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Did LIGO detect dark matter?

Simeon Bird ,∗ Ilias Cholis, Julian B. Mu˜noz, Yacine Ali-Haı̈moud, MarcKamionkowski, Ely D. Kovetz, Alvise Raccanelli, and Adam G. Riess 1

1 Department of Physics and Astronomy, Johns Hopkins University,3400 N. Charles St., Baltimore, MD 21218, USA

We consider the possibility that the black-hole (BH) binary detected by LIGO may be a signature

of dark matter. Interestingly enough, there remains a window for masses 10M

M

bh 100M

where primordial black holes (PBHs) may constitute the dark matter. If two BHs in a galactichalo pass sufficiently close, they can radiate enough energy in gravitational waves to become grav-itationally bound. The bound BHs will then rapidly spiral inward due to emission of gravitationalradiation and ultimately merge. Uncertainties in the rate for such events arise from our impreciseknowledge of the phase-space structure of galactic halos on the smallest scales. Still, reasonableestimates span a range that overlaps the 2 − 53 Gpc − 3 yr − 1 rate estimated from GW150914, thusraising the possibility that LIGO has detected PBH dark matter. PBH mergers are likely to bedistributed spatially more like dark matter than luminous matter and have no optical nor neutrinocounterparts. They may be distinguished from mergers of BHs from more traditional astrophysicalsources through the observed mass spectrum, their high ellipticities, or their stochastic gravitationalwave background. Next generation experiments will be invaluable in performing these tests.

The nature of the dark matter (DM) is one of themost longstanding and puzzling questions in physics.Cosmological measurements have now determined withexquisite precision the abundance of DM [1, 2], and weknow from a combination of observations and numericalsimulations quite a bit about its distribution in Galactichalos. Still, the nature of the DM remains a mystery.Given the efficacy with which weakly-interacting mas-sive particles—for many years the favored particle-theoryexplanation—have eluded detection, it may be warrantedto consider other possibilities for DM. Primordial blackholes (PBHs) are one such possibility [3, 4].

Here we consider whether the two ∼30 M black holes

detected by LIGO [5] could plausibly be PBHs. There isa window for PBHs to be DM if the BH mass is in therange 10 M M 100M [6, 7]. Lower masses areexcluded by microlensing surveys [8], and higher masseswould disrupt wide binaries [7, 9, 10]. It has been ar-gued that PBHs in this mass range are excluded by cos-mic microwave background (CMB) constraints [11, 12].However, these constraints require modeling of severalcomplex physical processes, including the accretion of gasonto a moving BH, the conversion of the accreted mass toa luminosity, the self-consistent feedback of the BH radi-ation on the accretion process, and the deposition of theradiated energy as heat in the photon-baryon plasma. A

signicant (and difficult to quantify) uncertainty shouldtherefore be associated with this upper limit, and it seemsworthwhile to examine whether PBHs in this mass rangecould have other observational consequences.

In this Letter , we show that if DM consists of ∼30 M BHs, then the rate for mergers of such PBHs falls withinthe merger rate inferred from GW150914. In any galac-tic halo, there is some small chance that two BHs willundergo a hard scatter and in so doing lose energy to asoft gravitational wave (GW) burst and thereby become

gravitationally bound. This binary will then merge viaemission of GWs in less than a Hubble time. Below werst estimate roughly the rate of such mergers and thenpresent the results of more detailed calculations. We thendiscuss uncertainties in the calculation and some possibleways to distinguish PBHs from BH binaries from moretraditional astrophysical sources.

