Neutrino Spectral Pinching in 3D Core-Collapse Supernovae: Late-Time Convergence, Failed-Explosion Signatures, and Viewing-Angle Dispersion

We present a systematic survey of the neutrino spectral pinching parameter αp (t, M, n̂) across 25 simulations spanning progenitor masses 8.1–100 M⊙ with durations up to 8.47 s post-bounce, computed with the Fornax code and the SFHo equation of state. We aim to characterize the secular evolution of αp with progenitor mass and post-bounce time, describe the spectral behavior observed in the two BH-forming models of the ensemble, quantify viewing-angle systematics for terrestrial detectors, and compute oscillation-corrected detection rates for next-generation experiments.

The pinching parameter is defined as:

αp = (2⟨E⟩2 −Erms) / (Erms − ⟨E⟩2)

Four results emerge from this survey:

1. The ν̄e pinching floor is αp = 1.92 ± 0.10 (N = 13 long-running models), lying 0.2–0.4 below the 1D Prometheus-Vertex prediction of Hüdepohl et al. [26].

2. Both BH-forming models (12.25, 14 M⊙) show anti-pinching (αp ≲ 0.9) before collapse, with deficit ∆αp ≈ 0.65 visible from t = 0.5 s.

3. Two of six long-running models exhibit a hierarchy reversal (⟨Eνe⟩ > ⟨Eνx⟩) after t = 5 s; leptonic flavors carry (40 ± 3)% of radiated energy.

4. The LESA dipole is suppressed by ≳ 3× in BH-forming models; viewing-angle spread ∆αp68% ≈ 0.8–1.5 dominates the spectral-inversion uncertainty. Mollweide sky maps of L(n̂), ⟨E⟩(n̂), and αp (n̂) reveal coherent angular structures driven by LESA and higher-order multipoles, with αp anticorrelated with luminosity and correlated with mean energy.

The analysis utilizes the quasi-thermal family parametrized by:

f (E; ⟨E⟩, αp) ∝ E / [⟨E⟩ (αp + 1) E exp −(αp + 1)]

where larger αp corresponds to a more sharply concentrated (“pinched”) spectrum with suppressed high- and low-energy tails. Values αp < 2 (“anti-pinching”) indicate a distribution broader than Maxwell–Boltzmann, indicative of a hard non-thermal tail from accretion or of a flattened temperature gradient at the decoupling surface.

The physical importance of αp is threefold: first, the inverse-beta-decay event rate at water-Cherenkov detectors scales as ⟨E2⟩ ∝ (αp + 2)/(αp + 1), so a shift ∆αp ≈ 0.3 alters predicted rates by ∼ 4–6%—hundreds of events at Hyper-Kamiokande for a Galactic supernova at 10 kpc—directly affecting equation-of-state and neutron-star-mass inference. Second, the boundaries of collective-oscillation spectral swaps and the onset of fast-flavor instabilities are controlled by the spectral widths of νe and ν̄e, so αp sets the post-oscillation spectra reaching terrestrial detectors and governs the detectability of the neutrino mass ordering. Third, charged-current captures in the neutrino-driven wind depend on the high-energy spectral tail; spectral pinching suppresses this tail, reducing the electron fraction Ye and shifting conditions for r-process nucleosynthesis.

The late-time ν̄e pinching floor αpν̄e = 1.92 ± 0.10 represents the first systematic 3D characterization of spectral convergence across a homogeneous model ensemble during Kelvin–Helmholtz cooling. The viewing-angle uncertainty in αp is quantified by the half-width of the 68% interval, σαgeom ≈ p(∆αp68%) = (Erms − ⟨E⟩2) / [2⟨E⟩2 − Erms], which ranges from 0.4–0.75, exceeding the statistical precision σαstat ≈ 0.12 by a factor of 3–6, thus making the geometric term dominant for spectral inversion analyses. The LESA dipole direction defines the axis of maximum νe /ν̄e contrast and is a key prior for spectral inversion.

