Gravitational waves from eccentric intermediate-mass black hole binaries
aa r X i v : . [ a s t r o - ph . S R ] J a n D RAFT VERSION N OVEMBER
8, 2018
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GRAVITATIONAL WAVES FROM ECCENTRIC INTERMEDIATE-MASS BLACK HOLE BINARIES P AU A MARO -S EOANE ,M. C OLEMAN M ILLER , M ARC F REITAG Draft version November 8, 2018
ABSTRACTIf binary intermediate-mass black holes (IMBHs; with masses between 100 and 10 M ⊙ ) form in dense stel-lar clusters, their inspiral will be detectable with the planned Laser Interferometer Space Antenna (LISA) outto several Gpc. Here we present a study of the dynamical evolution of such binaries using a combination ofdirect N -body techniques (when the binaries are well separated) and three-body relativistic scattering experi-ments (when the binaries are tight enough that interactions with stars occur one at a time). We find that forreasonable IMBH masses there is only a mild effect on the structure of the surrounding cluster even though thebinary binding energy can exceed the binding energy of the cluster. We demonstrate that, contrary to standardassumptions, the eccentricity in the LISA band can be in some cases as large as ∼ . - . Subject headings:
Black hole physics, gravitational waves, stellar dynamics, methods: N-body simulations INTRODUCTION
The existence of intermediate-mass black holes (IMBHs;masses M ∼ - M ⊙ ) is not as certain as that of stellar-massor supermassive black holes because there is as yet no con-clusively established dynamical mass for any candidate, al-though there is strong circumstantial evidence for this massrange in several cases (see Miller & Colbert 2004 and refer-ences therein for a review). Mergers of IMBHs would, how-ever, be strong sources of gravitational waves.The best studied scenario is the runaway growth of a star ina young cluster via physical collisions among the most mas-sive stars in the center, which have sunk through mass seg-regation (Portegies Zwart & McMillan 2000; Gürkan et al.2004; Portegies Zwart et al. 2004; Freitag et al. 2006). Re-cently, Gürkan et al. (2006) addressed the same configurationbut added a fraction of primordial binaries to the stellar sys-tem. Using a Monte-Carlo stellar-dynamics code, they foundthat not one but two very massive stars grow in rich clustersin which 10% or more of stars are in primordial hard binaries,suggesting the formation of two IMBHs. However, this re-sult has not been confirmed yet using more accurate direct N - body simulations. Portegies Zwart et al. (2004) have asimulation with primordial binaries but they do not see thisformation, though it is also currently unclear how differentcore concentrations will affect binary IMBH formation witha certain fraction of primordial binaries. It is also possiblethat wind losses may drive away mass more rapidly than itaccretes through further collisions (see Belkus, van Bever, &Vanbeveren 2007), although this relies on uncertain extrapo- Electronic address: [email protected], [email protected],[email protected] (PAS) Institut de Ciències de l’Espai, IEEC/CSIC, Campus UAB, TorreC-5, parells, 2 na planta, ES-08193, Bellaterra, Barcelona and Max PlanckInstitut für Gravitationsphysik (Albert-Einstein-Institut), Am Mühlenberg 1,D-14476 Potsdam, Germany (MCM) Department of Astronomy and Maryland Astronomy Center forTheory and Computation, University of Maryland, College Park, MD 20742-2421, USA (MF) Institute of Astronomy, University of Cambridge, Madingley Road,CB3 0HA Cambridge, UK lations from the ∼ M ⊙ that is the top of their range (seetheir Table 2) to the ∼ M ⊙ masses observed in N-bodysimulations.Fregeau et al. (2006) considered for the first time the pos-sibility that such a binary could be observed thanks to theemission of gravitational waves in the coalescence phase andestimated that one can expect the Laser Interferometer SpaceAntenna (LISA) to detect tens of them depending on the distri-bution of cluster masses and densities. Amaro-Seoane & Fre-itag (2006) addressed the evolution of a binary of two IMBHsformed as the result of the collision of two independent stel-lar clusters and followed the parameters of the binary orbitdown to the region in which it will emit gravitational wavesin the ∼ - - - Hz LISA domain. To do this, they com-bined direct-summation simulations with an analytical modelto evolve the binary from a point in which it was hard.Here we assume that an IMBH binary has been produced ina single dense stellar cluster, and study the subsequent sinkingof the IMBHs and the evolution and properties of the binarywhen it forms. In § 2 we discuss our numerical method, whichcombines direct N -body studies with three-body scattering in-tegrations. In § 3 we discuss the astrophysical implications ofour results. EVOLUTION OF THE IMBH PAIR: NUMERICAL METHOD
