Hybrid stars in the light of the merging event GW170817
Alessandro Parisi, C. Vásquez Flores, C. Henrique Lenzi, Chian-Shu Chen, Germán Lugones
PPrepared for submission to JCAP
Hybrid stars in the light of themerging event GW170817
Alessandro Parisi, C. Vásquez Flores, C. Henrique Lenzi, Chian-Shu Chen and Germán Lugones Department of Physics, Tamkang University, New Taipei 251, Taiwan Centro de Ciências Exatas, Naturais e Tecnológicas, UEMASUL, Rua Godofredo Viana1300, Centro CEP: 65901-480, Imperatriz, Maranhão, Brazil Department of Physics, Instituto Tecnologico de Aeronautica, DCTA , 12228-900, São Josédos Campos, SP, Brazil Universidade Federal do ABC, Centro de Ciências Naturais e Humanas, Avenida dos Esta-dos, 5001, CEP 09210-170, Santo André, SP, Brazil
Abstract.
We study quark-hadron hybrid stars with sharp phase transitions assuming thatphase conversions at the interface are slow. Hadronic matter is described by a set of equa-tions of state (EoS) based on the chiral effective field theory and quark matter by a genericbag model. Due to slow conversions at the interface, there is an extended region of stablehybrid stars with central densities above the density of the maximum mass star. We ex-plore systematically the role of the transition pressure and the energy-density jump ∆ (cid:15) atthe interface on some global properties of hybrid stars, such as the maximum mass, the laststable configuration, and tidal deformabilities. We find that for a given transition pressure,the radius of the last stable hybrid star decreases as ∆ (cid:15) raises resulting in a larger extendedbranch of stable hybrid stars. Contrary to purely hadronic stars, the tidal deformability Λ can be either a decreasing or an increasing function of the stellar mass M and for large valuesof the transition pressure has a very weak dependence on M . Finally, we analyze the tidaldeformabilities Λ and Λ for a binary system with the same chirp mass as GW170817. Inthe scenario where at least one of the stars in the binary is hybrid, we find that models withlow enough transition pressure are inside the credible region of GW170817. However,these models have maximum masses below M (cid:12) , in disagreement with observations. Wealso find that the LIGO/Virgo constrain (at level) and the M (cid:12) requirement can besimultaneously fulfilled in a scenario where all hybrid configurations have masses larger than . M (cid:12) and the hadronic EoS is not too stiff, such as several of our hybrid models involvinga hadronic EoS of intermediate stiffness. In such scenario hybrid stars may exist in Naturebut both objects in GW170817 were hadronic stars. Keywords: neutron star — tidal deformation — phase transition a r X i v : . [ a s t r o - ph . H E ] O c t ontents The composition and properties of neutron stars (NSs), which are among the densest objectsin the universe, are not completely understood theoretically. However, measurements of themasses and radii of these objects can strongly constrain their equation of state (EoS) andconsequently their internal composition (see [1, 2] for a complete review). In fact, the moststringent constraints for the EoS come from the two solar mass limit of the pulsars PSR J1614-2230 [3], PSR J0348+0432 [4], PSR J0740+6620 [5], and PSR J2215-5135 [6]. Mass and radiusdeterminations are receiving a strong boost thanks to the observational data coming from theNeutron Star Interior Composition Explorer (NICER) experiment by NASA. NICER hasprovided a simultaneous measurement of both mass and radius of the millisecond-pulsar PSRJ0030+0451 and also inferred some thermal properties of hot regions present in the star [7, 8].The gravitational wave signal from mergers of binary NSs [9] is also sensitive to the EoS.New limits on the EoS were posed by the LIGO/Virgo detection of gravitational waves (GWs)originating from the NS-NS merger event GW170817 [9], which allowed determining the tidaldeformabilities of the stars involved in the collision [10–13]. During the inspiral phase of aNS-NS merger, strong tidal gravitational fields deform the multipolar structure of the NSs,leaving a detectable imprint on