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arxiv: 2512.08497 · v2 · submitted 2025-12-09 · 🌀 gr-qc · astro-ph.HE

Recognition: no theorem link

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

Fiona H. Panther , Paul D. Lasky
Authors on Pith no claims yet

Pith reviewed 2026-05-17 00:26 UTC · model grok-4.3

classification 🌀 gr-qc astro-ph.HE
keywords binary neutron star mergerspost-merger gravitational wavessub-threshold signalshot nuclear equation of stateneutron star maximum massTolman-Oppenheimer-Volkoff massprompt collapse fractiongravitational wave populations
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The pith

The fraction of prompt collapses in sub-threshold post-merger signals, combined with inspiral mass distributions, constrains the maximum mass of hot neutron stars to 11-20 percent uncertainty with 50-70 events.

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The paper presents a statistical method for coherently combining information from populations of sub-threshold post-merger gravitational-wave signals from binary neutron star mergers. Although no single event reaches detection threshold, the fraction of mergers that promptly collapse to black holes versus those leaving a surviving neutron star remnant can be recovered. Paired with the neutron star mass distribution measured from the inspiral phase, this fraction supplies an indirect determination of the maximum mass supported by the hot, rapidly rotating nuclear equation of state. Conservative waveform models show that 50-70 events suffice for an 11-20 percent fractional uncertainty on the hot maximum mass, which can be mapped to a 12-21 percent constraint on the Tolman-Oppenheimer-Volkoff mass. The approach also enables comparison between hot and cold neutron star properties to probe temperature effects and possible phase transitions in dense matter.

Core claim

The authors establish that the prompt-collapse fraction recovered statistically from a population of sub-threshold post-merger gravitational-wave events, when combined with inspiral measurements of the neutron star mass distribution, provides an indirect measure of the maximum mass of rapidly rotating hot neutron stars. Using conservative post-merger waveform models, they demonstrate that 50-70 such events yield an 11-20 percent fractional uncertainty on this hot maximum mass, which can be converted into a 12-21 percent constraint on the Tolman-Oppenheimer-Volkoff mass. This supplies a route to studying the effects of temperature on the nuclear equation of state and to obtaining indirect证据证据

What carries the argument

The prompt-collapse fraction extracted from sub-threshold post-merger waveforms, which distinguishes immediate black-hole formation from neutron-star survival for tens of milliseconds and serves as a statistical proxy for whether the total mass exceeds the hot maximum-mass limit.

If this is right

  • 50-70 events with inspiral measurements produce 11-20 percent fractional uncertainty on the maximum mass of hot, rapidly rotating neutron stars.
  • This uncertainty can be mapped to a 12-21 percent fractional constraint on the Tolman-Oppenheimer-Volkoff mass.
  • The hot maximum-mass measurement can be compared directly to cold-neutron-star maximum-mass constraints to isolate temperature-dependent effects in the equation of state.
  • Such comparisons may yield indirect evidence for first-order phase transitions in the interiors of neutron stars.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

  • The population-level approach could be tested on simulated data sets with varying waveform models to quantify bias from unknown post-merger details.
  • Future improvements in waveform accuracy or detector sensitivity would likely reduce the number of events required for the same precision.
  • Combining the hot-mass constraint with electromagnetic or kilonova observations of the same events could provide cross-checks on the temperature dependence of dense matter.
  • The method offers a path to probing nuclear physics at densities and temperatures not directly accessible through cold neutron-star observations alone.

Load-bearing premise

The prompt-collapse fraction can be recovered accurately from the population of sub-threshold signals using conservative waveform models without large biases from unknown waveform details, population priors, or the mapping from hot maximum mass to collapse behavior.

What would settle it

A measured prompt-collapse fraction in a sample of 50-70 binary neutron star events that lies outside the range predicted by a given hot equation of state model, after accounting for the inspiral mass distribution, would falsify the claimed constraint.

Figures

Figures reproduced from arXiv: 2512.08497 by Fiona H. Panther, Paul D. Lasky.

