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\newcommand{\Rs}{R_{\star}}
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\newcommand{\peryr}{\mathrm{yr}^{1}}
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\shorttitle{Massive Perturbers in the Galactic Center}
\shortauthors{Perets and Alexander}
\usepackage{babel}
\makeatother
\begin{document}
\newcommand{\Mo}{M_{\odot}}
\newcommand{\Ro}{R_{\odot}}
\newcommand{\Lo}{L_{\odot}}
\newcommand{\SgrA}{\mathrm{Sgr\, A^{\star}}}
\newcommand{\Ms}{M_{\star}}
\newcommand{\Mbh}{M_{\bullet}}
\newcommand{\rMP}{r_{\mathrm{MP}}}
\newcommand{\aGW}{a_{\mathrm{GW}}}
\title{Massive perturbers and the efficient merger of binary massive black
holes}
\author{Hagai B. Perets and Tal Alexander\altaffilmark{1}}
\begin{abstract}
We show that dynamical relaxation in the aftermath of a galactic merger,
and the ensuing formation of a binary massive black hole (MBH), are
dominated by massive perturbers (MPs), such as giant molecular clouds
or clusters. MPs accelerate relaxation by orders of magnitude relative
to 2body stellar relaxation alone, and efficiently scatter stars
into the binary MBH's orbit. The 3body starbinary MBH interactions
shrink the binary MBH to the point where energy loss from the emission
of gravitational waves (GW) leads to rapid coalescence. We take into
account the decreased efficiency of the starbinary MBH interaction
due to acceleration in the galactic potential, and show that the observed
MP abundances in galactic nuclei imply binary MBH coalescence times
shorter than the Hubble time. These events are observable by their
strong GW emission. MPs thus increase the cosmic rate of these GW
events, increase the mass deficit in the stellar core, lead to the
ejection of hypervelocity stars, and suppress the formation of triple
MBH systems and the resulting ejection of MBHs into intergalactic
space.
\end{abstract}
\keywords{black hole physics  clusters  galaxies: nuclei  stars: kinematics
 giant molecular clouds}
\section{Introduction}
There is compelling evidence that massive black holes (MBHs) exist
in the centers of most galaxies \citep{fer+00,geb+03,shi+03}. It
is believed that galaxies grow by successive mergers, during which
the two MBHs sink to the center of the newly formed galaxy by dynamical
friction and form a {}``hard'' binary MBH (BMBH) \citep{beg+80}
with a semimajor axis of \begin{equation}
a_{h}=[Q/(1+Q)^{2}]r_{h}(M_{12})/4\,,\label{e:a_h}\end{equation}
where $M_{12}\!=\! M_{1}\!+\! M_{2}$ is the mass of the binary, $Q\!\equiv\! M_{2}/M_{1}\!\le\!1$
is the mass ratio and $r_{h}(M_{12})$ is the radius of dynamical
influence of the BMBH %
\footnote{Defined here as the radius that encloses a stellar mass of $2M_{12}$
\citep{mer+06}. The threshold semimajor axis for a hard BMBH is
sometimes defined in terms of of $\sigma^{2}$, the typical velocity
dispersion in the center, $a_{h}\!=\! GM_{1}M_{2}\mu/4\sigma^{2}$,
where $\mu=M_{1}M_{2}/M_{12}$ is the reduced mass. However, this
is illdefined since $\sigma^{2}$ usually varies with distance from
the BMBH.%
}, where typically, $a_{h}\!\sim\!110$ pc. After the BMBH hardens,
it continues to shrink by losing energy and angular momentum to stars
and gas with which it interacts dynamically. Once the separation further
decreases by 23 orders of magnitude, the BMBH rapidly decays by
the emission of gravitational waves (GWs) until the two MBHs coalesce.
Simulations show that such dynamical interactions with stars are typically
not efficient enough; the BMBH stalls before reaching a small enough
separation for efficient decay by GW emission, and fails to coalesce
in a Hubble time, $t_{H}$ (e.g. \citealt{ber+05}, see review by
\citealt{mer+05b}). This result appears to contradict the circumstantial
evidence that most Galactic nuclei contain only a single MBH \citep{ber+06,mer+05b},
and furthermore implies few such very strong GW sources, which future
GW detectors, such as the Laser Interferometric Space Antenna, hope
to detect.
Several mechanisms were suggested as means of accelerating BMBH coalescence,
either involving interactions with stars ({}``dry mergers'') or
with gas ({}``wet mergers''). These include reejection of stars
that had a previous interaction with the BMBH but were not ejected
out of the galactic core \citep{mil+03,ber+05}; BMBHs embedded in
dense gas \citep{esc+04,esc+05}; interactions of the BMBH with a
third MBH \citep{mak+94,bla+02,iwa+05}; BMBH coalescence due to accelerated
losscone replenishment in a nonaxisymmetric potential \citep{ber+06}
or in a steep cusp \citep{zie06a,zie06b}. It is still unclear whether
these mechanisms are efficient enough, or whether they occur commonly
enough to solve the stalling problem. Wet mergers require a very dense
concentration of gas in which the BMBH is embedded, gas which may
not exists there in the required quantities (e.g. the central $\sim\!2$
pc of the Galactic center (GC) are gasdepleted \citealt{chr+05};
some other galaxies show central gas cavities in their nuclei \citep{sak+99}).
They may also be dispersed by the accreting BMBH before the merger
is completed \citep{mer+05b}, and may not be efficient for minor
mergers \citep{esc+04,esc+05}. It is likewise unknown whether the
nonaxisymmetric potential assumed by the dry merger scenario of \citet{ber+06}
is generally present in the postmerger galaxy on the relevant scales.
Even if that is the case, actual demonstration of rapid BMBH coalescence
still awaits future $N$body simulations with realistically high
$N$ \citep{ber+06}.
Here we explore another possibility, which is likely to apply generally:
BMBH coalescence driven by massive perturbers (MPs) in the postmerger
galaxy \citep{zha+02,per+07}. MPs accelerate the relaxation of stars,
scatter them into the BMBH orbit, and extract orbital energy from
it. Efficient relaxation by MPs was first suggested by \citet{spi+51,spi+53}
to explain stellar velocities in the galactic disk. MPs remain an
important component in modern models of galactic disk heating (see
e.g. \citealt{vil83,vil85,lac84,jen+90,han+02} and references therein).
A similar mechanism was proposed to explain the spatial diffusion
of stars in the inner Galactic bulge \citep{kim+01}. In addition
to dynamical heating, efficient relaxation by MPs was suggested as
a mechanism for loss cone replenishment and relaxation, both in the
context of scattering of Oort cloud comets to the Sun \citep{hil81,bai83}
and the scattering of stars to a MBH in a galactic nucleus \citep{zha+02}.
\citet{zha+02} proposed MPs as a mechanism for establishing the $M_{\bullet}/\sigma$
relation \citep{fer+00,geb+00} by fast accretion of stars and dark
matter. They also noted the possibility of increased tidal disruption
flares and accelerated MBH binary coalescence due to MPs. Recently,
\citet*{per+07} (hereafter PHA07) studied in detail MPdriven interactions
of single and binary stars with a single MBH.
In this study we apply the methods developed in PHA07 to investigate
MPdriven interactions of stars with a BMBH, and the consequences
for BMBH coalescence. We explore different MP populations and merger
scenarios based on the available observations and simulations, and
estimate the BMBH coalescence rate for these scenarios.
This paper is organized as follows. The main concepts and procedures
of our losscone calculations, which are presented in detail in PHA07,
are summarized briefly in \S \ref{s:MPlosscone}. The observations
and theoretical predictions of MPs in the inner hundreds pc of galactic
nuclei are reviewed in \S \ref{s:MP_GC}. In \S \ref{s:merger_dyn}
we briefly review the dynamics of BMBH mergers; a detailed technical
discussion is presented in appendices \ref{a:stall} and \ref{a:energy}.
We present our procedure for modeling BMBH coalescence under various
assumptions in \S \ref{s:models} and analyze the results of our
calculations in \S \ref{s:Results}. We explore their implications
in \S \ref{s:Implications} and discuss and summarize our results
in \S \ref{s:summary}.
\section{Losscone refilling by massive perturbers}
\label{s:MPlosscone}
PHA07 analyzed in detail MPinduced deflections of stars to nearly
radial ({}``losscone'') orbits that bring them within some threshold
distance $q$ from the central mass. Here we present a brief qualitative
summary of the results.
