Twin stars as probes of the nuclear equation of state: Effects of rotation through the PSR J0952-0607 pulsar and constraints via the tidal deformability from the GW170817 event

In agreement with the constantly increasing gravitational wave events, new aspects of the internal structure of compact stars can be considered. A scenario in which a first-order transition takes place inside these stars is of particular interest, as it can lead, under certain conditions, to a third gravitationally stable branch (besides white dwarfs and neutron stars), the twin stars. The new branch yields stars with the same mass as normal compact stars but quite different radii. We focus on hybrid stars undergoing a hadron-to-quark phase transition near their core and how this new stable configuration arises. Emphasis is given on the aspects of the phase transition and its parametrization in two different ways—namely, with the Maxwell and Gibbs constructions. We systematically study the gravitational mass, the radius, and the tidal deformability, and we compare them with the predictions of the recent observation by the LIGO/VIRGO Collaboration, the GW170817 event, and the mass and radius limits, suggesting possible robust constraints. Moreover, we extend the study to include rotation effects on the twin star configurations. The recent discovery of the fast rotating supermassive pulsar PSR J0952-0607 triggered the efforts to constrain the equation of state and, moreover, to examine possible predictions related to the phase transition in dense nuclear matter. We pay special attention to relate the PSR J0952-0607 pulsar properties to the twin star predictions, and mainly to explore the possibility that the existence of such a massive object would rule out the existence of twin stars. Finally, we discuss the constraints on the radius and mass of the recently observed compact object within the supernova remnant HESS J1731-347. The estimations imply that this object is either the lightest known neutron star or a star with a more exotic equation of state.

Figure 1

Mass vs radius diagram for the $\mathrm{MDI}+\mathrm{APR}1$ EOS under the MC and for (a) $\mathrm{\Delta }{\mathcal{E}}_{\mathrm{cr}}$ and (b) $\mathrm{\Delta }\mathcal{E}=\mathrm{\Delta }{\mathcal{E}}_{\mathrm{cr}}+100\mathrm{MeV}{\mathrm{fm}}^{-3}$ (blue curves) and $\mathrm{\Delta }\mathcal{E}=\mathrm{\Delta }{\mathcal{E}}_{\mathrm{cr}}+200\mathrm{MeV}{\mathrm{fm}}^{-3}$ (green curves). The black curve indicates the original EOS. The shaded regions from bottom to top represent the HESS J1731-347 remnant [70], the GW170817 event [7], PSR J1614-2230 [83], PSR $\mathrm{J}0348+0432$ [84], PSR $\mathrm{J}0740+6620$ [85], and PSR J0952-0607 [68] pulsar observations for the possible maximum mass.

Figure 2

Mass vs radius diagram for the $\mathrm{MDI}+\mathrm{APR}1$ EOS under the GC and for (a) $\mathrm{\Delta }{\mathcal{E}}_{\mathrm{G}}$ and (b) $\mathrm{\Delta }\mathcal{E}=\mathrm{\Delta }{\mathcal{E}}_{\mathrm{G}}+100\mathrm{MeV}{\mathrm{fm}}^{-3}$ (blue curves) and $\mathrm{\Delta }\mathcal{E}=\mathrm{\Delta }{\mathcal{E}}_{\mathrm{G}}+200\mathrm{MeV}{\mathrm{fm}}^{-3}$ (green curves). The black curve indicates the original EOS. The shaded regions from bottom to top represent the HESS J1731-347 remnant [70], the GW170817 event [7], PSR J1614-2230 [83], PSR $\mathrm{J}0348+0432$ [84], PSR $\mathrm{J}0740+6620$ [85], and PSR J0952-0607 [68] pulsar observations for the possible maximum mass.

Figure 3

Mass vs radius diagram for the GRDF-DD2 EOS under the MC and for (a) $\mathrm{\Delta }{\mathcal{E}}_{\mathrm{cr}}$ and (b) $\mathrm{\Delta }\mathcal{E}=\mathrm{\Delta }{\mathcal{E}}_{\mathrm{cr}}+100\mathrm{MeV}{\mathrm{fm}}^{-3}$ (blue curves) and $\mathrm{\Delta }\mathcal{E}=\mathrm{\Delta }{\mathcal{E}}_{\mathrm{cr}}+200\mathrm{MeV}{\mathrm{fm}}^{-3}$ (green curves). The black curve indicates the original EOS. The shaded regions from bottom to top represent the HESS J1731-347 remnant [70], the GW170817 event [7], PSR J1614-2230 [83], PSR $\mathrm{J}0348+0432$ [84], PSR $\mathrm{J}0740+6620$ [85], and PSR J0952-0607 [68] pulsar observations for the possible maximum mass.

