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The Stochastic Siren: Astrophysical Gravitational-Wave Background Measurements of the Hubble Constant

This paper proposes a novel method to measure the Hubble constant using the stochastic gravitational-wave background from binary black hole mergers as a "stochastic siren," demonstrating that combining this approach with resolved merger data can improve measurement precision and potentially help resolve the Hubble tension.

Original authors: Bryce Cousins, Kristen Schumacher, Adrian Ka-Wai Chung, Colm Talbot, Thomas Callister, Daniel E. Holz, Nicolás Yunes

Published 2026-02-03

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CC BY 4.0

Original authors: Bryce Cousins, Kristen Schumacher, Adrian Ka-Wai Chung, Colm Talbot, Thomas Callister, Daniel E. Holz, Nicolás Yunes

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 is Growing, but We Can't Agree on How Fast

Imagine the Universe is a giant balloon being blown up. Scientists have known for decades that this balloon is expanding, and it's actually speeding up. However, there is a massive argument in the scientific community about how fast it is expanding right now.

Team Early Universe: They look at the "baby photos" of the universe (the Cosmic Microwave Background) and say, "It's expanding at about 67 units per second."
Team Late Universe: They look at "adult photos" (distant stars and supernovae) and say, "No, it's expanding faster, at about 73 units per second."

This disagreement is called the Hubble Tension. It's like two people measuring the same room with different tape measures and getting completely different numbers. Something is wrong, or we are missing a piece of the puzzle.

The New Tool: The "Stochastic Siren"

For a while, scientists have used gravitational waves (ripples in space-time caused by crashing black holes) to measure this expansion. They call individual crashing black holes "Standard Sirens." It's like hearing a single fire truck siren; if you know how loud it should be, you can tell how far away it is.

But this new paper introduces a new concept: the "Stochastic Siren."

Instead of listening to one specific fire truck, imagine you are standing in a city with thousands of fire trucks, but they are all too far away to hear individually. You can't pick out one siren, but you can hear a constant, low-level hum or roar coming from the whole city.

The Analogy: The "Stochastic Gravitational-Wave Background" is that cosmic hum. It is the combined noise of billions of black holes colliding across the history of the universe. We haven't heard a specific crash yet, but we are listening for that background noise.

How the "Hum" Tells Us the Speed

The paper argues that this cosmic hum is a secret code for the expansion rate of the universe. Here is the logic, simplified:

The Volume of the Room: The expansion rate (Hubble Constant) determines how much "space" (volume) exists in the universe at any given time.
The Crowd: If the universe is expanding slowly (a lower Hubble number), the "room" is bigger. A bigger room means there is more space for black holes to exist and collide. More collisions = a louder hum.
The Crowd (Fast): If the universe is expanding fast (a higher Hubble number), the "room" is smaller. There is less space for black holes to live and collide. Fewer collisions = a quieter hum.

The Twist:
Scientists haven't actually heard the hum yet. They are currently listening and saying, "It's too quiet to hear."

If the universe were expanding very slowly (low Hubble number), the room would be huge, and the hum should be very loud.
Since we haven't heard a loud hum, we can rule out the idea that the universe is expanding very slowly.
Therefore, the fact that the background is silent pushes the possible speed of expansion upward.

What They Found

The authors took data from the last few years of gravitational wave observations (which included 42 individual black hole crashes they could hear) and combined it with the fact that they couldn't hear the background hum.

Using only the individual crashes: Their measurement was very fuzzy and leaned toward the slower expansion rate (closer to the "Early Universe" team).
Using the "Silence" of the background: By adding the fact that the background hum is too quiet to be heard, they were able to rule out the slowest expansion rates.

The Result:
When they combined the two methods, their measurement shifted. It didn't solve the argument completely, but it moved the result closer to the "Late Universe" team's numbers (around 72). It made the measurement more accurate and consistent with other ways of measuring the universe.

Why This Matters

This is a unique tool because it doesn't rely on light (telescopes) or the "distance ladder" (measuring steps to the stars). It relies purely on the physics of space-time itself.

The paper suggests that as we keep listening and the background hum remains silent, our measurement of the universe's expansion speed will get even more precise. Eventually, when we finally do hear the hum clearly, this "Stochastic Siren" could be the tie-breaker that finally solves the Hubble Tension.

In short: By listening for a cosmic noise that isn't there, the scientists were able to prove that the universe isn't expanding as slowly as some thought, helping to narrow down the mystery of how fast our universe is growing.

Technical Summary: The Stochastic Siren: Astrophysical Gravitational-Wave Background Measurements of the Hubble Constant

Problem Statement
The paper addresses the "Hubble tension," a significant discrepancy ( $>5\sigma$ ) between early-universe measurements of the Hubble constant ( $H_0 \approx 67\text{--}68 \text{ km s}^{-1} \text{ Mpc}^{-1}$ ) and late-universe measurements ( $H_0 \approx 72\text{--}74 \text{ km s}^{-1} \text{ Mpc}^{-1}$ ). While gravitational wave (GW) "standard sirens" (individual resolved mergers) offer an independent method to measure $H_0$ , current constraints from the LIGO-Virgo-KAGRA (LVK) collaboration using spectral siren methods (mass spectrum features) remain broad ( $H_0 = 46^{+49}_{-26} \text{ km s}^{-1} \text{ Mpc}^{-1}$ ). Furthermore, the stochastic gravitational-wave background (GWB) arising from unresolved stellar-mass binary black hole (BBH) mergers has not yet been detected. The authors investigate whether the non-detection of this background can still provide cosmological constraints, specifically on $H_0$ , by leveraging the background's dependence on the Universe's expansion history.

