Universal EOS-Radius Inverse Mappings Govern Precision-Dependent Inference of the Neutron Star Equation of State

Technical Summary: Universal EOS–Radius Inverse Mappings Govern Precision-Dependent Inference of the Neutron Star Equation of State

Problem Statement
Current Bayesian inference of the neutron star (NS) equation of state (EOS) generally operates under the assumption that improved observational precision primarily reduces posterior uncertainties while leaving the inferred posterior means of EOS parameters unchanged. However, as next-generation X-ray timing missions and gravitational-wave detectors aim to reduce NS radius measurement uncertainties from $\sim 1$ km to $\sim 0.1$ km, the validity of this assumption is critical. The paper investigates whether increasing observational precision merely narrows distributions or systematically alters the inferred central values of EOS parameters due to the nonlinear relationship between NS observables and the underlying EOS.

Methodology
The study employs a meta-model framework to parameterize the energy per nucleon in neutron-rich matter using six empirical parameters: $J_0, K_0, J_{sym}, K_{sym}, L,$ and $E_{sym,0}$. These parameters characterize the properties of symmetric nuclear matter (SNM) and the density dependence of the nuclear symmetry energy.

• Mock Observations: The authors generate mock measurements for the radius of a canonical $1.4 M_\odot$ NS ($R_{1.4}$) with a fixed central value ($R_0 = 11.9$ km) but varying observational uncertainties ($\sigma_R$) ranging from 0.1 km to 1.2 km.
• Bayesian Inference: Independent Bayesian analyses are performed for each precision level using identical priors and astrophysical constraints (including causality, stability, and constraints from heavy nuclei and massive NSs).
• Inverse Mapping Construction: Posterior samples are sorted by their predicted $R_{1.4}$ values. Conditional posterior means of EOS parameters, defined as $H(R_{1.4}) \equiv \langle h | R_{1.4} \rangle$, are calculated within radius bins to construct inverse mappings.
• Nonlinear Filtering Analysis: The overall posterior mean of an EOS parameter is reconstructed by integrating these conditional means over the posterior radius distribution $P(R_{1.4}, \sigma_R)$. This is compared against a standard Jensen-type expansion (a local quadratic approximation) and a modified Jensen expansion centered on the posterior mean radius.

Key Contributions and Results

1. Precision-Dependent Shifts: The study demonstrates that inferred posterior means of EOS parameters shift systematically as measurement uncertainty changes. Specifically, as $\sigma_R$ increases, the posterior mean radius shifts toward larger values, and the inferred means of parameters like $L$ and $K_{sym}$ change significantly (by $\sim 37\%$ and $-47\%$, respectively) relative to the high-precision limit.
2. Nearly Universal Inverse Mappings: The authors identify a set of "nearly universal" inverse mappings between $R_{1.4}$ and the six empirical EOS parameters. These mappings, derived from posterior samples across ten different observational precisions, collapse onto nearly identical curves (accurately fitted by fifth-order polynomials). This indicates that the posterior samples occupy a low-dimensional, constrained manifold within the six-dimensional EOS parameter space, largely independent of the specific observational precision.
3. Nonlinear Filtering Mechanism: The precision dependence of the inferred parameters is attributed to the nonlinear filtering of the posterior radius distribution through these inverse mappings.
   • In the narrow-distribution limit, this effect reduces to a Jensen-type correction proportional to the local curvature of the mapping ($h''$).
   • For realistic uncertainties, the full nonlinear filtering relation (Eq. 7) accurately reproduces the posterior means, whereas the local quadratic approximation (Eq. 1) fails for broader distributions.
4. Parameter Sensitivity Hierarchy: The sensitivity of EOS parameters to radius precision varies. The symmetry energy parameters $L$ and $K_{sym}$, which dominate the pressure at densities relevant to the canonical NS radius ($\sim 1-2\rho_0$), exhibit the strongest nonlinear curvature and thus the largest precision-dependent shifts. Parameters like $K_0$ and $E_{sym,0}$ show weak sensitivity, while $J_0$ and $J_{sym}$ show intermediate sensitivity.

Significance and Claims
The paper claims to reveal a geometric origin for precision-dependent inference in NS EOS studies. The existence of these nearly universal inverse mappings suggests that the nonlinear filtering effect is not a technical artifact of Bayesian analysis or a specific EOS parameterization, but a generic consequence of the nonlinear relationship between NS structure (governed by the Tolman–Oppenheimer–Volkoff equations) and the EOS.

The authors assert that posterior means inferred from finite-precision observations should not be automatically interpreted as unbiased estimates of the underlying EOS parameters. Instead, they represent estimates filtered through the curvature of the observable-model mapping. Consequently, direct comparisons between current NS observational results and microscopic nuclear many-body theories must account for these nonlinear-filtering corrections. The framework provides a method to correct for precision-dependent biases, suggesting that the asymptotic high-precision limit may offer a more faithful estimate of the underlying physics than current posterior means. The authors further note that similar nonlinear filtering effects and Jensen-type shifts are expected in other nuclear physics contexts involving finite-precision observables, such as neutron skin thicknesses and heavy-ion collision dynamics.