Figure 1
Figure 1. Figure 1: Prior probability distributions used in this work. We assume a bimodal remnant mass distribution calculated from the expected distribution of merging binary neutron star pairs with masses drawn independently from the observed neutron star population (top panel; green histogram). We choose uniform priors on 𝑀TOV (top panel; purple filled histogram) and 𝑀Max = 1.5𝑀TOV (top panel; purple hollow histogram). Th… view at source ↗
Figure 2
Figure 2. Figure 2: Masses of simulated binary neutron star signals used in this work. Dashed lines show different values of 𝑀Max used to evaluate the robustness of the technique. The dotted line shows 𝑚1 = 𝑚2. The colour￾bar indicates the network SNR of the injected signals. For each 𝑀Max = [3.1, 3.25, 3.5] M⊙, the fraction of data segments that contain injected sig￾nals is 𝜉 = [0.56, 0.70, 0.84]. 𝑀Tot ≤ 𝑀Max, our simulated … view at source ↗
Figure 3
Figure 3. Figure 3: Distribution of luminosity distances and network postmerger SNRs for our injected signal population. Luminosity distances are drawn such that they are uniform in co-moving volume. The size of each point indicates the relative total mass of each binary neutron star system. More massive systems, and face-on systems (cos 𝜄 ≃ ±1, represented by higher color saturations) produce signals with larger amplitudes, … view at source ↗
Figure 5
Figure 5. Figure 5: Posterior probability distributions on 𝑀tot, mass ratio 𝑞 = 𝑚1/𝑚2 and tidal deformabilities Λ1, Λ2 for a post-merger signal with SNR = 3.7. In this case, 𝑀tot can be recovered well, however the posterior distributions of other parameters do not have strong predictive power. panel), we find 𝑀Max = 2.96+0.23 −0.12 M⊙, where the 90% confidence interval corresponds to a ∼ 12% uncertainty in the value of 𝑀Max. … view at source ↗
Figure 6
Figure 6. Figure 6: Recovered normalized posterior distributions for 𝑀Max with an increasing number of injected signals. Here, 𝑀Max,True = 3.1 M⊙ (vertical dashed line). Vertical dotted lines indicate the upper and lower bounds of the 90% confidence interval on 𝑀max [PITH_FULL_IMAGE:figures/full_fig_p007_6.png] view at source ↗
Figure 7
Figure 7. Figure 7: Same as [PITH_FULL_IMAGE:figures/full_fig_p007_7.png] view at source ↗
Figure 8
Figure 8. Figure 8: Same as [PITH_FULL_IMAGE:figures/full_fig_p007_8.png] view at source ↗
Figure 9
Figure 9. Figure 9: Recovered median 𝑀Max for each experiment. Error bars indicate 90% CI. With 70 events, the percentage error in estimation of 𝑀Max is 11, 21, and 15% respectively for 𝑀Max,true = 3.1, 3.25, 3.4 M⊙ [PITH_FULL_IMAGE:figures/full_fig_p008_9.png] view at source ↗
Figure 10
Figure 10. Figure 10: Number of simulated signals that are added to the injection set before a signal with SNR> 12 is detected for binary neutron star systems that are distributed uniform in co-moving volume out to 𝑑𝐿 = 200 Mpc. In 17% of our realizations, no signal with SNR> 12 is generated before the injection set is complete (violet hatched bin). of the number of events added to our sub-threshold sample before an event with… view at source ↗
read the original abstract

We show how to coherently combine information from a population of sub-threshold, gravitational-wave binary neutron star post-merger remnants. Although no individual event in our synthetic population can be claimed as a confident detection, we show how to statistically determine the fraction of merger events that promptly collapse to form a black hole, compared to those for which a neutron star survives the merger for at least tens of milliseconds. This fraction, when combined with information about the neutron star mass distribution gleaned from the inspiral portion of the signals, provides an indirect measure of the neutron star maximum mass. Using conservative measures of the post-merger waveforms, we show that 50-70 events with binary neutron star inspiral measurements can be combined to give an $11-20\%$ fractional uncertainty on the maximum mass of rapidly rotating, hot neutron stars, which can potentially be turned into a $12-21\%$ fractional constraint on the Tolman-Oppenheimer-Volkoff mass. We discuss how this measure of the hot nuclear equation of state can be combined with information of cold neutron stars to see the effect of temperature on physics in the densest regions of the Universe by providing indirect evidence for first-order phase transitions in neutron star interiors.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Referee Report

2 major / 1 minor

Summary. The paper proposes a statistical method to combine information from populations of sub-threshold post-merger gravitational-wave signals in binary neutron star mergers. It claims that the fraction of prompt collapses to black holes versus surviving neutron star remnants can be inferred even when no individual event is detectable, and that combining this fraction with inspiral mass measurements yields an 11-20% fractional uncertainty on the maximum mass of hot, rapidly rotating neutron stars (and potentially 12-21% on the TOV mass) from 50-70 events. The work discusses implications for constraining the hot nuclear equation of state and detecting temperature-driven effects such as first-order phase transitions.

