Equation of state for dense nucleonic matter from metamodeling. II. Predictions for neutron star properties

Employing recently proposed metamodeling for the nucleonic matter equation of state, we analyze neutron star global properties such as masses, radii, momentum of inertia, and others. The impact of the uncertainty on empirical parameters on these global properties is analyzed in a Bayesian statistical approach. Physical constraints, such as causality and stability, are imposed on the equation of state and different hypotheses for the direct Urca (dUrca) process are investigated. In addition, only metamodels with maximum masses above $2M_{\odot }$ are selected. Our main results are the following: the equation of state exhibits a universal behavior against the dUrca hypothesis under the condition of charge neutrality and $\beta$ equilibrium; neutron stars, if composed exclusively of nucleons and leptons, have a radius of $12.7\pm 0.4$ km for masses ranging from 1 up to $2M_{\odot }$; a small radius lower than 11 km is very marginally compatible with our present knowledge of the nuclear empirical parameters; and finally, the most important empirical parameters which are still affected by large uncertainties and play an important role in determining the radius of neutrons stars are the slope and curvature of the symmetry energy ($L_{\mathrm{sym}}$ and $K_{\mathrm{sym}}$) and, to a lower extent, the skewness parameters ($Q_{\mathrm{sat}/\mathrm{sym}}$).

Figure 1

Effect of the crust and of the matching between the crust and the core EoS. Left part: mass-radius relationship with different prescriptions for the choice of the crust EOS and of the matching procedure (see text). Right part: sensitivity analysis of the shift in the NS radius induced by the different choices given in the legend of the figure (see text also). For instance, the solid black line stands for $\mathrm{\Delta }R=R\left(\mathrm{SLY}0\right)-R\left(\mathrm{FPS}0\right)$. The largest impact is found for the modification of the low-density boundary of the matching between the crust and the core. Even for this extreme case, the uncertainty in the radius definition is less than 0.1 km.

Figure 2

$M-R$ diagram generated with meta-EOS where the isoscalar empirical parameters are varied. See text for more details.

Figure 3

Same as Fig. 2 for the variation of isovector empirical parameters.

Figure 4

One-parameter probability distributions $p\left(P_{\alpha }\right)$. See text for more details.

Figure 5

Two-parameter probability functions of $Q_{\mathrm{sat}}$ for DURCA-0, DURCA-1, and DURCA-2 (the prior is here taken flat). The units on the $x$ axis are MeV and the $y$ axes cover the range defined in Table 1 from bottom to top.

Figure 6

Two-parameter probabilities showing the correlation between $Q_{\mathrm{sym}}$ and $K_{\mathrm{sym}}$ for DURCA-0, 1, and 2 (the prior is here taken to be uniform). The units on the axes are MeV.

Figure 7

Correlation matrix for the eight empirical parameters and the three scenarios DURCA-0, 1, and 2. The color index goes from 0 (white) to 1 in absolute value (green) in a linear scale as shown on the right bars in each graphs. See text for discussion.

Figure 8

$1\sigma$-CL, $2\sigma$-CL and $4\sigma$-CL domains for the correlation between the total radius $R$, moment of inertia $I$ and surface red-shift $z$ function of the mass $M$ for DURCA-0, 1, and 2 scenarios. The dashed lines on the top panels represent the maximum and minimal values of the radius. See text for discussion.

Figure 9

Same as Fig. 8 for the correlation between the central proton fraction $x_p$, crust thickness $\mathrm{\Delta }{R}_{\mathrm{crust}}$, and central density $n_c$ function of the mass.

Figure 10

Compactness $(M/M_{\odot })/(R/km)$ as a function of the mass for the three hypotheses DURCA-0, 1, and 2.

Figure 11

Centroids and standard deviation of the NS radii function of the mass, with DURCA-0, 1, and 2.

Figure 12

Same as Fig. 11 for the moment of inertia.

Figure 13

Same as Fig. 11 for the central density.

Figure 14

Empirical relation between the pressure (in MeV ${\mathrm{fm}}^{-3}$) and the radius (in km) function of the mass. The pressure is calculated in NM and is defined at $1.0n_{\mathrm{sat}},1.5n_{\mathrm{sat}}$, or $2.0n_{\mathrm{sat}}$ as indicated in the plots. We used the same units on the $y$ axis as in Ref. [21].

Figure 15

Impact of the DURCA-0, 1, and 2 scenarios on the proton fraction density dependence. The dUrca threshold condition is also shown. See text for more details.

Figure 16

Same as Fig. 15 for the electron fraction $x_e$ and muon fraction $x_{\mu }$.

Figure 17

Same as Fig. 15 for the symmetry energy.

Figure 18

Total pressure function of the energy density $\rho$ (including the rest mass contribution).

Figure 19

Adiabatic index $\mathrm{\Gamma }\left(\rho \right)$ function of the energy density $\rho$ (including the rest mass contribution).

Figure 20

Same as Fig. 15 for the baryon sound velocity.

Figure 21

Effect of shifting the average radius by $-R_{\mathrm{shift}}$ on the empirical parameters. The purple 1-$\sigma$ bands shown for each empirical parameter are deduced from Table 1. See the text for more details.