Leveraging differentiable programming in the inverse problem of neutron stars

TL;DR

This work tackles the inverse problem of the neutron-star EOS from NS observations by employing differentiable programming to enable GPU-accelerated, emulator-free Bayesian inference in high-dimensional EOS spaces and by introducing a gradient-based inversion that recovers the EOS from a given mass–radius or mass–tidal-deformability curve. The authors present a metamodel plus CSE parametrization, differentiable through the TOV equations via enthalpy, and leverage flow-based gradient samplers alongside on-the-fly normalizing flows to efficiently sample EOS posteriors across multiple constraints, including $\chi$EFT, NICER, pulsar masses, and GW170817. They demonstrate that the breakdown density $n_{ m break}$ can be inferred from NS data, quantify degeneracies in the nuclear empirical parameters (NEP), and show that a gradient-based inversion can recover EOSs with errors below about $100$ meters in radius and $|\oldsymbol{\Lambda}|$ within roughly $10$ for a given mass range, while revealing parametric degeneracies that depend on the chosen EOS representation. These results suggest that future NS data from next-generation detectors can be analyzed efficiently and robustly with differentiable programming, enabling precise EOS constraints and systematic tests of nuclear physics against astrophysical observations.

Abstract

Neutron stars (NSs) probe the high-density regime of the nuclear equation of state (EOS). However, inferring the EOS from observations of NSs is a computationally challenging task. In this work, we efficiently solve this inverse problem by leveraging differential programming in two ways. First, we enable full Bayesian inference in under one hour of wall time on a GPU by using gradient-based samplers, without requiring pre-trained machine learning emulators. Moreover, we demonstrate efficient scaling to high-dimensional parameter spaces. Second, we introduce a novel gradient-based optimization scheme that recovers the EOS of a given NS mass-radius curve. We demonstrate how our framework can reveal consistencies or tensions between nuclear physics and astrophysics. First, we show how the breakdown density of a metamodel description of the EOS can be determined from NS observations. Second, we demonstrate how degeneracies in EOS modeling using nuclear empirical parameters can influence the inverse problem during gradient-based optimization. Looking ahead, our approach opens up new theoretical studies of the relation between NS properties and the EOS, while effectively tackling the data analysis challenges brought by future detectors.

Figures (5)

• Figure 1: Illustration of the gradient-based inversion scheme recovering a given mass-radius relation of NS. The color gradient shows the different iterations while performing gradient descent on the objective function of Eq. \ref{['eq: loss function EOS doppelgangers']}. The gradient is computed with respect to the EOS parameters throughout the TOV equations using automatic differentiation. Top panel: Trajectory of the EOS parameters, showing only $L_{\rm{sym}}$ and $K_{\rm{sat}}$ for simplicity. Middle panel: Mass-radius and mass-tidal deformability curves during the gradient descent, with the target shown in black. Lower panel: Pressure as a function of density curves during optimization.
• Figure 2: Scaling of the runtime divided by effective sample size (ESS) as a function of the number of EOS parameters when varying the number of grid points for the CSE parameterization, for two different GPU architectures. The right axis converts this number to total wall time to obtain $5000$ effective samples.
• Figure 3: Posterior on $n_{\rm{break}}$ of the metamodel parametrization inferred from a uniform prior between $1n_{\rm{sat}}$ to $4n_{\rm{sat}}$ (other priors are given in Tab. \ref{['tab: prior distributions for the EOS parameters']}) and given the NS observations from Tab. \ref{['tab: reproduction of Hauke']}.
• Figure 4: Recovery of an EOS in the metamodel parametrization from the $M$-$\Lambda$ curve using the gradient-based optimization algorithm as a function of the number of NEP being varied. The left panel shows the recovered range of $L_{\rm{sym}}$ when increasingly more NEP at higher orders in the Taylor expansion are varied freely. For the recovery in which all NEP were varied freely, we show the recovered EOS samples in the middle and right panels. The middle panel shows the pressure as a function of density, while the right panel shows the mass-radius profile of NS, color-coded by their $L_{\rm{sym}}$ value. The grey bands show mass ranges not considered by the loss function in Eq. \ref{['eq: loss function EOS doppelgangers']}. The EOS shown in this figure deviate from the target by less than $100$ meters in radius and less than $10$ in dimensionless tidal deformability across the mass range considered in the loss function of Eq. \ref{['eq: loss function EOS doppelgangers']}.
• Figure 5: Comparison of prior distributions between Ref. Koehn:2024set (Koehn+2024) and this work on the EOS quantities shown in Tab. \ref{['tab: reproduction of Hauke']} and Tab. \ref{['tab: reproduction of Hauke appendix']}. Since both use the EOS parametrization described in Sec. \ref{['sec: methods EOS parametrization']}, with priors given by Tab. \ref{['tab: prior distributions for the EOS parameters']}, remaining differences are due to our jax implementation.