On the Eccentricity Distribution of Coalescing Black Hole Binaries Driven by the Kozai Mechanism in Globular Clusters

Linqing Wen (Caltech)

REVIEW

On the Eccentricity Distribution of Coalescing Black Hole Binaries Driven by the Kozai Mechanism in Globular Clusters

Linqing Wen (Caltech)

Astrophysics · Submitted 22 Nov 2002 · 23 references

Pith's verdict: Sound Kozai-plus-GR machinery and a real general-mass ǫmin formula, but the 30% at 10 Hz is a readout of hand-picked priors, not a derived population.

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CitationLinqing Wen (Caltech). "On the Eccentricity Distribution of Coalescing Black Hole Binaries Driven by the Kozai Mechanism in Globular Clusters." arXiv:astro-ph/0211492 (2002). https://pith.science/paper/GR5FUM3E

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Worth your time, with one caveat about the headline number. The durable piece is the general-mass closed form for the per-cycle minimum ǫ in eqs. (8)–(9), which extends Miller & Hamilton's restricted m0≪m2≪m1 case, plus what I believe is the first calculation that carries Kozai-driven mergers all the way to fixed gravitational-wave frequency and reports e(f) rather than just e at merger. The machinery is standard — the equations of motion are explicitly Blaes, Lee & Socrates (2002), the Kozai and Peters pieces are classical — but the assembly is careful. Figures 3 and 6 check the numerical integrations against the analytic ǫmin estimate and against Peters' circularization law, and the two worked examples cleanly separate the many-cycle and one-cycle merger paths. The qualitative conclusion, that a non-negligible eccentric tail survives to 10 Hz and that circular templates are not universally adequate, is probably right and is the part I would cite. The soft spot is the 30%. It is a Monte Carlo average over the Table 1 box: a1 uniform in 0.2–30 AU, I0 uniform in 85–110 degrees, e0 uniform in 0.01–0.9, five discrete a2/a1 values, three equal 10 solar-mass black holes. None of that is derived from binary-binary scattering outcomes or cluster models; the text says only that the ranges were investigated. The a1 prior is load-bearing — Fig. 9 shows the merging region is a cone whose a1 extent shrinks with a2/a1, and e at fixed f depends on a1 through eq. (37). A log-uniform a1 prior would move the fraction. The paper also never says how the five a2/a1 values are weighted, which matters because the permitted a1 volume differs across them. The body is honest that the fraction is relative to systems that merge before disruption; the abstract drops that qualifier. The factor-of-three PN bound in eq. (31) is acknowledged and shown to affect mainly low-e systems, so it is not the issue. Who should read it: anyone doing template design or population modeling for ground-based detectors, and anyone working on Kozai dynamics in clusters. It deserves referees — the physics is sound and the qualitative claim is robust; the referee's job is to make the authors either derive the priors or quote a systematic band on the fraction.

Referee report

Significance

If the result holds, this is one of the few astrophysical channels that leaves a measurable eccentricity in the low end of the ground-based detector band, with direct consequences for circular-template searches. The paper deserves credit for several concrete strengths: the analytic epsilon_min formula (eqs. 8-9) is a parameter-free generalization of the restricted Kozai result to arbitrary masses and initial eccentricities, and it is checked against direct numerical integration rather than asserted; the peak-harmonic fit of eq. (36) is a compact, reusable result; the prediction is falsifiable, both by population-synthesis work on binary-binary scattering and, eventually, by gravitational-wave observations; and the paper is unusually transparent about the initial-condition box it samples, which is set out explicitly in Table 1. The qualitative conclusion, that a non-negligible eccentric tail survives to 10 Hz while the population is essentially circular by 40-200 Hz, appears robust to the choices I can test.

