Gravitational waveforms from periodic orbits around a Schwarzschild black hole embedded in a Dehnen-type dark matter halo

In this paper, we study the periodic orbits, characterized by zoom-whirl behavior, around a Schwarzschild-like black hole (BH) embedded within a Dehnen-type dark matter (DM) halo. We demonstrate how the DM halo modifies the gravitational dynamics of the black hole, influencing the energy and angular momentum of timelike particle geodesics and enhancing their interaction with the BH. We determine the radii of the marginally bound orbits (MBOs) and innermost stable circular orbits (ISCOs), showing that the DM halo increases both. This provides a deeper understanding of how the DM alters the behavior, energy, and angular momentum of timelike particle geodesics. Furthermore, we explore the gravitational waveforms emitted by a timelike particle in periodic orbits around a supermassive black hole (SMBH) within this BH-DM system. Using a semi-analytical approach, we calculate particle trajectories and derive the corresponding waveforms, demonstrating that the DM halo modifies the zoom-whirl orbital behavior, leading to distinct changes in the waveform structure. Our findings suggest that future gravitational wave (GW) observations could constrain the properties of DM halos surrounding BHs, providing new insights into the gravitational wave signatures arising from the interaction between BH gravity and DM.