MPs of mass $M_{p}$ and space density $n_{p}$ dominate dynamical
relaxation over scattering by stars of mass $M_{\star}$ and space
density $n_{\star}$, when the ratio of the 2nd moments of the mass
distributions satisfies $\mu_{2}\!\equiv\!\left.n_{p}M_{p}^{2}\right/n_{\star}M_{\star}^{2}\!\gg\!1$.
This can be shown by considering first close encounters at the {}``capture
radius'' $r_{c}\sim GM_{p}/v^{2}$, where $v$ is the typical relative
velocity. The {}``$nv\sigma$'' rate of such encounters with a test
star is then $t_{r}^{1}\!\sim\! nvr_{c}^{2}\!\propto\! n_{p}M_{p}^{2}/v^{3}$,
where $t_{r}$ is the relaxation time. Integration over all encounter
distances further decreases the relaxation time by a Coulomb logarithm
factor that depends on the size of the system and possibly also on
$R_{p}$, the size of the MP, if $R_{P}\!>\! r_{c}$. The impact of
fast, MPinduced relaxation on the rate at which stars enter the losscone
can be incorporated into standard losscone theory (e.g. \citealt{lig+77})
by replacing the relaxation time due to scattering by stars with that
due to MPs, with the only modification being the separate treatment
of rare scattering events.
The effects of MPs are most significant in situations where perturbations
by stars alone are not efficient enough to refill the losscone (i.e.
isotropize the orbits) on the orbital timescale. Since the larger
$q$, the larger the losscone and the larger the perturbations needed
to fill it, MPs are most effective for large$q$ processes, such
as close interactions between binaries and a single MBH (where $q$
is the tidal disruption radius of the binary), or interactions of
single stars with a BMBH (where $q$ is the semimajor axis, $a$,
of the BMBH).
\section{Massive perturbers in galactic nuclei}
\label{s:MP_GC}
The space density of MPs is much smaller than that of stars, so to
dominate relaxation ($\mu_{2}\!\gg\!1$) they must be significantly
more massive. Here we consider only MPs with masses $M_{p}\!\ge\!10^{4}M_{\odot}$,
such as stellar clusters and giant molecular clouds or clumps (GMCs).
Intermediate mass black holes (IMBHs) could be very effective MPs,
but these are not considered here since it is still unclear whether
they actually exist. A summary of the observed properties of MPs and
those derived from simulations is presented in tables \ref{t:MPs_prop}
and \ref{t:MPs_abun}.
\subsection{Massive perturbers in spiral galaxies}
\label{ss:spiral_MPs}
Observations of the Galactic center (GC) reveal the existence of $\sim\!100$
GMCs, and $\sim\!10$ clusters in the central $\sim\!100$ pc \citep{fig+99,oka+01,bor+05}.
The mass fraction of the GMCs is $\mathrm{few}\times0.01$ of the
total dynamical mass on the $\mathrm{few}\times100$ pc scale and
a $\mathrm{few}\times0.1$ in the central $\sim\!100$ pc (see PHA07
for an extended discussion of the properties of MPs in our GC). In
contrast, the central \textasciitilde{}2 pc of the GC contain negligible
amounts of gas.
The lifespan of the MPs is limited by dynamical friction, which makes
them sink to the center, where they are disrupted by the central galactic
tidal field. In the GC, the time to sink to the center from radius
$r$ is $t_{\mathrm{df}}\!\sim\!0.1r^{2}v_{c}/GM_{p}$ \citep{ale05},
where $v_{c}\!\sim\!150\,\mathrm{km\, s^{1}}$ is the circular velocity
at $r\!\sim\!100\,\mathrm{pc}$ \citep{ken92} (for example, $t_{\mathrm{df}}\sim3\!\times\!10^{9}\,\mathrm{yr}$
for a $10^{4}\, M_{\odot}$ MP originating from $r\!=\!100\,\mathrm{pc}$).
The total mass supply rate required to maintain the GMC population
within $r$ of the MBH in steady state is then approximately $\mathrm{d}M/\mathrm{d}t\!=\!\int(\mathrm{d}^{2}M/\mathrm{d}M_{p}\mathrm{d}t)\mathrm{d}M_{p}\!\sim\!\int(M_{p}/t_{\mathrm{df}})(\mathrm{d}N_{p}/\mathrm{d}M_{p})\mathrm{d}M_{p}$.
For the mass functions of the two extreme cases of heavy and light
GMCs, which were considered by PHA07, $\mathrm{d}M/\mathrm{dt}\!\sim\!0.05$
and $3\,\Mo\,\mathrm{yr^{1}}$, respectively. GMCs and young stellar
clusters are stages in the path of star formation, and so the star
formation rate can be used to estimate the MP mass supply rate. \citet{fig+04}
show that the formation history of stars in the central projected
30 pc of the GC is well described by continuous star formation over
10 Gyr at a rate of $0.02\,\Mo\,\mathrm{yr^{1}}$. Extrapolated out
to 100 pc in the $n_{\star}\!\sim\! r^{2}$ stellar distribution
of the inner bulge, this corresponds to $\mathrm{d}\Ms/\mathrm{d}t\!\sim\!0.05\,\Mo\,\mathrm{yr^{1}}$.
Since the mean star formation efficiency (fraction of mass turned
into stars) is on average very low, $f_{\star}\!\sim\!\mathrm{few}\times0.01$
\citep{mye+86}, the star formation rate is broadly consistent with
the required mass supply rate for MPs, $\mathrm{d}M/\mathrm{d}t\!\sim\!\mathrm{(d}\Ms/\mathrm{d}t)/f_{\star}\!\sim\!{\cal O}(1\,\,\Mo\,\mathrm{yr^{1}})$,
even for the highest estimates for the masses of GMCs in the GC.
The MP contents of the GC appear to be quite typical of spiral galaxies.
Single molecular clumps cannot be resolved in the nuclei of other
spiral galaxies, but the total fraction of gas and its distribution
are usually quite similar to those observed in the GC (e.g. \citealt{sak+99,saw+04};
see review by \citealt{hen+91}). Likewise, CO observations show that
the gas contains very dense large clumps that account for up to $\lesssim\!50\%$
of the total gas contents in these regions \citep{dow+93,dow+98}.
Simulations of such regions show a quasisteady state behavior, where
dense massive gaseous structures on the $\mathrm{few\times\!100}$
pc scale are constantly formed and destroyed \citep{wad01,wad+01}.
It is reasonable to assume, based on both the observational evidence
and the theoretical results, that the properties of MPs in galactic
nuclei of spiral galaxies: their mass function, spatial distribution
and mass fraction, resemble those observed in the GC (see PHA07 for
details).
In addition to GMCs, many globular clusters \citep{fri95,ash+98}
and open clusters may inspiral into, or form in the galactic nucleus
in the course of their evolution (e.g. \citealt{gne+99}). For example,
the Galaxy contains hundreds of $\sim\!10^{3}\, M_{\odot}$ open clusters
and $\mathrm{few}\times10^{5}M_{\odot}$ globular clusters \citep{mey+91,fri95}.
Many more are observed in other galaxies \citep{ash+98}. If some
of these clusters contain IMBHs, they will contribute to the MPs population
even after the disruption of the host cluster is disrupted \citep{ebi+01,mill+02},
and will sink all the way to the center. However, the existence of
IMBHs is still a matter of speculation.
\subsection{Massive perturbers in elliptical galaxies}
\label{ss:ellip_MPs}
The gas fraction in elliptical galaxies is typically $10100$ times
smaller than in spiral galaxies \citep{rup+97,kna+99}. However, in
some elliptical galaxies it is comparable or even larger than that
in spirals. Such gasrich ellipticals are thought to have been formed
recently in a merger of two late type galaxies (e.g. \citealt{wik+97}).
In particular, ultraluminous infrared galaxies (ULIRGs, see review
by \citealt{san+96}) have extreme amounts of gas, $10100$ times
more than in the Galaxy, and can have as much or more mass in gas
compared to the mass in stars. Elliptical galaxies may well be evolved
merger products, where most of the dense gas in the core formed stars
(e.g. \citealt{ben+99}). In that case, it is plausible that the main
type of MPs would be the stellar clusters that were born of the GMCs,
rather than the GMCs themselves. Observations of stellar rings and
disks in the cores of elliptical galaxies indeed suggest that presentday
stellar structures reflect earlier gaseous structures \citep{dow+98}.
