If binary intermediate-mass black holes (IMBHs; with masses between 100 and $10^4 \Msun$) form in dense stellar clusters, their inspiral will be detectable with the planned Laser Interferometer Space Antenna (LISA) out to several Gpc. Here we present a study of the dynamical evolution of such binaries using a combination of direct $N$-body techniques (when the binaries are well separated) and three-body relativistic scattering experiments (when the binaries are tight enough that interactions with stars occur one at a time). We find that for reasonable IMBH masses there is only a mild effect on the structure of the surrounding cluster even though the binary binding energy can exceed the binding energy of the cluster. We demonstrate that, contrary to standard assumptions, the eccentricity in the LISA band can be in {\em some} cases as large as $\sim 0.2 - 0.3$ and that it induces a measurable phase difference from circular binaries in the last year before merger. We also show that, even though energy input from the binary decreases the density of the core and slows down interactions, the total time to coalescence is short enough (typically less than a hundred million years) that such mergers will be unique snapshots of clustered star formation.
Deep Dive into Gravitational waves from eccentric intermediate-mass black hole binaries.
If binary intermediate-mass black holes (IMBHs; with masses between 100 and $10^4 \Msun$) form in dense stellar clusters, their inspiral will be detectable with the planned Laser Interferometer Space Antenna (LISA) out to several Gpc. Here we present a study of the dynamical evolution of such binaries using a combination of direct $N$-body techniques (when the binaries are well separated) and three-body relativistic scattering experiments (when the binaries are tight enough that interactions with stars occur one at a time). We find that for reasonable IMBH masses there is only a mild effect on the structure of the surrounding cluster even though the binary binding energy can exceed the binding energy of the cluster. We demonstrate that, contrary to standard assumptions, the eccentricity in the LISA band can be in {\em some} cases as large as $\sim 0.2 - 0.3$ and that it induces a measurable phase difference from circular binaries in the last year before merger. We also show that, even
The existence of intermediate-mass black holes (IMBHs; masses M ∼ 10 2-4 M ⊙ ) is not as certain as that of stellar-mass or supermassive black holes because there is as yet no conclusively established dynamical mass for any candidate, although there is strong circumstantial evidence for this mass range in several cases (see Miller & Colbert 2004 and references therein for a review). Mergers of IMBHs would, however, be strong sources of gravitational waves.
The best studied scenario is the runaway growth of a star in a young cluster via physical collisions among the most massive stars in the center, which have sunk through mass segregation (Portegies Zwart & McMillan 2000;Gürkan et al. 2004;Portegies Zwart et al. 2004;Freitag et al. 2006). Recently, Gürkan et al. (2006) addressed the same configuration but added a fraction of primordial binaries to the stellar system. Using a Monte-Carlo stellar-dynamics code, they found that not one but two very massive stars grow in rich clusters in which 10% or more of stars are in primordial hard binaries, suggesting the formation of two IMBHs. However, this result has not been confirmed yet using more accurate direct N-body simulations. Portegies Zwart et al. ( 2004) have a simulation with primordial binaries but they do not see this formation, though it is also currently unclear how different core concentrations will affect binary IMBH formation with a certain fraction of primordial binaries. It is also possible that wind losses may drive away mass more rapidly than it accretes through further collisions (see Belkus, van Bever, & Vanbeveren 2007), although this relies on uncertain extrapo-Electronic address: Pau.Amaro-Seoane@aei.mpg.de, miller@astro.umd.edu, freitag@ast.cam.ac.uk 1 (PAS) Institut de Ciències de l’Espai, IEEC/CSIC, Campus UAB, Torre C-5, parells, 2 na planta, ES-08193, Bellaterra, Barcelona and Max Planck Institut für Gravitationsphysik (Albert-Einstein-Institut), Am Mühlenberg lations from the ∼ 120 M ⊙ that is the top of their range (see their Table 2) to the ∼ 2000 M ⊙ masses observed in N-body simulations. Fregeau et al. (2006) considered for the first time the possibility that such a binary could be observed thanks to the emission of gravitational waves in the coalescence phase and estimated that one can expect the Laser Interferometer Space Antenna (LISA) to detect tens of them depending on the distribution of cluster masses and densities. Amaro-Seoane & Freitag (2006) addressed the evolution of a binary of two IMBHs formed as the result of the collision of two independent stellar clusters and followed the parameters of the binary orbit down to the region in which it will emit gravitational waves in the ∼ 10 -4 -10 -1 Hz LISA domain. To do this, they combined direct-summation simulations with an analytical model to evolve the binary from a point in which it was hard.
Here we assume that an IMBH binary has been produced in a single dense stellar cluster, and study the subsequent sinking of the IMBHs and the evolution and properties of the binary when it forms. In § 2 we discuss our numerical method, which combines direct N-body studies with three-body scattering integrations. In § 3 we discuss the astrophysical implications of our results.
Direct N-body codes integrate all gravitational accelerations in a stellar system without supposing any special symmetries. They are thus the most general and robust tools for numerical analysis of stellar clusters (Aarseth 1999(Aarseth , 2003)). The code we use, NBODY4, includes a variety of sophisticated approaches that improve speed and accuracy, including KS regularization (Kustaanheimo & Stiefel 1965), as well as triple (3-body subsystems), quad (4-body subsystems), and chain regularization (Aarseth 1999(Aarseth , 2003)). It also does not make use of any softening, which would lead to unrealistic evolution of the orbital parameters of the binary of massive black holes. The disadvantage of this or any direct N-body code is the required computational time. However, our calcu- lations are accelerated thanks to the special-purpose hardware GRAPE-6A single PCI cards of the AEI cluster TUFFSTEIN used for the simulations. Each card has a peak performance of 130 Gflops (Fukushige et al. 2005), so that a single node is comparable to a cluster of ∼100 individual CPUs working in parallel.
Table 1 gives the initial conditions for the different simulations that we feature. We ran six cases with varying number densities and concentrations, of which two had a Kroupa (2001) mass function instead of single-mass stars. The IMBHs have equal mass except in simulation F, which has a mass ratio of 5. In our simulations the individual time steps led to fractional energy errors that were always less than 10 -4 per N-body unit of time and, globally, the total energy error of the cluster (i.e. the accumulated error in the integration of all particles) is 0.015% in the case of our fiducial model (A).
Our approach is to evolve the
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