Revealing the formation of stellar-mass black hole binaries: The need for deci-Hertz gravitational wave observatories
Xian Chen Pau Amaro-Seoane





xian.chen@pku.edu.cn
Astronomy Department, School of Physics, Peking University, 100871 Beijing, China
Kavli Institute for Astronomy and Astrophysics at Peking University, 100871 Beijing, China

pau@ice.cat
Institut de Ciències de l'Espai (CSIC-IEEC) at Campus UAB, Carrer de Can Magrans s/n 08193 Barcelona, Spain

Kavli Institute for Astronomy and Astrophysics at Peking University, 100871 Beijing, China

Institute of Applied Mathematics, Academy of Mathematics and Systems Science, CAS, Beijing 100190, China

Zentrum für Astronomie und Astrophysik, TU Berlin, Hardenbergstraße 36, 10623 Berlin, Germany




             


The formation of compact stellar-mass binaries is a difficult, but interesting
problem in astrophysics.  There are two main formation channels: In the field
via binary star evolution, or in dense stellar systems via dynamical
interactions.  The Laser Interferometer Gravitational-Wave Observatory (LIGO)
has detected black hole binaries (BHBs) via their gravitational radiation.
These detections provide us with information about the physical parameters of
the system. It has been claimed that when the Laser Interferometer Space
Antenna (LISA) is operating, the joint observation of these binaries with LIGO
will allow us to derive the channels that lead to their formation.  However, we
show that  for BHBs in dense stellar systems dynamical interactions could
lead to high eccentricities such that a fraction of the relativistic mergers
are not audible to LISA.  A non-detection by LISA puts a lower limit of about
0.005 on the eccentricity of a BHB entering the LIGO band. On the other
hand, a deci-Hertz observatory, like DECIGO or Tian Qin, would significantly
enhance the chances of a joint detection, and shed light on the formation
channels of these binaries. 



Revealing the formation of stellar-mass black hole binaries:
The need
for deci-Hertz gravitational wave observatories
    Pau Amaro-Seoane
    December 30, 2023
=====================================================================================================================



Introduction.–The first LIGO events, GW150914 and GW151226
<cit.>, are consistent with mergers of General-Relativity
black holes (BHs). Data analysis reveal that the orbits
started at a semi-major axis of a∼10 Schwarzschild radii (R_S) with an
eccentricity of e<0.1. The BH masses are about M_1≃36 and
M_2≃29 M_⊙ for GW150914 and M_1≃14 and M_2≃7.5 M_⊙
for GW151226.  The detections can be used to infer new, more realistic event
rates, of about 9-240  Gpc^-3 yr^-1 <cit.>. This rate
agrees with two formation channels: (i) evolution of a binary of two stars in
the field of the host galaxy, where stellar densities are very low (e.g
<cit.>) or (ii) via exchange of energy and angular momentum in
dense stellar systems, where the densities are high enough for stellar close
encounters to be common (e.g. <cit.>).

LIGO and other ground-based gravitational wave (GW) observatories, such as
Virgo, are, however, blind with regarding the formation channels of BH binaries
(BHBs). Both channels predict populations in the 10-10^3  Hz detector
band with similar features, i.e. masses larger than the nominal
10 M_⊙, a mass ratio (q≡ M_2/M_1) of about 1, low spin, and
nearly circular orbits <cit.>.

It has been suggested that a joint detection with a space-borne observatory
such as LISA
<cit.> could
allow us to study different moments in the evolution of BHBs on their way to
coalescence:  LISA can detect BHBs when the BHs are still 10^2-10^3 R_S
apart, years to weeks before they enter the LIGO/Virgo band
<cit.>.  At such
a separation, the orbital eccentricity bears the imprint of the formation
channel because (i) BHBs in dense stellar systems form on systematically more
eccentric orbits and (ii) the GW radiation at this stage is too weak to
circularize the orbits <cit.>.
Therefore, circular binaries typically form in the field, while eccentric ones
through the dynamical channel.  Recent studies further predict that those BHBs
with an eccentricity of e>0.01 in the LISA band preferentially originate from
the dynamical channel
<cit.>.

In this letter we prove that eccentric BHBs originating in dense stellar
environments have a large chance to elude the LISA band.

Inaudible black hole binaries–Non-circular BHBs have two distinct
properties. (i) Eccentricity damps the characteristic amplitude (h_c) of each
GW harmonic, as compared to a circular BHB. In Figure <ref> we
depict two sources similar to GW150914 but originating from two distinct
channels, i.e. with two different initial eccentricities. In the
low-eccentricity case, the n=2 harmonic predominates and it is strong enough
to be jointly detected by LISA and LIGO/Virgo.  In the (very) eccentric case,
however, the amplitudes of the harmonics are orders of magnitude below the
noise level of LISA, so that a joint detection is ruled out. When the
eccentricity has been significantly damped, about one hour before the merger,
the dominant harmonic starts to converge to the n=2 one, and later, upon
entering the LIGO band, becomes indistinguishable from that in the circular
case.
Therefore, the imprint about the formation channel is lost.



(ii) Increasing the eccentricity shifts the peak of the relative power of the
GW harmonics towards higher frequencies (see Fig. 3 of <cit.>).
Hence, more eccentric orbits emit their maximum power at frequencies farther
away from LISA. More precisely, when e=0, all the GW power is
radiated through the n=2 harmonic, so that the GWs have a single
frequency of 2/P, where P=2π(GM_12/a^3)^-1/2 is the orbital period
and M_12=M_1+M_2.  On the other hand, when e≃1, the
n=2.16(1-e)^-3/2 harmonic becomes predominant <cit.>, so most GW
power is radiated at a frequency of f_ peak=2.16(1-e)^-3/2P^-1.



