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

Figures (20)

• Figure 1: Left: Plot of the horizon radius $r_{H}$ versus core density $\rho_s$ of a unit mass Schwarzschild BH-DM halo combined spacetime for fixed scale radius $r_s=0.15$. The inset shows a zoomed version of the horizon variation for central density $\rho_s$ in $0\leq \rho_s \leq 0.06$. Right: Plot of the horizon radius $r_{H}$ versus scale radius $r_s$ of a unit mass Schwarzschild BH-DM halo combined spacetime for fixed central density $\rho_s=0.01$. The inset shows a zoomed version of the horizon variation in the ranges of scale radius $r_s$ in $0\leq r_s \leq 0.3$.
• Figure 2: Left: The conserved angular momentum $L$ is plotted for circular orbits as a function of their circumferential radius $r$. Each curve corresponds to a different value of the central density $\rho_s$ while the scale radius of the DM halo is kept to be constant at $r_s=0.15$. The dotted segments of these curves indicate the regions where the orbits are stable. In contrast, the solid and dashed segments represent regions where the orbits are unstable. The locations of the ISCO for each configuration are marked by circular points, MBO radii are denoted by square points. Note that the displayed range for $L$ corresponds to units of $10^{-5}$. Right: The conserved angular momentum $L$ is plotted for circular orbits as a function of their circumferential radius $r$. Each curve corresponds to a different value of the scale radius $r_s$ while the central density of the DM halo is kept to be constant at $\rho_s=0.01$. The dotted segments of these curves indicate the regions where the orbits are stable. In contrast, the solid and dashed segments represent regions where the orbits are unstable. The locations of the ISCO for each configuration are marked by circular points, MBO radii are denoted by square points. Note that the displayed range for $L$ corresponds to units of $10^{-5}$.
• Figure 3: Left: The conserved energy $E$ is plotted for circular orbits as a function of their circumferential radius $r$. Each curve corresponds to a different value of the central density $\rho_s$ while the scale radius of the DM halo is kept to be constant at $r_s=0.15$. The dotted segments of these curves indicate the regions where the orbits are stable. In contrast, the solid and dashed segments represent regions where the orbits are unstable. The locations of the ISCO for each configuration are marked by circular points. The horizontal line $E=1$ marks the limit of bound orbits. Note that the displayed range for $E$ in vertical axis corresponds to units of $10^{-5}$. Right: The conserved energy $E$ is plotted for circular orbits as a function of their circumferential radius $r$. Each curve corresponds to a different value of the scale radius $r_s$ while the central density of the DM halo is kept to be constant at $\rho_s=0.01$. The dotted segments of these curves indicate the regions where the orbits are stable. In contrast, the solid and dashed segments represent regions where the orbits are unstable. The locations of the ISCO for each configuration are marked by circular points. The horizontal line $E=1$ marks the limit of bound orbits. Note that the displayed range for $E$ in vertical axis corresponds to units of $10^{-5}$.
• Figure 4: Left: The figures illustrate the dual branches of a complete homoclinic orbit of a stellar massive probe particle around a supermassive Schwarzschild BH immeresed in a Dehnen-type DM halo within the $(r, p_r)$ phase space. This is shown for multiple values of the DM halo's central density and the particle's angular momentum. For all cases presented, the scale radius of the DM halo is held constant at $r_s = 0.15$. The trajectory represented by the red curve is a homoclinic orbit occurring in a Schwarzschild spacetime (the specific case corresponds to the absence of DM halo's core density $\rho_s = 0$). Note that the displayed range for $p_r$ in vertical axis and the legend labels of $L$ are in units of $10^{-5}$. Right: The figures illustrate the dual branches of a complete homoclinic orbit of a stellar massive probe particle a supermassive Schwarzschild BH immeresed in a Dehnen-type DM halo within the $(r, p_r)$ phase space. This is shown for multiple values of the DM halo's scale radius and the particle's angular momentum. For all cases presented, the central of the DM halo is held constant at $\rho_s = 0.01$. The trajectory represented by the red curve is a homoclinic orbit occurring in a Schwarzschild spacetime (the specific case corresponds to the absence DM halo's radius $r_s = 0$). Note that the displayed range for $p_r$ in vertical axis and the legend labels of $L$ are in units of $10^{-5}$.
• Figure 5: A segment of a homoclinic trajectories is illustrated for various DM halo parameters ($\rho_s$, $r_s$) in the spacetime of a Schwarzschild BH surrounded by a Dehnen-type DM halo, within the EMRI limit. The analysis is conducted for two distinct values of angular momentum: $L = 3.75\times10^{-5}$ and $L = 4.25\times10^{-5}$. The boundaries of the unstable circular orbits, $r_{\rm un}$, and the maximum attainable radii, $r_{\rm max}$, are indicated by the red and green dotted circles, respectively. The central solid black circle denotes the location of the Schwarzschild BH surrounded by the DM halo. Note that the radius of the BH is depicted individually for each specific case in the parameter set.