Methodology
The authors propose a novel method termed the "stochastic siren," which combines the likelihood of observing resolved foreground (FG) BBH mergers with the likelihood of the unresolved background (BG) GWB signal.

Cosmological Dependence of the GWB: The dimensionless GW energy density, $\Omega_{\text{gw}}$ , is derived to depend inversely on $H_0$ . Specifically, $\Omega_{\text{gw}} \propto H_0^{-3}$ (assuming a fixed merger rate density). A lower $H_0$ implies larger comoving volumes, leading to a higher integrated number of mergers and thus a stronger $\Omega_{\text{gw}}$ . Conversely, a higher $H_0$ results in a weaker signal. Therefore, the non-detection of the GWB (an upper limit on $\Omega_{\text{gw}}$ ) can exclude lower values of $H_0$ that would predict a detectable background.
Likelihood Construction:
- Foreground ( $L_{\text{FG}}$ ): The likelihood of the observed catalog of 42 resolved BBHs from GWTC-3 is modeled using an inhomogeneous Poisson distribution dependent on population hyperparameters ( $\vec{\lambda}$ ) and $H_0$ .
- Background ( $L_{\text{BG}}$ ): The likelihood of the cross-correlation statistic $\hat{C}$ (used to search for the GWB) is modeled as a Gaussian. This likelihood depends on the predicted $\Omega_{\text{gw}}(\vec{\lambda}, H_0)$ . The authors utilize existing upper limits on $\Omega_{\text{gw}}$ from LVK searches to construct this likelihood, assuming a uniform prior on $\Omega_{\text{gw}}$ .
- Joint Likelihood: Assuming independence between the foreground and background signals (justified by the negligible contribution of resolved mergers to the total $\Omega_{\text{gw}}$ upper limit, $\sim0.3\%$ ), the joint likelihood is $L_{\text{joint}} = L_{\text{FG}} L_{\text{BG}}$ .
Population Models: The analysis employs 12 hyperparameters to model the BBH population: a PowerLaw+Peak model for the mass distribution (8 parameters) and a Madau-Dickinson functional form for the redshift evolution (4 parameters).
Inference: The authors perform Bayesian inference over 15 parameters (12 population + 3 cosmological: $H_0$ , $\Omega_{m,0}$ , and $w_0$ ) using nested sampling (via the dynesty sampler in bilby). They utilize kernel density estimators (KDEs) derived from previous GWTC-3 population analyses to evaluate the foreground likelihood efficiently.

Key Results

Improved Precision: The joint "stochastic siren" measurement yields a tighter constraint on $H_0$ compared to the spectral siren method alone. The maximum a-posteriori (MAP) probability for the joint measurement is $H_0 = 72^{+44}_{-37} \text{ km s}^{-1} \text{ Mpc}^{-1}$ , compared to $57^{+43}_{-35} \text{ km s}^{-1} \text{ Mpc}^{-1}$ for the foreground-only analysis and $46^{+49}_{-26} \text{ km s}^{-1} \text{ Mpc}^{-1}$ for the spectral siren.
Shift in Posterior: The inclusion of the GWB non-detection shifts the posterior distribution toward higher values of $H_0$ , bringing the GW measurement into better alignment with late-universe electromagnetic measurements (SH0ES) while remaining consistent with early-universe measurements (Planck) within the large uncertainties.
Lower Bound Mechanism: The authors demonstrate that the non-detection of the GWB effectively places a lower bound on $H_0$ . As observing time increases and the GWB remains undetected, this lower bound will progressively rise, improving the precision of the measurement.
Parameter Constraints: While the inclusion of the GWB non-detection improves $H_0$ constraints, it does not significantly tighten constraints on other population hyperparameters (mass distribution) or cosmological parameters ( $\Omega_{m,0}$ , $w_0$ ) given current data sensitivity.

Significance and Claims
The paper claims to present the first measurement of $H_0$ using the stochastic gravitational-wave background. Its primary significance lies in demonstrating that the absence of a detected background is a powerful cosmological probe. The "stochastic siren" offers a unique role in addressing the Hubble tension because:

It is entirely independent of electromagnetic observations and the cosmic distance ladder.
It provides a mechanism to progressively constrain $H_0$ from below as non-detections continue, potentially probing the tension before the background is directly detected.
It complements existing "standard siren" and "spectral siren" approaches.

The authors note that their results are conservative, as they considered only BBH contributions to the GWB, whereas the actual background may include other sources (e.g., neutron star binaries) that would increase $\Omega_{\text{gw}}$ and further tighten the $H_0$ constraints. Future projections suggest that with detector upgrades (e.g., A#), the GWB could be detected with a signal-to-noise ratio of 8 in less than a year, at which point the stochastic siren method would yield even stronger cosmological constraints.

The Big Problem: The Universe is Growing, but We Can't Agree on How Fast

The New Tool: The "Stochastic Siren"

How the "Hum" Tells Us the Speed

What They Found

Why This Matters

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