Final spins from the merger of precessing binary black holes
The inspiral of binary black holes is governed by gravitational radiation reaction at binary separations , yet it is too computationally expensive to begin numerical-relativity simulations with initial separations . Fortunately, binary evolution between these separations is well described by post-Newtonian equations of motion. We examine how this post-Newtonian evolution affects the distribution of spin orientations at separations where numerical-relativity simulations typically begin. Although isotropic spin distributions at remain isotropic at , distributions that are initially partially aligned with the orbital angular momentum can be significantly distorted during the post-Newtonian inspiral. Spin precession tends to align (anti-align) the binary black hole spins with each other if the spin of the more massive black hole is initially partially aligned (anti-aligned) with the orbital angular momentum, thus increasing (decreasing) the average final spin. Spin precession is stronger for comparable-mass binaries, and could produce significant spin alignment before merger for both supermassive and stellar-mass black hole binaries. We also point out that precession induces an intrinsic accuracy limitation ( in the dimensionless spin magnitude, in the direction) in predicting the final spin resulting from the merger of widely separated binaries.
pacs:04.25.dg, 04.25.Nx, 04.70.-s, 04.30.Tv
The existence of black holes is a fundamental prediction of general relativity. Isolated individual black holes are stationary solutions to Einstein’s equations, but binary black holes (BBHs) can inspiral and eventually merge. BBH mergers offer a unique opportunity to test general relativity in the strong-field limit, and as such are a primary science target for current and future gravitational-wave (GW) observatories like LIGO, VIRGO, LISA, and the Einstein telescope. BBH mergers are also important for cosmology, as they can serve as standard candles to help determine the geometry and hence energy content of the universe Schutz (1986); Holz and Hughes (2005). Astrophysical BBHs are found on at least two very different mass scales. Compact objects believed to be stellar-mass black holes have been observed in binary systems with more luminous companions. These black holes are the remnants of massive main-sequence stars, and binary systems with two such stars may ultimately evolve into BBHs. On larger scales, supermassive black holes (SBHs) with masses reside in the centers of most galaxies. They can be observed through their dynamical influence on surrounding gas and stars, and when accreting as active galactic nuclei (AGN). SBHs will form binaries as well, following the merger of two galaxies which each host an SBH at their center.
In order to merge, BBHs must find a way to shed their orbital angular momentum. At large separations, binary SBHs will be escorted inwards by dynamical friction between their host galaxies Begelman et al. (1980). The BBHs become gravitationally bound when the sum of their masses exceeds the mass of gas and stars enclosed by their orbit. The binary hardens further by scattering stars on “loss-cone” orbits that pass within a critical radius Frank and Rees (1976), though this scattering may stall at separations pc unless these orbits are refilled by stellar diffusion Milosavljevic and Merritt (2001). Unlike stars, gas can cool to form a circumbinary disk about the BBHs. A circumbinary disk of mass and radius will exert a tidal torque
on the binary in the limit that the BBH mass ratio is small and Lin and Papaloizou (1979); Goldreich and Tremaine (1980); Chang (2008). Throughout this paper we use relativists’ units in which Newton’s constant and the speed of light are unity. At a sufficiently small separation , the magnitude of this tidal torque will fall below that of the radiation-reaction torque Peters (1964)
where is the symmetric mass ratio. Once , the inspiral of the BBH is dominated by radiation reaction. The precise value of depends on the properties of the circumbinary disk, but an order-of-magnitude estimate is given by Begelman et al. (1980)
where is the mass of the larger black hole in units of , is the dynamical friction timescale for a hard binary, and is the evolution timescale from gaseous tidal torques.
General relativity completely determines the inspiral of BBH systems from separations less than . These systems are fully specified by 7 parameters: the mass ratio and the 3 components of each dimensionless spin . To a good approximation the individual masses and spin magnitudes remain constant during the inspiral, so only the precession of the two spin directions needs to be calculated. At an initial separation , the binary’s orbital speed and the spin-precession equations can therefore be expanded in this small post-Newtonian (PN) parameter. The PN expansion remains valid until the BBHs reach a final separation , after which their evolution can only be described by fully nonlinear numerical relativity (for more precise assessments of the validity of the PN expansion for spinning precessing binaries, see e.g. Buonanno et al. (2006); Campanelli et al. (2009)). Numerical relativists can simulate BBH mergers from separations Pretorius (2005); Campanelli et al. (2006); Baker et al. (2006), but these simulations are too computationally expensive to begin when the binaries are much more widely separated. The GWs produced in the merger and the mass, spin, and recoil velocity of the final black hole depend sensitively on the orientation of the BBH spins at , so it is important to determine what BBH spin orientations are expected at and whether these orientations are modified by the PN evolution between and .
