Spin alignment and differential accretion
in merging black hole binaries
Interactions between a supermassive black hole binary and the surrounding accretion disc can both assist the binary inspiral and align the black hole spins to the disc angular momentum. While binary migration is due to angular-momentum transfer within the circumbinary disc, the spin-alignment process is driven by the mass accreting on to each black hole. Mass transfer between different disc components thus couples the inspiral and the alignment process together. Mass is expected to leak through the cavity cleared by the binary, and preferentially accretes on to the lighter (secondary) black hole which orbits closer to the disc edge. Low accretion rate on to the heavier (primary) black hole slows the alignment process down. We revisit the problem and develop a semi-analytical model to describe the coupling between gas-driven inspiral and spin alignment, finding that binaries with mass ratio approach the gravitational-wave driven inspiral in differential misalignment: light secondaries prevent primaries from aligning. Binary black holes with misaligned primaries are ideal candidates for precession effects in the strong-gravity regime and may suffer from moderately large ( km/s) recoil velocities.
keywords:accretion, accretion discs - black hole physics - galaxies: evolution - galaxies: nuclei
Following a galaxy merger, the supermassive black holes (BHs) hosted (Kormendy & Richstone, 1995) by the two merging galaxies sink towards the centre of the newly formed stellar environment through dynamical friction, forming a binary (Begelman et al., 1980; Mayer, 2013). Binary BHs can merge if the astrophysical environment provides a way to dissipate their angular momentum in less than a Hubble time. Scattering from stars can bring the binary only down to parsec scales (Frank & Rees, 1976), below which the available phase space is quickly depleted, thus stalling the inspiral process (final-parsec problem, Milosavljević & Merritt 2001; Yu 2002). While triaxiality in the stellar potential may help driving the inspiral in elliptical gas-poor galaxies (Merritt & Poon, 2004; Berczik et al., 2006), the interaction with gaseous disc(s) may actually solve the final-parsec problem in gas-rich galaxies (Armitage & Natarajan, 2002; Cuadra et al., 2009). Indeed, dynamical friction against a gaseous background can promote the merger bringing the binary separation to distances of the order of 0.1 pc within a time-scale of 10-50 Myrs (Escala et al., 2005; Dotti et al., 2007, 2009). Further shrinking of the binary can proceed through what is known as type II migration in the context of planet-disc interaction. However, such disc-assisted migration can only be effective at separations smaller than 0.01 pc, beyond which the disc becomes self-gravitating and will likely fragment and form stars (Lodato et al., 2009). Finally, if the binary reaches separations close to , gravitational-waves (GWs) quickly become an extremely efficient way to drive the binary to a prompt merger (Peters & Mathews, 1963; Peters, 1964). Asymmetric emission of GWs in the late inspiral and merger may imprint recoil velocities to the remnant BHs (Redmount & Rees, 1989) which can be as high as km/s (Campanelli et al., 2007; González et al., 2007; Lousto & Zlochower, 2011), possibly larger than the escape speed of the host galaxies (Merritt et al., 2004).
A variety of electromagnetic signatures has been proposed to detect supermassive BH binaries, which however remain elusive (Dotti et al., 2012; Schnittman, 2013; Bogdanović, 2015). The most convincing evidence comes from double active galactic nuclei (AGN) imaging, with the notable example of the radio galaxy 0402+379 showing two compact cores with estimated separation of pc (Rodriguez et al., 2006). Indirect evidence for supermassive BH binaries at sub-parsec scales suffer from higher uncertainties; we mention in particular the case of the blazar OJ287 (Valtonen et al., 2008), where yr periodic outbursts have been interpreted as signature of a BH binary orbital motion. Identifying recoiling BHs trough observations is even more challenging (Komossa, 2012), but tentative candidates are nonetheless present (Komossa et al., 2008; Civano et al., 2012; Decarli et al., 2014; Koss et al., 2014). Direct measurements of supermassive BH inspirals and mergers are the main target of all future space-based GW observatories. The eLISA (evolved Laser Interferometer Space Antenna) mission, recently approved by the European Space Agency, is expected to detect hundreds merging binaries per year casting new light on our understating of such systems (Seoane et al., 2013; Barausse et al., 2015).
Hydrodynamical interactions can not only assist the binary inspiral, but are responsible for the reorientation of the two BH spins. BH spin alignment has a crucial impact on the merger dynamics and on the cosmic growth history of supermassive BHs. Strong recoil velocities can only be achieved if the merging BHs are highly spinning and the two spin vectors are strongly misaligned between each other. Highly recoiling BHs can be significantly displaced from the galactic nucleus or even ejected from it. This has strong consequences on the coevolution of BHs and their host galaxy, with mild recoil regulating the BH growth and large kicks velocities strongly affecting the feedback process (Blecha & Loeb, 2008). If large recoils causing BH ejections are present, this affects the fraction of galaxies hosting supermassive BHs (Schnittman, 2007; Volonteri et al., 2010; Gerosa & Sesana, 2015) and consequently the predicted (e)LISA event rates (Sesana, 2007). From the GW data analysis point of view, spin misalignments introduce a richer structure in the expected signals that carries precious information on the binary dynamics (Vecchio, 2004; Klein et al., 2009; O’Shaughnessy et al., 2013) and can improve the parameter estimation process by up to an order of magnitude (Chatziioannou et al., 2014). At the same time, since accurate waveform modelling is required in GW searches, spin precession makes the waveform generation more challenging (Pekowsky et al., 2013; Klein et al., 2014), dramatically increasing the parameter-space dimensions that need to be explored.
It is thus important for both electromagnetic and GW observations to understand in which region of the BH-binary parameter space we expect significant spin misalignment, which is the goal of this paper. The physical process responsible for the reorientation of BH spins during the long gas-driven inspiral is the so-called Bardeen-Petterson effect (Bardeen & Petterson, 1975), where the general relativistic Lense-Thirring torque between the BH and a misaligned disc warps the accretion disc and secularly aligns the BH spin with the disc angular momentum. The Bardeen-Petterson effect does not just affect the binary dynamics during the gas-driven phase, but leaves a deep imprint in the subsequent GW-driven inspiral, where precession effects are strongly dependent on the residual misalignments left by the astrophysical environment (Kesden et al., 2010a, b; Gerosa et al., 2013). It is therefore an essential ingredient to predict the spin configuration at merger.
The effectiveness of the Bardeen-Petterson effect in aligning the spins to the binary angular momentum within the time-scale of the merger has been recently investigated by multiple authors. Bogdanović et al. (2007) first made some order-of-magnitude estimates of the alignment time for a single BH with its own disc and found that it is much smaller than the merger time concluding that alignment is likely in a gaseous environment. A similar conclusion was obtained by Dotti et al. (2010), who found short alignment time-scales of yr. One notable achievement of the study performed by Dotti et al. (2010) is combination of smoothed-particle hydrodynamics (SPH) simulations and a semi-analytical treatment of the Bardeen-Petterson effect (Perego et al., 2009), through which they have been able to quantify the residual misalignment to either or depending on the gas temperature. In a previous work (Lodato & Gerosa, 2013), some of us revisited these estimates considering the previously neglected effects of non-linear warps in the misalignment propagation trough the disc. Their conclusion is that the alignment time can be significantly longer than yr if the initial misalignment are large, thus casting doubts on the ability of the disc to align the binary. Miller & Krolik (2013) made a sensible step forward, pointing out that the spin alignment process in BH binaries may actually be sensibly faster than for isolated BHs, because of the stabilizing effect of the companion that increases the degree of disc warping close to the holes.
In this paper, we argue that the binary mass ratio plays a key role in estimating the spin-alignment likelihood in merging BH binaries. We present a semi-analytical model to compute the inspiral and alignment processes from the properties of the circumbinary disc. On the one hand, the mass ratio strongly affects the binary inspiral rate, marking the onset of different disc-morphology regimes when either a gap or a proper cavity can be opened. On the other hand, and perhaps most importantly, the binary mass ratio sets the amount of differential accretion on to the two components of the binary system. The Bardeen-Peterson effectiveness in aligning the spins depends sensitively on the mass accretion rate through each single disc. We quantify this quantity constructing prescriptions based on results of hydrodynamical simulations. Gas is expected to preferentially accrete on to the lighter binary member that therefore aligns faster. Accretion rates on to the heavier BH is consequently smaller and may prevent it from aligning.
This paper is organized as follows. Our model to compute the relevant time-scales is introduced in Sec. 2, where both the inspiral and the alignment problems are treated. Sec. 3 describes our main findings, namely the role of the binary mass ratio in the spin-alignment process, and its relevance for cosmologically motivated binary distributions and kick predictions. Finally, in Sec. 4 we draw our conclusions and stress the possible caveats of our analysis.
