Orbital stability, relativistic precession, and QPO constraints for constant-curvature black holes in $f(R)$ gravity

Abstract

We analyze circular-orbit stability, epicyclic resonances, relativistic precession, and quasi-periodic-oscillation (QPO) constraints in the constant-curvature vacuum sector of metric $f(R)$ gravity. The Schwarzschild--(anti-)de~Sitter and Kerr--(anti-)de~Sitter backgrounds are characterized by curvature $R_0$ satisfying the algebraic trace equation. We derive stationary-coordinate-time orbital and epicyclic frequencies and determine stable timelike circular-orbit domains. Positive curvature confines stable motion between innermost and outermost stable circular orbits, which merge at a critical curvature, and permits two resonance radii for suitable frequency ratios. Negative curvature allows stable motion to arbitrarily large radii, with ${\Omega }_{\theta }/{\Omega }_r\to 1/2$, compared with unity in the asymptotically flat limit. We determine the $3:2$, $2:1$, and $3:1$ resonance branches and signed nodal precession, recovering the Lense--Thirring limit. We also evaluate angular-momentum-weighted rigid-flow precession and viscous alignment, with the outermost stable orbit bounding positive-curvature flows. A Bayesian Markov chain Monte Carlo analysis combining the simultaneous GRO J1655--40 QPO triplet with an independent dynamical-mass measurement yields $R_0{M}^2=-5.57_{-16.15}^{+12.94}\times 10^{-4}$ at $68\%$ credibility, conditional on the geodesic relativistic-precession prescription and adopted time normalization. Kerr remains allowed, with negligible fit improvement from nonzero curvature. Doubling the mass uncertainty broadens the curvature interval by approximately $2.1$; removing the mass likelihood reveals an extended, prior-dependent degeneracy. This is therefore a joint QPO and dynamical-mass constraint, with no significant evidence for departure from Kerr.

arXiv comment33 pages, 11 figures, 6 tables