The angular momentum barrier is central to SBH formation theories. Any model for how material is channeled into an accreting black hole must describe the mechanism by which angular momentum is removed from the material. Whatever this mechanism may be, it is expected that it operates universally during the epoch in which SBHs grew to their present masses by rapidly accreting material onto pre-existing black hole “seeds”. This is also the period when galaxy merging peaks [97, 83, 222, 216] . While still elusive to astronomical probes due to severe obscuration [190] , the nuclei of merging galaxies, which are also the sites for the formation of binary SBHs [18] , are expected to contain the largest concentration of dense gas anywhere in the universe. The inevitable abundance of gas motivates an inquiry into the role of gas dynamics as an alternative to stellar dynamics in the process of SBH coalescence. Some of the mechanisms that remove angular momentum of interstellar gas and thus channel it into the neighborhood of SBHs include the torquing of gas flow by the rapidly-fluctuating potential of merging galaxies [144] and by nested stellar bars [199] , angular momentum transport by hydrodynamical turbulence that might be driven by the onset of self-gravity [198, 60, 73] or by supernovae embedded within a large-scale toroidal circumnuclear flow [218] , angular momentum extraction by magnetohydrodynamical turbulence [11] or by magnetic braking [22] , and more speculatively, by Rossby vortex instabilities [118] .

Astronomical observations offer abundant evidence for both hot and cold gas flows in the immediate vicinity of SBH candidates. The origin and the dynamical impact of the two classes of gas flow are distinct and are discussed here separately.

8.1 Interaction with hot gas

Hot gas permeates the interstellar space in galaxies and the intergalactic space in groups of galaxies and galaxy clusters. Virial temperatures range between ${10}^6\text{–}{10}^{8}K$  and the hot gas is almost completely ionized. Primordial and secondary sources contribute to the pool of hot has.

During the early stages of galaxy formation, intergalactic space contains partially ionized gas inherited from the pregalactic, early universe. Hydrogen recombines at redshifts $z\sim 1000$  and is reionized at redshifts $z\sim 10$  by the radiation emitted by the earliest structures. The partially ionized gas cools within the confining gravitational potential of dark matter halos and filaments.

Cold gas accelerates toward the halos' centers of gravity and is shock-heated to about the virial temperature. Some of the coldest inflowing gas escapes heating by accreting along narrow channels that reach deep inside the primary halo. Cooling times in the halo centers where the gas is the densest are short compared to the dynamical time and thus most of the primordial gas is consumed in starbursts on a dynamical time scale.

Tenuous gas that remains after the cooling time has exceeded the dynamical time in the nascent galaxy might still be plentiful enough to feed a massive black hole growing at an Eddington-limited rate. The residual number density at the radius of influence of the SBH is

$$\begin{array}{ccc}n& \approx & \frac{{\sigma }^{3}kT}{GM\Lambda }\end{array}$$

$$\begin{array}{ccc}& \approx & 20c{m}^{-3}\mu {\left(\frac{M}{{10}^{8}M\odot }\right)}^{0.11}{\left(\frac{\Lambda }{2\times {10}^{-23}ergc{m}^3{s}^{-1}}\right)}^{-1},\end{array}$$

(44)

where $T$  is the virial temperature of the galaxy, $\mu$  is the average atomic mass in units of the proton mass, $\Lambda$  is the cooling function [30] , and we have employed the $M$  - $\sigma$  relation (Equation  9 ) to relate the virial temperature to the black hole mass. The thermal stability limit could in principle be exceeded if the gas kept at the Compton temperature by a continuum flux from an unobscured AGN [45] .

This so-called “cooling flow model of quasar fueling” [26, 160] is however plagued by many problems (see [108] and references therein). Most of the gas left over from star formation might be blown out by the mechanical feedback associated with the radiative and mechanical output of the accreting massive black hole [200, 100, 156] . A small amount of angular momentum in the gas results in circularization and settling into an accretion disk. This disk may be susceptible to fragmentation, thereby converting most of the gas mass into stars and effectively cutting off the supply of gas to the SBH [208] .

The geometry of the flow of a hot, magnetized gas near a binary black hole is unknown.