2 Binary and discs modelling
We model the gas environment surrounding merging BH binaries through three different accretion discs: mass may reach the binary from galactic scales forming a circumbinary disc, and later be accreted on to the individual BHs from circumprimary and circumsecondary discs respectively111The name minidiscs can also be found in the literature to indicate circumprimary and circumsecondary discs.. We define to be the binary separation, and to be the two BH masses (with ), to be the total mass of the binary, to be the binary mass ratio and to be the spin of any of two BHs (where and dimensionless spin ). Following Syer & Clarke (1995), we also define a measure of the local222For typical values of the surface density exponent, this is a reasonable estimate of the rigorous value obtained radially integrating the surface density (Shakura & Sunyaev, 1973). In addition, the migration rates are set by the properties of the disc in the vicinity of the planet, and the local nature of this parameter is thus more relevant than the total disc mass. circumbinary-disc mass at a generic radius as and finally to be the disc-to-binary mass ratio. When studying the spin alignment, we will refer to the mass of the aligning BH as (meaning either or ) and the mass of the other binary member as . While the accretion rate of the circumbinary disc determines the inspiral process, the alignment time-scales are only determined by the rates and at which mass reaches the circumprimary and circumsecondary discs respectively. It is also useful to define to be the dimensionless value of in terms of the Eddington accretion rate
where is the Salpeter (1964) time, and is the opacity for Thomson electron scattering.
In this section, we first present a new estimate for the inspiral time-scale by interpolating estimates computed in different regimes and we discuss the circumbinary disc self-gravity condition to evaluate such inspiral time-scale. Secondly, we summarize the main findings of Lodato & Gerosa (2013) on the spin-alignment time-scale and we explore the effect of the companion on the individual-discs structure. We finally model mass transfer and differential accretion on to the different discs.
2.1 Gas-driven inspiral
If BH mergers do happen in nature, it is likely that the gas-driven phase is the bottleneck of the whole binary evolutionary track. Therefore, the time spent by the binary in such phase gives us an estimate of the total time available to align the BH spins through environmental processes before merger. Although gas-driven inspiral is mediated by the torques exerted by the disc on to the binary, a detailed description of the torques is not necessary to correctly derive the migration rates. Ultimately, the migration rate is controlled by the rate at which the disc is able to redistribute the angular momentum gained from the binary, and the torques will adjust to give the correct rate (e.g., Armitage 2010). This same mechanism is called type II migration in the context of protoplanetary discs (Lin & Papaloizou, 1986). Depending on the ratio between the BHs and the circumbinary disc masses, we identify three possible regimes:
For small mass ratios (), the secondary BH perturbs the disc of the primary, which reacts opening a gap333Binaries with very small mass ratios (, see Armitage 2010) cannot open a gap; however, such low mass ratios are not expected to be relevant in the supermassive BH context. at the binary separation (Lin & Papaloizou, 1979; Artymowicz & Lubow, 1994). Tidal interactions between gas particles and the secondary BH transfer angular momentum to the disc, thus decreasing the binary separation. The secondary BH behaves like a fluid element in the disc, evolving at the viscous rate (Armitage & Natarajan, 2002)
where is kinematic viscosity coefficient of the disc, usually (Shakura & Sunyaev, 1973) rescaled to a dimensionless coefficient with the speed of sound and the disc height .
If the secondary BH mass becomes comparable to the disc mass (), the disc cannot efficiently redistribute the momentum acquired from the binary. The shrinking rate consequently decreases. An analytical expression for the inspiral time-scale in this regime can be computed directly from the angular-momentum conservation equation in the thin-disc approximation (Syer & Clarke, 1995; Ivanov et al., 1999; Lodato et al., 2009; Baruteau & Masset, 2013) and reads
which correctly reduces to in the limit .
For comparable mass binaries (), the secondary-BH potential cannot be neglected. The gap at the secondary location now becomes a proper cavity in the disc with radius cleared by both BHs (MacFadyen & Milosavljević, 2008). A rich phenomenology may be present, including disc asymmetry and growing eccentricity, and can only be captured using hydrodynamical simulations (Cuadra et al., 2009; Roedig et al., 2011, 2012; Shi et al., 2012). An approximate expression for the inspiral time-scale in the comparable mass regime has been presented by Rafikov (2013), assuming the binary potential to be represented by a Newtonian potential produced by the binary total mass. He obtains
where the correction factor models the expected speed up due to the higher angular momentum flux induced by the binary mass. The same mass-ratio dependence has been very recently obtained by Dotti et al. (2015) integrating the torque at the edge of a -wide cavity
Here, we propose a smooth analytical interpolation between the time-scales obtained in the different regimes given by
which correctly reduces to either Eq. (2), (3) or (4) in the relevant limits. Various numerical factors in Eq. (4) –as already acknowledged by Rafikov (2013) himself– and different possible definitions of the viscous time-scale may modify our this estimate of a factor few. The accretion rate of the circumbinary disc only enters in the merger time-scale through the viscous time-scale, which can be rewritten as
since both and are related to the surface density of the circumbinary disc in the stationary limit. Combining Eqs. (1), (5) and (6), we obtain our final estimate of the inspiral time-scale, to be compared to the individual alignment time-scales,
While the low mass ratio regime is relatively well tested in the planetary community (e.g. Nelson et al. 2000; Bate et al. 2003, but see below for possible caveats), the regime of high mass ratios has not been explored extensively. At the moment only few simulations of disc driven migration of a binary have been conducted (MacFadyen & Milosavljević, 2008; Cuadra et al., 2009; Shi et al., 2012), which test only a small part of the parameter space. For example, for a ratio , Cuadra et al. (2009) find in 3D SPH simulations a migration rate of , where is the orbital frequency of the binary. They compare this value with the analytical formula from Ivanov et al. (1999), our Eq. (3), which yields , in very good agreement. MacFadyen & Milosavljević (2008) find in 2D grid-based simulations of binaries that the inspiral time-scale is roughly the viscous time-scale, reduced by , which is consistent with Eq. (7). The simulations described so far neglect the details of the angular momentum redistribution mechanism, which in the standard picture is the magneto-rotational instability (MRI, Balbus & Hawley 1991), and typically adopt the Shakura & Sunyaev (1973) parametrization (e.g., MacFadyen & Milosavljević 2008) in order to reduce the computational cost. Only recently Shi et al. (2012) were able to perform global numerical simulations of migrating binaries including the MRI. They found that magnetohydrodynamics effects slightly enhance the migration rate with respect to the purely hydrodynamical case (a factor of when compared to MacFadyen & Milosavljević 2008). They also observed that the accretion of material with a higher specific angular momentum than the binary can make the binary gain angular momentum, which however is offset by the higher torques they measure from the disc. Given the number of other uncertainties present in the model, we are thus satisfied that our expressions can be used reliably.
Recent numerical simulations (Duffell et al., 2014; Dürmann & Kley, 2015) in the planetary community have questioned the validity of type II migration, casting doubts that a regime where the satellite behaves like a test particle exists at all. In particular, the simulations show that it is possible to achieve faster (up to a factor of 5) migration rates than what expected from Type II theory. These simulations have only been run for a fraction of a viscous time, and it is still unclear if this result holds on the time-scale of the merger. For this reason we neglect these results in what follows, and note that this makes our estimates an upper limit for the merger time-scales.
Finally, we note that the simulations conducted so far, to the best of our knowledge, have explored relatively thick discs, with an aspect ratio ranging from 0.05 to 0.1 (e.g., MacFadyen & Milosavljević, 2008; Cuadra et al., 2009; Roedig et al., 2012; Shi et al., 2012). This is significantly thicker than the value we derive in the next section and it is not clear how the results would change with more realistic values (cf. Sec. 4).
2.2 Self-gravity condition
The inspiral time-scale reported in Eq. (7) depends on the binary separation . For typical disc structures (Shakura & Sunyaev, 1973; Goodman, 2003), is a steep monotonically increasing function of (Haiman et al., 2009). Most time will be spent by the binary at large separations, while the remaining inspiral is completed rather quickly. The time available to align the spins –which the spin-alignment time must be compared to– is roughly the inspiral time-scale evaluated at the largest separation of the disc-driven evolution.