In Figure <ref> we display the a-(1-e) plane for a BHB.  The
boundaries of the stripes have been estimated by looking at the minimum and
maximum frequencies audible by the detectors, f_1 and f_2, and letting
f_1< f_ peak< f_2, with f_ peak defined before.  If a BHB is
evolving only due to GW emission, it will evolve parallel to the green lines.
These  track are parallel to the stripes because as long as e≃1, the
pericenter distance, r_p=a (1-e), is almost constant during the evolution
<cit.>, and a constant r_p corresponds to a constant f_ peak.
Because of this parallelism, a BHB cannot evolve into the band of a GW detector
if it initially lies below the detector stripe.

Hence, we can see that some binaries will fully miss the LISA/TJ range. A good
example is the eccentric BHB we chose for Figure <ref>. A
detector operating at higher frequencies, such as TQ or DECIGO, can however
cover the relevant part of the phase-space, so that a joint search is possible.
These detectors could alert LIGO/Virgo decades to hours before an event is
triggered, as one can read from the isochrones of Figure <ref>.

Dense stellar environments.–BHBs such as the one we have used for our
last example completely miss the LISA/TJ band. Eccentric binaries typically
originate from dense stellar systems such as globular clusters (GCs) and
nuclear star clusters (NSC), as shown by a number of authors in a number of
publications
<cit.>.
In these systems, BHs diffuse towards the center via a process called mass
segregation <cit.>.
To model it, we adopt a Plummer model <cit.>, and we assume that the
mean stellar density is ρ_*=5×10^5 M_⊙  pc^-3 and the
one-dimensional velocity dispersion is σ_*=15  km s^-1. These
values correspond to a typical GC with a final mass of M_
GC≈10^5 M_⊙ and a half-mass radius of R_h≈0.5 pc.  We note,
however, that the main conclusions derived in this work do not significantly
change for a NSC.

The two driving and competing mechanisms in the evolution of any BHB in the
center of the cluster are (i) interaction with other stars, “interlopers”,
which come in at a rate of Γ∼2π Gρ_*a(M_12/M_*)/σ_*, with
M_*=10 M_⊙ the mean mass of the interlopers because the cluster has gone
through mass segregation, and (ii) gravitational radiation, which shrinks the
orbital semi-major axis at a rate of


    ȧ_ gw=-8 c R_S^3q (1+q)/5a^3(1-e^2)^7/2(1+73/24e^2+37/96e^4),


<cit.>. We can readily separate the phase-space in two distinct
regimes according to these two competing processes by equating their associated
timescales: t_ int:=1/Γ and t_
gw:=(1/4) |a/ȧ_ gw|, which defines the threshold shown
as the thick, black line in Figure <ref>. The reason for the 1/4 factor
is given in <cit.>.  Below the curve, BHBs will evolve due to GW
emission. Above it, close encounters with interlopers are the main driving
mechanism, so that BHBs can be scattered in
both directions in angular momentum in a random-walk fashion. The
scattering in energy is less significant but also present (see
<cit.> and discussion in <cit.>).



Possible ways of forming relativistic BHBs.– Different mechanisms have
been proposed in the literature to form a BHB which eventually might end up
emitting detectable GWs.

(1) Primordial binaries: In stellar dynamics this term refers to binaries
already present in the cluster which form via stellar evolution.  Population
synthesis models predict that these binaries populate the area of phase-space
displayed as the grey thick-dashed box of Figure <ref> (see e.g.
<cit.>). We note that only a small fraction of them are in the
LISA/TJ band.

(2) Dynamics: (2.1) Close encounters of multiple single, i.e. initially not
bound, objects also form BHBs (see e.g.
<cit.>). Their formation
follows a thermal distribution in e (e.g. <cit.>), like
primordial binaries, but the distribution of a is better constrained: When
the binding energy of the binary, E_b=GM_1M_2/(2a) becomes smaller than the
mean kinetic energy of the interlopers, E_*=3M_*σ_*^2/2, the binary
ionizes <cit.>. The threshold condition E_b=E_* can be expressed in
terms of a “hard radius”, a_h=GM_1M_2/(3M_*σ_*^2).  These “hard”
binaries heat up the system, meaning that they deliver energy to the rest of
the stars interacting with them: Binaries with a<a_h impart on average an energy
of Δ E≃ kGμ M_*/a to each interloper, where μ is the reduced
mass of the binary and k is about 0.4 when M_1≃ M_2≃ M_*
<cit.>.  The interloper hence is re-ejected into the stellar system
with a higher velocity because of the extra energy,
v∼(3σ_*^2+2kGμ/a)^1/2, and the center-of-mass of the
BHB recoils at a velocity of v_b∼ M_*v/(M_1+M_2).  Occasionally, the BHB
will leave the system if this velocity exceeds the escape velocity of the GC,
v_ esc=√(2.6GM_ GC/R_h) <cit.>. The threshold for
this to happen is defined by the condition v_b=v_ esc, i.e. the binary
must have a semi-major axis smaller than the “ejection radius”, a_ ej.
Therefore, all of these BHBs are confined in a_h<a<a_ ej of
Figure <ref>. Because of their thermal distribution, we have that
95% of them have e<0.975. Therefore, they populate an even smaller area
than those primordial binaries.

