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A model-independent measurement of the Hubble constant from gravitational-wave standard sirens and electromagnetic observations

This paper employs a fully model-independent Gaussian process method combining gravitational-wave standard siren data from GWTC3 with electromagnetic observations to measure the Hubble constant, finding results consistent with the SH0ES measurement and suggesting no statistically significant discrepancy.

Original authors: Gaurav N. Gadbail, Kazuharu Bamba

Published 2026-06-03

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CC BY 4.0

Original authors: Gaurav N. Gadbail, Kazuharu Bamba

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 Big Problem: The Universe's Speedometer is Broken

Imagine the universe is a car driving down a highway. Cosmologists want to know exactly how fast it is going right now. This speed is called the Hubble Constant ( $H_0$ ).

For a long time, scientists have been measuring this speed using two different methods, and they are getting two very different answers. It's like one mechanic says the car is going 73 mph, while another says it's going 67 mph. Both mechanics are using high-quality tools, but the gap between their numbers is so big that it can't just be a mistake. This disagreement is called the "Hubble Tension," and it's one of the biggest mysteries in physics today.

The New Approach: A "Model-Independent" Map

Usually, to measure the universe's speed, scientists have to assume a specific story about how the universe works (like assuming the car has a specific engine type). If that story is wrong, the speed measurement is wrong.

The authors of this paper decided to try a different approach. Instead of guessing the story first, they used a mathematical tool called a Gaussian Process (GP).

The Analogy: Imagine you have a bunch of scattered dots on a piece of paper representing the universe at different distances. Instead of forcing a ruler or a specific curve through them based on a theory, the GP is like a flexible, stretchy rubber band. It connects the dots in the smoothest way possible, letting the data itself decide the shape of the curve. This way, they don't have to assume any specific "engine type" for the universe; they just let the observations speak.

The New Data: Listening to the Universe's "Sirens"

To make their rubber band map more accurate, the authors added a new type of data that hasn't been used much before: Gravitational Waves.

The Analogy: Think of electromagnetic light (like from stars or supernovas) as a flashlight. It tells you where something is, but to know how far away it is, you have to guess how bright the bulb actually is.
The New Tool: Gravitational waves are like sirens (think of an ambulance). When two massive objects (like black holes) crash together, they send out ripples in space-time. The shape of the "siren's wail" tells you exactly how loud the crash was. Because you know the true "loudness," you can calculate the distance perfectly without needing to guess. These are called "Standard Sirens."

The authors took data from the GWTC-3 catalog (a list of 90 recent gravitational wave events) and combined it with traditional light-based data (like supernovas and galaxy ages).

What They Found

By stretching their "rubber band" (the Gaussian Process) across both the light data and the siren data, they calculated the current speed of the universe.

The Result: Their measurement came out to be very close to the 73 mph side of the argument (the SH0ES result).
The Agreement: Their result is so close to the "73 mph" measurement that the difference is statistically tiny (less than 2 sigma). This means their new method strongly agrees with the local measurements and does not agree with the older "67 mph" measurements.
The Benefit: Adding the gravitational wave "sirens" made the measurement much sharper. It reduced the uncertainty (the "fuzziness" of the result) by about 15–20% compared to using just the light data.

The Catch

The authors are careful to note that while this method is great, the gravitational wave data they used is still a bit "noisy." For many of the wave events, they couldn't see the light (the host galaxy) directly, so they had to make some educated guesses about where the waves came from. This adds a little bit of uncertainty, but not enough to change their main conclusion.

The Bottom Line

This paper shows that by using a flexible, assumption-free math tool and combining traditional light data with new gravitational wave "sirens," we can measure the universe's expansion rate more precisely. The result leans heavily toward the "faster" expansion rate, suggesting that the "Hubble Tension" might not be solved by changing the rules of physics, but perhaps by refining how we measure the distance. It's a promising new step in solving the mystery of the universe's speed.

Technical Summary: A Model-Independent Measurement of the Hubble Constant from Gravitational-Wave Standard Sirens and Electromagnetic Observations

Problem Statement
The paper addresses the "Hubble tension," a significant discrepancy of approximately $5\sigma - 6\sigma$ between early-universe measurements of the Hubble constant ( $H_0$ ) derived from the Cosmic Microwave Background (CMB) under the $\Lambda$ CDM model ( $H_0 = 67.4 \pm 0.5$ km s $^{-1}$ Mpc $^{-1}$ ) and late-universe direct measurements using Type Ia supernovae (SNe Ia) and the local distance ladder ( $H_0 = 73.04 \pm 1.04$ km s $^{-1}$ Mpc $^{-1}$ ). Traditional methods often rely on specific cosmological models or assumptions regarding the sound horizon scale. The authors aim to provide a fully model-independent determination of $H_0$ that avoids these theoretical biases by relying solely on observational data, incorporating both electromagnetic (EM) and gravitational-wave (GW) datasets.