A natural physical limit on the size of the circumbinary disc is set by the disc self gravity. Local gravitational stability under axisymmetric disturbances is guaranteed up to the fragmentation radius where the Toomre’s (1964) parameter equals unity:
At separation , self gravity cannot be neglected and the disc is gravitationally unstable (cf. e.g. Lodato 2007 for a review). The evolution of gravitationally unstable discs has been investigated in great details in recent years (Lodato & Rice, 2004; Rice et al., 2005; Cossins et al., 2009). If the cooling time is smaller or of the order of the dynamical time (Gammie, 2001), the disc will fragment into gravitationally bound clumps, although the actual fragmentation threshold is debated (Meru & Bate, 2012). For values appropriate to AGN discs, the disc is expected to fragment, create stars and thus deplete the area surrounding the binary of gaseous material, possibly halting the inspiral (Lodato et al., 2009). Using the vertical-equilibrium equation , the self-gravity stability condition can be rewritten as
evaluated at . The fragmentation radius is likely to lie in the outer region of the circumbinary disc, dominated by gas-pressure and electron-scattering opacity (Shakura & Sunyaev, 1973). Assuming viscosity to be proportional to the gas pressure (-disc) and setting the mass of the accreting object to , one gets for the fragmentation radius (Shakura & Sunyaev, 1973; Goodman, 2003; Haiman et al., 2009)
For a separation in such region, the disc aspect ratio reads
In this paper, we evaluate the inspiral time-scale of Eq. (7) at the fragmentation radius: . This is a rather conservative assumption, being monotonically increasing with (Haiman et al., 2009) and being the largest separation at which gas can be found under the form of a circumbinary disc. We are therefore assuming –somehow overcoming the final parsec problem– that some previous mechanisms are efficient enough to shrink the binary separation down to .
From Eqs. (7-11) we find that the inspiral time scales only mildly with the viscosity and the binary total mass . As for the accretion rate , the implicit dependence from Eq. (11) and (10) is also mild; the explicit dependence in Eq. (7) is still present but will cancel when compared to the spin-aligment time (Sec. 3.1). On the other hand, the dependence on plays a crucial role when comparing each spin-alignment time with the inspiral time-scale, and is therefore the main subject of this study.
2.3 BH spin alignment
The circumprimary and cirbumsecondary discs interact with the BHs through the Bardeen-Petterson effect. Bardeen & Petterson (1975) showed that a viscous disc initially misaligned with the equatorial plane of a spinning BH naturally relaxes to a coplanar state in the inner regions, while the outer disc may retain its original misalignment. Rees (1978) realized that, by Newton’s third law, the outer disc must react by pulling the BH towards complete alignment (or antialignment) of the spin with the orbital angular momentum of the outer disc itself.
Angular momentum is initially transferred from the spin to the inner disc trough relativistic Lense-Thirring precession and finally to the outer disc by the propagation of warps, i.e. vertical shearing by close, misaligned, gas rings (Scheuer & Feiler, 1996; Lodato & Pringle, 2006; Martin et al., 2007). Warp propagation is ruled by a vertical viscosity coefficient , which is generally different than the kinematic viscosity coefficient introduced above. As done for , let us introduce a vertical-viscosity coefficient such that (Papaloizou & Pringle, 1983). In the small-warp –which in our case actually means small-misalignment– limit, the warp-propagation coefficient is related to the kinematic viscosity by (Pringle, 1992; Ogilvie, 1999)
and, in particular, it is independent of the misalignment between the inner disc and the outer disc. A full non-linear theory of warp propagation has been computed by Ogilvie (1999) and later verified numerically by Lodato & Price (2010). Non-linearities introduce a qualitatively new dependence444The warp-propagation coefficient actually depends on the radial derivative of the local inclination of the disc , see Eq. (1) in Lodato & Price (2010). Here we implement the same approximation as already done by Lodato & Gerosa (2013). on , which can lower the value of by a factor of for large misalignment angles (see Fig. 1 in Lodato & Gerosa 2013). In this paper we consider the full non-linear expression as derived by Ogilvie (1999), which reduces to Eq. (12) for .
Lense-Thirring precession efficiently aligns the disc up to the Bardeen-Petterson radius , defined to be the disc location where the inverse of the Lense-Thirring precession frequency (Wilkins, 1972)
equals the warp propagation time , i.e
For a single BH-disc system, coincides with the maximum warp location (warp radius) and marks the boundary between the (quickly aligned) inner disc and the (still aligning) outer disc. The time-scale over which the outer disc finally aligns the BH spin can be found by computing the torque acting on the disc at (Natarajan & Pringle, 1998). A single BH of mass and dimensionless spin aligns with the angular momentum of the surrounding accretion disc within (Scheuer & Feiler, 1996; Natarajan & Pringle, 1998; Lodato & Gerosa, 2013)
where is the accretion rate of the circumprimary/circumsecondary disc and its aspect ratio evaluated at the warp radius. We note here that the alignment time is sensibly smaller that the growth time for reasonable viscosities and aspect ratios . BH mass and spin magnitude can be therefore considered fixed during the alignment process (King & Kolb 1999; see Sec. 4).
The Bardeen-Petterson effect can drive the BH spin towards either alignment or antialignment with the outer disc. King et al. (2005) showed that the system antialigns if
where is the angle between the BH spin and the angular momentum of the outer disc, is the angular momentum of the inner disc (i.e. inside the warp radius) and is the BH spin. The BH spin aligns with the outer disc if any of the two conditions above is not satisfied. Once is provided (cf. Sec. 3), the misalignment between the inner-disc angular momentum and the outer-disc angular momentum is given by in the aligned case, while if the system tends towards antialignment. Note that even in cases where the BH spin antialigns with its own disc, the net effect is always to reduce the misalignment with the binary plane (King et al., 2005).
2.4 Effect of the companion on disc-spin alignment
So far we have only considered the alignment of a single BH with its surrounding accretion disc. Here we discuss the effect of a far () companion on the alignment process. Such effect –neglected in our previous study (Lodato & Gerosa, 2013)– has been recently pointed out by Miller & Krolik (2013) in the supermassive BH binary case, while Martin et al. (2009) have previously considered the same interaction for stellar-mass BHs with stellar companions. For a further study, see Tremaine & Davis (2014).
If the aligning BH is part of a binary system, the gravitational potential felt by an orbiting gas ring is perturbed by the presence of the companion (Lubow & Ogilvie, 2000; Ogilvie & Dubus, 2001). The binary gravitational potential can be expanded in series of (where is the distance of the gas ring from the BH and is the binary separation; see e.g. Katz et al. 1982): to leading order, the resulting torque is perpendicular to the angular momentum of the gas ring , causing precession about the angular momentum of the binary . The precessional frequency can be obtained by averaging the torque over the binary orbital period and reads555We do not quote the sign of the precession frequency, because it only sets the precession direction about which is not important for our order-of-magnitude estimate. (Petterson, 1977)
where , is the mass of the aligning BH and is the mass of the companion. Note that in our notation in the case of the primary BH, but when the alignment of the secondary is considered.
If a spinning BH is part of a binary system, both Lense-Thirring and companion-induced precession are present. The companion drives the system towards alignment with the angular momentum of the binary, which tracks the plane of the circumbinary disc (see Sec. 4 on this point). At the same time, the inner disc is being aligned to the BH spin by Lense-Thirring precession. In practice, the companion reduces the frame-dragging efficiency: material could stay misaligned with the BH spin at closer locations, thus speeding the alignment process up (Miller & Krolik, 2013). This effect can be quantified by computing the locations at which the two contributions are equally important. The Lense-Thirring time equals the warp propagation time at , as given by Eq. (14). On the other hand, the disc is now expected to be maximally warped at the warp radius , where the Lense-Thirring contribution matches the companion one (Martin et al., 2009). From Eq. (13) and (17) one gets (Martin et al. 2009; Miller & Krolik 2013)
If , the companion term can be neglected and the closer location at which misaligned material can be found is still . The alignment speed-up discussed by Miller & Krolik (2013) is relevant if , because warped regions are present closer to the hole. At (cf. Sec. 2.2), we find
Fig. 1 shows the dependences of on the binary mass ratio and the spin magnitude of the aligning BH, both for primaries and secondaries. Slowly rotating BHs are less affected by the presence of a companion because the spin set the magnitude of the frame-dragging term. For fixed total mass , primaries are more sensible to the companion than secondaries, because their gravitational radius is larger and Lense-Thirring precession can be matched more easily by the additional precession term.
In this paper we use the simple expression reported in Eq. (15) to compute the spin-alignment time, as formally obtained for an isolated BH-disc system. Our analysis [Eq. (19) and Fig. 1] shows that the position of the warp radius can be modified by a factor of if the BH is part of a binary system. The alignment time (Natarajan & Pringle, 1998; Miller & Krolik, 2013) can therefore only be lowered by a factor of few. From Eq. (19), this assumption may not be valid if (i) the binary is very massive , (ii) the individual discs are thinner than the circumbinary disc at the fragmentation radius , (iii) the binary accretion rate is close to the Eddington limit . A more complete understating of the alignment process in BH binary systems requires explicit integrations of the angular momentum equation (Martin et al., 2009). This goes beyond the scope of this work, which instead focuses on getting an estimate for the alignment time-scale.
2.5 Cavity pile-up and differential accretion
The accretions rates of the individual discs and depend on the circumbinary-disc accretion rate , since the formers are fed by the latter. Here we develop a simple prescription to link these three quantities.