Consider two PBHs that approach each other withsome impact parameter on a hyperbolic orbit, with rel-ative velocity vpbh . When the PBHs deect each other,there is a time-varying quadrupole moment and thus GWemission. The PBH pair becomes gravitationally boundif the GW emission is greater than the initial kinetic en-ergy. The cross section for this process is [ 13, 14],

σ = 2 3 / 7 π85 π6√ 2

2 / 7

R2svpbh

c

− 18 / 7

= 1 .37 ×10− 14 M 230 v− 18 / 7pbh − 200 pc

2 , (1)

where R s = 2GM pbh /c 2 is the Schwarzchild radius, M 30the PBH mass in units of 30 M , and vpbh − 200 the rela-tive velocity in units of 200 km sec − 1 .

We begin with a rough but simple and illustrative es-timate of the rate per unit volume of such mergers. Sup-pose that all DM matter in the Universe resided in Milky-Way like halos of mass M = M 12 1012 M and uniformmass density ρ = 0 .002 ρ0 .002 M pc− 3 with ρ0 .002 ∼ 1.The rate of mergers per halo, assuming a uniform-densityhalo of volume V = M/ρ , would then be

N (1/ 2)V (ρ/M pbh )2 σv3.10 ×10− 12 M 12 ρ0 .002 v

− 11 / 7pbh − 200 yr

− 1 , (2)

where the relative velocity vpbh − 200 is specied by a char-acteristic halo velocity. The mean cosmic DM mass den-sity is ρdm 3.6 ×1010 M Mpc− 3 , and so the spatial

a r X i v : 1 6 0 3 . 0 0 4 6 4 v 1 [ a s t r o - p h . C O ] 1 M a r 2 0 1 6

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density of halos is n 0.036 M − 112 Mpc− 3 . Thus, the rate

per unit comoving volume in the Universe is,

Γ 1.1 ×10− 4 ρ0 .002 v− 11 / 7pbh − 200 Gpc

− 3 yr− 1 . (3)

The factor M 12 drops out, as it should. The merger rateper unit volume also does not depend on the PBH mass,a consequence of the dependence of the capture cross

section on M 2bh .This rate is small compared with the 2 −53 Gpc− 3 yr− 1estimated by LIGO for a population of ∼30 M −30 M mergers [15], but it is a very conservative estimate. As

Eq. ( 3) indicates, the merger rate is higher in higher-density regions and also in regions of lower DM velocitydispersion. The DM in Milky-Way like halos is knownfrom simulations [16] and analytic models [ 17] to havesubstructure, regions of higher density and lower veloc-ity dispersion. DM halos also have a broad mass spec-trum, extending to very low masses where the densitiescan become far higher, and velocity dispersion far lower,than in the Milky Way. To get a very rough estimate

of the conceivable increase in the PBH merger rate dueto these smaller-scale structures, we can replace ρ andv in Eq. (3) by the values they would have had in theearliest generation of collapsed objects, where the DMdensities would have been largest and velocity disper-sions the smallest. If the primordial power spectrum isnearly scale invariant, then gravitationally bound halosof mass M c ∼ 500 M , for example, will form at red-shift zc 28 − log10 (M c / 500 M ). These objects willhave virial velocities v 0.2 km sec− 1 and densitiesρ 0.24 M pc− 3 [18]. Using these values in Eq. ( 3)increases the merger rate per unit volume to

Γ 1400 Gpc−

3 yr−

1 . (4)

This would be the merger rate if all the DM resided in thesmallest haloes. Clearly, this is not true by the presentday; substructures are at the least partially stripped asthey merge to form larger objects in the hierarchy, and soEq. ( 4) should be viewed as a conservative upper limit.

Having demonstrated that rough estimates contain themerger-rate range 2 −53 Gpc− 3 yr− 1 suggested by LIGO,we now turn to more careful estimates of the PBH mergerrate. As Eq. ( 3) suggests, the merger rate will depend ona density-weighted average, over the entire cosmic DMdistribution, of ρ0 .002 v

− 11 / 7pbh − 200 . To perform this average,

we will (a) assume that DM is distributed within galactichalos with a Navarro-Frenk-White prole [19] with con-centration parameters inferred from simulations; and (b)try several halo mass functions taken from the literaturefor the distribution of halos.

