Probing holonomy corrected Schwarzschild black holes with precessing and periodic orbits

To investigate the detectability of holonomy corrected Schwarzschild black holes, we first derive the metric and geodesics for a test particle, as well as the conditions for circular orbits. Then, we derive the exact solution for the precessing orbits of a test particle in the holonomy corrected Schwarzschild black hole spacetime. We also provide preliminary bounds on the orbital parameters, considering the effects of these corrections on orbital precession. The precession deviations of the stellar orbits of S2, S4716, and hypothetical stars are calculated using preliminary bounds on the loop quantum gravity parameter $b$. We consider an extreme-mass-ratio inspiral system, where a stellar-mass black hole is inspiraling into a supermassive black hole. The periodic and quasiperiodic orbits of timelike particle around holonomy corrected Schwarzschild black holes, along with their corresponding gravitational waveforms, are investigated using the Levin and Perez-Giz classification of periodic orbits. These results provide new insights into distinguishing Schwarzschild black holes from holonomy corrected Schwarzschild black holes, offering not only a better understanding of the orbital dynamics of black holes but also new ideas for gravitational wave detection. In addition, this paper investigates the periodic orbits based on orbital eccentricity and the latus rectum, thereby opening up new avenues for research in periodic orbits. Our findings suggest that the orbital eccentricity has a more pronounced effect on the periodic orbits, while the loop quantum gravity parameter $b$ has minimal impact.