Figure 4

Mass vs radius diagram for the GRDF-DD2 EOS under the GC and for (a) $\mathrm{\Delta }{\mathcal{E}}_{\mathrm{G}}$ and (b) $\mathrm{\Delta }\mathcal{E}=\mathrm{\Delta }{\mathcal{E}}_{\mathrm{G}}+100\mathrm{MeV}{\mathrm{fm}}^{-3}$ (blue curves) and $\mathrm{\Delta }\mathcal{E}=\mathrm{\Delta }{\mathcal{E}}_{\mathrm{G}}+200\mathrm{MeV}{\mathrm{fm}}^{-3}$ (green curves). The black curve indicates the original EOS. The shaded regions from bottom to top represent the HESS J1731-347 remnant [70], the GW170817 event [7], PSR J1614-2230 [83], PSR $\mathrm{J}0348+0432$ [84], PSR $\mathrm{J}0740+6620$ [85], and PSR J0952-0607 [68] pulsar observations for the possible maximum mass.

Figure 5

Tidal parameters (a) $k_2$ and (b) $\lambda$ as a relation of the neutron star mass for the MDI+APR1 EOS under the MC and for $\mathrm{\Delta }\mathcal{E}=\mathrm{\Delta }{\mathcal{E}}_{\mathrm{cr}}+100\mathrm{MeV}{\mathrm{fm}}^{-3}$ (blue curves) and $\mathrm{\Delta }\mathcal{E}=\mathrm{\Delta }{\mathcal{E}}_{\mathrm{cr}}+200\mathrm{MeV}{\mathrm{fm}}^{-3}$ (green curves). The black curve indicates the original EOS. The dashed part of the curves indicates their unstable region. The shaded regions from left to right represent the HESS J1731-347 remnant [70], PSR J1614-2230 [83], PSR $\mathrm{J}0348+0432$ [84], PSR $\mathrm{J}0740+6620$ [85], and PSR J0952-0607 [68] pulsar observations for the possible maximum mass.

Figure 6

${\mathrm{\Lambda }}_1-{\mathrm{\Lambda }}_2$ relation for the MDI+APR1 EOS and (a) the MC, and (b) the GC. The blue (green) curves correspond to $\mathrm{\Delta }\mathcal{E}=\mathrm{\Delta }{\mathcal{E}}_{cr}+100\mathrm{MeV}{\mathrm{fm}}^{-3}$ ($\mathrm{\Delta }\mathcal{E}=\mathrm{\Delta }{\mathcal{E}}_{cr}+200\mathrm{MeV}{\mathrm{fm}}^{-3}$) for the MC, while for the GC the curves correspond to $\mathrm{\Delta }\mathcal{E}=\mathrm{\Delta }{\mathcal{E}}_{\mathrm{G}}+100\mathrm{MeV}{\mathrm{fm}}^{-3}$ and $\mathrm{\Delta }\mathcal{E}=\mathrm{\Delta }{\mathcal{E}}_{\mathrm{G}}+200\mathrm{MeV}{\mathrm{fm}}^{-3}$, respectively. The black curve indicates the original EOS. The shaded region shows the acceptance values derived by the GW170817 event [7].

Figure 7

${\mathrm{\Lambda }}_1-{\mathrm{\Lambda }}_2$ relation for the GRDF-DD2 EOS and (a) the MC and (b) the GC. The blue (green) curves correspond to $\mathrm{\Delta }\mathcal{E}=\mathrm{\Delta }{\mathcal{E}}_{cr}+100\mathrm{MeV}{\mathrm{fm}}^{-3}$ ($\mathrm{\Delta }\mathcal{E}=\mathrm{\Delta }{\mathcal{E}}_{cr}+200\mathrm{MeV}{\mathrm{fm}}^{-3}$) for the MC, while the curves for the GC correspond to $\mathrm{\Delta }\mathcal{E}=\mathrm{\Delta }{\mathcal{E}}_{\mathrm{G}}+100\mathrm{MeV}{\mathrm{fm}}^{-3}$ and $\mathrm{\Delta }\mathcal{E}=\mathrm{\Delta }{\mathcal{E}}_{\mathrm{G}}+200\mathrm{MeV}{\mathrm{fm}}^{-3}$, respectively. The black curve indicates the original EOS. The shaded region shows the acceptance values derived by the GW170817 event [7].