(2.2) Binary-single interactions: Initially we have a hard BHB which interacts
with a single object in a chaotic way. During the interaction the
interloper might excite the eccentricity of the inner binary to such high
values that the binary is on an almost head-on-collision orbit, to soon merge
and emit a detectable burst of GWs <cit.>. This
happens only if t_ gw is shorter than the period of the captured
interloper P_ int. The event rate for BHBs has not been calculated for this
scenario but earlier calculations for neutron-star binaries find it to be
1  Gpc^-3 yr^-1 <cit.>.  We derive now the
eccentricities of these BHBs: Suppose the
semi-major axis of a BHB changes from a (with, of course, a_
ej< a< a_h) to a', and e to e' during the three-body interaction, and
the final orbit of the interloper around the center-of-mass of the BHB has a
semi-major axis of a_ int.  Energy conservation results in the following
relations,  a'>a and a_ int≃ 2a/(1-a/a') (see <cit.>),
where we neglect the initial energy of the interloper because the BHB is
assumed to be hard.  Then using a conservative criterion for a successful
inspiral, t_ gw(a', e')=P_ int(a_ int), we derive e' for the BHB,
which allows us to confine the range of eccentricities as the dashed, blue
curve of Figure <ref>.

(2.3) Hierarchical triple: This is similar to the previous configuration, but
now we only consider 1<a'/a<1.5, because this requires that a_ int >
6 a, in which case the  configuration is stable <cit.>.  This
leads to a secular evolution of the orbital eccentricity of the inner BHB which
is known as the Lidov-Kozai resonance (see <cit.> and also
<cit.>).  The inner BHB will
decouple via GW emission and merge at a critical eccentricity, and the merger
rate has been estimated to be 0.3-6  Gpc^-3 yr^-1
<cit.>.  We follow the scheme of
<cit.> of isolated hierarchical triples but impose four additional
requirements which are fundamental for a realistic estimation of the threshold
eccentricity in our work: (a) The BHB has a_ ej< a< a_h.  (b) The third body
orbiting the BHB has a mass of M_ int=10 M_⊙ because of mass
segregation, and an eccentricity of e_ int=2/3, which corresponds to the
mean of a thermal distribution <cit.>.  (c) The outer binary, i.e.
the third object and the inner BHB, is also hard, so that a_
int<GM_12/(3σ_*^2). (d) The pericenter distance of the outer binary,
a_ int(1- e_ int) should meet the criterion for a stable triple
(Eq. 90 in Ref.  <cit.>). These conditions delimit the range of
eccentricities as shown by the dashed, orange lines in Figure <ref>.

(3) Gravitational braking: There is a small probability that two single BHs
come to such a close distance that GW radiation dissipates a significant amount
of the orbital energy, leaving the two BHs gravitationally bound
<cit.>.  For GCs, and using
optimistic assumptions, these binaries contribute an event rate of
0.06-20  Gpc^-3 yr^-1 in the LIGO band <cit.>,
while in NSCs it has been estimated to range between 0.005-0.02 
Gpc^-3 yr^-1 <cit.>. The boundaries in Figure <ref> for
BHBs formed via this mechanism can be calculated using the formulae of
<cit.>.  For that, we choose an initial relative velocity v in the
range σ_*<v<3σ_* and an initial impact parameter b in the range
0.3b_ max<b<0.99b_ max to account for the majority of the
encounters, because the encounter probability is proportional to b^2, and
b_ max is the maximum impact parameter that leads to a bound binary.
The first LIGO detections, had they been originated via this mechanism, should
originate from the red area above the green line.




Discussions and conclusions.–A joint detection of BHBs with LIGO/Virgo
and LISA/TJ would be desirable because of the science payback. In this paper we
show that the actual number of BHBs to be coincidentally detected is very
uncertain.  As Figure <ref> shows, LISA/TJ is already deaf to
mildly eccentric BHBs: For example, a BHB at milli-Hertz orbital frequencies
starting at a ∼ 10^-3 AU and 0.7≲ e≲0.9 will also be
missed by LISA/TJ, but later be detectable by LIGO/Virgo.

BHBs can form via the five mechanisms which we discussed in the list of
possible formations. This allows us to pinpoint the regions in phase-space
which produce BHBs that eventually will merge via gravitational radiation. The
total area of these five regions is a small subset of phase-space.  It is an
error to assume that all binaries born in this subset are jointly detectable by
LIGO/Virgo and LISA/TJ.

Only a subset of that subset of phase-space will lead to successful joint
detections. This sub-subset depends on the masses of the BHBs.  We can see this
in Figures <ref> and <ref>. While in the first figure the
hierarchical triple gets into the LISA/TJ band, it does not in the second one.

On the other hand, up to 95% of primordial and dynamical binaries (1 and 2.1
in the list of possible formations) are produced in the box delimited by grey
dashed lines. In that box, and in principle, the BHBs can lead to sources
jointly detectable by LIGO/Virgo and LISA/TJ. However, exceptions might occur
if a scatter results in a BHB jumping towards high eccentricities. This
probability has not been fully addressed. It requires dedicated numerical
scattering experiments with relativistic corrections (e.g.  <cit.>), as
well as a proper star-cluster model to screen out BHBs that can decouple from
the stellar dynamics (e.g. our model as presented in Figures <ref> and
<ref>).

We have shown that mergers in GCs produced by the mechanisms (2.2), (2.3), and
(3) are inaudible to LISA.  The event rates corresponding to these mergers have
been largely discussed in the literature, but are uncertain, due to
questionable parameters, such as the cosmic density of GCs and the number of
BHs in them.  Nevertheless, it has been estimated that the rate could be as
large as 20  Gpc^-3 yr^-1 <cit.>, while the current LIGO
detections infer a total event rate of 9-240  Gpc^-3 yr^-1.
Moreover, these mergers could also originate in NSCs,
<cit.>,
and the event rates there are higher, up to 10^2  Gpc^-3 yr^-1
<cit.>. 

