arXiv:astro-ph/0609443v1 15 Sep 2006
Mon. Not. R. Astron. Soc. 000, 000–000 (0000) Printed 5 February 2008(MN LATEX style file v2.2)
The Role of Primordial Kicks on Black Hole Merger Rates
Miroslav Micic1⋆, Tom Abel2& Steinn Sigurdsson1
1Department of Astronomy & Astrophysics, Pennsylvania State University
2Department of Physics, Stanford University
5 February 2008
Primordial stars are likely to be very massive ≥30M⊙, form in isolation, and will
likely leave black holes as remnants in the centers of their host dark matter halos.
We expect primordial stars to form in halos in the mass range 106− 1010M⊙. Some
of these early black holes, formed at redshifts z∼>10, could be the seed black hole for
a significant fraction of the supermassive black holes found in galaxies in the local
universe. If the black hole descendants of the primordial stars exist, their mergers
with nearby supermassive black holes may be a prime candidate for long wavelength
gravitationalwavedetectors. We simulate formation and evolution of dark matter halos
in ΛCDM universe. We seed high-redshift dark matter halos with early black holes,
and explore the merger history of the host halos and the implications of black hole’s
kick velocities arising from their coalescence. The central concentration of low mass
early black holes in present day galaxies is reduced if they experience even moderate
kicks of tens of kms−1. Even such modest kicks allow the black holes to leave their
parent halo, which consequently leads to dynamical friction being less effective on the
low mass black holes that were ejected, compared to those still embedded in their
parent halos. Therefore, merger rates with central supermassive black holes in the
largest halos may be reduced by more than an order of magnitude. Using analytical
and illustrative cosmological N–body simulations, we quantify the role of kicks on the
merger rates of black holes formed from massive metal free stars with supermassive
black holes in present day galaxies.
Key words: IMBH, SMBH, Pop III, gravitational radiation, LISA, BBO
It is firmly established that most galaxies have super-massive black holes (SMBH) at their centers. Their observed masses
are in the range 106M⊙∼<M∼<109M⊙ and it appears that there are correlations (Kormendy & Gebhardt 2001, Gerhard
2001) between their masses and the bulk properties of the galactic bulges hosting them (Laor 2001, Merritt & Ferrarese 2001,
Tremaine et al. 2002, Gebhardt et al. 2001, Graham et al. 2001). These correlations point to a link between the formation
of SMBHs and the evolution of their hosts. It also appears that SMBHs are linked to the properties of the host dark matter
halos. If the SMBH precursors were present from very early on, then their mergers, together with growth by accretion, could
account for the abundance of the SMBHs today (Schneider et al. 2002).
Ab initio numerical simulations of the formation of the first luminous objects within the standard structure formation
framework, finds that metal free stars form in isolation, and may have masses 30 M⊙∼<m∼<300 M⊙ (Abel et al. 2000, Abel
et al. 2002, Bromm et al. 2002). In current models of structure formation, dark matter initially dominates and pregalactic
objects form from small initial density perturbations. As they assemble via hierarchical merging, the metal-free primordial
gas cools slowly through the rotational lines of hydrogen molecules. As the gas cools, it sinks to the center of the dark matter
potential well. However, as the cooling rates are small, the clumps fail to fragment as they collapse. This leads to a top-heavy
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Miroslav Micic, Tom Abel, Steinn Sigurdsson
initial stellar mass function (IMF) and to the production of very massive stars, unlike the modern stellar IMF which declines
rapidly with increasing mass.
Metal-free, high mass Population III stars lose only a small fraction of their mass in their lifetime of ∼3 Myr. This
suggests that a large population of primordial massive black holes (MBH, with mass∼<103M⊙) could be an end product
of such pregalactic star formation (Heger et al. 2003). Since they form in rare high-σ density peaks (Couchman & Rees
1986, Madau & Rees 2001), relic MBHs are predicted to cluster in the cores of more massive halos (Abel et al. 2002) which
in turn are formed by subsequent mergers of the original parent dark matter halos. It has been suggested (Volonteri et al.
2003, Islam et al. 2003) that mergers of massive halos and clustering of these MBHs should start at redshifts as large as
z∼20. Subsequent growth of MBHs would proceed through accretion of baryonic gas and coalescence, possibly leading to the
formation of intermediate mass black holes (IMBH, with mass in range 103M⊙ ∼<m∼<106M⊙). In the most optimistic
scenario (Haiman & Loeb 2001), 109M⊙ black holes could form as early as z∼10.
In this paper we will explore the evolution of the seed black hole in dark matter halos. Mergers of dark matter halos hosting
at least one black hole (Bromm & Loeb 2003) at high redshift form black hole binaries, and their subsequent coalescence
under gravitational radiation may give them a significant kick velocity (Favata et al. 2004, Merritt et al. 2004). These recoils
are caused by anisotropic emission of gravitational waves which carry away linear momentum. The recoil velocity is highly
controversial requiring full general relativistic calculations. However, recent estimates place most recoils in the range 10-100
kms−1, although kicks of a few hundred kms−1are not unexpected, and the largest recoils should not be above 500 kms−1
(Favata et al. 2004, Blanchet et al. 2005, see also Baker et al 2006). The amplitude of the kick determines if the black hole
will be ejected from its host dark matter halo. Even the most massive dark matter halo at redshift z≥11 can not retain a
black hole that receives ≥150kms−1kick, while at redshift of z≥8, kicks of 300kms−1are required to cause ejection (Merritt
et al. 2004, Fig 3.). This is valid for black holes with masses 103M⊙∼<m∼<108M⊙. Kicks of 40kms−1should be sufficient
to: eject IMBHs from globular clusters; displace IMBHs from the centers of dwarf galaxies; and perhaps provide a sufficient
population of IMBHs for merging at the centers of spiral galaxies (Merritt et al. 2004).
