Binary Mergers near a Supermassive Black Hole: Relativistic Effects in Triples

TL;DR

This paper investigates how relativistic effects near a spinning supermassive black hole influence the orbital and spin dynamics of merging black hole binaries in triple systems, potentially affecting merger rates and spin orientations.

Contribution

It introduces the concept of a 'GR-enhanced' merger channel where relativistic precessions expand the conditions for black hole binary mergers in hierarchical triples.

Findings

Relativistic effects increase the inclination window for mergers.

They produce a wide range of spin orientations at LIGO detection.

The 'GR-enhanced' channel may significantly impact black hole merger rates.

Abstract

We study the general relativitic (GR) effects induced by a spinning supermassive black hole on the orbital and spin evolution of a merging black hole binary (BHB) in a hierarchical triple system. A sufficiently inclined outer orbit can excite Lidov-Kozai eccentricity oscillations in the BHB and induce its merger. These GR effects generate extra precessions on the BHB orbits and spins, significantly increasing the inclination window for mergers and producing a wide range of spin orientations when the BHB enters LIGO band. This "GR-enhanced" channel may play an important role in BHB mergers.

Equations

$$t_{\mathrm{LK}}=\frac{1}{\Omega_{\mathrm{LK}}}=\frac{1}{n_{\mathrm{in}}}\frac{m_{12}}{m_{3}}\bigg{(}\frac{a_{\mathrm{out},\mathrm{eff}}}{a_{\mathrm{in}}}\bigg{)}^{3},$$

$$\frac{d\mathbf{e}_{\mathrm{in}}}{dt}\bigg{|}_{\mathrm{GR}}=\dot{\omega}_{\mathrm{GR}}\hat{\textbf{L}}_{\mathrm{in}}\times\mathbf{e}_{\mathrm{in}},~{}~{}\dot{\omega}_{\mathrm{GR}}=\frac{3Gn_{\mathrm{in}}m_{12}}{c^{2}a_{\mathrm{in}}(1-e_{\mathrm{in}}^{2})}.$$

$$\frac{d\hat{\textbf{S}}_{1}}{dt}\bigg{|}_{\mathrm{S_{1}L_{\mathrm{in}}}}=\Omega_{\mathrm{S_{1}L_{\mathrm{in}}}}\hat{\mathbf{L}}_{\mathrm{in}}\times\hat{\textbf{S}}_{1},~{}\Omega_{\mathrm{S_{1}L_{\mathrm{in}}}}=\frac{3Gn_{\mathrm{in}}(m_{2}+\mu_{\mathrm{in}}/3)}{2c^{2}a_{\mathrm{in}}(1-e_{\mathrm{in}}^{2})},$$

$$\displaystyle\frac{d\textbf{L}_{\mathrm{out}}}{dt}\bigg{|}_{\mathrm{L_{\mathrm{out}}S_{3}}}=$$

$$\displaystyle\frac{d\textbf{e}_{\mathrm{out}}}{dt}\bigg{|}_{\mathrm{L_{\mathrm{out}}S_{3}}}=$$

$${\Omega }_{{\mathrm{L}}_{\mathrm{out}}{\mathrm{S}}_3}=\frac{G{S}_3(4+3m_{12}/m_3)}{2c^2{a}_{\mathrm{out}}^3(1-e_{\mathrm{out}}^2{)}^{3/2}}.$$

$$\Omega_{\mathrm{L_{\mathrm{in}}L_{\mathrm{out}}}}^{(\mathrm{N})}=-\frac{3}{4}\Omega_{\mathrm{LK}}\big{(}\hat{\textbf{L}}_{\mathrm{out}}\cdot\hat{\textbf{L}}_{\mathrm{in}}\big{)}~{}~{}~{}~{}~{}(\mathrm{for}~{}e_{\mathrm{in}}=0).$$

$$\frac{d\textbf{L}_{\mathrm{in}}}{dt}\bigg{|}_{\mathrm{L_{\mathrm{in}}L_{\mathrm{out}}}}=\Omega_{\mathrm{L_{\mathrm{in}}L_{\mathrm{out}}}}\hat{\textbf{L}}_{\mathrm{out}}\times\textbf{L}_{\mathrm{in}},$$

