Extreme-Mass-Ratio Inspirals Embedded in Dark Matter Halo I:Existence of Homoclinic Orbit and Near-Horizon Chaos

TL;DR

This paper investigates how dark matter halos influence the dynamics of particles near black holes, revealing conditions for chaos and homoclinic orbits that could affect gravitational wave signals.

Contribution

It introduces a Hamiltonian framework to analyze the impact of dark matter halos on particle orbits around black holes, highlighting the emergence of chaos in realistic astrophysical environments.

Findings

Dark matter halos modify effective potential and spacetime curvature.

Transition from regular to chaotic motion occurs near the horizon.

Halo density and energy increase amplify chaos.

Abstract

We study the existence of homoclinic orbit and the onset of chaotic motion for a massive particle moving around a Schwarzschild-like black hole embedded in a Dehnen-(1,4,5/2) type dark matter halo, within the extreme-mass-ratio limit q = m/M << 1, where m and M are the masses of the particle and the central black hole, respectively. The presence of the halo modifies the spacetime curvature and consequently deforms the effective potential governing the particle's motion. Using the Hamiltonian formulation, we derive the conditions under which unstable circular orbit and the associated homoclinic trajectory arise, marking the separatrix between bound and plunging motion. By analyzing the effective potential and the corresponding phase-space structure, we identify the transition from regular to chaotic dynamics in the near-horizon region. Numerical analyses through Poincare sections and Lyapunov exponents calculations demonstrate that increasing the halo density, scale radius along with energy amplifies nonlinear effects which leads to chaos eventually. We demonstrate that within a dark matter halo environment, the dynamical stability of particle motion can be significantly altered without violating the universal surface gravity bound on chaos. This work provides a deeper understanding of horizon-induced chaos in astrophysically realistic environments and serves as a theoretical basis for exploring its possible imprints on gravitational wave signals in extreme-mass-ratio inspirals system.