the observed gravitational waveform of the merger [9, 14].This effect can be quantified in terms of the so-called dimensionless tidal deformability of thestar, defined as: Λ i = 23 k ( i )2 (cid:18) c R i GM i (cid:19) , (1.1)where k ( i )2 is the second Love number, R i the radius, and M i the mass of the i -th star [15]. Λ i describes the amount of induced mass quadrupole moment when reacting to a certain externaltidal field (see [16, 17] for more details).Previously, several authors have studied tidal effects trying to impose some constraintson the EoS [18–22]. Some works analyzed the tidal properties of NSs using relativistic mean-field models [23, 24] and Skyrme-type models [25–27]. Other focused on the role of the– 1 –ymmetry energy E sym ( ρ ) [28] and analyzed the role of the appearance of ∆ -isobars takinginto account the data from GW170817 [29].Among the open questions that may be explored with GW measurements, the possibleappearance of quark matter in compact stars is an issue that has attracted the attentionfor decades. In fact, it has been conjectured long ago [30–32] that a higher density class ofcompact stars may arise in the form of hybrid stars or quark stars (QSs), whose core or entirevolume consists of quark matter (see also [33–35] and references therein). Nowadays, it isgenerally believed that deconfined quarks may exist inside the core of compact stars, whichcorresponds to the high-density and low-temperature region of the QCD phase diagram.An important issue related with the hadron–quark phase transition in compact starinteriors, is the possible existence of mass twins in the mass-radius diagram. The term masstwins refers to the existence of doublets (and sometimes multiplets) of stellar configurationswith the same gravitational mass but different radii. The appearance of twins may be aconsequence of the occurrence of a third family of compact stars, besides white dwarfs andordinary NSs. As already recognized long ago [36], the third family is related to the behaviorof the high-density EoS, which may exhibit a phase transition [37–39]. In this context, EoSswith multiple phases which include a deconfined quark-matter segment were analysed as well[40–46] (see also [47] for a recent review). Moreover, an additional phase transition in thequark core can lead to a fourth family of compact stars [48]. The observation of twins ormultiplets may be regarded as a signature of the existence of hybrid stars but other ways ofidentifying them would be possible. For example, general relativistic simulations of mergingNSs including quarks at finite temperatures [49], show that the phase transition would leadto a post-merger signal considerably different from the one expected from the inspiral.Besides the standard mass twins and multiplets described before, it has been shownrecently that there is another mechanism that gives rise to an additional family of twins.In fact, in the case of hybrid stars having slow quark-hadron conversion rates at a sharp interface, there exists a new connected stable branch beyond the maximum mass star thathas ∂M/∂(cid:15) c < , being M the stellar mass and (cid:15) c the energy density at the stellar center[50–52]. In this paper we analyze the possibility of identifying these new hybrid objects bymeans of the observation of their tidal deformabilities.This paper is organized as follows. In Sec. 2, we describe the hadronic and quark matterEoSs used in this work and combine them in order to obtain hybrid EoSs with sharp firstorder phase transitions at some given pressures. In Sec. 3 we summarize the role of slowtransitions on the dynamical stability of hybrid stars and in Sec. 4 we present the equationsfor the tidal deformabilty used in the calculations. In Sec. 5 we present our results for thetidal deformations of hybrid stars. In Sec. 6 we present our main conclusions. In this section, we discuss the cold dense-matter models we use in our analysis. In Sec. 2.1we describe the hadronic model, in Sec. 2.2 the quark matter model, and in Sec. 2.3 weexplain the procedure used to construct first order phase transitions with a sharp densitydiscontinuity.
