Core-Collapse Waveform Morphologies

Core-collapse waveforms are conditional products of relativistic gravity, nuclear matter, neutrino transport, hydrodynamic instability, rotation, and numerical extraction. Their time-frequency morphology can still be interpreted: bounce, prompt convection, standing-accretion-shock activity, proto-neutron-star oscillation, and asymmetric explosion occupy different timescales. This page makes those mappings without promoting one simulation family to a universal template. Search pipelines, upper limits, and current detections are excluded.

Helpful background. Einstein Equations supplies gravity and quadrupolar radiation. Waves, Green Functions, and Geometric Optics supplies spectral localization. Gravitational Radiation and Waveforms supplies strain and flux normalization. Relativistic Matter and Compact Objects supplies collapse, transport, and the equation of state. Relativistic Cosmology supplies distance and redshift conventions.

Stages and their waveform signatures

Define bounce time $t_b=0$ and describe strain in a source-aligned polarization frame. The main conditional associations are:

• Rotational bounce: an axisymmetric spike followed by $10$–$20\,\mathrm{ms}$ of proto-neutron-star ringing; the frequency and amplitude depend non-monotonically on rotation because extreme centrifugal support lowers the bounce density.
• Prompt and neutrino-driven convection: stochastic broadband power driven by overturning flows and accretion downflows.
• Standing accretion shock instability (SASI): lower-frequency, comparatively coherent modulation of the shock and accretion flow, when that instability develops.
• Proto-neutron-star modes: a dominant ridge that rises as the star contracts; one reviewed $25M_\odot$ model moves from roughly $400\,\mathrm{Hz}$ at $150\,\mathrm{ms}$ to $1100\,\mathrm{Hz}$ at $600\,\mathrm{ms}$ after bounce.
• Asymmetric explosion and neutrino emission: slowly varying contributions from anisotropic mass motion and escaping neutrino energy, whose presence, sign, and relative importance vary across models. Matter-quadrupole extraction alone does not include the neutrino contribution.

These associations and their model dependence are synthesized in Abdikamalov, Pagliaroli, and Radice 2022, §§2–5, especially pp. 16–19, PDF.

A declared rotational benchmark

The review’s Fig. 8 and accompanying text compare a $12M_\odot$ progenitor with the same rotation-law family and pre-collapse central angular velocities $\Omega_c=2.0$, $5.5$, and $9.0\,\mathrm{rad\,s^{-1}}$. The reported peak frequencies are approximately $800$, $750$, and $372\,\mathrm{Hz}$, respectively.

The underlying 98-configuration study specifies the rotation law, collapse microphysics, waveform extraction, and how bounce features encode inner-core angular momentum; its model construction and waveform comparisons are documented in Abdikamalov et al. 2014, §§II–IV and Figs. 1–8.

model label used here	$\Omega_c$ ($\mathrm{rad\,s^{-1}}$)	reported peak frequency (Hz)	interpretation
R2.0	2.0	800	rotation weak at bounce; prompt convection follows
R5.5	5.5	750	strong rotational bounce and ringing
R9.0	9.0	372	extreme centrifugal support broadens and slows bounce

Only the printed frequency values are used here; no third-party waveform array is copied or redistributed. The quantities plotted in the source use distance-scaled strain $hD$ for an equatorial view and axisymmetric polarization. A distance and orientation would be needed to turn $hD$ into detector-frame strain. Gravity treatment, hydrodynamics, neutrino prescription, equation of state, progenitor, rotation law, and quadrupole extraction remain part of the model label, not nuisance details.

Reproducible frequency-recovery check

The following benchmark tests a frequency estimator, not a core-collapse simulation. It uses an illustrative unit-amplitude, linearly polarized, Gaussian-windowed carrier near the R5.5 frequency, with exact $f_*=750.3\,\mathrm{Hz}$ over $-10$ to $20\,\mathrm{ms}$, sampled at $16384\,\mathrm{Hz}$. The decimal value is deliberately chosen off the search grids and is not a more precise physical prediction for R5.5. A grid search fits sine and cosine coefficients by least squares. The normalized amplitude removes distance; $h_\times=0$ fixes the illustrative polarization.