Accretion from the outermost regions of the circumbinary disc on to the binary BH is suppressed because of either the perturbation of the secondary (low- regime) or the two-body central potential (high- regime). Therefore, the binary may only accrete at a lower accretion rate . Mass tends to pile up at the outer edge of the cavity created by the binary itself: accretion –and therefore spin alignment– is still possible if gas streams can penetrate the cavity and reach the BHs. We call the mass accretion rate that overcomes the cavity pile-up: this gas will sooner or later accrete on to either the primary or the secondary BH, i.e.
Gas-stream propagation is an intrinsically multi-dimensional non-linear phenomenon that requires dedicated hydrodynamical simulations to be studied in detail. In particular, the dynamics of gas accretion through the cavity is strongly dependent on the binary mass ratio , since qualitatively different regimes are present. MacFadyen & Milosavljević (2008) first discussed equal-mass binary simulations, while the case has been presented by Hayasaki et al. (2007) and extended to by Cuadra et al. (2009) and Roedig et al. (2012). A recent major improvement has been made by D’Orazio et al. (2013) and Farris et al. (2014) who extensively studied the dependence on of the mass rate overcoming the cavity pile-up.
D’Orazio et al. (2013) present 2D hydrodynamical simulations in the range assuming fiducial values and . Accretion on to the binary is indeed limited to narrow gas streams and it is typically suppressed by a factor of 2-5 when compared to a single-BH disc of the same mass. They detect the presence of two physical regimes:
For high mass ratios , the presence of the binary strongly modulates the streams. Streams are generated by deviations from spherical symmetry in the binary potential: more asymmetry is present for equal mass binaries that therefore show less mass pile-up at the cavity edge and more binary accretion. In such regime, the ratio is expected to increase with .
In the low-mass ratio regime , the secondary BH quickly swipes through the disc accreting most of the material coming from large distances: a single gas stream is present feeding the secondary BH. Such effect gets more pronounced when the mass ratio is lower and consequently the ratio decreases with .
Such results are shown in the left-hand panel of Fig. 2, were the “Mid-Lo” simulations by D’Orazio et al. (2013) are considered. The minimum in separates the two physical regimes just described above. We interpolate the results from the simulations performed by D’Orazio et al. (2013) with the ansatz
and best-fitting coefficients , and . For a given circumbinary-disc accretion rate , Eq. (21) specifies the mass rate which overcomes the cavity pile-up and accretes on to either the primary or the secondary BH. The subsequent study by Farris et al. (2014) found that the ratio between the total accretion rate on to either one of the two BHs and the accretion rate on to a single BH of the same total mass may exceed unity, thus casting doubts on whether such fraction can be interpreted as . To bracket this uncertainty, we use Eq. (21) as our reference model but we also study an additional variation where we fix (cf. Sec. 3.1).
Farris et al. (2014) recently performed 2D grid simulations (assuming and ), specifically addressing the feeding of the individual discs from streams penetrating the cavity. They systematically find that the secondary BH accretes faster than the primary, mainly because the former orbits closer to the cavity edge. Their results are here reported in the right-hand panel of Fig. 2, where the ratio of the accretion times () is showed as a function of the binary mass ratio . Symmetry implies for binaries with high mass ratios, while lower values of show pronounced differential accretion in favour of the secondary. A qualitatively different regime is detected for the lowest of their simulated cases : the cavity is not efficiently cleared by the secondary BH, and mass from the circumbinary disc directly flows inwards forming a large circumprimary disc. As pointed out in Sec. 2.1, such change in the dynamics of the system is expected for lower mass ratio, where the disc should form a small annular gap rather than a large hollow cavity. To directly reach the circumprimary disc, gas should be able to flow past the secondary escaping its gravitational attraction. As pointed out by Farris et al. (2014) themselves, the actual turning point in is likely to be highly dependent on the thickness of the disc and possibly on the viscosity. As shown recently by Young et al. (2015) in the context of binary stars, direct flowing from the circumbinary to the circumprimary disc is easier for thicker discs, where the stronger pressure forces can make part of the material “skirt” the Roche lobe of the secondary, eventually reaching the primary Roche lobe and being captured by its gravitational attraction. Due to such uncertainties, in this work we deliberately ignore the onset of such low- regime when considering differential accretion. The growth-time ratio presented by Farris et al. (2014) appear to be well approximated by (see Fig. 2, right-hand panel)
Due to such pronounced differential accretion, the prescription here presented may formally predict super-Eddington rates for the secondary BHs in the low- regime. This has no relevant impact on our model: defined the secondary Eddington ratio to be , the Eddington limit is only marginally reached for very high circumbinary-disc accretion rates () and low mass ratio (assuming a typical accretion efficiency ). Lower values of shift the critical mass ratio at which the Eddington limit is formally reached to even lower values. Finally, we note that, as in Sec. 2.1, the thickness values explored by the simulations considered in this section are significantly higher than those expected for massive-BH binaries (Sec. 2.2).
Fig. (3) combines the results presented in Eq. (21) and Eq. (23). For equal-mass binaries, of the incoming mass may accrete on to the binary and it is equally distributed between the two binary members. Unequal-mass binaries present differential accretion that grows stronger as is decreased. When is increased from to unity, gas streams start to flows towards the primary and consequently flattens. On the other hand, if is decreased from to , the secondary orbit gets closer to the inner edge of the cavity (D’Orazio et al., 2013): more mass can overcome the cavity pile-up and it is almost entirely accreted by the secondary.
3 Results: differential misalignment
In this section we compare the spin-alignment time and the inspiral time. We first outline the regions of the parameter space where misalignments are foreseen (Sec. 3.1); secondly, we fold our model into synthetic supermassive-BH binary populations (Sec. 3.2); and we finally present a preliminary study to address the impact of our findings on the occurrence of large post-merger kicks (Sec. 3.3).
3.1 Misaligned primary BHs
3.1.1 Fiducial values of the parameters
The circumbinary disc properties enter the inspiral time while the primary/secondary alignment times are set by individual-disc parameters. The ratio in general depends on the binary separation , the three disc aspect ratios , the gas viscosity , the accretion rates of the circumbinary and the individual discs , the BH masses and (or equivalently and ), the orientation angles and , and the BH spin magnitudes and . We first specify a fiducial model by taking likely values of all these parameters and we later perform a small parameter study around such model. Table 1 summarizes the values we assume for the parameters, highlighting the differences with the next section. We discuss our choices as follows.
|Parameter||Fiducial model||Synthetic distributions|
|Free parameter||Power-law distributions|
|Not relevant||Not relevant|
|,||1||Either 1 (E) or 0.1 (C)|
|,||Extremize over||Random variables|
|Not relevant||Not relevant|
|,||Eqs. (21) and (23)||Eqs. (21) and (23)|
As detailed in Sec. 2.2, a rather conservative assumption can be made by evaluating the inspiral time at the fragmentation radius . This is a measurement of largest separation where the inspiral can be driven by interaction with a gaseous environment and is typically believed to be the bottleneck of the whole binary evolution.
In our fiducial model we fix the aspect ratios of all discs to . As reported in Eq. (11) for the circumbinary disc, the aspect-ratio dependences on the other parameters (namely the viscosity, the accreting mass and the accretion rate) are not crucial to evaluate the inspiral time-scale, and will be here neglected for simplicity. For the same reason, we assume the individual discs to share the same aspect ratio of the circumbinary disc (cf. the analogous assumption made by Miller & Krolik 2013).
Unless specified, we fix . A parametric study on the viscosity has already been presented previously (Lodato & Gerosa, 2013) and the alignment process has been found to be overall quite independent of . Negative azimuthal viscosities are formally predicted by the non-linear warp propagation theory for and large misalignments (Ogilvie, 1999; Lodato & Gerosa, 2013): the evolution of the disc in these cases is unclear and out of the scope of this study (see Nixon & King 2012 and Nixon et al. 2013 for extensive discussions).
As described in Sec. 2.5, the BH accretion rates and are related to the circumbinary disc accretion rate , conveniently expressed through the dimensionless quantity in Eq. (1). In our fiducial model we implement Eqs. (21) and (23). Once is fixed, the alignment likelihood is independent of because both times scale as [cf. Eqs. (7) and (15)]. This is a point of improvement over our previous estimate (Lodato & Gerosa, 2013), where an effective dependence on was introduced when decoupling the inspiral and the alignment processes. For concreteness, the overall scale of Fig. 4 below is computed assuming .
Within our assumptions, both the inspiral and the alignment times are independent of the binary total mass . This is compatible with Haiman et al. (2009) when is evaluated at the fragmentation radius.
