Combining gravitational lensing and gravitational redshift to measure the anisotropic stress with future galaxy surveys

Galaxy surveys provide one of the best ways to constrain the theory of gravity at cosmological scales. They can be used to constrain the two gravitational potentials encoding time, $\Psi$, and spatial, $\Phi$, distortions, which are exactly equal at late time within general relativity. Hence, any small variation leading to a nonzero anisotropic stress, i.e. a difference between these potentials, would be an indication for modified gravity. Current analyses usually consider gravitational lensing and redshift-space distortions to constrain the anisotropic stress, but these rely on certain assumptions like the validity of the weak equivalence principle, and a specific time evolution of the functions encoding deviations from general relativity. In this work, we propose a reparametrization of the gravitational lensing observable, together with the use of the relativistic dipole of the correlation function of galaxies to directly measure the anisotropic stress with a minimum amount of assumptions. We consider the future Legacy Survey of Space and Time of the Vera C. Rubin Observatory and the future Square Kilometer Array, and show that combining gravitational lensing and gravitational redshift with the proposed approach we will achieve model-independent constraints on the anisotropic stress at the level of $\sim 20\,\%$.