Evolution of resonant binaries

# Probing evolution of binaries influenced by the spin-orbit resonances

A Gupta & A Gopakumar Department of Astronomy and Astrophysics, Tata Institute of Fundamental Research, Mumbai 400005, India
###### Abstract

We evolve isolated comparable mass spinning compact binaries experiencing Schnittman’s post-Newtonian spin-orbit resonances in an inertial frame associated with , the initial direction of the total angular momentum. We argue that accurate gravitational wave (GW) measurements of the initial orientations of the two spins and orbital angular momentum from should allow us to distinguish between the two possible families of spin-orbit resonances. Therefore, these measurements have the potential to provide direct observational evidence of possible binary formation scenarios. The above statements should also apply for binaries that do not remain in a resonant plane when they become detectable by GW interferometers. The resonant plane, characterized by the vanishing scalar triple product involving the two spins and the orbital angular momentum, naturally appears in the one parameter family of equilibrium solutions, discovered by Schnittman. We develop a prescription to compute the time-domain inspiral templates for binaries residing in these resonant configurations and explore their preliminary data analysis consequences.

###### pacs:
04.25.Nx, 04.30.-w, 97.60.Lf, 95.30.Sf
: Class. Quantum Grav.

## 1 Introduction

Coalescing comparable mass compact binaries containing spinning stellar mass black holes (BHs) are among the expected gravitational wave (GW) sources for the ground based interferometric GW detectors like advanced LIGO (aLIGO), advanced Virgo and KAGRA [1]. In comparison, comparable mass spinning supermassive BH binaries will be required to realize GW astronomy in the milli-Hertz and nano-Hertz regimes in the coming decades [2, 3]. The optimal data analysis method of matched filtering is being invoked to extract the weak GW signals that are deeply buried in the noisy data sets. In this method one cross-correlates the relevant data with several template banks that contain accurately modeled GW signals from a number of expected compact binary sources. The construction of these GW search templates requires one to model GW polarization states, and , associated with the inspiral phase of coalescing compact binaries in an accurate and efficient manner. Fortunately, GWs emitted during the inspiral phase can be accurately modeled by invoking the post-Newtonian (PN) approximation to general relativity. In the case of non-spinning compact binaries, inspiraling along quasi-circular orbits, theoretical inputs required to compute highly accurate inspiral phase templates are available to the 3.5PN order. These include inputs to compute the fully 3.5PN accurate orbital phase evolution and 3PN accurate expressions for and  [4, 5, 6]. Recall that PN corrections provide contributions that are accurate to the relative order beyond the âNewtonianâ estimate, where and are the orbital and light speeds, respectively. Interestingly, there are on-going efforts to describe the compact binary dynamics at the conservative 4PN order  [7]. In the case of binaries containing Kerr BHs, the spin effects should be incorporated while constructing appropriate search templates. This is the main motivation for the ongoing efforts to compute the higher PN order corrections to the dominant order spin-orbit and spin-spin contributions to the dynamics of spinning compact binaries, computed some four decades ago by Barker and O’Connell [8]. We observe that for compact binaries, containing maximally spinning BHs, the leading order spin-orbit and spin-spin interactions enter the orbital dynamics at 1.5PN and 2PN order, respectively [9]. At present, the next-to-next-to-leading order spin-orbit contributions that appear at 3.5PN order are available in [10] while the next-to-next-to-leading order spin-orbit and spin(1)-spin(2) Hamiltonians are available in [11]. Very recently, the next-to-next-to-leading order spin-orbit contributions to the gravitational wave flux and associated orbital phase evolutions were obtained for binaries in quasi-circular orbits [12]. Strictly speaking, we only have all the relevant inputs to perform GW phasing to 2.5PN order while incorporating all the spin effects associated with the two maximally spinning Kerr BHs. In comparison, the ready-to-use amplitude corrected GW polarization states are only available to 2PN order for binaries in quasi-circular orbits [13, 14, 15] and to 1PN order for eccentric binaries  [16]. In what follows, we explore the evolution of certain binary configurations that contain two spinning compact objects of comparable masses and () and having Kerr parameters and such that their spin angular momenta are given by and .