The results establish αp and the LESA dipole as complementary diagnostics for next-generation neutrino detectors. The late-time floor value (Section IV B) is computed as the mean of per-model time-averages over t = 3.0 s–tmax after smoothing, restricted to the N = 13 models with tmax > 3.5 s; the Spearman coefficients use single smoothed values at the target epoch; and f+ uses a 250 ms boxcar applied after the primary 25 ms smoothing. The viewing-angle percentile bands in Section IV F 2 are computed directly from the unsmoothed sky map at a single snapshot epoch.

The angular structure visible in Fig. 8 does not arise from αp alone: it is physically connected to concurrent variations in luminosity L(n̂) and mean energy ⟨E⟩(n̂) across the sky. For the BH-forming 14 M⊙ model (top row of Fig. 9), all three maps are nearly featureless: luminosity, mean energy, and spectral shape are essentially isotropic at t = 1.0 s, with Pearson correlation r(αp, L/⟨L⟩) ≈ −0.16. This confirms that the anti-pinching signature in BHforming models is not an artifact of angle averaging: every direction on the sky is anti-pinched. For the successful 17 M⊙ model (bottom row of Fig. 9), the three maps share a common large-scale pattern driven by LESA and SASI: directions with elevated luminosity L(n̂) > ⟨L⟩ tend to have lower αp (broader spectra), with r(αp, L/⟨L⟩) ≈ −0.53. This anti-correlation is physically expected: in the LESA picture, the ν̄e-excess hemisphere emits a higher luminosity fraction but with a harder spectral tail from the deeper, more opaque decoupling surface, which broadens the effective spectrum and reduces αp. The mean energy ⟨E⟩(n̂) shows a similar spatial pattern, with high-⟨E⟩ pixels (green in the scatter) preferentially falling in the high-L regime, confirming that luminosity, mean energy, and spectral shape are all modulated by the same LESA-SASI geometry. The key observational conclusion is that the viewing-angle uncertainty in αp is correlated with the uncertainties in L and ⟨E⟩: a detector in a bright direction simultaneously measures a lower αp, a higher ⟨E⟩, and a higher event rate.

The resulting ν̄e pinching-parameter distribution over the 128 × 256 sky at t = 0.5, 1.0, and 3.0 s post-bounce is shown in Fig. 7 panels (b)–(d). The BH-forming models (12.25 and 14 M⊙) are again outliers: their narrow, low-αp distributions (αp,med ≈ 1.1, ∆α68 < 0.3) reflect the nearly isotropic, pinched spectra of the accretion-dominated phase at which they terminate (Section IV D 1). The total uncertainty budget for any single-direction measurement is dominated by the geometric term: σαgeom ≈ p(∆αp68%) ≈ 0.4–0.75, factor 3–6 above p statistical precision. A meaningful spectral inversion therefore requires either multi-detector triangulation or a prior on the LESA orientation.

The detection rates discussed in Section V C and Figure 10 use the unoscillated source spectra. In practice, MSW flavor transformation in the stellar envelope modifies the ν̄e flux reaching the detector. The resulting Hyper-K IBD event rates for both Normal Mass Ordering (NMO) and Inverted Mass Ordering (IMO) are shown in Fig. 11, where the NMO/IMO asymmetry is ∼ 8–12% during Kelvin–Helmholtz cooling, rising toward the BH-formation cutoff. This interplay between spectral pinching evolution and flavor transformation makes αp (t) a key input for oscillation-sensitive analyses at Hyper-K.

The results of Section V E focus on the cooling phase, so the neglect of synchronized collective oscillations is unlikely to affect the bulk NMO/IMO discrimination, but FFI effects remain unquantified. The 8–12% NMO/IMO rate asymmetry in IBD rate is the dominant spectral discriminant in the cooling phase. However, the viewing-angle dispersion in αp (Section IV F 2, σαgeom ≈ 0.4–0.75) means that for an unknown source direction, the effective αp entering the MSW rate formula (Eq. 22) is uncertain at the ∆αp ∼ 0.5 level, suppressing the apparent rate asymmetry. Multi-detector triangulation [4] or an independent determination of the LESA dipole direction would reduce this systematic. The BH-forming models (12.25 and 14 M⊙) are again outliers: their narrow, low-αp distributions (αp,med ≈ 1.1, ∆α68 < 0.3) reflect the nearly isotropic, pinched spectra of the accretion-dominated phase at which they terminate (Section IV D 1).

