⚛️ general relativity

No parametrisation, No Problem: A Weakly Modelled Framework to Constrain the Luminosity Distance--Redshift Relation Using Gravitational Wave Sirens

This paper introduces a weakly modelled, quasi-model-independent framework to reconstruct deviations from General Relativity in the luminosity distance-redshift relation using gravitational wave sirens, which successfully constrains various deviation patterns in current GWTC-5 data while demonstrating that third-generation detectors will be required to detect sub-5% deviations.

Original authors: Elena Colangeli, Konstantin Leyde, Tessa Baker, Anson Chen

Published 2026-08-20

📖 5 min read🧠 Deep dive

CC BY 4.0

Original authors: Elena Colangeli, Konstantin Leyde, Tessa Baker, Anson Chen

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

The universe is expanding, and for decades, astronomers have mapped this growth by measuring how far away galaxies are and how fast they are moving away from us. To do this, they traditionally rely on light. When a star explodes or a galaxy shines, its light carries a fingerprint of its distance and its speed, allowing scientists to trace the history of cosmic expansion. However, light can be dimmed or distorted by dust and gas along its journey, introducing uncertainty. In recent years, a new messenger has arrived: gravitational waves. These are ripples in the fabric of space-time itself, created by the violent collision of massive objects like black holes. Unlike light, these ripples travel through the universe unimpeded, offering a pristine way to measure distance. By comparing the distance measured by these ripples with the distance measured by light, scientists can test whether the rules of gravity, as described by Albert Einstein, hold true across the vast expanse of the cosmos. If gravity behaves differently over time or distance, it would signal a fundamental shift in our understanding of physics.

A team of researchers has developed a new, flexible method to perform this test without forcing the data into a rigid, pre-defined shape. Traditionally, scientists testing gravity have had to assume a specific mathematical formula for how gravity might change, essentially guessing the answer before looking at the evidence. This new approach, however, acts more like a flexible ruler that can bend to fit any shape the data suggests, whether the change is smooth, jagged, or wavy. The researchers applied this method to a massive collection of gravitational wave events detected by the LIGO-Virgo-KAGRA collaboration. They found that, within the limits of current data, gravity behaves exactly as Einstein predicted, with no signs of deviation. While current instruments are not yet sensitive enough to spot very subtle changes, the team showed that future, more powerful detectors could confidently identify even small departures from standard gravity, opening a new window into the fundamental laws of the universe.

The core of this work lies in comparing two ways of measuring distance. One is the standard distance derived from light, which follows the known laws of physics. The other is the distance derived from gravitational waves. In Einstein's theory, these two measurements should match perfectly. However, in some alternative theories of gravity, the strength of gravity might change over time, causing the gravitational wave distance to drift away from the light-based distance. To catch this drift, the researchers needed a way to look for it without assuming exactly what the drift would look like. They created a framework that reconstructs the relationship between distance and the expansion of the universe using a mathematical technique that can describe almost any pattern. This allows them to search for monotonic changes, where the effect grows steadily, as well as irregular or oscillating features, where the effect might rise and fall like a wave.

To test this new framework, the scientists first simulated thousands of gravitational wave events that a future observing run might capture. They injected different types of fake deviations into the data, including smooth curves, sudden bumps, and wavy patterns, to see if their method could find them. When they analyzed these simulations with the current generation of detectors, the results were inconclusive; the data was too noisy to distinguish the fake deviations from normal background noise. This is expected, as current detectors measure distances with a margin of error that is too large to see subtle shifts. However, the researchers also simulated data for a next-generation detector, the Einstein Telescope, which will be far more sensitive. In these simulations, the new method successfully recovered the injected deviations with high confidence, proving that the approach works and that future instruments will be capable of detecting even small changes in gravity.

Finally, the team applied their method to real data from the latest catalog of gravitational wave events, which includes hundreds of binary black hole collisions. They fixed the background model of the universe to known values and let their flexible framework search for any anomalies in the gravitational wave distances. The result was a clean bill of health for Einstein's theory. The data showed no evidence of any deviation; the gravitational wave distances matched the light-based distances exactly as predicted. Furthermore, the researchers demonstrated that their method did not distort the understanding of the black holes themselves. They were able to accurately reconstruct the distribution of black hole masses in the universe, matching previous findings from other teams. This confirms that their new, flexible way of testing gravity does not introduce hidden errors or biases into the analysis.

The study concludes that while current data supports the standard model of gravity, the door is wide open for future discoveries. The researchers note that with the arrival of third-generation detectors, which will observe tens of thousands of events, the precision will improve dramatically. They estimate that these future instruments could detect deviations as small as five percent with high statistical certainty. Until then, the new framework stands ready, offering a way to listen to the universe without imposing preconceived notions on what the music should sound like. It represents a shift from asking "Does the data fit our specific guess?" to "What does the data actually tell us?" ensuring that if gravity is indeed changing, we will be ready to hear it.

