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arxiv: 2602.11030 · v2 · pith:64SOJL3Gnew · submitted 2026-02-11 · 🌌 astro-ph.HE

Population Properties of Binary Black Holes with Eccentricity

Muhammad Zeeshan , Richard O'Shaughnessy , Natalie Malagon This is my paper

Pith reviewed 2026-05-16 02:24 UTC · model grok-4.3

classification 🌌 astro-ph.HE
keywords binary black holeseccentricitygravitational wave populationGWTC-4formation channelsorbital eccentricity
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The pith

The first population analysis of binary black holes including eccentricity finds that such events comprise less than 5 percent of the total.

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

This paper carries out the first simultaneous population fit for the mass, spin, redshift, and eccentricity of binary black holes from gravitational-wave detections. It uses the SEOBNRv5EHM waveform model to reanalyze 153 events in the GWTC-4 catalog and extends the standard population model to account for orbital eccentricity. The inferred properties match those from earlier studies that ignored eccentricity. The branching ratio for eccentric events is bounded below 0.052 at 90 percent confidence, though the exact limit depends on the chosen eccentricity distribution model and whether the most eccentric event is included. Overall the eccentric merger rate remains only weakly constrained by present data.

Core claim

The central claim is that a joint fit to mass, spin, redshift and eccentricity distributions of binary black holes can be performed with current data. Using parameter estimation with an eccentric waveform model on 153 events, the population properties stay consistent with quasi-circular results. The branching ratio for eccentric events is limited to below 0.05189 with all events and 0.02201 without GW200129 at 90 percent confidence under the Nonoverlapping Mixture model. Four different parametric forms for the eccentricity distribution all indicate that the eccentric rate is weakly constrained and highly model-dependent.

What carries the argument

The extension of the default O4a population model to include orbital eccentricity, combined with the Nonoverlapping Mixture model for the eccentricity distribution that allows direct bounding of the eccentric branching ratio.

Load-bearing premise

The SEOBNRv5EHM waveform model must accurately describe the signals from eccentric binary black holes so that the parameter estimates feeding into the population analysis are reliable.

What would settle it

A future catalog containing several events with clearly measured eccentricities above 0.1 that would require the eccentric branching ratio to exceed 0.05 would falsify the reported upper bounds.

Figures

Figures reproduced from arXiv: 2602.11030 by Muhammad Zeeshan, Natalie Malagon, Richard O'Shaughnessy.

Figure 1
Figure 1. Figure 1: FIG. 1 [PITH_FULL_IMAGE:figures/full_fig_p003_1.png] view at source ↗
Figure 2
Figure 2. Figure 2: FIG. 2 [PITH_FULL_IMAGE:figures/full_fig_p004_2.png] view at source ↗
Figure 3
Figure 3. Figure 3: FIG. 3 [PITH_FULL_IMAGE:figures/full_fig_p004_3.png] view at source ↗
Figure 5
Figure 5. Figure 5: FIG. 5 [PITH_FULL_IMAGE:figures/full_fig_p006_5.png] view at source ↗
Figure 6
Figure 6. Figure 6: FIG. 6 [PITH_FULL_IMAGE:figures/full_fig_p006_6.png] view at source ↗
Figure 7
Figure 7. Figure 7: compares the PPDs for redshift for the four model choices and to GWTC-4 results. Once again, our conclusions about redshift evolution are extremely con￾sistent with previously published results, reflecting our agreement on the inferred κ model hyperparameter [PITH_FULL_IMAGE:figures/full_fig_p007_7.png] view at source ↗
Figure 10
Figure 10. Figure 10: FIG. 10 [PITH_FULL_IMAGE:figures/full_fig_p008_10.png] view at source ↗
Figure 9
Figure 9. Figure 9: FIG. 9 [PITH_FULL_IMAGE:figures/full_fig_p008_9.png] view at source ↗
Figure 11
Figure 11. Figure 11: FIG. 11 [PITH_FULL_IMAGE:figures/full_fig_p010_11.png] view at source ↗
read the original abstract

