Gravitational Waves from Core-Collapse Supernovae: Dependence on the Progenitor Star, Rotation Rate and Nuclear Equation of State

Two things to know before you read it. The genuinely new result is the stochasticity study: for models near the failed/successful boundary, changing only the random seed bifurcates the outcome (two of five s15 SFHx seeds fail, three explode) and shifts the GW amplitude dramatically. That is not in the cited literature—Cardall & Budiardja looked at SASI dynamics, not GW emission. Second, the headline attached to that result is sloppy. The abstract and conclusions say perturbations alter the SNR by as much as 62%, but Section 8 defines 62% as variation in gravitational-wave energy across models, and the Table 1 SNRs for s15 SFHx (84–163 at 10 kpc) give roughly a 48% max-relative range. The abstract is quoting one statistic under the name of another.

The rest is a competent parameter survey: 137 2D runs, up to 5.5 s post-bounce, three EoS, three rotation rates, 15 progenitors. The EoS trends (CMF lower amplitudes, SASI persisting to ~5 s from failed shock revival) and the rotation trends (later revival, lower energies, extra visible modes) are supported by the simulations shown. Including neutrino memory is a genuine plus—it dominates detectability for ET-class low-frequency sensitivity and is usually neglected. The recalibrated universal relation (Eq. 6) with the relativistic factor is new, fitted to their own 2D data, and the late-time check outside the 1.5 s fit window is an honest out-of-sample test. It is a fit to this code rather than a universal law, but they do not oversell it much.

Soft spots, in proportion. The 62% figure is the real presentation problem—pick energy or SNR and label it correctly. Five seeds per model is acknowledged (b369), so the remaining gap is that the spread is an extreme-value statistic over a tiny sample, and the authors themselves disclaim its 3D relevance (b381: variation 'not likely to be representative of what we would see in 3D'). The qualitative bifurcation is credible and consistent with Powell & Müller 2025a; the quantitative spread is not yet anchored. The 2D amplitude overestimate is acknowledged (b153, b179); b154's assertion that relative differences survive in 3D is an assertion, not a demonstration—a scope limit, not a flaw. Waveforms released only on request is a minor negative.

This is for people designing CCSN GW searches, building phenomenological templates, or projecting next-gen detector science. It deserves a serious referee; fix the 62% labeling, ideally release the waveforms, and it is publishable.

The most load-bearing spot is the quantitative anchor of the new stochasticity result: the abstract's claim (b4) that seed perturbations alter detection SNR by as much as 62% for models near the failed/successful boundary. The supports for that figure are under-built in three stacked ways. (i) The metric is ambiguous: Section 8 assigns the 13%-62% range to 'variation in gravitational-wave energy across models', while b4 attributes 62% to SNR alteration; the Table 1 SNRs for the driving case, s15 SFHx non-rotating (84-163 at 10 kpc), give only about 48% as a max-relative range, so the abstract's number depends on an unstated outcome-grouped comparison. (ii) It is an extreme-value statistic over five seeds per mass, a sample size the authors acknowledge is small (b369). (iii) Most load-bearing: the statistic measures within-model seed-to-seed variation, and b381 states that this variation 'is not likely to be representative of what we would see in 3D', while b154's robustness assertion refers to cross-model relative amplitudes. The headline quantity therefore sits on a spread the paper itself disclaims for 3D relevance, with no 3D or larger-seed calibration provided. The qualitative finding (two of five s15 SFHx seeds fail, three explode, with correspondingly large amplitude differences) is credible and consistent with Powell & Müller (2025a), but the 62% number as presented is not currently anchored.