For the hadronic phase (HP) we use the model presented in Ref. [53], which is based on nuclearinteractions derived from chiral effective field theory (EFT). In recent years, the development– 2 –f chiral EFT has provided the framework for a systematic expansion for nuclear forces atlow momenta, allowing one to constrain the properties of neutron-rich matter up to nuclearsaturation density to a high degree. However, our knowledge of the EOS at densities greaterthan 1 to 2 times the nuclear saturation density is still insufficient due to limitations on bothlaboratory experiments and theoretical methods. Because of this, the EoS at supranucleardensities is usually described by a set of three polytropes which are valid, respectively, in threeconsecutive density regions. For these polytropic EoSs it is required non-violation of causalityand consistence with the recently observed pulsars with ∼ M (cid:12) [3, 4]. In this work we willuse three representative EoSs labeled as soft, intermediate and stiff that have been presentedin Ref. [53] . To describe the structure of the crust we follow the formalism developed in Ref.[54] for the outer crust, while we employ the pioneering work of Ref. [55] for the descriptionof the inner crust. For the quark phase (QP) we adopt a generic bag model defined by the following grandthermodynamic potential [56]: Ω QM = − π a µ + 34 π a µ + B eff + Ω e , (2.1)where µ ≡ ( µ u + µ d + µ s ) / is the quark chemical potential, B eff , a , and a are three freeparameters independent of µ , and Ω e is the grand thermodynamic potential for electrons e .For hybrid systems, though, it happens that the contribution to the thermodynamic quantitiescoming from electrons is negligible (see discussion in [51]). The influence of strong interactionson the pressure of the free-quark Fermi sea is roughly taken into account by the parameter a , where ≤ a ≤ , and a = 1 indicates no correction to the ideal gas. The effect ofthe color superconductivity phenomenon in the Color Flavor Locked (CFL) phase can beexplored setting a = m s − , being m s the mass of the strange quark and ∆ the energygap associated with quark pairing. The standard MIT bag model is obtained for a = 1 and a = m s . The bag constant B eff is related to the confinement of quarks, representing in aphenomenological way the vacuum energy [57]. From Eq. (2.1), one can obtain the pressure p = − Ω QM , the baryon number density n = − ∂ Ω QM ∂µ = 12 π (2 a µ − a µ ) , (2.2)and the energy density (cid:15) = 3 nµ − p = 94 π a µ − π a µ + B eff = 3 p + 4 B eff + 3 a π µ . (2.3)An analytic expression of the form p = p ( (cid:15) ) for this phenomenological model can be simplyobtained if one solves Eq. (2.3) for µ and replaces it in Eq. (2.1). The final result is p ( (cid:15) ) = ( (cid:15) − B eff )3 − a π a (cid:34) (cid:115) π a a ( (cid:15) − B eff ) (cid:35) . (2.4)– 3 –odel p tr (cid:15) HPtr n HPB , tr ∆ (cid:15) a / B / a M max (cid:0) MeVfm (cid:1) (cid:0) MeVfm (cid:1) (fm − ) (cid:0) MeVfm (cid:1) (MeV) (MeV) ( M (cid:12) ) . . .
592 60 100 164 .
79 0 .
976 1 . . . .
639 50 100 153 .
15 0 .
848 1 . . . .
639 120 100 161 .
75 0 .
919 1 . . . .
639 200 100 170 .
15 0 .
999 1 . . . .
704 100 100 153 .
35 0 .
837 1 . . . .
704 250 100 170 .
30 0 .
960 1 . . . .
799 300 100 163 .
89 0 .
856 1 . . . .
799 400 100 173 .
85 0 .
916 1 . . . .
799 500 100 182 .
34 0 .
976 1 . . .
912 500 100 163 .
87 0 .
783 1 . . .
912 700 100 182 .
33 0 .
865 1 . . .
912 900 100 196 .
45 0 .
947 1 . . .
008 600 100 146 .
29 0 .
671 1 . . .
008 800 100 170 .
35 0 .
730 1 . . .
008 1000 100 187 .
13 0 .
788 1 . Table 1 . We list our choices for the parameters a , p tr and ∆ (cid:15) , which according to the discussiongiven in Sec. 2.3, define the localisation of the quark-hadron interface as well as the parameters B eff and a of the quark matter EoS. In this Table it is assumed that the hadronic EoS is described by the soft model of Ref. [53]. For completeness we include the maximum stellar mass M max of each model. Model p tr (cid:15) HPtr n HPB , tr ∆ (cid:15) a / B / a M max (cid:0) MeVfm (cid:1) (cid:0) MeVfm (cid:1) (fm − ) (cid:0) MeVfm (cid:1) (MeV) (MeV) ( M (cid:12) ) .