Therefore, future multi-band GW astronomy should prepare for LIGO/Virgo BHBs
that do not have LISA/TJ counterparts. A non-detection by LISA/TJ is also
useful in constraining astrophysics: It puts a lower limit on the
eccentricities of the LIGO/Virgo sources, which according to
Figures <ref> and <ref> is about 0.005. 

A deci-Hz detector, by covering the gap in frequencies between LISA/TJ and
LIGO/Virgo,  would drastically enhance the number of jointly detectable
binaries.

Acknowledgement.–This work is supported partly by the Strategic Priority
Research Program “Multi-wavelength gravitational wave universe” of the
Chinese Academy of Sciences (No. XDB23040100) and by the CAS President's
International Fellowship Initiative. PAS acknowledges support from the
Ramón y Cajal Programme of the Ministry of Economy, Industry and
Competitiveness of Spain.  We thank Bence Kocsis and Fukun Liu for many
fruitful discussions, and Eric Peng for a thorough reading of our manuscript. 









74
fxundefined [1]
 ifx#1

fnum [1]
 #1firstoftwo
 secondoftwo
 
fx [1]
 #1firstoftwo
 secondoftwo
 





noop [0]secondoftwo

ref[1]@startlink#1@href
href[1]#1@endlink
anitize@url [0]`
12`$12`&12`#12`1̂2`_12`%12
startlink[1]
endlink[0]

