------------------------------------------------------------------------ bmbh_ms.tex eprint arXiv:0705.2123 Content-Type: multipart/mixed; boundary="------------090801070308020408090503" X-ESAFE-STATUS: Mail clean X-ESAFE-DETAILS: X-MailScanner-Information: Please contact the postmaster@aoc.nrao.edu for more information X-MailScanner: Found to be clean X-MailScanner-SpamCheck: not spam, SpamAssassin (not cached, score=1, required 5, autolearn=disabled, SARE_OEM_OBFU 1.00) X-MailScanner-SpamScore: s X-MailScanner-From: hagai.perets@weizmann.ac.il X-Spam-Status: No This is a multi-part message in MIME format. --------------090801070308020408090503 Content-Type: text/plain; charset=ISO-8859-1; format=flowed Content-Transfer-Encoding: 7bit |%arXiv:0705.2123 | --------------090801070308020408090503 Content-Type: text/x-tex; name="bmbh_ms.tex" Content-Transfer-Encoding: 7bit Content-Disposition: inline; filename="bmbh_ms.tex" %% LyX 1.4.3 created this file. For more info, see http://www.lyx.org/. %% Do not edit unless you really know what you are doing. \documentclass[english]{emulateapj} \usepackage[T1]{fontenc} \usepackage[latin1]{inputenc} \setcounter{secnumdepth}{4} \setcounter{tocdepth}{4} \usepackage{array} \usepackage{color} \usepackage{graphicx} \usepackage{amssymb} \makeatletter %%%%%%%%%%%%%%%%%%%%%%%%%%%%%% LyX specific LaTeX commands. %% Because html converters don't know tabularnewline \providecommand{\tabularnewline}{\\} %%%%%%%%%%%%%%%%%%%%%%%%%%%%%% User specified LaTeX commands. \usepackage{graphicx} \usepackage{amssymb} \usepackage{amsmath} \usepackage{times} \newcommand{\Rs}{R_{\star}} \newcommand{\Ls}{L_{\star}} \newcommand{\ns}{n_{\star}} \newcommand{\Es}{E_{\star}} \newcommand{\peryr}{\mathrm{yr}^{-1}} \newcommand{\yr}{\mathrm{yr}} \newcommand{\pc}{\mathrm{pc}} \newcommand{\perpc}{\mathrm{pc}^{-1}} \newcommand{\nbh}{n_{\bullet}} \newcommand{\Md}{M_{\mathrm{disk}}} \newcommand{\half}{{1\over 2}} \newcommand{\scr}{s_{\mathrm{crit}}} \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 2-body stellar relaxation alone, and efficiently scatter stars into the binary MBH's orbit. The 3-body star--binary 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 star-binary 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 hyper-velocity 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 semi-major 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 semi-major 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 ill-defined since $\sigma^{2}$ usually varies with distance from the BMBH.% }, where typically, $a_{h}\!\sim\!1-10$ 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 2--3 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 re-ejection 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 loss-cone replenishment in a non-axisymmetric 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 gas-depleted \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 non-axisymmetric potential assumed by the dry merger scenario of \citet{ber+06} is generally present in the post-merger 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 post-merger 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 MP-driven interactions of single and binary stars with a single MBH. In this study we apply the methods developed in PHA07 to investigate MP-driven 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 loss-cone 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{Loss-cone refilling by massive perturbers} \label{s:MPlosscone} PHA07 analyzed in detail MP-induced deflections of stars to nearly radial ({}``loss-cone'') 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, MP-induced relaxation on the rate at which stars enter the loss-cone can be incorporated into standard loss-cone 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 loss-cone (i.e. isotropize the orbits) on the orbital timescale. Since the larger $q$, the larger the loss-cone 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 semi-major 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 quasi-steady 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 $10-100$ 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 gas-rich ellipticals are thought to have been formed recently in a merger of two late type galaxies (e.g. \citealt{wik+97}). In particular, ultra-luminous infrared galaxies (ULIRGs, see review by \citealt{san+96}) have extreme amounts of gas, $10-100$ 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 present-day 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 gas-poor 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 slow-down of the decay when $a\!\sim\! a_{h}$ and stalling, unless the the loss-cone is replenished by a process more efficient than diffusion due to 2-body 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 semi-major 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 loss-cone 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 semi-major 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 semi-major 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 loss-cone 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 loss-cone, 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 perturber-driven 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 2-body 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 gas-rich 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 post-merger 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 power-law 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 gas-rich 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 gas-poor 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 birth-rate and the massive cluster birth-rate 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 power-law 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 post-merger galaxy is due to stellar 2-body 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 MP-induced coalescence time, we compute the rate in which stars are scattered by MPs into the BMBH. Based on results from 3-body 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 re-ejection, 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 re-ejection 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 time-dependent loss-cone 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 non-negligible 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:M-t_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 GW-induced 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 (dashed-dotted line). Without MPs the decay times are longer than $t_{H}$ for all BMBH mass in this range. } \end{figure} Figure (\ref{f:M-t_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 2-body relaxation cannot replenish the loss cone fast enough. In the absence of MPs, the merger proceeds in the empty loss-cone 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 loss-cone is small enough (lower BMBH mass), the loss-cone is full, and the merger time is determined by the size of the loss-cone 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 loss-cone. 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 loss-cone is filled, and the scattering rate attains its maximal value. This fast MP-driven evolution continues until the BMBH shrinks to the point where stellar relaxation alone can fill the loss-cone. 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 gas-poor 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 re-ejection 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 3-body 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 dash-dotted 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 loss-cone 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 MP-induced 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, X-ray shaped radio galaxies and so called double-double radio galaxies (sources which exhibit pairs of symmetric double-lobed radio-structures, aligned along the same axis) were interpreted as traces of a merged BMBH. In addition, it was suggested that semi-periodic 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 MP-driven coalescence is efficient (Fig. \ref{f:M-t_decay}), with the possible exception of high-mass 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 MP-driven 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 high-velocity MBHs ejected by the slingshot mechanism should be rare% \footnote{Recent observations of an apparently host-less 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 B-type 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 MP-driven 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 ($3-5\, M_{\odot}$) unbound HVSs% \footnote{\label{ft:PMF}The number fraction of young $3-5\, M_{\odot}$ stars, $\sim\!10^{-3}$ is calculated assuming continuous star formation over 10 Gyr with a Miller-Scalo 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 low-mass ratio IMBH--MBH 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 3-body 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 MP-driven merger. % }. Taking into account the GC present-day mass function (footnote \ref{ft:PMF}), their results either under- or over-estimated 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 MBH--IMBH 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 MBH-IMBH 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). Dash-dotted 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 (dash-dotted). 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 MBH-IMBH 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 2--7 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 $\sim1-2r_{\mathrm{MP}}$ 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 30-40\% 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 non-axisymmetric 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 2-body 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 gas-poor 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 MP-driven 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. Low-mass 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 pre-merger 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 MBH-to-stellar 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 post-merger 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 loss-cone 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(