Tests of Loop Quantum Gravity from the Event Horizon Telescope Results of Sgr A$^*$

The Event Horizon Telescope (EHT) collaboration's image of the compact object at the galactic center is the first direct evidence of the supermassive black hole (BH) Sgr A$^*$. The shadow of Sgr A$^*$ has an angular diameter $d_{sh}= 48.7 \pm 7\,\mu$as with fractional deviation from the Schwarzschild BH shadow diameter $\delta= -0.08^{+0.09}_{-0.09}\,,-0.04^{+0.09}_{-0.10}$ (for the VLTI and Keck mass-to-distance ratios). Sgr A$^*$'s shadow size is within $~10\%$ of Kerr predictions, equipping us with yet another tool to analyze gravity in the strong-field regime, including testing loop quantum gravity (LQG). We use Sgr A$^*$'s shadow to constrain the metrics of two well-motivated LQG-inspired rotating BH (LIRBH) models characterized by an additional deviation parameter $L_q$, which recover the Kerr spacetime in the absence of quantum effects ($L_q \to 0$). When increasing the quantum effects through $L_q$, the shadow size increases monotonically, while the shape gets more distorted, allowing us to constrain the fundamental parameter $L_q$. We use the astrophysical observables shadow area $A$ and oblateness $D$ to estimate the BH parameters. It may be useful in extracting additional information about LIRBHs. While the EHT observational results completely rule out the wormhole region in the LIRBH-2, a substantial parameter region of the generic BHs in both models agrees with the EHT results. We find that the upper bounds on $L_q$ obtained from the shadow of Sgr A$^*$ -- $L_q \lesssim 0.0423$ and $L_q \lesssim 0.0821$ for the two LIRBHs, respectively -- are more stringent than those obtained from the EHT image of M87$^*$.