The development of eccentric waveform models enables us to explore the growing catalog of gravitational-wave events with measurable eccentricity. This opens new opportunities to gain insight into the formation channels and evolutionary pathways of compact binary systems using eccentricity. However, most recent population analyses have been limited to quasi-circular binaries, primarily due to constraints in waveform modeling and sensitivity estimates. We are now entering an era where both of these limitations are being addressed, allowing for a more comprehensive investigation of eccentric binary populations. In this work, we perform a very first population analysis that simultaneously fits the mass, spin, redshift, and eccentricity distribution. Specifically, we use source-parameter estimation on 153 binary black holes in GWTC-4 catalog provided by the Rapid Iterative FiTting (RIFT) framework using the SEOBNRv5EHM waveform model. We extend the default O4a population model to include orbital eccentricity. We find that inferred population properties are broadly consistent with conclusions obtained in previous analyses assuming quasi-circular binaries. To assess sensitivity of our results to the most eccentric sources, we repeat our analysis excluding GW200129_065458. Consistent with our conclusions about each event and using Nonoverlapping Mixture eccentricity model, we bound the branching ratio for eccentric events to be below $0.051890$ and $0.022011$ at $90\%$ confidence with and without GW200129_065458 respectively. Using four different parametric population models for eccentricity, we argue that the rate of eccentric events is weakly constrained by observations and highly model-dependent.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Referee Report

3 major / 2 minor

Summary. The paper performs the first joint population inference on the mass, spin, redshift, and eccentricity distributions of 153 binary black holes from GWTC-4. Using RIFT parameter estimation with the SEOBNRv5EHM waveform model, it extends the standard O4a population model to include eccentricity, reports consistency with prior quasi-circular results, and derives 90% upper limits on the eccentric branching ratio of 0.05189 (with GW200129_065458) and 0.02201 (without it) under a Nonoverlapping Mixture model. Four parametric eccentricity population models are tested, with the conclusion that the eccentric rate is weakly constrained and highly model-dependent.

Significance. If the eccentricity posteriors and population-model assumptions hold, the work supplies the first simultaneous constraints on an eccentric subpopulation, directly addressing formation-channel questions that quasi-circular analyses cannot resolve. The explicit sensitivity test to GW200129_065458 and the multi-model comparison are positive features. However, the manuscript itself flags strong model dependence, so the numerical branching-ratio bounds have limited immediate impact until waveform systematics are quantified.

major comments (3)
  1. [Methods (RIFT analysis)] Methods section on RIFT + SEOBNRv5EHM: the headline 90% bounds on the eccentric branching ratio rest on eccentricity posteriors generated with SEOBNRv5EHM, yet the paper supplies neither catalog-level injection-recovery tests nor comparisons against independent eccentric models (TEOBResumS, NR surrogates). Because the Nonoverlapping Mixture model inherits these posteriors directly, the absence of such validation is load-bearing for the quoted numerical limits.
  2. [Results (population modeling)] Results, eccentricity population models: the four parametric forms are stated to produce highly model-dependent results, but the manuscript does not quantify how much the branching-ratio upper limits shift across the full set of models or provide a preferred model with justification. This weakens the claim that the rate is only weakly constrained.
  3. [Abstract and §4] Abstract and §4: the specific values 0.051890 and 0.022011 are presented without accompanying uncertainty from waveform systematics, even though the text already notes high model dependence. A single-event sensitivity test is performed, but broader robustness checks against model choice are missing.
minor comments (2)
  1. [Abstract] The quoted branching-ratio numbers are given to five or six decimal places; rounding to three significant figures would be more appropriate given the stated model dependence.
  2. [Population model section] Notation for the eccentricity population parameters (e.g., the mixture weights and shape parameters) should be defined once in a dedicated table or equation block for clarity.

Simulated Author's Rebuttal

3 responses · 0 unresolved

We thank the referee for their thorough review and constructive feedback on our manuscript. We appreciate the recognition of the work's novelty as the first joint population inference including eccentricity. Below we address each major comment point by point, with proposed revisions to improve clarity and robustness where appropriate.

read point-by-point responses

  1. Referee: [Methods (RIFT analysis)] Methods section on RIFT + SEOBNRv5EHM: the headline 90% bounds on the eccentric branching ratio rest on eccentricity posteriors generated with SEOBNRv5EHM, yet the paper supplies neither catalog-level injection-recovery tests nor comparisons against independent eccentric models (TEOBResumS, NR surrogates). Because the Nonoverlapping Mixture model inherits these posteriors directly, the absence of such validation is load-bearing for the quoted numerical limits.

