Constraints on the hybrid equation of state with a crossover hadron-quark phase transition in the light of GW170817

In this paper, we use the recent updated source properties of GW170817 to constrain the hybrid equation of state (EOS) constructed by a three-window modeling between the hadronic EOS and quark EOS. Specifically, the hadronic EOS is described by the $\mathrm{NL}3\omega \rho$ model whose corresponding pure neutron star (NS) is already excluded by the constraint of tidal deformability (TD) from GW170817, and the quark EOS is calculated with the $2+1$ flavors Nambu-Jona-Lasinio (NJL) model. We also consider four other constraints on the hybrid EOS. As a result, we find the parameter set ($B^{\frac{1}{4}},\tilde{\mu },\mathrm{\Gamma }$) can be well constrained, indicating the possible existence of the hybrid star (HS) with a crossover inside. The type of the two stars in the binary system for nine representative hybrid EOSs is shown in this paper too. Furthermore, the HSs restricted by five constraints do not suggest a pure quark core but a mixed phase in the center.

Figure 1

Quark number density of $u$, $d$ and $s$ quark as a function of $\mu$ at $T=0$ with parameters fixed for $m_{\mathrm{u}}=3.3$, 3.4 MeV, respectively. Four lines (two red lines and two green lines) nearly coincide and so do the other two blue lines.

Figure 2

Considering the chemical equilibrium and electric charge neutrality of the quark system, density of $u$, $d$ and $s$ quark with parameters fixed for $m_{\mathrm{u}}=3.3$, 3.4 MeV are shown, respectively. Two red lines nearly coincide and so do two green lines and two blue lines.

Figure 3

Comparison of quark EOSs and hadronic EOS. The black solid line is the $\mathrm{NL}3\omega \rho$ EOS while the red dashed line, the green dot-dashed line and the blue dotted line are the quark EOSs with $B^{\frac{1}{4}}=167$, 170, and 171 MeV, respectively.

Figure 4

Comparison of binding energy of hadrons and quarks. The black solid line is for the $\mathrm{NL}3\omega \rho$ EOS while the red dashed line, the green dot-dashed line and the blue dotted line are for the quark EOSs with $B^{\frac{1}{4}}=167$, 170, and 171 MeV, respectively.

Figure 5

Constraints on parameter set ($B^{\frac{1}{4}},\tilde{\mu },\mathrm{\Gamma }$). The gray shaded region is the allowed space for the subparameter set (a) ($B^{\frac{1}{4}},\mathrm{\Gamma }$) and (b) ($B^{\frac{1}{4}},\tilde{\mu }$), respectively, with five constraints considered in Sec. 3.

Figure 6

Constraints on subparameter set ($\tilde{\mu },\mathrm{\Gamma }$) with (a) $B^{\frac{1}{4}}=167$, (b) $B^{\frac{1}{4}}=170\mathrm{MeV}$, and (c) $B^{\frac{1}{4}}=171\mathrm{MeV}$, respectively. The gray shaded region is the allowed parameter space for these three cases. The black solid line, green dashed line, red dotted line, and orange dot-dashed line correspond to the constraints (2), (3), (4), and (5) in Sec. 3, respectively. The mass constraint (1) does not appear in these graphs because this constraint is relatively loose. When $\tilde{\mu }>1.95\mathrm{GeV}$, the maximum masses of hybrid stars constructed by the hybrid EOS are already well beyond $1.97M_{\odot }$.

Figure 7

The sound velocities of the nine representative hybrid EOSs with parameter set of $(B^{\frac{1}{4}},\tilde{\mu },\mathrm{\Gamma })=(167,2.6,1.58)$, (167, 3.0, 1.98), (167, 3.5, 2.48), (170, 3.0, 1.9), (170, 3.4, 2.3), (170, 3.8, 2.7), (171, 3.45, 2.45), (171, 3.8, 2.75), and (171, 4.2, 3.19), corresponding to the red solid line, red dotted line, red dashed line, green dashed line, green dotted line, green solid line, blue dashed line, blue dotted line, and blue solid line, respectively.

Figure 8

The $M-R$ relation of the hybrid stars constructed by the nine representative hybrid EOSs with parameter set of $(B^{\frac{1}{4}},\tilde{\mu },\mathrm{\Gamma })=(167,2.6,1.58)$, (167, 3.0, 1.98), (167, 3.5, 2.48), (170, 3.0, 1.9), (170, 3.4, 2.3), (170, 3.8, 2.7), (171, 3.45, 2.45), (171, 3.8, 2.75), and (171, 4.2, 3.19), corresponding to the red solid line, red dotted line, red dashed line, green dashed line, green dotted line, green solid line, blue dashed line, blue dotted line, and blue solid line, respectively. The gray shaded area represents the mass constraint of PSR $\mathrm{J}0348+0432$.

Figure 9

Comparison of the tidal deformability of hybrid stars constructed by the nine representative hybrid EOSs with parameter set of $(B^{\frac{1}{4}},\tilde{\mu },\mathrm{\Gamma })=(167,2.6,1.58)$, (167, 3.0, 1.98), (167, 3.5, 2.48), (170, 3.0, 1.9), (170, 3.4, 2.3), (170, 3.8, 2.7), (171, 3.45, 2.45), (171, 3.8, 2.75), and (171, 4.2, 3.19), corresponding to the red solid line, red dotted line, red dashed line, green dashed line, green dotted line, green solid line, blue dashed line, blue dotted line, and blue solid line, respectively. For $B^{\frac{1}{4}}=167$ and 170 MeV, the tidal deformability is very similar. The orange solid line represents the tidal deformability (${\mathrm{\Lambda }}_1$, ${\mathrm{\Lambda }}_2$) calculated by $\mathrm{NL}3\omega \rho$ EOS. Both the black dashed line and gray dashed line are the 90% posterior probability enclosed inside for the low spin prior case in GW170817. The difference is that the gray one represents the former prediction and the black one is the recent prediction in the light of the additional waveform model SEOBNRT. The brown dotted line indicates the ${\mathrm{\Lambda }}_1={\mathrm{\Lambda }}_2$ boundary.