Direct N - body simulations Direct N - body codes integrate all gravitational accelera-tions in a stellar system without supposing any special sym-metries. They are thus the most general and robust tools fornumerical analysis of stellar clusters (Aarseth 1999, 2003).The code we use, NBODY4, includes a variety of sophisti-cated approaches that improve speed and accuracy, includingKS regularization (Kustaanheimo & Stiefel 1965), as well as triple (3-body subsystems), quad (4-body subsystems), and chain regularization (Aarseth 1999, 2003). It also does notmake use of any softening, which would lead to unrealisticevolution of the orbital parameters of the binary of massiveblack holes. The disadvantage of this or any direct N - bodycode is the required computational time. However, our calcu- Amaro-Seoane et al Model N ⋆ M bin / M ⊙ ρ ( M ⊙ / pc ) W IMF a (pc) A · B . · C . · D · E F NITIAL CONDITIONS FOR OUR FEATURED DIRECT - SUMMATION N - BODY MODELS . N ⋆ IS THE NUMBER OF STELLAR PARTICLES USED .T HE MASS OF THE BINARY , NORMALIZED TO SOLAR MASSES , IS GIVENIN THE THIRD COLUMN , ρ IS THE INITIAL MASS DENSITY AT ADISTANCE OF PC , W IS THE K ING PARAMETER (K ING
LLCASES ARE SINGLE - MASS BUT FOR MODEL C AND F , IN WHICH WEHAVE A MASS FUNCTION , SPECIFICALLY A
YRS EVOLVED K ROUPA
IMF
OF MASSES
AND EXPONENTS
AND
ROUPA HE TH COLUMN SHOWS THE INITIAL SEMI - MAJOR AXIS OFTHE BINARY IN PC . lations are accelerated thanks to the special-purpose hardwareGRAPE-6A single PCI cards of the AEI cluster T UFFSTEIN used for the simulations. Each card has a peak performanceof 130 Gflops (Fukushige et al. 2005), so that a single node iscomparable to a cluster of ∼
100 individual CPUs working inparallel.Table 1 gives the initial conditions for the different sim-ulations that we feature. We ran six cases with varyingnumber densities and concentrations, of which two had aKroupa (2001) mass function instead of single-mass stars.The IMBHs have equal mass except in simulation F , whichhas a mass ratio of 5. In our simulations the individual timesteps led to fractional energy errors that were always less than10 - per N-body unit of time and, globally, the total energy er-ror of the cluster (i.e. the accumulated error in the integrationof all particles) is 0 . A ). Evolution of the binary: gravitational radiation versusdynamics
Our approach is to evolve the cluster up to ∼ -
70 Myrsusing the direct
NBODY N - bodysimulations. We take the last point of these evolutions andthe number density of field stars as input to relativistic scat-tering experiments which we performed following Gültekinet al. (2006). The equations of motion we use for the three-body encounters include relativistic precession to first post- F IG . 1.— Lagrangian radii showing the evolution of different mass frac-tions in the cluster for Model D. The fractions are, from the bottom to the top,0 . . . . .
3% . . . 0 .
9% and 0 . Newtonian order, as well as radiation reaction caused by grav-itational waves. Between encounters, we evolve the semima-jor axis and eccentricity of the IMBH binary using the Pe-ters quadrupolar formulae (Peters 1964). The stars that in-teract with the binary are sent with a velocity at infinity of v =10 km s - , typical of cluster velocity dispersions. The in-teraction time is drawn from an exponential distribution witha mean time of τ = ( n Σ v ) - , where n is the stellar number den-sity (taken from the N - body simulations) and Σ is the scatter-ing cross section including gravitational focusing. The typicalregion in which the IMBH binary wanders is larger than itsradius of influence, hence there is no loss cone as there is forsupermassive black holes.We ran four sets of 40 simulations, two sets starting at largeseparations with zero eccentricity that established agreementwith the direct-summation N - body simulations and two setsat the endpoints of Models A and B. In none of these runs wasthe binary itself ejected from the cluster, as expected givenits large mass. For the Model A endpoint run (binary mass600 M ⊙ , initial semimajor axis a = 10 AU, and initial ec-centricity e = 0 . e LISA = 0 . ± . - Hz low end of the
LISA band;for the Model B endpoint run (binary mass 2000 M ⊙ , initialsemimajor axis a = 400 AU, and initial eccentricity e = 0 . e LISA = 0 . ± .