This is also consistent with the fact that elliptical have larger
numbers of globular clusters than spirals, and that mergers are associated
with the formation of massive clusters \citep{ash+98,zha+99,kra+05,lar06}.
\subsection{Formation of massive perturbers in galactic mergers }
\label{ss:merger_MPs}
Simulations of mergers of gas rich spirals indicate that $\gtrsim\!50\%$
of the total gas mass in both galaxies is driven into the central
$\mathrm{few}\times100$ pc of the newly formed galaxy \citep{bar+91,bar+96},
where it probably forms massive clumps. In mergers of two gaspoor
ellipticals, stellar clusters may play a similar role. Many of the
newly formed stellar clusters will probably survive in the merged
nucleus \citep{por+02a}. In addition, many old globular clusters
will fall directly into the nucleus in the course of the merger \citep{gne+06},
or sink in by dynamical friction \citep{cap93}. While most will probably
be disrupted (O. Gnedin, priv. comm.), a significant fraction could
survive \citep[e.g. simulations by ][]{mio+06}. This central accumulation
of young and old stellar cluster could significantly shorten the relaxation
time. Further simulations are needed to address these issues quantitatively.
%
\begin{table*}
\caption{\label{t:MPs_prop}Observed and simulated properties of massive perturbers}
\begin{tabular}{lcccc>{\raggedright}p{1.5in}}
\hline
{\footnotesize MP type}&
{\footnotesize $M_{p}$ ($M_{\odot}$) }&
{\footnotesize Mass Profile}&
{\footnotesize $\left\langle M_{p}^{2}\right\rangle ^{1/2}\,(M_{\odot})$ }&
{\footnotesize $R_{p}$ (pc)}&
{\footnotesize References}\tabularnewline
\hline
{\footnotesize GMCs in the GC}&
{\footnotesize $10^{4}10^{8}$}&
{\footnotesize Power law ($\beta=1.2$)}&
{\footnotesize $4\!\times\!10^{5}$}&
{\footnotesize 5}&
{\footnotesize \citet{oka+01,gus+04,per+07}}\tabularnewline
{\footnotesize Young clusters in the GC}&
{\footnotesize $10^{3}10^{5}$}&
{\footnotesize Power law ($\beta=1.2$)}&
{\footnotesize $3\!\times\!10^{4}$}&
{\footnotesize 1}&
{\footnotesize \citet{fig+99,fig+02,mai+04,bor+05,per+07}}\tabularnewline
{\footnotesize Globular clusters in the Galaxy}&
{\footnotesize $10^{2.5}10^{6.5}$}&
{\footnotesize Log normal }&
{\footnotesize $1.9\times10^{5}$}&
{\footnotesize 5}&
{\footnotesize \citet{man+91}}\tabularnewline
{\footnotesize Young clusters in galaxies}&
{\footnotesize $10^{4.5}10^{6.5}$}&
{\footnotesize Power law ($\beta=2$)}&
{\footnotesize $4.3\times10^{5}$}&
{\footnotesize 3}&
{\footnotesize \citet{zha+99,kra+05,lar06}}\tabularnewline
\hline
\end{tabular}
\end{table*}
\begin{center}%
\begin{table}
\caption{\label{t:MPs_abun}Mass fraction of observed and predicted\protect \\
massive perturbers in galactic nuclei }
\begin{tabular}{>{\centering}p{1in}cl>{\raggedright}p{1.4in}}
\hline
{\footnotesize Galaxy}&
{\footnotesize MP type}&
{\footnotesize $M_{p}^{tot}/M_{\mathrm{dyn}}$ }&
{\footnotesize References}\tabularnewline
\hline
{\footnotesize Milky Way}&
{\footnotesize GMCs}&
{\footnotesize $0.2$}&
{\footnotesize \citet{oka+01,gus+04}}\tabularnewline
\multicolumn{1}{c}{}&
{\footnotesize Clusters}&
{\footnotesize $10^{4}$}&
{\footnotesize \citet{fig+99,fig+02,mai+04,bor+05}}\tabularnewline
{\footnotesize Spirals}&
{\footnotesize GMCs}&
{\footnotesize $0.1$}$0.3$&
{\footnotesize \citet{kod+05,gus+04,sak+99,you+91}}\tabularnewline
{\footnotesize Ellipticals}&
{\footnotesize GMCs}&
{\footnotesize $10^{3}$}$10^{2}$&
{\footnotesize \citet{rup+97,kna+99}}\tabularnewline
{\footnotesize ULIRGs }&
{\footnotesize GMCs}&
{\footnotesize $0.3$}$0.6$&
{\footnotesize \citet{san+96}}\tabularnewline
{\footnotesize Merger (Obs.)}&
{\footnotesize GMCs}&
{\footnotesize $0.3$}$0.6$&
\citet{cul+07}
{\footnotesize \citet{eva+02,sak+06}}\tabularnewline
{\footnotesize Merger (Sim.)}&
\textcolor{red}{\footnotesize }\textcolor{black}{\footnotesize GMCs}&
{\footnotesize $0.3$}$0.6$&
{\footnotesize \citet{bar+92}}\tabularnewline
\hline
\end{tabular}
\end{table}
\par\end{center}
\section{BMBH merger dynamics}
\label{s:merger_dyn}
\textcolor{black}{A BMBH merger (with $Q\!\ll\!1$) progresses through
three stages \citep[See][]{mer06}. (1) Gradual decay by dynamical
friction to the point where the separation between the two MBHs is
$r_{12}\!\sim\! r_{h}(M_{1}$). (2) Formation of a bound Keplerian
pair, when $r_{12}\!<\! r_{h}(M_{1})$, through rapid decay, initially
by dynamical friction on $M_{2}$ and later by the slingshot effect.
This is followed by a slowdown of the decay when $a\!\sim\! a_{h}$
and stalling, unless the the losscone is replenished by a process
more efficient than diffusion due to 2body relaxation. (3) Ultimately,
the BMBH orbital decay rate is dominated by GW emission, leading to
final coalescence. The operational definition of the stalling separation
$a_{s}$ at time $t_{s}$ is the point where the decay rate sharply
decreases. Typically $a_{s}\!\sim\!{\cal O}(a_{h})$ (see appendix}
\ref{a:stall}\textcolor{black}{). }
The slingshot effect occurs when $q$, the periapse distance of the
star from the BMBH center of mass, is of the order of the BMBH semimajor
axis $a$. Such stars are ejected and lost from the system, either
directly or after several repeated interactions with the BMBH, and
on average extract energy $\Delta E(q)$ and angular momentum $\Delta J(q)$
from the BMBH. The evolution of the BMBH energy, or equivalently,
the decrease in $a$, is given by
\begin{equation}
\!\frac{\mathrm{d}}{\mathrm{d}t}\left(\frac{GM_{1}M_{2}}{2a}\right)\!=\!\int_{0}^{\infty}\!\frac{\mathrm{d}\Gamma}{\mathrm{d}q}\Delta E(q)\mathrm{d}q\equiv\Gamma(a)\left\langle \Delta E\right\rangle \!(a)\,,\label{e:adot_dyn}\end{equation}
where $\Gamma(a)$ is the supply rate of stars that approach the BMBH
on orbits with $q\!<\! a$, and $\left\langle \Delta E\right\rangle \!\propto\! a^{1}$
is the appropriately weighted mean extracted energy (\citealt{mil+03,mer+05b};
see detailed discussion in appendix \ref{aa:CH}). The supply rates
for stars deflected from a typical radius $r$ in the empty loss cone
regime (inefficient supply) and the full loss cone regime (efficient
supply) are respectively (e.g. PHA07) \begin{equation}
\frac{\mathrm{d}\Gamma_{e}}{\mathrm{d}\log r}\!\sim\!\frac{N_{\star}(<\! r)}{\log(r/a)t_{r}}\,,\qquad\frac{\mathrm{d}\Gamma_{f}}{\mathrm{d}\log r}\!\sim\!\frac{a}{r}\frac{N_{\star}(<\! r)}{P(r)}\,,\label{e:Gammaef}\end{equation}
where $N_{\star}(<\! r)$ is the number of stars enclosed within $r$,
and $P$ is the orbital period (dynamical time). It then follows that
the dynamical decay rate in the two regimes scales as $\dot{a}_{\mathrm{dyn}}\!\propto\!a/\log(r/a)$
or $\propto\!a^{2}$, respectively, so that in both cases the dynamical
hardening rate decreases as $a$ decreases. Note that in the hard
BMBH limit ($a\!\rightarrow\!0$), when the losscone is small and
therefore full, $\mathrm{d}(1/a)/\mathrm{d}t\!\sim\!\mathrm{const.}$
\citep[hereafter Q96]{qui96}.