Assuming spherical, non-rotating accretion, the time scale on which the hot gas is captured by the SBH is

$$\begin{array}{ccc}{t}_{\text{capt}}& \equiv & \frac{M}{\dot{M}}\end{array}$$

$$\begin{array}{ccc}& \approx & f_b\frac{{\sigma }^3}{G^{2}M\mu m_{p}n}\end{array}$$

$$\begin{array}{ccc}& \approx & {10}^{8}yr{f}_b{\mu }^{-2}{\left(\frac{M}{{10}^{8}M\odot }\right)}^{-0.44}\left(\frac{\Lambda }{2\times {10}^{-23}ergc{m}^3{s}^{-1}}\right),\end{array}$$

(45)

where $f_b\sim 1\text{–}10$  is a numerical factor that depends on the equation of state of the gas.

If a binary black hole is present, gravitational torques from the gas induce decay of the binary's semi-major axis on approximately the same time scale. This crude estimate is based on an analogy with binary-star interactions: The binary must eject of order its own mass in stars to decay an $e$  -folding in separation. Hot gas torquing the binary might be ejected in an outflow and thus the actual rate at which gas is accreting onto individual binary components might be severely suppressed compared to the accretion expected in an isolated black hole.

Galactic nuclei also contain hot gas produced by secondary sources. For example, observations with the Chandra X-ray Observatory have revealed tenuous ( $n\approx 10\text{–}100c{m}^{-3}$  ), hot ( $T\approx 1keV$  ) plasma within a parsec of the $\sim 4\times {10}^{6}M\odot$  Milky Way SBH [7] . This plasma is being generated by the numerous massive, evolved stars in the galactic region [66] through stellar wind and supernova activity. Since its temperature is higher than the virial, most ( $>$  99%) of the plasma escapes the neighborhood of the SBH [176] . While the hot gas densities in active galaxies might be transiently larger than that at the Galactic center, the tendency of the hot plasma to escape the neighborhood of the SBH reduces the likelihood that large quantities of virialized gas would remain enmeshed with the binary's orbit long enough to affect its dynamical evolution.

Recently, Escala et al. [41, 42] carried out smoothed particle hydrodynamical (SPH) simulations of binary point masses interacting with a massive, spherical cloud of hot gas initially centered on the binary. Gravitational drag from the gas induces decay in the binary's orbit. The relevance of spherical, hot initial conditions is contingent on the astrophysical plausibility that a compressed accumulation of hot gas comparable in mass to the SBH can be sustained.

8.2 Interaction with cold gas

The specific angular momentum of a cold flow might easily exceed that of the binary. The gas then tends to settle into rotationally supported, geometrically thin rings and disks (recall that “cold” gas is colder than the virial temperature but can be hot enough to be ionized).

Observations offer abundant evidence for the presence of dense gas in galactic nuclei. Thin, Keplerian molecular disks on scales $0.1\text{–}0.5pc$  have been seen in the water maser emission in the nuclei of Seyfert galaxies [154, 59, 81] . The Galactic nucleus contains a $4\times {10}^{6}M\odot$  black hole surrounded by a $\sim \left({10}^4\text{–}{10}^5\right)M\odot$  molecular gas torus at distances $>1pc$  from the SBH [94] .

Compact stellar disks on scales $\ge 20pc$  , which are fossil evidence of past gas circularization, are evident in the nuclei of many galaxies [167] . Massive accretion disks must be present in quasars and the Narrow-Line Seyfert I nuclei to account for what appears to be rapid accretion onto the central SBHs in these systems. However, the structure of these disks at radii comparable to the size of a hard SBH binary is unknown. The principal challenges to constructing extended disk models are the instabilities related to incomplete ionization and the susceptibility to gravitational fragmentation [103, 198, 149, 74] .

If a disk surrounding a binary SBH is initially inclined with respect to the binary's orbital plane, the quadrupole component of the binary's gravitational potential causes differential precession in the disk at the rate [111] 

$$\begin{array}{c}{\Omega }_{\text{prec}}\left(r\right)=\frac{3}{4}\frac{q}{(1+q{)}^2}\frac{(G{M}_{12}{)}^{1/2}{a}^2}{r^{7/2}},\end{array}$$

(46)

which results in a warping of the disk. As in the Bardeen–Petterson mechanism [15, 173, 193] , the warp either dissipates, or smears around the binary, resulting ultimately in a nearly axisymmetric disk in the binary's orbital plane.

Interest in co-planar, circumbinary disks stems from their ability to extract a binary's angular momentum via a form of tidal coupling. Two interrelated questions might be posed:

• What is the response of a circumbinary disk to the binary's tidal forcing?

and:

• How does such a disk affect the evolution of the binary's orbit?