Roughly a decade ago, Schnittman discovered certain equilibrium configurations for spinning and precessing compact binaries [17]. The binaries in such configurations have their two spins and the orbital angular momentum lying in the same plane and the definition of the total angular momentum, namely , implies that will also lie in this plane. Schnittman observed that spinning compact binaries in these equilibrium configurations are characterized by constant values of , and during the precessional time scale, where , and are unit vectors along the two spins and orbital angular momenta, respectively. Schnittman termed these equilibrium solutions as the “spin-orbit resonant configurations” as , and precess around with a constant frequency for such binaries in the absence of gravitational radiation reaction effects ( is a conserved quantity both in magnitude and direction in the absence of GW damping). With the inclusion of the reactive contributions in PN-accurate orbital dynamics, Schnittman observed that approaches unity on the inspiral timescale for certain type of equilibrium configurations. Therefore, the equilibrium configurations experience the spin alignment towards the end of their inspiral. Binaries, not initially in the neighborhood of these equilibrium configurations, can eventually get locked and librate around them during their inspiral. The spin-orbit resonances may have important astrophysical implications as noted in  [17, 18, 19, 20]. This is because of the ability of these resonances to align the spins of comparable mass supermassive BH binaries prior to their mergers [18, 20]. This will ensure that the massive BHs formed via BH coalescences will not experience large recoil velocities. Therefore, the merger remnants may be retained in their host galaxies that are hierarchically formed from the merger of smaller galaxies. Very recently, it was argued that the BH spins in comparable mass stellar mass BH binaries would lie preferentially in a resonant plane due the spin-orbit resonances when their GWs enter the aLIGO frequency window [19]. They demonstrated that PN evolution forces the two spins to lie in certain resonant planes, characterized by either or where is the relative angular separation between the two spins in an associated orbital plane (this requires that the tides are efficient during the formation of these binaries). The binaries belonging to these two families are forced to belong to what they termed as the standard mass ratio (SMR) and the reverse mass ratio (RMR) binary formation scenarios. In the SMR binary formation channel, the more massive star will evolve to form the more massive component of the BH binary and the compact binary during its inspiral will be influenced by the spin-orbit resonances. In contrast, the heavier BH forms during the second supernova explosion in the RMR binary formation scenario and this is essentially due to the substantial mass transfer during the Roche lobe overflow of the progenitor. Such BH binaries are expected to get influenced by the resonances. Gerosa et. al studied in detail the combined effects of efficient tides and supernova kicks (both isotopic and polar) on the above two binary formation channels. Their detailed explorations allowed them to provide several distributions for the misalignments between and the two spins at orbital separations of roughly 500 Schwarzschild radius (see the scatter plots in figures 5 and 6 in [19]). These scatter plots turned out to be very helpful in providing the initial conditions for our numerical investigations. Gerosa et. al also explored the binary formation scenarios involving inefficient tides and termed the resulting binaries as the freely precessing ones. It was pointed out that the accurate matched filtering measurements of and from a large sample of GW observations will constrain the various possible models of binary formation [19]. We note that it is customary to probe the dynamics of these inspiraling and precessing binaries in an orbital triad associated with the Newtonian orbital angular momentum , where and are the reduced mass, orbital separation and velocity, respectively.