rl [1]href #1












@bib@innerbibempty

[Abbott et al.(2016a)Abbott, Abbott, Abbott, Abernathy,
  Acernese, Ackley, Adams, Adams, Addesso, Adhikari, and et al.]ligo16a
  author author B. P. Abbott, author R. Abbott,
  author T. D. Abbott, author M. R. Abernathy, author F. Acernese, author
  K. Ackley, author
  C. Adams, author T. Adams, author P. Addesso, author R. X.  Adhikari,  and author et al., 10.1103/PhysRevLett.116.061102 journal journal  volume 116,  eid 061102 (year 2016a), http://arxiv.org/abs/1602.03837 arXiv:1602.03837 [gr-qc] NoStop
[Abbott et al.(2016b)Abbott, Abbott, Abbott, Abernathy,
  Acernese, Ackley, Adams, Adams, Addesso, Adhikari, and et al.]ligo16b
  author author B. P. Abbott, author R. Abbott,
  author T. D. Abbott, author M. R. Abernathy, author F. Acernese, author
  K. Ackley, author
  C. Adams, author T. Adams, author P. Addesso, author R. X.  Adhikari,  and author et al., 10.1103/PhysRevLett.116.241103 journal journal  volume 116,  eid 241103 (year 2016b), http://arxiv.org/abs/1606.04855 arXiv:1606.04855 [gr-qc] NoStop
[Abbott et al.(2016c)Abbott, Abbott, Abbott, Abernathy,
  Acernese, Ackley, Adams, Adams, Addesso, Adhikari, and et al.]ligo16rate
  author author B. P. Abbott, author R. Abbott,
  author T. D. Abbott, author M. R. Abernathy, author F. Acernese, author
  K. Ackley, author
  C. Adams, author T. Adams, author P. Addesso, author R. X.  Adhikari,  and author et al., 10.1103/PhysRevX.6.041015 journal
  journal Physical Review X volume 6,  eid 041015 (year 2016c), http://arxiv.org/abs/1606.04856 arXiv:1606.04856 [gr-qc] NoStop
[Belczynski et al.(2016)Belczynski, Holz, Bulik, and  O'Shaughnessy]belczynski16
  author author K. Belczynski, author D. E. Holz, author T. Bulik,   and author R. O'Shaughnessy,  10.1038/nature18322 journal journal  volume 534, pages 512
  (year 2016), http://arxiv.org/abs/1602.04531
  arXiv:1602.04531 [astro-ph.HE] NoStop
[Rodriguez et al.(2016)Rodriguez, Chatterjee, and Rasio]rodriguez16
  author author C. L. Rodriguez, author S. Chatterjee,  and author F. A.  Rasio, 10.1103/PhysRevD.93.084029
  journal journal  volume 93, eid 084029 (year 2016), http://arxiv.org/abs/1602.02444 arXiv:1602.02444 [astro-ph.HE]
  NoStop
[Abbott et al.(2016d)Abbott, Abbott, Abbott, Abernathy,
  Acernese, Ackley, Adams, Adams, Addesso, Adhikari, and et al.]ligo16astro
  author author B. P. Abbott, author R. Abbott,
  author T. D. Abbott, author M. R. Abernathy, author F. Acernese, author
  K. Ackley, author
  C. Adams, author T. Adams, author P. Addesso, author R. X.  Adhikari,  and author et al., 10.3847/2041-8205/818/2/L22 journal journal  volume 818,  eid L22 (year 2016d), http://arxiv.org/abs/1602.03846 arXiv:1602.03846 [astro-ph.HE]
  NoStop
[Amaro-Seoane and Chen(2016)]ama16
  author author P. Amaro-Seoane and author X. Chen, 10.1093/mnras/stw503 journal
  journal  volume 458, pages 3075 (year 2016), http://arxiv.org/abs/1512.04897 arXiv:1512.04897 NoStop
[Amaro-Seoane et al.(2012)Amaro-Seoane, Aoudia, Babak,
  Binétruy, Berti, Bohé, Caprini, Colpi,
  Cornish, Danzmann, and et al.]Amaro-SeoaneEtAl2012
  author author P. Amaro-Seoane, author S. Aoudia, author S. Babak,
  author P. Binétruy,
  author E. Berti, author A. Bohé, author
  C. Caprini, author
  M. Colpi, author N. J. Cornish, author K. Danzmann,  and author et al., @noop  journal journal GW
  Notes, Vol. 6, p. 4-110  (year 2012), http://arxiv.org/abs/1201.3621 arXiv:1201.3621 [astro-ph.CO] NoStop
[Amaro-Seoane et al.(2013)Amaro-Seoane, Aoudia, Babak,
  Binétruy, Berti, Bohé, Caprini, Colpi,
  Cornish, Danzmann, and et al.]Amaro-SeoaneEtAl2013
  author author P. Amaro-Seoane, author S. Aoudia, author S. Babak,
  author P. Binétruy,
  author E. Berti, author A. Bohé, author
  C. Caprini, author
  M. Colpi, author N. J. Cornish, author K. Danzmann,  and author et al., @noop  journal journal GW
  Notes, Vol. 6, p. 4-110 volume 6, pages
  4 (year 2013), http://arxiv.org/abs/1201.3621
  arXiv:1201.3621 [astro-ph.CO] NoStop
[Amaro-Seoane et al.(2017)Amaro-Seoane, Audley, Babak,
  Baker, Barausse, Bender, Berti, Binetruy,
  Born, Bortoluzzi, and et al.]Amaro-SeoaneEtAl2017
  author author P. Amaro-Seoane, author H. Audley, author S. Babak,
  author J. Baker, author E. Barausse, author
  P. Bender, author
  E. Berti, author P. Binetruy, author M. Born, author D. Bortoluzzi,  and author et al., @noop  journal journal ArXiv
  e-prints  (year 2017), http://arxiv.org/abs/1702.00786 arXiv:1702.00786 [astro-ph.IM]
  NoStop
[Miller(2002)]miller02
  author author M. C. Miller, 10.1086/344156 journal
  journal Astrophys. J. volume 581,  pages 438 (year 2002), http://arxiv.org/abs/arXiv:astro-ph/0206404 arXiv:astro-ph/0206404
  NoStop
[Amaro-Seoane and Santamaría(2010)]ama10
  author author P. Amaro-Seoane and author L. Santamaría, 10.1088/0004-637X/722/2/1197
  journal journal  volume 722, pages 1197 (year 2010), http://arxiv.org/abs/0910.0254 arXiv:0910.0254 NoStop
[Kocsis and Levin(2012)]kocsis12levin
  author author B. Kocsis and author J. Levin, 10.1103/PhysRevD.85.123005 journal journal  volume 85,  eid 123005 (year 2012), http://arxiv.org/abs/1109.4170 arXiv:1109.4170 [astro-ph.CO] NoStop
