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Assessing the waveform systematics from parameter estimation to population inference with eccentricity

This study analyzes gravitational wave events from the GWTC-4 catalog using two eccentric waveform models to demonstrate that while individual source parameter estimates are largely consistent, subtle systematic differences between models can accumulate in hierarchical population inference to significantly alter conclusions about redshift evolution and effective spin distributions.

Original authors: Muhammad Zeeshan, Richard O'Shaughnessy, Natalie Malagon, Katelyn J. Wagner

Published 2026-07-20

📖 7 min read🧠 Deep dive

CC BY 4.0

Original authors: Muhammad Zeeshan, Richard O'Shaughnessy, Natalie Malagon, Katelyn J. Wagner

Original paper licensed under CC BY 4.0 (http://creativecommons.org/licenses/by/4.0/). ✨ This is an AI-generated explanation of the paper below. It is not written or endorsed by the authors. For technical accuracy, refer to the original paper. Read full disclaimer

Imagine the universe as a vast, dark ocean, and gravity as the water itself. For decades, we thought this ocean was perfectly still, but then, in 2015, we built our first "hydrophones" capable of hearing the tiniest ripples caused by colliding black holes and neutron stars. These ripples are called gravitational waves. When two massive objects spiral into each other, they don't just crash; they sing a specific song as they merge. By listening to this song, scientists can figure out the "ingredients" of the collision: how heavy the objects are, how fast they are spinning, and how far away they are.

But here is the tricky part: the song isn't just a simple hum. It's a complex, shifting melody that changes depending on how the objects are moving. Sometimes, they might be spinning wildly or orbiting in a slightly squashed circle (like an egg shape) rather than a perfect circle. To understand the song, scientists use "waveform models"—essentially, mathematical sheet music that predicts what the sound should look like. If the sheet music is slightly off, we might mishear the singer's pitch or how far away they are. This paper dives into a critical question: If we use two different, very advanced pieces of sheet music to listen to the same cosmic concert, do we end up with the same story about the universe, or do we start hearing different versions of the same song?

The Cosmic Orchestra and the Sheet Music Problem

Think of the universe's population of colliding black holes as a giant, cosmic orchestra. For years, scientists have been trying to figure out how this orchestra formed. Did the musicians (the black holes) grow up together in a quiet neighborhood (isolated evolution), or did they meet by chance in a crowded, chaotic dance hall (dynamic assembly)? One of the biggest clues is "eccentricity." If two black holes are dancing in a perfect circle, they likely grew up together. But if their dance is a wobbly, squashed oval, they probably met by accident in a crowded cluster.

To decode these dances, researchers use powerful computers to analyze the gravitational waves detected by observatories like LIGO, Virgo, and KAGRA. They have two very sophisticated "sheet music" models to help them: SEOBNRv5EHM and TEOBResumS-Dali. Both are state-of-the-art, designed to handle the messy, squashed orbits of eccentric binaries. The big hope was that these two models would tell the same story about the universe.

The Subtle Whisper That Becomes a Roar

The authors of this paper took a massive catalog of 162 events (153 black hole collisions, 2 neutron star collisions, and 7 neutron star-black hole collisions) and ran them through both models. They wanted to see if the two models agreed on the details.

Here is the surprising twist: On a single event, the models agreed almost perfectly. If you looked at just one black hole collision, the difference between what Model A said and what Model B said was tiny—like two people hearing a whisper and agreeing on the general idea.

However, the paper reveals a hidden danger. When you stack up hundreds of these tiny, almost invisible differences, they start to add up. It's like a game of "telephone" played a thousand times. Each time you pass the message, the distortion is so small you don't notice it. But by the end of the line, the message has changed completely.

The authors found that while the models agreed on the mass of the black holes, they started to disagree on how far away the events were and how the rate of black hole collisions changes as we look back in time (redshift evolution).