The PBH merger rate Rwithin each halo can be com-puted using

R= 4 π R vir

0r 2

12

ρnfw (r )M pbh

2

σvpbh dr (5)

where ρnfw (r ) = 4 ρs (r/R s )(1 + r/R s )2− 1

is theNavarro-Frenk-White density prole with characteristicradius and density rs and ρs , respectively. Rvir is thevirial radius, at which the NFW prole reaches a value200 times the comoving mean cosmic DM density andis cutoff. Here, M pbh is the PBH mass and vpbh is therelative velocity of two PBHs. The angle brackets de-note an average over the PBH relative velocity distribu-tion in the halo. The merger cross section σ is that inEq. (1). The concentration parameter C is C = Rvir /R s .To determine the prole of each halo, we need to relateC to the halo mass M . We will compare results usingconcentration-mass relations from Ref. [ 20] and Ref. [21],both t from DM N-body simulations.

We now turn to the average of the cross section timesrelative velocity. The one-dimensional velocity dispersionof a halo is dened in terms of the escape velocity atradius Rmax = 2.1626 Rs , the radius of the maximumcircular velocity of the halo; i.e.,

vdm = GM (r < rmax )r max = v

vir√ 2 C C m g(C m )g(C ) , (6)

where g(C ) = ln (1+ C )−C/ (1+ C ), and C m = 2 .1626 =Rmax /R s . We approximate the relative velocity distri-bution of PBHs within a halo as a Maxwell-Boltzmann(MB) distribution with a cutoff at the virial velocity; i.e.,

P (vpbh ) = F 0 exp −v2pbhv2dm −exp −

v2virv2dm

, (7)

where F 0 is chosen so that 4 π

v vir0 P (v)v

2 dv = 1. Thismodel provides a reasonable match to N-body simula-tions, at least for the velocities substantially less thanthan the virial velocity that dominate the merger rate(see, e.g., Ref. [22]). Since the cross-section is indepen-dent of radius, we can integrate the NFW prole to ndthe merger rate in any halo:

R= 85π12√ 2

2 / 7 9G2 M 2vircR3s

1 − 1

(1 + C )3 D(vdm )

g(C )2 ,

(8)

where

D (vdm ) = vvir

0 P (v, vdm )2vc

3 / 7

dv, (9)

comes from Eq. (7).Eq. (1) gives the cross section for two PBHs to form a

binary. However, if the binary is to produce an observ-able GW signal, these two PBHs must orbit and inspiral;a direct collision, lacking an inspiral phase, is unlikelyto be detectable by LIGO. This requirement imposes aminimum impact parameter of roughly the Schwarzchildradius. The fraction of BHs which do not merge is ∼v

2 / 7

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3

1 0

3

1 0

6

1 0

9

1 0

1 2

1 0

1 5

M

v i r

( M

/ h )

- 1 5

- 1 4

- 1 3

- 1 2

- 1 1

- 1 0

L u d l o w

P r a d a

FIG. 1. The PBH merger rate per halo as a function of halo mass. The solid line shows the trend assuming theconcentration-mass relation from Ref. [21], and the dashedline that from Ref. [20]. To guide the eye, the dot-dashed lineshows a constant BH merger rate per unit halo mass.

and reaches a maximum of ∼ 3% for a relative velocityof 2000 km/s. Thus, direct mergers are negligible. Wealso require that once the binary is formed, the time un-til it merges (which can be obtained from Ref. [23]) isless than a Hubble time. Due to the different velocitydispersion between halos of different mass, the charac-teristic time it takes for a binary BH to merge can behours for M vir 1012 M to kyrs for M vir 106 M ,and thus effectively instantaneous. Given the small sizeof the binary, and rapid time to merger, disruption of the binary, once formed, by a third PBH is negligible.

