Strong-gravity precession resonances for binary systems orbiting a Schwarzschild black hole

TL;DR

This work investigates how strong-field tides from a Schwarzschild SMBH modify precession resonances in a hierarchical triple consisting of a compact inner binary and an external SMBH. By deriving a relativistic resonance condition $q\,\dot{\gamma} = k\Omega_{\hat{r}} + l\Omega_{\hat{\Psi}}$ and analyzing the quadrupole tidal coupling both perturbatively and numerically, the authors demonstrate a richer spectrum of resonances than in the Newtonian case. These resonances can drive significant eccentricity growth and alter gravitational-wave phasing, with distinctive signatures in the LISA band and dependence on the outer orbit's frequencies and inclination. The results motivate extensions to Kerr backgrounds and more complete GW modeling, offering a pathway to use GW observations to probe strong-gravity tidal effects near SMBHs.

Abstract

Binary systems of compact objects in close orbit around a supermassive black hole (SMBH) may form in galactic nuclei, providing a unique environment to probe strong-gravity tidal effects on the binary's dynamics. In this work, we investigate precession resonances arising between the periastron precession frequency of a binary system and its orbital frequencies around the SMBH. By modeling the SMBH as a Schwarzschild black hole, we find that relativistic effects in the tidal field give rise to a significantly richer resonance spectrum compared to the Newtonian case. This result is supported by both perturbative and numerical analyses of the quadrupolar tidal interaction in the strong-gravity regime. Our results reveal new signatures for strong-gravity effects in such triple systems, with potential implications for gravitational-wave astronomy.

Figures (3)

• Figure 1: Schematic representation of the triple system. Two compact objects with masses $m_1$ and $m_2$ form the inner binary (red orbit) with characteristic size $r$. The center of mass of the inner binary (cyan dot) follows a geodesic around a non-rotating SMBH of mass $M_*$ at a distance $\hat{r}$, defining the outer orbit (blue).
• Figure 2: Left panel: Comparison between the numerical results obtained with our approach in the regime of strong gravity (GR) and the standard Newtonian description (N) for an initial inclination $I_0=55^{\circ}$. Right panel: Numerical results in the regime of strong gravity for different values of the initial inclination. In both panels, we used the following parameters and initial conditions for the inner binary: total mass $M = 50~M_{\odot}$, reduced mass $\mu=12.5 ~M_{\odot}$, semi-major axis $a_0 \sim 0.0006~\rm AU$, inner eccentricity $e_0 = 0.001$ and $\gamma_0= \vartheta_0 = 0^{\circ}$. For the outer binary we used: $M_* = 4 \times 10^6~M_{\odot}$, semi-major axis $\hat{a} = 18~G M_* / c^2 \sim 0.7~\rm AU$ and outer eccentricity $\hat{e}=0.4$. With these parameters, in the left panel the inner binary merges in $\sim 1.3$ yrs. In the right panel, the merger time is about $1.37$ yrs for the red curve and $\sim 1.5$ yrs for the blue curve. Moreover, the peak frequency associated with the inner binary's emission of GWs lies in the LISA bandwidth for the entire lifetime of the inner binary, where we considered $0.001~\text{Hz} < f_{\rm GW}^{\rm LISA} < 0.1~\text{Hz}$. Finally, the ratio between the ZLK and inner precession timescales is $t_{\rm ZLK}/t_{\rm 1PN}\sim 40$.
• Figure 3: Comparison between the analytical model developed in Sec. \ref{['analytic']} and the numerical results obtained in the strong-gravity regime for small outer eccentricity ($\hat{e} \ll 1$). The parameters and initial conditions for the inner binary are as follows: total mass $M = 50~M_{\odot}$, reduced mass $\mu=12.5 ~M_{\odot}$, semi-major axis $a_0 \sim 0.0014~\rm AU$, eccentricity $e_0 = 0.001$, initial inclination $I_0 = 60^\circ$ and $\gamma_0= \vartheta_0 = 0^{\circ}$. The inner binary orbits a SMBH of mass $M_* = 5 \times 10^7~M_{\odot}$, with semi-major axis $\hat{a} = 15~G M_* / c^2 \sim 7~\rm AU$ and outer eccentricity $\hat{e}=0.05$. As shown in the plot, there is a very good agreement between the analytical prediction (pink curve) and the full numerical integration presented in this section (blue curve). In contrast, the numerical evolution obtained using a Newtonian point-particle approximation (gray curve) exhibits clear deviations, highlighting the importance of incorporating strong-gravity effects in modeling such systems.