In this paper, we evolve comparable mass spinning compact binaries from an initial epoch characterized by (we usually denote this specific value as ). The dimensionless PN expansion parameter is defined in terms of an orbital-like frequency and the total mass : . We use the orbital angular momentum rather than its Newtonian counterpart to describe the binary orbits. Additionally, we invoke an inertial frame associated with , the unit vector along the initial direction of the total angular momentum of the binary, to specify both the orbital and spin angular momenta. In contrast, it is common to invoke an -based non-inertial triad to specify the two spins at the initial epoch. We observe that is customary to numerically evolve by invoking the precessional equation appropriate for while incorporating the effects of dominant order spin-orbit coupling [15]. The initial value makes sure that these compact binaries inspiral essentially due to the emission of GWs from orbital separations where being the Schwarzschild radius. We terminate these numerical integrations when reaches either of the following two fiducial values: or . In our numerical integrations, where Hz for ground-based interferometers like aLIGO while Hz for eLISA. The value is essentially influenced by the earlier investigations [17, 18]. We evolve comparable mass spinning compact binaries that satisfy the set of equilibrium spin configurations at , discovered by Schnittman. These one parameter family of equilibrium configurations, characterized either by or , can be obtained by demanding that both and its time derivative should vanish at the initial epoch [17]. In our approach, provides the relative angular separation of the two spins in a plane perpendicular to . Recall that these equilibrium configurations may be viewed as spin-orbit resonances as the precession frequencies of and around are rather identical. We argue that accurate matched filtering measurements of the orientations of and from at should allow us to distinguish between binaries under the influence of either or spin-orbit resonances. Therefore, these accurate GW measurements from an inspiraling comparable mass spinning binary should provide, in principle, the direct observational evidence of binary formation channels involving the SMR or RMR scenarios that also involve efficient tides, as detailed in [19]. To illustrate the above statement, let and stand for the orientations of and from at . We show that the binaries that are influenced by resonances tend to have . The typical values are usually lie below . However, negligible values suggest very efficient tides during the binary formation. In contrast, resonant binaries tend to have and typical values are . Non-negligible values in the range of few degrees indicate efficient tides during the binary formation. We show that the above inferences also apply for binaries that do not remain in a resonant plane when they become detectable by GW interferometers. The resonant plane, characterized by either or restrictions, naturally appears in the above mentioned one parameter family of equilibrium solutions.

It turns out that the two black hole spins and the orbital angular momentum do not remain in a plane during the late stages of inspiral for binaries that were not in Schnittman’s equilibrium configurations at . Indeed, these binaries are influenced by the spin-orbit resonances and get locked into a nearby resonant plane during their inspiral. However, this may not be sufficient to force the above three vectors to share a common plane when these binaries inspiral to . For such binaries, the above listed angular variables librate around their resonant values and the plots for and can have non-negligible amplitudes during the late stages of inspiral. We also emphasize the importance of measuring accurately the values of at . The accurate measurements turned out to be crucial to distinguish the freely precessing binaries from those under the influence of the spin-orbit resonances. Following Gerosa et. al, the freely precessing binaries are expected to have as the tidal interactions play no significant role during their formation. Additionally, such binaries are not affected by the spin-orbit resonances during their inspiral from to [19]. Our numerical integrations show that these binaries can mimic the constraints on the and values that are satisfied by the two resonant families. However, the values of freely precessing binaries will not obey two specific relations, involving and values, that are fulfilled by binaries affected by the spin-orbit resonances. This is relevant as binaries under the influence of () spin-orbit resonances are expected to have (). Therefore, the accurate measurements of and values are crucial to distinguish the three possible types of inspiraling comparable mass spinning binaries. These three possible types, as expected, include binaries that are either freely precessing or influenced by one of the two spin-orbit resonances (the or resonances).