$${\Omega }_{{\mathrm{L}}_{\mathrm{in}}{\mathrm{L}}_{\mathrm{out}}}^{(\mathrm{GR})}=\frac{3}{2}\frac{G(m_3+{\mu }_{\mathrm{out}}/3)n_{\mathrm{out}}}{c^2{a}_{\mathrm{out}}(1-e_{\mathrm{out}}^2)},$$

$$\gamma \equiv \frac{{\Omega }_{{\mathrm{L}}_{\mathrm{in}}{\mathrm{L}}_{\mathrm{out}}}}{{\Omega }_{{\mathrm{L}}_{\mathrm{out}}{\mathrm{S}}_3}}=\frac{{\Omega }_{{\mathrm{L}}_{\mathrm{in}}{\mathrm{L}}_{\mathrm{out}}}^{(\mathrm{N})}+{\Omega }_{{\mathrm{L}}_{\mathrm{in}}{\mathrm{L}}_{\mathrm{out}}}^{(\mathrm{GR})}}{{\Omega }_{{\mathrm{L}}_{\mathrm{out}}{\mathrm{S}}_3}}.$$

$$\frac{d\hat{\textbf{S}}_{1}}{dt}\bigg{|}_{\mathrm{S_{1}L_{\mathrm{out}}}}=\Omega_{\mathrm{S_{1}L_{\mathrm{out}}}}\hat{\mathbf{L}}_{\mathrm{out}}\times\hat{\textbf{S}}_{1},$$

$${\Omega }_{{\mathrm{S}}_1{\mathrm{L}}_{\mathrm{out}}}=\frac{3}{2}\frac{G(m_3+{\mu }_{\mathrm{out}}/3)n_{\mathrm{out}}}{c^2{a}_{\mathrm{out}}(1-e_{\mathrm{out}}^2)}.$$

$$\mathcal{A}\equiv \left|\frac{{\Omega }_{{\mathrm{S}}_1{\mathrm{L}}_{\mathrm{in}}}}{{\Omega }_{{\mathrm{L}}_{\mathrm{in}}{\mathrm{L}}_{\mathrm{out}}}}\right|,\mathcal{B}\equiv \frac{{\Omega }_{{\mathrm{S}}_1{\mathrm{L}}_{\mathrm{in}}}}{{\Omega }_{{\mathrm{S}}_1{\mathrm{L}}_{\mathrm{out}}}}.$$

$${\Omega }_{{\mathrm{L}}_{\mathrm{in}}{\mathrm{S}}_3}={\Omega }_{{\mathrm{S}}_1{\mathrm{S}}_3}={\Omega }_{\mathrm{LT}}=\frac{G{S}_3}{2c^2{a}_{\mathrm{out}}^3(1-e_{\mathrm{out}}^2{)}^{3/2}}.$$

$$T_{\mathrm{m}}\simeq T_{\mathrm{m},0}(1-e_{\max }^2{)}^3,$$

Full text

• stars: black holes - stars: kinematics and dynamics

1. Introduction

The detections of gravitational waves from merging binary black holes (BHs) (e.g., Abbott et al., 2018a, b; Zackay et al., 2019; Venumadhav et al., 2019) have motivated many recent studies on the dynamical formation of such compact black-hole binaries (BHBs). Dynamical formation channels include mergers arising from strong gravitational scattering in dense clusters (e.g., Portegies Zwart & McMillan, 2000; O’Leary et al., 2006; Miller & Lauburg, 2009; Banerjee et al., 2010; Downing et al., 2010; Ziosi et al., 2014; Samsing & Ramirez-Ruiz, 2017; Samsing et al., 2018; Samsing & D’Orazio, 2018; Rodriguez et al., 2018; Gondán et al., 2018) and more gentle “tertiary-induced mergers” – the latter can take place either in isolated triple/quadrupole systems (e.g., Antonini et al., 2017; Silsbee & Tremaine, 2017; Liu & Lai, 2017, 2018; Liu et al., 2019) or in nuclear clusters dominated by a central supermassive BH (SMBH) (e.g., Antonini & Perets, 2012; VanLandingham et al., 2016; Petrovich & Antonini, 2017; Hoang et al., 2018; Hamers et al., 2018; Randall, & Xianyu, 2018; Fragione et al., 2019).

