Are NICER and GW170817 constraints suggesting a compactified scenario for Neutron stars?

Technical Summary: Are NICER and GW170817 Constraints Suggesting a Compactified Scenario for Neutron Stars?

Problem Statement
The equation of state (EoS) of matter at densities exceeding nuclear saturation remains a fundamental unsolved problem in physics. While chiral effective field theory ($\chi$EFT) reliably models sub-nuclear densities and perturbative QCD (pQCD) describes asymptotic high-density regimes, the intermediate density region within neutron star (NS) cores is poorly understood. Recent astrophysical observations from the Neutron Star Interior Composition Explorer (NICER) and the gravitational wave event GW170817 have provided precise constraints on NS masses, radii, and tidal deformabilities. However, it remains unclear whether these constraints favor a specific class of EoS (monotonic, non-monotonic, or discontinuous) or reveal a universal structural pattern in dense matter. Specifically, the paper investigates whether current data suggests a "compactified" scenario where gravity naturally favors configurations that maximize stellar compactness over those that merely maximize mass.

Methodology
The authors employ a comprehensive Bayesian inference framework to analyze three distinct, class-agnostic ensembles of EoS:

1. Monotonic: Characterized by a speed of sound ($c_s^2$) that increases monotonically with density (typical of hadronic EoS).
2. Non-monotonic: Characterized by a peak in $c_s^2$ followed by a decrease (associated with smooth phase transitions).
3. Discontinuous: Characterized by a density jump ($\Delta n$) at a phase transition (associated with strong first-order phase transitions).

EoS Construction:
The EoS is constructed using a hybrid approach. Low-density matter ($< 0.5 n_0$) is defined by the BPS tabulated EoS. The region from $0.5 n_0$ to $1.1 n_0$ is modeled via a polytrope with an adiabatic index $\Gamma$ constrained by $\chi$EFT uncertainty bands. For densities above $1.1 n_0$, the squared speed of sound $c_s^2$ is parametrized as a piecewise-linear function of chemical potential $\mu$ across four randomized segments, extending up to the pQCD regime ($\mu \approx 2.6$ GeV).

Inference Framework:
Unlike previous rejection-sampling methods that generate millions of EoS samples and filter them, this study utilizes the UltraNest nested sampling algorithm to directly explore the posterior distribution. The model contains 10 free parameters (chemical potentials and squared speed of sound values for the segments, plus $\Gamma_{CET}$ and a density jump parameter $\delta n$ for the discontinuous class).
The analysis incorporates:

• NICER Data: Joint posterior distributions for mass and radius of PSR J0030+0451, PSR J0740+6620, PSR J0437−4715, and the recently reported PSR J0614−3329.
• GW170817 Data: Posterior samples of component masses and tidal deformabilities from the binary neutron star merger.
• $\chi$EFT Constraints: Gaussian likelihoods for pressure at low densities.

The study compares the posterior distributions of the EoS parameters and derived stellar properties (Mass-Radius sequences) under different combinations of observational constraints.

Key Results

1. Constraining Power: The Bayesian analysis confirms that observational data effectively constrain the low-density region of the EoS (specifically $\Gamma_{CET}$ and the first speed-of-sound segment $c_{s,1}^2$), which remains robust across all EoS classes. In contrast, the high-density parameters ($c_{s,2}^2, c_{s,3}^2$) exhibit broader posterior distributions, indicating significant remaining uncertainty.
2. Impact of New Data: The inclusion of PSR J0614−3329 and GW170817 systematically shifts the chemical potential parameters toward higher values and reduces the maximum mass bounds. However, the Bayesian evidence ($\ln Z$) does not strongly favor one EoS class over another; the differences in evidence fall within the "inconclusive to weak" range.
3. Maximum Mass vs. Maximum Compactness: A critical distinction is drawn between the EoS that yields the maximum possible mass ($M_{TOV}$) and the EoS that yields the maximum compactness ($C = GM/Rc^2$).
   • The Maximum Mass EoS is stiff from very low densities, providing uniform pressure support to sustain the largest masses. These configurations occupy the large-radius edge of the M-R posterior.
   • The Maximum Compactness EoS features a softer low-to-intermediate density EoS (minimizing radius) followed by pronounced stiffening at high densities (supporting mass).
4. Observational Preference: The EoS most favored by the combined NICER and GW170817 data (highest $\ln Z$) structurally mirrors the Maximum Compactness EoS, not the Maximum Mass EoS. The observationally favored EoS exhibits relative softness at intermediate densities, transitioning to stiffness at high densities.
5. Phase Transitions: For the discontinuous class, the data prefer small density discontinuities ($\delta n \approx 0.09-0.10$ fm$^{-3}$) and late phase transitions (occurring at higher densities). While a low-density phase transition could theoretically produce a highly compact star with a large quark core, the Bayesian predictions favor configurations where the phase transition occurs later, confining quark matter to the cores of sufficiently massive stars.

Significance and Claims
The paper claims that current astrophysical constraints are selecting for EoS configurations that produce compact stars rather than maximally massive ones. The convergence of the observationally favored EoS with the maximum-compactness sequence suggests a fundamental property of dense matter: the EoS likely undergoes significant softening at intermediate densities (potentially signaling the onset of new degrees of freedom) before stiffening again to support the observed $\sim 2 M_\odot$ neutron stars.

The authors conclude that this "soft-to-stiff" structure is what gravity naturally favors to achieve compactification. While the data do not definitively rule out specific EoS classes, they point toward a universal pattern where the equation of state transitions from softness to stiffness, driving neutron stars toward their compactness limits. The paper modestly notes that future constraints on low-mass pulsars (to better define the low-density EoS) and high-mass pulsars (to constrain the high-density EoS) will be necessary to further refine these conclusions and potentially rule out specific scenarios, such as early phase transitions.