We also develop a prescription to compute the time domain GW polarization states for comparable mass spinning compact binaries experiencing spin-orbit resonances in the aLIGO/eLISA frequency windows. Our approach invokes to describe the binary orbits and the -based inertial frame to specify the two spins and is based on [15]. Therefore, our approach can easily incorporate various expressions that are required to analyze the spin-orbit resonances in a -based inertial frame. We show that the temporally evolving are uniquely characterized by only six parameters at the fiducial values for binaries that reside in the resonant planes. These six parameters include the four basic ones, namely , , , and the two angular parameters, and , that specify the orientation of more massive spin at . The requirement that and its time derivative should be zero at forces the orientation of to become dependent parameters for such binaries. In comparison, one requires to specify eight parameters to obtain for binaries not residing in the resonant plane. This is essentially due to the non-vanishing and values at for such binaries. Invoking the match computations, detailed in [21], we compare inspiral templates for binaries residing in and librating around the resonant configurations in aLIGO frequency window. Binaries in ‘near-resonance’ configurations tend to have estimates while the estimates are for binaries in ‘far-resonance’ configurations. The rather high estimates point to the possibility that a computationally cheaper resonant inspiral template bank may provide the desirable fitting factor for binaries influenced by spin-orbit resonances. This is because are obtained by maximizing the over all the templates present in a certain bank of inspiral waveforms.

The paper is organized in the following way. In the next section, we briefly describe the spin-orbit resonances, detailed in  [17] and the way to analyze the spin-orbit resonances in the inertial frame associated with . Various implications of our approach are probed in section 2.2. Our prescription to compute time-domain GW polarization states for inspiraling binaries experiencing the spin-orbit resonances is presented in section 3 along with certain preliminary data analysis implications. Conclusions are presented in section 4.

## 2 PN-accurate Equilibrium Configurations and their GW emission induced evolution

We first summarize [17] that probed the evolution of comparable mass precessing compact binaries initially residing in and around certain equilibrium spin configurations while invoking an orbital triad associated with . Section 2.2 contains our approach to describe the evolution of such binaries in an inertial frame based on along with various inferences.

### 2.1 Spin-orbit resonances in an orbital triad

Schnittman invoked an orbital triad based on to describe the dynamics of comparable mass spinning compact binary configurations as evident from figure 1 in [17]. In what follows, we use an orbital triad based on rather than to describe these binaries. For generic spinning compact binaries, the two spins are freely specified at the initial epoch by four angles, namely and . Therefore, the unit vectors along the two spins read

 s1(xi) =sin~θ1cos~ϕ1a+sin~θ1sin~ϕ1b+cos~θ1k, (1a) s2(xi) =sin~θ2cos~ϕ2a+sin~θ2sin~ϕ2b+cos~θ2k, (1b)

where and may be identified with unit vectors and of [17]. Additionally, Schnittman equated at the initial epoch to zero by noting that the orbital dynamics should be preserved under a rotation around . This implies that the orientations of these binaries, characterized by certain and (or ) values, are specified by just three angular variables. These variables are where specifies the relative angular separation of the two spins in the orbital plane while and define the orientations of and from , respectively. It is important to note that these angular variables vary over precessional and reactive time scales. Further, it may be recalled that the dynamical evolutions of such binaries involve three time-scales associated with the orbital, precessional and inspiral aspects of their dynamics and we denote these timescales by and , respectively. It is not very difficult to infer that as they are associated with the Newtonian, 1PN and 2.5PN order terms in the PN-accurate orbital dynamics.

The equilibrium spin configurations, detailed in  [17], are obtained by demanding that the first and second derivatives of should be zero. Invoking the precessional equations for and , given by equations (7) below, it is easy to show that the above requirements are identical to equating and its time derivative to zero [17]. The expression for in the orbital triad reads

 γ=sin~θ1sin~θ2sinΔ~ϕ, (2)

and the requirement that implies that can take only one of the following two values: or . The constraints, namely , allow us to numerically obtain in terms of for a binary characterized by specific values of , and . In other words, the solutions to the above two constraints trace out one-dimensional curves in () space [17]. These solutions, having or , stand for specific configurations where and lie in a plane such that and precess around with a roughly constant angular frequency on a precessional time scale. This prompted, as noted earlier, Schnittman to term these equilibrium configurations as certain spin-orbit resonant configurations. The requirement that may be written as an algebraic constraint invoking the precessional equations for and . The constraint reads

 (Ω1×S1)⋅[S2×(L+S1)]=(Ω2×S2)⋅[S1×(L+S2)], (3)

where and provide precessional frequencies of and . Schnittman incorporated the contributions arising from the leading order spin-orbit and spin-spin interactions in to the above precessional frequencies and these contributions may be extracted from [8, 9].

