Light bending and gravitational lensing in energy-momentum-squared gravity

Abstract

In the present work, we derive the motion of light in the weak-field limit of energy-momentum-squared gravity (EMSG). To do so, we introduce the post-Newtonian expansion of this modified theory of gravity. It is shown that, in addition to the Newtonian potential, a new EMSG potential affects the trajectory of photons. As a result, in this theory, photons do not behave as predicted by general relativity (GR). To evaluate the EMSG theory by the Solar System tests, we study light deflection and Shapiro time delay. Regarding the results obtained in Bertotti et al. [Nature (London) 425, 374 (2003)] and Shapiro et al. [Phys. Rev. Lett. 92, 121101 (2004)], we restrict the free parameter of the theory and show that it lies within the range $\ensuremath{-}4.0\ifmmode\times\else\texttimes\fi{}{10}^{\ensuremath{-}27}\text{ }\text{ }\mathrm{m}\text{ }{\mathrm{s}}^{2}\text{ }{\mathrm{kg}}^{\ensuremath{-}1}<{f}_{0}^{\ensuremath{'}}<8.7\ifmmode\times\else\texttimes\fi{}{10}^{\ensuremath{-}26}\text{ }\text{ }\mathrm{m}\text{ }{\mathrm{s}}^{2}\text{ }{\mathrm{kg}}^{\ensuremath{-}1}$. This interval is in agreement with those derived in Nazari et al. [Phys. Rev. D 105, 044014 (2022)] and Akarsu et al. [Phys. Rev. D 97, 124017 (2018)]. This consistency manifests that this theory passes these Solar System tests with flying colors. Interestingly, it turns out that the magnitude of the EMSG correction strongly depends on the density of the deflector. So, we investigate the possible effects of EMSG on images of a light source microlensed by a compact dense object such as neutron stars. It is estimated that the EMSG correction to the position of lensed images could be as large as (1--0.1) microarcseconds, which may be detected by future high-resolution missions. Moreover, the total magnification and the shape of light curves are obtained in the EMSG theory. It is revealed that except for a small deviation, the overall behavior of the EMSG light curves is similar to that in GR. We also show that as long as the light source and the dense lens are aligned, the EMSG correction is effective, and the combined light of the lensed images is different from the GR case. This issue makes it possible to observe signatures of this theory in the microlensing regime.