Figure 8

$\tilde{\mathrm{\Lambda }}-q$ relation for the MDI+APR1 EOS and (a) the MC and (b) the GC. The blue (green) curves correspond to $\mathrm{\Delta }\mathcal{E}=\mathrm{\Delta }{\mathcal{E}}_{cr}+100\mathrm{MeV}{\mathrm{fm}}^{-3}$ ($\mathrm{\Delta }\mathcal{E}=\mathrm{\Delta }{\mathcal{E}}_{cr}+200\mathrm{MeV}{\mathrm{fm}}^{-3}$) for the MC, while the curves for the GC correspond to $\mathrm{\Delta }\mathcal{E}=\mathrm{\Delta }{\mathcal{E}}_{\mathrm{G}}+100\mathrm{MeV}{\mathrm{fm}}^{-3}$ and $\mathrm{\Delta }\mathcal{E}=\mathrm{\Delta }{\mathcal{E}}_{\mathrm{G}}+200\mathrm{MeV}{\mathrm{fm}}^{-3}$, respectively. The black curve indicates the original EOS. The shaded region shows the acceptance values derived from the GW170817 event [7].

Figure 9

$\tilde{\mathrm{\Lambda }}-q$ relation for the GRDF-DD2 EOS and (a) the MC and (b) the GC. The blue (green) curves correspond to $\mathrm{\Delta }\mathcal{E}=\mathrm{\Delta }{\mathcal{E}}_{cr}+100\mathrm{MeV}{\mathrm{fm}}^{-3}$ ($\mathrm{\Delta }\mathcal{E}=\mathrm{\Delta }{\mathcal{E}}_{cr}+200\mathrm{MeV}{\mathrm{fm}}^{-3}$) for the MC, while the curves for the GC correspond to $\mathrm{\Delta }\mathcal{E}=\mathrm{\Delta }{\mathcal{E}}_{\mathrm{G}}+100\mathrm{MeV}{\mathrm{fm}}^{-3}$ and $\mathrm{\Delta }\mathcal{E}=\mathrm{\Delta }{\mathcal{E}}_{\mathrm{G}}+200\mathrm{MeV}{\mathrm{fm}}^{-3}$, respectively. The black curve indicates the original EOS. The shaded region shows the acceptance values derived from the GW170817 event [7].

Figure 10

${\mathrm{\Lambda }}_{1.4}-\mathrm{\Delta }\mathcal{E}$ diagram for a single $1.4M_{\odot }$ neutron star for (a) the MDI+APR1 and (b) GRDF-DD2 EOS for both the MC (circles) and the GC (diamonds). The squares correspond to the $\mathrm{\Delta }\mathcal{E}=\mathrm{\Delta }{\mathcal{E}}_{\mathrm{cr}}-5\mathrm{MeV}{\mathrm{fm}}^{-3}$ case under the MC and, for the GRDF-DD2 EOS, the dark blue marks correspond to the $\mathrm{\Delta }{\mathcal{E}}_{\mathrm{cr}}$ ($\mathrm{\Delta }{\mathcal{E}}_{\mathrm{G}}$) case, the light blue marks correspond to $\mathrm{\Delta }\mathcal{E}=\mathrm{\Delta }{\mathcal{E}}_{\mathrm{cr}}+100\mathrm{MeV}{\mathrm{fm}}^{-3}$ ($\mathrm{\Delta }\mathcal{E}=\mathrm{\Delta }{\mathcal{E}}_{\mathrm{G}}+100\mathrm{MeV}{\mathrm{fm}}^{-3}$), and the green ones correspond to $\mathrm{\Delta }\mathcal{E}=\mathrm{\Delta }{\mathcal{E}}_{\mathrm{cr}}+200\mathrm{MeV}{\mathrm{fm}}^{-3}$ ($\mathrm{\Delta }\mathcal{E}=\mathrm{\Delta }{\mathcal{E}}_{\mathrm{G}}+200\mathrm{MeV}{\mathrm{fm}}^{-3}$) for the MC (the GC). The empty colored circle in each panel indicates the original EOS. The light purple shaded region corresponds to the GW170817 event [6].

