Constraints on self-dual black hole in loop quantum gravity with S0-2 star in the Galactic Center

One of remarkable features of loop quantum gravity (LQG) is that it can provide resolutions to both the black hole and big bang singularities. In the mini-superspace approach based on the polymerization procedure in LQG, a quantum corrected black hole metric is constructed. This metric is also known as self-dual spacetime since the form of the metric is invariant under the exchange $r \to a_0/r$ with $a_0$ being proportional to the minimum area in LQG and $r$ is the standard radial coordinate at asymptotic infinity. It modifies the Schwarzschild spacetime by the polymeric function $P$, purely due to the geometric quantum effects from LQG. Here $P$ is related to the polymeric parameter $\delta$ which is introduced to define the paths one integrates the connection along to define the holonomies in the quantum corrected Hamiltonian constraint in the polymerization procedure in LQG. In this paper, we consider its effects on the orbital signatures of S0-2 star orbiting Sgr A* in the central region of our Milky Way, and compare it with the publicly available astrometric and spectroscopic data, including the astrometric positions, the radial velocities, and the orbital precession for the S0-2 star. We perform Monte Carlo Markov Chain (MCMC) simulations to probe the possible LQG effects on the orbit of S0-2 star. No significant evidence of the self-dual spacetime arisIng from LQG is found. We thus place an upper bounds at 95\% confidence level on the polymeric function $P < 0.043$ and $P < 0.056$, for Gaussian and uniform priors on orbital parameters, respectively.