Sub-threshold post-merger gravitational waves can constrain the hot nuclear equation of state

This paper demonstrates that coherently combining a population of sub-threshold gravitational-wave signals from binary neutron star mergers can statistically constrain the maximum mass of hot, rapidly rotating neutron stars and the nuclear equation of state, potentially revealing evidence for first-order phase transitions in dense matter.

The Gist

The Big Picture: Listening to the "Whispers" of Colliding Stars

Imagine two neutron stars—cities made of pure, super-dense matter, heavier than the sun but squeezed into a space the size of a city—crashing into each other. When they collide, they create a massive explosion of gravitational waves (ripples in space-time).

Scientists have a big question: What happens right after the crash?

1. Does the new, super-heavy object immediately collapse into a black hole? (A "prompt collapse").
2. Or does it survive for a little while as a giant, spinning, super-hot neutron star before eventually collapsing? (A "survivor").

The answer to this question tells us about the nuclear equation of state. Think of this as the "recipe" for how matter behaves when it's squeezed to the absolute limit. It's like trying to figure out the ingredients of a cake just by listening to the sound of the oven door opening, without ever seeing the cake.

The Problem: The Signal is Too Quiet

The problem is that the "survivor" phase emits a very specific, high-pitched sound (gravitational waves) that is currently too quiet for our detectors to hear clearly on its own. It's like trying to hear a single person whispering in a stadium full of cheering fans. Even with our best future telescopes (like the Cosmic Explorer), we might only hear a few of these whispers clearly.

Most of the time, the signal is "sub-threshold." It's there, but it's buried in the noise. We can't say, "Yes, that was definitely a survivor!" for any single event.

The Solution: The "Crowd Source" Approach

This paper proposes a clever statistical trick. Instead of waiting for one loud whisper, the authors suggest we listen to many quiet whispers at once.

The Analogy: The "Crowd Vote"
Imagine you are in a room with 100 people. You ask them, "Did you see a ghost?"

• If 99 people say "No" and 1 says "Yes," you might think the "Yes" person is just imagining things.
• But, if you ask 1,000 people and 600 say "Yes" and 400 say "No," you can be statistically confident that ghosts do exist, even if you can't point to exactly which person saw one.

The authors do the same with gravitational waves. They take a population of about 25 to 70 neutron star collisions. Even though none of them are loud enough to be a "confident detection" on their own, they combine the data mathematically.

By looking at the pattern of the noise across all these events, they can calculate:

• What percentage of these collisions resulted in a "survivor" (a hot neutron star)?
• What percentage collapsed immediately into a black hole?

The "Magic Number": The Maximum Mass

Once they know the percentage of survivors, they can figure out the Maximum Mass a neutron star can have before it gives up and becomes a black hole.

• The Analogy: Imagine a stack of bricks. If you know that 60% of the time you can stack 10 bricks before it falls, but 40% of the time it falls at 8 bricks, you can calculate the exact weight limit of the bricks.
• In this case, if they know how many collisions created a survivor, they can calculate the "weight limit" (maximum mass) of these hot, spinning stars.

What Did They Find?

The authors ran computer simulations to test their idea. They pretended to have a future detector network and simulated 70 collisions.

• The Result: By combining the "whispers" of these 70 events, they could determine the maximum mass of a neutron star with about 11% to 20% accuracy.
• Why it matters: This accuracy is good enough to tell us if the "recipe" for matter inside these stars involves exotic things like phase transitions (where matter suddenly changes state, like water turning to ice, but inside a star). This could prove that quarks (tiny particles) break free from protons and neutrons in the core of these stars.

The Catch: The "Loud" vs. The "Crowd"

The paper also asks a funny question: Will we hear one loud, clear scream (a single loud detection) before we can finish counting the crowd of whispers?

Their simulation suggests it's a toss-up. There's a roughly 50/50 chance we'll hear one loud event first, or we'll need to wait until we've collected enough quiet ones to do the math.

The Bottom Line

This paper is a "proof of concept." It says: "Don't give up just because the signals are too quiet to hear individually. If we listen to enough of them together, we can still solve the mystery of how matter behaves at the universe's most extreme limits."

It's a shift from looking for a "smoking gun" to looking for a "pattern in the smoke." Even if the gun is hidden, the smoke tells us it was fired.

Technical Summary

1. Problem Statement

Binary neutron star (BNS) mergers produce high-frequency ($\sim$kHz) gravitational waves (GWs) from the post-merger remnant. These signals encode the hot nuclear equation of state (EOS), specifically the maximum mass of a rapidly rotating, hot neutron star ($M_{\text{Max}}$), which relates to the non-rotating Tolman-Oppenheimer-Volkoff (TOV) mass ($M_{\text{TOV}}$).

However, current and near-future GW detectors (like LIGO/Virgo/KAGRA) lack the sensitivity in the kHz band to confidently detect these post-merger signals individually. Even with third-generation (3G) detectors (e.g., Cosmic Explorer, Einstein Telescope), only a small fraction of events ($\sim$2–10 per year) are expected to have a signal-to-noise ratio (SNR) $>8$. Consequently, the vast majority of post-merger signals will remain sub-threshold, rendering them undetectable via standard single-event analysis. The authors address the challenge of extracting physical constraints on the hot nuclear EOS from this population of undetected, sub-threshold signals.

