Periodic orbits and their gravitational waves in EMRIs: supermassive black hole affected by galactic dark matter halos

Abstract

Periodic orbits exhibiting zoom-whirl behavior have become attractive topics for studying particle dynamics and gravitational wave emission in extreme-mass-ratio inspirals (EMRIs). This study systematically investigates periodic orbits around black holes and their gravitational wave radiation in three dark matter halo environments: NFW, Beta, and Moore models. The dark matter distribution in these models can be effectively incorporated using two parameters -- the dark matter characteristic mass and halo characteristic radius. Our results reveal that for a larger dark matter mass and a smaller characteristic radius, the shapes of the periodic orbits and the corresponding gravitational waveforms show more significant deviations from the Schwarzschild case. As the halo characteristic radius increases, the orbital shapes and waveform characteristics gradually converge with the Schwarzschild black hole results. Furthermore, our results also suggest that the NFW and Beta models produce nearly indistinguishable results, while the Moore model shows distinct and resolvable signatures compared with Beta/NFW models. These findings deepen our understanding of dark matter halo effects on periodic motions and gravitational wave signatures, providing guidance on future space-based observation for EMRIs.

Figures (18)

• Figure 1: The influence of dark matter on the effective potential of particles. (a) The left subplot highlights the effects of different dark matter halo models, with $k = 10^4 M$. (b) The right subplot emphasizes the role of the dark matter mass $k$. In both subplots, the particle angular momentum is set to $L = 3.8$, and the dark matter halo characteristic radius is $h = 10^7 M$. The black curve corresponds to the case without dark matter (i.e., the Schwarzschild black hole).
• Figure 2: The orbital angular momentum of MBO and ISCO affected by dark matter mass $k$ in different dark matter halo models (with characteristic radius $h = 10^7 M$). (a) Marginally bound orbit (MBO); (b) Innermost stable circular orbit (ISCO). The black curve represents the case of a Schwarzschild black hole without dark matter.
• Figure 3: Schematic of periodic orbits. (a) The $(z~w~v) = (2~1~1)$ orbit, where z = 2 is visualized by the blue and black curves (representing 2 leaves in one full periodic orbit), and w = 1 is characterized by the purple dotted curve (indicating 1 additional whirl in the inner part of periodic orbit during particles drift outward to apoapsis). (b) The $(z~w~v)$ = (3 1 2) orbit illustrating the vertex labeling scheme: vertices are numbered 0, 1, 2 along successive apastra, where v = 2 indicates the orbit skips one vertex and moves to the vertex 2 (at the next apoapsis) after leaving the initial apoapsis at vertex 0.
• Figure 4: Precession angle parameter $q$ changes with orbital energy. The dark matter halo scale is fixed at $h=10^7M$, and the effects of different dark matter masses $k$ (ranging from $1\times10^3M \sim 2\times10^4M$) on the precession angles are compared within each halo model: (a) NFW model; (b) Beta model; (c) Moore model. The Schwarzschild black hole results (dashed black lines) serve as reference baselines.
• Figure 5: Comparison of precession angles for different dark matter halo models: the dark matter halo scale is fixed at $h=10^7M$, and the subfigures illustrate results under four distinct dark matter masses, including (a) $k=1\times10^3M$; (b) $k=3\times10^3M$; (c) $k=1\times10^4M$; and (d) $k=2\times10^4M$.