How Well Do We Know the Neutron-Matter Equation of State at the Densities Inside Neutron Stars? A Bayesian Approach with Correlated Uncertainties

We introduce a new framework for quantifying correlated uncertainties of the infinite-matter equation of state derived from chiral effective field theory ($\chi \mathrm{EFT}$). Bayesian machine learning via Gaussian processes with physics-based hyperparameters allows us to efficiently quantify and propagate theoretical uncertainties of the equation of state, such as $\chi \mathrm{EFT}$ truncation errors, to derived quantities. We apply this framework to state-of-the-art many-body perturbation theory calculations with nucleon-nucleon and three-nucleon interactions up to fourth order in the $\chi \mathrm{EFT}$ expansion. This produces the first statistically robust uncertainty estimates for key quantities of neutron stars. We give results up to twice nuclear saturation density for the energy per particle, pressure, and speed of sound of neutron matter, as well as for the nuclear symmetry energy and its derivative. At nuclear saturation density, the predicted symmetry energy and its slope are consistent with experimental constraints.

Figure 1

Order-by-order predictions with 68% bands for (a) the energy per particle $(E/N)(n)$ and (b) the pressure $P(n)$ of PNM; (c) the symmetry energy $S_2(n)$ and (d) its (rescaled) density dependence $L(n)$; (e) the energy per particle $(E/A)(n)$ of SNM; and (f) the speed of sound $c_s^2(n)$ of PNM, each as a function of density. Dots denote every fifth interpolation point, where $n=0.05,0.06,\dots ,0.34{\mathrm{fm}}^{-3}$. The gray box in (e) depicts the empirical saturation point, $n_0=0.164\pm 0.007{\mathrm{fm}}^{-3}$ with $(E/A)(n_0)=-15.86\pm 0.57\mathrm{MeV}$, obtained from a set of energy density functionals [17, 43]. The vertical lines are located at $n=0.164{\mathrm{fm}}^{-3}$. See the main text for details.

Figure 2

Constraints on the $S_v–L$ correlation. Our results (“GP–B”) are given at the 68% (dark-yellow ellipse) and 95% level (light-yellow ellipse). Experimental constraints are derived from heavy-ion collisions (HIC) [72], neutron-skin thicknesses of Sn isotopes [73], giant dipole resonances (GDR) [74], the dipole polarizability of ${}^{208}\mathrm{Pb}$ [75, 76], and nuclear masses [77]. The intersection is depicted by the white area, which only barely overlaps with constraints from isobaric analog states and isovector skins ($\mathrm{IAS}+\mathrm{\Delta }R$) [78]. In addition, theoretical constraints derived from microscopic neutron-matter calculations by Hebeler et al. (H) [79] and Gandolfi et al. (G) [80] as well as from the unitary gas (UG) limit by Tews et al. [69]. The figure has been adapted from Refs. [70, 71]. A Jupyter notebook that generates it is provided in Ref. [42].