity and cosmic distances in the lensing system. Gravitational waves (GWs) from binary neutron star mergers (standard sirens) provide a direct way to break this degeneracy by providing self-calibrated measurements of the luminosity distance. We investigate the possibility of estimating γ_PPN by combining well-measured strongly lensed systems with GW signals from coalescing neutron stars. Such combination provides a cosmological-model independent, relatively pure and unbiased method for the inference of γ_PPN parameter, avoiding the influence of the above factors and the mass-sheet degeneracy in the lens. Based on the simulated future 55 lensed quasar systems we demonstrated that the precision of γ_PPN parameter obtained by our method could be of order of ∼10^-2. One may reasonably expect that our approach will play an increasingly important role in precise testing the validity of general relativity at galactic scales and high redshifts.