Event Rates of Gravitational Waves from merging Intermediatemass Black Holes: based on a Runaway Path to a SMBH
Hisaaki Shinkai1 ,⋆1

Department of Information Systems, Osaka Institute of Technology, Hirakata City, Osaka 573-0196, Japan

Abstract. Based on a dynamical formation model of a supermassive black hole (SMBH),
we estimate the expected observational profile of gravitational wave at ground-based detectors, such as KAGRA or advanced LIGO/VIRGO. Noting that the second generation ofdetectors have enough sensitivity from 10 Hz and up, we are able to detect the ring-down
gravitational wave of a BH with the mass M < 2 × 103 M⊙ . This enables us to check thesequence of BH mergers to SMBHs via intermediate-mass BHs. We estimate the number density of galaxies from the halo formation model and estimate the number of BH
mergers from the giant molecular cloud model assuming hierarchical growth of mergedcores. At the designed KAGRA (and/or advanced LIGO/VIRGO), we find that the BH
merger of its total mass M ∼ 60M⊙ is at the peak of the expected mass distribution. Withits signal-to-noise ratio ρ = 10(30), we estimate the event rate R ∼ 200(20) per year in
the most optimistic case, and we also find that BH mergers in the range M < 150M⊙ areR > 1 per year for ρ = 10. Thus, if we observe a BH with more than 100M⊙ in future
gravitational-wave observations, our model naturally explains its source.

1 Introduction
The LIGO group reported three detections of gravitational waves (GW150914, GW151226,GW170104) and one transient event (LVT151012) so far (see Table. 1), all three of which are regarded as the events of coalescence of binary black holes (BBHs). These announcements were not
only valuable on the point of the direct detections of the gravitational wave, but also the first results ofconfirming the existence of BHs, the existence of BHs of this mass range, and the existence of BBHs.
In this article, as a possible sources of BHs of 30M⊙ –60M⊙ , we consider simple sequencesof mergers of BHs, which start from 10M⊙ BHs and end with a supermassive BH (SMBH) via
intermediate-mass BHs (IMBHs). After counting the numbers of galaxies in the Universe from ahalo formation model, and modeling the numbers of BHs in a galaxy from a giant molecular cloud
scenario, we estimate their observational detectability at ground-based gravitational-wave detectors.
More details are reported in [1].
The formation process of an SMBH is one of the unsolved problems in galaxy evolution history.
Many possible routes were suggested by Rees [5] long ago, but we still debate a plausible route. In
this article, we take the scenario that SMBHs are formed by accumulations of BHs. This route wascame to be believed when an IMBH (102 –103 M⊙ ) was first discovered in a starburst galaxy M82. So
far, many IMBHs have been found in the center of galaxies.
⋆ e-mail: hisaaki.shinkai@oit.ac.jp

Table 1. GW events (as of July 2017)

GW150914
ref [2]
LVT151012
GW151226
ref [3]
GW170104ref [4]

Primary & Secondary
m1 /M⊙
m2 /M⊙
36.2
29.1+5.2−3.8+3.7
−4.4

23

13+18
−6+4
−5

14.2

7.5+8.3−3.7+2.3
−2.3

31.2

19.4+8.4−6.0+5.3−5.9

Final BHMfinal /M⊙ ∆M/M⊙
62.3−3+3.7(4.59%)−3.1
35−1+14(2.78%)−420.8−0.9+6.1(4.15%)
−1.7
48.7
−1.9+5.7(3.75%)
−4.6

SNRρ
24

Mpc
z410Mpc
0.09

0.74

130.6413440Mpc
0.09
880Mpc
0.18

a/M⊙
0.68

deg2
600

0.66

1300

This runaway path was first proposed by Ebisuzaki et al.[6]. The scenario consists of three steps:(1) formation of IMBHs by runaway mergers of massive stars in dense star clusters, (2) accumulationsof IMBHs at the center region of a galaxy due to sinkages of clusters by dynamical friction, and(3) mergers of IMBHs by multibody interactions and gravitational radiation. Successive mergers ofIMBHs are likely to form an SMBH with a mass of heavier than > 106 M⊙ . Numerical simulations
support the above first step and the second step is also confirmed in a realistic mass-loss model, while
the third step is not yet investigated in detail. The discovery of an SMBH binary system, together witha simulation of an eccentric evolution of SMBH binaries, supports this formation scenario throughmerging of IMBHs. (See references in [1]).
In Matsubayashi et al.(2004)[7], we pointed out that gravitational waves from IMBHs can be atrigger to prove this process. Later, Fregeau et al.[8] discussed the event rates of IMBH–IMBH binary
observations at advanced LIGO and VIRGO and concluded that we can expect ∼ 10 mergers per year.
This work was followed by Gair et al.[9], including the Einstein Telescope project. Amaro-Seoane &Santamaría [10] also discussed the IMBH–IMBH system, including the pre-merger phase.

