Abstract
Abstract We critically reanalyze the relativistic precession model of quasi-periodic oscillations, exploring its natural extension beyond the standard harmonic approximation. To do so, we show that the perturbed geodesic equations must include anharmonic contributions arising from the higher-order expansion of the effective potential that cannot be neglected a priori , as commonly done in all the approaches pursued so far. More specifically, independently of the underlying spacetime geometry, we find that in the radial sector the non-negligible anharmonic correction is quadratic in the radial displacement, i.e. ∝ δr 2 , and significantly affects the radial epicyclic frequency close to the innermost stable circular orbit. Conversely, polar oscillations δθ remain approximately decoupled from radial ones, preserving their independent dynamical behavior. To show the need of anharmonic corrections, we thus carry out Monte Carlo-Markov chain analyses on eight neutron star sources of quasi-periodic oscillations. Afterwards, we first work out the outcomes of the harmonic approximation in Schwarzschild, Schwarzschild-de Sitter, and Kerr spacetimes. Subsequently, we apply the anharmonic corrections to them and use it to fit the aforementioned neutron star sources. Our findings indicate that the standard paradigm requires a systematic generalization to include the leading anharmonic corrections that appear physically necessary, although still insufficient to fully account for the observed phenomenology of quasi-periodic oscillations. Accordingly, we speculate on possible refinements of the relativistic precession model, showing the need to revise it at a fundamental level.