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O5 dark-siren forecasts for modified GW propagation: background robustness of the $\Xi$ posterior

This study demonstrates that forecasts for modified gravitational-wave propagation using O5 dark sirens yield a robust posterior for the modification parameter $\Xi$ that is insensitive to variations in the matter density $\Omega_m$ , though achieving sub-percent precision on $\Xi$ will still require measured host redshifts to overcome the large uncertainties inherent in galaxy-catalog-free analyses.

Original authors: Zhaorui Zhang, Hong-Bo Jin

Published 2026-07-14

📖 5 min read🧠 Deep dive

CC BY 4.0

Original authors: Zhaorui Zhang, Hong-Bo Jin

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 giant, expanding trampoline. For decades, scientists have been trying to measure how fast this trampoline is stretching (a speed called the Hubble constant, or $H_0$ ) and checking if the fabric of the trampoline follows the standard rules of physics or if it's behaving a bit weirdly (a test for modified gravity, represented by a number called $\Xi$ ).

Usually, to measure this stretch, scientists need to know exactly where the "bump" on the trampoline is. In the world of gravitational waves, these bumps are collisions of black holes. If we can see the light from the crash (like a flashbulb going off), we know the distance perfectly. These are called "bright sirens." But most of the time, especially in the upcoming O5 observing run, the black holes collide in the dark with no light to see. These are "dark sirens."

The big question this paper asks is: If we don't know exactly where the dark sirens are, does our guess about the universe's ingredients (specifically the amount of matter, called $\Omega_m$ ) mess up our measurement of whether gravity is acting weird?

The Great Cosmic Tug-of-War

To answer this, the authors ran a massive simulation using a computer program called CHIMERA 2.0. They created a fake universe with 300 fake black hole collisions (dark sirens) and tried to solve the mystery of gravity without any galaxy catalogs (no maps to tell them where the sirens are).

Here is the clever trick they used: They played a game of "What if?" They changed the amount of matter in their fake universe ( $\Omega_m$ ) from 0.20 to 0.35 (a range that includes the value measured by the Planck satellite, 0.315).

The Forecast Result:
No matter how much they changed the amount of matter in the simulation, the answer for the gravity test ( $\Xi$ ) stayed exactly the same.

The result was $\Xi = 0.9783 \pm 0.3548$ .
This means the uncertainty is about 36.3%.
The value is consistent with standard gravity (where $\Xi = 1.0$ ), but the "fuzziness" is huge.

Why did this happen?
Think of it like a seesaw. The gravitational waves tell us the distance to the black hole, but that distance depends on both the expansion speed ( $H_0$ ) and the amount of matter ( $\Omega_m$ ).

When the authors increased the matter ( $\Omega_m$ ), the "expansion speed" ( $H_0$ ) automatically shifted to compensate, sliding along the seesaw to keep the total distance the same.
Because the gravity test ( $\Xi$ ) is a ratio of distances, and the distance stayed locked in place by this seesaw effect, the result for $\Xi$ didn't budge. It was robust.

What This Suggests

The paper explicitly argues against the idea that you need to worry about picking the "perfect" value for the amount of matter ( $\Omega_m$ ) when testing for weird gravity with dark sirens in the O5 era.

Some previous studies fixed $\Omega_m$ to a single value (like 0.3) and assumed that was fine.
This paper suggests that even if you pick a different valid value (like 0.25 or 0.35), your conclusion about weird gravity ( $\Xi$ ) does not change in these simulations.
So, the "systematic error" of guessing the wrong amount of matter is not the problem here.

The Real Problem: Missing Redshifts

If the amount of matter isn't the problem, what is? The paper points the finger at missing redshifts (the speed at which the host galaxy is moving away from us).

Without redshifts (Dark Sirens only): The precision on the gravity test is $\pm 36.3\%$ . It's like trying to guess the weight of a cat while wearing thick mittens; you can tell it's a cat, but you can't be sure if it's a kitten or a tiger.
With redshifts (Galaxy Catalogs): The paper compares their result to a study that did have galaxy maps. With those maps, the precision jumped to $\sim 7.5\%$ .
The Conclusion: The huge uncertainty isn't because we guessed the wrong amount of matter in the universe. It's because we don't know exactly where the black holes are. To get a "sub-percent" (super precise) test of gravity, we must have measured redshifts, even if dark sirens dominate the detection rate.

How Sure Are We?

It is important to remember that these results come from simulations, not real data yet.

The authors used 300 mock events generated by a computer to forecast what might happen during the O5 observing run.
They tested different scenarios (changing $\Omega_m$ , changing the number of events) and the pattern held up every time in the model.
They even cross-checked their math using two different methods (a grid search and a method called MCMC) and got the same answer.

So, while we haven't measured this in the real world yet, the simulation strongly suggests that for the upcoming O5 run, scientists can stop worrying about whether their guess for the amount of matter ( $\Omega_m$ ) is slightly off. That guess won't break their gravity test. Instead, they should focus their energy on finding the host galaxies to get those crucial redshift measurements.

In short: The "background" of the universe (how much matter is in it) is a flexible friend that won't ruin your gravity test. But the lack of a map (redshifts) is the real villain keeping us from seeing the truth clearly.

Technical Summary: O5 Dark-Siren Forecasts for Modified GW Propagation

Problem Statement
Gravitational-wave (GW) standard sirens offer an independent path to measure the Hubble constant ( $H_0$ ) and test General Relativity (GR) through the phenomenological parameter $\Xi$ , which characterizes modified GW propagation ( $D_L^{GW} = \Xi D_L^{EM}$ ). While the upcoming O5 observing run is expected to yield hundreds of binary black hole (BBH) detections, the majority will be "dark sirens" lacking electromagnetic counterparts and host galaxy redshifts. Previous forecasts, such as those by Tagliazucchi et al. (2025) using the CHIMERA 2.0 framework, have constrained $\Xi$ to $\sim 7.5\%$ precision but relied on two potentially limiting assumptions: (1) the use of spectroscopic galaxy catalogs to identify hosts, and (2) fixing the matter density parameter $\Omega_m$ to a single CMB-informed value (e.g., 0.3).