In this paper we are interested in stellar-mass BHB mergers induced by a SMBH. Such BHBs may exist in abundance in the nuclear cluster around the SMBH due to various dynamical processes, such as scatterings and mass segregation (e.g., O’Leary et al., 2009; Leigh et al., 2018). Gravitational perturbation from the SMBH induces Lidov-Kozai (LK) eccentricity oscillations of the BHB, which leads to enhanced gravitational radiation and merger of the BHB. Our paper examines several general relativistic (GR) effects that are overlooked in previous studies, but significantly impact the efficiency and outcomes of LK-induced mergers. We focus on isolated BHB-SMBH systems, and do not consider other processes related to scatterings and relaxation with surrounding stars in the cluster (e.g., VanLandingham et al., 2016; Petrovich & Antonini, 2017; Hamers et al., 2018), which may also change the character of LK-induced mergers.

In the Standard LK-Induced Merger scenario, a BHB with masses $m_{1}$, $m_{2}$, semimajor axis $a_{\mathrm{in}}$ and eccentricity $e_{\mathrm{in}}$, moves around a tertiary ($m_{3}$) on a wider orbit with $a_{\mathrm{out}}$ and $e_{\mathrm{out}}$. The angular momenta of the inner and outer binaries are denoted by $\textbf{L}_{\mathrm{in}}\equiv\mathrm{L}_{\mathrm{in}}\hat{\textbf{L}}_{\mathrm{in}}$ and $\textbf{L}_{\mathrm{out}}\equiv\mathrm{L}_{\mathrm{out}}\hat{\textbf{L}}_{\mathrm{out}}$ (where $\hat{\textbf{L}}_{\mathrm{in}}$ and $\hat{\textbf{L}}_{\mathrm{out}}$ are unit vectors). If the mutual inclination between $\hat{\textbf{L}}_{\mathrm{in}}$ and $\hat{\textbf{L}}_{\mathrm{out}}$ (denoted as $I$) is sufficiently high, the inner binary would experience LK eccentricity oscillations on the timescale

[TABLE]

where $m_{12}\equiv m_{1}+m_{2}$, $n_{\mathrm{in}}=(Gm_{12}/a_{\mathrm{in}}^{3})^{1/2}$ is the mean motion of the inner binary, and $a_{\mathrm{out},\mathrm{eff}}\equiv a_{\mathrm{out}}\sqrt{1-e^{2}_{\mathrm{out}}}$ is the effective outer binary separation.

GR introduces pericenter precession of the inner binary, which can be described by the first-order post-Newtonian (PN) theory

[TABLE]

This precession competes with $\Omega_{\mathrm{LK}}$, and tends to suppress LK oscillations or limit the maximum eccentricity $e_{\mathrm{max}}$ (e.g., Fabrycky & Tremaine, 2007; Liu et al., 2015). The general secular and quasi-secular equations of motion (see vector form in Liu et al., 2015; Liu & Lai, 2018; Petrovich, 2015), combined with the gravitational wave (GW) radiation, completely determine the evolution of triple system. Such LK-induced mergers have been extensively studied (e.g., Miller & Hamilton, 2002; Blaes et al., 2002; Wen, 2003; Antonini & Perets, 2012; Silsbee & Tremaine, 2017; Liu & Lai, 2017, 2018; Liu et al., 2019).

The spin vector ( $\textbf{S}_{1}\equiv\mathrm{S}_{1}\hat{\textbf{S}}_{1}$) of the BH is also coupled to the orbital angular momentum vector $\textbf{L}_{\mathrm{in}}$ through de-Sitter precession (1.5 PN effect) (e.g., Barker & O’Connell, 1975):

[TABLE]

where $\mu_{\mathrm{in}}\equiv m_{1}m_{2}/m_{12}$ is the reduced mass for the inner binary. Similar equation applies to the spinning body 2. To determine the final spin-orbit misalignments of the BHBs, it is essential to include this spin-orbit coupling effect in the scenario of LK-induced merger. Our recent works (e.g., Liu & Lai, 2017, 2018; Liu et al., 2019), focusing on the BHB mergers induced by stellar-mass tertiary ($m_{3}$ comparable to $m_{1}$, $m_{2}$), have shown that LK-induced mergers can give rise to unique signatures for the final spin-orbit misalignment angle $\theta_{\rm sl}^{\rm f}$ (see also Antonini et al., 2018; Rodriguez & Antonini, 2018). In particular, for initially close BHBs (with $a_{0}\lesssim 0.2\mathrm{AU}$), which can merge without the aid of the tertiary companion, modest ($\lesssim 40^{\circ}$) $\theta_{\rm sl}^{\rm f}$ can be produced in the majority of triples (e.g., Liu & Lai, 2017). For wide binaries (with $a_{0}\gtrsim 10\mathrm{AU}$), the distribution of $\theta_{\rm sl}^{\mathrm{f}}$ is peaked around $90^{\circ}$ if the BHs have comparable masses (negligible octupole effect), while a more isotropic distribution of final spin axis is produced as the octupole effect increases (e.g., Liu & Lai, 2018; Liu et al., 2019).