2. Methodology

The authors propose a Bayesian hierarchical framework to coherently combine information from an ensemble of sub-threshold events. The method is adapted from the "Bayesian Search" (TBS) technique used for stochastic backgrounds.

• Mixture Model: For each data segment following a BNS inspiral detection, the authors define a mixture model where the data $s_i$ is either:
  1. Signal + Noise: The system formed a long-lived remnant emitting post-merger GWs (probability $\xi$).
  2. Noise Only: The system underwent prompt collapse to a black hole (probability $1-\xi$).
• Likelihood Construction:
  • The likelihood for a signal segment is calculated using a Gaussian noise assumption and a specific post-merger waveform model (NRPMw_pmonly).
  • The likelihood for a noise segment is calculated assuming only Gaussian noise.
  • The total likelihood for a population of $N$ events is the product of individual likelihoods, marginalized over waveform parameters ($\theta$) and the fraction parameter $\xi$.
• Connecting $\xi$ to Physics:
  • $\xi$ represents the fraction of mergers where the total mass ($M_{\text{Tot}}$) is below the maximum mass threshold ($M_{\text{Max}}$).
  • $M_{\text{Max}}$ is related to the TOV mass by $M_{\text{Max}} = \chi M_{\text{TOV}}$, where $\chi$ is a factor dependent on the EOS and rotation profile (typically $1.3 \le \chi \le 1.6$).
  • By assuming a prior distribution for progenitor masses (based on observed Galactic BNS and GW190425), the authors calculate the probability that a random merger results in a remnant, thereby linking the observable $\xi$ to $M_{\text{TOV}}$.
• Simulation Setup:
  • Waveforms: Injected using the NRPMw_pmonly model (calibrated to numerical relativity).
  • Detector: Simulated for a "third-generation" network (two Cosmic Explorer detectors).
  • Population: 70 synthetic BNS events with inspiral SNR $\sim$300, but post-merger SNR in the range $[2.5, 7]$ (sub-threshold).
  • Priors: Conservative priors on $M_{\text{TOV}}$ (Uniform 1.9–2.5 $M_\odot$) and $\chi$ (fixed at 1.5 for the proof-of-concept). Extrinsic parameters (sky location, time) are fixed using "inspiral-informed" priors to reduce computational cost.

3. Key Contributions

• Statistical Framework: Introduction of an optimal statistical technique to combine sub-threshold post-merger signals, distinct from previous methods that relied on stacking oscillation modes or empirical relations for radius constraints.
• Indirect $M_{\text{TOV}}$ Measurement: Demonstration that the fraction of prompt collapses vs. stable remnants, derived from sub-threshold data, provides a direct measure of the maximum mass threshold for hot neutron stars.
• Conservative Approach: The method avoids assuming quasi-universal relations between post-merger frequencies and radii, which can break down if exotic physics (e.g., phase transitions) occurs in the hot remnant. Instead, it relies on the mass distribution and the existence/non-existence of the signal.

4. Results

The authors performed simulations with varying true values of $M_{\text{Max}}$ ($3.1, 3.25, 3.4 M_\odot$) and varying population sizes ($N=5$ to $70$).

• Convergence: The posterior distribution for $M_{\text{Max}}$ converges as the number of events increases.
  • With 25–35 events, the method can constrain $M_{\text{Max}}$ to within 10–20%.
  • With 70 events, the constraint tightens to approximately 11–20% fractional uncertainty on $M_{\text{Max}}$.
• TOV Mass Constraint: Assuming $\chi = 1.5$, this translates to a 12–21% fractional constraint on $M_{\text{TOV}}$.
• Bias Analysis:
  • The method tends to slightly overestimate $M_{\text{Max}}$ when the true value is high (due to noise realizations mimicking signals) and underestimate it when the true value is low (due to finite sample size effects).
  • However, the true value consistently falls within the 90% credible interval.
• Single vs. Population Detection:
  • The authors analyzed the probability of detecting a single "loud" event (SNR > 12) before accumulating the necessary population of sub-threshold events.
  • Results show a 50–57% chance that a single loud event will be detected before 25–35 sub-threshold events are collected.
  • Crucially, in 17% of realizations, no loud event is detected even after 70 sub-threshold events, highlighting the necessity of the population approach.

5. Significance and Implications

• Hot vs. Cold EOS: This method provides a unique probe of the hot nuclear EOS (relevant to merger remnants), complementing constraints from cold neutron stars (pulsars, inspiral tidal deformability).
• Phase Transitions: By comparing the $M_{\text{TOV}}$ derived from cold inspirals with the $M_{\text{Max}}$ derived from hot remnants, scientists can indirectly test for first-order phase transitions (e.g., to quark matter) that might occur at the high temperatures and densities of merger remnants.
• Future Readiness: The technique is designed for 3G detectors but utilizes sub-threshold data, meaning it can be applied immediately as soon as a sufficient number of BNS inspirals are detected, even if individual post-merger signals remain undetected.
• Complementarity: The authors argue that single loud detections and population statistics are complementary; a loud event provides a precise point estimate, while the population method provides a robust, EOS-agnostic constraint on the maximum mass.

In conclusion, Panther & Lasky demonstrate that even in the absence of confident individual detections, the collective statistical power of sub-threshold post-merger signals can yield significant constraints on the maximum mass of neutron stars and the nature of dense nuclear matter.