Gravitational Wave Signatures of Schwarzschild Black Hole in a Generalized Dehnen-Type $(1,4,γ)$ Dark Matter Halo

In this paper, we investigate timelike geodesic motion, periodic orbits, and the associated gravitational-wave signals around a Schwarzschild-like black hole (BH) embedded in a generalized Dehnen-type dark matter (DM) halo. We show that the Dehnen-type $(1,4,γ)$ DM halo profile modifies test-particle dynamics, with increasing the parameter of density profile, $γ$, leading to larger marginally bound orbit (MBO) and innermost stable circular orbit (ISCO) radii and angular momenta, together with a higher ISCO energy. These findings provide further insight into the role of the DM distribution in modifying the orbital dynamics, energy, and angular momentum of timelike test particles near the BH. Furthermore, we investigate the gravitational-wave signals produced by a stellar-mass compact object moving along periodic orbits around a supermassive BH embedded in a generalized Dehnen-type DM halo. Using the numerical kludge approach, we calculate the orbital trajectories and the corresponding gravitational-wave polarizations. We find that increasing the halo parameters $γ$, $ρ_s$, and $r_s$ produces larger periodic orbits, longer orbital periods, and lower waveform amplitudes. The resulting spectra lie mainly in the millihertz frequency range, while several characteristic-strain peaks lie above the sensitivity curves of future space-based gravitational-wave detectors such as LISA, Taiji, and TianQin. These results suggest that the surrounding DM halo may leave observable imprints on extreme mass-ratio inspiral (EMRI) gravitational-wave signals.

Hali tarjima qilinmagan