25 295 . .
304 100 100 156 .
68 0 .
784 1 . .
25 295 . .
304 150 100 162 .
58 0 .
874 1 . .
25 295 . .
304 200 100 167 .
90 0 .
964 1 . .
02 380 . .
384 150 100 161 .
74 0 .
791 1 . .
02 380 . .
384 220 100 169 .
16 0 .
882 1 . .
02 380 . .
384 280 100 174 .
82 0 .
961 1 . . . .
464 200 100 158 .
36 0 .
683 1 . . . .
464 300 100 169 .
27 0 .
767 1 . . . .
464 400 100 178 .
41 0 .
851 1 . . . .
592 400 100 158 .
46 0 .
569 2 . . . .
592 550 100 174 .
09 0 .
633 2 . . . .
592 700 100 186 .
38 0 .
698 2 . . . .
704 600 100 145 .
08 0 .
445 2 . . . .
704 800 100 169 .
59 0 .
492 2 . . . .
704 1000 100 186 .
56 0 .
540 2 . Table 2 . Same as in Table 1 but assuming that the hadronic EoS is described by the intermediate model of Ref. [53]. – 4 –odel p tr (cid:15) HPtr n HPB , tr ∆ (cid:15) a / B / a M max (cid:0) MeVfm (cid:1) (cid:0) MeVfm (cid:1) (fm − ) (cid:0) MeVfm (cid:1) (MeV) (MeV) ( M (cid:12) ) .
39 249 . .
256 80 100 147 .
08 0 .
622 1 . .
39 249 . .
256 130 100 154 .
11 0 .
705 1 . .
39 249 . .
256 180 100 160 .
28 0 .
789 1 . .
49 302 . .
304 100 100 142 .
52 0 .
535 2 . .
49 302 . .
304 200 100 156 .
81 0 .
645 1 . .
49 302 . .
304 300 100 168 .
01 0 .
756 1 . .
63 361 . .
352 250 100 154 .
54 0 .
547 2 . .
63 361 . .
352 300 100 160 .
67 0 .
584 2 . .
63 361 . .
352 350 100 166 .
17 0 .
621 2 . . . .
416 300 100 147 .
16 0 .
443 2 . . . .
416 400 100 160 .
35 0 .
489 2 . . . .
416 500 100 170 .
91 0 .
536 2 . . . .
496 500 100 145 .
66 0 .
359 2 . . . .
496 600 100 159 .
19 0 .
385 2 . . . .
496 700 100 169 .
95 0 .
411 2 . Table 3 . Same as in Table 1 but assuming that the hadronic EoS is described by the stiff model ofRef. [53].
