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Neutron Star Merger Universality Relations for a Quark--Hadron Crossover Equation of State

This paper investigates how a crossover transition from hadronic to quark matter in neutron star mergers, particularly within the QHC21 equation of state, alters post-merger gravitational wave signals and causes measurable deviations from established universality relations, potentially offering a unique observational signature for such phase transitions.

Original authors: Wei Sun, Grant J. Mathews, Atul Kedia

Published 2026-09-16

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CC BY 4.0

Original authors: Wei Sun, Grant J. Mathews, Atul Kedia

Original paper licensed under CC BY 4.0 (http://creativecommons.org/licenses/by/4.0/). ✨ This is an AI-generated explanation of the paper below. It is not written or endorsed by the authors. For technical accuracy, refer to the original paper. Read full disclaimer

Deep within the crushing cores of neutron stars, matter exists in a state far more extreme than anything found on Earth. These stellar remnants are so dense that a single teaspoon of their material would weigh billions of tons. Under such immense pressure, the protons and neutrons that make up normal atomic nuclei are thought to break apart, dissolving into a soup of their constituent parts called quarks. This transition from solid nuclear matter to a fluid of free quarks is a fundamental question in physics, yet it remains hidden from direct view. While scientists have observed hints of this change in laboratory experiments using heavy ions, they have never seen it happen in the cosmos. The only way to probe the interior of a neutron star is to listen to the gravitational waves—ripples in the fabric of space-time—emitted when two of these stars collide. By analyzing the frequency and duration of these ripples, researchers can infer the stiffness of the matter inside the stars and determine whether a phase change to quark matter occurs during the violent crash.

A team of researchers at the University of Notre Dame and North Carolina Central University has used powerful computer simulations to explore this possibility, specifically looking for the signature of a smooth transition, known as a crossover, rather than a sudden, sharp jump. In their study, published in September 2026, they modeled the collision of two neutron stars using a variety of theoretical descriptions for how matter behaves under extreme pressure. Some of these models described matter that remained purely nuclear, while others included a gradual shift where hadronic matter slowly transforms into quark matter at densities roughly two to five times the nuclear saturation density. The researchers found that when this smooth crossover occurs, it creates a distinct and prolonged burst of high-frequency gravitational waves after the stars merge. Specifically, the emission of waves in the range of 2 to 3.5 kilohertz lasts longer than it does in models without this transition, because the crossover provides extra pressure support that keeps the merged remnant from collapsing immediately into a black hole.

The team compared their results against established patterns, known as universality relations, which link observable quantities like the frequency of the gravitational waves to the properties of the neutron stars. These relations have been found to hold true across many different models of purely nuclear matter. The researchers discovered that while some of these patterns remain valid even when a crossover occurs, others break down in a predictable way. For instance, the relationship between the peak frequency of the gravitational waves and the star's tidal deformability—a measure of how easily the star's shape can be distorted—deviates systematically when a crossover transition is present. In their simulations, the crossover models produced a distinct cluster of data points that fell below the line established by purely nuclear models. This deviation is significant because it offers a potential observational signal: if future detectors like LIGO capture a post-merger signal that does not fit the standard nuclear pattern, it could indicate that the matter inside the stars underwent a smooth transition to quarks.

Interestingly, the study also highlighted that not all deviations are the same. Previous research suggested that a sudden, first-order phase transition would cause the gravitational wave frequency to jump higher, pushing the data points above the standard line. In contrast, the smooth crossover transition studied here pushes the data points downward. This difference in direction provides a way for astronomers to distinguish between a sudden phase change and a smooth one, should the data ever become clear enough to see. The researchers noted that the QHC21 model, which features a crossover occurring at a lower density, showed the most pronounced effects, extending the life of the merged remnant and making the high-frequency signal more detectable. However, they also found that some relationships, such as the one linking the maximum density reached during the merger to the peak frequency, did not show a clear deviation for the crossover models. This is because the factors influencing both sides of that specific relationship shifted in a way that partially canceled each other out, masking the transition.

Ultimately, this work suggests that the key to unlocking the secrets of neutron star interiors lies in the details of the gravitational wave signal that follows the initial collision. By looking for specific deviations in the frequency patterns, particularly in the high-frequency range of 2 to 3.5 kilohertz, scientists may soon be able to confirm whether quark matter exists in the hearts of these dense stars. The simulations provide a roadmap for what to look for: a signal that lingers longer than expected and follows a different mathematical path than the one predicted for purely nuclear matter. While the paper relies on computer models rather than direct observation, it establishes a clear theoretical expectation that could guide the interpretation of future data from gravitational wave detectors, bringing us closer to understanding the fundamental nature of matter at its most extreme.

Technical Summary: Neutron Star Merger Universality Relations for a Quark–Hadron Crossover Equation of State

Problem Statement
The internal composition of neutron stars (NS) at densities exceeding nuclear saturation ( $n_{sat} \approx 0.16 \text{ fm}^{-3}$ ) remains uncertain, particularly regarding the potential transition from hadronic matter to deconfined quark matter. While ground-based heavy-ion collisions suggest a crossover transition, astrophysical evidence is lacking. A critical open question is whether a smooth crossover phase transition (PT) or a strong first-order phase transition (FOPT) occurs in NS cores. Previous studies have established "universality relations"—correlations between observable gravitational wave (GW) quantities and NS properties—for purely hadronic equations of state (EoS). While deviations from these relations have been proposed as signatures of a FOPT, it remains unclear whether a crossover transition produces identifiable deviations in these same relations. This work addresses the gap in systematically exploring how a quark-hadron crossover affects post-merger GW observables and universality relations.

