Orbital Dynamics and Gravitational Wave Signatures of Extreme Mass Ratio Inspirals in Galactic Dark Matter Halos

Abstract

In astrophysics, extreme mass ratio inspiral (EMRI) systems, which consist of a central supermassive black hole and a stellar-mass compact object (SCO), are typically embedded in galactic dark matter (DM) halos. This dark matter environment inevitably affects the orbital dynamics of the SCO and the gravitational wave (GW) signals emitted by the system. In this work, we select two typical dark matter halo profiles -- the Navarro-Frenk-White (NFW) and Beta models -- to systematically investigate their specific impacts on the long-term orbital evolution of the SCO. By incorporating three dissipative mechanisms -- dynamical friction, accretion, and gravitational radiation reaction -- our results demonstrate that, compared to a pure vacuum medium, the presence of a dark matter halo significantly alters the trajectories of precessing orbits, the dynamical evolution of orbital parameters, and the waveforms and phases of the emitted gravitational waves. Due to the strong accretion effect within the NFW model, the energy flux exhibits a distinctive "cusp" feature, marking a reversal from net energy loss to gain at a specific semi-latus rectum, which is a phenomenon absent in the Beta model. Although short-term observations may not be sufficient to distinguish between the NFW and Beta models, their differences become evident over long-term orbital evolution. The gravitational waveforms computed using the NFW and Beta models exhibit a phase shift, which could be detectable in high-density DM environments. This phase shift becomes even more pronounced for higher eccentric orbits and longer observation times. These results offer a theoretical framework for probing environmental effects on EMRIs across different dark matter models using future space-based gravitational wave observatories.

Figures (13)

• Figure 1: The orbital period $T$ (in hours) as a function of the semi-latus rectum $p$ ranging from $10M$ to $200M$ with $e=0.6$. The panels correspond to different halo mass parameters $k \in \{1000M, 5000M, 10000M, 20000M\}$ with a fixed scale $h=10^7M$. The black solid, red dashed, and blue dash-dotted lines represent the results for the pure black hole case without dark matter, Beta model, and NFW model respectively.
• Figure 2: Comparison of the orbital precession $\Delta\phi$ (in rad/orbit) versus $p$ for $e=0.6$ and $p \in [10M, 200M]$. It is observed that the NFW and Beta profiles yield nearly identical results, rendering them indistinguishable even in the high-density scenario.
• Figure 3: Comparison of orbital trajectories under different dark matter halo profiles. The orbits are computed with fixed parameters $p=200M$ and $e=0.6$. The subplots correspond to different combinations of halo mass $k$ and characteristic radius $h$. The black dot at the center represents the location of SMBH.
• Figure 4: The absolute values of the time-averaged energy flux $|\langle dE/dt \rangle|$ for the NFW model, plotted on a logarithmic scale. The orbital parameters are $e=0.1$, $M=10^{6}M_{\odot}$, and $\mu=10M_{\odot}$. The total flux is represented by black lines, where the solid segment indicates net energy loss ($dE/dt<0$) and the dashed segment indicates net energy gain ($dE/dt>0$). The sharp cusp in the total flux indicates the competition and balance between the energy injection from accretion and the energy loss from GW radiation and dynamical friction.
• Figure 5: The absolute values of the time-averaged angular momentum flux $|\langle dL/dt \rangle|$ for the NFW model on a logarithmic scale ($e=0.1$, $M=10^{6}M_{\odot}$, $\mu=10M_{\odot}$). The black solid curves (total flux) and red solid curves (GW flux) perfectly overlap. Since accretion is assumed to transfer no angular momentum, the total flux is persistently negative across the entire range of parameter $p$. It is dominated by GW radiation and dynamical friction, and is thus represented by solid lines.