We use N-body TREEcode simulations (GADGET, Springel et al. 2001) to investigate the observational implications of
black hole kicks through the asymmetric emission of gravitational radiation during the merger. In this academic exercise, we
assume that Population III stars collapse to form massive black holes at high redshifts. These black holes grow through gas
accretion and mergers at the centers of dark matter halos. We use collisionless particles at the centers of dark matter halos
as tracers of IMBHs. If kicks are important, the growth of IMBHs into supermassive black hole will be suppressed. We test
two extreme scenarios for this behaviour and whether there are predictable consequences. In the first scenario, kicks do not
occur in any dark matter halo at any redshift. In the second, kicks occur in every dark matter halo at an initial redshift
and we introduce a prescription for assigning kicks to our tracer particles. As a result, these two scenarios set limits to the
IMBHs’ merger rate. The first scenario provides a maximum merger rate, and the second scenario provides a lower bound
on the merger rate (the absolute lower bound is that no mergers occur, for example if black hole formation is suppressed
and primordial stars leave no compact remnants). A better approach would be to track every individual merger, and assign
kicks when mergers occur. Our approach is computationally less expensive and provides only constraints. We have chosen
z=8 as an initial redshift for assigning kicks to our tracers for IMBHs. At redshift z∼12 supernovae rates for Pop III stars
fall to zero (Wise & Abel 2005). The massive black holes resulting from Pop III collapse have already settled in the centers
of dark matter halos by redshift z∼8, the initial redshift for kicks. As mentioned before we are calculating constraints for the
merger rates and not studying individual mergers. The MBH tracers are tracked from the initial to the final redshift and used
to quantify the changes in distributions of IMBHs in these two scenarios. Through detailed analysis of their trajectories, we
study how IMBHs respond to the dynamical evolution of dark matter halos, testing extreme ranges of outcome to check for
the plausible scope for detection. Our results also apply to black holes ejected by the gravitational slingshot mechanism. The
slingshot velocities fall in the recoil range (Aarseth 2005).
Our numerical simulations are followed by an analytical treatment of dynamical friction. As long as the dark matter halos
have a bound core with density greater than the local galactic density and have ”many” particles in them, the numerical
dynamical friction will mimic ”real” dynamical friction with reasonable fidelity (Vine & Sigurdsson 1998). If the halo density
becomes smaller than the local galactic density, or if the secondary halo structure is down to being resolved by only ”few”
bound particles, then the dynamical friction prescription breaks down and relaxation due to numerical noise and large scale
potential fluctuations dominate; integration of the dynamical friction term becomes meaningless. This happens basically where
we stop the simulations, and that is where we analytically estimate a conservative lifetime to coalescence.
We are exploring the effects of kicks on IMBH merger rates and the formation of SMBHs that are detected in the nuclei
of luminous galaxies. Knowledge of IMBH merger rates is essential for the future experiments such as LISA (Danzmann 2003)
and BBO (Big Bang Observer) (Cornish & Crowder 2005). LISA will be able to study much of the last year of inspiral of
IMBH into SMBHs, as well as the waves from the final collision and coalescence of IMBH binaries. For binary black holes with
masses in the range 100 M⊙ ∼<M∼<104M⊙, LISA can observe the last few years of inspiral, but not the final collision even
at cosmological distances. Equal-mass black hole binaries enter LISA’s frequency band roughly 1000 years before their final
coalescence, independent of their masses, for the range 100M⊙∼<M∼<106M⊙ (Thorne 1995, Cornish & Levin 2002, Vecchio
The Role of Primordial Kicks on Black Hole Merger Rates
Figure 4. Number of black holes as a function of primary halo radius. No kick case (thick); [0,150] km/s kick centered at 75 km/s (dash)
and [125,275] km/s kick centered at 200 km/s (dash-dot). Although the number of IMBHs entering primary halo is comparable in all
three cases, the difference in their interior distribution is well pronounced.
Figure 5. Fraction of SIM1 black holes in SIM2a and SIM2b as a function of radius. ∆N is N - Nkickwhere N is number of black holes
in SIM1 (no kick). Thick line for [0,150] km/s kicks centered at 75 km/s and dots for [125,275] km/s kicks centered at 200 km/s.
Miroslav Micic, Tom Abel, Steinn Sigurdsson
Figure 6. Density of the primary halo at z=1 (thick dash) and its most massive progenitor from z=8.16 (dots) as a function of radius.
Also, density in hosted black holes for no kick (thick); [0,150] km/s kick centered at 75 km/s (dash) and [125,275] km/s kick centered at
200 km/s (dash-dot).
Figure 7. Local density of dark matter traced by black holes as a function of redshift. No kick (thick); [0,150] km/s kick centered at 75
km/s (dash) and [125,275] km/s kick centered at 200 km/s (dash-dot). Ejection of IMBHs from gas enriched regions of galaxy infuences
AGNs formation rates, reduces their numbers and their contribution to the ionizing background.