Methodology
The authors performed numerical relativity simulations of equal-mass binary neutron star (BNS) mergers using the Einstein Toolkit.

Equations of State (EoS): The study utilized two classes of EoSs:
1. Hybrid Crossover EoSs: Based on the QHC19 and the updated QHC21 models. These models thermodynamically interpolate between low-density hadronic EoSs (Togashi or chiral effective field theory, $\chi$ EFT) and high-density quark matter (Nambu–Jona-Lasinio model). The QHC21 models feature a stiffer EoS at intermediate densities ( $\sim 2\text{--}5 n_{sat}$ ) and a lower onset density for the crossover compared to QHC19.
2. Purely Hadronic EoSs: A broad set of models (including SLy, BL, APR, PCGS, and various XMLSLZ variants) spanning a range of stiffness to establish baseline universality relations.
Simulation Setup: Simulations involved equal-mass binaries with baryonic masses of 1.45, 1.50, and 1.55 $M_\odot$ (gravitational masses $\sim 1.30\text{--}1.40 M_\odot$ ). The hydrodynamic evolution employed a piecewise-polytropic fit for the cold EoS and a thermal contribution ( $\Gamma_{th} = 1.8$ ).
Analysis: The authors extracted GW signals using the Newman–Penrose formalism ( $\Psi_4$ ) at a radius of 700 $M_\odot$ . Key observables included the dominant post-merger GW frequency ( $f_{peak}$ ), the frequency at maximum strain amplitude ( $f_{max}$ ), the dimensionless tidal deformability ( $\Lambda$ ), the maximum rest-mass density ( $\rho_{max}^{max}$ ), and the characteristic mean-density scale.

Key Contributions and Results

Post-Merger Lifetime and Stiffness: The study confirms that the QHC21 crossover EoSs, which stiffen at densities of $\sim 2\text{--}5 n_{sat}$ , provide additional pressure support. This leads to significantly longer-lived hypermassive neutron star (HMNS) remnants compared to softer hadronic EoSs or QHC19 models, which often collapse promptly to black holes.
Spectral Properties: A correlation was observed between EoS stiffness and $f_{peak}$ . Stiffer EoSs (including the crossover models) produce larger remnant radii, resulting in lower characteristic dynamical frequencies and thus lower $f_{peak}$ values (typically in the 2–3.5 kHz range).
Universality Relations: The authors tested several established universality relations to determine if crossover transitions cause deviations:
- Relations that Hold: The relations between $\log_{10}(M_{tot}f_{max})$ and $\Lambda^{1/5}$ , and between $\rho_{max}^{max}$ and $(M_{tot}/R_{max}^3)^{1/2}$ , remain valid for crossover EoSs. The crossover data points generally align with soft hadronic EoSs in these plots.
- Relations with Deviations: Significant deviations from hadronic universality were found in:
  - The $f_{peak}$ – $\Lambda$ relation.
  - The $f_{peak}$ – $f_{max}$ relation.
  - The $f_{max}$ – $\rho_{max}^{max}$ relation.
  - The $\rho_{max}^{max}M_{tot}$ – $\Lambda$ relation.
- Direction of Deviation: Unlike FOPT models, which typically shift $f_{peak}$ to higher frequencies (upward deviation), the crossover EoSs in this study exhibit a systematic downward deviation in $f_{peak}$ -related relations. This is attributed to the stiffening of the EoS at intermediate densities, which lowers the characteristic frequency.
- The $\rho_{max}^{max}$ – $f_{peak}$ Relation: The authors found no clear deviation for crossover EoSs in the $\rho_{max}^{max}$ – $f_{peak}$ relation. They explain this by noting that both quantities are shifted to lower values by the crossover transition, causing the shifts to partially compensate for each other, thereby masking the deviation.

Significance and Claims
The paper claims that while some universality relations are robust against the presence of a crossover transition, specific relations involving $f_{peak}$ and $\rho_{max}^{max}$ offer a potential observational signature to distinguish a crossover PT from a FOPT.

Differentiation: The study suggests that the direction of deviation is a key discriminator: FOPTs tend to shift relations upward (higher frequencies), whereas crossover transitions shift them downward (lower frequencies) due to EoS stiffening.
Observational Potential: The authors propose that future high-precision measurements of the post-merger GW power spectral density in the 2–3.5 kHz region could reveal these deviations. Such observations could provide evidence for the existence of a quark-hadron crossover and help distinguish it from a first-order phase transition.
Modesty: The authors acknowledge that the detection of these deviations depends on the accuracy of GW data and simulation resolution, particularly for quantities like $\rho_{max}^{max}$ . They do not claim that these relations definitively prove a crossover exists, but rather that deviations in specific directions could serve as indicators warranting further investigation.

Technical Summary: Neutron Star Merger Universality Relations for a Quark–Hadron Crossover Equation of State

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