The answer to the first of these questions comes from astrophysics, not general relativity. At very large separations, the two black holes are unaffected by each other and one would therefore expect an isotropic distribution of spin directions. However, an isotropic distribution of spins at would imply that most mergers would result in a gravitational recoil of km/s for the final black hole Gonzalez et al. (2007); Campanelli et al. (2007); Dotti et al. (2009). Recoils this large would eject SBHs from all but the most massive host galaxies Merritt et al. (2004), in seeming contradiction to the observed tight correlations between SBHs and their hosts Magorrian et al. (1998); Ferrarese and Merritt (2000); Tremaine et al. (2002). This problem can be avoided if Lense-Thirring precession and viscous torques align the spins of the BBHs with the accretion disk responsible for their inwards migration Bardeen and Petterson (1975); Bogdanovic et al. (2007); Berti and Volonteri (2008). The efficiency of this alignment depends on the properties of the accretion disk, but -body simulations using smoothed-particle hydrodynamics (SPH) suggest that the residual misalignment of the BBH spins with their accretion disk at could typically be for cold (hot) accretion disks Dotti et al. (2009).
The second question, does the distribution of spin directions change as the BBHs inspiral from to , can be answered by evolving this distribution over this interval using the PN spin-precession equations. We will describe these PN equations and our numerical solutions to them in Sec. II. The precession of a given spin configuration in the PN regime can be understood in terms of the proximity of that configuration to the nearest spin-orbit resonance. Schnittman Schnittman (2004) identified a set of equilibrium spin configurations in which both black hole spins and the orbital angular momentum lie in a plane, along with the total angular momentum . In the absence of radiation reaction, is conserved. For these equilibrium configurations, the spins and orbital angular momentum remain coplanar and precess jointly about with the angles between and remaining fixed. The equilibrium configurations can thus be understood as spin-orbit resonances since the precession frequencies of and about are all the same. Once radiation reaction is added, the spins and orbital angular momentum remain coplanar as the BBHs inspiral, although and slowly change on the inspiral timescale. Not only do resonant configurations remain resonant, but configurations near resonance can be captured into resonance during the inspiral. The resonances are thus very important for understanding the evolution of generic BBH systems, although the resonances themselves only occupy a small portion of the 7-dimensional parameter space characterizing generic mergers. We shall review these spin-orbit resonances in more detail in Sec. III.
Bogdanović et al. Bogdanovic et al. (2007) briefly considered whether spin-orbit resonances could effectively align SBH spins with the orbital angular momentum following the merger of gas-poor galaxies. They found that for a mass ratio and maximal spins , an isotropic distribution of spins at remains isotropically distributed when evolved to . They therefore concluded that an alternative mechanism, such as the accretion torques considered later in their paper, is needed to align the BBH spins with . This conclusion is supported by a much larger set of PN inspirals presented by Herrmann et al. Herrmann et al. (2009) who found that for equal-mass BBHs, an isotropic distribution of spins at yields a flat distribution in at . Here and in this paper is the angle between the two spins and . In the final plot of their paper, Herrmann et al. Herrmann et al. (2009) revealed their discovery of an anti-correlation between the initial and final values of for BBHs with equal dimensionless spins . Investigation of this anti-correlation was left to future work. Lousto et al. Lousto et al. (2009a) also found indications that an initially isotropic distribution of spins can become non-isotropic during the PN stage of the inspiral. For a range of mass ratios and equal spins , they found that an isotropic spin distribution at develops a slight but statistically significant tendency towards anti-alignment with the orbital angular momentum . This amplitude of anti-alignment scales linearly in the BBH spin magnitudes and appears to decrease as .
We perform our own study of PN spin evolution from to for several reasons. BBHs get locked into spin-orbit resonances at a separation
which can become large in the equal-mass limit Schnittman (2004). This limit is important, as the largest recoil velocities occur for nearly equal-mass mergers. Numerical integration of the PN equations has shown that for a mass ratio , spin-orbit resonances affect spin orientations at separations . This is a much larger separation than was considered in previous studies Herrmann et al. (2009); Lousto et al. (2009a) of spin alignment, which may therefore have failed to capture the full magnitude of the effect. These studies also focused on whether an initially isotropic distribution of spins becomes anisotropic just prior to merger. However, as discussed above, tidal torques from a circumbinary disk partially align spins with the orbital angular momentum at separations before relativistic effects become important. As we will show in Sec. IV, such partially aligned distributions can be strongly affected by spin-orbit resonances despite the fact that isotropic distributions remain nearly isotropic. We will consider how spin precession affects the final spin magnitudes and directions in Sec. V. The evolution of the distribution of BBH spin directions between and changes the distribution of final spin magnitudes and directions from what it would have been in the absence of precession. In addition, spin precession introduces a fundamental uncertainty in predicting the final spin of a given BBH system. At large separations, a small uncertainty in the separation leads to an uncertainty in the predicted time until merger that exceeds the precession time. In this case, one cannot predict at what phase of the spin precession the merger will occur and thus the resulting final spin. We will explore this uncertainty in Sec. VI. A brief discussion of the chief findings of this paper is given in Sec. VII.