Major objections (3)

[Abstract (s3); §4.2; Table 1; Fig. 9] The ~30% figure is a direct readout of the Table 1 prior box, and the weighting over a2/a1 is not specified. §4.2 states that the authors consider 'a uniform distribution of ... ratios of the semimajor axis a2/a1', but Table 1 lists five discrete values {3, 5, 10, 20, 30}, so 'uniform' is ambiguous between uniform over the set and uniform over the interval [3, 30]. This matters because Fig. 9 shows that the permitted a1 volume differs strongly across the five values, and because the eccentricity at fixed f^m_GW depends on a1 through eq. (37) and on the post-Kozai evolution through eq. (38). Please state the weighting explicitly and quote the >0.1 fraction under at least one alternative, for example a log-uniform a1 prior or an a2/a1 distribution taken from the binary-binary scattering outcomes of Miller & Hamilton (2002). At minimum, the abstract should say that the 30% is conditional on the Table 1 priors, since as written it will be quoted as a population-independent number.
[§4.2; Table 1; Fig. 10] Table 1 states that the numbers of steps for e0 and g1 are 4 and 4 respectively, over the ranges [0.01, 0.9] and [0, 90] degrees. The 30% figure is therefore a quadrature over a very coarse grid in two of the four parameters that control the Kozai cycle. Please demonstrate convergence of the Fig. 10 histogram with respect to this sampling, for example by doubling the e0 and g1 grids and quoting the change in the fraction with e > 0.1, or state the resulting uncertainty on the 30% explicitly. As it stands the reader cannot tell whether the quoted fraction is stable at the few-percent level or not.
[§4.1; eqs. (40)-(41)] The merger criterion tau_GR/sqrt(epsilon_min) < tau_enc rests on the heuristic that the system spends a fraction sqrt(epsilon_min) of its Kozai cycle near epsilon_min. This criterion determines which systems enter the Fig. 10 histogram and is therefore load-bearing for the population fraction. The paper verifies the epsilon_min value itself against the numerical integrations (Figs. 3 and 6), but the residence-time scaling is not checked. Please extract the fraction of time spent below, say, 10 epsilon_min directly from the integrations of Fig. 2 and confirm that it scales as sqrt(epsilon_min); if the coefficient differs from unity by more than a modest factor, the boundary of the permitted cone in Fig. 9 shifts accordingly.

Minor objections (11)

[Table 1] The entry for e0 reads '0.01-0.901'; this appears to be a typographical error for 0.9.
[Fig. 10] The caption states that the x-axes of the two histograms in the left panel are shifted for a better view. This makes the grey (no PN) and dark (with PN) distributions hard to compare directly; please overlay them or tabulate the two fractions with e > 0.1.
[Abstract (s3)] The phrase '~30% of the systems' should specify that the denominator is the set of systems that merge before disruption by field stars, as §4.2 makes clear; otherwise the number reads as a fraction of all triples.
[§3.4; eq. (36)] The fitted exponent 1.1954 is quoted without residuals or the range of epsilon over which the fit was calibrated. Please quote the maximum fractional error of eq. (36) over epsilon = 10^-6 to 1, since eq. (37) propagates that fit directly into the frequency-eccentricity relation used throughout §4.
[§4.2; eq. (42)] The intermediate algebra leading from eq. (8) to the numerical coefficient 399 in eq. (42) is not shown, and the direction of the constraint is not transparent from the text. Please give the steps. It would also help to state eq. (42) alongside the separate requirement beta >= sqrt(epsilon_0)/2 implied by eq. (3), which is a lower bound on a2/a1, so that the reader sees the two constraints together.
[§2.1; eq. (2)] As printed, the denominator of eq. (2) reads '2 beta0 sqrt(a1 epsilon)' with the 'a1 epsilon' run together as 'a1 epsilon^2 beta0'; please check the typesetting of this equation, since it is the basis for eqs. (3), (4), and (11).
[§3.2; eqs. (23)-(24)] The quantity gamma_0 in eq. (24) carries a numerical coefficient 4.253040 x 10^-19 and a stated unit of AU^5.5; please state the units of gamma_0 explicitly in the text and confirm the a1^{-5.5} exponent in eq. (23) is consistent with the definition.
[§4.1] The lower limit a1 >= 0.2 AU is justified by the requirement that the recoil velocity from binary hardening be low enough for the system to remain in the cluster. Please give the recoil-velocity estimate or cite the specific section of Miller & Hamilton (2002) from which the 0.2 AU value follows.
[§5] The paper correctly notes that the Newtonian quadrupole approximation breaks down near 200 Hz. Since a quantitative statement (e < 0.02) is quoted at 200 Hz, please state explicitly in §5 that this statement is unaffected because the orbits are already circular well before that frequency.
[References] Moreno-Garrido, Mediavilla, & Buitrago 1994 is listed in the references but the text cites only the 1995 entry; please reconcile.
[§1; §4.1] The astrophysical premise and the disruption time scale of eq. (39) are taken from Miller & Hamilton (2002), listed as 'ApJ submitted'. Since the merger criterion of eqs. (40)-(41) inherits that time scale, please confirm that the final published version agrees with the assumptions used here.