The density at which the hadron-quark phase transition occurs is not known, but it is expectedto occur at some times the nuclear saturation density. In a NS, such a transition can lead totwo possible types of internal structure, depending on the surface tension between hadronicand quark matter [58, 59]. If the surface tension between hadronic and quark matter islarger than a critical value, which is estimated to be around
60 MeV / fm [60, 61], thenthere is a sharp interface and we have a Maxwell phase transition. If the surface tension isbelow the critical value, then there is a mixed phase of pure nuclear matter and pure quarkmatter usually called Gibbs phase transition. In this work we assume that the interfacebetween hadronic and quark matter is a sharp interface, which is a possible scenario giventhe uncertainties in the value of the surface tension [62–65].At the quark-hadron interface the following conditions are verified:• the pressure is continuous (mechanical equilibrium): p QPtr = p HPtr ≡ p tr . (2.5)• The energy density has a jump ∆ (cid:15) , i.e.: (cid:15) QPtr = (cid:15) HPtr + ∆ (cid:15). (2.6)• The Gibbs free energy per baryon at zero temperature, g = ( p + (cid:15) ) /n , is continuous(chemical equilibrium): g QPtr = g HPtr . (2.7)– 5 –rom Eq. (2.3) we see that g QP = 3 µ , and for hadronic matter we write g HP = ( p HP + (cid:15) HP ) /n HP . Therefore, at the interface we have: µ tr = p HPtr + (cid:15) HPtr n HPtr . (2.8)Now using Eq. (2.1), we can write Eq. (2.5) as: p HPtr = 34 π a µ − π a µ − B eff . (2.9)Similarly, using Eq. (2.3), we can write Eq. (2.6) as: (cid:15) HPtr + ∆ (cid:15) = 3 p HPtr + 4 B eff + 3 a π µ , (2.10)where µ tr is given by Eq. (2.8). From Eqs. (2.9)–(2.10) we can write the parameters B eff and a in terms of p HPtr , (cid:15) HPtr , n HPtr , µ tr , ∆ (cid:15) , and a in the form: (cid:26) B eff = β + β ∆ (cid:15) + β a a = α + α ∆ (cid:15) + α a (2.11)with β = 14 ( (cid:15) HPtr − p HPtr ) , β = 14 , β = − µ π ,α = π µ ( (cid:15) HPtr + p HPtr ) , α = π µ , α = 12 µ . These coefficients are fixed once the transition point has been chosen.In this work we fix the parameter a to the typical value a / = 100 MeV and exploreseveral values of the transition pressure p tr and the density jump ∆ (cid:15) at the interface, spanninga wide range of values for these quantities. Our choices for p tr and ∆ (cid:15) are shown in Tables1, 2 and 3, together with other properties of matter at the discontinuity, the correspondingvalues of the quark matter EoS parameters, and the maximum mass M max for each model.The values presented in Tables 1, 2 and 3 correspond, respectively, to the soft, intermediateand stiff hadronic EoSs of Ref. [53]. The curves of the resulting hybrid EoSs are shown inFig. 1. When matter in the neighborhood of the quark-hadron interface is perturbed and oscillatesaround an equilibrium position, reactions between both phases may occur due to compressionand rarefaction of fluid elements. The physics of these reactions can be classified in slow orfast depending on their timescale. Slow transitions are those with a timescale greater thanthe timescale of the oscillations. In the opposite case, when the timescale of the reactions islower than the timescale of the oscillations, we have a fast transition. In the slow case, whenfluid elements near the interface are perturbed and displaced from their equilibrium position,they maintain their composition and co-move with the motion of the interface. In this workwe will focus our discussion on the consequences of slow transitions. A slow phase transitionhas significant consequences at the macroscopic level, when the dynamical behavior of thestar against perturbations is investigated [51, 52]. The effect can be encoded in the junction– 6 –onditions for the radial eigenfunctions ξ and ∆ p . In particular, in the case of slow transitionsthey are continuous variables, which can be written as [ ξ ] = ξ + − ξ − = 0 , [∆ p ] = (∆ p ) + − (∆ p ) − = 0 (3.1)where [ x ] ≡ x + − x − , with x + and x − defined as the values after and before the interface,taking a reference system centered at the star. One direct consequence of the discontinuityconditions for the slow transition case is that the region of stable stars can be extendedbeyond the maximum mass point. This can be shown by determining