Figure 11

${\mathrm{\Lambda }}_{1.4}-n_{\mathrm{tr}}$ diagram for a single $1.4M_{\odot }$ neutron star for (a) the MDI+APR1 and (b) GRDF-DD2 EOS for both the MC (solid lines) and the GC (dashed lines). The dark purple line corresponds to the $\mathrm{\Delta }\mathcal{E}=\mathrm{\Delta }{\mathcal{E}}_{\mathrm{cr}}-5\mathrm{MeV}{\mathrm{fm}}^{-3}$ case under the MC and, for the GRDF-DD2 EOS, the dark blue lines correspond to the $\mathrm{\Delta }{\mathcal{E}}_{\mathrm{cr}}$ ($\mathrm{\Delta }{\mathcal{E}}_{\mathrm{G}}$) case, the light blue lines correspond to $\mathrm{\Delta }\mathcal{E}=\mathrm{\Delta }{\mathcal{E}}_{\mathrm{cr}}+100\mathrm{MeV}{\mathrm{fm}}^{-3}$ ($\mathrm{\Delta }\mathcal{E}=\mathrm{\Delta }{\mathcal{E}}_{\mathrm{G}}+100\mathrm{MeV}{\mathrm{fm}}^{-3}$), and the green ones correspond to $\mathrm{\Delta }\mathcal{E}=\mathrm{\Delta }{\mathcal{E}}_{\mathrm{cr}}+200\mathrm{MeV}{\mathrm{fm}}^{-3}$ ($\mathrm{\Delta }\mathcal{E}=\mathrm{\Delta }{\mathcal{E}}_{\mathrm{G}}+200\mathrm{MeV}{\mathrm{fm}}^{-3}$) for the MC (the GC). The light purple shaded region corresponds to the GW170817 event [6].

Figure 12

Mass vs radius for the GRDF-DD2 EOS under the MC and for $\mathrm{\Delta }\mathcal{E}=\mathrm{\Delta }{\mathcal{E}}_{\mathrm{cr}}-5\mathrm{MeV}{\mathrm{fm}}^{-3}$. The dashed part of the curves indicates their unstable region. The black curve indicates the original EOS. The shaded regions from bottom to top represent the HESS J1731-347 remnant [70], the GW170817 event [7], PSR J1614-2230 [83], PSR $\mathrm{J}0348+0432$ [84], PSR $\mathrm{J}0740+6620$ [85], and PSR J0952-0607 [68] pulsar observations for the possible maximum mass. Inset: the case with $n_{\mathrm{tr}}=0.38{\mathrm{fm}}^{-3}$, where the square represents the phase transition.

Figure 13

(a) Gravitational mass as a function of the energy jump for transition densities in the range $[0.2,0.3]{\mathrm{fm}}^{-3}$ at the maximum mass configuration. The shaded regions from bottom to top represent the PSR J1614-2230 [83], PSR $\mathrm{J}0348+0432$ [84], PSR $\mathrm{J}0740+6620$ [85], and PSR J0952-0607 [68] pulsar observations for the possible maximum mass. (b) Equatorial radius as a function of the energy jump for transition densities in the range $[0.2,0.3]{\mathrm{fm}}^{-3}$ at the $1.4M_{\odot }$ configuration. The shaded region represents the constraints extracted through the GW170817 event [86]. Normal neutron stars are presented with the diamonds and crosses, corresponding to the rotation in the 709 Hz configuration and the nonrotating one, respectively, while twin stars use the stars and plus signs. Both panels correspond to the Maxwell construction method.

Figure 14

(a) Gravitational mass as a function of the transition density for energy jumps in the range $[\mathrm{\Delta }{\mathcal{E}}_{\mathrm{cr}},\mathrm{\Delta }{\mathcal{E}}_{\mathrm{cr}}+150]\mathrm{MeV}{\mathrm{fm}}^{-3}$ at the maximum mass configuration. The shaded regions from bottom to top represent the PSR J1614-2230 [83], PSR $\mathrm{J}0348+0432$ [84], PSR $\mathrm{J}0740+6620$ [85], and PSR J0952-0607 [68] pulsar observations for the possible maximum mass. (b) Equatorial radius as a function of the transition density for energy jumps in the range $[\mathrm{\Delta }{\mathcal{E}}_{\mathrm{cr}},\mathrm{\Delta }{\mathcal{E}}_{\mathrm{cr}}+150]\mathrm{MeV}{\mathrm{fm}}^{-3}$ in the $1.4M_{\odot }$ configuration. The shaded region represents the constraints extracted through the GW170817 event [86]. Normal neutron stars are presented with diamonds and crosses corresponding to the rotation in the 709 Hz configuration and the nonrotating one, respectively, while twin stars use the stars and plus signs. Both panels correspond to the Maxwell construction method.