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The paper

The paper claims

In globular clusters, the Kozai mechanism can drive a black-hole binary's eccentricity close to one; Wen asks whether any of that eccentricity survives gravitational-wave emission long enough to be measurable when the waves reach 10 Hz, and

The reading

The paper asks a follow-up question to the recent proposal that hierarchical triple black holes in globular clusters can be driven to merge by the Kozai mechanism: if the inner binary really is whipped up to near-extreme eccentricity, does any of that eccentricity survive the circularizing pull of gravitational radiation long enough to be seen? Wen couples three effects in one set of equations of motion — the secular Kozai oscillation, post-Newtonian periastron precession, and quadrupole gravitational radiation reaction — and follows the inner binary from its first Kozai cycle through to merger. The answer is that the eccentricity is largely erased but not entirely: about 30 percent of mergers should still carry $e_1>0.1$ when their waves sweep through 10 Hz, and about 2 percent should be near $e_1=1$ there because they merge inside their first Kozai cycle. The stake is search strategy: if a non-trivial fraction of the expected sources are measurably non-circular in the low part of the ground-based band, then templates built for circular orbits are not the optimal filter for them.

What carries the argument

The argument is carried by a doubly-averaged Hamiltonian for the hierarchical triple, $W(2e_1,g_1)$, built from the quadrupole Kozai term plus a post-Newtonian periastron-precession term $2e_1^{-1/2}$, which stays conserved along the secular evolution. Conservation of $W$ across a Kozai cycle is what lets Wen solve for the minimum $2e_1$ the system can reach, first analytically for negligible radiation reaction and then numerically with a downhill simplex search once radiation reaction is included. That minimum, together with the conserved semi-major axis and total angular momentum, fixes how much time the binary spends near pericenter, and hence whether merger happens within one Kozai cycle or after many. A second piece of machinery converts between eccentricity and observable frequency: a fitted peak-harmonic formula, $n_m=2(1+e_1)^{1.1954}/(1-e_1^2)^{1.5}$, combined with the Kepler frequency, gives the peak gravitational-wave frequency as a function of eccentricity and semi-latus rectum. The numerical work integrates the four coupled equations for $2e_1$, $g_1$, $a_1$ and the total angular momentum, using the same hybrid Kozai-plus-radiation-reaction system as Blaes, Lee, and Socrates.

Extended reading notes

Core claim

Wen's central claim is that the Kozai mechanism does not deliver its mergers with their extreme eccentricities intact, but it does leave a residue that matters at the bottom of the ground-based detector band. The competition is between two tendencies: the Kozai cycle pumps the inner binary toward $e_1=1$, or $e_1^2=1-2e_1+2e_1^2+2e_1^3-2e_1^4-2e_1^5+e_1^6$ in the paper's notation, once per cycle, while gravitational radiation removes energy and angular momentum most efficiently precisely at those low-$e_1$ pericenter passages, so the system is circularized where it is most vulnerable. Which effect wins depends on which reaches the inner binary first. If radiation reaction takes over at the first eccentricity minimum, the system plunges while still eccentric and can arrive at 10 Hz with $e_1$ around 0.9. If post-Newtonian precession grows large first, it scrambles the phase of the Kozai oscillation, quenches the cycle, and the binary then circularizes like an isolated system; the eccentricity at 10 Hz then falls roughly as $f_{m}^{19/18}$ in frequency, leaving almost everything below 0.2 by 40 Hz and below 0.02 by 200 Hz.

Load-bearing premise

The 30 percent figure follows directly from a hand-picked set of initial conditions — uniform ranges in semi-major axis, mutual inclination, starting eccentricity and argument of pericenter, with three equal 10 solar mass black holes and a discrete set of outer-to-inner axis ratios — rather than from any distribution derived from binary-binary scattering, cluster simulations, or observation.

