Tests of Loop Quantum Gravity from the EHT Results of Sgr A$^*$

Abstract: The Event Horizon Telescope (EHT) collaboration's image of the compact object at the galactic centre is the first direct evidence of the supermassive black hole 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 black hole 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 the Kerr predictions, equipping us with yet another tool to analyse the 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 black holes (LIRBHs) models characterized by an additional deviation parameter $L_q$, which recover the Kerr spacetime in the absence of quantum effects ($L_q \to 0$). We use the astrophysical observables shadow area $A$ and oblateness $D$ to estimate the black hole parameters. When increasing the size of the quantum effects through $L_q$, the black hole shadow size increases monotonically, while the shape gets more distorted, allowing us to constrain the fundamental parameter $L_q$. While the EHT observational results completely rule out the wormhole region in the second LIRBH, a substantial parameter region of the generic black holes in both models agree with the EHT results. We find upper bounds on $L_q$ from the shadow of Sgr A$^*$: $L_q \lesssim 0.0423$ and $L_q \lesssim 0.0821$ for the two LIRBHs respectively, both more stringent than those obtained with the EHT image of M87$^*$.