Strong gravitational lensing by Bardeen black holes in 4D EGB gravity: constraints from supermassive black holes

Abstract

Observation indicates that many nearby galaxies host supermassive central black holes. Modelling Bardeen models in four-dimensional Einstein-Gauss-Bonnet (4D EGB) gravity, with additional parameters $\tilde{\alpha}$ and charge $q$, as central black holes in various galaxies, we investigate gravitational lensing properties in strong deflection limits. Interestingly, the spherical photon orbit radius $x_m$, the critical impact parameter $u_m$, the lensing coefficient $\bar{b}$, the deflection angle $\alpha_D(\theta)$, angular position $\theta_{\infty}$ are decreasing with $q$ and $\alpha$ whereas the other lensing coefficient $\bar{a}$ and angular separation $s$ have opposite behaviour. Taking the supermassive black holes Sgr A* and M87* as the lens, we also compare observable signatures of 4D EGB Bardeen black holes with those of the Schwarzschild black holes. The angular position $\theta_\infty$ for Sgr A* $\in$ (23.1853, \; 25.56427) $\mu$as, whereas for M87* it is $\in$ ( 17.941,\; 19.7819) $\mu$as. Further, the angular separation $s$, which is an increasing function of $\tilde{\alpha}$ and $q$ for Sgr A* and M87* differs significantly, respectively, in (0.031997,0.14895) $\mu$as and (0.0247, 0.1152) $\mu$as. The deviations of the lensing observables $\Delta \theta_{\infty}$ and $\Delta s$ for 4D EGB Bardeen black hole ($\tilde{\alpha}=0.9,~q=0.09$) from the Schwarzschild black hole, respectively, can reach up to $2.3789~\mu$as and $0.11695~\mu$as for Sgr A* , $1.84084~\mu$as and $0.0905~\mu$as for M87*. On the other hand, the relative magnification $\in$ (4.65751,\; 6.82173). Considering twenty-two massive central black holes as lens, we also estimate the time delay $\Delta T^s_{2,1}$ between the first and second relativistic image to find that, e.g., the time delay for Sgr A* and M87*, respectively, can reach $\sim9.86088$~min and $\sim16023.93$~min.