When the BMBH separation becomes small enough, the orbital decay rate
due to GW emission, $\dot{a}_{\mathrm{GW}}$, becomes higher than
the dynamical decay rate. The decay rate on a circular orbit due to
the emission of GW is \citep{pet64} \begin{equation}
\dot{a}_{\mathrm{GW}}=\frac{64}{5}\frac{G^{3}\mu M_{12}^{2}}{c^{5}a^{3}}\,,\label{e:adot_gw}\end{equation}
which increases as $a$ decreases. The time to decay to $a\!=\!0$
from an initial semimajor axis $a$ is \begin{equation}
t_{\mathrm{GW}}=\frac{5}{256}\frac{c^{5}}{G^{3}}\frac{a^{4}}{\mu M_{12}^{2}}\,,\label{e:t_gw}\end{equation}
Since $\dot{a}_{\mathrm{dyn}}$ decreases with $a$, while $\dot{a}_{\mathrm{GW}}$
increases, there exists a transition BMBH separation, $\aGW$, such
that $\dot{a}_{\mathrm{dyn}}(\aGW)=\dot{a}_{\mathrm{GW}}(\aGW)$.
Once the BMBH shrinks to $a_{\mathrm{GW}}$, the coalescence is inevitable
as long as $t_{\mathrm{GW}}(a_{\mathrm{GW}})\!<\! t_{H}$ and as long
as the BMBH remains unperturbed. The total time from the hardening
semimajor axis $a_{h}$ to the coalescence is then
\begin{equation}
t_{c}=t_{\mathrm{dyn}}(a_{h}\rightarrow a_{\mathrm{GW}})+t_{\mathrm{GW}}(a_{\mathrm{GW}}\rightarrow0)\,.\label{e:equal_times}\end{equation}
The dynamical decay timescale $t_{\mathrm{dyn}}(a_{h}\rightarrow a_{\mathrm{GW}})$
is of the order of the time it takes the BMBH to intercept and interact
with stars whose total mass equals its own , $t_{\mathrm{dyn}}\sim M_{12}/[\Ms\Gamma(a_{\mathrm{GW}})]$,
where $\Gamma$ is evaluated $a_{\mathrm{GW}}$, where the rate is
slowest%
\footnote{\label{ft:dEdt}Every star that passes near the binary MBH extracts
from it binding energy of order $\Ms\varepsilon_{12}$, where $\varepsilon_{12}\!=\! G\mu/2a$
is the specific energy of the BMBH, so that $\mathrm{d}E\!=\!GM_{1}M_{2}/2a^{2}\mathrm{d}a\!=\!(\Ms G\mu/2a)\Gamma(a)\mathrm{d}t$.
Integrating between $a_{h}\!\gg\! a_{\mathrm{GW}}$ with $\Gamma(a)\sim\left(a/r)N_{\star}(<\, r)/P(r)\right_{r_{\mathrm{MP}}}$
(when the losscone is filled by MPs that orbit as close as $r_{\mathrm{MP}}$
from the MBH) yields $t_{\mathrm{dyn}}\!\simeq\! M_{12}/\Ms\Gamma(a_{\mathrm{GW}})$.%
}. This estimate is conservative, since it neglects the possibility
that a fraction of the stars are not ejected from the losscone, but
return to interact again with the BMBH. This can further accelerate
the decay, but is not enough in itself to prevent stalling \citep{mil+03}.
\section{Modeling massive perturberdriven BMBH coalescence}
\label{s:models}
Based on the observations and simulations described above, we formulate
three representative merger scenarios that include MPs, and compare
them to a merger scenario where only stellar 2body relaxation plays
a role. The model parameters are listed in table \ref{t:models}.
The major merger scenario consists of a $Q\!=\!1$ merger of two gasrich
galaxies. It is assumed that the merger a major gas inflow to the
center, increasing the amount of gas there to $\sim\!1/2$ of the
total dynamical mass ($\sim\!5$ times more than presently in the
center of the Milky Way; the mass of the cold gas in postmerger galaxies
can be even higher, but we take into account only the densest regions
that correspond to the more massive MPs). It is further assumed that
the MPs are similar to massive GMCs in our GC, that they have a powerlaw
mass function, $\mathrm{d}N_{p}/\mathrm{d}M_{p}\propto M_{p}^{\beta}$
with $\beta=1.2$ (see MP model GMC1 in PHA07 for details), and that
their spatial distribution is isotropic%
\footnote{While the geometry of central molecular zone of the Galaxy is flattened,
its height of $\mathrm{few}\times10$ pc implies that it is nearly
isotropic of the scale of interest of $\sim100$ pc.%
}.
The minor merger scenario consists of a $Q\!=\!0.05$ merger between
a large, massive gasrich galaxy and a much smaller galaxy, which
only slightly perturbs the large galaxy and triggers only a moderate
gas inflow to the center. It is assumed that the nuclear gas mass
is $\sim\!1/3$ of the total dynamical mass ($\sim\!1.5$ times more
than presently in the center of the Milky Way). The MP properties
are the same as in the major merger scenario.
In the elliptical merger scenario we attempt to model a $Q\!=\!1$
merger of two equal mass gaspoor elliptical galaxies. We assume that
the MPs are mostly stellar systems such as clusters or spiral structures.
Lacking secure observations, we model the MPs after results from simulations
\citep{li+04,pri+06}. These simulations show that both the total
cluster birthrate and the massive cluster birthrate peak at the
center of the galaxy \citep{li+04}. We assume that the MP mass fraction
is $0.2$ of the total dynamical mass and that the cluster mass function
is a powerlaw with $\beta=2$ for $10^{5}\,\Mo\le M_{p}\le10^{7}\,\Mo$,
following the results of \citet{pri+06}.
Finally, we consider, for comparison, a model that assumes that relaxation
in the postmerger galaxy is due to stellar 2body interactions only.
In our calculations we assume that the stellar distribution over the
entire relevant distance range can be approximated by a singular isothermal
stellar distribution \begin{equation}
\rho(r)=\frac{\sigma_{\infty}^{2}}{2\pi Gr^{2}}\,,\label{e:iso_dist}\end{equation}
where the velocity dispersion $\sigma_{\infty}$, and hence the normalization,
is determined by the empirical $\Mbh/\sigma$ relation \citep[e.g.][]{wan+04}.
The MP distribution is assumed to follows the stars, down to a minimal
radius $r_{\mathrm{MP}}$, where the MPs are destroyed either by the
central tidal field, the radiation of the accreting BMBH, or the outflows
associated with the accretion or star formation triggered by the merger.
The exact value of $r_{\mathrm{MP}}$ is uncertain, since the processes
involved in the destruction of the MPs are complex. Here it is assumed
that $r_{\mathrm{MP}}\!=\!2r_{h}$ %
\footnote{Note that the $\Mbh/\sigma$and $\Mbh/M_{b}$ relations ($M_{b}$
is the mass of the bulge, with typical length scale $r_{b}$) imply
then the assumption that $r_{\mathrm{MP}}\!\sim\! r_{b}$.%
}. This is probably a conservative estimate, since transient dense
clumps and dense cluster cores can survive even at smaller distances
(e.g. observations in our GC, \citealt{mai+04,chr+05}, and theoretical
predictions from simulations, \citealt{wad+01,por+03}).