This study addresses a critical gap: the robustness of the $\Xi$ posterior when applied to pure dark sirens (without galaxy catalogs) and the sensitivity of these constraints to variations in the assumed cosmological background ( $\Omega_m$ ). Specifically, it investigates whether fixing $\Omega_m$ biases the $\Xi$ constraint when host redshifts are unavailable.

Methodology
The authors utilize the public CHIMERA 2.0 hierarchical Bayesian pipeline and a mock O5 dataset consisting of 300 BBH events with signal-to-noise ratios $>20$ . The injected cosmology is $H_0 = 70$ km s $^{-1}$ Mpc $^{-1}$ , $\Omega_m = 0.25$ , and $\Xi = 1.0$ (GR). No galaxy catalogs or host redshifts are used.

The analysis employs a grid-based inference approach to evaluate the likelihood across parameter spaces:

Fixed-Background Analysis: The authors vary $\Omega_m$ over the CMB-allowed range $[0.20, 0.35]$ (including Planck 0.315). At each fixed $\Omega_m$ , they evaluate the joint posterior of $H_0$ and $\Xi$ on a $24 \times 24$ grid.
Joint Inference: A fully marginalized joint inference is performed over $(H_0, \Omega_m, \Xi)$ on a $12 \times 12 \times 12$ grid to compare with the fixed-background results.
Validation: The grid-based results are cross-checked against an independent Markov Chain Monte Carlo (MCMC) run at $\Omega_m = 0.25$ to ensure the discretization method does not introduce artifacts.
Scaling Tests: The analysis includes event-number scaling (30 to 300 events) and a comparison with the galaxy-catalog scenario from Tagliazucchi et al. (2025) to quantify the loss of precision due to missing redshifts.

Key Results

Robustness of $\Xi$ : The marginalized posterior for $\Xi$ is remarkably stable across the entire tested range of $\Omega_m$ . For pure dark sirens, the result is $\Xi = 0.9783 \pm 0.3548$ (68% credible interval) regardless of whether $\Omega_m$ is fixed at 0.20, 0.25, 0.30, 0.315, or 0.35. The median and width of the $\Xi$ distribution show no variation (span $< 10^{-8}$ ) across this interval.
Degeneracy Behavior: The stability of $\Xi$ is driven by the distance degeneracy. The likelihood constrains the combination $\Xi D_L^{EM}(H_0, \Omega_m)$ . When $\Omega_m$ is varied, the posterior compensates by shifting the marginalized $H_0$ median. For example, moving $\Omega_m$ from 0.20 to 0.35 shifts the $H_0$ median by $\approx 4.2$ km s $^{-1}$ Mpc $^{-1}$ (a slope of $\approx -2.8$ km s $^{-1}$ Mpc $^{-1}$ per 0.1 $\Omega_m$ ), while the $\Xi$ marginal remains unchanged. The correlation between $H_0$ and $\Xi$ is weak ( $|\rho| \lesssim 0.05$ ).
Joint Inference: In the fully joint $(H_0, \Omega_m, \Xi)$ analysis, the result is $\Xi_{joint} = 0.9550 \pm 0.3710$ , with negligible correlation between $\Omega_m$ and $\Xi$ ( $|\rho| \lesssim 0.05$ ).
Impact of Missing Redshifts: The primary limitation for precision is the lack of host redshifts, not the choice of $\Omega_m$ . Compared to the galaxy-catalog analysis of Tagliazucchi et al. (2025) which achieved $\sim 7.5\%$ precision on $\Xi$ , the pure dark-siren analysis yields $\pm 36.3\%$ precision. This represents a $\sim 5\times$ degradation in precision solely due to missing redshift information.
$H_0$ Constraints: The $H_0$ constraint is highly sensitive to the assumed $\Omega_m$ . At $\Omega_m = 0.25$ , $H_0 = 67.9 \pm 3.1$ km s $^{-1}$ Mpc $^{-1}$ . Without external constraints on $\Omega_m$ or measured redshifts, $H_0$ measurements from dark sirens carry a systematic floor of several km s $^{-1}$ Mpc $^{-1}$ .

Significance and Claims
The paper claims that for O5 sensitivity and pure dark-siren populations, the choice of a CMB-informed $\Omega_m$ (within the range 0.20–0.35) does not bias the constraint on modified gravity ( $\Xi$ ). This validates the use of fixed-background pipelines for monitoring modified gravity in the absence of host redshifts, as the systematic uncertainty introduced by $\Omega_m$ variation is absorbed entirely by $H_0$ .

The authors emphasize that the dominant systematic for dark-siren cosmology is the $\Omega_m$ - $H_0$ degeneracy, while the dominant systematic for modified-gravity tests is the lack of redshift information. Consequently, achieving sub-percent precision on $\Xi$ will require measured host redshifts, even if dark sirens dominate the detection rate. The study confirms that previous fixed- $\Omega_m$ forecasts for $\Xi$ are robust against background variations, but highlights that the precision gap between dark sirens and bright-siren/galaxy-catalog analyses is driven by redshift availability, not cosmological model assumptions.

The Great Cosmic Tug-of-War

What This Suggests

The Real Problem: Missing Redshifts

How Sure Are We?

Technical Summary: O5 Dark-Siren Forecasts for Modified GW Propagation

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