Existing attempts to answer these questions have employed ad hoc models for the form of the binary-disk torque coupling [172, 93] , or have been restricted to binaries with components of very unequal mass [4] , where an array of neighboring Lindblad resonances facilitates binary-disk coupling [71] , much like the coupling between a massive planet and its natal gas disk [72] . Early numerical simulations of circumbinary disks with nearly equal masses [5] , however, suggested that the disks are truncated exterior to the resonances, which was interpreted as a consequence of a collisionless nonlinear parametric instability [189, 39] . Fluid dynamical theory of circumbinary disk truncation is still lacking.

In a circular binary the outer Lindblad resonances (OLR) are located at radii $r_m=(1+1/m{)}^{2/3}$  , where $m=1,2,\dots$  is the order in the decomposition of the binary's gravitational potential into multipoles:

$$\begin{array}{c}\phi (r,\theta )={\sum }_{m=0}^{\infty }{\phi }_m\left(r\right)cos\left[m\right(\theta -{\Omega }_{\text{bin}}t\left)\right].\end{array}$$

(47)

The outermost OLR is located at $r\approx 1.6a$  . The resonances are radii in the disk where the natural, epicyclic frequency of radial oscillations in the disk is an integer multiple of the rate at which a packet of disk gas receives tidal “kicks” by the binary. The forcing near a resonance, as well as at a radius where surface density in the disk exhibits a large gradient, excites nonaxisymmetric propagating disturbances, or “density waves”, in the disk.

The gravitational potential of eccentric binaries contains low-frequency components that are absent in circular binaries. These low-frequency components activate resonances located at larger radii than in the circular case, and might lead to mutual excitation and reinforcement of the binary and the disk eccentricities [162, 70] . Many extrasolar planets, which are thought to form in circumstellar disks, are notably eccentric ^$\text{2}$  , suggesting that dynamical coupling between a binary point mass (a star and a planet, or a pair of black holes) and a gas disk is conducive to eccentricity growth. The observed circumbinary disks in young stellar binaries such as GG Tau, which are typically eccentric, are truncated at radii a few times the semimajor axis [130] , which lends support to this hypothesis. Eccentricity in SBH binaries accelerates coalescence due to gravitational wave emission (Equation  7 ) and might be detectable in gravitational wave trains.

Density waves transport angular momentum outward through the circumbinary disk. Angular momentum flux carried by the waves is extracted from the binary's angular momentum. The binary experiences a negative torque equal and opposite to the total angular momentum flux transferred to the disk. The location of the inner edge of the disk reflects a balance between the angular momentum flux deposited into the disk, and the angular momentum flux transported through the disk by another, possibly viscous mechanism. Wave momentum is deposited into the disk material via a form of dissipative damping. The location in the disk where the waves are damped can be separated by many wavelengths from the location where they are excited. The damping could take place in the nonlinear steepening and the breaking of wave crests [192, 179] . In marginally optically thick disks, radiation damping might also play a role [24] . Yet another form of damping could be due to the dissipation of wave shear if the disk is strongly viscous [207] . The amplitude of the density waves is a steeply decreasing function of the radius of excitation. The amplitude is diminished if the waves are nonlinear at excitation and damp in situ, but then one expects the inner edge to recede where in situ damping shuts off.

The intricate and insufficiently understood nature of binary-disk interactions calls for grid-based hydrodynamical simulations with a shock-capturing capability. The necessity that the radial wavelength, which is smaller than the vertical scale height of the disk, be resolved by multiple cells, places severe demands on the computational resources, especially if a three dimensional representation of the disk is required. It should also be noted that the radiative and thermal structure of accretion disks around single SBHs are not adequately understood on any radial scale.

As a binary's semimajor axis decreases due to stellar, gas dynamical, or gravitational radiation processes, a circumbinary disk's inner edge spreads inward viscously while maintaining constant edge-to-semimajor axis ratio, e.g. $r_{\text{edge}}/a\sim 2$  . In the final stages of the gravitational radiation-driven inspiral, however, the time scale on which the semimajor axis decays becomes shorter than the viscous time scale, and the disk can no longer keep up with the binary, resulting in binary-disk detachment. On the relevant length scales the disk might be dominated by radiation pressure and the electron scattering opacity; the structure and the stability of such disks is an active research area [210] .