Editorial extensions

If this is right

About 30 percent of the simulated mergers retain $e_1>0.1$ at 10 Hz, so circular-orbit search templates are suboptimal for a sizeable minority of these sources.
Roughly 2 percent arrive at 10 Hz with eccentricity near one, having merged within their first Kozai cycle; these are the cases with initial inclination closest to the critical angle.
By 40 Hz essentially all systems are below $e_1=0.2$, and by 200 Hz below 0.02, so the eccentricity problem is confined to the low-frequency end of the ground-based band.
The same mergers should be strongly eccentric in the lower-frequency band, making them candidate sources for a space-based detector as well as ground-based ones.
For the low-eccentricity tail, post-Newtonian precession raises the minimum eccentricity above the value radiation reaction alone would give, which sets an upper bound on how circular these systems can become.

Reading between the lines

Editorial extensions of the paper, not claims the author makes directly.

The 30 percent figure is a readout of a chosen prior, so the more useful deliverable is arguably the mapping from initial conditions to final eccentricity, which a population-synthesis study could convolve with any scattering-derived distribution.
Because the eccentric cases are those that merge fastest, a detector's effective reach for them differs from the circular-case reach; the implied correction to predicted event rates is not computed here.
The peak-harmonic fit suggests an observational handle: a measured eccentricity at 10 Hz would constrain the initial mutual inclination relative to the critical angle, and hence indirectly the triple-formation channel.
The same competition between Kozai pumping and radiation-reaction circularization should apply to neutron-star and mixed-mass triples, where the merger would be accompanied by electromagnetic emission and the eccentricity could leave a signature in the pre-merger orbit.

Original abstract

In a globular cluster, hierarchical triple black hole systems can be produced through binary-binary interaction. It has been proposed recently that the Kozai mechanism could drive the inner binary of the triple system to merge before it is interrupted by interactions with other field stars. We investigate qualitatively and numerically the evolution of the eccentricities in these binaries under gravitational radiation (GR) reaction. We predict that ~ 30 % of the systems will possess eccentricities >0.1 when their emitted gravitational waves pass through 10 Hz frequency. The implications for gravitational wave detection, especially the relevance to data analyses for broad-band laser-interferometer gravitational wave detectors, are discussed.

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The abstract, sentence by sentence

arXiv listing abstract v1. Results and the main claim are underlined; show every role.

In a globular cluster, hierarchical triple black hole systems can be produced through binary-binary interaction. It has been proposed recently that the Kozai mechanism could drive the inner binary of the triple system to merge before it is interrupted by interactions with other field stars. We investigate qualitatively and numerically the evolution of the eccentricities in these binaries under gravitational radiation (GR) reaction. We predict that ~ 30 % of the systems will possess eccentricities >0.1 when their emitted gravitational waves pass through 10 Hz frequency. The implications for gravitational wave detection, especially the relevance to data analyses for broad-band laser-interferometer gravitational wave detectors, are discussed.

background contribution method result interpretation main claim

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#	sentence	role	the authors' own work	states the main conclusion	strength
s0	In a globular cluster, hierarchical triple black hole systems can be produced through binary-binary interaction.	background 1.00	0.08	0.03	asserted 0.74
s1	It has been proposed recently that the Kozai mechanism could drive the inner binary of the triple system to merge before it is interrupted by interactions with other field stars.	background 1.00	0.04	0.05	hedged possibility 0.98
s2	We investigate qualitatively and numerically the evolution of the eccentricities in these binaries under gravitational radiation (GR) reaction.	contribution 0.77	0.94	0.04	asserted 0.87
s3	We predict that ~ 30 % of the systems will possess eccentricities >0.1 when their emitted gravitational waves pass through 10 Hz frequency.	result 1.00	0.91	0.80	asserted 0.74
s4	The implications for gravitational wave detection, especially the relevance to data analyses for broad-band laser-interferometer gravitational wave detectors, are discussed.	interpretation 0.99	0.43	0.05	unclear 0.54

What kind of paper the classifier reports this is: simulation study 0.99 · evidence from simulation or computation 1.00.

