Speed of sound in dense matter and two families of compact stars
Silvia Traversi, Prasanta Char, Giuseppe Pagliara, Alessandro Drago
SSpeed of sound in dense matter and two families of compact stars
Silvia Traversi , , Prasanta Char , , Giuseppe Pagliara , , Alessandro Drago , Dipartimento di Fisica e Scienze della Terra, Universit`a di Ferrara, Via Saragat 1, 44122 Ferrara, Italy, INFN Sezione di Ferrara, Via Saragat 1, 44122 Ferrara, Italy, Space sciences, Technologies and Astrophysics Research (STAR)Institute, Universit´e de Li`ege, Bˆat. B5a, 4000 Li`ege, Belgium
We investigate the possibility of having massive compact stars, (
M > M (cid:12) ), and at the sametime fulfilling the theoretical bounds on the speed of sound c s in dense matter suggesting that theconformal limit of c s = 1 / c s is peaked around 0.3, and the maximum mass of the most probableequation of state is ∼ . M (cid:12) . We finally discuss also the possibility that the maximum mass islarger than 2 . M (cid:12) as it could be the case if the secondary component of GW190814 is a compactstar and not a black hole. PACS numbers:
I. INTRODUCTION
After the discovery of neutron stars with masses ∼ M (cid:12) [1–3] it became clear that the equation of state (EoS) ofdense baryonic matter must be rather stiff to support such a large mass against gravitational collapse. The stiffnessof the equation of state for nucleonic matter is regulated by the adiabatic index or equivalently by the speed of sound c s i.e. the derivative of pressure with respect to the energy density at fixed entropy. A remarkable result obtained in[4] (see also the more recent [5]) is that while it is expected that at asymptotically high density the speed of soundmust reach the conformal limit of c s = 1 / c s must increase to values significantly larger than the conformalbound and then should decrease and reach the conformal limit from above . Actually, when considering the nowvery popular possibility of twin stars [8–10], namely stars having a core of quark matter separated, via a large energydensity jump, from a nucleonic layer, c s must basically be set to the causal limit c s = 1, see also [11]. Notice thatthis inferred behaviour is different from the case of finite temperature and zero density matter for which lattice QCDhas definitely established that c s is always below the conformal limit [12]. A behaviour predicted also in several weakcouplings and strong couplings theories, see [4].There are two possible ways to solve the tension between the existence of massive stars (from astrophysical data)and the theoretical expectations to recover the high density conformal limit: i) there is some physical mechanismwhich is responsible for a rapid increase of c s at densities close to twice saturation density and which then ”switchesoff” at larger densities allowing to recover the conformal limit from above. There are several examples in the literaturefor this kind of explanation, see [7, 13–17].ii) the conformal limit is never violated but there exist two types of stars, Hadronic Stars (HSs) and Quark Stars(QS) within the so-called two-families scenario [18–22]. QSs are self-bound objects that can be though as giant nucleardrops having a pressure profile which ends at the surface of the star with a finite (and large) value of the energy density e . It is well known that for this kind of objects, the maximum mass does depend on the speed of sound as ordinaryneutron stars but also on the value of e . In particular, when adopting the most simple prescription i.e. the constantspeed of sound (CSS) EoS p = c s ( e − e ) where p and e are pressure and energy density [23–26] one can showthat the maximum mass m max ∝ e − / [28]. Unfortunately, there is no such a simple scaling with c s . One can shownumerically that, to good approximation, m max ( x, e ) = ( e n /e ) / (0 . . x − . x +0 . x ) where Notice that while in Ref.[4] a constant speed of sound EoS with c s = 1 / . Notice that this simple prescription provides EoSs very similar to the ones obtained within the popular MIT bag model in which therole of e is played by the bag constant and c s ∼ / ∼
100 MeV and thus small with respect to the chemical potential, see [27]. a r X i v : . [ a s t r o - ph . H E ] F e b s2 /(1/3)11.21.41.61.82 e / e n sun sun sun FIG. 1: Lines of constant maximum mass in the parameter space of the CSS model. Indicated are also the radii of maximummass configurations at the edges of the constant maximum mass lines. x = c s / (1 /
3) and e n = 150 MeV fm − is the nuclear matter energy density. We display in Fig.1 the contour plot of m max . One can notice that in the conformal limit, x = 1, 2 M (cid:12) < ∼ m max < ∼ . M (cid:12) provided that 1 < ∼ e /e n < ∼ . c s exceeds the conformal limit in order to have masses larger than 2 M (cid:12) when QSs areconsidered. Actually, one can also notice from the same plot that even a value of x ∼ .