[Kocsis(2013)]kocsis13
  author author B. Kocsis, 10.1088/0004-637X/763/2/122 journal journal  volume 763,  eid 122 (year 2013), http://arxiv.org/abs/1211.6427 arXiv:1211.6427 [astro-ph.HE] NoStop
[Sesana(2016)]sesana16
  author author A. Sesana, 10.1103/PhysRevLett.116.231102 journal journal Physical Review Letters volume 116, eid 231102 (year 2016), http://arxiv.org/abs/1602.06951 arXiv:1602.06951 [gr-qc] NoStop
[Seto(2016)]seto16
  author author N. Seto, 10.1093/mnrasl/slw060 journal
  journal  volume 460, pages L1 (year 2016), http://arxiv.org/abs/1602.04715 arXiv:1602.04715 [astro-ph.HE]
  NoStop
[Vitale(2016)]vitale16
  author author S. Vitale, 10.1103/PhysRevLett.117.051102 journal journal Physical Review Letters volume 117, eid 051102 (year 2016), http://arxiv.org/abs/1605.01037 arXiv:1605.01037 [gr-qc] NoStop
[Wen(2003)]wen03
  author author L. Wen, 10.1086/378794 journal journal  volume 598, pages 419
  (year 2003), http://arxiv.org/abs/astro-ph/0211492
  astro-ph/0211492 NoStop
[Gültekin et al.(2004)Gültekin, Miller, and Hamilton]gultekin04
  author author K. Gültekin, author M. C. Miller,  and author D. P. Hamilton, 10.1086/424809 journal
  journal  volume 616, pages 221 (year 2004), http://arxiv.org/abs/astro-ph/0402532 astro-ph/0402532 NoStop
[Gültekin et al.(2006)Gültekin, Miller, and Hamilton]gultekin06
  author author K. Gültekin, author M. C. Miller,  and author D. P. Hamilton, 10.1086/499917 journal
  journal  volume 640, pages 156 (year 2006), http://arxiv.org/abs/astro-ph/0509885 astro-ph/0509885 NoStop
[O'Leary et al.(2006)O'Leary, Rasio, Fregeau,
  Ivanova, and O'Shaughnessy]oleary06
  author author R. M. O'Leary, author F. A. Rasio, author J. M. Fregeau, author N. Ivanova,  and author R. O'Shaughnessy, 10.1086/498446 journal
  journal  volume 637, pages 937 (year 2006), http://arxiv.org/abs/astro-ph/0508224 astro-ph/0508224 NoStop
[Kyutoku and Seto(2016)]kyutoku16
  author author K. Kyutoku and author N. Seto, 10.1093/mnras/stw1767 journal
  journal  volume 462, pages 2177 (year 2016), http://arxiv.org/abs/1606.02298 arXiv:1606.02298 [astro-ph.HE]
  NoStop
[Nishizawa et al.(2016)Nishizawa, Berti, Klein, and  Sesana]nishizawa16a
  author author A. Nishizawa, author E. Berti, author A. Klein,   and author A. Sesana, 10.1103/PhysRevD.94.064020 journal journal  volume 94, eid 064020
  (year 2016), http://arxiv.org/abs/1605.01341
  arXiv:1605.01341 [gr-qc] NoStop
[Nishizawa et al.(2017)Nishizawa, Sesana, Berti, and  Klein]nishizawa16b
  author author A. Nishizawa, author A. Sesana, author E. Berti,   and author A. Klein, 10.1093/mnras/stw2993 journal journal
   volume 465, pages 4375
  (year 2017), http://arxiv.org/abs/1606.09295
  arXiv:1606.09295 [astro-ph.HE] NoStop
[Breivik et al.(2016)Breivik, Rodriguez, Larson,
  Kalogera, and Rasio]breivik16
  author author K. Breivik, author C. L. Rodriguez, author S. L. Larson, author V. Kalogera,  and author F. A. Rasio, 10.3847/2041-8205/830/1/L18 journal journal  volume 830,  eid L18 (year 2016), http://arxiv.org/abs/1606.09558 arXiv:1606.09558 NoStop
[Barack and Cutler(2004)]barack04
  author author L. Barack and author C. Cutler, 10.1103/PhysRevD.69.082005 journal journal  volume 69,  eid 082005 (year 2004), http://arxiv.org/abs/gr-qc/0310125 gr-qc/0310125 NoStop
[Peters(1964)]peters64
  author author P. C. Peters, @noop  journal journal
  Physical Review volume 136, pages
  1224 (year 1964)NoStop
[Peters and Mathews(1963)]Peters63
  author author P. C. Peters and author J. Mathews, 10.1103/PhysRev.131.435 journal journal Physical Review volume
  131, pages 435 (year 1963)NoStop
[Farmer and Phinney(2003)]farmer03
  author author A. J. Farmer and author E. S. Phinney, 10.1111/j.1365-2966.2003.07176.x journal journal  volume 346,  pages 1197 (year 2003), http://arxiv.org/abs/astro-ph/0304393 astro-ph/0304393 NoStop
[Abbott et al.(2009)Abbott, Abbott, Adhikari,
  Ajith, Allen, Allen,
  Amin, Anderson, Anderson, Arain, and et al.]abbott09
  author author B. P. Abbott, author R. Abbott,
  author R. Adhikari, author P. Ajith, author
  B. Allen, author G. Allen, author R. S.  Amin, author S. B.  Anderson, author W. G.  Anderson, author M. A.  Arain,  and author et al.,  10.1088/0034-4885/72/7/076901 journal
  journal Reports on Progress in Physics volume 72, eid 076901 (year 2009), http://arxiv.org/abs/0711.3041 arXiv:0711.3041 [gr-qc] NoStop
[Accadia et al.(2010)Accadia, Swinkels, and Virgo
  Collaboration]accadia10
  author author T. Accadia, author B. L. Swinkels,  and author Virgo
  Collaboration, 10.1088/0264-9381/27/8/084002 journal journal Classical and Quantum Gravity volume 27, eid 084002 (year
  2010)NoStop
[Kawamura et al.(2011)Kawamura, Ando, Seto, Sato, Nakamura, Tsubono,
  Kanda, Tanaka, Yokoyama, and et al.]kawamura11
  author author S. Kawamura, author M. Ando,
  author N. Seto, author S. Sato, author
  T. Nakamura, author
  K. Tsubono, author
  N. Kanda, author T. Tanaka, author J. Yokoyama,  and author et
  al., 10.1088/0264-9381/28/9/094011 journal journal Classical and Quantum Gravity volume 28, eid 094011 (year
  2011)NoStop
[Luo et al.(2016)Luo,
  Chen, Duan, Gong,