the frequencies of thefundamental radial oscillation modes [50–52]. In general, the frequency ω of the fundamentalmode verifies ( ω > for the dynamically stable configurations, and the last stable objecthas ω = 0 . Coincidentally, for homogeneous (one phase) stars, the last stable star is theone that is just localized at the top of the mass versus radius curve (the maximum masspoint). This fact justifies the wide use of the following static stability criterion instead of thedynamical one: ∂M∂(cid:15) c < ⇒ (unstable configuration) , (3.2) ∂M∂(cid:15) c > ⇐ (stable configuration) . (3.3)In practice, this criterion is used to identify the last stable stellar configuration because, fora one-phase star, when ω = 0 the derivative ∂M/∂(cid:15) c changes its sign.However, in the case of two-phase stars with slow conversions at the interface, zerofrequency modes happen in general after the point of maximum mass (for further details seee.g. Ref. [51]). This gives rise to an extended region of stable hybrid stars with centraldensities that are larger than the density of the maximum mass star, as can be seen in Fig.2. The last stable star, which has a zero frequency fundamental mode, has been identifiedin Fig. 2 with an asterisk. Notice that hybrid models with low enough transition pressureare not able to reach a maximum mass above the observed value M max ≈ M (cid:12) . As apparentfrom Fig. 2, for a given transition pressure, the largest the density jump ∆ (cid:15) , the smallest theradius of the last stable hybrid star. The theory of relativistic tidal effects in binary systems has been focus of intense research inrecent years [15–17, 19, 66–70]. In this section we will summarize our procedure for computingthe important physical quantities.The first step in order to investigate the tidal deformability of binary systems is the com-putation of the compact star structure by means of the Tolman-Oppenheimer-Volkoff (TOV)equations, because some equilibrium quantities are needed during the numerical solution ofthe differential equations corresponding to the physics of tidal deformations.The tidal Love number k is found using the following expression [16]: k = 8 C − C ) [2 − y R + 2 C ( y R − ×{ C [6 − y R + 3 C (5 y R − C [13 − y R + C (3 y R −
2) + 2 C (1 + y R )]+3(1 − C ) [2 − y R +2 C ( y R − − C ) } − (4.1)– 7 – igure 1 . Hybrid EoSs used in this work. For each of the three hadronic EoS labeled as soft, inter-mediate and stiff we consider fifteen possible phase transitions using different values of the transitionpressure p tr and the density jump ∆ (cid:15) at the interface (see Sec. 2.3). The numerical values of theparameters for each model can be found in Tables 1, 2 and 3. – 8 – igure 2 . Mass-radius relations for the hybrid EoSs shown in Fig. 1. The asterisks represent the laststable star of each model, having a zero frequency fundamental radial mode. As explained in Sec. 3,since we assume that quark-hadron phase conversions at the interface of a hybrid star have a slowtimescale, there are parts of the curves that are dynamically stable in spite of having ∂M/∂(cid:15) c < . – 9 –here C ≡
M/R is the dimensionless compactness parameter and y R ≡ y ( R ) , being y ( r ) thesolution of the following first-order differential equation (see Ref. [71] for more details): r dy ( r ) dr + y ( r ) + y ( r ) F ( r ) + r Q ( r ) = 0 . (4.2)In this equation, the coefficients are: F ( r ) = [1 − πr ( (cid:15) − p )] (cid:20) − mr (cid:21) − , (4.3) Q ( r ) = 4 π (cid:20) (cid:15) + 9 p + (cid:15) + pc s − πr (cid:21) (cid:20) − mr (cid:21) − − m r (cid:20) πr pm (cid:21) (cid:20) − mr (cid:21) − , (4.4)where c s ≡ dp/d(cid:15) is the squared speed of sound. The boundary condition for Eq. (4.2) at r = 0 is given by y (0) = 2 . In summary, the tidal Love number can be obtained once an EoSis supplied and the TOV equations together with Eq. (4.2) are integrated. Several models of NSs with first-order phase transitions have been recently analyzed (see e.g.Refs. [52, 72–80]). In the presence of a finite energy density discontinuity, the last term in Eq.