For simplicity, we consider maximally spinning BHs () unless specified. The status of supermassive-BH spin measurements has been recently reviewed by Reynolds (2013): some highly spinning BHs are found, but the current statistic is too low to provide a complete picture of the spin magnitude distributions. The effect of the spin magnitude on the alignment likelihood can however be easily predicted, because the alignment time scales as , cf. Eq. (15).
3.1.2 Predicted time-scales
The key dependence of the problem is the one on the binary mass ratio , which both marks the onset of different inspiral regimes and sets the importance of differential accretion.
Fig. 4 shows the inspiral and the alignments times as a function of for our fiducial set of parameters. The uncertainty in the initial misalignments causes the alignment times to appear as stripes in the figure, rather than lines. For comparison, we also show (dotted lines) the behaviour predicted by the linear warp-propagation of Eq. (12) where is independent of . The linear theory underestimates the alignment time by up to a factor of , as already pointed out by Lodato & Gerosa (2013). Fig. 4 illustrates the main result of this paper: while secondaries are found aligned () for every value of , primary BHs only align if . Light secondaries may prevent primaries from aligning. If such BHs were misaligned before the disc interactions, these misalignments are carried over to the next stages of the binary evolution. As briefly explored in Sec. 3.3 this differential alignment between the two binary members will affect the subsequent GW-driven inspiral, the merger phase and the properties of the remnant BHs opening for the possibilities of large kicks.
|Variation||Primary BH||Secondary BH|
|Fiducial||[ - ]||()||[n-n]||(n)|
|[ - ]||()||[n-n]||(n)|
|[ - ]||()||[n-n]||(n)|
|[ - ]||()||[n-n]||(n)|
|[ - ]||()||[n-n]||(n)|
|[ - ]||()||[n-n]||(n)|
|[ - ]||()||[ - n]||(n)|
|[ - ]||()||[n-n]||(n)|
|[ - ]||()||[n-n]||(n)|
|[ - ]||()||[n-n]||(n)|
|[ - ]||()||[n-n]||(n)|
A short parametric study around our fiducial model is shown in Table 2, where we compute the values of which mark the onset of the misaligned regime (i.e. where ). As expected (Lodato & Gerosa, 2013), the alignment process is rather independent on with thresholds varying from to if is increased from to . Notably, the alignment likelihood is also rather independent on the spin magnitudes and , because of the mild scaling of [cf. Eq. (15)]. Alignments times are longer for maximally spinning BHs chosen for our fiducial model, but misaligned primaries are predicted for mass ratios even when moderately spinning BHs are considered. Perhaps more surprisingly, the alignment process appear to be strongly dependent on the discs aspect ratio which enters linearly in and with a lower power in . Only primaries with have enough time to align their spins in thicker discs , even when maximally rotating BHs are considered. Moreover, if is large enough, the inspiral time may become comparable with the secondary alignment time within the physical uncertainty due to initial spin orientation. As already pointed out (Sec. 2.1, se also Sec. 4 below), the disc thickness is one of the main uncertainties in the current modelling of binary-disc interactions. Details of the gas streams leaking through the disc cavity have also a notable effect: the largest value of where misalignment is foreseen drops down to if all of the gas of the circumbinary disc ends up being accreted by either one of the two BHs (i.e. if , cf. Farris et al. 2014).
3.2 Cosmogically-motivated distributions
Our findings are relevant if supermassive-BH binaries with spins and mass ratios in the misaligned regime are present in nature and detectable. While electromagnetic observations already constrained almost a hundred supermassive BH masses (McConnell & Ma, 2013) and a handful BH spins (Reynolds, 2013), the measurements of the global properties of the supermassive-BH binary population is the main goal of future space-based GW observatories. eLISA (Seoane et al., 2013) will detect hundreds binaries per year up to redshift with sufficient signal-to-noise ratio to measure accurately both individual-source parameters and their statistical distribution (Barausse et al., 2015).
Here we present a simplified analysis to address whether the misaligned-spin regime highlighted above is relevant in such context. Publicly available666http://www2.iap.fr/users/volonter/LISA_catalogs.html synthetic distributions of merging BH binaries have been developed by the LISA collaboration in the context of the LISA Parameter Estimation Taskforce (Arun et al., 2009) and later updated by Sesana et al. (2011). The authors developed four merger-tree models of BH evolution, varying over only two ingredients, considered to be the main sources of uncertainty.
The mass of the BH seeds. In the small seed scenario (S), first BHs of mass are initialized as remnants of Population III stars at and evolved according to Volonteri et al. (2003). In the large seed scenario (L), BH with mass are formed from gaseous protogalactic discs at to as developed by Begelman et al. (2006) (see also Lodato & Natarajan 2006)
The accretion geometry. If accretion efficiently (E) occurs on few long episodes, the BHs will generally be spun up during their cosmic evolution (Thorne, 1974). On the other hand, accretion may also happen to be chaotic (C), on many short episodes (King & Pringle, 2006). In this case, lumps of material accreted in random directions spin on average the holes down.
This approach results into four models, referred to as SE, SC, LE and LC. Fig. 5 shows the extracted mass ratio distributions, together with power-law fits in the range . We obtain for LC, for LE, for SC and for SE. The spin-magnitude distributions presented by Arun et al. (2009) are strongly peaked towards slowly spinning BHs for the C models and for the E models. This is a direct consequence of their simplified accretion treatment, which is either completely coherent or completely chaotic; broader distributions are predicted for more realistic evolutionary models where such assumption is relaxed (Barausse, 2012; Dotti et al., 2013; Sesana et al., 2014). Spin orientations are not tracked during the cosmic evolution by Arun et al. (2009): spins are assumed to efficiently align in models E, while their directions are kept isotropic in models C.
Fig. 6 show the cumulative fraction of aligned BH using these four synthetic BH-binary populations777We are aware of the inconsistency of our procedure, being the binary mass ratio distributions used here coupled to the spin orientations: at each merger tree level, the properties of the daughter BHs do depend on the spin orientations of their progenitors [cf. Sec. 2.1 in Gerosa & Sesana (2015) and references therein].. We sample the mass ratio over the fitted power-law distributions from Fig. 5; spin magnitudes are set to in the models and in the models, to mimick the strongly peaked distributions of Arun et al. (2009). For simplicity, we fix the disc properties to our fiducial values [, , given by Eq. (21)] and we sample over a uniform distribution in to extract values of the alignment time within the initial-orientation uncertainty presented in Sec. 3.1. Fig. 6 shows, for each value of , the fraction of binaries P for which . Sections at correspond to our current model: while all secondaries align during the inspiral, up to of the primaries may not have time to align their spin before merger. This statement appear to be rather independent of the population synthesis model chosen. In particular we find for LC, for SC, for LE and for SE. Two main effects are here combined: while the E models present higher spin magnitudes (hence longer alignment times) than the C models, they also predict a steeper profile in the mass ratio (Fig. 5), with fewer small- binaries (hence, on average, shorter alignment times). Those misaligned, rapidly rotating, primaries BHs predicted by the E models are ideal targets for strong-gravity precession effects in the late inspiral and merger (Schnittman 2004; Kesden et al. 2015; Sec. 3.3).
Fig. 6 also provides intuitions on the consequences of systematic errors in our time-scale estimates. If the inspiral (alignment) time is larger (smaller) of a factor , all binaries in the sample align by the end of the gas-driven inspiral. On the other hand, if the inspiral (alignment) process is times faster (slower), i.e. , only 60%-80% of the primaries aligns.
3.3 Differential misalignment and kick velocity
The most notable consequence of our findings is a clear prediction for the spin-orientation angles at the onset of the GW-driven inspirals: a non-negligible fraction of supermassive BH binaries approaches the GW-driven phase with and .
If the binary lies in the same plane of the circumbinary disc (Ivanov et al. 1999; Miller & Krolik 2013; see Sec. 4), the angles may be taken as estimates of the misalignment between the BH spins and the binary angular momentum , and used to estimate the properties of the post-merger BH. While final mass (Barausse et al., 2012) and spin (Barausse & Rezzolla, 2009) do not critically depend on the spin misalignments, these are crucial to predict the final recoil (Campanelli et al., 2007; González et al., 2007). The largest kick velocities (up to Km s) are attained for maximally spinning, equal-mass BH mergers with moderately large misalignments (Lousto & Zlochower, 2011, 2013).
