Estimation of the MTOV precision for ET, CE, and NEMO from the post-merger of BNS coalescences

This study demonstrates that even with optimistic signal-to-noise ratios and merger rates, next-generation gravitational wave detectors like Cosmic Explorer can only achieve marginal precision in estimating the maximum neutron star mass ($M_{\rm TOV}$) from post-merger signals, highlighting the critical need for improved high-frequency sensitivity to significantly reduce uncertainties.

The Gist

Imagine two neutron stars as the universe's most extreme weightlifters. They are city-sized balls of matter so dense that a single teaspoon of them would weigh a billion tons on Earth. When these two giants crash into each other, they create a massive explosion of ripples in space-time called gravitational waves.

This paper is essentially a "reality check" for scientists who are building the next generation of super-powerful microphones (detectors) to listen to the aftermath of these crashes.

Here is the breakdown of the paper using simple analogies:

1. The Big Question: How Heavy is Too Heavy?

Neutron stars have a strict weight limit. If they get too heavy, gravity wins, and they collapse into a black hole. This limit is called the TOV Mass (named after three physicists).

• The Analogy: Think of a neutron star like a balloon. You can blow more air into it (add mass), but there is a point where the rubber can't stretch anymore, and it pops (collapses into a black hole). Scientists want to know exactly how much air is in that balloon right before it pops.

2. The "After-Party" Signal

When two neutron stars merge, they don't just disappear instantly. Sometimes, they form a temporary, super-heavy "remnant" star that spins wildly for a split second before collapsing.

• The Analogy: Imagine two ice skaters spinning and grabbing hands. They spin together for a moment, wobbling and vibrating, before one of them falls down. That "wobble" and "vibration" creates a specific sound (a gravitational wave signal) that lasts for a tiny fraction of a second.
• The Problem: Current microphones (like LIGO) are great at hearing the "approach" of the skaters (the inspiral), but they are too deaf to hear the high-pitched "wobble" of the crash itself. We need next-generation detectors (named ET, CE, and NEMO) that are sensitive enough to hear these high-frequency squeaks.

3. The Simulation Game

The authors didn't wait for the real event; they ran computer simulations. They created two "test groups" of crashing stars:

• Group 1: Stars of equal weight, but made of different "materials" (Equations of State). Some are stiff like steel, others are soft like jelly.
• Group 2: Stars of different weights, all made of the same "material."

They calculated what the "sound" (the gravitational wave) would look like for each scenario.

4. The "Signal-to-Noise" Challenge

The universe is loud with background static (noise). To hear the crash, the signal needs to be loud enough to stand out. The scientists used a metric called SNR (Signal-to-Noise Ratio).

• The Analogy: Imagine trying to hear a whisper in a rock concert. If the whisper is too quiet, you miss it. The scientists set a rule: "We need the whisper to be at least 8 times louder than the background noise to count it as a real discovery."

5. The Results: A Reality Check

The authors asked: "If we build these new detectors (ET, CE, NEMO), how many of these crashes can we actually hear, and how accurately can we measure the weight limit?"

• The Good News: The Cosmic Explorer (CE) is the best microphone. It hears the loudest and clearest signals.
• The Bad News: Even with the best microphone, the "whispers" are very faint.
  • In the most optimistic scenario (where the universe is full of these crashes and our detectors are perfect), the Cosmic Explorer might only catch one or two of these events per year.
  • Because the sample size is so small, the measurement isn't very precise.

6. The "Precision" Verdict

The paper concludes that even with the best future technology, our estimate of the weight limit will still be a bit fuzzy.

• The Result: They estimate the uncertainty (the "fuzziness") to be about 0.3 to 0.8 solar masses.
• The Analogy: Imagine trying to guess the exact weight of a person. If you say, "They weigh between 140 and 150 pounds," that's a decent guess. But if you say, "They weigh between 140 and 180 pounds," that's a very wide range. The authors are saying that even with our best tools, we might only be able to say, "The weight limit is somewhere in a range of about 1 to 2 tons."

7. The Conclusion: We Need Better Ears

The paper ends with a clear message: We aren't there yet.
To get a precise answer (like knowing the weight limit is exactly 2.15 tons, not "somewhere between 2 and 3"), we need to improve the sensitivity of these detectors even further, specifically for those high-pitched, high-frequency sounds.

In a nutshell:
This paper is a cautionary tale for the future of astronomy. It says, "We have built amazing new tools to listen to the universe, but the signals from crashing neutron stars are so faint and rare that, for now, we can only get a rough guess at their maximum weight. We need to make our ears even more sensitive to get the exact answer."

Technical Summary

1. Problem Statement

The primary goal of the paper is to estimate the precision with which next-generation gravitational wave (GW) detectors can determine the Tolman-Oppenheimer-Volkoff mass ($M_{TOV}$)—the maximum mass a non-rotating neutron star can support before collapsing into a black hole.

While current detectors (LIGO/Virgo) successfully observe the inspiral phase of Binary Neutron Star (BNS) mergers, they lack the sensitivity to detect the post-merger (PM) high-frequency signals ($>1$ kHz). These PM signals contain critical information about the remnant object (Hypermassive or Supermassive Neutron Star) and its equation of state (EoS). The authors investigate whether future third-generation (3G) detectors (Einstein Telescope, Cosmic Explorer) and 2.5-generation detectors (NEMO) can detect these signals with sufficient Signal-to-Noise Ratio (SNR) to constrain $M_{TOV}$ with meaningful precision.