(4.4) involves a singularity ∝ ( (cid:15) + p ) /c s . This was first discussed by addressing the surfacevacuum discontinuity for incompressible stars [17]. Then, this approach was extended to thecontext of possible first-order transitions inside the stars and a junction condition was derived[71] that was corrected more recently [81, 82]. In summary, the following junction conditionfor the y ( r ) function has to be imposed: y ( r d + (cid:15) ) = y ( r d − (cid:15) ) − ∆ ρ ˜ ρ/ p ( r d ) (5.1)where r d represents the coordinate radius of the interface, i.e. the point where the phasetransition takes place. The region r < r d represents the internal core containing quark matterand r > r d is the external hadronic region of the star. We also have that ˜ ρ = m ( r d ) / (4 πr d / and ∆ ρ = ρ ( r d + (cid:15) ) − ρ ( r d − (cid:15) ) .The accumulated phase contribution due to the deformation from both of the stars isincluded in the inspiral signal as the combined dimensionless tidal deformability, which isgiven by ˜Λ = 1613 (cid:20) ( m + 12 m ) m Λ + ( m + 12 m ) m Λ ( m + m ) (cid:21) , (5.2)where Λ ( m ) and Λ ( m ) are the tidal deformabilities of the individual binary compo-nents [83]. The quantity ˜Λ is usually evaluated as a function of the chirp mass M c =( m m ) / / ( m + m ) / , for various values of the mass ratio q = m /m . From the event Eq. (5.1) contains an extra p ( r d ) term in the denominator as compared to Eq. (14) of Ref. [71] as shownin Refs. [81, 82]. We have checked that in our case the original and the corrected formulae lead to differencesin the tidal deformabilities that are less that . . – 10 – igure 3 . Tidal deformability Λ as a function of the stellar mass for the hybrid EoSs presented inFig. 1. – 11 –W170817, it was possible to set the first upper limit on the dimensionless tidal deformabil-ity Λ . of a NS with a mass . M (cid:12) such that Λ . ≤ at confidence level (for thecase of low-spin priors) [9]. Independently of the waveform model or the choice of priors,the source-frame chirp mass was constrained to the range M = 1 . +0 . − . M (cid:12) and the massratio q = m /m for low spin priors to the interval between . − . [9]. Working under thehypothesis that both NSs are described by the same EoS and have spins within the rangeobserved in Galactic binary NSs, the tidal deformability Λ . of a . M (cid:12) NS was estimatedthrough a linear expansion of Λ( m ) m around . M (cid:12) to be in the range − at the level [84].In Fig. 3, we show the dimensionless tidal deformability Λ as a function of the stellarmass for all the hybrid models presented in Fig. 1. Notice that Λ is a decreasing function of M for the hadronic branch. However, for hybrid configurations, Λ can be either a decreasingor an increasing function of M . Moreover, for large values of the transition pressure, Λ showsa very weak dependence on the stellar mass (the curves are quite horizontal).In Fig. 4, we show the dimensionless tidal deformabilities Λ and Λ for a binary NSsystem with the same chirp mass as GW170817. The curves are obtained as follows: oncea value of m is chosen one can calculate m by fixing M = 1 . M (cid:12) . Running m in therange of . ≤ m /M (cid:12) ≤ . we obtain m (with . ≤ m /M (cid:12) ≤ . ). The values of Λ and Λ associated with m and m are depicted in Fig. 4.We have analysed the three possibilities for the internal composition of the two stars inthe binary system: hadron-hadron, hybrid-hadron and hybrid-hybrid. In the case with twopurely hadronic stars, the presence of these binaries in the detection region depends stronglyon the stiffness of the EoS. As can be seen in Fig. 4 the soft hadronic EOS is inside the region of GW170817, the intermediate one is inside the region and the stiff one is outsidethe . Note that the hybrid models that we do not show in Figure 4 are the cases thatdo not present hybrid configurations in the range 1.365 - 1.6 M (cid:12) (these cases are completelyrepresented in the Λ – Λ plane by the hadronic curve with circle dots). In the scenario whereat least one of the stars in the binary is hybrid, we find that only models T1, T2, T3, T4, T5and T6 fall inside the detection region. Some of these curves have a lower part correspondingto hybrid-hybrid mergers and an upper part corresponding to hybrid-hadron mergers. Thereare also curves where the lower part represents hadron-hadron mergers and the upper parthybrid-hadron mergers. In general, we observe that as the density jump ∆ (cid:15) increases, thecurves shift to lower values of Λ .Notice that binary systems with at