%
\begin{table}
\caption{\label{t:models}Massive perturber models in major and minor mergers
and in major merger of late type galaxies}
\begin{centering}{\footnotesize }\begin{tabular}{lccccrcc}
\hline
\multicolumn{1}{l}{{\footnotesize Merger model}}&
{\footnotesize $Q$}&
{\footnotesize $r/r_{h}\,^{a}$ }&
{\footnotesize $M_{p}^{tot}/M_{\mathrm{dyn}}^{tot}$}&
{\footnotesize $M_{p}(M_{\odot})$}&
{\footnotesize $\beta\,^{b}$}&
{\footnotesize $R_{p}$ (pc)}&
{\footnotesize $\mu_{2}\,^{c}$}\tabularnewline
\hline
{\footnotesize Major}&
{\footnotesize 1}&
{\footnotesize $2$$30$ }&
{\footnotesize $1/2$}&
{\footnotesize $5\!\times\!10^{4}$$1\!\times\!10^{7}$}&
{\footnotesize $1.2$}&
{\footnotesize 5}&
{\footnotesize $3\!\times\!10^{7}$}\tabularnewline
{\footnotesize Minor}&
{\footnotesize 0.05}&
{\footnotesize $2$$30$ }&
{\footnotesize $1/3$}&
{\footnotesize $5\!\times\!10^{4}$$1\!\times\!10^{7}$}&
{\footnotesize $1.2$}&
{\footnotesize 5}&
{\footnotesize $5\!\times\!10^{6}$}\tabularnewline
{\footnotesize Elliptical }&
{\footnotesize 1}&
{\footnotesize $2$$30$ }&
{\footnotesize $1/5$}&
{\footnotesize $1\!\times\!10^{5}$$1\!\times\!10^{7}$}&
{\footnotesize $2$}&
{\footnotesize 3}&
{\footnotesize $5\!\times\!10^{5}$}\tabularnewline
{\footnotesize Stars}&
{\footnotesize }&
{\footnotesize $1$$30$ }&
{\footnotesize $1$}&
{\footnotesize $1$}&
{\footnotesize }&
{\footnotesize $0$}&
{\footnotesize 1}\tabularnewline
\hline
\multicolumn{8}{l}{\textcolor{black}{\footnotesize \rule{0em}{1.5em}$^{a}$} {\footnotesize Assuming
$N_{p}(r)\!\propto\! r^{2}$.}}\tabularnewline
\multicolumn{8}{l}{\textcolor{black}{\footnotesize \rule{0em}{1.5em}$^{b}$} Assuming
$\mathrm{d}N_{p}/\mathrm{d}M_{p}\!\propto\! M_{p}^{\beta}$}\tabularnewline
\multicolumn{8}{l}{\textcolor{black}{\footnotesize \rule{0em}{1.5em}$^{c}$} {\footnotesize $\mu_{2}\!\equiv\! N_{p}\left\langle M_{p}^{2}\right\rangle \left/N_{\star}\left\langle M_{\star}^{2}\right\rangle \right.$,
where} \textcolor{black}{\footnotesize $\left\langle M^{2}\right\rangle \!=\!\int M^{2}(\mathrm{d}N/\mathrm{d}M)\mathrm{d}M/N$. }}\tabularnewline
\hline
\end{tabular}\par\end{centering}
\end{table}
In order to calculate the MPinduced coalescence time, we compute
the rate in which stars are scattered by MPs into the BMBH. Based
on results from 3body scattering experiments \citep{hil83,qui96,ses+06a,ses+06b},
we assume that a star whose periapse distance $q$ from the BMBH's
center of mass is smaller than the BMBH separation $a$, interacts
strongly with the MBH and is then ejected out of the system. We omit
the possibility of reejection, and we neglect soft scattering events
($q\!>\! a$) since these are inefficient in extracting energy from
the BMBH (see \citealt{ses+06a} and appendix \ref{aa:CH}). Since
reejection and soft scattering increase the energy extraction rate,
we obtain a conservative upper limit on the coalescence time.
Beginning with a hard BMBH of separation $a(t\!=\!0)\!=\! a_{s}$
(appendix \ref{a:stall}), we define the timedependent losscone
periapse as $q\!=\! a(t)$ and calculate the loss cone rate $\Gamma(q)$,
using the methods described in PHA07. We follow the evolution of the
BMBH separation by numerically integrating the evolution equation
with small enough time steps such that $\mathrm{d}a\!\ll\negmedspace a$
(see \citealt{mil+03} and \citealt{ses+06a,ses+06b} for a similar
approach) until the orbital decay is dominated by GW emission (Eq.
\ref{e:equal_times}), which is effectively the coalescence time $t_{c}$.
In simplified notation, the evolution equation (Eq. \ref{e:dlogadt})
is
\begin{equation}
\frac{\mathrm{d}\log a}{\mathrm{d}t}=2\frac{M_{\star}}{M_{12}}\int\bar{C}(a,r)\frac{\mathrm{d}\Gamma(a)}{\mathrm{d}r}\mathrm{d}r\,,\label{e:s_evol}\end{equation}
where $d\Gamma/dr$ is the differential loss cone replenishment rate
and $\bar{C}$ is the mean value of the dimensionless energy, $C\!\equiv\!\left.M_{12}\Delta E\right/2M_{\star}E_{12}$,
exchanged between the scattered star and the BMBH (see detailed derivation
and numeric estimation in appendix \ref{aa:CH}; $C\!=\!1$ corresponds
to the case where the specific energy carried by the star equals twice
that of the BMBH). The quantity $\bar{C}(a,r)$ depends on the hardness
parameter of the encounter $\zeta\!\equiv\!\sigma(r)/V_{12}(a)$,
defined as the ratio between the typical initial velocity of the scattered
star far from the BMBH, $\sigma(r)$ and the orbital velocity of the
BMBH, $V_{12}\!=\!\sqrt{GM_{12}/a}$. An additional $r$dependence
is introduced by the acceleration of the star toward the BMBH by galactic
potential, which increases the relative velocity between the BMBH
and the star at the point of encounter over what it would have been
if the star fell toward an isolated BMBH (see appendix \ref{aa:galpot})
and decreases the efficiency of the slingshot effect (Figure \ref{f:Ceff}).
This nonnegligible effect, taken into account here, was neglected
in previous estimations of the BMBH coalescence times \citep{ses+06a,qui96}.
\section{Results }
\label{s:Results}
%
\begin{figure}
\begin{tabular}{c}
\includegraphics[clip,width=1\columnwidth,keepaspectratio]{f1}\tabularnewline
\end{tabular}
\caption{\label{f:Mt_decay} The dynamical decay times, $t_{\mathrm{dyn}}$
of BMBHs from $a_{h}$ to $\aGW$ as function of the BMBH mass (the
time to final GWinduced decay, $t_{\mathrm{GW}}$, from $a_{\mathrm{GW}}$
to $a\!=\!0$ is negligible compared to the initial dynamical decay
phase). Different merger scenarios are shown (table \ref{t:models}):
major mergers (solid line), minor mergers (dashed line) and elliptical
mergers (dasheddotted line). Without MPs the decay times are longer
than $t_{H}$ for all BMBH mass in this range. }
\end{figure}
Figure (\ref{f:Mt_decay}) shows the total decay time of BMBHs in
the mass range $M_{12}\!=\!10^{6}$$10^{9}$$M_{\odot}$ for different
merger scenarios. Stellar 2body relaxation cannot replenish the loss
cone fast enough. In the absence of MPs, the merger proceeds in the
empty losscone regime, where the timescale is set by the slow relaxation
time (Eq. \ref{e:Gammaef}), leading to merger times orders of magnitude
longer than $t_{H}$. In contrast, when the MP number density is high
enough, or the losscone is small enough (lower BMBH mass), the losscone
is full, and the merger time is determined by the size of the losscone
and by the dynamical time (Eq. \ref{e:Gammaef}). These conditions
hold for major mergers across almost the entire mass range, and are
also true for the lower mass BMBHs in minor mergers and the mergers
of elliptical galaxies. However, for higher BMBH masses in minor and
elliptical mergers, there are not enough MPs to refill the losscone.
Nevertheless, the merger evolves faster by a factor of $\mu_{2}$
than it would with stellar relaxation alone (table \ref{t:models}),
until the BMBH separation decreases, the losscone is filled, and
the scattering rate attains its maximal value. This fast MPdriven
evolution continues until the BMBH shrinks to the point where stellar
relaxation alone can fill the losscone. Since the BMBH spends most
of its time in those late stages, the overall decrease in the merger
time is $1\ll t_{\mathrm{dyn}}^{\mathrm{MP}}/t_{\mathrm{dyn}}^{\star}\ll\mu_{2}^{1}$.
The results indicate that MPs drive rapid coalescence of BMBHs in
less than $t_{H}$, in most minor and major mergers. Moreover, for
most BMBHs coalescence occurs in less than a Gyr, which is comparable
to the dynamical timescale of the galactic merger itself \citep{bar+92}.
