The Environmental Effects on Inspiraling Binary Black Hole Systems in the Centers of the LMC and M31

Abstract

Binary black hole (BBH) systems residing in the centers of galaxies evolve within complex astrophysical environments. These environments, comprising dark matter (DM) halos and baryonic accretion disks, can significantly alter the orbital dynamics of the binaries and their resulting gravitational wave (GW) emission. In this study, we investigate the dynamical evolution and GW waveforms of BBH systems embedded in the centers of the Large Magellanic Cloud (LMC) and the Andromeda Galaxy (M31). We construct a comprehensive analytical framework that jointly incorporates GW radiation reaction, DM spike effects (including dynamical friction and accretion, derived from the Navarro-Frenk-White profile), and accretion disk perturbations. Using this framework, we track the long-term evolution of the binary's semi-latus rectum $p$ and orbital eccentricity $e$. Our simulations reveal that the coexistence of a DM spike and an accretion disk significantly accelerates the inspiral process compared to pure DM or vacuum scenarios. Crucially, to assess the observability of these environmental effects, we calculate the Signal-to-Noise Ratio (SNR) and waveform Mismatch for future Pulsar Timing Arrays (PTAs). Our analysis demonstrates that these systems can achieve robust detectability thresholds ($\text{SNR} \ge 8$) within specific parameter spaces. Furthermore, the substantial Mismatch (reaching $\sim 0.7$ over a 20-year observation in the LMC scenario) indicates that the phase deviations induced by these environmental effects are highly distinguishable from vacuum templates. These findings predict the prospect of using future GW detections to probe complex galactic environments.

Figures (16)

• Figure 1: Schematic illustration of a BBH system, comprising a central massive BH embedded within a spherically symmetric DM halo, and a secondary BH undergoing an eccentric inspiral. For visual clarity, the secondary BH's orbit is depicted as circular.
• Figure 2: NFW density profiles for the LMC and M31, computed using the parameters detailed in Tables \ref{['tab:galaxy']} and \ref{['tab:mpri']}. The radial extent of each curve is bounded by the ISCO ($r_{\mathrm{ISCO}}$) of the central BH and the virial radius ($R_\mathrm{vir}$) of the respective galaxy. The larger $r_{\mathrm{ISCO}}$ cutoff for M31 reflects the greater mass of its central BH. The DM density is assumed to vanish in the region $r < r_{\mathrm{ISCO}}$.
• Figure 3: The initial DM distribution around the central BH follows the NFW profile. The strong gravitational potential and adiabatic growth of the central BH induce the formation of a dense DM spike extending up to a radius $r_{sp}$. Inward from $r_{sp}$, the DM density is substantially enhanced relative to the unperturbed NFW profile. Furthermore, a steeper spike index $\gamma_{sp}$ corresponds to a higher central DM density. The DM distribution is assumed to vanish within the ISCO.
• Figure 4: Orbital geometry of the BBH system, modeled as two Schwarzschild BHs restricted to the equatorial plane. The secondary BH follows a perturbed Keplerian orbit governed by the central BH's gravitational potential.
• Figure 5: Evolution of the semi-latus rectum $p$ and eccentricity $e$ as a function of time $t$ for a BBH system at the center of the LMC, computed using Eqs. (\ref{['eq17']}) and (\ref{['eq18']}). GW radiation reaction drives the secular decay of both the orbital radius and eccentricity. For initial orbital parameters $p_0 = \{500, 500, 500, 750\}R_s$ and $e_0 = \{0.3, 0.5, 0.7, 0.3\}$, the inspiral times required for the secondary BH to reach the ISCO ($r_{\text{ISCO}}$) of the central BH are $1.82\times10^5\,\mathrm{yr}$, $1.985\times10^5\,\mathrm{yr}$, $2.452\times10^5\,\mathrm{yr}$, and $9.21\times10^5\,\mathrm{yr}$, respectively. The corresponding terminal parameters are $p \approx \{2.936, 2.978, 2.464, 2.961\}R_s$ and $e \approx \{0.00008983, 0.000158754, 0.000173085, 0.0000479251\}$.