Constraint on phase transition with the multimessenger data of neutron stars

The equation of state (EoS) of the neutron star (NS) matter remains an enigma. In this work we perform the Bayesian parameter inference with the gravitational wave data (GW170817) and mass-radius observations of some NSs (PSR $\mathrm{J}0030+0451$, PSR $\mathrm{J}0437-4715$, and 4U 1702-429) using the phenomenologically constructed EoS models to search for a potential first-order phase transition. Our phenomenological EoS models take the advantages of current widely used parametrizing methods, which are flexible enough to resemble various theoretical EoS models. We find that the current observation data are still not informative enough to support/rule out phase transition, due to the comparable evidences for models with and without phase transition. However, the bulk properties of the canonical $1.4M_{\odot }$ NS and the pressure at around $2{\rho }_{\mathrm{sat}}$ are well constrained by the data, where ${\rho }_{\mathrm{sat}}$ is the nuclear saturation density. Moreover, strong phase transition at low densities is disfavored, and the $1\sigma$ lower bound of transition density is constrained to $1.84{\rho }_{\mathrm{sat}}$.

Figure 1

Fitting results of the rest-mass density versus the adiabatic index (left panel) and the rest-mass density versus the pressure (right panel) relations of some widely-studied/adopted theoretical EoSs using the causal spectral representation. In the top panels, the scatter points with different markers represent the data given by the EoS tables, and the solid lines are the corresponding best fit result for each EoS. The relative fitting errors are shown in the bottom panels.

Figure 2

The 90% uncertainties of the rest-mass density versus the pressure ($\rho -p$) and the rest-mass density versus the squared speed of sound ($\rho -c_{\mathrm{s}}^2$) relations. The left (right) panel is the case of the PT (NPT) model. The gray and blue regions represent, respectively, the prior and the posterior constrained with the GW data and the $M-R$ measurements. The red curves are representative EoSs constructed using the associated models, while the horizontal dashed black lines in the bottom panels represent the asymptotic limit $c_{\mathrm{s}}^2=1/3$.

Figure 3

Distributions of priors (represented by the gray and green colors) and posteriors (represented by the blue and red colors) of the parameters $\{p_1,v_0,v_1,v_2\}$ (left panel), and some inferred microscopic quantities and bulk properties of NS $\{P_{2{\rho }_{\mathrm{sat}}},R_{1.4},{\mathrm{\Lambda }}_{1.4},p_2\}$ (right panel) for PT and NPT models. The values above the diagonal corner plots represent the 90% credible intervals.

Figure 4

Distributions of priors (represented by the gray and green colors) and posteriors (represented by the blue and red colors) of the parameters $\{{\rho }_2,\mathrm{\Delta }\rho ,{\mathrm{\Gamma }}_{\mathrm{m}},c_{\mathrm{q}}^2\}$ for PT (left panel) and NPT (right panel) models. The values above the diagonal corner plots represent the 90% credible intervals.

Figure 5

Posterior $M-R$ distributions for PT (left panel) and NPT (right panel) models. The black dashed line and orange solid line denote respectively the 90% uncertainty region of $M-R$ relations of the prior and the posterior obtained with data set $\mathcal{D}$. The $M-R$ measurements of PSR $\mathrm{J}0030+0451$, PSR $\mathrm{J}0437-4715$, and 4U 1702-429 are represented by the blue dot-dashed contour, purple error bar, and gray density region, respectively. The associated reconstructed $M-R$ of these sources are represented by the colored solid contours.