To probe the effect of gravitational radiation reaction on these equilibrium spin configurations, Schnittman wrote down PN-accurate differential equations for the following four variables: , , and . It is straightforward to figure out that these differential equations arise from the PN-accurate precessional equations for and and therefore contain . This implies that the differential equation for that allows orbital frequency to slowly vary over may be invoked to incorporate the effects of gravitational radiation reaction on these variables. Therefore, the differential equations required to describe the dynamics of precessing compact binaries spiraling in from in an orbital triad read

 ˙z1 = c3Gmx3γχ2{δ2q−X222−32x1/2ηχ1z1}, (4a) ˙z2 = c3Gmx3γχ1{−δ2q+X212+32x1/2ηχ2z2}, (4b) ˙β = (4c) ˙γ = c3Gmx5/2{32δmm(β−z1z2)+x1/2[δ1qχ1(z2−βz1) (4d) +δ2qχ2(βz2−z1)+12X21χ1(−z2−2βz1+3z21z2) +12X22χ2(z1+2βz2−3z1z22)]+32xηχ1χ2(z21−z22)}, ˙x = 645c3Gmηx5, (4e)

where and are specified by the angles and , respectively and we have verified that these equations are comparable to equations (A10) in [17]. In the above equations, and stand for and , respectively while . The presence of in the above expression indicates that these spin-orbit resonances can sweep through a substantial portion of the space during the GW emission induced inspiral. The fact that of a generic spinning compact binary can vary over the precessional timescale implies that the binary may approach the resonant values, or , at some point during its lengthy PN-accurate inspiral regime, characterized by . It was argued that the orbital evolution of such generic spinning compact binaries will be heavily influenced by these spin-orbit resonances [17]. In practice, it is convenient to numerically solve the following four differential equations, namely and to probe how these binary configurations evolve under the combined influences of precessional and reactive dynamics from to the late stages of inspiral prior to their coalescence.

Numerical integration of the above equations allowed  [17] to infer that the initial equilibrium configurations, defined by at , remain in their resonant plane during the inspiral regime. Moreover, gravitational radiation reaction forces binary spins, initially not in the resonant plane, to get locked and then librate about the equilibrium configurations during its inspiral from to . This was demonstrated by showing that the instantaneous phase difference oscillates around with steadily decreasing amplitude as evident from figure 5 in  [17]. This spin alignment prompted Kesden et al. [18] to re-analyze these spin-orbit resonances in great detail and explore its implications for merging spinning BH binaries. Very recently, it was pointed out that the BH spins in comparable mass stellar mass BH binaries would preferentially lie in a resonant plane, characterized by or , when GWs from such binaries enter the aLIGO frequency window [19]. The above conclusion requires an admissible assumption that the spins of the BH progenitors should be partially aligned with the orbital angular momentum due to efficient tidal interactions. The ability of such binaries to stay essentially in a resonant plane is due to the combined effects of spin-orbit resonances and GW emission induced inspiral [19]. The authors also stated that it will be desirable to construct templates for inspiraling binaries influenced by the spin-orbit resonances.

In the next subsection, we evolve comparable mass binaries having spin configurations at that are influenced by Schnittman’s one parameter family of equilibrium solutions. We invoke an inertial frame associated with to specify both the orbital and spin angular momenta of such binaries and to describe their PN-accurate evolution. This is partly influenced by the observation that temporally evolving , associated with spinning compact binaries, are usually computed in such an inertial frame. Therefore, the various inputs that are required to describe the spin-orbit resonances in the frame may be invoked while constructing inspiral templates associated with such binaries.