BH binaries can also form through non-dissipative three-body encounters. The rate of these binary captures isnon-negligible in small halos [13, 24], but they generi-cally lead to the formation of wide binaries that will notbe able to harden and merge within a Hubble time. Thisformation mechanism should not affect our LIGO rates.The merger rate is therefore equal to the rate of binaryBH formation, Eq. ( 8).

Fig. 2 shows the contribution to the merger rate,Eq. (8), for two concentration-mass relations. As canbe seen, the dependence on the particular concentration-mass relation is quite small. An increase in halo mass

produces an increased PBH merger rate. However, lessmassive halos have a higher concentration (since they aremore likely to have virialized earlier), so that the mergerrate per unit mass increases signicantly as the halo massis decreased.

To compute the expected LIGO event rate, we con-volve the merger rate R per halo with the mass func-tion dn/dM . Since the redshifts ( z 0.3) detectable byLIGO are relatively low we will neglect redshift evolutionin the halo mass function. The total merger rate per unit

1 0

3

1 0

6

1 0

9

1 0

1 2

1 0

1 5

M

v i r

( M

/ h )

- 8

- 7

- 6

- 5

- 4

- 3

- 2

- 1

0

1

L u d l o w c o n c e n t r a t i o n

P r a d a c o n c e n t r a t i o n

P r e s s - S c h e c h t e r m a s s f u n c t i o n

J e n k i n s m a s s f u n c t i o n

FIG. 2. The total PBH merger rate as a function of halomass. Dashed and dotted lines show different prescriptionsfor the concentration-mass relation and halo mass function.

volume is then,

V = (dn/dM )(M ) R(M ) dM. (10)Given the exponential falloff of dn/dM at high masses,despite the increased merger rate per halo suggested inFig. 1, the precise value of the upper limit of the inte-grand does not affect the nal result.

At the lower limit, discreteness in the DM particlesbecomes important, and the NFW prole is no longer agood description of the halo prole. Furthermore, thesmallest halos will evaporate due to periodic ejection of objects by dynamical relaxation processes. The evapora-tion timescale is [25]

tevap ≈(14 N / ln N ) [Rvir / (C vdm )] , (11)where N is the number of individual BHs in the halo, andwe assumed that the PBH mass is 30 M . For a halo of mass 500 M , the velocity dispersion is 0 .2 km sec− 1 , andthe evaporation timescale is ∼5 Gyr. In practice, duringmatter domination, halos which have already formed willcontinuously grow through mergers or accretion. Evapo-ration will thus be compensated by the addition of newmaterial, and as halos grow new halos will form frommergers of smaller objects. However, during dark-energy

domination at z 0.3, 5 Gyr ago, this process slowsdown. Thus, we will neglect the signal from halos withan evaporation timescale less than 5 Gyr, correspondingto M < 500 M . This is in any case 13 PBHs, and thusclose to the point where the NFW prole is no longervalid.

The halo mass function dn/dM is computed using bothsemi-analytic ts to N-body simulations and with ana-lytic approximations. Computing the merger rate in thesmall halos discussed above requires us to extrapolate

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both the halo mass function and the concentration-massrelation around six orders of magnitude in mass beyondthe smallest halos present in the calibration simulations.Note, however, that the mass functions depend on thehalo mass through the perturbation amplitude σ(Rvir ) atthe virial radius Rvir of a given halo. Due to the scale in-variance of the window functions on small scales, σ(Rvir )varies only by a factor of two between M vir = 10 9 M and103 M . Thus the extrapolation in the mass function isless severe than it looks. We also note that the scale-invariant nature of the initial conditions suggests thatthe shape of the halo mass function should not evolveunduly until it reaches the scale of the PBH mass, orevaporation cutoff.