Ii Post-Newtonian Evolution
We evolve spinning BBH systems along a sequence of quasi-circular orbits according to the PN equations of motion for precessing binaries first derived by Kidder Kidder (1995), and later used by Buonanno, Chen and Vallisneri to build matched-filtering template families for GW detection Buonanno et al. (2003). The adiabatic evolution of the binary’s orbital frequency is described including terms up to 3.5PN order, and spin effects are included up to 2PN order. These evolution equations were chosen for consistency with previous work, in particular with the study by Barausse and Rezzolla Barausse and Rezzolla (2009) of the final spin resulting from the coalescence of BBHs and with the statistical investigation of spinning BBH evolutions using Graphics Processing Units by Herrmann et al. Herrmann et al. (2009). Lousto et al. Lousto et al. (2009a) evolved a large sample of spinning BBH systems using a non-resummed, PN expanded Hamiltonian. The convergence properties of non-resummed Hamiltonians for spinning BBH systems are somewhat problematic (see e.g. Fig. 1 of Ref. Buonanno et al. (2006)), and it will be interesting to repeat these statistical investigations of precessing BBH systems using the effective-one-body resummations of the PN Hamiltonian recently proposed by Barausse et al. Barausse et al. (2009); Barausse and Buonanno (2009).
In our simulations, the spins evolve according to
are the spin precession frequencies averaged over a circular orbit, including the quadrupole-monopole interaction Racine (2008),
is the Newtonian orbital angular momentum, and
is the orbital frequency. In the absence of gravitational radiation, and are constant, implying that the direction of the orbital angular momentum evolves according to
where . Once radiation reaction is included, the orbital frequency slowly evolves as
where is Euler’s constant, , and we have defined
The two terms in square parentheses on the third line of Eq. (10) are due to the quadrupole-monopole interaction Poisson (1998) and to the spin-spin self interaction Mikoczi et al. (2005), respectively, and they were neglected in the statistical study of Ref. Herrmann et al. (2009). Their sum agrees with Eq. (5.17) of Ref. Racine et al. (2009).
The numerical integration of this system of ordinary differential equations is performed using the adaptive stepsize integrator StepperDopr5 Press et al. (2007). The evolution of any given BBH system is specified by the following parameters: the initial orbital frequency , the binary’s mass ratio , the dimensionless magnitude of each spin , and the relative orientation of each spin with respect to the orbital angular momentum at time (). To monitor the variables along the whole evolution we output all quantities using a constant logarithmic spacing in the orbital frequency at low frequencies, and the stepsize as used in the integrator at high frequencies. Typically this results in a total of about points in the range , where and . Numerical experimentation indicates that a tolerance parameter atol in the adaptive stepsize integrator is sufficient for a pointwise accuracy of order or better in the final quantities. Therefore the error induced by the numerical integrations of the PN equations of motion is subdominant with respect to the errors induced by precessional effects and by fits of the numerical simulations, which will be one of the main topics of this paper.
Iii Spin-orbit Resonances
In this Section, we review the equilibrium configurations of BBH spins first presented in Schnittman Schnittman (2004) for which the Newtonian orbital angular momentum and individual spins all precess at the same resonant frequency. As discussed briefly in the Introduction, at a given binary separation fully general quasi-circular BBHs are described by 7 parameters: the mass ratio and the 3 components of each black hole spin. In spherical coordinates with defining the -axis, each spin is given by its magnitude and direction . In the PN limit for which this analysis is valid, a clear hierarchy
exists between the orbital time , the precession time , and the radiation time . This hierarchy implies that the BBH spins will precess many times before merger leaving only their relative angular separation in the orbital plane well defined. This reduces the BBH parameter space to 6 dimensions. Since the mass ratio and individual spin magnitudes are preserved during the inspiral, a given BBH evolves through the 3-dimensional parameter space on the precession timescale . This evolution is governed by the spin precession equations (5).
Schnittman Schnittman (2004) discovered a one-parameter family of equilibrium solutions to these equations for which remain fixed on the precession timescale . These solutions have or , implying that , and all lie in a plane and precess at the same resonant frequency about the total angular momentum , which remains fixed in the absence of gravitational radiation. The values of for these resonances can be determined by requiring the first and second time derivatives of to vanish. This is equivalent to satisfying the algebraic constraint
Since appears in Eq. (13) both explicitly and implicitly through , the resonant values of vary with the binary separation. This is crucial, as otherwise these one-parameter families of resonances would affect only a small portion of the 3-dimensional parameter space through which generic BBH configurations evolve. As gravitational radiation slowly extracts angular momentum from the binary on the radiation time , the resonances sweep through a significant portion of the plane. The angular separation of a generic BBH is varying on the much shorter precession time , and thus has a significant chance to closely approach the resonant values or at some point during the long inspiral. Such generic BBHs will be strongly influenced or even captured by the spin-orbit resonances, as we will see in detail in Sec. IV.
