Periodic orbits and their gravitational wave radiations in a black hole with a dark matter halo

This study investigates the gravitational dynamics of a spherically symmetric black hole embedded within a dark matter halo, focusing on the impact of the dark matter halo on timelike particle geodesics. We analyze key orbital parameters, including marginally bound orbits, innermost stable circular orbits, and periodic orbits, demonstrating how the dark matter halo alters their energy and angular momentum compared to an isolated black hole. Furthermore, we calculate the gravitational waveforms emitted by these periodic orbits, providing potential observational signatures for such systems. Our results demonstrate a clear correlation between the gravitational waveforms emitted by a small object orbiting a supermassive black hole and the object's zoom-whirl orbital behavior. Higher zoom numbers correspond to more complex waveform substructures. The presence of a dark matter halo significantly impacts these waveforms. We have conducted a detailed analysis of the frequency spectra of the gravitational waveforms arising from these periodic orbits and evaluated their potential detectability. This research contributes to a deeper understanding of the interplay between black holes and dark matter halos, offering insights into their complex gravitational interactions.