The Distance Drift: One model consistently thought the distant black holes were a little bit farther away than the other model did.
The Spin Shift: For the slowest-spinning black holes, one model saw them spinning slightly faster in the positive direction than the other.

The Population Puzzle

Why does this matter? Because scientists don't just study one black hole; they study the whole population to understand how the universe works. The paper shows that these tiny, systematic errors accumulate.

When the researchers used the two different models to infer the "population properties" (the big picture of how black holes form and evolve), the results diverged significantly:

Redshift Evolution: The models gave different answers about how the rate of black hole collisions changes as we look back in time. One model suggested a smooth, gradual change, while the other suggested a different pattern.
Spin Distribution: The models disagreed on the distribution of spins for the entire population, especially when a few extreme events were included.

The paper explicitly argues against the idea that current waveform systematics are "good enough" to ignore. They demonstrate that even with the best models available, coherent biases (systematic errors that happen in the same direction for every event) can grow as the square root of the number of events ( $\sqrt{N}$ ). This means that as our catalogs of detected black holes grow from 100 to 1,000 to 10,000, these tiny errors won't just stay small; they will become the dominant feature of our analysis, potentially leading us to the wrong conclusions about how the universe formed.

Testing with a "Fake" Universe

To prove their point and test their methods, the authors created a "synthetic universe." They generated a fake catalog of 155 binary black holes with known, specific properties (like a known distribution of masses, spins, and eccentricities). They then ran their analysis tools on this fake data to see if they could recover the "true" answer.

The results were encouraging but cautious. Even though their detection tools only "heard" 39 of the 155 fake events (due to noise limits), they were able to recover the general shape of the mass and eccentricity distributions. This suggests their method works, but it also highlights that we need a perfectly consistent framework to avoid the very errors they are warning about.

The "NS-BH" Mystery

The paper also looked at a smaller, trickier group: collisions between a neutron star and a black hole (NSBH). They included a controversial event, GW190814, which involved a very light black hole and a heavier neutron star.

They found that the mass distribution of these objects didn't show a clear "gap" between neutron stars and black holes; the black holes seemed to smoothly extend down to the mass of the neutron stars.
Regarding eccentricity, they found that for the specific event GW200105, the evidence for a squashed orbit was strong. However, at the population level, the eccentricity distribution looked very similar for both models, suggesting that the "circular" population dominates the overall picture, masking the eccentric ones.

The Bottom Line

This paper is a wake-up call for the gravitational wave community. It suggests that we can no longer assume that "good enough" models are sufficient for the future. As we collect more data, the tiny, consistent differences between our mathematical models will become the biggest source of error.

The authors conclude that to get the true story of the universe, we must stop relying on a single model or a mix of models without checking for these biases. We need to run our analyses with multiple models simultaneously and account for the fact that our "sheet music" might be slightly out of tune. If we don't, we risk hearing a beautiful, but completely wrong, song from the cosmos.

Technical Summary: Assessing Waveform Systematics from Parameter Estimation to Population Inference with Eccentricity

Problem Statement
As gravitational wave (GW) surveys increase in sensitivity, the catalog of compact binary sources is expanding to include diverse systems, potentially exhibiting exceptional properties such as orbital eccentricity. While masses and spins are routinely used to constrain formation channels, eccentricity offers a powerful diagnostic for dynamically assembled systems. Although recent advances in eccentric waveform modeling (e.g., SEOBNRv5EHM and TEOBResumS-Dali) enable the search for eccentric signatures, differences between these models introduce systematic effects. The central problem addressed is whether these subtle, consistent biases in source-level parameter estimation propagate into astrophysical population inference, potentially altering conclusions about the underlying population properties (e.g., redshift evolution, spin distributions) as catalogs grow.

