Probing a NED inspired Magnetically Charged Black Hole in the Hernquist Dark Matter Halo

Abstract: With an intent to examine the combined effect of non-linear electrodynamics (NED) and dark matter (DM), we obtain a static and spherically symmetric solution with the black hole (BH) magnetically charged and immersed in the Hernquist DM halo (MHDM). The position of the event horizon $r_h$ and the critical impact parameter $b_m$ are then probed to gauge the extent of influence magnetic charge $g$ and halo parameters $α$, $β$ have on them. A recurring outcome of our analysis with respect to different BH observables is the nullification of competing effects of charge and halo parameters, leading to observables obtaining values equal to those for a Schwarzschild BH. This is also observed for $r_h$ and $b_m$. We delve into unraveling the impact of NED and DM combined on the strong gravitational lensing (GL) and its related observables, such as the angular separation, relative magnification, and the angular position of the inner, closely packed bright ring. Interestingly, we find combinations of charge and halo parameters that leave the deflection angle unchanged from the Schwarzschild case, thereby leading to a situation where an MHDM BH and a Schwarzschild BH become indistinguishable. Similar results are also observed for lensing observables. Finally, utilizing observations related to the angular diameter of super-massive BHs (SMBHs) $M87^*$ and $SgrA^*$ and employing the $χ^2$ test, we extract bounds on $g$, $α$, and $β$ signifying the viability of our BH model as an SMBH.