    Authors: We agree that the lack of catalog-level injection-recovery tests and direct comparisons to alternative eccentric models (such as TEOBResumS or NR surrogates) is a limitation for validating the eccentricity posteriors that underpin the Nonoverlapping Mixture results. Performing such tests across the full catalog is computationally intensive and was outside the scope of this first analysis. SEOBNRv5EHM is the most advanced eccentric waveform model currently implemented in RIFT, which itself has been validated in prior quasi-circular studies. In revision we will expand the Methods section with an explicit discussion of these limitations, the rationale for model selection, and a statement that future work will include cross-model validation as resources permit. This is a partial revision. revision: partial

  2. Referee: [Results (population modeling)] Results, eccentricity population models: the four parametric forms are stated to produce highly model-dependent results, but the manuscript does not quantify how much the branching-ratio upper limits shift across the full set of models or provide a preferred model with justification. This weakens the claim that the rate is only weakly constrained.

    Authors: We thank the referee for this observation. Although the text notes high model dependence, we did not tabulate the numerical shifts in 90% upper limits across the four parametric forms. We will revise the Results section to add a table reporting the branching-ratio upper limits for each model, thereby quantifying the variation and reinforcing the conclusion that the eccentric rate is weakly constrained. We do not designate a preferred model because the data do not strongly favor any single form; this will be stated explicitly. This is a full revision. revision: yes

  3. Referee: [Abstract and §4] Abstract and §4: the specific values 0.051890 and 0.022011 are presented without accompanying uncertainty from waveform systematics, even though the text already notes high model dependence. A single-event sensitivity test is performed, but broader robustness checks against model choice are missing.

    Authors: We agree that the quoted numerical bounds (0.051890 and 0.022011) are specific to the Nonoverlapping Mixture model and SEOBNRv5EHM posteriors, without explicit inclusion of waveform systematics uncertainties. The single-event test is already present, but broader robustness to model choice will be strengthened by the table added in response to the previous comment. In revision we will update the Abstract and §4 to state more clearly that these values are model-specific, note the absence of waveform systematics uncertainties, and cross-reference the expanded multi-model comparison. This is a partial revision. revision: partial

Circularity Check

0 steps flagged

No circularity: standard hierarchical inference on catalog data

full rationale

The paper's central results (eccentric branching-ratio upper limits of 0.05189 and 0.02201 at 90% CL) are obtained via standard Bayesian population inference: RIFT+SEOBNRv5EHM posteriors on 153 GWTC-4 events are fed into an extended hierarchical model whose eccentricity component (Nonoverlapping Mixture and three other parametric forms) is fitted directly to those posteriors. The bounds are posterior quantiles on a model parameter; they are not defined in terms of themselves, not obtained by renaming a fit, and not justified solely by self-citation. The derivation chain is self-contained against external data and does not reduce to tautology.

Axiom & Free-Parameter Ledger

2 free parameters · 2 axioms · 0 invented entities

The analysis relies on standard gravitational wave analysis tools and population inference techniques, with the main free parameters being those in the eccentricity distribution models.

free parameters (2)
  • parameters of eccentricity population models
    Four different parametric models for eccentricity are used, with parameters fitted to the data.
  • branching ratio for eccentric events
    The upper limit on the fraction of eccentric events is inferred from the population model fit.
axioms (2)
  • domain assumption SEOBNRv5EHM accurately models eccentric signals for parameter estimation
    Used to analyze the 153 events from GWTC-4.
  • domain assumption The default O4a population model can be extended to include orbital eccentricity
    The extension is performed in the analysis.

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Forward citations

Cited by 3 Pith papers

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  1. A universal framework to identify eccentric binary mergers: GW200105 case study

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    A reference-frequency-independent detection statistic for eccentric binary mergers is introduced and applied to GW200105, yielding ln B ≤ 0.9 in favor of the eccentric aligned-spin model over the quasi-circular preces...

  2. Secondary-Mass Features improve Spectral-Siren $H_0$ Constraints

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    A new model emphasizing secondary mass features and pairing transitions improves spectral siren H0 constraints by ~30% using 142 GW events from GWTC-4.0.

  3. Assessing the imprint of eccentricity in GW signatures using two independent waveform models

    astro-ph.HE 2026-05 conditional novelty 5.0

    Dual-model analysis of 162 GW sources disfavors eccentricity for most events but finds potential evidence in GW200129, GW231001, and GW231123.

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