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Every quoted sentence on this record is the authors'. Every label is from Pith's closed vocabulary, with the probability shown beside it. Every number in the measurements table is the authors', checked by code. Record as JSON

Figures

**Figure 1.** — Geometry of a hierarchical triple system.

**Figure 2.** — Secular evolution of ǫ = 1 − e 2 1 of the inner binary and mutual inclination angle I between the inner and outer binaries. This evolution was computed by integrating equations (16)–(19) which include contributions from Kozai mechanism, PN periastron precession, and gravitational radiation reaction. The initial system parameters are: m0 = m1 = m2 = 10 M⊙, e1 = e2 = 0.1, a1 = 10 AU, a2/a1 = 10, g1 = 0, and I0 = 95.3 o . This is a typical case in which the PN effect becomes important before the GR effect dominates in the evolution of ǫ, and that the system merges after many Kozai cy

**Figure 3.** — Evolution of ǫ (solid line) and the time scales of gravitational merger and the PN effect (dashed-dotted lines) with the semimajor axis a1 for the same initial parameters and evolution equations as in

**Figure 4.** — Secular evolution for ǫ of the inner binary and of I, as computed by integrating the evolution equations (16)–(19) which include contributions from the Kozai mechanism, PN periastron precession, and gravitational radiation reaction. The initial system parameters are: m0 = m1 = m2 = 5 M⊙, e1 = 0.01, e2 = 0.51, a1 = 2.1920 AU, a2/a1 = 5, g1 = 0, and I0 = 99.2 o . This is a typical case where the GR effect dominates in the evolution within one Kozai cycle

**Figure 5.** — Phase diagram of ǫ vs g1 for the same initial parameters as in

**Figure 6.** — Evolution of ǫ with the semimajor axis a1 (open circles) for the same initial parameters and evolution equations as in

**Figure 7.** — Evolution of e1 with gravitational wave frequency f m GW for the system shown in

**Figure 8.** — Evolution of e1 with f m GW for the system shown in

**Figure 9.** — Parameter space for a1 (upper set) and corresponding values of ǫmin (lower set) vs I0 for a2/a1 = 20, 10, 5, 3 from left to right (see text)

**Figure 10.** — The distribution of eccentricities at f m GW = 10 Hz (left panel) and 40 Hz, 200 Hz (right panel) for black hole mergers driven by the Kozai effect in a globular cluster. See

Figures reproduced from arXiv:astro-ph/0211492 by the authors.

References

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gr-qc 2019-08 accept novelty 5.0 of 10

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Are merging black holes born from stellar collapse or previous mergers?
gr-qc 2017-03 conditional novelty 5.0 of 10

Simulated LIGO populations show that roughly 20 to 200 detections at design sensitivity suffice to distinguish first-generation black holes from those built by earlier mergers at 5 sigma.
First Law of Mechanics for Compact Binaries on Eccentric Orbits
gr-qc 2015-06 conditional novelty 5.0 of 10

The circular-orbit first law of binary mechanics is extended to eccentric orbits as delta M = omega delta L + n delta R + sum <z_a> delta m_a, verified to 3PN order and used to extract effective-one-body potentials fr...
PRECESSION 2.1: black-hole binary spin precession on eccentric orbits
gr-qc 2025-08 unverdicted novelty 4.0 of 10

Version 2.1 of the precession code extends post-Newtonian modeling of spinning black-hole binaries to eccentric orbits via a decorator-based adaptation and new evolutionary equations.
Hierarchical mergers of stellar-mass black holes and their gravitational-wave signatures
astro-ph.HE 2021-05 accept novelty 3.0 of 10

Hierarchical black hole mergers should leave a distinctive high-mass, high-spin signature, and GW190521 is the best current candidate, though population-level evidence remains inconclusive.
Binary Black Hole Coalescence
gr-qc 2007-10 accept novelty 3.0 of 10

By 2007 numerical relativity had produced stable binary black hole merger simulations whose waveforms are simple enough to be stitched together from analytic inspiral and ringdown models.
Numerical Relativity and Astrophysics
astro-ph.HE 2014-05 accept novelty 2.0 of 10

Numerical relativity now delivers quantitative predictions for compact binary mergers and core collapse, including remnant masses, spins, recoil velocities, and electromagnetic counterpart estimates.

How later papers used it

164 later papers cite it

Evidence: substantial. 164 of the citing papers share no author with it, and 42 of those carry a confident label from Pith's citation classifier.