The Standard LK-Induced Merger mechanism, as outlined above (and studied in all previous works), includes the key GR effects associated with the inner binaries, but neglects the GR effects associated with the tertiary companion. This is adequate when the tertiary mass $m_{3}$ is not much larger than the masses of the inner BHB. However, for BHB-SMBH triples, with $m_{3}\gg m_{1},m_{2}$, several GR effects involving the SMBH can qualitatively change the efficiency and outcomes of LK-induced mergers.

2. New GR Effects Involving SMBH Tertiary

We start by examining how various GR effects associated with the SMBH tertiary affect the LK oscillations and spin evolution of the inner BHB (see Figure 1).

(i)* Effect I: Lense-Thirring Precession of $\textbf{L}_{\mathrm{out}}$ around $\textbf{S}_{3}$*. For a SMBH, the spin angular momentum $\mathrm{S}_{3}=\chi_{3}Gm_{3}^{2}/c$ (where $\chi_{3}\leqslant 1$ is the Kerr parameter) can be easily larger than $L_{\mathrm{out}}=\mu_{\mathrm{out}}\sqrt{Gm_{\mathrm{tot}}a_{\mathrm{out}}(1-e_{\mathrm{out}}^{2})}$ [where $\mu_{\mathrm{out}}\equiv(m_{12}m_{3})/m_{\mathrm{tot}}$ and $m_{\mathrm{tot}}=m_{12}+m_{3}$]. Thus $\textbf{L}_{\mathrm{out}}$ experiences Lense-Thirring precession around $\textbf{S}_{3}$ if the two vectors are misaligned (1.5 PN effect)(e.g., Barker & O’Connell, 1975; Fang, & Huang, 2019):

[TABLE]

where the orbit-averaged precession rate is

[TABLE]

The back-reaction of Equation (4) implies that $\textbf{S}_{3}$ precesses around $\textbf{L}_{\mathrm{out}}$ at the rate $\Omega_{\mathrm{L_{\mathrm{out}}S_{3}}}\mathrm{L_{\mathrm{out}}}/\mathrm{S_{3}}$.

As shown in Hamers & Lai (2017) in a different context, the variation of $\hat{\textbf{L}}_{\mathrm{out}}$ can significantly affect LK eccentricity excitation when $\Omega_{\mathrm{L_{\mathrm{out}}S_{3}}}$ becomes comparable to $\Omega_{\rm LK}$. As shown in Figure 1, $\Omega_{\mathrm{L_{\mathrm{out}}S_{3}}}/\Omega_{\mathrm{LK}}\sim 1$ can be satisfied for sufficiently large $m_{3}$ ($\gtrsim 10^{9}M_{\odot}$). More precisely, LK oscillations can be affected or triggered due to an inclination resonance, which occurs when $\Omega_{\mathrm{L_{\mathrm{out}}S_{3}}}$ matches $\Omega_{\mathrm{L_{\mathrm{in}}L_{\mathrm{out}}}}$, the precession rate of $\hat{\textbf{L}}_{\mathrm{in}}$ around $\hat{\textbf{L}}_{\mathrm{out}}$ (see below).

Figure 2 depicts an example of how various relativistic effects associated with the SMBH modify LK oscillations. The results are obtained by integrating the double-averaged (DA) secular equations of motion (averaging over both the inner and outer orbits; e.g., Liu et al., 2015; Liu & Lai, 2018). We see that the BHB eccentricity exhibits regular oscillations in the “standard LK” case (black lines), but the inclusion of Effect I (Equations 4-5) (purple lines) makes the eccentricity evolve chaotically and extend to higher values.