^$\text{2}$  See http://exoplanets.org

9 Spin Evolution during Mergers

Coalescence of a binary black hole results in a spinning remnant. ^$\text{3}$  Angular momentum conservation implies

$$\begin{array}{c}{\mathbf{S}}_1+{\mathbf{S}}_2+{\mathbf{L}}_{\text{orb}}=\mathbf{S}+{\mathbf{J}}_{\text{rad}}\end{array}$$

(48)

where ${\mathbf{S}}_1$  and ${\mathbf{S}}_2$  are the spin angular momenta of the two SBHs just before the final plunge, ${\mathbf{L}}_{\text{orb}}$  is the orbital angular momentum of the binary before the plunge, $\mathbf{S}$  is the spin of the resulting black hole, and ${\mathbf{J}}_{\text{rad}}$  is the angular momentum carried away by gravitational waves during and after the coalescence [54] . The simplest case to treat is extreme mass ratio mergers, $q\equiv m_2/m_1\ll 1$  , for which the binary can be described as a test particle of mass $m_2\equiv m$  orbiting a black hole of mass $m_1\equiv M\gg m$  , and both ${\mathbf{S}}_2$  and ${\mathbf{J}}_{\text{rad}}$  can be ignored. The change in the larger hole's spin is computed by adding the smaller hole's energy and orbital angular momentum at the last stable orbit (LSO).

The latter varies from $L_{\text{LSO}}/m=\sqrt{12}M$  for circular equatorial orbits around a non-spinning hole to $L_{\text{LSO}}=M\left(9M\right)$  for prograde (retrograde) orbits around a maximally-spinning hole, ${\mathbf{S}}_1=M^2$  .

The much larger value of $L_{\text{LSO}}$  in the case of retrograde capture implies that a rapidly-rotating hole will typically spin down if capture occurs from random directions [34, 69, 226, 227, 91] . The change in spin assuming $q\ll 1$  is

$$\begin{array}{c}\delta \hat{a}=q\left(-2\hat{a}+{\hat{L}}_{\text{LSO},z}\right)+O\left(q^2\right)\end{array}$$

(49)

where $\hat{a}\equiv \left|{\mathbf{S}}_1\right|/M^2$  , $L_z$  is the orbital angular momentum parallel to $\mathbf{S}$  , and $\hat{L}\equiv L/mM$  . The first term in Equation ( 49 ) describes conservation of spin angular momentum of the larger hole as its mass grows, $\hat{a}\propto M^{-2}$  , while the second term describes the increase in spin due to torquing by the smaller body.

The change in spin after a single coalescence is illustrated in Figure  13 as a function of $q$  and initial spin; the upper (lower) curves represent prograde (retrograde) captures from equatorial orbits, and the dashed lines are for capture over the pole. The bias toward spin-down is evident; retrograde capture from the equatorial plane produces a nearly (but never completely) non-spinning remnant when $q\approx 2.5\hat{a}$  , $q\le 0.23$  , and rapid final rotation ( $\hat{a}\gtrsim 0.9$  ) requires both a large initial spin and a favorable inclination. On the other hand, if the larger hole is slowly rotating initially, $\hat{a}\lesssim 0.5$  , mass ratios $q\gtrsim 0.3$  always result in spin-up. The oft-repeated statement that “mergers spin down black holes” reflects a preconception that SBHs are likely to be formed in a state of near-maximal rotation (e.g. [14, 61] ). ^$\text{4}$ 

Figure 13 : Final spin $\hat{a}$  of a remnant black hole in terms of its original spin, for mass ratios $q=0.1$  (red), $0.3$  (green) and $0.5$  (blue). The change in spin was computed using the test-particle approximation for $L_{\text{LSO}}$  [16, 91, 227] . Upper (lower) curves correspond to prograde (retrograde) capture from the equatorial plane; dashed curves are for capture over the pole. Capture of a low-mass secondary is likely to spin down the larger hole unless the latter is slowly rotating initially. Capture of a massive secondary results in spinup unless infall is nearly retrograde or the original spin is large.

Successive mergers from random directions with fixed $q$  (i.e. secondary mass grows proportionately to primary mass) lead to a steady-state spin distribution $N\left(\hat{a}\right)$  that is uniquely determined by $q$  .