The neutrino spectral pinching parameter and the LESA dipole are established as complementary diagnostics for next-generation neutrino observatories. The late-time ν̄e pinching floor αpν̄e = 1.92 ± 0.10 represents the first systematic 3D characterization of spectral convergence across a homogeneous model ensemble during Kelvin–Helmholtz cooling.

The total radiated energy increases with mass from ≈ 2.1 × 1053 erg (8.1 M⊙) to ≈ 4.7 × 1053 erg (18.5 M⊙), consistent with the Eb ≈ (2.4–3.4) × 1053 erg binding energy range quoted in Section IV A [35]. The leptonic fraction flep = (Eνe + Eν̄e) / Etot across all 23 successful models (integrated to tmax) is 0.40 ± 0.03, consistent with Choi et al. [14]. The two BH-forming models have flep ≈ 0.47–0.48, slightly elevated because the accretion-phase νe /ν̄e luminosity is sustained until BH formation."

The LESA dipole amplitude ε(t) for all 25 models is shown in Fig. 7 panel (a). All successful explosions develop a LESA dipole with a peak amplitude in the range ε ≈ 0.02–0.23 over the full simulation duration (mean = 0.11, median = 0.10), with the highest values typically reached within the first ∼ 1–4 s post-bounce. The BH-forming models (12.25 and 14 M⊙) are strikingly different: ε < 0.05 throughout their entire evolution, a factor of ≳ 3 below typical successful explosions at the same epoch. This is consistent with the LESA suppression in BH-forming models reported by Walk et al. [70, 71] using the PrometheusVertex code; their work first identified that rapid accretion in failed explosions quenches the convective Ledouxunstable region that drives LESA.

Sky maps of L(n̂)/⟨L⟩, ⟨E⟩(n̂), and αp (n̂) on the 128 × 256 angular grid (Figs. 8 and 9) reveal coherent multipolar structures in all three observables. At t = 1 s, αp (n̂) is anticorrelated with L(n̂) (Pearson r ≈ −0.5 to −0.7) and positively correlated with ⟨E⟩(n̂) (r ≈ +0.3 to +0.6), confirming that the spectral pinching anisotropy is not an isolated artefact but traces the same hydrodynamic structures that modulate the luminosity and mean energy across the sky. These results establish αp and the LESA dipole as complementary diagnostics for next-generation neutrino detectors.

The late-time ν̄e pinching floor αpν̄e = 1.92 ± 0.10 represents the first systematic 3D characterization of spectral convergence across a homogeneous model ensemble during Kelvin–Helmholtz cooling. This is consistent with the quasi-thermal floor measured in Table III: αp = 1.92 ± 0.10 (Eq. 7).

The total radiated energy increases with mass from ≈ 2.1 × 1053 erg (8.1 M⊙) to ≈ 4.7 × 1053 erg (18.5 M⊙), consistent with the Eb ≈ (2.4–3.4) × 1053 erg binding energy range quoted in Section IV A [35]. The leptonic fraction flep = Eνe + Eν̄e / Etot across all 23 successful models (integrated to tmax) is 0.40 ± 0.03, consistent with Choi et al. [14]. The two BH-forming models have flep ≈ 0.47–0.48, slightly elevated because the accretion-phase νe /ν̄e luminosity is sustained until BH formation."

Based on a rigorous review of this scientific paper, here are the specific improvements that can be made to AI systems, along with what those improved systems could achieve:

) Improvements for AI Systems derived from the Paper:

1. The ability to perform Spectral Pinching Anisotropy analysis by correlating spectral moments across angular grids.

2. The capacity to reconstruct viewing-angle dependent physical parameters (like the spectral pinching parameter, αp(n̂)) from simulated data sets that include directional information (L(E, n̂), ⟨E⟩(n̂)).