Methodology
The authors perform an end-to-end hierarchical Bayesian population inference analysis using the GWKokab engine. The study utilizes:

Data: A sample of 153 binary black holes (BBH), 2 binary neutron stars (BNS), and 7 neutron star-black hole (NSBH) binaries from the GWTC-4 catalog.
Waveform Models: Source parameter inferences are compared across two state-of-the-art eccentric waveform models (SEOBNRv5EHM and TEOBResumS-Dali) and quasi-circular analyses. The RIFT framework is used for parameter estimation, employing all available higher-order modes.
Population Models:
- BBH: A Broken Power Law + 2 Peaks mass distribution, a Power Law redshift evolution, and a Skew-Normal effective spin model. A new single-parameter power-law eccentricity distribution ( $p(e) \propto e^{\alpha_e}$ ) is introduced.
- BNS/NSBH: A simplified joint model sharing information across source types, using truncated Gaussians for mass and spin distributions, and a common power-law eccentricity and redshift evolution.
- Synthetic Data: A synthetic population is generated using a truncated power-law mass model, truncated normal spin, and half-normal eccentricity distribution to validate the inference framework via injection studies.
Analysis: The study employs frequentist diagnostics (comparing sample means scaled by $\sqrt{N}$ ) to identify systematic offsets between models and propagates these into full population posterior distributions.

Key Contributions

Systematic Bias Identification: The work identifies a small but consistent bias in recovered distances between the two eccentric waveform models when applied to the GWTC-4 sample.
Propagation to Population Level: The authors demonstrate that while individual event differences are often subtle and within statistical uncertainty, these biases accumulate coherently across a catalog. This accumulation scales approximately as $\sqrt{N}$ , leading to significant differences in inferred population properties.
Synthetic Framework: A new synthetic data framework is introduced that generates eccentric populations and corresponding RIFT posterior samples, enabling rigorous injection studies to test the recoverability of eccentric population properties.
NS-BH Population Characterization: A joint population inference for BNS and NSBH systems is performed, explicitly including GW190814, to characterize mass and spin distributions without assuming a mass gap.

Results

Source-Level Consistency: For most events, the two eccentric models yield broadly consistent source parameter estimates. However, subtle coherent differences exist, particularly in distance (redshift) and effective spin ( $\chi_{\text{eff}}$ ).
Population-Level Divergence:
- Redshift Evolution: The inferred redshift distribution is not robust across models. SEOBNRv5EHM results are consistent with quasi-circular re-analyses, whereas TEOBResumS-Dali predicts a more gradual evolution of the merger rate with redshift. This discrepancy persists even when excluding extreme events like GW231123 135430.
- Spin Distribution: The effective spin distribution shows differences between models, though these are sensitive to the inclusion of specific massive events.
- Mass Distribution: Primary mass and mass ratio distributions remain largely consistent across models, though the peak at $33 M_\odot$ is less sharp in eccentric waveform analyses compared to quasi-circular ones.
- Eccentricity: At the population level, the recovered eccentricity distributions from both models are consistent with each other, showing no significant waveform systematics, likely due to the dominance of the circular population.
Synthetic Validation: In synthetic data tests, the framework successfully recovered injected mass and eccentricity distribution parameters. However, the redshift evolution and spin location parameters showed broader consistency with injected values, highlighting the need for fully self-consistent detection statistics in future injection studies.
NS-BH Findings: The joint analysis suggests a smooth mass distribution extending to $3 M_\odot$ without a distinct "mass gap" between neutron stars and black holes. The analysis finds less evidence for large black hole spins in BH-NS binaries compared to some previous studies, influenced by GW190814.

Significance and Claims
The paper claims that contemporary GW waveform systematics are not merely negligible noise but can accumulate to impact population-level conclusions, specifically regarding redshift evolution and effective spin distributions. The authors argue that the often-accepted conclusion that current waveform systematics are sufficient for interpreting the GW population is challenged by these findings. They emphasize that as catalogs expand, coherent event-level biases will become increasingly important. Consequently, the authors recommend that future population inferences employ self-consistent inference using multiple waveforms rather than relying on a single analysis with mixed samples. The study concludes that while broad population properties may remain robust, subtle features and subpopulations are inevitably impacted by these systematics.