To quantify the uncertainty induced by the dn/dM extrapolation, we obtained results with two differentmass functions: the classic analytic Press-Schechter cal-culation [ 26] and one calibrated to numerical simula-tions from Tinker et al. [27]. The agreement betweenthe two small-scale behaviors suggests that the small-

scale extrapolation of the numerical mass functions isnot as blind as it might otherwise seem. We also in-clude a third mass function, due to Jenkins [ 28], thatincludes an articial small-scale mass cutoff at a halomass M vir ∼ 106 M . This cutoff is inserted to model,roughly speaking, the mass function that arises if there isno power on scales smaller than those that can somehowcurrently be probed observationally. We include it to pro-vide a very conservative lower limit to the merger rate if,for some reason, small-scale power were suppressed. Wedo not, however, consider it likely that this mass functionaccurately represents the distribution of halo masses inour Universe.

Fig. 2 shows the merger rate per logarithmic intervalin halo mass. In all cases, halos with M vir 109 M dominate the signal, due to the increase in concentra-tion and decrease in velocity dispersion with smaller halomasses. The Tinker mass function, which asymptotes toa constant number density for small masses, produces themost mergers. Press-Schechter has ∼ 50% fewer eventsin small halos, while the Jenkins mass function resultsin merger rates nearly four orders of magnitude smaller(and in rough agreement with Eq. ( 3)).

We integrate the curves in Fig. 2 to compute the to-tal merger rate

V . All mass functions give a similar re-

sult, ∼(4 ±1) ×10−

3 Gpc−

3 yr−

1 , from halos of masses 109 M , which represents for the Tinker and Press-

Schechter mass function a small fraction of the events.When we include all halos with M vir > 500M , thenumber of events increases dramatically, and dependsstrongly on the lower cutoff mass M c for the halo mass.Both the Press-Schechter and Tinker mass functions arefor small halos linear in the integrated perturbation am-plitude ∝ 1/σ (Rvir ) at the virial radius Rvir of the col-lapsing halo. In small halos, 1 /σ (Rvir ) is roughly con-

stant. Thus for a mass function MF( σ), we have

(dn/dM )∼(C log σ/dM ) [MF( σ)/M vir ]∼M − 2vir . (12)

The concentration is also a function of 1 /σ (Rvir ) and ittoo becomes roughly constant for small masses. Assum-ing a constant concentration, the merger rate per haloscales as R ∼ M 10 / 21 . Thus, Eq. ( 10) suggests thatV ∼M

− 11 / 21c . This compares well to numerical differen-tiation of Fig. 2, which yields V ∼M − 0 .51c .The integrated merger rate is thusV = 5 f (M c / 500 M )− 11 / 21 Gpc − 3 yr− 1 , (13)

with f 1 for the Press-Schechter mass function, andf 1.4 for the Tinker mass function (the Jenkins massfunction results in an event rate V 0.05 Gpc− 3 yr− 1 ,independent of M c 106 M ).A variety of astrophysical processes may alter the mass

function in some halos, especially within the dwarf galaxyrange, 10 9 − 1010 M . However, halos with M vir 109 M are too small to form stars against the ther-mal pressure of the ionized intergalactic medium [29] andare thus unlikely to be affected by these astrophysicalprocesses. Inclusion of galactic substructure, which ourcalculation neglects, should boost the results. However,since the event rate is dominated by the smallest halos,which should have little in the way of substructure, weexpect this to make negligible difference to our nal re-sult. There is also the issue of the NFW density proleassumed. The results are fairly insensitive to the detaileddensity prole as long as the slope of the density proleas r → 0 is close to r − 1 . If, however, the inner proleincreases more like ρ ∝ r − 1 .5 , as some simulations sug-gest [16], then the merger rates could be enhanced byas much as an order of magnitude. Our assumption of an isotropic MB-like velocity distribution in the halo mayalso underestimate the correct answer, as any other veloc-ity distribution would have lower entropy and thus largeraveraged v− 11 / 7 . Finally, our estimates assume a nearlyscale-invariant spectrum of primordial perturbations allthe way down to the ∼500 M mass scale. However, newearly-Universe physics that gives rise to PBHs is likely toinvolve an enhancement of power on small scales. At thebare minimum, the discreteness of PBH DM will pro-vide some Poisson enhancement of power on ∼ 500M scales. More small-scale power would then lead to ear-lier collapse and thus denser structures and probably anenhancement, beyond Eq. ( 13), of the event rate.

