Neutron star crust and outer core equation of state from chiral effective field theory with quantified uncertainties

TL;DR

This paper develops a two-dimensional Gaussian-process emulator to quantify EFT truncation uncertainties in the neutron-star EOS for asymmetric nuclear matter up to about $2n_0$, training on chiral NN and 3N interactions up to N$^3$LO. By introducing an $x$-dependent reference energy, the authors achieve consistent, stationarized expansion coefficients across density and proton fraction, enabling reliable uncertainty propagation to $E(n,x)$, $P(n,x)$, and chemical potentials in $\beta$-equilibrium. The EOS is extended to the neutron-star inner crust via a compressible liquid-drop model with surface and Coulomb corrections, yielding crust-core transition densities in the range $0.062$–$0.088$ fm$^{-3}$ and highlighting proton drip phenomena. Overall, the work provides a robust, uncertainty-aware description of the non-uniform and uniform phases of neutron-star matter, with potential for finite-temperature generalization and integration into Bayesian astrophysical analyses.

Abstract

We study the order-by-order expansion of the energy per particle of asymmetric nuclear matter up to twice saturation density in chiral effective field theory (EFT) within a Bayesian framework. For this, we develop a two-dimensional Gaussian process (2D GP) that is trained using many-body perturbation theory results based on chiral two- and three-nucleon interactions from leading to next-to-next-to-next-to-leading order (N$^3$LO). This allows for an efficient evaluation of the equation of state (EOS) and thermodynamic derivatives with EFT truncation uncertainties. After benchmarking our 2D GP against Bayesian uncertainties for pure neutron matter and symmetric matter, we study the energy per particle, pressure, and chemical potentials of neutron star matter in $β$-equilibrium including EFT uncertainties. We investigate the phase diagram of neutron-rich matter from neutron- to proton-drip and to the uniform phase, including surface and Coulomb corrections. Based on this, we construct EOSs for the inner crust of neutron stars that are consistent with the chiral EFT results for uniform matter at N$^3$LO.

Figures (15)

• Figure 1: Expansion coefficients at NLO ($c_2$, left), N$^2$LO ($c_3$, middle), and N$^3$LO ($c_4$, right panel) extracted from the asymmetric matter calculations with reference energy $E_\text{ref}^\text{BQ}$ as a function of density $n$. Results are shown for different proton fractions from $x=0$ to $x=0.5$ in steps of $0.1$ (from light to darker).
• Figure 2: Expansion coefficients analogous to Fig. \ref{['fig:c_n']}, here as function of proton fraction $x$ shown for selected fixed densities between $n = 0.08 \, \text{fm}^{-3}$ and $n = 0.32 \, \text{fm}^{-3}$ in steps of $0.04 \, \text{fm}^{-3}$ (from light to darker). Note that the kink at $x=0.5$ is due to asymmetric matter results being on a grid of proton fractions, the actual behavior around $x=0.5$ is smooth.
• Figure 3: Corrections to the energy per particle from LO to N$^3$LO divided by the respective powers of the expansion parameter $Q$ as function of density $n$ at different proton fractions from PNM ($x = 0$, left) to SNM ($x = 0.5$, right). The 95% credibility ranges of the EFT uncertainties from the GP are shown for the reference energies $E_\mathrm{ref}^\mathrm{BQ + 3B}$ (gray region) and $E_\mathrm{ref}^\mathrm{BQ}$ (dashed gray lines).
• Figure 4: Optimized marginal variance ${\bar{c}}$ for the three training sets (see text for details), the two different reference energies $E_\mathrm{ref}^\mathrm{BQ}$ and $E_\mathrm{ref}^\mathrm{BQ + 3B}$, and optimization choices (inverse-$\chi^2$ prior or uniform prior, when not stated).
• Figure 5: Model-checking diagnostics for the GP constructed with the training set $\mathbf{y}^{(1)}$, the reference energy $E_\mathrm{ref}^\mathrm{BQ + 3B}$, and the inverse-$\chi^2$ prior. Left: credibility interval diagnostics. Middle: Mahalanobis distance. Right: Pivoted Cholesky errors. For each diagnostics, the dark (light) gray areas/lines give 68% (95%) credibility intervals of the corresponding reference distributions. See text for details.