I read the paper as a 2D parameter study whose two headline contributions are the first multi-seed stochasticity analysis of CCSN gravitational-wave emission and a recalibrated universal relation. The comparative claims about EoS and rotation are supported within the axisymmetric scope: the CMF non-explosion trend and lower amplitudes, the rotation-induced later shock revival and lower energies, and the SASI-frequency tracking against Eq. (2) all hang together and are consistent with the cited literature; I do not see a load-bearing flaw there. The stochasticity section is genuinely new, and the qualitative result (seed perturbations bifurcate boundary models between failed and successful shock revival, with large amplitude differences) is credible, including its consistency with the precursor dynamics study of Powell & Müller (2025a). Credit is also due for explicitly flagging the axisymmetric amplitude overestimate (b153) and the detector-distance caveat in Section 7. The place the central claim is least secure is the quantitative anchor of the novelty claim: the abstract's 'as much as 62%' SNR alteration (b4). Three stacked weaknesses: (1) metric ambiguity — Section 8 attributes the 13%-62% range to gravitational-wave energy variation, while b4 attributes 62% to SNR; the Table 1 SNRs for the driving model (s15 SFHx, 84-163) only reach about 48% as a max-relative range, so the abstract's figure requires an unstated outcome-grouped definition; (2) the estimate rests on five seeds per mass (acknowledged in b369); (3) it is a seed-to-seed variation statistic, and b381 explicitly disclaims that seed-to-seed variation in this 2D setup is representative of 3D, whereas b154's robustness assertion concerns cross-model relative amplitudes, not the within-model variation the headline uses. The paper therefore carries its own strongest caveat against its own headline number, without a calibration of the statistic's magnitude in 3D or with more seeds. A secondary observation: the stacked-spectrogram robustness claim (Figure 28) uses |A|^6 weighting, under which the quieter failed-seed signals contribute almost nothing, so it mainly establishes track stability among the loud exploding seeds. A bounded 3D check over the one-second window used for Table 1 can settle the main question without a full 3D redo. I recommend keeping the reader's CONDITIONAL verdict: the missing data release and the in-sample, post-hoc-excluded character of the Equation (6) recalibration remain unsatisfied, and the 62% claim needs either verification or requalification. This is partial agreement with the reader's weakest assumption: the same 2D/5-seed root, but I would not extend the concern to the EoS/rotation comparative claims, which are well-protected by the within-2D scope and consistency with prior 3D work.

If the qualitative results hold, this is a valuable systematic survey of a parameter regime that is computationally prohibitive in 3D: 137 long-duration runs, three EoS, several rotation rates, inclusion of both matter and neutrino memory, and an explicit seed-perturbation study. The strengths include the out-of-sample-in-time validation of the recalibrated universal relation of Equation (6), the stable stacked-spectrogram morphology across seeds (Figure 28), and the credible qualitative trend that CMF models have lower amplitudes and persistent SASI. The paper is also honest about the 2D amplitude overestimate (Section 4.1) and about the limited number of seeds (Section 8). However, the headline quantitative stochasticity claim, the 62% SNR alteration in the abstract, is not currently anchored to the reported Table 1 values, and it is presented without the paper's own 3D caveat. The qualitative finding that two of five s15 SFHx seeds fail while three explode is credible, but the specific 62% number needs to be either redefined, justified, or removed from the abstract.

This paper tries to establish how the gravitational-wave signal of a core-collapse supernova depends on three things observers cannot choose — the progenitor star's mass, its rotation rate, and the nuclear equation of state — and to quantify one further ingredient: the random seed of the convection perturbations. On the basis of 137 axisymmetric simulations covering 15 masses from 9.71 to 36.61 solar masses, three equations of state, and three rotation rates, extended up to 5.5 seconds after bounce, it claims that the CMF equation of state suppresses gravitational-wave amplitude and leaves a standing-accretion-shock-instability mode visible even at ~5 seconds post bounce, while rapid rotation delays shock revival and adds visible modes. The paper further claims that a recalibrated universal relation between the high-frequency mode and the proto-neutron-star mass and radius cuts the late-time fit error to roughly 100 Hz at the 95% level, and that neutrino memory dominates detectability for observatories with improved sub-10 Hz sensitivity. The payoff for a reader is concrete: a parameter-space map of supernova gravitational-wave detectability, a first quantification of how seed stochasticity alone shifts detection prospects, and a testable relation for inferring proto-neutron-star properties from a real detection.