Here we perform a preliminary study to estimate the impact of our findings on the kick velocity distribution. To maximize the effect, we consider maximally spinning BHs in binaries with mass ratio , right at the onset of the misaligned regime highlighted in Sec. 3.1 (cf. Fig. 4). Numerical-relativity fitting formulae are available to compute kick velocities, but precession effects during the GW-driven inspiral must be taken into account, especially for configurations with sensibly different spin tilts (Schnittman, 2004; Kesden et al., 2010a, b, 2015; Berti et al., 2012; Gerosa et al., 2013). GWs start driving the merger at the decoupling radius (Armitage & Natarajan 2002; Gold et al. 2014; see also Milosavljević & Phinney 2005)
where the angular momentum losses in GWs dominate over the viscous evolution of the disc. We first transfer the spin orientations from the initial separation to using the precession-averaged formalism recently presented by Kesden et al. (2015); and we finally apply888We refer the reader to Sec. 2.1 in Gerosa & Sesana (2015) for details on the fitting formulae implementation. In their notation, we assume a random initial phase at . In order to disentangle the dependence of the kick velocity on the spin orientations, we maximize over the orbital phase at merger (thus only showing the maximum kick allowed in each configuration). The relevance of on the results presented in this section can be easily predicted because the kick velocity scales roughly as [c.f. e.g. Eq. (2) in Campanelli et al. (2007)]. The kick velocity is independent of . the numerical-relativity fitting formula by Lousto & Zlochower (2013) at .
Fig. 7 relates the spin orientation at the decoupling radius to the maximum kicks velocities allowed in each configuration. Notably, higher kicks are found in the , region, which we predict to be populated by the Bardeen-Petterson effect ( of the cases from the models used in Sec. 3.2). The evolution of the spin orientation in the GW-driven inspiral can be qualitatively understood in terms of two special families of configurations (spin-orbit resonances, Schnittman 2004), in which the projections of the two spins on the orbital plane are either aligned or anti-aligned to each other. Kicks are suppressed (enhanced) by spin-precession effects for binaries in the aligned (anti-aligned) family (Kesden et al., 2010b). Configurations lying in the , region of the parameter space are likely to be attracted into the anti-aligned family (cf. e.g. Fig. 5 in Gerosa et al. 2013): as binaries approach the merger phase, most of their spin-precession cycles is spent with the two spins forming an angle when projected on to the orbital plane. These configurations are qualitatively similar to the standard superkick configuration (Campanelli et al., 2007; González et al., 2007) and notably predict high kick velocities.
Our prediction is that, if merging rapidly rotating BHs are present in gas-rich environments, kicks as large as km/s can happen. Such kicks can make the BH wander in the galaxy outskirt for times as long as Myr with displacements of pc (Gualandris & Merritt, 2008; Komossa & Merritt, 2008; Sijacki et al., 2011; Gerosa & Sesana, 2015), possibly at the level of observational consequences (Komossa, 2012). Larger values of are only possible in merging binaries with mass ratio closer to the equal-mass case (for more quantitative information see e.g. Fig. 3 in Lousto & Zlochower 2013). Both BHs in these binaries are predicted to be found aligned at merger (Sec. 3.1), which limits the kick velocity to km/s. Our analysis shows that superkicks with km/s are disfavoured in gas-rich environments where the Bardeen-Petterson effect comes into play.
4 Discussion and conclusions
Alignment of BH spins in merging BH binaries may be differential. Using a semi-analytical model, we find that light secondaries may accrete almost all mass leaking tough the binary gap and prevent primary BHs from alignment. In particular, such differential alignment occurs for binary with mass ratio . Gaseous interactions have enough time to align both spins in binaries with mass ratio closer to the equal-mass case (Sec. 3.1). We implement our analysis trough a time-scale argument, comparing the time needed to align the BH spins in the Bardeen-Petterson effect to the total time available in the gas-driven inspiral phase . The alignment and the inspiral processes are coupled by the accretion rates: while the binary migration is set by the circumbinary disc mass rate, alignment is powered by the mass accreting on to each BH. Mass from the circumbinary disc is expected to pile up at the outer edge of the cleared cavity, suppressing the alignment process. On top of this, mass leaking trough the cavity is found to preferentially accrete on to the secondary BH which orbits closer to the disc edge. This causes the alignment time of the primary BH to be several orders of magnitudes longer than that of the secondary, and possibly even longer than the inspiral time. Differential accretion is a key, previously neglected, feature to tackle the spin-alignment problem: for comparison, Miller & Krolik (2013) only quoted a factor of between the alignment times of the two BHs. While powerful for its simplicity, our time-scale argument fails to capture the dynamics of the alignment process: more elaborate models involving numerical simulations are needed to predict the residual misalignment of primary BHs that cannot be aligned with the Bardeen-Petterson effect, and to estimate how close to complete alignment secondaries can be found in realistic environments.
We present preliminary results to address the relevance of our findings on to the supermassive-BH cosmic history. Using publicly available synthetic populations, we find that binaries in differential misalignment are expected in realistic cosmological scenarios (Sec. 3.2). A fraction of of the BH primaries are found misaligned at merger even in models predicting large spin magnitudes, opening for the possibility of large kick velocities. Merging BHs with spin angles and are subject to the largest kicks velocities available for their mass ratio and spin magnitudes. In particular, misaligned primaries in BH binaries with may suffer kicks as large as km/s, while higher mass ratios are needed to succeed in achieving proper superkicks (Sec. 3.3). Binaries approaching the merger phase with differentially misaligned spins will exhibit pronounced precession effects in the later GW-driven inspiral phase (Schnittman, 2004; Kesden et al., 2015). Orbital plane precession modulates the amplitude of the GW cycles, encoding information of the astrophysical environment on to the emitted GW pattern (Gerosa et al. 2014; Vitale et al. 2014; Trifirò et al. in preparation). These features may in principle be used to recall and constrain our models using future space-based GW observations, although more work is needed to quantify these statements.
Several assumptions have been made in developing our models, some of them worth of future improvements. First and perhaps most importantly, our model estimates whether spin misalignments, if present, are carried over towards merger. The same dynamical processes that bring the binary together may play a role in determining the spin directions before Type II migration takes place. While star scattering is unlikely to affect the spin orientations because it does not present any preferred direction, this may not be the case for previous larger-scale gas interactions. Dynamical friction against gaseous environment may be crucial to promote the binary formation: even if short ( Myr; Escala et al. 2005; Dotti et al. 2007), this phase presents interesting dynamics involving tidal shocks and nuclear cusp disruption (Van Wassenhove et al., 2014) whose possible consequences on the spin directions still need to be explored. Secondly, our model only estimates whether the BH spins aligns to the angular momentum of the disc, while strong-gravity effects in latest inspiral and merger phase depend on the misalignments between the BH spins and the binary angular momentum. The further assumption of alignment between the binary orbital plane and the circumbinary disc (Ivanov et al., 1999) is necessary to estimate the properties of the post-merger BHs, in particular the kick velocity. Thirdly, we have neglected the BH mass growth during the alignment process. Differential accretion brings binary towards larger mass ratios on time-scales . While this effect can be safely neglected on the time-scale of the alignment process [cf. Eq. (15)], it may not be negligible on the time-scale of the inspiral. However, this point may only be important for aligned binaries which do not present large kick velocities anyway. As extensively discussed is Sec. 2.4, we also neglect the presence of the companion when estimating the alignment time (cf. Miller & Krolik 2013). However, Fig. 1 shows that this effect (a factor of in ) mostly affects binaries where both BHs aligns anyway. This point is worth further investigation, but sensible modelling efforts are likely to be required because the presence of two external torques (Lense-Thirring precession and the companion) cannot be fully captured within a time-scale argument (Martin et al., 2009). Finally, we have assumed that all values of are allowed on cosmological grounds. Assuming the BH mass correlates with the galaxy mass, galaxy pairs with may fail in forming close binaries because of strong tidal interactions before the galactic merger (Taffoni et al., 2003; Callegari et al., 2009; Van Wassenhove et al., 2014). Mass stripped away from the secondary galaxy may sensibly increase the delay time between the galaxy and the BH mergers, possibly even preventing the BH binary formation.
We stress that the impact of the disc aspect ratio on the Bardeen-Petterson effect is still not understood and can potentially be crucial. Both migration process (Sec. 2.1) and gas streaming through the binary cavity (Sec. 2.5) have only been simulated with sensibly thicker disc (typically ) than those predicted for discs surrounding supermassive BHs (, see Sec. 2.2). In particular, a sensibly lower amount of gas may be able to leak trough the cavity in thinner discs, possibly slowing down the alignment process. Although we are aware of the computational constraints in simulating thin discs, we stress that such simulations are needed to validate the analytical expressions assumed here, and we point towards the importance of pushing these numerical efforts to lower values of the aspect ratio.
Merging supermassive BH binaries are unique systems where gravity and astrophysics both play together to shape the dynamics. BH spin alignment (or misalignment) is an imprint of angular momentum transfer between the astrophysical and the relativistic side of BH binaries whose potential still need to be fully uncovered.