(ii)* Effect II: de-Sitter-like Precession of $\textbf{L}_{\mathrm{in}}$ around $\textbf{L}_{\mathrm{out}}$*. The standard LK mechanism already includes the Newtonian precession of $\textbf{L}_{\mathrm{in}}$ around $\textbf{L}_{\mathrm{out}}$ (driven by the tidal potential of $m_{3}$ on the inner orbit), at the rate given by (to quadrupole order) 111 The general equation for finite $e_{\mathrm{in}}$ can be found in Liu et al. (2015). Note that for BHB-SMBH systems ($m_{3}\gg m_{12}$), dynamical stability requires $a_{\mathrm{out}}\gg a_{\mathrm{in}}$. Thus, the octupole LK is negligible since $\varepsilon_{\mathrm{oct}}\equiv[(m_{1}-m_{2})/(m_{1}+m_{2})](a_{\mathrm{in}}/a_{\mathrm{out}})[e_{\mathrm{out}}/(1-e_{\mathrm{out}}^{2})]\ll 1$.

[TABLE]

In GR, $\textbf{L}_{\mathrm{in}}$ experiences an additional de-Sitter like (geodesic) precession in the gravitational field of $m_{3}$, such that the net precession of $\textbf{L}_{\mathrm{in}}$ around $\textbf{L}_{\mathrm{out}}$ is governed by

[TABLE]

with $\Omega_{\mathrm{L_{\mathrm{in}}L_{\mathrm{out}}}}\equiv\Omega_{\mathrm{L_{\mathrm{in}}L_{\mathrm{out}}}}^{(\mathrm{N})}+\Omega_{\mathrm{L_{\mathrm{in}}L_{\mathrm{out}}}}^{(\mathrm{GR})}$, and

[TABLE]

where $n_{\mathrm{out}}=(Gm_{\mathrm{tot}}/a_{\mathrm{out}}^{3})^{1/2}$. To keep $\textbf{L}_{\mathrm{in}}\cdot\textbf{e}_{\mathrm{in}}=0$, we also need to add $d\textbf{e}_{\mathrm{in}}/dt=\Omega_{\mathrm{L_{\mathrm{in}}L_{\mathrm{out}}}}^{(\mathrm{GR})}\hat{\textbf{L}}_{\mathrm{out}}\times\textbf{e}_{\mathrm{in}}$ to the eccentricity evolution equation. We can safely neglect the feedback from $\hat{\mathbf{L}}_{\mathrm{in}}$, $\mathbf{e}_{\mathrm{in}}$ on $\hat{\mathbf{L}}_{\mathrm{out}}$ and $\mathbf{e}_{\mathrm{out}}$. Equation (9) has the same form as Equation (3), but can also be reproduced through the “cross terms” in the PN equations of motion of hierarchical triple systems (Private communication with Clifford Will; see also Will, 2014, 2018).

Note that for the standard LK mechanism (and with negligible octupole effect, as valid for the $m_{3}\gg m_{12}$ case considered in this paper), the nodal precession of $\textbf{L}_{\mathrm{in}}$ around $\textbf{L}_{\mathrm{out}}$ is decoupled from the LK exccentricty/inclination oscillations. Therefore adding $\Omega_{\mathrm{L_{\mathrm{in}}L_{\mathrm{out}}}}^{(\mathrm{GR})}$ (Effect II) to $\Omega_{\mathrm{L_{\mathrm{in}}L_{\mathrm{out}}}}$ by itself does not alter the $e_{\mathrm{in}}$-excitation (although it can affect the spin evolution). However, when combined with Effect I, it can significantly affect LK oscillation (see Figure 2, dotted green line). We quantify this behavior by defining the dimensionless ratio

[TABLE]

Since $\Omega_{\mathrm{L_{\mathrm{in}}L_{\mathrm{out}}}}^{(\mathrm{N})}$ depends on $I$ [where $\hat{\textbf{L}}_{\mathrm{out}}\cdot\hat{\textbf{L}}_{\mathrm{in}}=\cos I$], $\gamma$ ranges from $\gamma_{\mathrm{min}}=\gamma~{}(I=0^{\circ})$ to $\gamma_{\mathrm{max}}=\gamma~{}(I=180^{\circ})$.