For small $q$  , this distribution can be derived from the Fokker–Planck equation [91] :

$$\begin{array}{c}N\left(\hat{a}\right)d\hat{a}\approx N_0{\hat{a}}^2{e}^{-3{\hat{a}}^2/2{\hat{a}}_{\text{rms}}^2}d\hat{a},{\hat{a}}_{\text{rms}}\approx 1.58\sqrt{q}.\end{array}$$

(50)

Figure  14 shows $N\left(\hat{a}\right)$  for various values of $q$  , computed via Monte-Carlo experiments (not from the Fokker–Planck equation) using the test-mass approximation for $L_{\text{LSO}}$  . The Gaussian form of Equation ( 50 ) is seen to be accurate only for $q\lesssim 0.1$  . For $q\gtrsim 1/8$  , the distribution is skewed toward large spins.

Figure 14 : Steady-state spin distributions produced by successive capture from random directions at fixed mass ratio $q$  , for $q=(1/32,1/16,1/8,1/4,1/2)$  . Curves were generated using Monte-Carlo experiments based on the test-particle approximation, $q\ll 1$  ; hence the curve for $q=1/2$  should be viewed as illustrative only.

Accurate calculation of spin-up during a merger of comparably massive black holes requires a fully general-relativistic numerical treatment. Adopting various approximations for the radius of the innermost stable circular orbit (ISCO) for comparably massive binaries [28, 17, 80, 31] , and assuming that mass and angular momenta are conserved during coalescence, gives a remnant spin in equal-mass mergers of $\hat{a}\approx 0.8\text{–}0.9$  . Baker et al. [9, 10] present full numerical calculations of equal-mass mergers with and without initial spins. In the absence of initial spins, 3% of the system's mass-energy and 12% of its angular momentum are lost to gravitational radiation, and the final spin is $\hat{a}\approx 0.72$  . Coalescence of initially spinning holes from circular orbits in the equatorial plane yields $\hat{a}\approx 0.72+0.32\hat{s}$  with $\hat{s}$  the initial spin parameter of the two holes (assumed equal); Baker et al. considered initial spins in the range $-0.3\le \hat{s}\le 0.2$  , where negative/positive values indicate spins aligned/counteraligned with the orbital angular momentum. Extrapolating this result toward $\hat{s}=1$  suggests that prograde mergers of black holes with initial spins $\hat{s}\ge 0.85$  will result in a maximally-spinning remnant.

Confronting these predictions with observation is problematic for a number of reasons: Merger histories of observed SBHs are not known, SBH spins are difficult to determine observationally, and other mechanisms, such as gas accretion, can act efficiently to spin up SBHs [14] . However there is circumstantial evidence that mergers played a dominant role in determining the spins of at least some SBH. If SBH spins were the product of gas accretion, the jets in active galaxies should point nearly perpendicularly to the disks of their host galaxies [182] . In fact, there is almost no correlation between jet direction and galaxy major axis in Seyfert galaxies [211, 101, 194] . Among the possible explanations [101] for the misalignment, perhaps the most natural is that SBH spins were determined by the same merger events that formed the bulge, long before the formation of the gaseous disk, and that subsequent spin-up by gas accretion from the disk plane has been minimal [134] .

Coalescence of two black holes during a merger should result in a “spin-flip”, a reorientation of the spin axis of the more massive black hole. In the test-particle limit, the reorientation angle is

$$\begin{array}{c}\delta \theta =\frac{q}{\hat{a}}{\hat{L}}_{\text{LSO}}\sqrt{1-{\mu }^2}+O\left(q^2\right)\end{array}$$

(51)

with $\mu$  the cosine of the angle between the orbital angular momentum vector and the spin axis of the larger hole. When $q\gtrsim 0.2$  , the spin orientation is overwhelmed by the plunging body in a retrograde merger, even if the initial spin of the larger hole was close to maximal. Hence, even “minor mergers” (defined, following galactic dynamicists, as mergers with $q\le 0.3$  ) are able to produce a substantial reorientation. In fact there is a class of active galaxies which exhibit radio lobes at two, nearly-orthogonal orientations, and in which the production of plasma along the fainter lobes appears to have ceased [115, 32] . These “X-shaped” or “winged” radio galaxies, of which about a dozen are known, are plausible sites of recent (within the last $\sim {10}^{8}yr$  ) black-hole coalescence [137, 233] . Furthermore the implied coalescence rate is roughly consistent with the expected merger rate for the host galaxies of luminous radio sources [137] . Alternative models have been proposed for the X-shaped sources, including a warping instability of accretion disks [174] , backflow of gas along the active lobes [116] , and binary-disk interactions before coalescence [120] . ^$\text{5}$  It is likely that all of these mechanisms are active at some level and that the time scale for realignment influences the radio source morphology, with the most rapid realignments producing the classical X-shaped sources, while slower realignment would cause the jet to deposit its energy into a large volume, leading to an S-shaped FRI radio source [137] .