  Hu, Ji, Liu,
  Mei, Milyukov, Sazhin,
  Shao, Toth, Tu,
  Wang, Wang, Yeh,
  Zhan, Zhang, Zharov,  and Zhou]luo16
  author author J. Luo, author L.-S. Chen,
  author H.-Z. Duan, author Y.-G. Gong, author
  S. Hu, author J. Ji, author Q. Liu, author J. Mei,
  author V. Milyukov, author M. Sazhin, author
  C.-G. Shao, author
  V. T. Toth, author
  H.-B. Tu, author
  Y. Wang, author Y. Wang, author H.-C.  Yeh, author M.-S. Zhan, author Y. Zhang,
  author V. Zharov,  and  author Z.-B. Zhou, 10.1088/0264-9381/33/3/035010 journal journal CQG volume 33, eid 035010
  (year 2016), http://arxiv.org/abs/1512.02076
  arXiv:1512.02076 [astro-ph.IM] NoStop
[Gong et al.(2015)Gong,
  Lau, Xu, and et al.]gong15
  author author X. Gong, author Y.-K. Lau,
  author S. Xu,  and author et al., 10.1088/1742-6596/610/1/012011 journal journal
  Journal of Physics Conference Series volume 610,  eid 012011 (year 2015), http://arxiv.org/abs/1410.7296 arXiv:1410.7296 [gr-qc] NoStop
[Peebles(1972)]Peebles72
  author author P. J. E.  Peebles, @noop  journal journal Astrophys. J. volume 178, pages 371 (year 1972)NoStop
[Bahcall and Wolf(1976)]BW76
  author author J. N. Bahcall and author R. A. Wolf, @noop  journal journal
  Astrophys. J. volume 209, pages 214
  (year 1976)NoStop
[Amaro-Seoane et al.(2004)Amaro-Seoane, Freitag, and Spurzem]ASEtAl04
  author author P. Amaro-Seoane, author M. Freitag,  and author R. Spurzem, 10.1111/j.1365-2966.2004.07956.x journal journal  volume 352,  pages 655 (year 2004), http://arxiv.org/abs/astro-ph/0401163 astro-ph/0401163 NoStop
[Freitag et al.(2006)Freitag, Amaro-Seoane, and Kalogera]FAK06a
  author author M. Freitag, author P. Amaro-Seoane,  and author V. Kalogera, 10.1086/506193 journal
  journal Astrophys. J. volume 649,  pages 91 (year 2006), http://arxiv.org/abs/arXiv:astro-ph/0603280 arXiv:astro-ph/0603280
  NoStop
[Alexander and Hopman(2009)]AlexanderHopman09
  author author T. Alexander and author C. Hopman, 10.1088/0004-637X/697/2/1861 journal journal Astrophys. J. volume
  697, pages 1861 (year 2009)NoStop
[Preto and Amaro-Seoane(2010)]PretoAmaroSeoane10
  author author M. Preto and author P. Amaro-Seoane, 10.1088/2041-8205/708/1/L42  volume 708, pages L42 (year
  2010), http://arxiv.org/abs/0910.3206 arXiv:0910.3206
  NoStop
[Plummer(1911)]Plummer11
  author author H. C. Plummer, @noop   volume 71, pages 460 (year 1911)NoStop
[Alexander(2017)]alexander17
  author author T. Alexander, @noop  journal journal
  ArXiv e-prints  (year 2017), http://arxiv.org/abs/1701.04762 arXiv:1701.04762 NoStop
[Amaro-Seoane(2012)]Amaro-SeoaneLRR2012
  author author P. Amaro-Seoane, @noop  journal journal
  ArXiv e-prints  (year 2012), http://arxiv.org/abs/1205.5240 arXiv:1205.5240 [astro-ph.CO] NoStop
[Belczynski et al.(2004)Belczynski, Sadowski, and Rasio]belczynski04
  author author K. Belczynski, author A. Sadowski,  and author F. A. Rasio, 10.1086/422191 journal journal  volume 611, pages
  1068 (year 2004), http://arxiv.org/abs/astro-ph/0404068 astro-ph/0404068 NoStop
[Kulkarni et al.(1993)Kulkarni, Hut, and McMillan]kulkarni93
  author author S. R. Kulkarni, author P. Hut,   and author S. McMillan,  10.1038/364421a0 journal journal
   volume 364, pages 421 (year 1993)NoStop
[Sigurdsson and Hernquist(1993)]sigurdsson93
  author author S. Sigurdsson and author L. Hernquist, 10.1038/364423a0 journal
  journal  volume 364, pages 423 (year 1993)NoStop
[Miller and Hamilton(2002)]miller02hamilton
  author author M. C. Miller and author D. P. Hamilton, 10.1086/341788 journal
  journal  volume 576, pages 894 (year 2002), http://arxiv.org/abs/astro-ph/0202298 astro-ph/0202298 NoStop
[Ivanova et al.(2005)Ivanova, Belczynski, Fregeau, and  Rasio]ivanova05
  author author N. Ivanova, author K. Belczynski, author J. M. Fregeau,  and author F. A. Rasio, 10.1111/j.1365-2966.2005.08804.x journal journal  volume 358,  pages 572 (year 2005), http://arxiv.org/abs/astro-ph/0501131 astro-ph/0501131 NoStop
[Antognini and Thompson(2016)]antognini16
  author author J. M. O.  Antognini and author T. A.  Thompson, 10.1093/mnras/stv2938
  journal journal  volume 456, pages 4219 (year 2016), http://arxiv.org/abs/1507.03593 arXiv:1507.03593 [astro-ph.SR]
  NoStop
[Binney and Tremaine(2008)]BT08
  author author J. Binney and author S. Tremaine, @noop  title Galactic Dynamics:
  Second Edition, by James Binney and Scott Tremaine. ISBN 978-0-691-13026-2
  (HB). Published by Princeton University Press, Princeton, NJ USA, 2008.  (publisher Princeton University Press, year
  2008)NoStop
[Heggie(1975)]heggie75
  author author D. C. Heggie, 10.1093/mnras/173.3.729 journal
  journal  volume 173, pages 729 (year 1975)NoStop
[Samsing et al.(2014)Samsing, MacLeod, and Ramirez-Ruiz]samsing14
  author author J. Samsing, author M. MacLeod,  and author E. Ramirez-Ruiz, 10.1088/0004-637X/784/1/71 journal journal  volume 784,  eid 71 (year 2014), http://arxiv.org/abs/1308.2964 arXiv:1308.2964 [astro-ph.HE] NoStop
[Mardling and Aarseth(2001)]mardling01
  author author R. A. Mardling and author S. J. Aarseth, 10.1046/j.1365-8711.2001.03974.x journal journal  volume 321,  pages 398 (year 2001)NoStop
[Lidov(1962)]lidov62