As explained in Hamers & Lai (2017), when $\gamma\sim 1$, an inclination resonance generates larger $I$ even from a small initial $I_{0}$, leading to a wider range of initial inclinations for extreme eccentricity excitation. Figure 3 explores these new GR effects by showing the $e_{\mathrm{in}}$-excitation window as a function of $I_{0}$ for BHB-SMBH systems with given $m_{1},~{}m_{2},~{}a_{\mathrm{in}},~{}a_{\rm out}$ but different values of $m_{3}$ (thus different $\gamma$’s). The misalignment angle between $\hat{\textbf{S}}_{3}$ and $\hat{\textbf{L}}_{\mathrm{out}}$ is set to $30^{\circ}$, but with a random azimuthal phase angle (i.e., the initial $\hat{\textbf{L}}_{\mathrm{in}}$, $\hat{\textbf{L}}_{\mathrm{out}}$ and $\hat{\textbf{S}}_{3}$ are not in the same plane 222 Note that in examples shown in Hamers (2018); Liu & Lai (2019), the phase angle is set to be fixed, where $\hat{\textbf{L}}_{1}$, $\hat{\textbf{L}}_{2}$ and $\hat{\textbf{L}}_{\mathrm{out}}$ initially lie in the same plane. ). By evolving the triple system using the DA secular equations, we record $e_{\mathrm{max}}$ achieved over an integration timespan of 500 $t_{\mathrm{LK}}$ for each system with and without Effects I, II and IV. In each panel, the cyan dots are the “standard LK” results; these can be calculated analytically (e.g., Liu et al., 2015). Note that since the octupole-order effects are negligible, systems with finite $e_{\mathrm{out}}$ should exhibit a similar behavior as the cyan dots. We see that including Effects I-II (purple dots) can dramatically widen the eccentricity excitation window. As $\gamma$ approaches unity with increasing $m_{3}$, overlapping inclination and LK resonances give rise to the widespread chaos (e.g., Hamers & Lai, 2017), causing systems with modest $I_{0}$ to attain extreme eccentricity growth.

When $e_{\mathrm{max}}$ becomes sufficiently close to unity, the timescale the inner BHB spends in high-$e_{\mathrm{in}}$ phase ($t_{\mathrm{LK}}\sqrt{1-e_{\mathrm{max}}^{2}}$; e.g., Anderson et al., 2016) becomes less than the period of the outer binary, the DA approximation breaks down, and the system enters semi-secular regime (e.g., Luo et al., 2016). If it is shorter than the inner orbital period, the evolution of triples can only be resolved correctly by N-body integration. In Figure 3, the systems in the bottom-right panel belong to the semi-secular regime. To better address the orbital evolution, we also integrate the single-averaged (SA) secular equations (only averaging over the inner orbital period; e.g., Liu & Lai, 2018). The result (light blue dots) shows that the eccentricity in SA integrations can undergo excursions to even more extreme values.

(iii)* Effect III: de-Sitter Precession of $\textbf{S}_{1}$ around $\textbf{L}_{\mathrm{out}}$*. The “standard LK” already includes de-Sitter precession of $\textbf{S}_{1}$ around $\textbf{L}_{\mathrm{in}}$. With a SMBH tertiary, $\textbf{S}_{1}$ also experiences a precessional torque from $m_{3}$:

[TABLE]

with

[TABLE]

Note that $\Omega_{\mathrm{S_{1}L_{\mathrm{out}}}}=\Omega_{\mathrm{L_{\mathrm{in}}L_{\mathrm{out}}}}^{(\mathrm{GR})}$ (Equation 9). The back-reaction torques on $\hat{\mathbf{L}}_{\mathrm{out}}$ and $\hat{\mathbf{e}}_{\mathrm{out}}$ can be safely neglected since $\mathrm{L}_{\mathrm{out}}\gg\mathrm{S}_{1}$. Although Equation (11) does not affect the orbital evolution of the inner binary, it does affect the evolution of $\textbf{S}_{1}$ and the spin-orbit misalignment angle $\theta_{\mathrm{sl}}$.