Figure 15

(a) Gravitational mass as a function of the energy increase for transition densities in the range $[0.2,0.3]{\mathrm{fm}}^{-3}$ at the maximum mass configuration. The shaded regions from bottom to top represent the PSR J1614-2230 [83], PSR $\mathrm{J}0348+0432$ [84], PSR $\mathrm{J}0740+6620$ [85], and PSR J0952-0607 [68] pulsar observations for the possible maximum mass. (b) Equatorial radius as a function of the energy increase for transition densities in the range $[0.2,0.3]{\mathrm{fm}}^{-3}$ at the $1.4M_{\odot }$ configuration. The shaded region represents the constraints extracted through the GW170817 event [86]. Normal neutron stars are presented with the diamonds and crosses, corresponding to the rotation in the 709 Hz configuration and the nonrotating one, respectively, while twin stars use the stars and plus signs. Both panels correspond to the Gibbs construction method.

Figure 16

(a) Gravitational mass as a function of the transition density for energy increases in the range $[\mathrm{\Delta }{\mathcal{E}}_{\mathrm{G}},\mathrm{\Delta }{\mathcal{E}}_{\mathrm{G}}+150]\mathrm{MeV}{\mathrm{fm}}^{-3}$ at the maximum mass configuration. The shaded regions from bottom to top represent the PSR J1614-2230 [83], PSR $\mathrm{J}0348+0432$ [84], PSR $\mathrm{J}0740+6620$ [85], and PSR J0952-0607 [68] pulsar observations for the possible maximum mass. (b) Equatorial radius as a function of the transition density for energy increases in the range $[\mathrm{\Delta }{\mathcal{E}}_{\mathrm{G}},\mathrm{\Delta }{\mathcal{E}}_{\mathrm{G}}+150]\mathrm{MeV}{\mathrm{fm}}^{-3}$ in the $1.4M_{\odot }$ configuration. The shaded region represents the constraints extracted through GW170817 event [86]. Normal neutron stars are presented with diamonds and crosses corresponding to the rotation in the 709 Hz configuration and the nonrotating one, respectively, while twin stars use the stars and plus signs. Both panels correspond to the Gibbs construction method.

Figure 17

Gravitational mass as a function of the equatorial radius for two representative cases of the GC: (a) $n_{\mathrm{tr}}=0.25{\mathrm{fm}}^{-3}$ and $\mathrm{\Delta }\mathcal{E}=\mathrm{\Delta }{\mathcal{E}}_{\mathrm{G}}+120\mathrm{MeV}{\mathrm{fm}}^{-3}$ and (b) $n_{\mathrm{tr}}=0.3{\mathrm{fm}}^{-3}$ and $\mathrm{\Delta }\mathcal{E}=\mathrm{\Delta }{\mathcal{E}}_{\mathrm{G}}+60\mathrm{MeV}{\mathrm{fm}}^{-3}$. The circles represent the maximum mass configuration, while the diamonds correspond to the twin stars, assuming a mass equal to the first maximum mass configuration. Inset: twin star branch for the case in which $n_{\mathrm{tr}}=0.3{\mathrm{fm}}^{-3}$ and $\mathrm{\Delta }\mathcal{E}=\mathrm{\Delta }{\mathcal{E}}_{\mathrm{G}}+60\mathrm{MeV}{\mathrm{fm}}^{-3}$. The dash-dotted lines represent the nonrotating configuration, while the solid lines correspond to the 709 Hz configuration. As a guide for the eye, the unstable region is marked with dashed lines. The shaded regions from bottom to top represent the PSR J1614-2230 [83], PSR $\mathrm{J}0348+0432$ [84], PSR $\mathrm{J}0740+6620$ [85], and PSR J0952-0607 [68] pulsar observations for the possible maximum mass.