05. We show the envelope of theModel B endpoint runs in the left panel of Figure 3. We alsodid scattering experiments corresponding to the endpoint ofModel C, which had a Kroupa mass function. The resultsare shown in the right panel of Figure 3. Compared to thesingle-mass runs we see considerably greater variance in theeccentricity as a function of semimajor axis, and although therange of eccentricities in the
LISA band e LISA = 0 . ± . - Hz. This couldbe a general feature of scattering interactions when there is aravitational waves from binary IMBHs 3 F IG . 2.— Left panel:
Inspiral of the IMBH binary of Model E followedin the eccentricity–semi-major axis plane. The irregular line shows the re-sults of the N - body simulation. The smooth black solid curves are the esti-mated trajectories due to gravitational wave emission following the approxi-mation of Peters (1964), and the dashed curves show the corresponding inspi-ral timescale, t GW . The dark dashed area depicts the region of unstable orbits.The lightly shaded area corresponds to the phase in the evolution in whichthe n = 2 harmonic of the gravitational wave signal is in the LISA band. Thedashed irregular line starting after the last point of the results of the N - bodysimulation (in the color version depicted in magenta), are the results fromthe scattering experiments. See text for further details. Near the beginningthe eccentricity temporarily exceeds unity because the black holes are not yetbound to each other. Right panel:
Same for Model F, which is one case inwhich we have initially a Kroupa IMF (see Table 1). When star with a bigmass interacts with the binary, the eccentricity change is substantial. The ec-centricity therefore wanders up and down, and when it becomes large enoughthe binary has a greater chance to spiral together by gravitational radiationF IG . 3.— Left panel:
Average evolution of eccentricity as a function ofsemimajor axis. This figure shows the results of 40 three-body scatteringexperiments, starting with IMBH masses of 10 M ⊙ each at an initial semi-major axis of 400 AU and an initial eccentricity of 0.55, corresponding to theend point of Model B. We see the - σ to + σ ranges of eccentricity. The hor-izontal line shows a semimajor axis of 0.093 AU, which is where the orbitalfrequency is 5 × - Hz and thus the dominant gravitational wave frequencyis 1e-4 Hz.
Right panel:
Similar but assuming a Kroupa IMF and using the end point ofModel C. There were no ejections of the binary from the cluster, which wasassumed to have an escape speed of 50 km/s. The horizontal line is lowerthan in the left panel because the binary mass is less. broad mass function, but we have not performed enough runsto determine this with confidence. DISCUSSION AND CONCLUSIONS
We have addressed the inspiral of two massive black holesin a single young stellar cluster. Our three main results are:(1) the cluster itself experiences only mild structural changesas a result of the inspiral, (2) the coalescence takes a shortenough time (typically <
100 Myrs) that mergers occurs closeto the time of formation of the cluster, and (3) there is a sig-nificant residual eccentricity by the time the binary enters theLISA band. We now discuss these conclusions in order.The stability of clusters against IMBH mergers is consis-tent with analytic expectations even though the binding en-ergy of the IMBH binary can exceed the total binding en-ergy of the cluster by a significant factor. To see this, con- sider a circular IMBH binary of component masses m and m ≤ m , with total mass M = m + m and reduced mass µ = m m / M . As shown by Quinlan (1996), a star withlow speed at infinity that interacts with the binary will typi-cally be ejected with a speed v ej ≈ . p m / M V orb , where V orb is the relative speed of the two objects. For equal masses m = m , this is v ej ≈ . V bin . Suppose now that the clusterhas an escape speed v esc . If v ej < v esc then the star will beretained and share its kinetic energy with the cluster. Other-wise, the star will be ejected from the cluster without deposit-ing significant energy, because the dynamical time of escapeis much less than the relaxation time (which is the time re-quired for the