If the black holes are spinning prior to coalescence, they will experience spin-orbit precession, on a time scale that is intermediate between $tgr$  and the orbital period. To PN2.5 order, the spin angular momentum of either hole evolves as [98] 

$$\begin{array}{c}\frac{d{\mathbf{S}}_1}{dt}=\frac{1}{a^3}\left[\left(2+\frac{3m_2}{2m_1}\right){\mathbf{L}}_{\text{orb}}-{\mathbf{S}}_2+3\left(\hat{n}\cdot {\mathbf{S}}_2\right)\hat{n}\right]\times {\mathbf{S}}_1\end{array}$$

(52)

where $\hat{n}$  is a unit vector in the direction of the displacement vector between the two black holes.

The evolution equation for ${\mathbf{S}}_2$  is given by interchanging the indices. The magnitude of each spin vector remains fixed (to this order), and each spin precesses around the total angular momentum vector $\mathbf{J}={\mathbf{L}}_{\text{orb}}+{\mathbf{S}}_1+{\mathbf{S}}_2$  . When the two black holes are comparably massive, the orbital angular momentum greatly exceeds the spin angular momentum of either hole until just prior to coalescence.

As the binary shrinks, the spins have a tendency to unalign with ${\mathbf{L}}_{\text{orb}}$  . Ignoring spin-spin effects, the precession rate for equal-mass holes in a circular orbit is

$$\begin{array}{c}{\Omega }_1=\frac{7}{2}\frac{m_{12}^{1/2}\mu }{a^{5/2}}.\end{array}$$

(53)

The precession rate is lower than the orbital frequency:

$$\begin{array}{c}\frac{{\Omega }_{\text{orb}}}{{\Omega }_1}=\frac{2}{7}\frac{a}{\mu }=\frac{16}{7}\frac{a}{R}\end{array}$$

(54)

where $R=2m_{12}$  , but higher than the radiation reaction time scale:

$$\begin{array}{c}\frac{1}{2\pi }{\Omega }_{1}tgr=\frac{35}{1024\pi }\frac{a^{3/2}}{m_{12}^{3/2}}=\frac{35\sqrt{2}}{512\pi }{\left(\frac{a}{R}\right)}^{3/2},\end{array}$$

(55)

unless the binary is close to coalescence. If the dominant source of energy loss during the late stages of infall is gravitational radiation, the spin direction will undergo many cycles of precession before the black holes coalesce. This does not seem to happen in the X-shaped radio sources, based on the apparently sudden change in jet direction; if the coalescence model for the X-sources is correct, the final stages of infall must occur on a shorter time scale than $tgr$  . This might be seen as evidence that shrinkage of the binary is usually driven by gas dynamics, not gravitational radiation losses, prior to the final coalescence.

When the black hole masses are very different, $q\ll 1$  , the ratio of spin of the larger hole to ${\mathbf{L}}_{\text{orb}}$  is

$$\begin{array}{c}\frac{S_1}{L_{\text{orb}}}=\frac{f_1}{q}{\left(\frac{m_1}{a}\right)}^{1/2}=\frac{f_1}{q}{\left(\frac{R}{a}\right)}^{1/2}\end{array}$$

(56)

where $S_1=f_1{m}_1^2$  . The two quantities are approximately equal when the separation measured in units of the larger hole's Schwarzschild radius is equal to $q^{-2}$  . When this separation is reached, the binary orbit rapidly changes its plane, and a new regime is reached where the spin of the smaller black hole precesses about the spin of the larger hole. The precession rate of the larger hole is given by

$$\begin{array}{c}{\Omega }_1=\frac{2m_1^{1/2}{m}_2}{a^{5/2}}\end{array}$$

(57)

in both regimes, and

$$\begin{array}{c}\frac{{\Omega }_{\text{orb}}}{{\Omega }_1}=\frac{2a}{R},\frac{1}{2\pi }{\Omega }_{1}tgr=\frac{5}{128\pi }{\left(\frac{a}{R}\right)}^{3/2}.\end{array}$$

(58)