2 Black Hole Merger Model
Ring-down Frequency from BHs
The gravitational waveform of binary-star mergers which ends up with a single BH, has three typical
phases: inspiral phase, merging phase, and ring-down phase. The ring-down frequency is simply
estimated by the quasi-normal frequency of BHs, which is determined from the mass and spin of thefinal BH and is estimated to be higher frequency than in its inspiral phase.
Supposing that advanced GW interferometers can detect fqnm above 10 Hz, then BHs less than
1200 M⊙ are within the target if BHs are nonrotating (a = 0), while BHs less than 2500M⊙ are in thedetectable range for highly rotating cases (a = 0.98).
Number of Galaxies in the Universe
In order to model the typical mass of galaxies and its distribution, we apply the halo mass function
given by Vale & Ostriker [11], in which they discuss an empirically based, nonparametric model for
galaxy luminosities with halo/subhalo masses. They apply the Sheth-Tormen mass function for halo
number density. Integrating it by the volume as a function of redshift z (we use the standard cosmologymodel), we can derive the number density of halos (Figure 1(left)). Combining the average number ofgalaxies and the number density of halos, we get the number density of galaxies ngalaxy (M, z), which
we show in Fig.1(right).

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Figure 1. (left) Global massfunctions for halos (halo andsubhalo), nH (M), forz = 0, 0.1, 0.5, 2, 4, 5. nH (M) is inunits of h4 /Mpc3 /M⊙ . M is inunits of h−1 M⊙ , with h = 0.7.
(right) Number density ofgalaxies, ngalaxy (M).

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Number of BHs in a Galaxy
We next estimate the number of BH candidates in a galaxy. Recently, Inutsuka et al. [12] developed ascenario of galactic-scale star formation from a giant molecular cloud. Their model includes both thegrowth of molecular clouds and the destruction of magnetized molecular clouds by radiation. Simulations and steady-state analysis show that the mass density function of molecular clouds, ncl (Mg ),converges at the Schechter-like function, ncl (Mcl ) ∼ Mcl−1.7 exp (−Mcl /Mcut ) where the cutoﬀ mass
Mcut = 106 M⊙ . On the other hand, many N-body simulations report that there is a simple relation
between the mass of the most massive cluster mmax and the total mass of the molecular cloud Mcl ,mmax = 0.20Mcl0.76 .
We therefore combine these results, and we suppose that each molecular cloud forms a single
BH in its core if it is more than 10M⊙ , and we suppose that these BHs become “building blocks"
for forming stellar-sized and intermediate-mass BHs. Many N-body simulations suggest that massiveobjects will accumulate in the center of a galaxy owing to dynamical friction, so that we modeledthat these seed BHs accumulate and merge repeatedly (as we model below), resulting in IMBHs andSMBHs. We do not specify where these mergers occur, but we count our BH mergers after we set up
the initial seeds. We show the number density of BHs in a galaxy, nBH (MBH ) in Fig. 2(left).

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Figure 2. (left) Number density of
BHs per galaxy as a function of BH
mass for diﬀerent total mass ofgalaxies Mgalaxy = 109 M⊙ –1012 M⊙ .
(right) Cumulative distributionfunction of the number of BH
mergers Nmerger (MBH ) as a function of
the redshift z. We express number
with binned one, of which we binned
20 for one order in MBH .

Number of BH Mergers in a Galaxy
We simply assume that BHs formed at cores of clouds will accumulate each other hierarchically, i.e.
the mass and the number of BHs at steps from k to k + 1 can be expressed simply by Mk+1 = 2Mk ,and Nk+1 = Nk /2. The mass of a BH merger, then, obeys the distribution M −1 . On the other hand,
we know empirically that the mass of the central BH of the galaxy, MSMBH , and the total mass of thegalaxy, Mgalaxy , has a relation MSMBH = 2 × 10−4 Mgalaxy (or equal to 10−3 of the bulge mass).

