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 ${\gamma }_{\mathrm{PPN}}$ is a robust test of general relativity (GR). In some modified theories of gravity, ${\gamma }_{\mathrm{PPN}}$ may evolve with the redshift and deviate from one at high redshifts. This means that precise constraints on ${\gamma }_{\mathrm{PPN}}$ acquired in the solar system experiments could not be sufficient to test such theories and it is necessary to constrain ${\gamma }_{\mathrm{PPN}}$ 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 ${\gamma }_{\mathrm{PPN}}$ 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 ${\gamma }_{\mathrm{PPN}}$ 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 ${\gamma }_{\mathrm{PPN}}$ 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 ${\gamma }_{\mathrm{PPN}}$ parameter obtained by our method could be of order of $\sim {10}^{-2}$. 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.

Figure 1

Redshift distribution of lenses and sources in a sample of 55 strong lensing systems expected to be observed in current and upcoming projects.

Figure 2

Posterior distributions for PPN parameters obtained by the combination of the future LSST survey and ET detector yields. Three panels illustrate the following parametrizations: ${\gamma }_{\mathrm{PPN}}={\gamma }_0$ constant case (left), ${\gamma }_{\mathrm{PPN}}=1+{\gamma }_1*z$ varying with redshift (middle), and ${\gamma }_{\mathrm{PPN}}=1+{\gamma }_1*a$ varying with the scale factor (right).

Figure 3

The reconstructed ${\gamma }_{\mathrm{PPN}}$ for the three parametrizations with their $1\sigma$ uncertainties. The purple dashed line represents the nonlocal Gauss-Bonnet theory. The green, pink, and blue bands correspond to the $1\sigma$ regions of the best-fitting in the constant, evolution with the redshift, and evolution with the scale factor cases, respectively.