2. Methodology

The study employs a multi-step numerical and statistical approach:

• Simulation Data (CoRe Database):
  • The authors utilized two datasets from the Computational Relativity (CoRe) database, simulating BNS coalescences at a distance of 40 Mpc.
  • Dataset 1: 10 simulations with equal mass components ($m_1=m_2=1.35 M_\odot$) but varying Equations of State (EoS), including hadronic, hybrid, and exotic matter models (e.g., SLy, H4, DD2, 2B).
  • Dataset 2: 12 simulations with a fixed SLy EoS but varying mass ratios ($q = m_1/m_2$ from 1.0 to 1.5) and total masses ($2.44 - 3.00 M_\odot$).
• Signal Processing:
  • The post-merger strain signals were extracted (starting 1.5 ms after the merger).
  • Amplitude Spectral Density (ASD) was calculated via Fast Fourier Transform (FFT) to identify dominant peak frequencies ($f_2$ or $f_{peak}$).
  • Matched Filtering: Theoretical templates were compared against detector sensitivity curves (ET-C, CE, and NEMO) to calculate the optimal SNR.
  • Averaging: SNR values were averaged over all-sky angles ($\theta, \phi$) and inclination/polarization angles ($\iota, \psi$) using Monte Carlo sampling to ensure robustness against source orientation.
• Detection Rate Estimation:
  • Using a reference SNR threshold of SNR $\ge$ 8, the authors calculated the maximum detection distance ($d_m$) for each simulation and detector.
  • The expected number of detections per year ($N$) was estimated by integrating the comoving volume defined by $d_m$ with three merger rate densities ($R_0$): 7.6, 129, and 250 Gpc$^{-3}$yr$^{-1}$.
• Mass Precision Calculation:
  • A Bayesian analysis was performed on 122 electromagnetic neutron star mass observations to derive a bimodal Gaussian distribution for neutron star masses.
  • The total mass distribution of the binary was derived via self-convolution of the component mass PDF.
  • A 5% mass loss (due to gravitational radiation and winds) was applied to the total mass to estimate the final remnant mass distribution.
  • The precision of $M_{TOV}$ ($\delta M$) was estimated using the relation:
    $$\delta M(M_F) \approx \frac{1}{N \cdot \text{PDF}(M_F)}$$
    Where $N$ is the number of detections and $\text{PDF}(M_F)$ is the probability density of the final mass.

3. Key Contributions

• Quantitative Precision Limits: The paper provides the first quantitative estimate of the uncertainty ($\delta M$) in determining $M_{TOV}$ specifically from post-merger signals of next-generation detectors.
• Detector Comparison: It rigorously compares the performance of ET, CE, and NEMO, demonstrating that Cosmic Explorer (CE) significantly outperforms the others due to superior high-frequency sensitivity.
• Impact of Mass Asymmetry: The study highlights that mass-asymmetric systems (Dataset 2) produce less coherent post-merger oscillations, resulting in lower SNRs and making them harder to detect compared to equal-mass systems.
• Realistic Constraints: Unlike previous optimistic theoretical studies, this work incorporates realistic merger rates and SNR thresholds, revealing that current projections for $M_{TOV}$ precision are still limited.

4. Key Results

• SNR Performance:
  • CE consistently achieved the highest SNRs. For Dataset 1 (equal mass), CE SNRs ranged from 4.30 (SLy EoS) to 15.30 (H4 EoS).
  • ET and NEMO showed lower SNRs, with many simulations falling below the SNR=8 threshold, particularly for softer EoSs or high-frequency peaks (>3.3 kHz).
  • Dataset 2 (Mass Asymmetry): SNRs were generally lower. The highest SNR for CE was 9.67, but many configurations yielded SNRs $< 2.3$.
• Detection Rates ($N$):
  • Even under the most optimistic scenario (CE, highest SNR, maximum merger rate $R_0 = 250$ Gpc$^{-3}$yr$^{-1}$), the expected number of detections per year is $N \approx 0.47 \pm 0.76$.
  • For lower merger rates or other detectors, $N$ drops significantly, often resulting in zero expected detections per year for the post-merger phase.
• $M_{TOV}$ Precision ($\delta M$):
  • Due to the low number of expected detections, the precision is limited.
  • In the most optimistic scenario (CE, $N \approx 0.47$), the minimum uncertainty is estimated at $\delta M/2 \approx 0.3 - 0.8 M_\odot$.
  • This precision corresponds to the most probable remnant mass (the main peak of the distribution at $\approx 2.57 M_\odot$). Uncertainties would be larger for masses in the tails of the distribution.
  • For Dataset 2, no valid number of detections ($N \ge 1$) was found, meaning no precision could be estimated.

5. Significance and Conclusion

The paper concludes that while next-generation detectors like CE are essential for observing post-merger signals, current sensitivity levels are insufficient to achieve high-precision measurements of $M_{TOV}$ solely from these events.

• Current Limitations: The uncertainty of $0.3 - 0.8 M_\odot$ is too large to tightly constrain the Equation of State or definitively distinguish between different nuclear matter models.
• Future Requirements: To achieve greater precision, future ground-based observatories must improve their sensitivity specifically in the high-frequency range (3–4 kHz). This improvement is crucial not only for detecting the post-merger signal but also for determining whether a black hole formed immediately (prompt collapse) or after a delay.
• Optimistic Outlook: The study suggests that with the most optimistic merger rates and the superior sensitivity of Cosmic Explorer, a marginal constraint on $M_{TOV}$ might be possible, but significant technological advancements in high-frequency detection are still required to make this a routine measurement.

In summary, the paper serves as a realistic "reality check" for the field, indicating that while the potential exists, the precision of $M_{TOV}$ estimation from post-merger signals remains a challenge for the near future.