We show the dependence of the spin-orbit resonances on for maximally spinning BBHs in Figs. 1 and 2. Those resonances with (shown in Fig. 2 of Schnittman (2004)) always have , and thus appear below the diagonal in our Figs. 1 and 2. Those resonances with (shown in Fig. 3 of Schnittman (2004)) have and therefore appear above the diagonal in our Figs. 1 and 2. We plot rather than like Schnittman (2004) because isotropically oriented spins should have a flat distribution in these variables.
In the limit , so that also , the resonant configurations either have or aligned or anti-aligned with (either or equals to or ). This corresponds to the four edges of the plot in Fig. 1. For smaller fixed values of , the values for the one-parameter families of resonant configurations approach the diagonal . BBHs in spin-orbit resonances at large values of (large ) remain resonant as they inspiral. As gravitational radiation carries away angular momentum, decreases and for individual resonant BBHs evolves towards this diagonal along the red long-dashed curves in Fig. 1. For resonances with (those below the diagonal), this evolution aligns the two spins with each other. Symmetry implies that aligning the spins with each other will lead to larger final spins and smaller recoil velocities Boyle et al. (2008); Boyle and Kesden (2008).
of the total spin parallel to the orbital angular momentum is constant along the short-dashed blue lines in Figs. 1 and 2. These blue lines have steeper slopes than the red lines along which the resonant binaries inspiral. This implies that the total spin becomes anti-aligned (aligned) with the orbital angular momentum for resonant configurations with , leading to smaller (larger) final spins. The families of resonances with (below the diagonal) sweep through a larger area of the plane as the BBHs inspiral, and approach the diagonal more closely. This implies that anti-alignment may be more effective than alignment, which might explain the “small but statistically significant bias of the distribution towards counter-alignment” in noted in Lousto et al. Lousto et al. (2009a). However, Table IV of Lousto et al. (2009a) indicates that both and individually become anti-aligned with , whereas the spin-orbit resonances would align one black hole while anti-aligning the other. All of the PN evolutions in Lousto et al. Lousto et al. (2009a) begin at separations of , which corresponds to the curve in Fig. 1 that is second closest to the diagonal. The resonances sweep through most of the plane below the diagonal at larger separations, suggesting that these short-duration PN evolutions may have failed to capture the full magnitude of the anti-alignment. We will investigate this possibility in Sec. IV.
Another interesting feature of Figs. 1 and 2 is that the red long-dashed curves along which the BBHs inspiral are nearly parallel to the dot-dashed green lines along which the projection of the effective-one-body (EOB) spin Damour (2001)
is constant. The conservation of this quantity at 2PN order was first noted in Ref. Racine (2008) and follows directly from Eqs. (5), (6), and (9). The conservation of rather than itself allows for the possible alignment of the total spin discussed in the previous paragraph.
We conclude this Section by briefly discussing how the spin-orbit resonances vary with the mass ratio , as can be seen by comparing the resonances in Fig. 1 with the resonances in Fig. 2. The most pronounced differences are that the resonances sweep away from the edges of the plane at much smaller values of the separation , and do not approach the diagonal as closely. This is consistent with the decreasing value of in Eq. (4) as . In this limit both and are proportional to , implying that generic BBHs will be less likely to be affected by the resonances as they sweep through the plane over a smaller range in . BBHs already in a resonant configuration will also be less affected since the resonant curves do not approach the diagonal as closely. The red long-dashed curves showing the inspiral of resonant configurations have steeper slopes for , consistent with the larger black hole being immune to its smaller companion in the limit . This seems to contradict the puzzling result presented in Table IV of Lousto et al. Lousto et al. (2009a) that it is the smaller companion that remains randomly distributed during the inspiral. We will examine this behavior as well in the next Section.
Iv Spin Alignment
In this Section, we examine the extent to which the spins of generic (i.e. misaligned) BBH configurations become aligned with the orbital angular momentum and each other as the BBHs inspiral from to . Although we use maximally spinning BBHs to demonstrate this alignment, the magnitude of the alignment is comparable for all BBHs with as shown in Fig. 11 of Schnittman (2004). We first consider initial spin configurations given by a uniform grid evenly spaced in . This distribution is isotropic, and would be expected in the absence of an astrophysical mechanism to align the spins. BBHs with isotropically oriented spins might form in gas-poor mergers of SBHs and mergers of stellar-mass black holes in dense clusters.