The bottom panel of Figure 2 shows several examples of the evolution of $\theta_{\mathrm{sl}}$ during LK oscillations, with and without various GR effects. The evolution of ${\bf S}_{1}$ is governed by two “adiabaticity parameters”:

[TABLE]

We expect (i) When $\mathcal{A},\mathcal{B}\ll 1$ (“nonadiabatic”), the spin axis $\hat{\bf S}_{1}$ cannot “keep up” with the rapidly changing $\hat{\textbf{L}}_{\mathrm{in}}$, and thus effectively precesses around $\textbf{L}_{\mathrm{out}}$, keeping $\theta_{\mathrm{S_{1}L_{\mathrm{out}}}}\simeq$ constant [Note that since $\Omega_{\mathrm{S_{1}L_{\mathrm{out}}}}=\Omega_{\mathrm{L_{\mathrm{in}}L_{\mathrm{out}}}}$ is only a few times larger than $\Omega_{\mathrm{L_{\mathrm{out}}S_{3}}}$ (see Figure 1), $\theta_{\mathrm{S_{1}L_{\mathrm{out}}}}$ is only approximately constant as $\hat{\textbf{L}}_{\mathrm{out}}$ precesses around $\hat{\textbf{S}}_{3}$]; (ii) When $\mathcal{A},\mathcal{B}\gg 1$ (“adiabatic”), $\hat{\textbf{S}}_{1}$ closely “follows” $\hat{\textbf{L}}_{\mathrm{in}}$, maintaining an approximately constant $\theta_{\mathrm{sl}}$. (iii) In the regime between (i) and (ii) (“trans-adiabatic”), the evolution of $\hat{\textbf{S}}_{1}$ can be quite complicated and chaotic, because of its dependence on $e_{\mathrm{in}}$ during the LK cycles (see Storch et al., 2014; Storch & Lai, 2015; Anderson et al., 2016, 2017; Liu & Lai, 2017, 2018).

As the BHB orbit decays, the system may transitions from “nonadiabatic” at large $a_{\mathrm{in}}$ to “adiabatic” at small $a_{\mathrm{in}}$, where the final spin-orbit misalignment angle $\theta_{\mathrm{sl}}^{\mathrm{f}}$ is “frozen”. From Figure 1, we see that, because of the contribution of $\Omega_{\mathrm{L_{\mathrm{in}}L_{\mathrm{out}}}}^{(\mathrm{GR})}$ to $\Omega_{\mathrm{L_{\mathrm{in}}L_{\mathrm{out}}}}$, the conditions $\mathcal{A},\mathcal{B}\ll 1$ can be easily satisfied initially for systems with $m_{3}\gtrsim 10^{8}M_{\odot}$. As these systems experience LK-induced orbital decay, they must go through the “trans-adiabatic” regime and therefore may attain a wide range of $\theta_{\mathrm{sl}}^{\mathrm{f}}$ (see below).

(iv)* Effects IV*. Both $\hat{\textbf{L}}_{\mathrm{in}}$ and $\hat{\textbf{S}}_{1}$ (and $\hat{\textbf{S}}_{2}$) experience Lens-Thirring precession around $\hat{\textbf{S}}_{3}$ at the rate

[TABLE]

Since $\Omega_{\mathrm{L_{\mathrm{in}}S_{3}}}/\Omega_{\mathrm{L_{\mathrm{in}}L_{\mathrm{out}}}}^{(\mathrm{GR})}=\Omega_{\mathrm{S_{1}S_{3}}}/\Omega_{\mathrm{S_{1}L_{\mathrm{out}}}}\sim V_{\mathrm{out}}/c$ (where $V_{\mathrm{out}}$ is the orbital velocity of the outer binary), they can be neglected when $V_{\mathrm{out}}/c\ll 1$.

3. Binary BH Mergers Induced by SMBH

We now add gravitational radiation in our fiducial example (Figure 3 with $m_{3}=2.3\times 10^{9}M_{\odot}$). Since the Effect IV is not important in this example, we perform two sets of calculations with and without Effects I-III, evolve the system until the BHB enters the LIGO band (i.e., when the peak GW frequency reaches 10 Hz). The results are summarized in Figure 4.

