7 Evidence for Cusp Destruction

A potentially powerful constraint on models of binary SBH evolution is the observed central density structure of galaxies. Figure 5

shows that a massive binary must eject of order its own mass in reaching a separation at which $10 tgr <~ 10 yr$ if $m2 ~~ m1$ , or several times $m2$ if $m2 « m1$ . These numbers should be interpreted with caution since:

Binaries might not decay this far - they may stall - or the final stages of decay might be driven by gas dynamics rather than energy exchange with stars.
The definition of “ejection” used in Figure 5 is escape of a star from an isolated binary, and does not take into account the confining effect of the nuclear potential.
The effect of repeated mergers on nuclear density profiles, particularly mergers involving very unequal-mass binaries, is poorly understood.

Nevertheless, even the initial formation of a hard binary displaces a mass of order $m2$ (Figure 8 ).

The luminosity profile data can probably be used to rule out one model of binary evolution. In a “collisionless” galaxy (Table 2), the binary’s loss cone never refills, and decay of the binary would stall. The binary carves out a “hole” in both phase space and configuration space; the radius of the latter would be $~ 3a h$ [233 ]. While central minima may have been seen in the luminosity profiles of a few galaxies [113 ], these are likely due to dust obscuration, and the great majority of galaxies show a clearly rising stellar density into radii $<~ rinfl$ . The non-existence of true “cores” suggests either that some degree of loss-cone refilling occurs, or that the final decay of the binary takes place via a more efficient process than ejection of stars.

Nevertheless there is a well-defined trend for the central densities of bright galaxies to decrease with increasing luminosity [52 , 138 , 44 , 65 , 78 ]. Nuclear densities in elliptical galaxies and spiral bulges with $MV <~ - 20$ follow $r ~ r -g$ , $g <~ 1$ , while in fainter spheroids, $1 <~ g <~ 2.5$ . A natural interpretation is that the brightest galaxies - which presumably formed via one or more mergers - have experienced more cusp destruction than fainter galaxies. (An alternative possibility, discussed below, is that the nuclei in faint galaxies re-formed after being destroyed.)

In practice, this hypothesis is difficult to test since it requires knowledge of the pre-merger density profiles. A reasonable guess is that all galaxies originally had steep power-law density cusps, since these are generic in the faintest galaxies known to harbor SBHs. For instance, both M32 and the bulge of the Milky Way have $r ~ r- 1.5$ at $r <~ rinfl$ and $r ~ r-2$ just outside [112 , 66 ].

Figure 11:

Observed surface brightness profile of NGC 3348. The dashed line is the best-fitting Sersic model to the large-radius data. Solid line is the fit of an alternative model, the “core-Sersic” model, which fits both the inner and outer data well. The mass deficit is illustrated by the area designated “depleted zone” and the corresponding mass is roughly $3 × 108Mo .$ [76

The “mass deficit” [153

] is defined as the difference in integrated mass between the observed density profile and the primordial (pre-merger) profile. For instance, if the primoridal profile is a power law of index $g0$ inward of some radius $rb$ , then

Mass deficits in samples of bright elliptical galaxies were computed in three recent studies [153 , 181 , 76 ]. In the first two studies, the authors assumed power-laws of various slopes for the pre-merger profiles, and found $<Mdef /M •> ~~ 1$ for $g0 = 1.5$ with $M •$ the current mass of the SBH. The latter study made use of the fact that the light profiles of bright galaxies show an abrupt downward deviation relative to a Sersic [197 ] profile fit to the outer regions (Figure 11

). Mass deficits inferred in this study were slightly larger, $Mdef /M • ~~ 2$ .¹ These numbers are within the range predicted by the binary SBH model, particularly given the uncertainties associated with the effects of multiple mergers. In small dense galaxies, a destroyed cusp would be expected to re-form via the Bahcall-Wolf [8 , 171 ] process, on a timescale of order the star-star relaxation time measured at $rinfl$ . This time is of order $109 yr$ in the Milky Way bulge and the nucleus of M32. This may be the explanation for the steep power-law profiles observed at the centers of these galaxies. Alternatively, the steep cusps may be due to star formation that occurred after the most recent merger [97

Figure 12:

Effect on the nuclear density profile of SBH ejection. The initial galaxy model (black line) has a $r ~ r-1$ density cusp. (a) Impulsive removal of the SBH. Tick marks show the radius of the black hole’s sphere of influence $rinfl$ before ejection. A core forms with radius $~ 2rinfl$ . (b) Ejection at velocities less than escape velocity. The black hole has mass 0.3% that of the galaxy; the galaxy is initially spherical and the black hole’s orbit remains nearly radial as it decays via dynamical friction. The arrow in this panel marks $rinfl$ in the initial galaxy. [140

More rigorous tests of the binary SBH model will require a better understanding of the expected effect of massive binaries on stellar density profiles. As discussed above, while the best current $N$ -body simulations suggest $r ~ r-1$ following binary formation [150 ], the simulations are dominated by noise over the long term.

A number of other processes could compete with binary SBHs in the destruction of nuclear density cusps. A population of three or more SBHs in a galactic nucleus would undergo a complicated set of close encounters resulting ultimately in coalescence and/or ejection of some or all of the SBHs (Section 5). In the process, the stellar background would be heated and a mass of order five times the combined mass in SBHs removed [139 ]. This model reproduces the observed time dependence of core radii in globular clusters very well [142 ] but its relevance to galactic nuclei is less clear; the model requires binary coalescence times long enough that an uncoalesced binary is present when a third SBH falls in [217 ]. If a binary SBH does eventually coalesce, the gravitational radiation carries a linear momentum leading to a recoil of the coalesced hole [19 , 53 ]. Recoil velocities are estimated to be as large as $~ 400 km s- 1$ [48 , 140 ], although with considerable uncertainty. A SBH ejected from a galactic nucleus with a velocity of $~ 102 km s-1$ would quickly fall back to the center, but its displacement and infall would heat the stellar fluid and lower its density. Figure 12 shows the effects of ejection on nuclear density profiles. Mass deficits produced by this mechanism can be comparable in amplitude to those predicted by the binary SBH model.

A major focus of future work should be to calculate the evolution of $r(r)$ as predicted by the various scenarios for binary decay discussed in this article.