Particle dynamics and quasi-periodic oscillations in the non-minimally coupled Horndeski black hole

We study equatorial timelike geodesics and high-frequency quasi-periodic oscillations (HF QPOs) in the spacetime of a non-minimally coupled Horndeski black hole belonging to the quartic square-root sector of the theory. Starting from the geodesic equations, we derive the specific energy, specific angular momentum, effective potential, and the characteristic frequencies governing circular motion. We then determine the innermost stable circular orbit (ISCO) together with the Keplerian, radial, and vertical epicyclic frequencies, and we quantify their dependence on the Horndeski parameters η and β . We show that the non-minimal coupling shifts the binding properties of circular orbits, displaces the ISCO, and modifies the epicyclic-frequency structure relevant to QPO phenomenology. Owing to spherical symmetry, the vertical epicyclic frequency coincides with the orbital one, whereas the radial epicyclic mode vanishes at the ISCO and develops a maximum at larger radii. Using the ER, ER1, ER3, and ER4 resonance prescriptions, we map these frequency shifts into the observable HF QPO plane and compare the model phenomenologically with the microquasar sources GRO J1655–40, XTE J1550–564, XTE J1859+226, GRS 1915+105, and H1743–322. In addition, for GRS 1915+105 and XTE J1550–564—the two systems for which full posterior results are presented in this manuscript—we perform a Markov Chain Monte Carlo analysis to constrain the parameter combinations ( M, η, β, r ) favored by the observed QPO pairs. Our results show that HF QPO measurements can place nontrivial constraints on non-minimally coupled Horndeski black holes and provide a useful probe of strong-field deviations from general relativity.