Pith read 25 of the 241 citing sentences, choosing only those whose citation markers point at this paper alone, and wrote the paraphrases from them. The summary, the counts, the evidence level, the links and the quotes come from code; the summary counts the papers listed below. A later paper is listed under a label when Pith's citation classifier gives one of its sentences that label at probability 0.8 or more (0.95 for support), Pith's reading finds that the citing paper itself did it, and a second check confirms the paraphrase against the sentence. This section lists uses only. It does not list criticism or disagreement, so their absence here says nothing about how the paper has been received. Papers that share an author with this one are listed apart, under "By the same authors". A quote is the citing authors' own sentence, shown when their paper's licence is CC BY, CC BY-SA or CC0. JSON

11 later papers by other authors are listed below: 11 used its method or tool.

Uses its method or tool 11 of 37 papers with this label

Used the peak gravitational-wave frequency defined by this paper to determine the residual eccentricity at 10 Hz.

High Eccentricities and High Masses Characterize Gravitational-wave Captures in Galactic Nuclei as Seen by Earth-based Detectors (2020) · 0.92 · paraphrased: the licence does not allow quoting
Numerically solved for the orbital properties at an eccentric peak frequency as in this paper.

Implications of Eccentric Observations on Binary Black Hole Formation Channels (2021) · 1.00 · paraphrased: the licence does not allow quoting
Combined another work's orbit-averaged equations with this paper's estimate for the gravitational-wave peak frequency to calculate the eccentricity at 10 Hz.

First and second-generation black hole and neutron star mergers in 2+2 quadruples: population statistics (2021) · 1.00 · paraphrased: the licence does not allow quoting
The eccentricites at a GW frequency of $10$ Hz [ref] are determined by integrating the time-averaged change in semi-major axis and eccentricity [ref] from the point of binary formation until the peak of the rest-frame GW frequency [ref] equals $10 Hz$.

Dynamical Formation Scenarios for GW190521 and Prospects for Decihertz Gravitational-Wave Astronomy with GW190521-Like Binaries (2020) · GW Captures during Three-Body Encounters · 0.97
Evolved the frequency of the emitted gravitational waves according to this paper.

Measuring eccentricity in binary black hole inspirals with gravitational waves (2018) · 1.00 · paraphrased: the licence does not allow quoting
Used the peak gravitational-wave frequency defined by this paper to determine the residual eccentricity at 10 Hz.

Parameter Distributions of Binary Black Hole Mergers Near Supermassive Black Holes as Seen by Advanced Gravitational Wave Detectors (2022) · 0.93 · paraphrased: the licence does not allow quoting
Adopted this paper's definition of the gravitational-wave frequency.

Measurement Accuracy of Inspiraling Eccentric Neutron Star and Black Hole Binaries Using Gravitational Waves (2018) · 0.98 · paraphrased: the licence does not allow quoting
Calculated the approximate gravitational peak frequency following this paper.

High rate of gravitational waves mergers from flyby perturbations of wide black-hole triples in the field (2020) · 0.95 · paraphrased: the licence does not allow quoting
Used this paper's expression for the peak-power gravitational-wave frequency.

Stellar-mass black holes in young massive and open stellar clusters IV: updated stellar-evolutionary and black hole spin models and comparisons with the LIGO-Virgo O1/O2 merger-event data (2020) · 1.00 · paraphrased: the licence does not allow quoting
Calculated the frequency as the highest harmonic produced by the inner binary as defined by this paper.

Black Hole Mergers from Hierarchical Triples in Dense Star Clusters (2020) · 0.99 · paraphrased: the licence does not allow quoting
Calculated the gravitational-wave frequency using this paper's equation.

Formation and merging of Mass Gap Black Holes in Gravitational Wave Merger Events from Wide Hierarchical Quadruple Systems (2019) · 0.95 · paraphrased: the licence does not allow quoting

How the literature uses it

79 arXiv papers cite this in 123 sentences

Each citing sentence is taken from the citing paper's LaTeX source and typed by Pith's classifier. A label appears at probability 0.8 or more (0.95 for support), a contest at 0.7 or more; the quoted words are the citing authors'. How each label was checked · JSON

Papers citing it · 89 sentences without a confident label

Contests it 1 paper

The Wen prescription [ref] gives $n_peak^Wen≃18$ at the same eccentricity and therefore substantially underestimates the cutoff required by the present detector-weighted criterion, especially for the earlier observation.