We especially thank Gordon Ogilvie for providing us the code to compute the warp-propagation coefficient , Brian Farris for the data that enter in Fig. 2, Cole Miller for insightful discussion on the role of the companion-induced precession and Alberto Sesana for help with the synthetic binary populations. We thank Cathie Clarke and the participants of the 2014 Milan Christmas workshop for stimulating discussions on our preliminary results; together with Michael Kesden, Emanuele Berti, Ulrich Sperhake, Richard O’Shaughnessy and the gravitational-physics group at the University of Mississippi for early discussions on the inspiral-time-scale interpolation and the kick predictions. DG is supported by the UK Science and Technology Facility Council and the Isaac Newton Studentship of the University of Cambridge; partial support is also acknowledged from FP7-PEOPLE-2011-CIG Grant No. 293412, FP7-PEOPLE-2011-IRSES Grant No.295189, SDSC and TACC through XSEDE Grant No. PHY-090003 by the NSF, Finis Terrae through Grant No. ICTS-CESGA-249, ERC-2013-ADG Grant No. 341137, STFC Roller Grant No. ST/L000636/1 and DiRAC’s Cosmos Shared Memory system through BIS Grant No. ST/J005673/1 and STFC Grant Nos. ST/H008586/1, ST/K00333X/1. Figures have been generated using the Python-based matplotlib package (Hunter, 2007).
- Armitage (2010) Armitage P. J., 2010, Astrophysics of Planet Formation. Cambridge University Press.
- Armitage & Natarajan (2002) Armitage P. J., Natarajan P., 2002, ApJ, 567, L9
- Artymowicz & Lubow (1994) Artymowicz P., Lubow S. H., 1994, ApJ, 421, 651
- Arun et al. (2009) Arun K. G. et al., 2009, Classical and Quantum Gravity, 26, 094027
- Balbus & Hawley (1991) Balbus S. A., Hawley J. F., 1991, ApJ, 376, 214
- Barausse (2012) Barausse E., 2012, MNRAS, 423, 2533
- Barausse et al. (2015) Barausse E., Bellovary J., Berti E., Holley-Bockelmann K., Farris B., Sathyaprakash B., Sesana A., 2015, Journal of Physics Conference Series, 610, 012001
- Barausse et al. (2012) Barausse E., Morozova V., Rezzolla L., 2012, ApJ, 758, 63
- Barausse & Rezzolla (2009) Barausse E., Rezzolla L., 2009, ApJ, 704, L40
- Bardeen & Petterson (1975) Bardeen J. M., Petterson J. A., 1975, ApJ, 195, L65
- Baruteau & Masset (2013) Baruteau C., Masset F., 2013, in Lecture Notes in Physics, Berlin Springer Verlag, Souchay J., Mathis S., Tokieda T., eds., Vol. 861, p. 201
- Bate et al. (2003) Bate M. R., Lubow S. H., Ogilvie G. I., Miller K. A., 2003, MNRAS, 341, 213
- Begelman et al. (1980) Begelman M. C., Blandford R. D., Rees M. J., 1980, Nature, 287, 307
- Begelman et al. (2006) Begelman M. C., Volonteri M., Rees M. J., 2006, MNRAS, 370, 289
- Berczik et al. (2006) Berczik P., Merritt D., Spurzem R., Bischof H.-P., 2006, ApJ, 642, L21
- Berti et al. (2012) Berti E., Kesden M., Sperhake U., 2012, Phys. Rev. D, 85, 124049
- Blecha & Loeb (2008) Blecha L., Loeb A., 2008, MNRAS, 390, 1311
- Bogdanović (2015) Bogdanović T., 2015, Astrophysics and Space Science Proceedings, 40, 103
- Bogdanović et al. (2007) Bogdanović T., Reynolds C. S., Miller M. C., 2007, ApJ, 661, L147
- Callegari et al. (2009) Callegari S., Mayer L., Kazantzidis S., Colpi M., Governato F., Quinn T., Wadsley J., 2009, ApJ, 696, L89
- Campanelli et al. (2007) Campanelli M., Lousto C. O., Zlochower Y., Merritt D., 2007, Physical Review Letters, 98, 231102
- Chatziioannou et al. (2014) Chatziioannou K., Cornish N., Klein A., Yunes N., 2014, Phys. Rev. D, 89, 104023
- Civano et al. (2012) Civano F. et al., 2012, ApJ, 752, 49
- Cossins et al. (2009) Cossins P., Lodato G., Clarke C. J., 2009, MNRAS, 393, 1157
- Cuadra et al. (2009) Cuadra J., Armitage P. J., Alexander R. D., Begelman M. C., 2009, MNRAS, 393, 1423
- Decarli et al. (2014) Decarli R., Dotti M., Mazzucchelli C., Montuori C., Volonteri M., 2014, MNRAS, 445, 1558
- D’Orazio et al. (2013) D’Orazio D. J., Haiman Z., MacFadyen A., 2013, MNRAS, 436, 2997
- Dotti et al. (2007) Dotti M., Colpi M., Haardt F., Mayer L., 2007, MNRAS, 379, 956
- Dotti et al. (2013) Dotti M., Colpi M., Pallini S., Perego A., Volonteri M., 2013, ApJ, 762, 68
- Dotti et al. (2015) Dotti M., Merloni A., Montuori C., 2015, MNRAS, 448, 3603
- Dotti et al. (2009) Dotti M., Ruszkowski M., Paredi L., Colpi M., Volonteri M., Haardt F., 2009, MNRAS, 396, 1640
- Dotti et al. (2012) Dotti M., Sesana A., Decarli R., 2012, Advances in Astronomy, 2012, 3
- Dotti et al. (2010) Dotti M., Volonteri M., Perego A., Colpi M., Ruszkowski M., Haardt F., 2010, MNRAS, 402, 682
- Duffell et al. (2014) Duffell P. C., Haiman Z., MacFadyen A. I., D’Orazio D. J., Farris B. D., 2014, ApJ, 792, L10
- Dürmann & Kley (2015) Dürmann C., Kley W., 2015, A&A, 574, A52
- Escala et al. (2005) Escala A., Larson R. B., Coppi P. S., Mardones D., 2005, ApJ, 630, 152
- Farris et al. (2014) Farris B. D., Duffell P., MacFadyen A. I., Haiman Z., 2014, ApJ, 783, 134
- Frank & Rees (1976) Frank J., Rees M. J., 1976, MNRAS, 176, 633
- Gammie (2001) Gammie C. F., 2001, ApJ, 553, 174
- Gerosa et al. (2013) Gerosa D., Kesden M., Berti E., O’Shaughnessy R., Sperhake U., 2013, Phys. Rev. D, 87, 104028
- Gerosa et al. (2014) Gerosa D., O’Shaughnessy R., Kesden M., Berti E., Sperhake U., 2014, Phys. Rev. D, 89, 124025
- Gerosa & Sesana (2015) Gerosa D., Sesana A., 2015, MNRAS, 446, 38
- Gold et al. (2014) Gold R., Paschalidis V., Etienne Z. B., Shapiro S. L., Pfeiffer H. P., 2014, Phys. Rev. D, 89, 064060
- González et al. (2007) González J. A., Hannam M., Sperhake U., Brügmann B., Husa S., 2007, Physical Review Letters, 98, 231101
- Goodman (2003) Goodman J., 2003, MNRAS, 339, 937
- Gualandris & Merritt (2008) Gualandris A., Merritt D., 2008, ApJ, 678, 780
- Haiman et al. (2009) Haiman Z., Kocsis B., Menou K., 2009, ApJ, 700, 1952
- Hayasaki et al. (2007) Hayasaki K., Mineshige S., Sudou H., 2007, PASJ, 59, 427
- Hunter (2007) Hunter J. D., 2007, Computing in Science and Engineering, 9, 90
- Ivanov et al. (1999) Ivanov P. B., Papaloizou J. C. B., Polnarev A. G., 1999, MNRAS, 307, 79
- Katz et al. (1982) Katz J. I., Anderson S. F., Grandi S. A., Margon B., 1982, ApJ, 260, 780
- Kesden et al. (2015) Kesden M., Gerosa D., O’Shaughnessy R., Berti E., Sperhake U., 2015, Phys.Rev.Lett., 114, 081103
- Kesden et al. (2010a) Kesden M., Sperhake U., Berti E., 2010a, Phys. Rev. D, 81, 084054
- Kesden et al. (2010b) Kesden M., Sperhake U., Berti E., 2010b, ApJ, 715, 1006
