Impact of the equation of state on core collapse supernovae I: the low-$T/|W|$ instability

This study demonstrates through three-dimensional simulations that while the low-$T/|W|$ instability robustly occurs in rapidly rotating core-collapse supernovae regardless of the nuclear equation of state, its specific multimessenger signatures—particularly the gravitational wave frequency—vary systematically with the equation of state, offering a potential diagnostic tool for probing dense-matter physics.

The Gist

Imagine a massive star, about 35 times heavier than our Sun, running out of fuel. Like a deflating balloon, its core collapses inward at incredible speeds. Usually, this collapse creates a shockwave that bounces back, blowing the star apart in a supernova. But if that star is spinning very fast, things get even more chaotic and interesting.

This paper is like a high-speed, 3D movie simulation of that exact moment. The researchers wanted to see how the "rules of physics" inside the star's core—specifically how matter behaves under crushing pressure—change the way the explosion happens and what signals it sends out.

Here is the story of their findings, broken down into simple concepts:

1. The "Recipe" for the Core (The Equation of State)

Think of the star's core as a giant, super-dense soup. In physics, the "Equation of State" (EOS) is like the recipe for that soup. It tells us how the ingredients (protons, neutrons, etc.) react when you squeeze them.

• The Experiment: The researchers took the same spinning star and ran the simulation five times. Each time, they used a different "recipe" (five different theoretical models of dense matter).
• The Goal: They wanted to see if changing the recipe changed the outcome of the explosion.

2. The "Wobbly Top" Instability

Because the star is spinning so fast, the new core that forms (called a Proto-Neutron Star) doesn't stay perfectly round. It starts to wobble like a spinning top that's about to fall over.

• The Low-T/|W| Instability: This is a fancy name for a specific kind of wobble. Unlike other instabilities that need the star to be spinning extremely fast, this one happens even at moderate speeds.
• The Result: In all five of their different "recipes," this wobble happened. It was a robust feature. The core didn't just stay round; it developed giant, swirling spiral arms, like a pinwheel made of star-stuff.

3. The "Fingerprint" of the Wobble

While the wobble happened in every model, the way it wobbled depended on the recipe.

• The Analogy: Imagine five different people spinning a hula hoop. They all spin it, but one person makes it spin fast and tight, while another makes it spin slowly and loosely.
• The Finding: The "stiffness" of the recipe determined the speed of the wobble.
  • Stiffer recipes (where the matter is harder to squish) made the core smaller and tighter. This made the spiral spin faster, creating a higher-pitched signal.
  • Softer recipes (where matter squishes easier) made the core larger and looser. This made the spiral spin slower, creating a lower-pitched signal.

4. The Cosmic "Radio Stations" (Gravitational Waves and Neutrinos)

When the star wobbles, it broadcasts two types of signals across the universe:

1. Gravitational Waves: Ripples in space-time itself.
2. Neutrinos: Tiny, ghostly particles that stream out of the core.

The Gravitational Wave Signal:
The paper found that the "pitch" (frequency) of the gravitational waves is a direct dial-in to the stiffness of the core's recipe.

• If we hear a high-pitched hum from a supernova, it tells us the core is made of "stiff" matter.
• If we hear a low-pitched hum, the core is "soft."
• This is huge because it means gravitational waves could act as a tool to "weigh" and "measure" the physics of matter that we can't recreate in any lab on Earth.

The Neutrino Signal:
The wobble also makes the neutrino light flicker.

• The light doesn't just shine steadily; it pulses in rhythm with the spiral arms.
• These pulses are strongest if you look at the star from its "equator" (the side), much like how a lighthouse beam is brightest when you are in the path of the rotating light.
• The paper suggests that if we have big enough neutrino detectors, we might be able to see these flickers, confirming the wobble is happening.

5. The Big Picture

The researchers concluded that:

• The wobble is real: No matter which "recipe" of physics you use, a fast-spinning star will develop these giant spiral arms.
• The wobble is a messenger: The specific sound (frequency) of the gravitational waves and the flickering pattern of the neutrinos act as a diagnostic tool. They tell us exactly how "stiff" or "soft" the matter inside the dying star is.
• It's detectable: If a star like this exploded in our neighborhood (the Milky Way or nearby galaxies), our current and future detectors (like LIGO for sound and huge tanks of water for neutrinos) would be able to hear and see these signals clearly.

In short, the paper shows that the "music" a dying star plays isn't random; it's a direct reflection of the fundamental laws of physics holding its core together. By listening to the music, we can learn about the ingredients of the universe.

Technical Summary

Technical Summary: Impact of the Equation of State on Core Collapse Supernovae I: The Low-T/|W| Instability

Problem Statement
Rapidly rotating core-collapse supernovae (CCSNe) are promising sources of multimessenger emission, where non-axisymmetric dynamics in the newly formed proto-neutron star (PNS) can imprint characteristic signatures on gravitational waves (GWs) and neutrinos. A critical uncertainty in modeling these events is the dense-matter nuclear equation of state (EOS), which governs the PNS's contraction, thermal evolution, and maximum mass. While previous studies have explored the impact of the EOS on explosion dynamics and GW emission, the specific influence of different nuclear EOSs on the development of the low-$T/|W|$ instability (LTWI)—a shear-driven non-axisymmetric instability in rapidly rotating PNSs—and its associated multimessenger signatures remains to be systematically characterized. The LTWI is distinct from the classical bar-mode instability as it can develop at significantly lower ratios of rotational to gravitational energy ($T/|W| \sim 0.04-0.06$), conditions achievable in rapidly rotating CCSNe.