A parametrized test of general relativity for inspiralling eccentric binaries in LISA · Convergence of the radial-harmonic expansion · contest 0.82

Uses its method or tool 9 papers

We also compute the peak GW frequency following [ref] , who showed that an eccentric binary emits GW power across multiple harmonics of the orbital frequency, with the peak harmonic at [equation] As shown in the bottom row of Figure [ref], most binaries have $f_GW$ in the range $10^-3$–$10^-1 Hz$ at formation, placing them not merely within the LISA band but near its sensitivity sweet spot at $∼10^-2$–$10^-1 Hz$ [ref] .

Dynamical formation of high-eccentricity compact binaries through BH--BH*/TZO collisions · Merger timescale and GW frequency · 0.81
The harmonic corresponding to the maximum GW emitted power can be estimated as a function of the PBH initial eccentricity $e_0$ as (see e.g. [ref] ) [equation] The top left and top right panels of Fig. [ref] show the probability distributions of the number of sources resolved by LISA and $μ$Ares, respectively, while the bottom left and bottom right panels show the corresponding distributions of SNRs.

Gravitational waves from an eccentric population of primordial black holes orbiting Sgr A$^{\star}$ · Resolved sources · 0.90
Refs. [ref] , have provided two fitting functions that determine the peak harmonic for a given value of $e$.

On the orbital eccentricities of primordial black hole binaries inside and outside of dark matter halos · Characteristic strain · 0.82

Uses its data or values 1 paper

Simulations which include the post-Newtonian (PN) description of the orbital dynamics in globular star clusters show that $≈5%$ of dynamical mergers have eccentricities greater than 0.1 This estimate refers to an astrophysical definition of eccentricity [ref] , not the quasi-Keplerian definition. at 10 Hz (in the detector frame) [ref] .

Evidence for eccentricity in the population of binary black holes observed by LIGO-Virgo-KAGRA · Introduction · 0.88

Compares with it 3 papers

This is different from the binaries in GCs where the mass has little impact on eccentricity distribution [ref] , and is in contrast to what is claimed in [ref] who considered an alternative eccentric BBH formation channel with direct two-body encounter and found that the binary mass and the eccentricity is positively correlated.

An Analytical Portrait of Binary Mergers in Hierarchical Triple Systems · Examples of Eccentricity Distribution · 0.90
This is also why we find smaller residual eccentricities than previous studies that focused on lighter inner binaries (see, e.g., Refs. [ref] ).

Spin and Eccentricity Evolution in Triple Systems: from the Lidov-Kozai Interaction to the Final Merger of the Inner Binary · Limiting eccentricity obtained during the LK oscillation · 0.97
On the other hand, let us compare the amplitude of the binary-binary force with respect to the Kozai-Lidov interaction $S_KL$ also present in the expansion of the Newtonian action [ref] in the center-of-mass frame of the binaries (see e.g. [ref] for definite expressions), and responsible for many interesting astrophysical phenomena [ref] : [equation] Thus, the binary-binary force is actually a subdominant many-body conservative interaction in the Newtonian limit, and can affect the system only on timescales much longer than the Kozai-Lidov time.

A supplementary radiation-reaction force between two binaries · Newtonian limit · 0.86

Cites it as support 1 paper

But measuring eccentricity in a BBH can serve as a reliable indicator of a dynamical formation channel [ref] .

Evidence for eccentricity in the population of binary black holes observed by LIGO-Virgo-KAGRA · Introduction · 0.98

Cites it as background 15 papers

These can be further decomposed into binary-single or binary-binary interactions [ref] , single-single gravitational-wave captures [ref] , and von Zeipel-Kozai-Lidov (ZKL) oscillations [ref] .

Eccentricity constraints disfavor single-single capture in nuclear star clusters as the origin of all LIGO-Virgo-KAGRA binary black holes · Introduction · 0.98
The relativity precession of the inner orbital pericenter is known to suppress the Kozai-Lidov oscillation [ref] if the time scale of the former is shorter than the later.

The three body first post-Newtonian effects on the secular dynamics of a compact binary near a spinning supermassive black hole · introduction · 0.84
Similarly, the dynamical channel has sub-channels, e.g., binaries arising from triple interactions [ref] , von Zeipel-Kozai-Lidov (ZKL) oscillations [ref] , binary-binary interactions [ref] , and binary-single interactions [ref] .

Evidence for eccentricity in the population of binary black holes observed by LIGO-Virgo-KAGRA · Introduction · 0.89

Discussion

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