3. The implementation of viewing-angle systematic uncertainty quantification using multipole decomposition (LESA dipole and SASI quadrupole).

4. The ability to perform Oscillation-Corrected Detection Rate calculations conditioned on both spectral shape evolution and neutrino mass ordering scenarios (NMO vs. IMO).

5. The capability to utilize a simulation-motivated spectral floor as a prior for constraining the equation of state (EOS) and testing nonstandard neutrino physics.

) What the Improved AI System Can Do:

1. This improved AI system can move beyond simple parameter fitting to perform high-fidelity, 3D diagnostic analysis of supernova neutrino signals. It can analyze raw or simulated directional data (like those from a future detector) and immediately determine the most likely physical interpretation by decomposing observed spectral variations into their underlying hydrodynamic drivers (e.g., separating effects due to Lepton-Number Emission Self-sustained Asymmetry (LESA, LESA dipole) from those caused by Standing Accretion Shock Instability (SASI)).

2. It can quantify the viewing-angle systematic uncertainty in spectral inversion analyses with high precision (e.g., determining that the geometric term dominates the uncertainty, resulting in a systematic error factor of 3–6 above statistical precision). This allows it to issue robust confidence intervals for inferred physical parameters, such as the neutrino mass ordering or the equation of state of neutron stars.

3. It can provide real-time assessments of supernova evolution by tracking secular trends (like the late-time convergence to a spectral floor, αp = 1.92 ± 0.10) and identifying critical physical signatures (like early anti-pinching in failed explosions). This allows it to rapidly classify simulation outcomes based on their proximity to known physical benchmarks, such as the transition boundary for black hole formation.

4. It can perform complex Bayesian inference where the prior is not just a simple constant but a sophisticated, simulation-derived distribution (the N=13 floor) that accounts for model mass and time dependence. This allows it to produce statistically rigorous constraints on unknown parameters (like PNS temperature or lepton fraction gradients) that are otherwise degenerate in simpler models.

5. It can assess the impact of observational biases (like detector energy thresholds, e.g., 5 MeV threshold), quantify their specific bias on derived spectral shape parameters (e.g., biasing αp upward by δαp ≈ 0.15), and correct these biases before comparing different mass-ordering hypotheses (NMO vs. IMO).

We present a systematic survey of the neutrino spectral pinching parameter alpha p(t, M, n-hat) across the Princeton Fornax ensemble of 3D core-collapse supernova simulations. We analyze 25 simulations spanning progenitor masses 8.1-100 M sun with durations up to 8.47 s post-bounce, computed with the Fornax code and the SFHo equation of state. The pinching parameter alpha p = (2 squared - E rms 2)/(E rms squared - 2) is derived from 12-bin spectral moments on a 128x256 sky grid for three neutrino species, enabling time- and angle-resolved spectral characterization. Four results emerge. (1) The nu-bar e pinching floor is alpha p = 1.92 +/- 0.10 (N=13 long-running models), lying 0.2-0.4 below 1D predictions due to 3D PNS convection. (2) Both BH-forming models (12.25, 14 M sun) show anti-pinching (alpha p < 0.9) before collapse, with deficit Delta alpha p 0.65 visible from t = 0.5 s. (3) Two of six long-running models exhibit a hierarchy reversal (>) after t = 5 s; leptonic flavors carry (40 +/- 3)% of radiated energy. (4) The LESA dipole is suppressed by >3x in BH-forming models; viewing-angle spread Delta alpha p(68%) 0.8-1.5 dominates spectral-inversion uncertainty. Mollweide sky maps reveal coherent angular structures with alpha p anticorrelated with luminosity and correlated with mean energy. Detection rates at Hyper-Kamiokande, DUNE, JUNO, and IceCube yield 8-12% NMO/IMO discrimination during Kelvin-Helmholtz cooling. The late-time nu-bar e pinching floor represents the first 3D characterization of spectral convergence during Kelvin-Helmholtz cooling.