In Fig. 3, we show how the distribution of evolves as maximally spinning BBHs with a mass ratio inspiral from slightly beyond to . The top left panel shows our initial evenly spaced grid. The points are colored to indicate their initial value of : blue squares begin with (), green triangles with , and red circles with . The dotted vertical lines denote these boundaries. Only 100 points are visible in the top left panel, as the different values of cannot be distinguished in this two-dimensional projection. Spin precession reveals all 1000 points after the BBHs have inspiraled to as seen in the top right panel. Notice that the spins of all 1000 BBHs precess in a way that conserves the projection of onto (parallel to the dot-dashed green lines in Fig. 1). This is not a special feature of the spin-orbit resonances, but occurs for generically oriented spins as well. These generic spin configurations do not individually preserve over a precession time like the resonant configurations do, but they do preserve the combination . This precession continues as the BBHs inspiral to and as shown in the bottom left and bottom right panels of Fig. 3. By the time they reach the green points have diffused to fill most of the plane, while the blue (red) points have diffused into the upper right (lower left) portion of the middle region. The bottom right panel, if the points had not been colored, would reproduce Fig. 1 of Bogdanovic et al. Bogdanovic et al. (2007) and therefore support their conclusion that isotropically distributed spins remain isotropic as they inspiral. However, the colors reveal that PN evolution can drastically alter spin distributions that have been partially aligned by a circumbinary disk. For example, if the spin of the more massive black hole was aligned so that at (shown by our blue points), by the time the binary reached the larger spin could easily lie in the orbital plane and thus give rise to a smaller final spin and potentially large “superkick” Gonzalez et al. (2007); Campanelli et al. (2007).
For comparison, we show the inspiral of the same grid of maximally spinning BBHs with a mass ratio in Fig. 4. The points diffuse along the steeper lines that preserve for this less equal mass ratio. This inhibits their ability to diffuse across the boundaries, again shown by the vertical dotted lines. Even at only a few points have trickled between the three regions. Since the spin of the more massive black hole remains aligned with the orbital angular momentum, one would expect a large final spin and an absence of superkicks for such small mass ratios. We will examine in detail how spin alignment affects recoil-velocity distributions in future work.
In Fig. 5 we show how the joint probability distribution function for and evolves for our evenly spaced grid of initially isotropic BBH spin configurations. As defined in the Introduction, is the cosine of the angle between and . It can be expressed in terms of the individual spin angles as
and has a flat distribution between -1 and 1 for isotropic, uncorrelated spins such as those given by our grid. However, as seen in Eq. (16), the values of and are correlated; for a given value of the distribution of is peaked about for flat distributions of and . This can be seen in Fig. 5 from the clustering of points about the curve . Although and are correlated even for isotropic spins, geometry implies that both are initially uncorrelated with the value of . This is revealed by the identical distributions of the red, green, and blue points in the top left panel of Fig. 5 to within the resolution of our grid. These distributions do not remain identical as the BBHs inspiral from to . Influenced by the spin-orbit resonances below the diagonal in Fig. 1, the blue points become concentrated about by the time they reach . The red points, similarly influenced by the resonances above the diagonal in Fig. 1, become concentrated about . The effect of this spin alignment on the spin of the final black hole will be explored in detail in the next Section, while the effect on recoil velocities will be examined in future work. Qualitatively, alignment of the spins with each other () increases the final spin and reduces the recoil velocity, while anti-alignment () does the opposite.
The magnitude of this spin alignment is greatly reduced for smaller mass ratios as seen in Fig. 6 for the case . Although the clustering of all the points about is again apparent, the distributions of the red, green, and blue points remain similar all the way down to as seen in the lower right panel. The weaker influence of the spin-orbit resonances for follows from the smaller value of in Eq. (4), and is similarly reflected by the smaller fraction of the plane occupied by the resonant curves in Fig. 2.
We have provided histograms of and in Fig. 7 to clarify the differences between Figs. 5 and 6. We see that the distributions of and are initially flat for both mass ratios, but evolve considerably for while remaining nearly flat for within the limits set by Poisson fluctuations. The open blue (red) curves in the left panels of Fig. 7 clearly show distributions peaked at (). Such trends are barely noticeable in the right panels. We will explore the implications of these findings for the final spins in the next Section.