Our results indicate that the coalescence of massive BMBHs ($M_{12}\!\gtrsim\!10^{9}\,\Mo$)
in gaspoor ellipticals may take $\gg t_{H}$ to coalesce. Although
we omit here processes that could shorten the coalescence time by
an additional factor of a few, such as reejection of loss cone stars
\citep{mil+03,ber+05}, it is possible that BMBH in massive elliptical
galaxies do stall. In that case, GW emission rate from such sources
will be suppressed in the absence of other efficient merger mechanisms.
%
\begin{figure}
\begin{tabular}{c}
\includegraphics[clip,width=1\columnwidth,keepaspectratio]{f2}\tabularnewline
\end{tabular}
\caption{\label{f:evolution} Evolution of the BMBH separation from $a_{s}$
to $a_{\mathrm{GW}}$ in a major merger due to 3body scatterings
of stars. The evolution in the major merger MP scenario (solid line)
is compared to that in the stellar relaxation scenario (dashed line)
for BMBH masses of $10^{6},\,10^{7},\,10^{8}$ and $10^{9}$ $\Mo$
(from bottom up). }
\end{figure}
%
\begin{figure}
\includegraphics[width=1\columnwidth]{f3}
\caption{\label{f:Ceff}The dependence of the mean dimensionless extracted
energy $\bar{C}_{\mathrm{eff}}\!=\!\bar{C}(Q,\zeta_{\mathrm{eff}})$
(Eq. \ref{e:eCmaxwell}) for $Q\!=\!1$, on the point of origin of
the deflected star, $r_{\star}/a_{s}$, for different stages of the
BMBH evolution $a/a_{s}$ (indicated by the numbers adjacent to the
lines), taking into account the acceleration in the Galactic potential.
The dashed line at the top is the asymptotic value of $\bar{C}_{\mathrm{eff}}$
in the hard limit. The dashdotted line is the value of $\bar{C}$
for the case $a/a_{s}\!=\!1$, when the Galactic potential is neglected
($\bar{C}$ approaches the hard limit when $a/a_{s}\rightarrow0$).
The vertical lines indicate the BMBH's radius of dynamical influence
$r_{h}$ and the inner cutoff of the MP distribution $r_{\mathrm{MP}}$.
Most of the stars are deflected toward the BMBH from $r_{\star}\gtrsim r_{\mathrm{MP}}$. }
\end{figure}
Figure (\ref{f:evolution}) shows the evolution of the BMBH separation
for $M_{12}\!=\!10^{6}$, $10^{7}$, $10^{8}$ and $10^{9}\, M_{\odot}$
in major mergers ($Q\!=\!1$), with and without MPs. The BMBH separation
is evolved up to the point where the decay is dominated by GW and
coalescence follows soon after (the transition criterion $\dot{a}_{dyn}\!=\!\dot{a}_{\mathrm{GW}}$
and Eq. \ref{e:adot_gw} imply that the evolution curves steepen sharply
beyond the transition point). The evolution of BMBHs with MP relaxation
exhibits a short initial stalled phase, where the initially large
losscone is empty even in the presence of MPs, followed by a phase
of rapid decay. It should be noted that the decay phase does not display
the $a\propto t^{1}$ evolution of a hard BMBH, expected when $\bar{C}\!\simeq\!\mathrm{const}$.
The acceleration of the infalling stars in the Galactic potential
softens the encounter with the BMBH and substantially reduces the
energy extraction efficiency. Figure (\ref{f:Ceff}) shows this efficiency
strongly depends on both the distance from which stars are deflected
to the BMBH and the BMBH separation. It should be emphasized that
acceleration by the galactic potential will substantially reduce the
efficiency of any BMBH slingshot mechanism, in particular those where
the potential gradient is steep (e.g. \citealp{zie06a,zie06b}) or
those where stars are deflected to the MBH from very large distances
(e.g. \citealp{ber+06}), and therefore should not be neglected.
\section{Implications of MPinduced BMBH coalescence}
\label{s:Implications}
\subsection{Observations of BMBHs}
\label{s:Observations}
BMBH mergers progress through three stages (\S \ref{s:merger_dyn}):
(1) decay to the center by dynamical friction, (2) formation of a
hard binary ($a_{h}\!\sim\!0.1$$10$ pc) and its subsequent decay
by the slingshot effect, if stars are supplied to the center, or else
stalling at $a_{s}\!\sim\! a_{h}$. \textcolor{black}{(3) ultimate
orbital decay by GW emission, leading to final coalescence. }
The first stage could appear as a galaxy with a resolved double nucleus
($a\!>\! a_{h}$). The second stage could still be resolved for the
largest separations (most massive BMBHs). The last stage could possibly
be observed indirectly through phenomena associated with the last
stages of the BMBH merger (see review by \citealt{kom06}), and directly
observed through its GW emission, with future GW detectors such as
the \emph{Laser Interferometer Space Antenna} (LISA). The prospects
of observing BMBHs with $a\!\sim\! a_{h}$ depends on whether relaxation
is driven stars or MPs (Fig. \ref{f:evolution}). With MPs, the BMBH
rapidly decays to $a_{\mathrm{GW}}\!<\! a\ll\! a_{h}$, making it
less likely to be observed at $a\!\lesssim\! a_{h}$ .
Various observed phenomena in galactic nuclei were suggested as evidence
for unresolved close BMBHs ($a\ll1$ pc). For example, Xray shaped
radio galaxies and so called doubledouble radio galaxies (sources
which exhibit pairs of symmetric doublelobed radiostructures, aligned
along the same axis) were interpreted as traces of a merged BMBH.
In addition, it was suggested that semiperiodic signals in light
curves or double peaked emission lines from AGN are due to BMBHs \citep{kom06}.
However, this interpretation is not unique. There are only two direct
observations of resolved double active MBHs in galactic nuclei. One
in NGC6240 \citep{kom+03} with $a\!=\!1.4$ kpc, and another in radio
galaxy 0402+379 with $a\!=\!7$ pc \citep{rod+06}. Interestingly,
the observed compact BMBH is just outside its hardening separation
($a_{h}\!\sim\!3.5$ pc $M_{12}\!\sim\!1.5\times10^{8}\, M_{\odot}$).
Thus these two systems are observed in the first stage of the BMBH
merger (dynamical friction).
The detection of the GW signal from coalescing MBHs would constitute
direct evidence of such events. Our calculations show that for most
galaxy mergers, the BMBH would coalesce within $t_{H}$, and so the
BMBH coalescence rate should follow the galaxy merger rate. In that
case the cosmic rate of these GW events could be as high as $10^{2}\,\mathrm{yr^{1}}$
\citep{hae94,ses+04,eno+04}.
\subsection{Triple MBHs and MBH ejection}
The galaxy merger rate in dense clusters may be high enough ($>10^{9}\,\mathrm{yr}^{1}$;
\citealt{mam06}) so that a second merger could occur before the first
BMBH coalesces. This would result in the formation of an unstable
triple MBH system, which will eject one of the MBHs at high velocity
\citep{sas+74}. This scenario was suggested as a possible solution
for the stalling problem, as the third component may drive the BMBH
to high eccentricities and to much more rapid coalescence \citep{bla+02,iwa+05,hof+06b}.
Because MPs accelerate most BMBH coalescence, it follows from our
results that triple MBH systems should be relatively rare wherever
MPdriven coalescence is efficient (Fig. \ref{f:Mt_decay}), with
the possible exception of highmass mergers, and in particular such
minor or dry elliptical mergers. Because of the rapid BMBH decay,
in those cases where a triple MBH is formed, it is expected that it
will be hierarchical. This would typically lead to fast coalescence
of the inner BMBH \citep{mak+94,iwa+05}, followed by the MPdriven
decay and coalescence of the newly formed central MBH with the outer
MBH. Thus, the MP scenario of BMBH coalescence implies that triple
MBH systems and highvelocity MBHs ejected by the slingshot mechanism
should be rare%
\footnote{Recent observations of an apparently hostless quasar \citep{mag+05}
were interpreted as an ejected MBH \citep{hae+06,hof+06a,hof+06b},
but see \citet{mer+06b} for an opposing view.%
} .
\subsection{Hypervelocity stars}
\label{ss:HVS}
Several Btype hypervelocity stars (HVSs) have been observed in our
Galaxy \citep{bro+05,hir+05,bro+06a,bro+06b,ede+06,bro+07}, implying
a Galactic population of $43\pm31$ such unbound HVSs \citep{bro+06}.
