Nearly model-independent constraints on dense matter equation of state in a Bayesian approach

We apply a Bayesian approach to construct a large number of minimally constrained equations of state (EOSs) and study their correlations with a few selected properties of a neutron star (NS). Our set of minimal constraints includes a few basic properties of saturated nuclear matter and low-density pure neutron matter EOS which is obtained from a precise next-to-next-to-next-to-leading-order (${\mathrm{N}}^3\mathrm{LO}$) calculation in chiral effective field theory. The tidal deformability and radius of a NS with mass $1–2M_{\odot }$ are found to be strongly correlated with the pressure of $\beta$-equilibrated matter at densities higher than the saturation density (${\rho }_0=0.16{\mathrm{fm}}^{-3}$) in a nearly model-independent manner. These correlations are employed to parametrize the pressure for $\beta$-equilibrated matter, around $2{\rho }_0$, as a function of neutron star mass and the corresponding tidal deformability. The maximum mass of the neutron star is also found to be strongly correlated with the pressure of $\beta$-equilibrated matter at densities $\sim 4.5{\rho }_0$.

Figure 1

Corner plots for the nuclear matter parameters (in MeV) obtained for Taylor expansions for the EOS of asymmetric nuclear matter. The one-dimensional marginalized posterior distributions (salmon) and the prior distributions (green lines) are displayed along the diagonal plots. The vertical lines indicate a 68% confidence interval of nuclear matter parameters. The confidence ellipses for two-dimensional posterior distributions are plotted with $1\sigma$, $2\sigma$, and $3\sigma$ confidence intervals along the off-diagonal plots. The distributions of nuclear matter parameters are obtained by subjecting them to minimal constraints (see the text for details).

Figure 2

The same as Fig. 1, but for $\frac{n}{3}$ expansions for the EOS of asymmetric nuclear matter.

Figure 3

Corner plots for the marginalized posterior distributions (salmon) of the tidal deformability ${\mathrm{\Lambda }}_{1.4}$, radii $R_{1.4}$ (km) and $R_{2.07}$ (km), and the maximum mass $M_{\max }$ ($M_{\odot }$) for Taylor (top) and $\frac{n}{3}$ (bottom) expansions. The green lines represent effective priors obtained using the priors for nuclear matter parameters (see also Table 1).

Figure 4

Plot for joint probability distribution $P(M,R)$ as a function of the mass and radius of the neutron star obtained for $\frac{n}{3}$ expansion. The red dashed line represents the 90% confidence interval. The outer and inner gray shaded regions indicate the 90% (solid) and 50% (dashed) confidence interval of the LIGO-Virgo analysis for the BNS component from the GW170817 event [100, 101, 102]. The rectangular regions enclosed by dotted lines indicate the constraints from the millisecond pulsar PSR $\mathrm{J}0030+0451$ (purple and black) NICER x-ray data [103, 104] and PSR $\mathrm{J}0740+6620$ (green) [92].

Figure 5

The correlation coefficients $\mathrm{r}(\mathrm{x},P_{\mathrm{BEM}}(\rho ))$, as approximated by both Taylor and $\frac{n}{3}$ expansion along with the mean-field theory calculations, are shown in this figure. Here, x represents either of the tidal deformability ${\mathrm{\Lambda }}_{1.4}$, radii $R_{1.4}$ and $R_{2.07}$, or maximum mass $M_{\max }$ of the neutron star, whereas $P_{\mathrm{BEM}}(\rho )$ represents the pressure for $\beta$-equilibrated matter at a density $\rho$. The calculations are performed with neutron star properties obtained using marginalized posterior distributions of nuclear matter parameters in Taylor and $\frac{n}{3}$ expansions. For comparison, the results are also displayed for a diverse set of nonrelativistic and relativistic microscopic MFMs.

Figure 6

The variations of pressure for $\beta$-equilibrated matter [$P_{\mathrm{BEM}}(\rho )$] at selected densities versus tidal deformability ${\mathrm{\Lambda }}_{1.4}$, radii $R_{1.4}$ and $R_{2.07}$, and maximum mass $M_{\max }$ of the neutron star. The red dashed lines are obtained by linear regression [see Eq. (17) in Sec. 3d].

Figure 7

Dependence of correlation coefficients between tidal deformability (${\mathrm{\Lambda }}_M$) and the pressure of $\beta$-equilibrated matter [$P_{\mathrm{BEM}}(\rho )$] on neutron star mass ($M$) and density ($\rho$) is depicted in this plot. Here, ${\rho }_0=0.16{\mathrm{fm}}^{-3}$ is used only for scaling purposes.

Figure 8

The median values of pressure for $\beta$-equilibrated matter is shown here as a function of the neutron star mass and its tidal deformability at densities $1.5{\rho }_0$ (top), $2.0{\rho }_0$ (middle), and $2.5{\rho }_0$ (bottom).