Periodic orbits and gravitational radiation from extreme mass-ratio inspirals as probes of black hole quantum hair

Abstract The classical no-hair theorem states that stationary black holes in general relativity can be completely described by only a small set of global parameters. Within this framework, no additional geometric structures are expected to persist outside the event horizon. However, quantum vacuum polarization may introduce small modifications to the near-horizon geometry, effectively giving rise to what is known as quantum hair. Such corrections may provide a possible window into the microscopic structure and thermodynamic properties of black holes. In this work, we examine how the quantum hair parameter $$\gamma $$ γ influences the periodic orbital dynamics of test bodies in extreme mass-ratio inspirals (EMRIs) and their associated gravitational-wave emission. We find that $$\gamma $$ γ significantly modifies the characteristic radii and angular momenta of two important circular orbits, namely the marginally bound orbit (MBO) and the innermost stable circular orbit (ISCO), leading to a shift in the allowed region of the energy–angular momentum ( E – L ) phase space. Based on the rational number q classification, we further show that quantum corrections tend to enhance the zoom–whirl orbital behavior. Within the conservative Numerical Kludge framework adopted in this work, the quantum hair parameter modifies the effective potential and induces small shifts in the fundamental orbital frequencies, which in turn produce cumulative phase differences in long-duration waveform evolutions. These results provide an illustrative example of how quantum-corrected geometry can affect conservative orbital dynamics and waveform phasing, and they offer a phenomenological basis for future, more complete studies of possible quantum-gravity imprints in EMRI signals.