The recent LIGO detection of two merging ∼ 30M black holes suggests a 90% C.L. event rate [15] of 2−53 Gpc− 3 yr− 1 if all mergers have the masses, nal spin,and emitted energy of GW150914. It is interesting that— although there are theoretical uncertainties—our best es-timates of the merger rate for 30M PBHs, obtained with canonical models for the DM distribution, fall in the LIGO window .

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The possibility that LIGO has seen DM thus cannot beimmediately excluded. Even if the predicted merger ratesturn out, with more precise treatments of the small-scalegalactic phase-space distribution, to be smaller, conser-vative lower estimates of the merger rate for PBH DMsuggests that the LIGO/VIRGO network should see aconsiderable number of PBH mergers over its lifetime.

We have assumed a population of PBHs with the samemass. The basic results obtained here should, how-ever, remain unaltered if there is some small spread of PBH masses, as expected from PBH-formation scenar-ios, around the nominal value of 30 M .

PBH mergers may also be interesting forLIGO/VIRGO even if PBHs make up only a frac-tion f pbh of the DM, as may be implied by Refs. [11, 12]or the limits in Ref. [ 6]. In this case, the number densityof black holes will be reduced by f pbh . The cutoff masswill increase as M c ∼ f − 1pbh if we continue to require> 10 PBHs in each halo to avoid halo evaporation.The overall event rate will be V ∼ 6f

53 / 21pbh Gpc

− 3 yr− 1 .Advanced LIGO will reach design sensitivity in 2019[30, 31]. It will then probe z < 0.75, an increase involume to ≈500 Gpc3 . Thus over the six planned yearsof aLIGO operation, while we should expect to detect∼ 2 ×104 events with f pbh = 1, we will expect at leastone event if f pbh > 0.04.

Distinguishing whether any individual GW event, oreven some population of events, are from PBH DM ormore traditional astrophysical sources will be daunting.Still, there are some prospects. Most apparently, PBHmergers will be distributed more like small-scale DM ha-los and are thus less likely to be found in or near luminousgalaxies than BH mergers from more traditional astro-

physical sources. Moreover, PBH mergers are expectedto have no electromagnetic/neutrino counterparts what-soever. A DM component could conceivably show up inthe BH mass spectrum as an excess of events with BHmasses near 30 M over a more broadly distributed massspectrum from astrophysical sources.

Since the binary is formed on a very elongated orbit,the GW waveforms will tend to indicate some ellipticity,exhibited by higher frequency harmonics [ 23]. We haveveried that the ellipticities become unobservably smallby the time the inspiral enters the LIGO band, but theymay be detectable in future experiments [32].

Another potential source of information is the stochas-

tic GW background. Models for the stochastic back-ground due to BH mergers usually entail a mass distribu-tion that extends to smaller BH masses and a redshift dis-tribution that is somehow related to the star-formationhistory. Given microlensing limits, the PBH mass func-tion cannot extend much below 30 M . Moreover, thePBH merger rate per unit comoving volume is likelyhigher for PBHs than for traditional BHs at high red-shifts. Together, these suggest a stochastic backgroundfor PBHs that has more weight at low frequencies and less

at higher ones than that from traditional BH sources.The results of this work provide additional motivation

for more sensitive next-generation GW experiments suchas the Einstein Telescope [33], DECIGO [34] and BBO[35], which will continuously extend the aLIGO frequencyrange downwards. These may enable the tests describedabove for excesses in the BH mass spectrum, high ellip-ticity and low-frequency stochastic background that arerequired to determine if LIGO has detected dark matter.