Methodology
The study employs a Gaussian Process (GP) regression, a non-parametric Bayesian method, to reconstruct the luminosity distance function, $D_L(z)$ , and its first derivative directly from observational data without assuming a predefined functional form (e.g., $\Lambda$ CDM).

Framework: The authors utilize the GaPP Python code. They model $D_L(z)$ as a Gaussian process defined by a mean function and a covariance kernel. The squared exponential kernel is selected over Matérn kernels due to its stability and well-behaved derivatives.
Derivation of $H_0$ : The Hubble constant is derived analytically from the reconstructed function using the relation:
$H_0 = c \left[ \frac{d}{dz} \left( \frac{D_L(z)}{1+z} \right) \bigg|_{z=0} \right]^{-1}$
This requires accurate reconstruction of both $D_L(z)$ and its derivative at $z=0$ .
Data Integration: The analysis combines:
- Gravitational Waves: Data from the GWTC-3 catalog (90 candidates with $p_{astro} > 0.5$ ), providing absolute luminosity distances independent of the cosmic distance ladder.
- Electromagnetic Probes:
  - Cosmic Chronometers (CC): Direct $H(z)$ measurements converted to $D_L(z)$ .
  - DESI BAO DR2: Baryon Acoustic Oscillation data, where the sound horizon $r_d$ is treated as a nuisance parameter and marginalized over to maintain model independence.
  - Type Ia Supernovae: Three compilations (Pantheon+SH0ES, Union 3.0, and DES-SN5YR).
  - Gamma-Ray Bursts (GRBs): Used as high-redshift indicators via the Amati relation.
Error Handling: The method accounts for peculiar velocity errors at low redshifts ( $z < 0.05$ ) by adding them in quadrature to observational errors. Crucially, the GP framework explicitly computes and propagates the covariance between the reconstructed function $D_L(z)$ and its derivative $D'_L(z)$ to ensure statistically consistent uncertainty estimation.

Key Contributions

Hybrid Model-Independent Approach: This work is among the first to integrate GWTC-3 standard siren data into a GP framework alongside diverse EM datasets to constrain $H_0$ without relying on the sound horizon scale or specific dark energy models.
Quantification of GW Impact: The study explicitly isolates the contribution of GW data to the precision of $H_0$ . By comparing GP runs with and without GW data, the authors demonstrate that adding GWTC-3 data reduces the $1\sigma$ uncertainty on $H_0$ by approximately 15–20% (from $1.02$ to $0.58$ km s $^{-1}$ Mpc $^{-1}$ in a specific combination).
Covariance Treatment: The implementation of analytical differentiation within the GP framework allows for the rigorous handling of correlations between the distance function and its derivative, a critical factor in minimizing bias at $z=0$ .

Results
The model-independent reconstruction yields $H_0$ values that lie between the Planck and SH0ES results but show a strong preference toward the SH0ES value ( $H_0 = 73.04 \pm 1.04$ km s $^{-1}$ Mpc $^{-1}$ ).

For most data combinations, the tension between the derived $H_0$ and the SH0ES result is less than $2\sigma$ , indicating no statistically significant discrepancy.
The inclusion of GW data tightens the constraints significantly compared to EM-only analyses.
One specific combination (GWTC3+CC+DESI+DESY5) slightly favors the Planck measurement, but the overall trend aligns with local distance ladder results.

Significance and Claims
The authors modestly claim that while this study does not definitively resolve the Hubble tension, it provides robust, consistent $H_0$ estimates that favor the SH0ES determination using only observational inputs. The significance of the work lies in:

Validation of GW Data: Demonstrating that GW standard sirens, even with current limitations (low redshift coverage and statistical uncertainties), provide an absolute calibration that meaningfully improves the precision of $H_0$ measurements when combined with EM data.
Consistency Check: Confirming that GW and EM measurements are consistent with each other within the model-independent framework.
Future Utility: Establishing a methodology that can be extended to future, larger GW catalogs from next-generation detectors and more precise EM datasets, offering a pathway to investigate the origin of the Hubble tension without theoretical bias.

The paper acknowledges current limitations, such as the reliance on population-level assumptions for redshift inference in BBH events without electromagnetic counterparts, noting that a rigorous treatment of these systematics remains a direction for future work.