  author author M. L. Lidov, 10.1016/0032-0633(62)90129-0 journal journal Planetary and Space Science volume 9, pages 719 (year
  1962)NoStop
[Kozai(1962)]kozai62
  author author Y. Kozai, @noop  journal journal AJ  volume 67, pages 591 (year
  1962)NoStop
[Naoz et al.(2013)Naoz,
  Kocsis, Loeb, and Yunes]naoz13
  author author S. Naoz, author B. Kocsis,
  author A. Loeb,  and author N. Yunes, 10.1088/0004-637X/773/2/187 journal journal
   volume 773, eid 187 (year 2013), http://arxiv.org/abs/1206.4316 arXiv:1206.4316
  [astro-ph.SR] NoStop
[Antognini et al.(2014)Antognini, Shappee, Thompson, and  Amaro-Seoane]antognini14
  author author J. M. Antognini, author B. J. Shappee, author T. A. Thompson,  and author P. Amaro-Seoane, 10.1093/mnras/stu039 journal journal  volume 439,  pages 1079 (year 2014), http://arxiv.org/abs/1308.5682 arXiv:1308.5682 [astro-ph.HE] NoStop
[Liu et al.(2015)Liu,
  Muñoz, and Lai]liu15
  author author B. Liu, author D. J. Muñoz,  and author D. Lai, 10.1093/mnras/stu2396 journal
  journal  volume 447, pages 747 (year 2015), http://arxiv.org/abs/1409.6717 arXiv:1409.6717 [astro-ph.EP] NoStop
[Antonini et al.(2014)Antonini, Murray, and Mikkola]antonini14
  author author F. Antonini, author N. Murray,  and author S. Mikkola, 10.1088/0004-637X/781/1/45 journal journal  volume 781,  eid 45 (year 2014), http://arxiv.org/abs/1308.3674 arXiv:1308.3674 [astro-ph.HE] NoStop
[Antonini et al.(2016)Antonini, Chatterjee, Rodriguez,
  Morscher, Pattabiraman, Kalogera, and Rasio]antonini16BS
  author author F. Antonini, author S. Chatterjee, author C. L. Rodriguez, author M. Morscher, author B. Pattabiraman, author V. Kalogera,  and author F. A. Rasio, 10.3847/0004-637X/816/2/65 journal journal  volume 816,  eid 65 (year 2016), http://arxiv.org/abs/1509.05080 arXiv:1509.05080 NoStop
[Kimpson et al.(2016)Kimpson, Spera, Mapelli, and  Ziosi]kimpson16
  author author T. O. Kimpson, author M. Spera,
  author M. Mapelli,  and  author B. M. Ziosi, 10.1093/mnras/stw2085 journal journal
   volume 463, pages 2443
  (year 2016), http://arxiv.org/abs/1608.05422
  arXiv:1608.05422 NoStop
[Silsbee and Tremaine(2017)]silsbee16
  author author K. Silsbee and author S. Tremaine, 10.3847/1538-4357/aa5729 journal journal  volume 836,  eid 39 (year 2017), http://arxiv.org/abs/1608.07642 arXiv:1608.07642 [astro-ph.HE]
  NoStop
[Turner(1977)]turner77
  author author M. Turner, 10.1086/155501 journal
  journal  volume 216, pages 610 (year 1977)NoStop
[Quinlan and Shapiro(1987)]quinlan87
  author author G. D. Quinlan and author S. L. Shapiro, 10.1086/165624 journal
  journal  volume 321, pages 199 (year 1987)NoStop
[Kocsis et al.(2006)Kocsis, Gáspár, and Márka]kocsis06
  author author B. Kocsis, author M. E. Gáspár,  and author S. Márka, 10.1086/505641
  journal journal  volume 648, pages 411 (year 2006), http://arxiv.org/abs/astro-ph/0603441 astro-ph/0603441 NoStop
[O'Leary et al.(2009)O'Leary, Kocsis, and Loeb]oleary09
  author author R. M. O'Leary, author B. Kocsis,
   and author A. Loeb, 10.1111/j.1365-2966.2009.14653.x journal journal  volume 395, pages
  2127 (year 2009), http://arxiv.org/abs/0807.2638
  arXiv:0807.2638 NoStop
[Lee et al.(2010)Lee,
  Ramirez-Ruiz, and van de Ven]lee10
  author author W. H. Lee, author E. Ramirez-Ruiz,  and author G. van de Ven, 10.1088/0004-637X/720/1/953 journal journal  volume 720,  pages 953 (year 2010), http://arxiv.org/abs/0909.2884 arXiv:0909.2884 [astro-ph.HE] NoStop
[Hong and Lee(2015)]hong15
  author author J. Hong and author H. M. Lee, 10.1093/mnras/stv035 journal
  journal  volume 448, pages 754 (year 2015), http://arxiv.org/abs/1501.02717 arXiv:1501.02717 NoStop
[Tsang(2013)]tsang13
  author author D. Tsang, 10.1088/0004-637X/777/2/103 journal journal  volume 777,  eid 103 (year 2013), http://arxiv.org/abs/1307.3554 arXiv:1307.3554 [astro-ph.HE] NoStop
[Miller and Lauburg(2009)]MillerLauburg09
  author author M. C. Miller and author V. M. Lauburg, 10.1088/0004-637X/692/1/917 journal journal Astrophys. J. volume
  692, pages 917 (year 2009), http://arxiv.org/abs/0804.2783 arXiv:0804.2783 NoStop
[Antonini and Perets(2012)]antonini12
  author author F. Antonini and author H. B. Perets, 10.1088/0004-637X/757/1/27 journal journal  volume 757,  eid 27 (year 2012), http://arxiv.org/abs/1203.2938 arXiv:1203.2938 NoStop
[Addison et al.(2015)Addison, Laguna, and Larson]addison15
  author author E. Addison, author P. Laguna,
   and author S. Larson,  @noop  journal journal ArXiv e-prints 
  (year 2015), http://arxiv.org/abs/1501.07856
  arXiv:1501.07856 [astro-ph.HE] NoStop
[Antonini and Rasio(2016)]antonini16
  author author F. Antonini and author F. A. Rasio, 10.3847/0004-637X/831/2/187 journal journal  volume 831,  eid 187 (year 2016), http://arxiv.org/abs/1606.04889 arXiv:1606.04889 [astro-ph.HE]
  NoStop
[VanLandingham et al.(2016)VanLandingham, Miller, Hamilton,  and Richardson]VL16
  author author J. H. VanLandingham, author M. C. Miller, author D. P. Hamilton,  and author D. C. Richardson, 10.3847/0004-637X/828/2/77 journal journal  volume 828,  eid 77 (year 2016), http://arxiv.org/abs/1604.04948 arXiv:1604.04948 [astro-ph.HE]
  NoStop

