Gravitational waveforms from periodic orbits around a charged black hole with scalar hair*

Abstract We investigated geodesic motion and gravitational-wave signatures of charged black holes with scalar hair. Using the effective potential approach, we analyzed marginally bound orbits and innermost stable circular orbits, showing how their positions and energy thresholds are modified by the scalar hair parameter . These results demonstrate the role of scalar hair in altering the boundary of stable motion. We further explored periodic orbits characterized by rational frequency ratios, labeled by the index , and quantified how scalar hair affects their orbital energy and angular momentum. Based on these orbital properties, we computed gravitational waveforms from extreme mass-ratio inspirals where a stellar-mass compact object orbits a supermassive charged black hole with scalar hair. Using the numerical kludge method, we generated waveforms that exhibit clear zoom-whirl patterns with morphology visibly affected by . Our results show that scalar hair leaves distinguishable imprints on waveforms, suggesting that future space-based detectors could probe deviations from classical black hole spacetimes through extreme mass-ratio inspiral observations.