V Final Spin Distributions
Several attempts have been made to predict the final dimensionless spin of the black hole resulting from a BBH merger. Initial attempts focused on finding simple phenomenological fitting formulae for the final spin resulting from non-spinning, unequal-mass BBH merger simulations Buonanno et al. (2007a); Berti et al. (2007); Buonanno et al. (2007b). A group at the Albert Einstein Institute (AEI) developed a fitting formula that provides the magnitude and direction of in terms of the initial spins , and the mass ratio Rezzolla et al. (2008a, b, c). They assumed that the final spin magnitude could be expressed as a polynomial in , , and the symmetric mass ratio , then made some additional assumptions about the symmetries of this polynomial dependence and how energy and angular momentum are radiated to reduce the number of terms in their expression. The coefficients of the remaining terms were calibrated using numerical-relativity (NR) simulations of BBH mergers in which the initial spins were either aligned or anti-aligned with the orbital angular momentum. We shall refer to this older AEI formula as “AEIo”. A more recent paper Barausse and Rezzolla (2009) by members of this group uses newer NR simulations to recalibrate their coefficients, and replaces earlier assumptions with the conjecture that the final spin points in the direction of the total angular momentum of the initial BBH at any separation. For consistency, this requires the further assumption that angular momentum is always radiated in the direction of the total angular momentum. We shall refer to this newer AEI formula as “AEIn”. An alternative fitting formula was proposed by a group at Florida Atlantic University (FAU) Tichy and Marronetti (2008). Following the procedure outlined in Boyle et al. (2008); Boyle and Kesden (2008), the FAU group performed 10 equal-mass misaligned simulations to calibrate the coefficients of fitting formulae for the Cartesian components of . They then made additional assumptions about the mass-ratio dependence of these formulae, and found good agreement between their predictions and independent NR simulations with mass ratios as small as . We shall refer to the formula of this group as “FAU”. The Rochester Institute of Technology (RIT) group proposed yet another fitting formula during the preparation of this paper Lousto et al. (2009b). This formula includes higher-order terms in the initial spins that may ultimately be needed to describe future high-accuracy NR simulations. However, current simulations are inadequate to calibrate all the terms appearing in the RIT formula, so we will not consider its predictions in this paper.
Other groups have predicted final spins by extrapolating analytical test-particle calculations to finite mass ratios, rather than calibrating fitting formulae with NR simulations. Buonanno, Kidder, and Lehner (BKL) Buonanno et al. (2008) derived a formula for the final spin by assuming, as is true in the test-particle limit, that the angular momentum radiated during the inspiral stage of a BBH merger exceeds that radiated during the plunge and ringdown. Using this assumption, they equated the final spin with the total angular momentum , where is the orbital angular momentum at the innermost stable circular orbit (ISCO) of a test particle of mass orbiting a black hole of mass and dimensionless spin equal to that of the final black hole. This counterintuitive but inspired choice correctly provides in the limit and respects the symmetry of BBH mergers under exchange of the labels of the two black holes. Though derived only from test-particle calculations, the BKL formula is remarkably successful at predicting final spins even for equal-mass BBH mergers. Kesden Kesden (2008) slightly modified the BKL spin formula to account for the energy radiated during the inspiral stage of the merger. This change makes the formula accurate to linear order in in the test-particle limit. It generically increases the magnitude of the predicted dimensionless final spin by reducing the predicted final mass below in the denominator of the expression . This increase improves the agreement with NR simulations of non-spinning BBH mergers, but leads to somewhat larger final spins than the other formulae for mergers of maximally spinning BBHs, such as those considered in this paper. The predictions of this formula are refered to as “Kes” in this paper.
We now present the predictions of the spin formulae summarized above for various distributions of BBH spins that are allowed to inspiral from to .
v.1 Spin Magnitudes
In the top panel of Fig. 8, we show the final spin magnitude predicted by the AEIn formula for the evenly spaced grid of maximally spinning BBHs with described in Sec. IV. The other spin formulae give very similar results; the mean and variance of the final spin distributions predicted by the other formulae for some of the initial distributions described below are provided in Table 1. As in Figs. 3-7, the black curves in Fig. 8 refer to all 1000 BBHs, the blue curves to the subset of 300 BBHs with the lowest values of , and the red curves to the subset of 300 BBHs with the highest values of . The dotted curves give the final spin distribution predicted for the BBH spin configurations at their initial separation , while the solid curves give the final spin distribution predicted when these same BBHs are allowed to inspiral to according to the PN evolution described in Sec. II. The AEIn formula is unique in that it claims to accurately predict final spins at all separations; separations as large as were considered in Barausse and Rezzolla (2009). The other fitting formulae were intended to apply at , the starting point for the NR simulations with which their coefficients were calibrated. The BKL and Kes formulae were designed for use at the ISCO. Although strictly speaking the formulae other than AEIn cannot be applied to widely separated BBHs, one can imagine that the BBHs inspiral to without spin precession where these formulae are valid. It is in this sense that we consider the predictions of these other formulae when we claim in this Section to apply them to BBH spin configuration at .
The dotted and solid black curves in the top panel of Fig. 8 are identical to within the Poisson noise of our limited number of BBH inspirals, confirming the finding of Refs. Bogdanovic et al. (2007); Herrmann et al. (2009); Lousto et al. (2009a) that isotropic distributions of BBH spins remain nearly isotropic as they inspiral. Even at , the blue (red) subset of spin configurations yields the largest (smallest) predicted final spins, because for these configurations the spin of the more massive black hole is aligned (anti-aligned) with the orbital angular momentum. The spin-orbit resonances further enhance (reduce) the final spins predicted for these subsets by aligning (anti-aligning) the BBH spins with each other during the inspiral for small (large) initial values of . As a result, the solid blue (red) distribution at has a larger (smaller) mean final spin than the initial dotted distribution at . This can be seen in the displacement of predicted final spins for the colored subsets away from towards larger and smaller values.