In the “standard LK” mechansim (without Effects I-III; cyan circles in the top two panels of Figure 4), for systems with negligible octupole effects, the merger time can be well approximated by (e.g., Liu & Lai, 2018)

[TABLE]

where $T_{\mathrm{m},0}\equiv(5c^{5}a_{\mathrm{in},0}^{4})/(256G^{3}m_{12}^{2}\mu_{\mathrm{in}})$ is the merger time due to GW emission for an isolated circular BHB (e.g., Peters, 1964) ($T_{\mathrm{m},0}\simeq 10^{9}$ yrs for the systems considered in Figure 4), and $e_{\mathrm{max}}$ is the maximum eccentricity achieved in the LK cycle (see Figure 3). When the GR effects associated with the SMBH are taken into account (purple dots), the range of inclinations for rapid mergers (shorter $T_{\rm m}$) becomes much larger, a direct consequence of the widened LK eccentricity excitation window (see Figure 3). Note that in a dense nuclear cluster, the orbits of a BHB-SMBH triple system can be perturbed or disrupted by close fly-bys of other objects. If we introduce upper limits of the survival time for the triples, the “standard LK” would give the merger fraction of $f_{\mathrm{merger}}\simeq 12\%,20\%,30\%$ for $T_{\mathrm{m}}\lesssim 10^{5},10^{6},10^{7}$ yrs, respectively, while including Effects I-III would increase the corresponding merger fraction to $f_{\mathrm{merger}}\simeq 58\%,63\%,70\%$.

The middle panel of Figure 4 shows the distribution of $\theta_{\mathrm{sl}}^{\mathrm{f}}$ as a function of $\cos I_{0}$ 333 In a nuclear cluster, the initial binary BHs may have nontrivial spin orientations due to the complicated scattering processes. In order to have an intuitive understanding of the spin dynamics, here we assume that the BH spin axis is initially aligned with the orbital axis.. In the “standard LK” (as studied in Liu & Lai (2017, 2018); Liu et al. (2019)), the final spin axis shows a regular distribution when the octupole effects are negligible (as in the BHB-SMBH case studied here); for the systems that do not experience eccentricity excitation, an analytical expression for $\theta_{\mathrm{sl}}^{\mathrm{f}}$ can be obtained (Liu & Lai, 2017) (see the dashed line). However, when the GR effects associated with the SMBH are included, the final BH spin orientation is significantly “randomized”. Given the wide distribution of $\theta_{\mathrm{sl}}^{\mathrm{f}}$, we find the large spread in $\chi_{\mathrm{eff}}$ in the bottom panel of Figure 4, where $\chi_{\rm eff}=(m_{1}\boldsymbol{\chi}_{1}+m_{2}\boldsymbol{\chi}_{2})\cdot\hat{\mathbf{L}}_{\mathrm{in}}/m_{12}$ [with $\boldsymbol{\chi}_{1,2}=c\mathbf{S}_{1,2}/(Gm_{1,2}^{2})$] is the effective binary spin parameter that can be directly measured from GW observations. Note that the two spins in the merging binary BHs are strong correlated (see also Liu et al. (2019); Fig. 10); this is different for the scenarios involving strong scattering, which expectedly produce uncorrelated isotropic spins.

Due to the negligible octupole effect in BHB-SMBH systems, the “residual” eccentricities of merging BHBs (when they enter the 10 Hz LIGO band) are all below $0.1$ in our simulations. This is in contrast to binary mergers induced by a stellar-mass tertiary studied in Liu et al. (2019).

4. Summary and Discussion

We have identified the impacts of several GR effects in BHB-SMBH triples that have been little explored. Effect I (Equations 4-6) allows the BHB eccentricity to reach extremely high values even with modestly inclined or nearly coplanar outer orbits. Effect II (Equations 8, 9) modifies the eccentricity growth (when combined with Effect I) and BH spin evolution indirectly. Effect III (Equations 11, 12) only affects the spin evolution. The overall dynamics of the BHB and BH spin around a SMBH can be characterized by the dimensionless rates (Equations 10, 13). Effects I and II generally require very massive SMBH ($m_{3}\gtrsim 10^{8}-10^{9}M_{\odot}$) to be effective, while Effect III can be important for a wide range of SMBH masses (see Figure 1). Overall, these GR effects can significantly widen the LK-induced merger window and increase the merger fraction. They also produce a broad distribution of the final BH spin-orbit misalignment angles, leading to a wide range of the effective BHB spin parameter $\chi_{\mathrm{eff}}$.

Our proof-of-concept calculations have demonstrated the importance of the GR effects in BHB-SMBH systems. However, we have not thoroughly explored the relevant parameter space, nor considered various “environmental” effects associated with BHBs in nuclear cluster. We leave these to future works.

5. Acknowledgments

We thank Jean Teyssandier and Clifford Will for useful discussion and communication. This work is supported in part by the NSF grant AST-1715246 and NASA grant NNX14AP31G.