Significance. If the inference of the prompt-collapse fraction proves robust against waveform and prior uncertainties, the approach would offer a useful complement to direct post-merger detections by exploiting the larger number of sub-threshold events. It could provide indirect access to the hot EOS at densities relevant to neutron star interiors and help bridge cold and hot constraints. The quantitative claims, however, rest on the fidelity of the synthetic population study and external waveform models, limiting the immediate impact until those elements are more thoroughly validated.

major comments (2)
  1. [Methods / population synthesis and inference procedure] The statistical recovery of the prompt-collapse fraction (central to the 11-20% uncertainty claim) relies on conservative waveform measures applied to synthetic populations. The manuscript does not present a systematic bias test in which injected post-merger signals are generated with durations, frequency content, or amplitudes that differ from the template family; without such a test, it is unclear whether a 10% systematic shift in the recovered fraction could arise and dominate the reported error budget.
  2. [Results and discussion of uncertainty propagation] The mapping from the inferred collapse fraction to the hot maximum mass (and onward to the TOV mass) incorporates an external mass-distribution prior and a collapse-threshold relation. The results section does not quantify how plausible variations in these inputs propagate into the final 11-20% and 12-21% fractional uncertainties; this propagation step is load-bearing for the headline constraint and requires explicit sensitivity checks.
minor comments (1)
  1. [Abstract and §2] The abstract and introduction refer to 'conservative measures' without a concise, self-contained definition; adding a short paragraph or table that lists the exact waveform cuts or statistics employed would improve clarity and reproducibility.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for their constructive and detailed report. We address each major comment below and have incorporated additional analyses to strengthen the robustness of our results.

read point-by-point responses

  1. Referee: The statistical recovery of the prompt-collapse fraction (central to the 11-20% uncertainty claim) relies on conservative waveform measures applied to synthetic populations. The manuscript does not present a systematic bias test in which injected post-merger signals are generated with durations, frequency content, or amplitudes that differ from the template family; without such a test, it is unclear whether a 10% systematic shift in the recovered fraction could arise and dominate the reported error budget.

    Authors: We agree that explicit tests for waveform mismatch are valuable for validating the recovery of the prompt-collapse fraction. While the original analysis employed conservative measures of post-merger signal strength to reduce sensitivity to detailed waveform morphology, we have now performed additional injection-recovery studies using alternative post-merger waveform families with varied durations, frequency evolution, and amplitudes. These tests show that the recovered fraction shifts by at most 4-6% under plausible mismatches, remaining within the statistical uncertainties quoted in the manuscript. A new subsection and supplementary figures documenting these tests will be added to the revised version. revision: yes

  2. Referee: The mapping from the inferred collapse fraction to the hot maximum mass (and onward to the TOV mass) incorporates an external mass-distribution prior and a collapse-threshold relation. The results section does not quantify how plausible variations in these inputs propagate into the final 11-20% and 12-21% fractional uncertainties; this propagation step is load-bearing for the headline constraint and requires explicit sensitivity checks.

    Authors: We acknowledge that the propagation of uncertainties from the external priors and collapse-threshold relation to the final mass constraints is an important step that merits explicit quantification. In the revised manuscript we have added a dedicated sensitivity analysis that varies the mass-distribution prior (including changes to the power-law index and cutoff) and the collapse-threshold relation (including shifts of ±0.1 M⊙ in the critical mass). The resulting fractional uncertainties on the hot maximum mass vary by no more than 3 percentage points, and the TOV-mass uncertainties remain within the reported 12-21% range. These checks and the associated error-budget table will be included in the results section. revision: yes

Circularity Check

0 steps flagged

No significant circularity; derivation relies on external models and synthetic populations

full rationale

The paper's central claim is a statistical method to infer the prompt-collapse fraction from a stack of sub-threshold post-merger signals using conservative waveform templates and synthetic populations, then combine it with inspiral-derived mass distributions to bound hot neutron-star maximum mass. No step reduces by construction to a fitted parameter or self-citation chain inside the paper; the quoted uncertainties (11-20%) emerge from Monte-Carlo injections against external waveform models rather than from re-deriving the input fraction. The derivation is therefore self-contained against external benchmarks and does not exhibit any of the enumerated circularity patterns.

Axiom & Free-Parameter Ledger

0 free parameters · 1 axioms · 0 invented entities

The central claim rests on standard domain assumptions about binary neutron star populations, post-merger waveform morphology, and the mapping from remnant lifetime to maximum mass; no new free parameters or invented entities are introduced in the abstract.

axioms (1)
  • domain assumption Standard assumptions about binary neutron star population distributions and post-merger waveform models from prior literature
    Used to generate the synthetic population and to define conservative measures of the post-merger signals.

pith-pipeline@v0.9.0 · 5518 in / 1403 out tokens · 53317 ms · 2026-05-17T00:26:42.349261+00:00 · methodology

discussion (0)

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