We thank Liang Dai for useful discussions. SBwas supported by NASA through Einstein Postdoc-toral Fellowship Award Number PF5-160133. This workwas supported by NSF Grant No. 0244990, NASANNX15AB18G, the John Templeton Foundation, and theSimons Foundation.

∗ E-mail: sbird4@jhu.edu[1] G. Hinshaw et al. [WMAP Collaboration], Astrophys. J.

Suppl. 208 , 19 (2013) [arXiv:1212.5226 [astro-ph.CO]].[2] P. A. R. Ade et al. [Planck Collaboration],

arXiv:1502.01589 [astro-ph.CO].[3] B. J. Carr and S. W. Hawking, Mon. Not. Roy. Astron.

Soc. 168 , 399 (1974).[4] B. J. Carr, Astrophys. J. 201 , 1 (1975).[5] B. P. Abbott et al. [LIGO Scientic and Virgo Col-

laborations], Phys. Rev. Lett. 116 , 061102 (2016)[arXiv:1602.03837 [gr-qc]].

[6] B. J. Carr, K. Kohri, Y. Sendouda and J. Yokoyama,Phys. Rev. D 81 , 104019 (2010) [arXiv:0912.5297 ].

[7] M. A. Monroy-Rodrguez and C. Allen, Astrophys. J. 790 ,159 (2014) [arXiv:1406.5169 [astro-ph.GA]].

[8] R. A. Allsman et al. [Macho Collaboration], Astrophys.

J. 550 , L169 (2001) [astro-ph/0011506] .[9] J. Yoo, J. Chaname and A. Gould, Astrophys. J. 601 ,311 (2004) [astro-ph/0307437] .

[10] D. P. Quinn et al. , Mon. Not. Roy. Astron. Soc. 396 , 11(2009) [arXiv:0903.1644 [astro-ph.GA]].

[11] M. Ricotti, J. P. Ostriker and K. J. Mack, Astrophys. J.680 , 829 (2008) [arXiv:0709.0524 [astro-ph]].

[12] M. Ricotti, Astrophys. J. 662 , 53 (2007)[arXiv:0706.0864 [astro-ph]].

[13] G. D. Quinlan and S. L. Shapiro, Astrophys. J. 343 , 725(1989).

[14] H. Mouri and Y. Taniguchi, Astrophys. J. 566 , L17(2002) [astro-ph/0201102 ].

[15] B. P. Abbott et al. [LIGO Scientic and Virgo Collabo-rations], arXiv:1602.03842 [astro-ph.HE].

[16] B. Moore et al. , Astrophys. J. 524 , L19 (1999) [astro-ph/9907411] .

[17] M. Kamionkowski and S. M. Koushiappas, Phys. Rev. D77 , 103509 (2008) [arXiv:0801.3269 [astro-ph]].

[18] M. Kamionkowski, S. M. Koushiappas and M. Kuhlen,Phys. Rev. D 81 , 043532 (2010) [arXiv:1001.3144 ].

[19] J. F. Navarro, C. S. Frenk and S. D. M. White, Astro-phys. J. 462 , 563 (1996) [astro-ph/9508025] .

[20] F. Prada et al. , Mon. Not. Roy. Astron. Soc. 428 , 3018(2012) [arXiv:1104.5130 [astro-ph.CO]].

[21] A. D. Ludlow et al. , arXiv:1601.02624 [astro-ph.CO].