- King & Kolb (1999) King A. R., Kolb U., 1999, MNRAS, 305, 654
- King et al. (2005) King A. R., Lubow S. H., Ogilvie G. I., Pringle J. E., 2005, MNRAS, 363, 49
- King & Pringle (2006) King A. R., Pringle J. E., 2006, MNRAS, 373, L90
- Klein et al. (2014) Klein A., Cornish N., Yunes N., 2014, Phys. Rev. D, 90, 124029
- Klein et al. (2009) Klein A., Jetzer P., Sereno M., 2009, Phys. Rev. D, 80, 064027
- Komossa (2012) Komossa S., 2012, Advances in Astronomy, 2012
- Komossa & Merritt (2008) Komossa S., Merritt D., 2008, ApJ, 689, L89
- Komossa et al. (2008) Komossa S., Zhou H., Lu H., 2008, ApJ, 678, L81
- Kormendy & Richstone (1995) Kormendy J., Richstone D., 1995, ARA&A, 33, 581
- Koss et al. (2014) Koss M. et al., 2014, MNRAS, 445, 515
- Lin & Papaloizou (1986) Lin D., Papaloizou J., 1986, ApJ, 309, 846
- Lin & Papaloizou (1979) Lin D. N. C., Papaloizou J., 1979, MNRAS, 186, 799
- Lodato (2007) Lodato G., 2007, Nuovo Cimento Rivista Serie, 30, 293
- Lodato & Gerosa (2013) Lodato G., Gerosa D., 2013, MNRAS, 429, L30
- Lodato & Natarajan (2006) Lodato G., Natarajan P., 2006, MNRAS, 371, 1813
- Lodato et al. (2009) Lodato G., Nayakshin S., King A. R., Pringle J. E., 2009, MNRAS, 398, 1392
- Lodato & Price (2010) Lodato G., Price D. J., 2010, MNRAS, 405, 1212
- Lodato & Pringle (2006) Lodato G., Pringle J. E., 2006, MNRAS, 368, 1196
- Lodato & Rice (2004) Lodato G., Rice W. K. M., 2004, MNRAS, 351, 630
- Lousto & Zlochower (2011) Lousto C. O., Zlochower Y., 2011, Physical Review Letters, 107, 231102
- Lousto & Zlochower (2013) Lousto C. O., Zlochower Y., 2013, Phys. Rev. D, 87, 084027
- Lubow & Ogilvie (2000) Lubow S. H., Ogilvie G. I., 2000, ApJ, 538, 326
- MacFadyen & Milosavljević (2008) MacFadyen A. I., Milosavljević M., 2008, ApJ, 672, 83
- Martin et al. (2007) Martin R. G., Pringle J. E., Tout C. A., 2007, MNRAS, 381, 1617
- Martin et al. (2009) Martin R. G., Pringle J. E., Tout C. A., 2009, MNRAS, 400, 383
- Mayer (2013) Mayer L., 2013, Classical and Quantum Gravity, 30, 244008
- McConnell & Ma (2013) McConnell N. J., Ma C.-P., 2013, ApJ, 764, 184
- Merritt et al. (2004) Merritt D., Milosavljević M., Favata M., Hughes S. A., Holz D. E., 2004, ApJ, 607, L9
- Merritt & Poon (2004) Merritt D., Poon M. Y., 2004, ApJ, 606, 788
- Meru & Bate (2012) Meru F., Bate M. R., 2012, MNRAS, 427, 2022
- Miller & Krolik (2013) Miller M. C., Krolik J. H., 2013, ApJ, 774, 43
- Milosavljević & Merritt (2001) Milosavljević M., Merritt D., 2001, ApJ, 563, 34
- Milosavljević & Phinney (2005) Milosavljević M., Phinney E. S., 2005, ApJ, 622, L93
- Natarajan & Pringle (1998) Natarajan P., Pringle J. E., 1998, ApJ, 506, L97
- Nelson et al. (2000) Nelson R. P., Papaloizou J. C. B., Masset F., Kley W., 2000, MNRAS, 318, 18
- Nixon et al. (2013) Nixon C., King A., Price D., 2013, MNRAS, 434, 1946
- Nixon & King (2012) Nixon C. J., King A. R., 2012, MNRAS, 421, 1201
- Ogilvie (1999) Ogilvie G. I., 1999, MNRAS, 304, 557
- Ogilvie & Dubus (2001) Ogilvie G. I., Dubus G., 2001, MNRAS, 320, 485
- O’Shaughnessy et al. (2013) O’Shaughnessy R., London L., Healy J., Shoemaker D., 2013, Phys. Rev. D, 87, 044038
- Papaloizou & Pringle (1983) Papaloizou J. C. B., Pringle J. E., 1983, MNRAS, 202, 1181
- Pekowsky et al. (2013) Pekowsky L., O’Shaughnessy R., Healy J., Shoemaker D., 2013, Phys. Rev. D, 88, 024040
- Perego et al. (2009) Perego A., Dotti M., Colpi M., Volonteri M., 2009, MNRAS, 399, 2249
- Peters (1964) Peters P. C., 1964, Physical Review, 136, 1224
- Peters & Mathews (1963) Peters P. C., Mathews J., 1963, Physical Review, 131, 435
- Petterson (1977) Petterson J. A., 1977, ApJ, 214, 550
- Pringle (1992) Pringle J. E., 1992, MNRAS, 258, 811
- Rafikov (2013) Rafikov R. R., 2013, ApJ, 774, 144
- Redmount & Rees (1989) Redmount I. H., Rees M. J., 1989, Comments on Astrophysics, 14, 165
- Rees (1978) Rees M. J., 1978, Nature, 275, 516
- Reynolds (2013) Reynolds C. S., 2013, Classical and Quantum Gravity, 30, 244004
- Rice et al. (2005) Rice W. K. M., Lodato G., Armitage P. J., 2005, MNRAS, 364, L56
- Rodriguez et al. (2006) Rodriguez C., Taylor G. B., Zavala R. T., Peck A. B., Pollack L. K., Romani R. W., 2006, ApJ, 646, 49
- Roedig et al. (2011) Roedig C., Dotti M., Sesana A., Cuadra J., Colpi M., 2011, MNRAS, 415, 3033
- Roedig et al. (2012) Roedig C., Sesana A., Dotti M., Cuadra J., Amaro-Seoane P., Haardt F., 2012, A&A, 545, A127
- Salpeter (1964) Salpeter E. E., 1964, ApJ, 140, 796
- Scheuer & Feiler (1996) Scheuer P. A. G., Feiler R., 1996, MNRAS, 282, 291
- Schnittman (2004) Schnittman J. D., 2004, Physical Review D, 70, 124020
- Schnittman (2007) Schnittman J. D., 2007, ApJ, 667, L133
- Schnittman (2013) Schnittman J. D., 2013, Classical and Quantum Gravity, 30, 244007
- Seoane et al. (2013) Seoane P. A. et al., 2013, ArXiv e-prints
- Sesana (2007) Sesana A., 2007, MNRAS, 382, L6
- Sesana et al. (2014) Sesana A., Barausse E., Dotti M., Rossi E. M., 2014, ApJ, 794, 104
- Sesana et al. (2011) Sesana A., Gair J., Berti E., Volonteri M., 2011, Phys. Rev. D, 83, 044036
- Shakura & Sunyaev (1973) Shakura N. I., Sunyaev R. A., 1973, A&A, 24, 337
- Shi et al. (2012) Shi J.-M., Krolik J. H., Lubow S. H., Hawley J. F., 2012, ApJ, 749, 118
- Sijacki et al. (2011) Sijacki D., Springel V., Haehnelt M. G., 2011, MNRAS, 414, 3656
- Syer & Clarke (1995) Syer D., Clarke C. J., 1995, MNRAS, 277, 758
- Taffoni et al. (2003) Taffoni G., Mayer L., Colpi M., Governato F., 2003, MNRAS, 341, 434
- Thorne (1974) Thorne K. S., 1974, ApJ, 191, 507
- Toomre (1964) Toomre A., 1964, ApJ, 139, 1217
- Tremaine & Davis (2014) Tremaine S., Davis S. W., 2014, MNRAS, 441, 1408
- Valtonen et al. (2008) Valtonen M. J. et al., 2008, Nature, 452, 851
- Van Wassenhove et al. (2014) Van Wassenhove S., Capelo P. R., Volonteri M., Dotti M., Bellovary J. M., Mayer L., Governato F., 2014, MNRAS, 439, 474
- Vecchio (2004) Vecchio A., 2004, Phys. Rev. D, 70, 042001
- Vitale et al. (2014) Vitale S., Lynch R., Veitch J., Raymond V., Sturani R., 2014, Physical Review Letters, 112, 251101
- Volonteri et al. (2010) Volonteri M., Gültekin K., Dotti M., 2010, MNRAS, 404, 2143
- Volonteri et al. (2003) Volonteri M., Haardt F., Madau P., 2003, ApJ, 582, 559
- Wilkins (1972) Wilkins D. C., 1972, Phys. Rev. D, 5, 814
- Young et al. (2015) Young M. D., Baird J. T., Clarke C. J., 2015, MNRAS, 447, 2907
- Yu (2002) Yu Q., 2002, MNRAS, 331, 935