Epicyclic frequencies around Proca-induced black holes with primary hair

In this paper, we investigate the motion of test particles in the vicinity of a static, non-rotating black hole (BH) described in the framework of Proca–Gauss–Bonnet (PGB) gravity endowed with primary hair, focusing on how the parameters of the BH model affect their trajectories. The geometry of this BH is determined by four quantities: the hair parameter q , the BH mass M , and the two coupling constants α and β . The stability conditions for equatorial circular trajectories are analyzed using the effective potential approach. Explicit analytical formulae for both the angular momentum and the conserved energy of the test particles are obtained in terms of the BH parameters. The behavior of the innermost stable circular orbits (ISCOs) is also addressed in detail. Furthermore, we examine small perturbations of circular motion and derive analytic forms of the oscillatory frequencies, such as radial, vertical, and orbital frequencies, close to the equatorial plane. The periastron precession frequency is also evaluated. The results show how the BH parameters influence the orbital dynamics of the test particles, providing new insights into their motion in the context of PGB gravity.

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