mailto:E-mail:%20sbird4@jhu.eduhttp://arxiv.org/abs/1212.5226http://arxiv.org/abs/1502.01589http://arxiv.org/abs/1602.03837http://arxiv.org/abs/0912.5297http://arxiv.org/abs/1406.5169http://arxiv.org/abs/astro-ph/0011506http://arxiv.org/abs/astro-ph/0307437http://arxiv.org/abs/0903.1644http://arxiv.org/abs/0709.0524http://arxiv.org/abs/0706.0864http://arxiv.org/abs/astro-ph/0201102http://arxiv.org/abs/1602.03842http://arxiv.org/abs/astro-ph/9907411http://arxiv.org/abs/astro-ph/9907411http://arxiv.org/abs/0801.3269http://arxiv.org/abs/1001.3144http://arxiv.org/abs/astro-ph/9508025http://arxiv.org/abs/1104.5130http://arxiv.org/abs/1601.02624http://arxiv.org/abs/1601.02624http://arxiv.org/abs/1104.5130http://arxiv.org/abs/astro-ph/9508025http://arxiv.org/abs/1001.3144http://arxiv.org/abs/0801.3269http://arxiv.org/abs/astro-ph/9907411http://arxiv.org/abs/astro-ph/9907411http://arxiv.org/abs/1602.03842http://arxiv.org/abs/astro-ph/0201102http://arxiv.org/abs/0706.0864http://arxiv.org/abs/0709.0524http://arxiv.org/abs/0903.1644http://arxiv.org/abs/astro-ph/0307437http://arxiv.org/abs/astro-ph/0011506http://arxiv.org/abs/1406.5169http://arxiv.org/abs/0912.5297http://arxiv.org/abs/1602.03837http://arxiv.org/abs/1502.01589http://arxiv.org/abs/1212.5226mailto:E-mail:%20sbird4@jhu.edu

8/18/2019 1603.00464v1

6/6

6

[22] Y. Y. Mao et al. , Astrophys. J. 764 , 35 (2013)[arXiv:1210.2721 [astro-ph.CO]].

[23] R. M. O’Leary, B. Kocsis and A. Loeb, Mon. Not. Roy.Astron. Soc. 395 , 2127 (2009) [arXiv:0807.2638 ].

[24] M. H. Lee, Astrophys. J. 418 , 147 (1993).[25] J. Binney and S. Tremaine, Galactic Dynamics (Prince-

ton University Press, Princeton, 1987), p. 747.[26] W. H. Press and P. Schechter, Astrophys. J. 187 , 425

(1974).[27] J. L. Tinker et al. , Astrophys. J. 688 , 709 (2008)

[arXiv:0803.2706 [astro-ph]].[28] A. Jenkins et al. , Mon. Not. Roy. Astron. Soc. 321 , 372

(2001) [astro-ph/0005260 ].

[29] G. Efstathiou, Mon. Not. Roy. Astron. Soc. 256 , 43P(1992).

[30] J. Aasi et al. [LIGO Scientic and VIRGO Collabora-tions], Living Rev. Rel. 19 , 1 (2016) doi:10.1007/lrr-2016-1 [arXiv:1304.0670 [gr-qc]].

[31] B. P. Abbott et al. [LIGO Scientic and Virgo Collabo-rations], arXiv:1602.03844 [gr-qc].

[32] I. Cholis et al. , in preparation (2016)[33] Einstein Telescope, design at: http://www.et-gw.eu/[34] N. Seto, S. Kawamura and T. Nakamura, Phys. Rev.

Lett. 87 , 221103 (2001) [astro-ph/0108011] .[35] E. S. Phinney et al. , Big Bang Observer Mission Concept

Study (NASA), (2003)

http://arxiv.org/abs/1210.2721http://arxiv.org/abs/0807.2638http://arxiv.org/abs/0803.2706http://arxiv.org/abs/astro-ph/0005260http://arxiv.org/abs/1304.0670http://arxiv.org/abs/1602.03844http://www.et-gw.eu/http://arxiv.org/abs/astro-ph/0108011http://arxiv.org/abs/astro-ph/0108011http://www.et-gw.eu/http://arxiv.org/abs/1602.03844http://arxiv.org/abs/1304.0670http://arxiv.org/abs/astro-ph/0005260http://arxiv.org/abs/0803.2706http://arxiv.org/abs/0807.2638http://arxiv.org/abs/1210.2721