Bound orbits and gravitational wave radiation around the hairy black hole

Abstract The hairy black hole model provides a new theoretical framework for exploring phenomena in strong gravitational fields. This paper systematically investigates the influence of the hair parameter $$\beta $$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mi>β</mml:mi> </mml:math> on the timelike geodesics of the regular hairy black hole, including the radius of the event horizon, the properties of bound orbits, and the characteristics of gravitational wave radiation over a single period. The study reveals that $$\beta $$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mi>β</mml:mi> </mml:math> has a significant impact on the event horizon but only a minor effect on the innermost stable circular orbit (ISCO), the marginally bound orbit (MBO), and periodic orbits. Moreover, the trajectories of the periodic orbits are nearly identical to those of the Schwarzschild black hole. In addition, the parameter $$\beta $$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mi>β</mml:mi> </mml:math> was constrained by simulating the precession observational data of the S2 star orbiting the supermassive black hole Sgr A*. The results indicate that the correction effects of $$\beta $$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mi>β</mml:mi> </mml:math> comply with existing observational constraints, without providing stricter limitations. Furthermore, by considering periodic orbits as transitional orbits in the extreme-mass-ratio inspiral (EMRI) system, it is found that the presence of $$\beta $$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mi>β</mml:mi> </mml:math> introduces subtle effects on the amplitude, phase, and period of the gravitational wave signal for a single orbit. Although these effects appear minor within a single cycle, they may accumulate into significant effects over long-term evolution. In the future, space-based gravitational wave detectors are expected to further investigate the properties of the hair parameter, enhancing our understanding of the spacetime structure and dynamical behavior of black holes.