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

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.