Methodology
The authors performed a series of three-dimensional (3D) neutrino-magnetohydrodynamics (MHD) simulations using the Aenus-ALCAR code. The study focused on the collapse of a single, rapidly rotating Wolf–Rayet progenitor (35OC, $M_{\text{ZAMS}} = 35 M_\odot$) to isolate the effects of the EOS.

• Progenitor and Magnetic Field: The simulations utilized the original rotation profile of the 35OC model but replaced the pre-supernova magnetic field with a weak, analytically prescribed dipolar field ($B \sim 10^6$ G) to ensure a regular magnetic topology while avoiding the suppression of the LTWI observed in previous models with stronger fields.
• Equations of State: Five distinct finite-temperature nuclear EOSs were employed: LS220, SLy4, BHB$\Lambda\phi$, BBSk3, and TSO. These models span a range of stiffness, symmetry energy parameters, and include different treatments of hyperons and nuclear interactions (Skyrme vs. Relativistic Mean Field approaches).
• Numerical Setup: Simulations evolved in axisymmetry until 5 ms before core bounce, then mapped to 3D grids. The grid extended to $4 \times 10^{10}$ cm with high resolution in the inner regions ($5 \times 10^4$ cm). Neutrino transport was handled via a spectral two-moment scheme.
• Analysis Tools: GW signals were extracted using the standard quadrupole formula with shell decomposition. The instability was characterized using spherical harmonics decomposition of density, Fourier analysis, and Ensemble Empirical Mode Decomposition (EEMD) to isolate intrinsic mode functions (IMFs). The "stiffness" of the PNS was tracked using the tidal Love number ($\kappa_2$) computed from spherically averaged profiles.

Key Contributions and Results

1. Robustness of the LTWI: The LTWI developed in all five EOS models considered, confirming its occurrence is robust for this rapidly rotating progenitor. The instability manifests as large-scale spiral modes ($m=1, 2$) that generate quasi-periodic GW emission and modulate neutrino luminosities.
2. EOS-Dependent Variations: While the instability's onset is robust, its specific characteristics vary significantly with the EOS:
   • Onset and Morphology: Onset times ranged from $\sim 50$ ms to $\sim 100$ ms post-bounce. The dominant azimuthal structure varied; for instance, model SL exhibited a persistent two-armed spiral, while others showed transitions between one-, two-, and three-armed configurations.
   • Frequency Correlation: A clear linear correlation was found between the dominant GW frequency associated with the LTWI and the effective stiffness (inverse of the tidal Love number $\kappa_2$) of the PNS. Stiffer EOSs (lower $\kappa_2$) produced higher GW frequencies (up to $\sim 370$ Hz), while softer EOSs produced lower frequencies.
   • Neutrino Modulation: The LTWI induced characteristic quasi-periodic modulations in neutrino luminosities, particularly along directions close to the equatorial plane. These modulations appeared at frequencies ($\sim 150-180$ Hz) distinct from prompt convection and SASI signals.
3. Pre-Bounce Evolution: The choice of EOS influenced the stratification and rotational structure of the inner core prior to bounce, leading to variations in the inner core radius, mass, and differential rotation profiles at the moment of core bounce, even with identical initial rotation profiles.
4. Signal Decomposition: The study demonstrated that EEMD effectively isolates the GW signal component associated with the LTWI (specifically IMF 6), separating it from bounce signals and other oscillatory modes.
5. Detectability:
   • Gravitational Waves: The GW signals are detectable by current-generation detectors (LIGO, Virgo, KAGRA) for Galactic events and by next-generation detectors (Einstein Telescope, Cosmic Explorer) for events throughout the Local Group. The equatorial symmetry of the system suppresses the $\times$-polarization for equatorial observers, reducing detection distances by a factor of two compared to polar views.
   • Neutrinos: While the total neutrino energy varies only modestly ($\sim 10\%$) across EOSs, the EOS affects the neutrinosphere temperature and the amplitude/duration of LTWI-driven modulations. The modulation signatures are expected to be observable in high-statistics detectors like IceCube for Galactic events.

Significance and Claims
The paper claims that the frequency of the low-$T/|W|$ instability-driven gravitational wave signal serves as a potential diagnostic for the dense-matter equation of state in rapidly rotating CCSNe. This offers a complementary constraint to information carried by neutrino signals. The study establishes that while the LTWI is a robust feature of rapidly rotating cores, its specific multimessenger fingerprints (frequencies, lifetimes, and morphologies) are sensitive to the microphysics of dense matter encoded in the EOS.

The authors note limitations, including the use of a single progenitor model, the assumption of weak magnetic fields, and the relatively short simulation time (250 ms post-bounce). They emphasize that future work must explore a broader range of progenitors, rotation rates, and magnetic field strengths to disentangle EOS effects from other dynamical parameters and to assess the generality of the identified trends. The results suggest that future multimessenger observations of rapidly rotating CCSNe could provide direct constraints on the properties of matter at supranuclear densities.