### 2.2 Spin-orbit resonances in an inertial frame defined by j0

We begin by describing how we specify the generic comparable mass spinning compact binaries, characterized by specific values of and , in an inertial frame associated with at large orbital separations ( or ). We invoke the Cartesian coordinate system associated with this inertial frame such that the unit vectors along and have the following components:

 s1 =(sinθ1cosϕ1,sinθ1sinϕ1,cosθ1), (5a) s2 =(sinθ2cosϕ2,sinθ2sinϕ2,cosθ2), (5b) k =(sinιcosα,sinιsinα,cosι). (5c)

Therefore, it appears that we will require six angles, namely , , , , and , to specify the orientation of our binary in the invariant frame as displayed in figure 1. However, the fact that the invariant frame is defined such that the total angular momentum at the initial epoch points along the -axis allows us to estimate the initial and components of in terms of , , , and the values of , , and at the initial epoch. In other words, the initial and components of become dependent variables as we equate the and components of to zero. The resulting expression for the initial and read

 kx(xi)=sinιcosα = −Gm2cLi{X21χ1sinθ1cosϕ1+X22χ2sinθ2cosϕ2}, (6a) ky(xi)=sinιsinα = −Gm2cLi{X21χ1sinθ1sinϕ1+X22χ2sinθ2sinϕ2}, (6b)

where we employ the Newtonian accurate expression for at , given by . Additionally, the initial value of is uniquely given by as almost point along at . Therefore, the initial orientation of the orbital angular momentum are uniquely given by equations (6). This implies that the dynamics of such binaries are uniquely obtained by freely specifying the initial values of the four angular variables that provide the orientations of the two spins in the invariant frame.

We invoke the following set of three vectorial and one scalar differential equations to describe the inspiral dynamics of precessing spinning compact binaries, extracted from  [8, 22, 23]. We first list the precessional equations for and , given by

 ˙s1 = c3Gmx5/2{δ1(k×s1)+12x1/2[X22χ2(s2×s1) (7a) ˙s2 = c3Gmx5/2{δ2(k×s2)+12x1/2[X21χ1(s1×s2) (7b)

where the terms proportional to and incorporate the dominant order spin-orbit and spin-spin interactions, respectively, for binaries moving in circular orbits. The terms that are proportional to are due to the quadrupole-monopole self interaction [23]. These terms were not included in the original analysis of Schnittman while they are present in the detailed analysis of [18]. The precessional motion of is described by

 ˙k = c3Gmx3{δ1qχ1(s1×k)+δ2qχ2(s2×k) (8) −32x1/2[ηχ1χ2(k⋅s1)(s2×k)+ηχ1χ2(k⋅s2)(s1×k) +X21χ21(k⋅s1)(s1×k)+X22χ22(k⋅s2)(s2×k)]},

and this equation arises from the conservation of total angular momentum which leads to . We incorporated the effects of gravitational radiation reaction which causes the binaries to inspiral from via

 dxdt=645c3Gmηx5. (9)

This equation requires the quadrupolar order GW luminosity along with the energy balance argument [22, 24]. We have verified that our inferences about the values of various angular variables at are rather insensitive to the inclusion of PN corrections to the above expression for .

To describe PN-accurate evolution of comparable mass spinning compact binaries, we employ the Cartesian components of the precessional equations while numerically integrating equations (7), (8) and (9). In practice, we numerically solve the nine Cartesian components of the following three equations, namely and where, for example, . This is how we follow the orientations of the two spins and the orbital angular momenta from as these binaries inspiral from to . During our numerical integrations, the values of and are extracted at the stipulated values from the three Cartesian components of and (we invoke equations (6) only once to estimate the initial Cartesian components of ). For example, the angular variables of the dominant spin, namely and , are obtained via and . Similar expressions are employed to obtain and values while the orbital inclination and values are also uniquely extracted from the three Cartesian components of . We are now in a position to explore PN evolution of binaries having spin configurations, at , that are influenced by Schnittman’s one parameter family of equilibrium solutions.

We begin by listing the expression for in terms of angular variables defined in the inertial frame:

 γ = sinθ1cosθ2sinιsin(ϕ1−α)−sinθ2cosθ1sinιsin(ϕ2−α) (10) −sinθ1sinθ2cosιsin(ϕ1−ϕ2).

