Strong Gravitational Lensing by Loop Quantum Gravity Motivated Rotating Black Holes and EHT Observations

We investigate gravitational lensing in the strong deflection regime by loop quantum gravity (LQG)-motivated rotating black hole (LMRBH) metrics with an additional parameter $l$ besides mass $M$ and rotation $a$. The LMRBH spacetimes are regular everywhere, asymptotically encompassing the Kerr black hole as a particular case and, depending on the parameters, describe black holes with one horizon only (BH-I), black holes with an event horizon and a Cauchy horizon (BH-II), black holes with three horizons (BH-III), or black holes with no horizons (NH) spacetime. It turns out that as the LQG parameter $l$ increases, the unstable photon orbit radius $x_{ps}$, the critical impact parameter $u_{ps}$, the deflection angle $\alpha_D(\theta)$ and angular position $\theta_{\infty}$ also increases. Meanwhile, the angular separation $s$ decreases, and relative magnitude $r_{mag}$ increases with increasing $l$ for prograde motion but they show opposite behaviour for the retrograde motion. For Sgr A*, the angular position $\theta_{\infty}$ is $\in$ (16.4, 39.8) $\mu$as, while for M87* $\in$ (12.33, 29.9) $\mu$as. The angular separation $s$, for SMBHs Sgr A* and M87*, differs significantly, with values ranging $\in$ (0.008-0.376) $\mu$as for Sgr A* and $\in$ (0.006-0.282) $\mu$as for M87*. We estimate the time delay between the first and second relativistic images using twenty supermassive galactic centre black holes as lenses. Our analysis concludes that, within the $1 \sigma$ region, a significant portion of the BH-I and BH-II and for a small portion of BH-III parameter space agrees with the EHT results of M87* and Sgr A* whereas NH is completely ruled out. We discover that the EHT results of Sgr A* place more stringent limits on the parameter space of LMRBH black holes than those established by the EHT results of M87*.