To clarify the magnitude of this effect, we have performed 6 additional sets of BBH inspirals, each of which consists of a fixed value of and a grid evenly spaced in and . Three of these sets have the spin of the more massive black hole nearly aligned with the orbital angular momentum (), while the other 3 sets have nearly anti-aligned with (). The choice of aligned distributions was partly motivated by the finding of Ref. Dotti et al. (2009) that accretion torques will align BBH spins to within () of the orbital angular momentum for a cold (hot) disk. The predicted final spins for these distributions, both at and , are shown in the bottom panel of Fig. 8. The final spins for the initially aligned () BBH distributions are significantly larger when predicted at than at , undermining the claim of Barausse and Rezzolla (2009) that the AEIn formula can accurately predict final spins at large separations without the need for PN evolutions. The predicted final spins for the initially anti-aligned () BBH distributions conversely shift to lower values as the predictions are made later in the inspiral. We provide the mean and standard deviation of the final spins predicted for these 6 new sets of partially aligned BBH distributions for all 5 formulae in Table 1.
To explore the dependence of these effects on the mass ratio, we have provided histograms of the predicted final spins for these same BBH spin distributions with in Fig. 9. The discrete peaks at low values of in the histograms in the top panel are an artifact of the 10 discrete values of in our grid. Each peak contains 100 points with the same initial value of . The decrease in the width of each peak as the BBHs inspiral from to is a consequence of the anti-alignment of the BBH spins for large , but the gaps between the peaks would be filled in if we used a finer grid. The shifts in the mean values of the peaks should be robust with respect to the grid spacing. These shifts for the initially aligned BBH distributions are provided in Table 1 for all 5 formulae for , as well as for the intermediate mass ratio .
v.2 Spin Directions
Before providing quantitative results, we need to clarify what is meant by the direction of the spin of the final black hole. In what reference frame is this direction defined? Most of the fitting formulae calibrated with NR simulations attempt to predict the angle
between the BBH orbital angular momentum at the separation where the NR simulations were performed and the final spin predicted from the BBH spin configuration at this same separation. The analytical predictions of BKL and Kes were designed to apply to BBH spin configurations at . If one assumed that neither the orbital angular momentum nor the BBH spins (upon which the prediction depends) precessed during the inspiral, one could insert these quantities at any separation into the right-hand side of Eq. (17) to predict . The angle is physically interesting because it quantifies the post-merger alignment between and the inner edge of the accretion disk if one assumes that torques have aligned the circumbinary disk with . However, one might also be interested in the alignment between and a feature like the galactic disk that is assumed to be aligned with at some larger scale . In that case, one would need to compute the angle
between at this larger separation and the final spin predicted from the BBH spins at this same separation.
The proper way to predict from the BBH spins at would be to use PN equations like those specified in Sec. II to propagate those spins and to down to , then insert them into the fitting formula of one’s choice. The AEIn formula is based on the conjecture that points in the direction of the total angular momentum at any separation, since angular momentum is always radiated parallel to , thus preserving its direction. This conjecture is plausible because at large separations, the precession time is much shorter than the inspiral time . If the vectors associated with the BBHs precess rapidly enough, all components except those parallel to (which varies on the longer timescale ) will average to zero. The AEIn conjecture is very useful because it allows to be computed without solving any PN equations. However, the approximation upon which it depends breaks down at small separations. This may lead to incomplete cancellation of the angular momentum radiated perpendicular to .
We test this possibility by calculating
the angle between the total angular momentum at and that after the BBHs have inspiraled to . If the direction of really was preserved during the inspiral, would vanish. We present histograms of for mass ratio in Fig. 10. The upper panel shows the grid of BBH spin configurations evenly spaced in () that we have discussed previously. The direction of changes by during most of the inspirals, though a tail extends to larger values for large initial values of . This tail can be seen more clearly in the bottom panel for the BBHs with initially anti-aligned with (). We agree with Barausse and Rezzolla (2009) that these large changes in the direction of are likely a consequence of the transitional precession first identified in Ref. Apostolatos et al. (1994). This transitional precession occurs to an even greater extent for smaller mass ratios, as can be seen in Fig. 11 for . As in the upper panel of Fig. 9, discrete peaks resulting from the grid spacing in can be seen in the left panel of Fig. 11. The middle panel shows that the direction of remains nearly constant () when in closely aligned with (). However, the right panel shows that the assumption of constant fails badly for the BBHs with , that comprise of isotropically distributed BBH mergers. The mass ratio is not extreme compared to the majority of astrophysical mergers, so caution should be taken when assuming that points in the direction of such as in Eq. (18).