Use CPython 3.12.7 and NumPy 2.1.3. Save the following program as core_collapse_resolution.py. The script writes core_collapse_resolution.csv and prints the same refinement rows.

import numpy as np

f_true = 750.3
sample_rate = 16384.0
t = np.arange(-0.010, 0.020, 1.0 / sample_rate)
window = np.exp(-0.5 * (t / 0.006) ** 2)
h = window * np.sin(2.0 * np.pi * f_true * t)
rows = []
for grid_n in (101, 201, 401):
    frequencies = np.linspace(700.0, 800.0, grid_n)
    residuals = []
    for frequency in frequencies:
        design = window[:, None] * np.column_stack((
            np.sin(2.0 * np.pi * frequency * t),
            np.cos(2.0 * np.pi * frequency * t),
        ))
        fit = design @ np.linalg.lstsq(design, h, rcond=None)[0]
        residuals.append(np.linalg.norm(h - fit))
    estimate = frequencies[int(np.argmin(residuals))]
    rows.append((grid_n, estimate, abs(estimate - f_true)))
np.savetxt("core_collapse_resolution.csv", rows, delimiter=",",
           header="grid_n,frequency_hz,error_hz", comments="")
for row in rows:
    print(f"{row[0]:4d} {row[1]:8.2f} {row[2]:6.3f}")

Create an isolated environment and rerun with:

python3.12 -m venv .venv-core-collapse
.venv-core-collapse/bin/python -m pip install numpy==2.1.3
.venv-core-collapse/bin/python core_collapse_resolution.py

The expected grid-refinement result is:

frequency-grid resolution	recovered frequency (Hz)	absolute error (Hz)
101	750.00	0.300
201	750.50	0.200
401	750.25	0.050

The exact carrier is an independent benchmark. Decreasing error verifies the estimator and grid, but it does not validate transport, gravity, or extraction in any simulation.

Convergence and cross-model comparisons

A published waveform product should ideally compare at least three spatial resolutions and, where possible, an independent code or extraction method. Track bounce time, peak frequency, peak $hD$, ridge frequency versus time, and radiated energy separately. Phase convergence can coexist with amplitude disagreement, while turbulent convection may not converge pointwise even when ensemble spectra stabilize.

Cross-family agreement that the contracting proto-neutron star generates an upward frequency ridge is stronger than any one amplitude prediction. Conversely, the presence and strength of SASI, explosion offset, and stochastic convection depend sensitively on progenitor, dimensionality, transport, and resolution. The review explicitly warns that different models can explode with very different instability histories.

Failure modes and scope

Axisymmetry suppresses non-axisymmetric modes and makes the early rotational signal artificially restricted. Approximate neutrino transport changes heating and contraction. An effective relativistic potential is not identical to full general relativity. Finite resolution alters turbulence, and a quadrupole extraction formula is not the same as asymptotic curvature extraction. Equation-of-state and progenitor variations can shift the ridge as much as numerical refinement. Orientation and distance rescale the observed polarizations.

Accordingly, a morphology should be stated as “in the declared model family” rather than “core collapse predicts.” Matter dynamics belongs to Relativistic Matter and Compact Objects; production algorithms belong to Computational Relativity; observational verdicts belong to Gravitational-Wave Astronomy.

Exercises

1. Interpret the non-monotonic benchmark. Why does increasing $\Omega_c$ from $5.5$ to $9.0\,\mathrm{rad\,s^{-1}}$ lower the reported peak frequency rather than raising it?

Solution

At moderate rotation the core reaches a compact bounce and undergoes rapid oscillations. At extreme rotation, centrifugal support slows collapse and halts it at lower central density. The characteristic dynamical frequency scales roughly as $\sqrt{G\bar\rho}$, so lower mean density produces a broader, lower-frequency bounce. The table is conditional on the common progenitor and rotation-law family; it is not a monotonic universal map from spin to frequency.

2. Read the numerical refinement. Estimate the frequency uncertainty from the last two grid rows and name one uncertainty that this estimate omits.

Solution

The estimates differ by $\lvert750.50-750.25\rvert=0.25\,\mathrm{Hz}$, while the known synthetic error at the finest grid is $0.05\,\mathrm{Hz}$. A conservative grid-discretization scale is therefore about $0.25\,\mathrm{Hz}$. It omits waveform-model error entirely: real core-collapse frequencies also depend on numerical hydrodynamics, equation of state, transport, extraction, and time-window choice.

References

• Abdikamalov, Ernazar, Giulia Pagliaroli, and David Radice. “Gravitational Waves from Core-Collapse Supernovae.” In Handbook of Gravitational Wave Astronomy. Singapore: Springer, 2022. Open PDF.
• Abdikamalov, Ernazar, Sarah Gossan, Alexandra M. DeMaio, and Christian D. Ott. “Measuring the Angular Momentum Distribution in Core-Collapse Supernova Progenitors with Gravitational Waves.” Physical Review D 90 (2014): 044001. DOI. Open PDF.