Neural Posterior Estimation of Neutron Star Equations of State

TL;DR

This paper tackles the challenge of constraining the neutron-star equation of state at supranuclear densities by adopting a simulation-based, likelihood-free inference approach. It combines Neural Posterior Estimation with Conditional Normalizing Flows and a physics-informed monotonicity loss to infer full posteriors over $p(n)$, $c_s^2(n)$, and $\Delta(n)$ from synthetic mass, radius, and tidal-deformability data, trained on agnostic PT and GP EoS priors. The study demonstrates accurate, well-calibrated posteriors and shows that including tidal deformability tightens high-density constraints, while predictive dispersion encodes the maximum central density $n_{c,\max}$ probed by observations. Importantly, the method generalizes to unseen, phenomenological EoSs and maintains meaningful uncertainty quantification, highlighting SBI-CNF as a promising tool for future multimessenger neutron-star EoS inference and planning for next-generation observations.

Abstract

We present a simulation-based inference (SBI) framework to constrain the neutron star (NS) equation of state (EoS) from astrophysical observations of masses, radii and tidal deformabilities, using Neural posterior estimation (NPE) with Conditional Normalising Flows (CNF). To ensure that the model conforms with reality, physics-informed constraints are embedded directly into the training loss. This enables efficient, likelihood-free inference of full posterior distributions for key thermodynamic quantities-including pressure, squared speed of sound, and the trace anomaly-conditioned on observational data. Our models are trained on synthetic datasets generated from two agnostic EoS priors: polytropic parametrizations (PT) and gaussian process (GP) reconstructions. These datasets span various scenarios, including the presence or absence of tidal deformability information and observational noise. Across all settings, the method produces accurate and well-calibrated posteriors, with uncertainties reduced when tidal deformability constraints are included. Furthermore, we find that the behavior of normalized predictive dispersions is strongly correlated with the maximum central density inside NSs, suggesting that the model can indirectly infer this physically meaningful quantity. The approach generalizes well across EoS families and accurately reconstructs derivative quantities such as the polytropic index, demonstrating its robustness and potential for probing dense matter in NS cores.

Figures (18)

• Figure 1: Top: Pressure versus baryonic density. Gray curves correspond to the GP dataset; blue curves represent the PT dataset. Middle: Squared speed of sound versus baryonic density, shown only for the GP dataset. Bottom: Trace anomaly versus baryonic density, shown only for the GP dataset. In all plots, the vertical shaded region marks the 90% quantile range of the maximum central density across the dataset.
• Figure 2: Range of mass-radius and mass-tidal deformability relations. The gray and blue regions show the band of the full extent of all relations generated from the GP and PT ensembles, respectively. Top: Mass–radius relations. Bottom: Mass–tidal deformability relations. Gray curves correspond to the GP dataset; blue curves represent the PT dataset. The shaded gray horizontal (top) and vertical (bottom) bands labeled a) from 1 to 1.4 $M_\odot$, b) from 1.4 to 1.7 $M_\odot$, and c) from 1.7 to $M_{\text{max}}(\text{EoS})$$M_\odot$ denote the three mass ranges used in the uniform sampling procedure for mock data generation described in Sec. \ref{['sec:ge_MD']}
• Figure 3: Compilation of neutron star mass measurements from rocha2023mass, supplemented with additional entries from J0030+0451 Riley_2019Miller19, J1231-1411 salmi2024nicer, and J1731-347 doroshenko2022strangely. Black points with error bars denote the reported central values and uncertainties of individual measurements. The dashed horizontal line marks the mean of all central mass values, $1.58,M_\odot$. The pink histogram on the right shows the distribution of these central values. The three shaded histogram bars, labeled a), b), and c), indicate the mass intervals adopted for uniform sampling in this work (see Sec. \ref{['sec:gm_test']}).
• Figure 4: Schematic illustration of CNFs. A base sample $\boldsymbol{z} = h_0\sim \mathcal{N}(0, I)$ is transformed via a learned, invertible mapping $f_{\boldsymbol{\phi}}(\boldsymbol{z};\boldsymbol{d})$ across $K$ coupling layers, denoted as $f_{\phi_K}(h_{K-1};d) \circ \cdots \circ f_{\phi_1}(z;d)$, conditioned on observations $\boldsymbol{d}$, to yield posterior samples $\boldsymbol{\theta} = h_K \sim q_\phi(\theta|d)$, more details can be seen in Sec. \ref{['sec:cnf']}. The normalizing direction, $f^{-1}_{\boldsymbol{\phi}}(\boldsymbol{\theta};\boldsymbol{d})$, is the training direction, while the generative direction stands for the inference direction.
• Figure 5: Reconstruction of two PT EoSs (top) and GP EoSs (bottom) from the $R\Lambda_2$ dataset. Solid lines: ground truth; shaded regions: 90% CIs; dots: $n_{c,\max}$; vertical bands: 68% and 90% CI for $n_{c,\max}$ distribution.