These stars have probably been ejected from the GC, either following
a disruption of a binary by the MBH \citep{hil88}, an interaction
of a single star with a coalescing BMBH \citep{yuq+03}, or with another
stellar object in the GC \citep{yuq+03}. The velocities and space
distributions of observed HVSs could, in principle, discriminate between
these scenarios and place constraints on their physical parameters
(\citealt{hol+05,lev05,ses+06,bau+06,bro+06b,ses+07,ole+06}). \texttt{\textbf{\textcolor{red}{}}}However,
more data is still needed for a decisive conclusion. The scenario
of HVS ejection by encounters with stellar black holes could, under
realistic assumptions, explain only a fraction of the HVSs \citet{ole+06}.
For the binary / MBH exchange scenario, recent calculations show that
both the number of observed HVSs and their velocities are consistent
with theoretical predictions \citep{per+07,bro+06b,kol+07}. Here
we explore an additional route, the MPdriven BMBH merger scenario
for HVS ejection.
The observed velocity of an HVS depends on its ejection velocity,
$v_{ej}$, and its position in the Galactic potential. Following \citet{bro+07},
we consider stars as unbound HVSs if their observed radial velocity
at a distance of tens of kpc from the GC is $v>450\,\mathrm{km\, s^{1}}$.
This corresponds to $v_{ej}\!=\!920\,\mathrm{km\, s^{1}}$(for an
estimated Galactic potential difference between the center and 55
kpc of $v_{55}\!\sim\!800\,\mathrm{km}\,\mathrm{s}^{1}$, \citealt{Car+87}).
HVSs with smaller velocities $275\!<\! v\!<\!450\,\mathrm{km\, s^{1}}$
far away from the GC (bound HVSs; corresponding to ejection velocities
of $840\le v_{ej}\!\le\!920\,\mathrm{km\, s^{1}}$) remain in the
Galaxy. Using these criteria, we estimate the integrated number of
HVSs in each of these classes that were ejected in the course of a
merger in the GC, taking $M_{12}=3.6\times10^{6}\, M_{\odot}$ \citep{pau+06},
and $5\times10^{4}\le Q\le1.3\times10^{2}$.
We find that the inferred Galactic population of young massive ($35\, M_{\odot}$)
unbound HVSs%
\footnote{\label{ft:PMF}The number fraction of young $35\, M_{\odot}$ stars,
$\sim\!10^{3}$ is calculated assuming continuous star formation
over 10 Gyr with a MillerScalo IMF \citep{mil+79} and using a stellar
population synthesis code \citep{ste+03} with the Geneva stellar
evolution tracks \citep{sch+92a}. See details in \citet{ale05}.%
} could be explained by a recent lowmass ratio IMBHMBH coalescence
with $Q\!\sim\!0.007$$0.013$ (see Fig. \ref{f:HVSs}), that occurred
$5\times10^{7}\,\mathrm{yr}\lesssim t\lesssim\!2\times10^{8}\,\mathrm{yr}$
ago. This time of flight roughly corresponds to a distance of $D\sim\sqrt{v_{\mathrm{ej}}^{2}v_{55}^{2}}t\!\ge\!450\,\mathrm{km\, s^{1}}t\!\sim\!20$$120$
kpc from the GC, as is observed.
We note that the constraints on the time when the hypothetical BMBH
merger could have occurred are satisfied by our model for the relevant
values of $a$ and $Q$. We find that most of the HVSs are ejected
toward the end of the dynamically driven BMBH decay phase, since typically
high ejection velocities, $\left\langle v_{\mathrm{ej}}\right\rangle \!\sim\!\sqrt{3.2G\mu/a}$
\citep{yuq+03}, are attained only when $a$ is smaller than a threshold
separation $a_{\mathrm{HVS}}$ (note that the galactic potential does
not play a role since $v_{\mathrm{ej}}^{2}\propto CV_{12}^{2}$, and
$C$ already reaches its maximal value for $a_{\mathrm{HVS}}\ll a_{s}$,
Fig. \ref{f:Ceff}). An upper limit on the time is set by the lifespan
of the young massive HVSs, $\sim\!2\times10^{8}\,\mathrm{yr}$, which
is known to agree with the observed range of their velocities ($\sim\!450$$800\,\mathrm{km\, s^{1}}$)
and distances ($D\sim\!20$$120$ kpc), corresponding to a time
of flight from the GC of $2\!\times\!10^{7}$$2\!\times\!10^{8}$
yr. A lower limit on the time can be deduced from the apparent absence
of a tight BMBH in the GC today. Therefore, the remaining lifetime
of the BMBH after the burst of ejected HVSs (due to GW decay, Eq.
\ref{e:t_gw}), must have been shorter than the time of flight of
the HVSs to their observed positions%
\footnote{Plus the travel time of the HVS light to earth, $\lesssim\mathrm{few\times10^{5}}$
yr.%
}, $\gtrsim10^{7}$ yr.
Previous predictions of the number of young massive HVSs used analytic
calculations based on 3body scattering experiments \citep{ses+06},
or on $N$body simulations \citep{bau+06}, and modeled low \citep{bau+06}
and high mass ratio (\citealt{ses+06}) BMBH mergers. Their estimates
for the \emph{total} number of ejected HVSs (all stellar types, bound
and unbound, with $v_{ej}\ge840\,\mathrm{km\, s^{1}}$) are consistent
with our results where the $Q$values overlap (Fig. \ref{f:HVSs}).
This lends confidence to the robustness of the calculations%
\footnote{The comparison is possible because the total \emph{}mass of ejected
stars depends only on the initial and final BMBH separation, and not
on the decay rate (Eq. \ref{eq:m_lost}), which is much faster for
MPdriven merger. %
}. Taking into account the GC presentday mass function (footnote \ref{ft:PMF}),
their results either under or overestimated the inferred HVS number
in the Galaxy, because the Q values they studied were too low or too
high. It should be emphasized that while we show that the number of
observed HVSs could be explained by a recent MBHIMBH merger, it
is still unknown whether there are enough such IMBHs in the GC (if
any) for this scenario to be realistic. Moreover, it is still not
clear whether the velocity and spatial distribution of the HVSs in
this scenario are consistent with those of the observed ones \citep{ses+07}.
%
\begin{figure}
\begin{tabular}{c}
\includegraphics[clip,width=1\columnwidth,keepaspectratio]{f4}\tabularnewline
\end{tabular}
\caption{\label{f:HVSs} The number of Galactic HVSs ejected by a hypothetical
recent MBHIMBH coalescence, as function of $Q$. Solid line: the
number of unbound ($v\!>\!450\,\mathrm{km\, s^{1}}$, $v_{ej}\!>\!920\,\mathrm{km\, s^{1}}$)
young massive ($3$$5\,\Mo$) HVSs (taking into account the present
day stellar mass function in the GC, their finite stellar lifespan,
and hence their maximal distance from the GC). Dashdotted line: the
same, for the total number of young HVSs (both bound and unbound;
$v\!>\!275\,\mathrm{km\, s^{1}}$, $v_{ej}\!>\!840\!\mathrm{km\, s^{1}}$).
Dashed line: the integrated number of HVSs ejected from the nucleus
(all stellar types, at all distances, disregarding stellar evolution).
Rectangles: total number of HVSs calculated by \citet{bau+06} (dashed),
and the corresponding number of young massive HVSs derived here from
their results (dashdotted). Circles: the same, for the HVS calculations
of \citet{ses+06a}. Shaded region shows the best estimate for the
number of unbound young massive HVSs in the Galaxy, based on current
observations \citep{bro+06b}. This number is consistent with a MBHIMBH
coalescence with $0.007\!\lesssim\! Q\!\lesssim\!0.013$.}
\end{figure}
\subsection{Mass deficits}
The large number of stars ejected from the system during the BMBH
coalescence could change the stellar distribution of the BMBH environment.
It has been suggested that the mass deficit observed in some bright
elliptical galaxies is the result of such events \citep{mil+02,rav+02,gra04,fer+06}.
The total \emph{}mass of ejected stars in the dynamical decay phase
depends only on the initial and final BMBH separations,
\begin{equation}
M_{\mathrm{ej}}(t)\equiv\Ms\int_{0}^{t}\mathrm{d}t^{\prime}\int\mathrm{d}E{\cal F}(E,t^{\prime})\sim{\cal \mathcal{J}}M_{12}\ln\frac{a(0)}{a(t)}\,,\label{eq:m_lost}\end{equation}
where ${\cal F}$ is flux of stars supplied to the loss cone, and
$\mathcal{J}$ is a numerical factor approximately equal to $1/2\bar{C}$
\texttt{\textbf{\textcolor{red}{}}}\citep{qui96,mil+03,ses+06b}.