What carries the argument. The machinery is a two-dimensional, general-relativistic supernova simulation code (CoCoNuT-FMT) run on a grid of 550 by 144 cells, delivering waveforms through the quadrupole formula and neutrino-memory strain through the Epstein formula. The load-bearing piece for the detectability claim is the seed-perturbation ensemble: four masses are rerun four extra times each, changing only the random radial-velocity perturbations, and the spread in shock revival time, explosion energy, and gravitational-wave energy is read off. For the universal-relation claim, the central object is the generalized frequency relation of Equation (6), which replaces the pure $(M/R^2)$ scaling of the Torres-Forné et al. (2019) relation with free power laws and a relativistic factor $1-2GM/Rc^2$; the stacked spectrogram, with amplitude weight $|A_i|^6$, is what shows that the f/g-mode time-frequency morphology is stable across perturbations even while amplitudes vary.

Core claim. The central claim is that stochasticity and astrophysical parameters act on different layers of the signal. For stars at the threshold between failed and successful shock revival — the SFHx s15 model being the clearest case, where two of five seeds fail to explode — the random velocity perturbations change the gravitational-wave amplitude enough to shift a detection's signal-to-noise ratio by up to 62%, while the same perturbations leave the time-frequency track of the dominant proto-neutron-star mode nearly unchanged, so that a stacked spectrogram keeps a sharply defined f/g-mode ridge. On top of that, a refit of the universal relation between the high-frequency mode and the proto-neutron-star mass and radius, generalized to $f_{\rm high}=3134.64\,\mathrm{Hz}\,(M/M_\odot)^{0.81}(R/10\,\mathrm{km})^{-1.43}(1-2GM/Rc^2)^{1.54}$, reduces the late-time fit error to about 100 Hz at the 95% confidence level, where the earlier Torres-Forné et al. (2019) relation drifts by up to 200 Hz and keeps growing. The same long-duration runs show that the CMF equation of state suppresses amplitudes and leaves a standing-accretion-shock-instability signature visible even at ~5 seconds post bounce, and that neutrino memory reaches amplitudes above 10,000 cm in the most extreme models — enough to dominate detectability for observatories with improved low-frequency sensitivity.

The premise it rests on. The central detectability claims assume the axisymmetric two-dimensional setup preserves the relative differences in gravitational-wave amplitude and mode structure that the comparative and detectability conclusions depend on; the authors acknowledge that total amplitudes are overestimated in 2D (b153, b179) and assert without direct 3D verification that relative differences would remain the same in 3D (b154), and the stochasticity spread rests on only five seeds per mass (b369), which they acknowledge is small-number-limited.

For stars straddling the explosion boundary, a detection's signal-to-noise ratio depends on which stochastic seed was realized, so detection-rate forecasts should treat near-threshold progenitors as a distribution of waveforms rather than a single template.

Neutrino memory dominates detectability below roughly 10 Hz, so for next-generation observatories with improved low-frequency sensitivity, the supernova detection reach is carried largely by the memory component rather than by the core-convection modes.

The recalibrated universal relation of Equation (6) reduces late-time 95% fit error to roughly 100 Hz for non-rotating and slowly rotating models, improving the prospects for inferring proto-neutron-star mass and radius from a real detection; rapidly rotating models remain the worst-fit case.

Because the f/g-mode time-frequency track is stable across seed perturbations, the proto-neutron-star mode frequency is a robust observable even when amplitude and explosion outcome are not.

In failed explosions, the standing-accretion-shock-instability mode persists to ~5 seconds post bounce and becomes detectable above signal-to-noise ratio of about 35, giving a gravitational-wave signature of imminent black hole formation.

The neutrino-memory amplitude approximately doubles across seeds in some near-threshold models, so uncertainties in next-generation detection distances are at least as large as in the convective-mode amplitudes; forecasters should marginalize over explosion outcome, which is the dominant source of scatter.

Because the seed-to-seed variation in gravitational-wave energy is dominated by whether or not the shock revives, a single astrophysical parameter — shock revival time or final explosion energy — may collapse most of the detected scatter, which would let a revival-time-tabulated signal-to-noise ratio stand in for full ensembles in rate estimates.

The neutrino-memory variation across seeds suggests that a memory-dominated low-signal detection, or a mode-dominated one, would itself constrain whether the shock revived, connecting gravitational-wave memory directly to black-hole-formation-time inference.

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