[Paper Review] Bayesian nonparametric inference of neutron star equation of state via neural network

Ming-Zhe Han, Jin-Liang Jiang+2 more|arXiv (Cornell University)|Mar 9, 2021

Pulsars and Gravitational Waves ResearchPhysics and Astronomy98 references44 citations

TL;DR

This paper introduces a Bayesian nonparametric method for reconstructing the neutron star equation of state (EoS) using a feed-forward neural network (FFNN) with one hidden layer and sigmoid activation, enabling flexible, model-independent EoS fitting. The method successfully recovers theoretical EoSs and parameters from simulated data, and when applied to real multimessenger data (GW170817 and NICER PSR J0030+0451), it yields R₁.₄ = 11.83⁺¹.²⁵₋₁.⁰⁸ km and Λ₁.₄ = 323⁺³³⁴₋₁₆₅ at 90% credible interval, while revealing that the conformal limit (c²ₛ/c² < 1/3) is violated in high-density regions of massive neutron stars.

ABSTRACT

We develop a new nonparametric method to reconstruct the Equation of State (EoS) of Neutron Star with multimessenger data. As an universal function approximator, the Feed-Forward Neural Network (FFNN) with one hidden layer and a sigmoidal activation function can approximately fit any continuous function. Thus we are able to implement the nonparametric FFNN representation of the EoSs. This new representation is validated by its capabilities of fitting the theoretical EoSs and recovering the injected parameters. Then we adopt this nonparametric method to analyze the real data, including mass-tidal deformability measurement from the Binary Neutron Star (BNS) merger Gravitational Wave (GW) event GW170817 and mass-radius measurement of PSR J0030+0451 by {\it NICER}. We take the publicly available samples to construct the likelihood and use the nested sampling to obtain the posteriors of the parameters of FFNN according to the Bayesian theorem, which in turn can be translated to the posteriors of EoS parameters. Combining all these data, for a canonical 1.4 $M_\odot$ neutron star, we get the radius $R_{1.4}=11.83^{+1.25}_{-1.08}$ km and the tidal deformability $\Lambda_{1.4} = 323^{+334}_{-165}$ (90\% confidence interval).Furthermore, we find that in the high density region ($\geq 3 ho_{ m sat}$), the 90\% lower limits of the $c_{ m s}^2/c^2$ ($c_{ m s}$ is the sound speed and $c$ is the velocity of light in the vacuum) are above $1/3$, which means that the so-called conformal limit (i.e., $c_{ m s}^2/c^2<1/3$) is not always valid in the neutron stars.

Motivation & Objective

• To develop a nonparametric, model-agnostic method for reconstructing the neutron star equation of state (EoS) without assuming a specific parametric form.
• To overcome limitations of parametric models, such as model dependence and restricted inference scope, especially in the medium-density transition region.
• To enable robust Bayesian inference of EoS using multimessenger data (gravitational waves and X-ray timing) while avoiding prior biases from training sets.
• To test whether the conformal limit (c²ₛ/c² < 1/3) holds in neutron star cores using a flexible, data-driven EoS representation.

Proposed method

• Uses a one-hidden-layer feed-forward neural network (FFNN) with sigmoid activation as a universal function approximator to represent the EoS as P(ρ) = f(ρ; θ), where θ are the network parameters.
• Applies Bayesian inference via nested sampling to compute posteriors over the FFNN parameters θ using likelihoods constructed from multimessenger data (GW170817 tidal deformability and NICER PSR J0030+0451 mass-radius measurements).
• Employs a finite-dimensional approximation of the EoS by sampling the network output at discrete densities, reducing the infinite-dimensional nonparametric problem to a finite-dimensional parameter inference task.
• Validates the method by injecting and recovering parameters from theoretical EoSs (WFF1, H4, MS1, ALF2), demonstrating high-fidelity reconstruction of both macroscopic and microscopic EoS properties.
• Translates the posterior over FFNN parameters into posterior distributions for EoS observables such as radius, tidal deformability, and sound speed squared.
• Uses the PolyChord nested sampling algorithm to efficiently explore the parameter space, balancing model complexity and computational cost.

Experimental results

Research questions

• RQ1 Can a feed-forward neural network with a single hidden layer serve as a flexible, nonparametric representation of the neutron star equation of state without assuming a specific functional form?
• RQ2 How accurately can this FFNN-based Bayesian nonparametric method recover known theoretical EoSs and injected parameters from simulated multimessenger data?
• RQ3 What are the constraints on the radius and tidal deformability of a canonical 1.4 M⊙ neutron star when combining GW170817 and NICER PSR J0030+0451 data using this method?
• RQ4 Does the speed of sound in neutron star cores violate the conformal limit (c²ₛ/c² < 1/3) at high densities, as suggested by the inferred EoS?
• RQ5 How does the performance of this method compare to traditional parametric or Gaussian process-based EoS inference in terms of computational efficiency and model robustness?

Key findings

• The FFNN-based nonparametric method achieves high-fidelity reconstruction of theoretical EoSs, with fitting errors in pressure and sound speed on the order of 10⁻⁴ to 10⁻¹.
• For a canonical 1.4 M⊙ neutron star, the method infers a radius of R₁.₄ = 11.83⁺¹.²⁵₋₁.⁰⁸ km and tidal deformability of Λ₁.₄ = 323⁺³³⁴₋₁₆₅ at 90% credible interval, consistent with prior multimessenger analyses.
• The 90% credible lower limit on c²ₛ/c² exceeds 1/3 in the high-density region (≥3ρsat), indicating that the conformal limit is violated in the cores of massive neutron stars.
• The method successfully recovers injected parameters from simulated data, demonstrating robustness and reliability in reconstructing both macroscopic (mass, radius, tidal deformability) and microscopic (sound speed) EoS properties.
• The approach avoids biases from training-set-dependent priors used in Gaussian process methods, offering a more flexible and transparent inference framework.
• The results suggest that the speed of sound in neutron star cores may exceed the conformal limit, challenging assumptions in some theoretical models based on perturbative QCD.