What about the less ambitious predictions of from BBH spins at , assuming that NR simulations correctly describe spin precession from this separation until merger? Spin-orbit resonances have significant implications for these predictions as well. We show predictions of by the AEIn formula for a mass ratio of in Fig. 12. The other formulae predict very similar results. As in Figs. 8 and 9, the dotted curves show predictions assuming that the initial BBH spin distribution is preserved down to . The solid curves include spin precession from to according to the PN equations of Sec. II. The difference between the dotted and solid black curves in the top panel is below the Poisson fluctuations, another consequence of the finding of Refs. Bogdanovic et al. (2007); Herrmann et al. (2009); Lousto et al. (2009a) that isotropically oriented BBH spins remain nearly isotropic as they inspiral. Careful examination of the upper panel reveals that spin precession has shifted the BBHs with initially aligned with (blue distribution) to larger , while the anti-aligned BBHs have conversely shifted to smaller .
This trend is much more pronounced in the middle and bottom panels of Fig. 12. Spin precession actually results in the initially aligned BBHs () having larger values of at than the anti-aligned BBHs (), a reversal of what would be predicted from the initial spin distributions shown by the dotted curves. The spin-orbit resonances explain this highly counterintuitive result. The BBHs initially with are influenced by the resonances which align the BBH spins with each other and anti-align with . Both effects lead to larger predicted values of . Conversely, the BBHs initially with are influenced by the resonances, which greatly decrease the magnitude of and align it with . This explains the reduced values of for these BBHs seen in the bottom panel of Fig. 12. This same effect can be seen for a mass ratio of in Fig. 13, albeit with less significance owing to the weaker resonances at this smaller mass ratio. Figs. 12 and 13 again illustrate the importance of accounting for spin precession between and when attempting to predict final spins.
Vi Spin Precession Uncertainty
So far, we focused on how spin precession between and alters the expected distribution of final spins. In this Section, we show that spin precession introduces a fundamental uncertainty in predicting the final spin. An uncertainty in the BBH separation leads to an uncertainty in the time until merger. If this uncertainty is comparable to the precession time , the phase of the spin precession at which the merger occurs will be uncertain as well. This new uncertainty is independent of and may exceed that associated with the NR simulations themselves. Readers only interested in astrophysical distributions of final spins may wish to proceed to the discussion in Sec. VII.
It is often useful to define the final spin direction relative to the orbital angular momentum at different separations. We therefore generalize the angles defined in Eqs. (17) and (18) to the separation-dependent quantities
Note that these quantities reduce to the previously defined angles in the appropriate limit: , . These definitions address two ambiguities; (i) the choice of the reference orbital angular momentum and (ii) the separation at which a given fitting formula is evaluated.
Before we discuss the uncertainties in determining these angles and the final spin magnitude, we illustrate the evolution of these quantities during the PN inspiral for a few characteristic examples. In Fig. 14 we display the final spin magnitude and the angle as predicted by the AEIn and the Kesden formulae for a binary with mass ratio , extremal spins, and initial spin orientation specified by the angles , , . The behavior of the AEIo, FAU and BKL formulae is quite similar to the Kesden formula. The different curves in each panel correspond to slightly different initial frequencies or separations, , , and . The spin precession generically manifests itself in the oscillatory character of the curves; these oscillations would be absent for the resonant configurations described in Sec. III. The thin solid lines represent envelope functions obtained by fitting fourth-order polynomials to the maxima and minima, respectively, of the evolutions starting with . Note that these fits contain no information on the results obtained by using different values of , and yet they still provide excellent envelopes in all cases.
This figure illustrates two ambiguities in predicting : (i) the initial frequency at which the BBH parameters are specified, and (ii) the final separation at which the given formula for should be applied. Uncertainty in the separation at which the binary decouples from external interactions could lead to ambiguity in in theoretical studies, while uncertainty in the observed distance, projected separation, or line-of-sight velocity could lead to uncertainty in for models of particular systems. Gauge-dependent definitions of could lead to uncertainty in the separation at which fitting formulae should be applied. Our task in evaluating the resulting uncertainties for the fitting formulae AEIn, AEIo, FAU, BKL and Kes introduced in Sec. V is somewhat simplified because both ambiguities are rooted in the rapid variations of the phase and in the resulting oscillations in the final quantities. These precession-induced oscillations are a clear manifestation of the hierarchy of time scales introduced in Eq. (12): .
In the upper panels of Fig. 15 we show the angle for the same binary configuration illustrated in Fig. 14. In the lower panel of Fig. 15 we consider instead, for comparison, a system with lower mass ratio and initial spin orientation , , . As before, different curves correspond to different initial frequencies. The predicted spin direction as described by shows little variation with . On the other hand, the figure demonstrates a strong dependence of on the separation at which we apply the fitting formulae.