Technical Summary: No parametrisation, No Problem: A Weakly Modelled Framework to Constrain the Luminosity Distance–Redshift Relation Using Gravitational Wave Sirens

Problem Statement
Current cosmological tests of General Relativity (GR) using gravitational waves (GWs) typically rely on parametrised forms of the luminosity distance–redshift ( $d_L - z$ ) relation to constrain deviations from GR. These parametrizations, such as the $\Xi_0 - n$ ansatz or the $\alpha_M$ description of Horndeski theory, are often restricted to monotonic functions governed by one or two parameters. The authors argue that these forms fail to capture the full variety of modified gravity (MG) theories, which may exhibit non-trivial, oscillatory, or irregular deviations in the GW propagation distance. Furthermore, existing non-parametric reconstruction methods often rely on Gaussian Processes, scaling anchored by electromagnetic (EM) standard rulers, or piecewise interpolations. The paper seeks to address the need for a framework that can reconstruct deviations from GR without being biased by specific theoretical functional forms, while simultaneously inferring the population properties of GW sources.

Methodology
The authors introduce a "quasi-model-independent" (or "weakly modelled") framework implemented in a new gradient-based pipeline named CosmoPyro. The methodology combines hierarchical Bayesian inference with a Fourier expansion to reconstruct the ratio of GW luminosity distance to EM luminosity distance, defined as $r(z) = d_{GW}(z) / d_{EM}(z)$ .

Key methodological components include:

Fourier Reconstruction: The deviation ratio $r(z)$ is modeled via a Fourier expansion. The method draws independent samples from a Gaussian field, transforms them into Fourier space, and reweights the modes by a power spectrum $P(k)$ . This allows the reconstruction of any functional form (monotonic, oscillatory, or irregular) as the number of modes increases.
Physical Constraints: To ensure physical viability, the model imposes two constraints:
1. A suppression factor $1 - \exp(-z/z^*)^2$ forces $r(z) \to 1$ as $z \to 0$ , satisfying local constraints.
2. The GW distance is computed via a cumulative sum of its derivative to ensure $d_{GW}$ remains monotonic, preventing unphysical turn-overs, though this introduces mild asymmetry in the prior favoring $r(z) > 1$ .
Spectral Sirens: The analysis utilizes the "spectral sirens" method, leveraging the population mass distribution of binary black holes (BBHs) to break the redshift-distance degeneracy in dark siren events (those without EM counterparts). The authors jointly infer the parameters of the source-frame mass distribution (using a Multi-Peak model, MLTP) and the MG parameters.
Statistical Framework: The pipeline employs the No-U-Turn Sampler (NUTS) for efficient sampling of high-dimensional posteriors. It accounts for selection bias via a detection probability term $\beta(\Lambda)$ and utilizes a hierarchical likelihood that integrates over source-frame masses and redshifts.

Key Contributions

Framework Development: The paper presents a novel, weakly-modelled approach to reconstruct $r(z)$ that does not assume a specific MG theory, capable of detecting monotonic, irregular, and oscillatory features.
Joint Inference: The method successfully jointly constrains deviations from GR and the BBH population mass distribution parameters without introducing bias to the population inference.
Pipeline Implementation: The authors provide CosmoPyro, a gradient-based inference code capable of handling modified gravity analyses of hundreds of events efficiently (e.g., 500 events in ~7 hours on GPUs).
Validation: The framework is validated against mock data representing the LIGO-Virgo-KAGRA (LVK) O5 observing run and future Einstein Telescope (ET) scenarios, as well as applied to real data from the GWTC-5 catalog.

Results

Mock LVK O5 Data (Dark Sirens): Applying the method to 500 mock BBH events simulating the O5 run, the authors found that current detector sensitivities cannot resolve deviations from GR smaller than approximately 30% at $3\sigma$ significance. The reconstructed $r(z)$ was consistent with GR for all injected non-GR cosmologies (including wiggly, bump, damped, and monotonic $c_M$ models), with $1\sigma$ errors averaging 10%. Crucially, the method recovered the true source-frame mass distribution parameters accurately, demonstrating that the MG analysis does not bias population inference.
Mock ET Data (Bright Sirens): For 1000 mock bright siren events (with known redshifts) simulating the Einstein Telescope, the method successfully reconstructed the injected deviations from GR with $3\sigma$ significance. The average $3\sigma$ error on the ratio dropped to 9%, allowing the detection of oscillatory and sudden features (e.g., the "bump" model) that were undetectable with O5-like data.
GWTC-5 Real Data: Applied to 231 BBH events from the GWTC-5 catalog, the analysis found no deviations from GR. The reconstructed $r(z)$ was consistent with unity ( $r(z)=1$ ) for both Planck ( $H_0 = 67.4$ km/s/Mpc) and SH0ES ( $H_0 = 73.04$ km/s/Mpc) background cosmologies. The inferred mass distribution matched official LVK findings, confirming the robustness of the method against population biases.

Significance and Claims
The paper claims that its primary significance lies in moving beyond the restrictive assumptions of parametric MG models. By using a Fourier-based, weakly-modelled approach, the authors demonstrate that it is possible to search for a broader class of deviations from GR without pre-selecting a specific functional form.

The authors modestly note that while their method is ready for deployment on current data (as evidenced by the GWTC-5 analysis), the current generation of detectors (LVK O5) lacks the statistical power to resolve small deviations (order 5-10%). They assert that third-generation detectors (like the Einstein Telescope), particularly when utilizing bright sirens, will be required to confidently detect deviations of order 5% at $3\sigma$ significance. The work establishes a foundation for future non-parametric constraints on gravity that can be combined with other methods (e.g., galaxy catalogs or cross-correlations) to further improve precision.

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