Combining these facts, for a certain galaxy with Mgalaxy , we pick up BHs with total mass MSMBH(equation above), obeying the mass distribution of Fig.2 (left). We suppose that picked-up BHs will
form an SMBH in its series of mergers in the hierarchical model. Together with galaxy distributionfunction ngalaxy (M, z), we are able to count the possible events of BH mergers, Nmerger (MBH , z), in theuniverse, which we show in Fig.2 (right).

3 Signal-to-noise Ratio, Detectable Distance, and Event Rate
The standard procedure for the detection is judged by the optimal S/N ratio, ρ, which is given byρ=2[∫ ∞0h̃( f ) h̃∗ ( f )dfS n( f )

]1/2,

(1)

where h̃( f ) is the Fourier-transformed quantity of the wave, and S n ( f ) is the (one-sided) power spectral density of strain noise of the detector. For KAGRA (bKAGRA), we use a fitted function()√S n ( f ) = 10−26 6.5 × 1010 f −8 + 6 × 106 f −2.3 + 1.5 f ,(2)
where f is measured in Hz, as was used in [13].
Following Flanagan & Hughes [14], we use the energy spectrum formula for the ring-down wave
which results in Eringdown ≈ 81 A2 M 2 fringdown Q, where M is the total mass of the binary, M = m1 + m2 .
Let ϵr (a) ≡ Eringdown /M, which expresses the energy fraction of the emitted gravitational wave
to the total mass. As we cited in the introduction, three GW events suggest us that the 4.0% of thetotal mass is emitted before the merger. The associate numerical simulations show that the ring-downpart emits the energy around 0.6 % of the total mass. If we use A ∼ 0.4, then we recover the ratioϵr (0.67) = 0.58%. The magnitude of this A is also consistent with the quadrupole formula.
The S/N is, then, expressed using the inertial mass µ = m1 m2 /M and the redshift of the source z,()2 ( )28 ϵr (a) (1 + z)M(1 + z)M4µ2
ρ =.
(3)
25 fR S h ( fR /(1 + z))
dL (z)
M
By specifying the BH mass and spin, together with ρ, we can then find the distance dL that satisfies
eq. (3). We call this distance the detectable distance, D(M, a, ρ). We show them in Fig.3(left) for
KAGRA for S/R=10 and 100. Using D(M, a, ρ), we set the upper limit of z for integrating the numberof galaxies, Ngalaxy , and then obtain the number of BH mergers, Nmerger . We show Nmerger in Figures 3(a1) and (b1) for S/R=10 and 30, respectively.
The event rate R, then, is estimated by R[/yr] = Nmerger (z)/V(D/2.26), where the factor 2.26 is foraveraging the distance for all directions. We show them in Figs 3 (a2) and (b2). These figures are for
specifying the BH spin parameter a, but if we assume that a is homogeneously distributed, then theaveraged R is estimated as its peak R ∼ 7.13[/yr] at M ∼ 59.1M⊙ (200–375 [Hz] for a = 0–0.9). The
mergers of the range above R > 1 [/yr] have mass 40M⊙ < M < 150M⊙ . The total number of events
above R > 1 [/yr] is ∼ 211.

4 Summary
Our event rate sounds similar to that of other groups. For example, the LIGO-Virgo group updated
their estimated event rates after the detection of GW150914 as 2–600 Gpc3 yr−1 assuming BH mass
distribution models as flat or power law (∼ M −2.35 ; [15]). Kinugawa et al. [16] estimate as 70–140

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Figure 3. (left) Detectable distance D of the ring-down signal at KAGRA. S/R is set to (a) 10 and (b) 100.
(middle and right) Number of BH mergers within the detectable distances (a1, b1) and event rate R (a2, b2) asa function of BH mass M with S/N ratio ρ = 10 and 30 for KAGRA. Three distributions for each figure are of
a = 0.9, 0.5, and 0.0 (from largest to lowest), respectively.

yr−1 from their Population III model. Inoue et al.[17] estimate as < 60 yr−1 from their BH merger
model inferred from the luminosity function of ultraluminous X-ray sources. However, our modelpredicts BH mergers with M > 100M⊙ , which will be a key to test our model in the future.
This work was supported in part by the Grant-in-Aid for Scientific Research Fund of the JSPS (C)No. 25400277, MEXT KAKENHI Grants No. 17H06357 and No. 17H06358.

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