Quasi-Periodic Oscillations and Particle Dynamics Around Regular Scale-Dependent Black Holes

Regular scale-dependent gravity (SDG) eliminates the Schwarzschild singularity by introducing a dimensionless running parameter E that modifies strong-field geodesics. We derive analytic Keplerian and epicyclic frequencies in the exact SDG metric, trace the shift of the innermost stable circular orbit (ISCO) with E, and embed these results in the relativistic-precession (RP) model of twin peak High-frequency Quasi-periodic Oscillations (HF-QPOs). Using Markov chain Monte Carlo fits to the 3:2 HF-QPO pairs observed in four microquasars — GRO J1655–40, XTE J1550–564, GRS 1915+105, and H1743–322 — we constrain the scale-dependent parameter E. The best-fit values indicate source-dependent constraints on E, with all selected sources favoring positive values of the scale-dependent parameter within the considered parameter range. This bound tightens contemporary iron-line spectral limits by a factor of two and demonstrates that current X-ray timing already probes quantum-inspired deformations of black hole (BH) spacetimes, paving the way for joint tests combining QPOs, spectral lines, and forthcoming shadow-diameter measurements.