Astrophysical Constraints on the Symmetry Energy and the Neutron Skin of ${}^{208}\mathrm{Pb}$ with Minimal Modeling Assumptions

The symmetry energy and its density dependence are crucial inputs for many nuclear physics and astrophysics applications, as they determine properties ranging from the neutron-skin thickness of nuclei to the crust thickness and the radius of neutron stars. Recently, PREX-II reported a value of $0.283\pm 0.071\mathrm{fm}$ for the neutron-skin thickness of ${}^{208}\mathrm{Pb}$, implying a slope parameter $L=106\pm 37\mathrm{MeV}$, larger than most ranges obtained from microscopic calculations and other nuclear experiments. We use a nonparametric equation of state representation based on Gaussian processes to constrain the symmetry energy $S_0$, $L$, and $R_{\mathrm{skin}}^{{}^{208}\mathrm{Pb}}$ directly from observations of neutron stars with minimal modeling assumptions. The resulting astrophysical constraints from heavy pulsar masses, LIGO/Virgo, and NICER clearly favor smaller values of the neutron skin and $L$, as well as negative symmetry incompressibilities. Combining astrophysical data with PREX-II and chiral effective field theory constraints yields $S_0=33.0_{-1.8}^{+2.0}\mathrm{MeV}$, $L=53_{-15}^{+14}\mathrm{MeV}$, and $R_{\mathrm{skin}}^{{}^{208}\mathrm{Pb}}=0.17_{-0.04}^{+0.04}\mathrm{fm}$.

Figure 1

Correlations between the symmetry energy $S_0$, the slope parameter $L$, and the neutron-skin thickness of ${}^{208}\mathrm{Pb}$ $R_{\mathrm{skin}}^{{}^{208}\mathrm{Pb}}$. We show the nonparametric prior (gray), the nonparametric posterior conditioned on astrophysical observations (green), and the nonparametric posterior conditioned on an average over four $\chi \mathrm{EFT}$ calculations (up to $\approx n_0$) and astrophysical observations (blue). Joint distributions show the 68% (shaded) and 90% (solid lines) credible regions. Shaded bands (pink) show the approximate 68% credible region for parameters constrained by PREX-II: $R_{\mathrm{skin}}^{{}^{208}\mathrm{Pb}}$ [19] and the resulting constraints on $L$ using the correlation from Ref. [50]. Note how the inclusion of the astrophysical observations shifts the peak in the marginal distributions for $S_0$, $L$, and $R_{\mathrm{skin}}^{{}^{208}\mathrm{Pb}}$, a trend that is reinforced by the addition of $\chi \mathrm{EFT}$ information. We also show the one-dimensional marginal distributions for the symmetry incompressibility $K_{\mathrm{sym}}$ in a separate panel.

Figure 2

Prior (gray, unshaded), Astro posterior (green, left-unshaded), and $\text{Astro}+\text{PREX-II}$ posterior (red, right-shaded) distributions for $L$ (top) and $R_{\mathrm{skin}}^{{}^{208}\mathrm{Pb}}$ (bottom) as a function of the maximum pressure (top axis) or density (bottom axis) up to which we trust theoretical nuclear-physics predictions from $\chi \mathrm{EFT}$ (see text for details). Shaded bands show the approximate 68% credible region from PREX-II [19] (pink) and from Ref. [13] based on the electric dipole polarizability ${\alpha }_D$ (light blue).

Figure 3

Correlations between $R_{\mathrm{skin}}^{{}^{208}\mathrm{Pb}}$, $L$, and the radius of a $1.4M_{\odot }$ NS, $R_{1.4}$. In addition to the priors and posteriors shown in Fig. 1, we show the nonparametric (red) and $\chi \mathrm{EFT}$ (trusted up to $n_0$; light blue) posteriors conditioned on both astrophysical observations and PREX-II. $\text{Astro}+\text{PREX-II}$ posteriors are shaded in the one-dimensional distributions to distinguish them from the Astro-only posteriors. Joint distributions show the 68% (shaded) and 90% (solid lines) credible regions. Shaded bands (pink) show the approximate 68% credible region from PREX-II.

Figure 4

Median and 90% one-dimensional symmetric posterior credible regions for $c_s^2$ at each density $n$ with astrophysical observations for all $L$ (shaded green), $30\mathrm{MeV}<L\le 70\mathrm{MeV}$ (unshaded blue hatches), and $100\mathrm{MeV}<L$ (shaded purple).