As above, so below: assessing extremeness of the neutron-star equation of state based on the unstable branch

Technical Summary: Assessing Extremeness of the Neutron-Star Equation of State Based on the Unstable Branch

Problem Statement
Microphysical models of neutron-star (NS) matter are essential for interpreting astrophysical observations, yet they are typically constructed and validated only up to the maximum densities reached in stable neutron stars. The behavior of the Equation of State (EoS) at densities exceeding these limits has historically been treated as unconstrained or assumed to follow the model's original trajectory. However, the EoS must ultimately connect to the perturbative Quantum Chromodynamics (pQCD) regime at asymptotically high densities ($n \sim 20\text{--}40 n_{\text{sat}}$). The requirement that this connection be causal, stable, and thermodynamically consistent imposes constraints on the EoS at intermediate densities, even in regions where neither the phenomenological models nor pQCD calculations are directly applicable. This paper investigates whether standard hadronic models, when extended to the maximal central densities of stable NSs, imply physically plausible continuations to the pQCD regime or if they necessitate "extreme" and unlikely behaviors.

Methodology
The authors employ a non-parametric, model-agnostic approach to sample the space of valid EoS extensions. The methodology involves:

1. Framework: Utilizing a Gaussian-process bridge (GPB) to generate a large ensemble of EoS extensions connecting a model's termination density ($n_{\text{term}}$) to the pQCD limit at $n_{\text{pQCD}} = 30 n_{\text{sat}}$.
2. Constraints: The sampling enforces mechanical stability, causality (sound speed $c_s \leq 1$), and thermodynamic consistency. The extensions are diffused to impose correlation lengths, allowing for sharp features at the transition from the model to the extension while ensuring smooth behavior toward the pQCD limit.
3. pQCD Tension Index ($I_{\text{pQCD}}$): To quantify the "extremeness" of a connection, the authors use the index defined in Komoltsev & Kurkela (2022) and Komoltsev et al. (2024). This index measures how close a valid EoS extension is to the minimal or maximal pressure-difference constructions required to bridge the low- and high-density points. $I_{\text{pQCD}} \approx 1$ indicates a connection requiring maximal stiffness or strong phase transitions.
4. Model Selection: A representative set of cold, $\beta$-equilibrium EoSs from the CompOSE database was selected, categorized into:
   • Purely Hadronic: Microscopic calculations (APR, BL(chiral)), non-relativistic density functionals (Sly4, BSk24), and relativistic density functionals (DD2, SFHo, FSU2R).
   • Models with Additional Degrees of Freedom: Models including quark matter (QHC18, V-QCD), hyperons (DDHdeltaY4, CMF-7), and quarkyonic matter.
5. Analysis: The study examines extensions from two termination points: the central density of a canonical $1.4 M_\odot$ star and the maximal central density ($n_{\text{TOV}}$) of the most massive stable star supported by each model.

Key Results

1. Canonical Density ($1.4 M_\odot$): When models are trusted only up to the central density of a $1.4 M_\odot$ star, the required extensions to the pQCD regime are relatively unconstrained. A broad range of behaviors, including various sound speed profiles and trace anomalies, are consistent with pQCD constraints for all renormalization scales ($X$).
2. Maximal Density ($n_{\text{TOV}}$): The situation changes drastically when models are assumed valid up to the maximal stable mass density.
   • Purely Hadronic Models: Most purely nucleonic models (Sly4, BSk24, DD2, SFHo, BL) are incompatible with higher values of the pQCD renormalization scale ($X_{\text{high}}$). For the allowed lower $X$ values, the extensions require an abrupt, dramatic softening immediately above the TOV density. This manifests as a strong, first-order phase-transition-like jump in density ($\Delta n \approx 15 n_{\text{sat}}$) and a kink in the mass-radius relation entering the unstable branch.
   • The FSU2R Exception: The FSU2R model permits extensions across the full range of $X$ because it exhibits softer behavior and a positive trace anomaly up to $n_{\text{TOV}}$. However, the authors note this model predicts radii $>12$ km, which conflicts with NICER observations.
   • Models with Additional Degrees of Freedom: Models incorporating quarks, hyperons, or quarkyonic matter (V-QCD, Quarkyonic) generally soften before reaching the TOV density. Consequently, they do not require the dramatic, coincidental phase transition immediately above $n_{\text{TOV}}$ that purely hadronic models do. Their extensions to pQCD are less restrictive.
3. Quantitative Trend: Analysis of the entire CompOSE database via the $I_{\text{pQCD}}$ index reveals that purely hadronic EoSs trusted up to $n_{\text{TOV}}$ cluster near $I_{\text{pQCD}} = 1$, indicating a generic requirement for extreme, first-order-like behavior. Models with softening degrees of freedom cluster at lower $I_{\text{pQCD}}$ values.

Significance and Claims
The paper argues that the requirement to connect stable neutron star matter to the pQCD regime in a causal and thermodynamically consistent manner acts as a powerful filter for microphysical models. The authors claim that:

• Purely nucleonic models, if assumed valid up to the maximal mass density, imply a highly specific and "artificial" behavior: a strong first-order phase transition occurring coincidentally just as the star reaches its maximum mass limit. The authors deem this coincidence highly unlikely.
• Consequently, the study disfavors purely nucleonic descriptions for all stable massive neutron stars.
• The results support the appearance of additional degrees of freedom (such as quarks or hyperons) that cause the EoS to soften before the TOV density, thereby avoiding the need for extreme, coincidental extensions to the pQCD regime.
• The work provides a tool (via the GPB framework and $I_{\text{pQCD}}$ index) for constructing and visualizing these constraints, offering guidance for future EoS model building that respects fundamental QCD constraints.

The authors do not claim to identify a specific type of new matter but rather assert that some mechanism for softening below $M_{\text{TOV}}$ is necessary to avoid the extreme constraints imposed by pQCD consistency on purely hadronic models.