Shadow behavior of an EMSG charged black hole

Recent shadow images of Sgr A* and M87* captured by Event Horizon Telescope (EHT) collaboration confirm the existence of black holes or their possible alternatives in the center of galaxies. On the other hand the new image of Sgr A* in polarized light suggests a Magnetic field spiraling at the Edge of the Milky Way’s Central Black Hole. Due to gravitational lensing effect, bending of light in the background geometry of the black hole casts a shadow. In recent years, black holes and their properties have been vastly studied in the framework of General Relativity and other modified theories of gravity. One of the possibilities to generalize GR is Energy–Momentum Squared Gravity (EMSG) which is constructed by adding a term proportional to $T^2=T^{\alpha \beta }{T}_{\alpha \beta }$ (where $T_{\alpha \beta }$ is the energy–momentum tensor) in the gravitational action. It is important to mention that EMSG modifies all matter field’s equation which leads to add some non-linear terms to Maxwell equations. EMSG theory as a modified theory of gravity predicts an asymptotically de Sitter charged black hole whose shadow cast and other related characteristics have not been examined yet. Hence we consider the EMSG charged black hole and investigate the shadow shape of this kind of black hole solution in confrontation with EHT results. In the case of non-linear electrodynamics the photon’s path is null on some effective metric. by deriving the effective metric of EMSG charged black hole we study the null geodesics of the effective metric in Hamilton–Jacobi method. we find the photon orbits and compute the shadow size of this black hole. Then we examine how electric charge and the coupling constant of the EMSG affect the shadow size of the black hole in a positively accelerated expanding universe (with a positive cosmological constant). We explore the viable values of these parameters constrained by EHT data by comparing the shadow radius of EMSG charged black hole with the shadow size of Sgr A*. We show for instance that for $Q=0.1$ in appropriate units, the coupling constant should be in the range of $0.01\le \eta \le 0.02$ in order to EMSG charged black hole to be the Sgr A*. Consecutively we obtain that in the case of $\eta =0.01$ the range of the electric charge could be $0.01\le Q\le 0.3$ in the adopted units. We observe that by enhancing the effect of the electric charge, the shadow size of this EMSG charged black hole increases accordingly By treating the energy emission rate of EMSG charged black hole, we demonstrate that for the small amount of the electric charge and large values of the coupling constant, the black hole evaporates faster.