Previous studies of the mass deficit \citep{mil+02a,mer06} took into
account only the stars evacuated from the core before the BMBH stalled
at $a\!\sim\! a_{h}$ because of inefficient stellar relaxation. We
note that that there are between 27 further $e$foldings between
$a_{h}$ and $a_{\mathrm{GW}}$ (Fig. \ref{f:evolution}). As a result,
when the BMBH merger is driven all the way to $a_{\mathrm{GW}}$ by
MPs, the mass deficit will grow substantially on the $\sim12r_{\mathrm{MP}}$
scale, where most of the stars are scattered from. We calculated the
total mass of stars that ejected from the core during coalescence,
which originated at such distances. We found that these constitute
approximately 3040\% of the total stellar mass in these regions.
Note that the magnitude and spatial scale of the mass deficit could,
in principle, discriminate between different proposed solutions for
the stalling problem. In mergers driven by nonaxisymmetric potentials,
most stars originate from large radii \citet{ber+06} where the enclosed
number of stars is very large. The fractional spatially averaged mass
deficit will therefore be very small, and harder to detect. In contrast,
scenarios that assume very steep cusps \citep{zie06a,zie06b} lead
to substantial central mass depletion. Even in gas induced mergers
\citep{esc+04,esc+05}, where stars play a minor role, there may be
an indirect mass deficit effect caused by the inhibition of star formation
due to heating of the gas by the inspiraling BMBH.
%
\begin{figure}
\noindent \begin{centering}\includegraphics[width=1\columnwidth]{f5}\par\end{centering}
\caption{\label{f:rh_stall}The stalling influence radius $r_{h}^{\prime}$,
as function of $Q$ for initial Dehnen density profiles with $\gamma=0.5,1.0,1.5$
(top to bottom), derived by \citet{mer06} in $N$body simulations
($\times$'s), and by the approximate analytical expression Eq. (\ref{e:rhstall_approx})
(lines).}
\end{figure}
\section{Discussion and Summary}
\label{s:summary}
We have shown that MPs play a dominant role in the aftermath of galactic
mergers. They shorten the relaxation timescale in the galactic nuclei
by orders of magnitudes relative to 2body stellar relaxation alone,
and drive the newly formed BMBH to rapid coalescence. The MP mechanism
requires only the existence of large enough inhomogeneities in the
galactic mass distribution. Since these occur naturally over a wide
range of conditions, the MP mechanism is robust and likely to accelerate
most BMBHs mergers. The one possible exception could be mergers of
two gaspoor elliptical galaxies, where GMCs are less common. However,
simulations indicate that stellar clusters can play the role of MPs
and drive an efficient merger even in most of these cases.
We conclude that most BMBHs are expected to coalesce within $t_{H}$,
even in cases where previous theoretical modeling, which did not consider
accelerated relaxation by MPs, predicts that the merging BMBH stalls.
This conclusion is strengthened by the facts that is based on conservative
assumptions. We considered only circular BMBHs, whereas eccentric
BMBHs coalesce even faster in the GW emission dominated phase, and
we neglected the possible concurrent effects of any of the other orbital
decay mechanisms proposed in the literature. It thus appears likely
the BMBH coalescence is in fact achieved on timescales $\ll\! t_{H},$
which implies that BMBH coalescence GW events occur at the cosmic
rate of galactic mergers.
Efficient MPdriven BMBH coalescence have additional implications,
which were discussed here briefly. Fast BMBH mergers decrease the
probability of nuclei containing triple MBHs, and hence of ejected
MBHs, since in most cases, the BMBH coalescence time is shorter than
the mean time between galactic collisions. During the final stage
of the merger, when the BMBH separation shrinks from the hardening
radius to the final GW radius, a large number of stars will be ejected
from the nuclei. We find that this additional ejection stage could
appreciably increase the mass deficit of the newly formed nucleus,
beyond what is predicted taking into account only the earlier stages
of the merger \citep{mer06}. We also analyzed the ejection of Galactic
HVSs by an inspiraling IMBH merging with the nuclear MBH. We model
the present day mass function of the GC and show that a recent merger
with a $\sim\!\mathrm{few}\times10^{4}\,\Mo$ IMBH can explain the
observed population of HVSs.
%\centerline{\rule{\columnwidth}{1pt}}
In summary, we have shown that the plausible existence of MPs in galactic
nuclei shortens the relaxation time by orders of magnitude. In particular,
MPs accelerate the dynamical decay of BMBHs by efficiently supplying
stars for the slingshot mechanism. This prevents stalling (the {}``last
parsec problem'') and allows the final coalescence of the BMBH by
GW emission within a Hubble time. Lowmass BMBHs, which are prospective
LISA targets, will coalesce even faster, within $10^{8}$$10^{9}$
yr.
\acknowledgements{TA is supported by ISF grant 928/06, Minerva grant 8563 and a New
Faculty grant by Sir H. Djangoly, CBE, of London, UK. HP would like
to thank the Israeli Commercial \& Industrial Club for their support
through the Ilan Ramon scholarship.}
\appendix
\section{A. The stalling radius}
\label{a:stall}
This appendix presents a simple analytic approximation for the stalling
separation, $a_{s}$, \textcolor{black}{as function of the premerger
galactic density profile and the BMBH mass ratio $Q$}, which is based
on the $N$body simulations of \textcolor{black}{\citet{mer06}.
Typically,} $a_{s}\!\sim\! a_{h}$ (Eq. \ref{e:a_h}), up to a factor
of a few. Assuming the \emph{ansatz} $a_{s}\rightarrow a_{h}$ in
the evaluation of the BMBH coalescence time can lead to inaccuracies
of up to a factor of a few, in particular for $Q\rightarrow1$.
\textcolor{black}{\citet{mer06} modeled typical galactic cores in
large $N$body simulations of BMBH coalescence by Dehnen configurations
\citep{deh+93}, \begin{equation}
\rho=\frac{M}{[4\pi/(3\gamma)]d^{3}}\frac{1}{{(r/d)}^{\gamma}[1+r/d)]^{4\gamma}}\,,\end{equation}
where $M$ is the total stellar mass, $d$ a scale length and $\gamma$
the logarithmic slope at $r\!\ll\! d$. A central MBH of mass $M_{1}/M\!=0.01$
was added to the initial density distribution. We assume here that
the results derived for this particular class of models also apply,
at least approximately, to other initial density distributions and
MBHtostellar cluster mass ratios. }
\textcolor{black}{\citet{mer06} finds that the stalling radius can
be described to a good approximation, independently of $\gamma$,
by\begin{equation}
a_{s}=0.2Q/(1+Q)^{2}r_{h}^{\prime}(M_{12})=0.8\left[r_{h}^{\prime}(M_{12})/r_{h}(M_{12})\right]a_{h}\,,\label{e:a_stall}\end{equation}
where $r_{h}^{\prime}(M_{12})$ is the radius of influence of the
BMBH at the stalling time $t_{s}$, after} \textcolor{black}{\emph{}}\textcolor{black}{the
scouring effect of the binary formation, which is estimated as follows.
The ejected mass at $t_{s}$ can be approximated analytically. \begin{equation}
\frac{\Delta M}{M_{12}}\simeq0.7Q^{0.2}\,.\end{equation}
The postmerger radius of influence $r_{h}^{\prime}$ can be estimated
to better than $3\%$ typically (Fig. \ref{f:rh_stall}), by assuming
that the post density profile resembles the original profile, except
for the removal of $\Delta M$ from the center further out, so that
\begin{equation}
M(\!4M_{\bullet}$ in stars. On that scale the potential
is dominated by the stars. For a $r^{2}$ stellar density distribution
far from the MBH, the velocity dispersion is $\sigma^{2}(r)\!\simeq\! GM_{\star}(\!\zeta$,
depends on the star's point of origin (see \S \ref{aa:galpot}).
Thus, the BMBH total decay rate is given by integrating over the contribution
of stars originating from all radii, with the losscone size expressed
in terms of the periapse, \begin{equation}
\frac{\mathrm{d}\log a}{\mathrm{d}t}=2\frac{M_{\star}}{M_{12}}\int\overline{C}\left[Q,\zeta_{\mathrm{eff}}(r;a)\right]\frac{\mathrm{d}\Gamma(