star to give up energy). The binding energyof the IMBH binary when v ej = v esc will be E bin = µ V ≈ µ ( v esc / . ( M / m ) ≈ . m v . In comparison, if thecluster has a three-dimensional velocity dispersion σ and amass M cl , the binding energy of the cluster is E cl ≈ M cl σ .The ratio is then E bin / E cl ≈ (cid:0) m / M cl (cid:1) (cid:0) v esc /σ (cid:1) . Typically v esc ∼ - × σ , so only if the larger black hole mass is m > . - . M cl could the release of energy unbind the clus-ter. We also note that subsequent to this point, the loss ofmass from stars being thrown out would also soften the clus-ter. However, since typically interaction with of order the bi-nary mass changes the semimajor axis by a factor of ∼
2, just ∼ M in stars will shrink the binary by enough of a fac-tor to produce coalescence. Therefore, as verified by our nu-merical simulations, hardening of an IMBH binary has only aminor effect on the cluster.For the time to merger, we note that hardening from largeseparations to a few hundred AU takes ∼
50 Myr, based onour simulations. Our three-body runs then indicate that thetotal time from that point to merger is virtually always lessthan 10 Myr, meaning that conservatively the total time fromformation to merger is less than 10 yr. This is significantlyshorter than the age of the universe. One consequence of thisis that if star formation in massive clusters was more commonat redshift z ∼ ∼ . - . - Hz. Consistent with Quinlan (1996) wefind that the eccentricity does not undergo a random walk,but instead tends to higher eccentricities when the binary ishard but before gravitational radiation circularization is im-portant. As discussed in section 4 of Amaro-Seoane & Fre-itag (2006), a residual eccentricity will induce a differencein the phase evolution of the second harmonic compared to acircular orbit, even if it is as small as 0.07, as Amaro-Seoane& Freitag (2006) found. In our case, if we use an eccentric-ity e - Hz = 0 . ∆Ψ e ≥ π ifobservations cover a time of at least t mrg ∼ . · (1 + z ) yr (1)before merger, where z is the redshift. If we set z = 1, thenwe have to cover a time t mrg = 17 days before merger. Thismeans that if we are able to observe the system during thatperiod of time before the final coalescence, we will recoverenough information to determine that the orbit is not circular. Amaro-Seoane et alOn the other hand, if we use a residual eccentricity of 0.07, asin Amaro-Seoane & Freitag (2006), we would need 3-4 yearsof observation for a 300 + M ⊙ binary before merger.In conclusion, if young massive clusters form binaryIMBHs then they will be strong and moderately eccentricLISA sources that could serve as unique signposts of clus-tered star formation. The non-zero residual eccentricity hasan impact on the detection of such sources, since it is gener-ally assumed that an equal-mass massive binary will have azero eccentricity when entering the LISA band. Our resultsshow that in our scenario e is non-negligible for certain cases-though for some other models it is very low but detectable-;notably, case C and F , which are the only models in which wehave a mass fraction and thus, they are the more realistic ones.The process of formation must of course be studied carefullyfrom the standpoints of stellar dynamics and merger productevolution, but if binary IMBHs can form then their mergers are promising sources for future LISA detections.We thank Vanessa Lauburg for valuable discussions. Weare also indebted to Yuri Levin, Ed Porter, Matt Benacquista,Jonathan Gair and Stas Babak for enlightening conversations.The authors are grateful to the Aspen Center for Physics forits hospitality during part of this project. PAS work was par-tially supported by the MEC (Ministerio de Educación y Cien-cia) at the Institut de Ciències de l’Espai (IEEC/CSIC) andthe DLR (Deutsches Zentrum für Luft- und Raumfahrt) at theMax-Planck Institut für Gravitationsphysik (Albert Einstein-Institut, AEI). MCM appreciates support from NASA un-der ATFP grant NNX08AH29G. The work of MF is fundedthrough the PPARC rolling grant at the Institute of Astronomy(IoA) in Cambridge. The simulations have been performed atthe GRAPE cluster T UFFSTEIN of the AEI.of the AEI.