Constraining a modified gravity theory in strong gravitational lensing and black hole shadow observations

We study the strong gravitational lensing effect around rotating black holes in different gravity theories. By calculating the deflection angle of strong gravitational lensing, we evaluate the lensing observables including the image position, separation, magnification, and the time delays between the relativistic images of different rotating black holes. We argue that the differences in image positions, separations between the rotating black hole in modified gravity (MOG) theory and the Kerr black hole in general relativity (GR) are more significant in SgrA* than those in M87*; however the differences in time delays between rotating black holes in MOG and GR are shorter in SgrA* than that in M87*. Our evaluations on lensing observables in the strong gravity regime can help to distinguish the MOG from GR. Furthermore, we investigate the shadow observables of different rotating black holes. Employing the Event Horizon Telescope observations on the angular shadow radius for supermassive M87* and SgrA* black holes, respectively, we estimate the ranges of MOG parameter and obtain its upper limit constraint $0.350\ensuremath{\lesssim}{\ensuremath{\alpha}}_{\mathrm{up}}\ensuremath{\lesssim}0.485$ and $0.162\ensuremath{\lesssim}{\ensuremath{\alpha}}_{\mathrm{up}}\ensuremath{\lesssim}0.285$ correspondingly, relating to black hole spins. This is the first constraint on the MOG parameter for rotating supermassive black holes from Event Horizon Telescope observations on the angular shadow radius. Our constraint on the MOG parameter is much tighter compared with the result obtained from the orbital precession of the S2 star.

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