New Constraints on Radii and Tidal Deformabilities of Neutron Stars from GW170817

We explore in a parameterized manner a very large range of physically plausible equations of state (EOSs) for compact stars for matter that is either purely hadronic or that exhibits a phase transition. In particular, we produce two classes of EOSs with and without phase transitions, each containing one million EOSs. We then impose constraints on the maximum mass ($M<2.16M_{\odot }$) and on the dimensionless tidal deformability ($\tilde{\mathrm{\Lambda }}<800$) deduced from GW170817, together with recent suggestions of lower limits on $\tilde{\mathrm{\Lambda }}$. Exploiting more than $10^9$ equilibrium models for each class of EOSs, we produce distribution functions of all the stellar properties and determine, among other quantities, the radius that is statistically most probable for any value of the stellar mass. In this way, we deduce that the radius of a purely hadronic neutron star with a representative mass of $1.4M_{\odot }$ is constrained to be $12.00<R_{1.4}/\mathrm{km}<13.45$ at a $2\sigma$ confidence level, with a most likely value of ${\overline{R}}_{1.4}=12.39\mathrm{km}$; similarly, the smallest dimensionless tidal deformability is ${\tilde{\mathrm{\Lambda }}}_{1.4}>375$, again at a $2\sigma$ level. On the other hand, because EOSs with a phase transition allow for very compact stars on the so-called “twin-star” branch, small radii are possible with such EOSs although not probable, i.e., $8.53<R_{1.4}/\mathrm{km}<13.74$ and ${\overline{R}}_{1.4}=13.06\mathrm{km}$ at a $2\sigma$ level, with ${\tilde{\mathrm{\Lambda }}}_{1.4}>35.5$ at a $3\sigma$ level. Finally, since these EOSs exhibit upper limits on $\tilde{\mathrm{\Lambda }}$, the detection of a binary with a total mass of $3.4M_{\odot }$ and ${\tilde{\mathrm{\Lambda }}}_{1.7}>461$ can rule out twin-star solutions.

Figure 1

PDFs of stellar radii. Top-left panel: PDF with only the observational constraints on the observed maximum mass and tidal deformability for pure hadronic EOSs; top right: PDF when also an upper limit is set on the maximum mass; bottom left: PDF with the combined constraints on maximum mass and tidal deformability; bottom right: the same as in the bottom left but for EOSs with a phase transition; the thick black line at 12 km distinguishes the PDFs of hadronic twin stars, which represent only 5% of the total sample with phase transitions. In all panels, the solid and dashed lines indicate the $2\sigma$ and $3\sigma$ confidence levels, respectively.

Figure 2

PDFs of stellar radii for a neutron star with mass 1.4. Reported with different lines are the PDFs with different constraints on the maximum mass and tidal deformability (see the legends); the left and middle panels refer to pure hadronic EOSs, while the right one to EOSs with a phase transition (cf. Fig. 1).

Figure 3

PDF of the tidal deformability $\tilde{\mathrm{\Lambda }}$ for pure hadronic EOSs satisfying the constraint $M_{\mathrm{TOV}}>2.01$. The white solid and dashed lines show where the corresponding cumulative distributions at a fixed mass reach a value of $2\sigma$ and $3\sigma$, respectively. Also shown are the $3\sigma$ regions for an EOS featuring a phase transition. Shown with an arrow is the upper limit deduced from GW170817, while several cuts at fixed masses are shown in the top panel.

Figure 4

Left panel: The same as in bottom-right panel in Fig. 1, but when the neutron matter in the outer core is treated following the approach of Ref. [46]. Right panel: The same as in the left panel but when considering the more recent prescription of Ref. [31] for the outer core. Shown as light shaded lines are the $2/3\sigma$ values reported in the bottom-right panel in Fig. 1.