Extreme-Mass-Ratio Inspirals Embedded in Dark Matter Halo II: Chaotic Imprints in Gravitational Waves

TL;DR

Chaos in extreme-mass-ratio inspirals is studied for a stellar-mass secondary orbiting a Schwarzschild-like BH embedded in a Dehnen-type DM halo using numerical-kludge gravitational waveforms. By analyzing both orbital dynamics and GW signals through spectral and recurrence methods, the work shows that chaotic orbits produce broader, quasi-continuous GW spectra and higher energy emission, with DM-halo parameters $(\rho_s,r_s)$ modulating the onset of chaos. A detectability assessment for LISA, TianQin, and Taiji indicates that chaotic imprints could be observable, especially by TianQin, highlighting the potential to use GW data to probe BH environments and DM halos. The study lays groundwork for DM- and environment-informed EMRI templates and motivates future work on rotating BHs, alternative DM profiles, and higher-multipole GW modeling.

Abstract

We investigate the imprints of chaos in gravitational waves from extreme-mass-ratio inspirals configuration, where a stellar massive object, confined in a harmonic potential, orbits a supermassive Schwarzschild-like black hole embedded in a Dehnen-type dark matter halo. In our first paper [1], we demonstrated the system's transition from non-chaotic to chaotic dynamics by analyzing Poincaré sections, orbital evolution, and Lyapunov exponents across different energies and dark matter halo parameters. In this work, we compute the gravitational waveforms of the small celestial object along different chaotic and non-chaotic orbits by implementing the numerical kludge scheme. We further perform a spectral analysis of the gravitational waveforms from such orbits. In particular, we show that when the system is in a chaotic state, the gravitational wave signals are characterized by broader frequency spectra with finite widths, enhanced amplitude and energy emission rate, distinctly differentiating them from the signals generated during the system's non-chaotic state. Through recurrence analysis we also show that the time series of gravitational waveforms strain carry unique information on the motion of chaotic dynamics, which can be used to distinctly differentiate from non-chaotic to chaotic motion of the source. Furthermore, we discuss the potential detectability of these orbits for upcoming observatories like LISA, TianQin, and Taiji, emphasizing the significant potential for detecting chaotic imprints in gravitational waves to substantially enhance our understanding of chaotic dynamics in black hole physics and the dark matter environments of galactic nuclei.

Figures (20)

• Figure 1: The gravitational waveforms of the $+$ and $\times$ modes corresponding to the non-chaotic, onset-of-chaos and chaotic orbits for the four different cases (Table \ref{['tab1']}). The time range is taken from 0 to 0.25 in unit of $t/(10^3 M)$, whereas the rectangular box inserted in each figure display of an extended view of the corresponding waveforms over an extended time range (from 0 to 1.5 in unit of $t/(10^3 M)$.
• Figure 2: Gravitational waveforms of the $+$ and $\times$ modes corresponding to the non-chaotic, onset-of-chaos and chaotic orbits for fixed $r_s=0.15,~E=90$ with variations in density parameter $\rho_s$ as our considered Case-III. Here the time range is shown from 0 to 0.25 in unit of $t/(10^3 M)$.
• Figure 3: Gravitational waveforms of the $+$ and $\times$ modes corresponding to the non-chaotic, onset-of-chaos and chaotic orbits for fixed $\rho_s=0.01,~E=115$ with variations in DM halo's scale radius $r_s$ as our considered Case-IV. Here the time range is shown from 0 to 0.25 in unit of $t/(10^3 M)$.
• Figure 4: The averaged GWs amplitude across all directions and the energy emission rate of the GWs corresponding to the non-chaotic, onset-of-chaos and chaotic orbits for all of the cases, mentioned in Table \ref{['tab1']}.
• Figure 5: Absolute values of Fourier transforms of the time-domain gravitational waveforms corresponding to the Non-chaotic, onset-of-chaos and chaotic orbits for all of the cases, mentioned in Table \ref{['tab1']}.