The fact that the equilibrium configurations are characterized by implies for such configurations. Therefore, these special configurations are also characterized by or in our inertial frame. Note that forces certain restriction on the initial value and it turned out to be consistent with initial value via equations (6). For numerically obtaining Schnittman equilibrium configurations in the inertial frame, we employ the following expression for involving the Cartesian components of

 γ = kx(sinθ1sinϕ1cosθ2−sinθ2sinϕ2cosθ1)+ky(sinθ2cosϕ2cosθ1 (11) −sinθ1cosϕ1cosθ2)+kzsinθ1sinθ2sin(ϕ2−ϕ1),

where we express , and in terms of the Cartesian components of and invoking equations (6). The associated expression for is given by

 ˙γ = c3Gmx5/2{32δmm(β−z1z2)+x1/2[δ1qχ1(z2−βz1) (12) +δ2qχ2(βz2−z1)+12X21χ1(−z2−2βz1+3z21z2) +12X22χ2(z1+2βz2−3z1z22)+32η(β−z1z2)(χ1z1−χ2z2)] +32x[ηχ1χ2(z21−z22)+X21χ21z1(z1β−z2) +X22χ22z2(z1−βz2)]},

where, , and are functions of , , , . These three dot products in the invariant frame may be written as

 z1 = k⋅s1=kxsinθ1cosϕ1+kysinθ1sinϕ1+kzcosθ1 (13a) z2 = k⋅s2=kxsinθ2cosϕ2+kysinθ2sinϕ2+kzcosθ2 (13b) β = s1⋅s2=sinθ1sinθ2cos(ϕ1−ϕ2)+cosθ1cosθ2. (13c)

It should be noted that the above expression for is not identical to equation  (4d) and this is due to the use of fully 2PN-accurate precessional equations that include the quadrupole-monopole self interaction terms in this subsection [23] . Let us note that we need to express , and , appearing in the above expressions for and , in terms of the Cartesian components of and with the help of equations (6) while dealing with the above equation for . We obtain Schnittman’s equilibrium configurations by simultaneously equating equations (11) and (12) for and to zero. This allows us to numerically obtain values in terms of for two specific values of , namely or and define the equilibrium configurations in the invariant frame. These configurations are such that initial value, obtained via equations (6), is consistent with the requirement arising from equating , given by equation (10), to zero. Additionally, the extracted values of the angular variables are also consistent with the requirement that the initial and components of should be zero. It should be noted that in our approach Schnittman’s equilibrium configurations are specified by three angular variables, namely (, , ), compared to two in  [17]. This is merely a consequence of invoking the Cartesian components to obtain these special configurations and we have verified that our results do not depend on the initial value of . In what follows, we consider maximally spinning BH binaires with and while choosing the initial to be . We are now in a position to explore and evolve binaries having spin configurations that are influenced by Schnittman’s one parameter family of equilibrium solutions.

We begin by displaying Schnittman’s equilibrium configurations in our inertial frame for maximally spinning BH binaries with as one dimensional curves in the (, ) plane (see figure 2). These curves, influenced by figures 2 and 3 in  [17], are for binaries residing at various orbital separations starting from . The equilibrium solutions having are displayed in the left panel and the right panel plots are for equilibrium configurations. These plots are fairly similar to the - plots in figure 1 of [18] that depict Schnittman’s equilibrium configurations in an orbital triad. We observe that resonances have and therefore the associated plots appear above the diagonal while the reverse holds good for the equilibrium configurations. For binaries having at large orbital separations like , the equilibrium solutions lie along two curves. This allows many more spin configurations to lie in the close neighborhood of such resonant configurations. Similar statement apply for binaries having while dealing with equilibrium configurations. We also observe that as these binaries inspiral from , the equilibrium solutions sweep through a larger area of the (, ) plane compared to the configurations. All these conclusions are consistent with the studies that invoked -based non-inertial orbital triad to describe these equilibrium solutions [17, 18].