least one hybrid star that fall inside the area of the credible region for Λ – Λ (T1, T2, T3, T4, T5 and T6) have maximum masses below M (cid:12) , and are therefore inconsistent with stringent constrains on the maximum NS masscoming from PSR J0348+0432 with . ± . M (cid:12) [4], PSR J0740+6620 with . +0 . − . M (cid:12) [5], and PSR J2215-5135 with . +0 . − . M (cid:12) [6]. Therefore, for our model to agree with boththe level determined by LIGO/Virgo and the M (cid:12) pulsars, both objects in GW170817must be hadronic stars and hybrid configurations cannot fall in the range 1.365 - 1.6 M (cid:12) .Viable candidates for this scenario would be the models from T7 to T15 linked to the hadronicEoS of intermediate stiffness. These models have M max > M (cid:12) (see central panel of Fig. 2)and are represented in the Λ – Λ plane by the hadronic curve with circle dots shown in thecentral panel of Fig. 4. – 12 – igure 4 . Dimensionless tidal deformabilities Λ and Λ for a binary NS system with masses m and m and the same chirp mass as GW170817 [9]. By definition we only plot combinations with m > m . The diagonal blue line indicates the Λ = Λ boundary. The black dashed line denotes the credibility level and the red dashed line the level determined by LIGO/Virgo in the low-spinprior scenario. – 13 – Summary and conclusions
In this work we have analyzed hybrid stars containing sharp phase transitions betweenhadronic and quark matter assuming that phase conversions at the interface are slow. Hadronicmatter has been described by an EoS based on nuclear interactions derived from chiral effec-tive field theory [53] and quark matter by a generic bag model [56].In the case of slow conversions at the quark-hadron interface, zero frequency radial modesarise in general beyond the point of maximum mass, giving rise to an extended region of stablehybrid stars with central densities that are larger than the density of the maximum mass star[51]. This effect can be seen clearly in Fig. 2 where large extended branches arise. We haveexplored systematically the role of the transition pressure p tr and the energy-density jump ∆ (cid:15) on the maximum mass object, the last stable configuration, and the tidal deformabilities. Asexpected, hybrid models with low enough p trans are not able to reach a maximum mass abovethe observed value M max ≈ M (cid:12) . Also, for a given transition pressure, the radius of the laststable hybrid star decreases as ∆ (cid:15) raises resulting in a larger extended branch of stable hybridstars (see Fig. 2).We also found that for hybrid configurations, the tidal deformability Λ can be eithera decreasing or an increasing function of M . Moreover, for large values of the transitionpressure, Λ shows a very weak dependence on the stellar mass and the curves become almosthorizontal in some cases (see Fig. 3). This is in contrast with the case of purely hadronicstars for which Λ is always a decreasing function of M .Finally, we analyzed the tidal deformabilities Λ and Λ for a binary NS system with thesame chirp mass as GW170817 (see Fig. 4). In the scenario where at least one of the stars inthe binary is hybrid, we found that only models with low enough transition pressure fall insidethe credible region for Λ – Λ . Unfortunately, these models have maximum masses below M (cid:12) , i.e. in disagreement with the observation of PSR J0348+0432, PSR J0740+6620 andPSR J2215-5135. However, our model can explain the level determined by LIGO/Virgotogether with the M (cid:12) constrain if both objects in GW170817 are hadronic stars and allhybrid configurations have masses larger than . M (cid:12) . We find that several hybrid modelsinvolving the hadronic EoS of intermediate stiffness are in agreement with observations. Acknowledgments
Alessandro Parisi is grateful to Professor Feng-Li Lin for many helpful discussions. GermánLugones acknowledges the Brazilian agency Conselho Nacional de Desenvolvimento Científicoe Tecnológico (CNPq) for financial support. César H. Lenzi is thankful to the Fundação deAmparo à Pesquisa do Estado de São Paulo (FAPESP) under thematic project 2017/05660-0for financial support. Chian-Shu Chen is supported by Ministry of Science and Technology(MOST), Taiwan, R.O.C. under Grant No. 107-2112-M-032-001-MY3.
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