The spin direction of a black hole formed via binary coalescence is also affected by torques that reorient the binary prior to coalescence. The role of torques from gaseous accretion disks was discussed above; another source of torques is perturbations from passing stars or gas clouds [134] . A single star that passes within a distance $\sim 3a$  of the binary will exchange orbital angular momentum with it, leading both to a change in the binary's orbital eccentricity as well as a change in the orientation of the binary's spin axis, as discussed above. Referring to Equations ( 17 ) and ( 20 ), the reorientation rate is related to the hardening rate via

$$\begin{array}{c}\langle \Delta {\vartheta }^2\rangle =\frac{L}{H}\frac{m◆}{m_{12}}{t}_{\text{harden}}^{-1}\end{array}$$

(59)

where $t_{\text{harden}}^{-1}=a(d/dt)(1/a)$  . Scattering experiments [134] give $L/H\approx 4$  for a hard, equal-mass binary. The implied change in the binary's orientation after shrinking from $a\approx ah$  to $a\approx {10}^{-3}ah$  is

$$\begin{array}{c}\delta \theta \approx \sqrt{\frac{30m◆}{m_{12}}}.\end{array}$$

(60)

The reorientation begins to be significant if $m◆/m_{12}\gtrsim {10}^{-3}$  , which may be the case for intermediate-mass black holes.

^$\text{3}$  In this section, we set $G=c=1$  .

^$\text{4}$  Note the error in Figure 1 of Hughes & Blandford (2003), which shows the change in spin for mergers with mass ratio $q=0.5$  : The darkest contour on that plot should be labelled $\hat{a}=0.5$  , not $\hat{a}=0$  .

^$\text{5}$  Liu (2004) criticized the black hole coalescence model on the grounds that

• calculations based on general relativity show that the change in inclination of a rotating central SMBH is negligible in a minor merger and a significant reorientation of the active SMBH requires a comparatively rare major merger (Hughes & Blandford 2003).

This erroneous statement probably had its origin in the final sentence of the Hughes & Blandford paper, which states that

• An abrupt change in inclination [. . . ] requires a comparatively rare major merger.

Hughes & Blandford defined a “major merger” as having a mass ratio $q\ge 0.1$  , in conflict with the standard definition among galactic dynamicists, $q\ge 0.3$  . In fact Hughes & Blandford conclude, in agreement with Merritt & Ekers (2002), that mass ratios exceeding $\sim 0.2$  can result in spin-flips.

10 Summary

• 1. No completely convincing case of a bound, binary SBH has yet been discovered. “Dual” SBHs, i.e. two, widely-separated SBHs in a single system, have been seen in some interacting galaxies and binary quasars. However none has a projected separation less than $\sim 1kpc$  .
  Strong limits can be placed on the binarity of the Milky Way SBH.
• 2. The evolution of binary SBHs in gas-poor galaxies is dominated by the gravitational slingshot ejection of stars that pass near the binary, carrying away energy and angular momentum. Once the binary has ejected all stars on intersecting orbits, continued hardening depends on a refilling of the binary's “loss cone”. Possible mechanisms for loss-cone refilling include star-star gravitational scattering, chaotic orbits in non-axisymmetric potentials, and perturbations from additional massive objects.
• 3. Most $N$  -body simulations of binary evolution have been based on such small particle numbers that the binary's loss cone was refilled at a spuriously high rate by gravitational encounters, Brownian motion of the binary and other finite- $N$  effects. As a result, many results from these studies can not usefully be extrapolated to the large- $N$  regime; in particular, they do not make useful predictions about the binary hardening rates expected in real galaxies.
• 4. $N$  -body studies do appear to make robust predictions about the “mass deficit”, the mass in stars ejected from a galactic nucleus by an evolving binary. Observed mass deficits are of order 1–2 times the SBH mass, consistent with $N$  -body predictions.
• 5. Studies of the interaction of binary SBHs with gas are in their infancy. Major uncertainties are the amount, distribution and thermodynamic state of gas very near the centers of galaxies containing massive binaries.
• 6. Binary coalescence can have a large influence on SBH spins; even mass ratios as extreme as 10:1 can substantially spin up or re-orient a SBH. Evidence for spin-flips may have been observed in the so-called X-shaped radio sources.

11 Acknowledgements

This work was supported by grants AST-0071099, AST-0206031, AST-0420920 and AST-0437519 from the NSF, grant NNG04GJ48G from NASA, and grant HST-AR-09519.01-A from STScI. M. M. was supported at Caltech by a postdoctoral fellowship from the Sherman Fairchild Foundation. References

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