We move on to probe the effect of GW induced damping on these equilibrium configurations. The plots of figure 3 probe the evolution of three equilibrium solutions while terminating the inspiral at . These configurations are characterized by three values of , namely and . The values are influenced by the inference that the traditional formation scenarios for comparable mass BH binaries likely to result in spin-orbit misalignments [25]. Note that this investigation actually provided estimates for spin-orbit misalignments, namely values, at large orbital separations. The large orbital separations allowed us to let . Secular increments/decrements in the values are clearly visible while experiences secular increase during the inspiral. It turned out that the secular evolution of various angular variables is a characteristic of binaries lying in the resonant planes, specified by either or . More importantly, we find that at the initial frequencies of GW detectors and typical values of are .

In figure  4, we explore how the above angular variables evolve for values lying below . These configurations can arise in the formation scenarios involving rather efficient tides along with isotopic or polar supernovae kicks. This should be evident from the green dots in the scatter plots in figures 5 and 6 of [19]. For the plots in figure  4, we let to take the three values, namely and . These spin configurations can have multiple resonant values and this is due to the presence of two equilibrium curves at in the plane. The evolution of and turned out to be quite different for resonant values lying below and above . For example, and values are noticeably higher for binaries having their resonant values that are above . We note that binaries with can have formation scenarios involving either isotopic or polar kicks along with efficient tides. However, binaries with substantially higher and values demand a binary formation channel involving efficient tides and isotopic supernovae kicks. This is mainly because such a formation channel naturally allows the less massive spin to lie in the neighborhood of from at large orbital separations (see figure 5 of [19]).

We also explored the PN-accurate evolution of binaries influenced by the equilibrium solutions and the results are displayed in figure 5. The chosen values of are , and . These initial choices are clearly influenced by the SMR binary formation channel that also experience efficient tides and isotopic kicks [19] (see their red scatter plots in figure 5). These plots clearly show that and typical values are in the range. We also explored the inspiral dynamics of binaries having lying below and the results are shown in figure 6. Interestingly, we find values are essentially negligible for initial values below . It should be noted that these and values are consistent with formation channels involving efficient tides while having both isotropic and polar supernovae kicks.

A close inspection of the and plots of figures 3, 4, 5 and 6 reveals the following interesting point. We infer that GWs from inspiraling resonant binaries should allow us, in principle, to distinguish between the models of their formation involving the SMR or RMR scenarios, detailed in [19]. This requires accurate measurements of and values that provide the orientations of and from when GWs enter interferometric frequency windows. We find that the SMR scenario binaries tend to have . The typical values of are expected to lie below . However, negligible values are likely for binaries whose formation scenario involved very efficient tides. In contrast, in the RMR scenario binaries likely to have and typical values are . Binaries with non-negligible values in the range of few degrees demand RMR formation scenario supplemented by efficient tides. The above deductions obviously require the crucial inference of [19] that the SMR and RMR formation channels lead to and resonances, respectively. We also gather from our numerical integrations that and essentially remain constant in the interval for the above two families of equilibrium solutions. The negligible evolution of these quantities validate Schnittman’s observation that the equilibrium configurations remain in their associated resonant plane during the inspiral. Note that the expression for allows us to follow the variations in the angular part of , given by equation (12).

It turns out that the above listed constraints also apply for binary configurations that do not satisfy Schnittman’s equilibrium solutions at . In figure 7, we present and plots for binaries that do not lie in the and resonant planes. This is implemented by choosing values that are noticeably different from their resonant values at . The left panel plots clearly show that and we clearly notice the influence of the spin-orbit resonances. The values of plotted quantities at in the right panel plots are consistent with the SMR formation scenario for these binaries. We again observe that as required for binaries influenced by the spin-orbit resonances. We have evolved a number of similar binary configurations and the resulting values of and at indeed follow the constraints satisfied by binaries that lie in the and