75 can provide m max ∼ M (cid:12) provided that e coincides with the nuclear matter density e n . Such a low value of e is most probably unlikely butthe important point here is that there is a window for c s and e that allows for massive stars even if c s is belowthe conformal limit. Indeed, in [29] by adopting the chiral color dielectric model for computing the EoS of quarkmatter, it has been found that it is possible to obtain values of m max close to the 2 M (cid:12) limit with an EoS for which c s approaches the conformal limit from below. The hypothesis of existence of two-families of compact stars (CSs)would then remove the tension between astrophysical observations and the theoretical bounds on c s in dense matter[4] unless future observations will discover stars with masses above ∼ . M (cid:12) .A correlated issue concerns the radii of CSs. One can show that the radius of the maximum mass configurationscales as r max ( x, e ) = ( e n /e ) / (7 . . x − . x + 0 . x ). In Fig.1 we indicate the range ofvalues of r max for fixed values of m max . Clearly, the larger the value of m max the larger the value of r max (up to ∼ e /e n = 1 and m max = 3 M (cid:12) ). In the two-families scenario indeed it is possible to interpret stars withlarge radii as massive QSs. This will be the starting point of the discussion on the astrophysical data presented inSec.II.Finally, concerning HSs: as shown in [4] they can reach m max ∼ . M (cid:12) (if the conformal limit is assumed to holdtrue in hadronic matter). Actually, the possible formation of hyperons and delta resonances could reduce the valueof c s in hadronic matter with a consequent reduction of the values of the maximum mass and the radii of HSs. Verycompact stars, with R . < ∼ m max in the two-familiesscenario (by adopting the CSS model for quark matter). Those predictions need to be confronted with the availableastrophysical data. It is the aim of this work to obtain the posterior distributions of e and c s by performing aBayesian analysis on a selected sample of data which are interpreted as QSs within the two-families scenario. We willinvestigate the case in which the m max = 2 . M (cid:12) as results from the observation of MSP J0740+6620[3].Finally, we will discuss the possibility that the source of GW190814 [31] is a BH-CS system implying that m max islarger than 2 . M (cid:12) . II. OBSERVATIONAL DATA
The two-families scenario is based on the coexistence of two classes of CSs. The first one, due to a soft hadronicEoS, is composed by light and very compact HSs, with R . of the order of [10 . −
11] km and the second one bymassive QSs with larger radii [19]. The formation of new degrees of freedom inside HSs softens the EoS leading to amaximum mass m HSmax of ∼ . − . M (cid:12) . At fixed baryonic mass, QSs have a lower gravitational mass with respectto HSs, of the order of 0 . M (cid:12) smaller. Thus, the transition from HSs to QSs is energetically favoured [32–36] andit can proceed only when there are enough hyperons in the system to form the first droplet of strange quark matterwhich can then trigger the conversion, see [37–40]. In turn, this implies the existence of a minimum mass for the QSsbranch, m QSmin ∼ m HSmax − . M (cid:12) and the coexistence of both HSs and QSs in the interval [ m QSmin , m
HSmax ]. In this range,HSs and QSs can have the same mass but different radii.We use the simultaneous measurements of masses and radii for several X-ray sources, and also the masses andtidal deformabilities from the gravitational wave events reported by LIGO-VIRGO collaboration (LVC) [41, 42].Specifically, for our sample of possible QSs candidates we have chosen 4U 1724-07, SAX J1748.9 2021, 4U 1820–30,4U 1702–429, J0437–4715, the high-mass component of GW170817 and both the components of GW190425 [43–48].For 4U 1702–429 [44] and J0437–4715 [45], we take a bivariate Gaussian distribution to resemble the M − R posteriorsince the full distribution is not available. For all the other sources we have instead used directly the posteriordistributions which are publicly available. Our data refer to CSs which can be interpreted, within the two-familiesscenario, as being QSs, as we explain in the following.First, we select J0437–4715 as QS, because its radius is larger than ∼
13 km and HSs in the two-families scenariocannot reach such large radii. The central value of its mass is ∼ . M (cid:12) and we will choose this value as a guess for m QSmin . Thus m HSmax is fixed to m QSmin + 0 . M (cid:12) = 1 . M (cid:12) and all the sources with masses > ∼ . M (cid:12) can be interpretedas QSs. For making this comparison, the masses have been chosen to be the mean values of the marginalizeddistribution of the sources. This criterion is fulfilled by all the sources which we have selected in this study apartfrom GW170817 1, whose mass ( ∼ . M (cid:12) ) falls in the coexistence region [ m QSmin , m
HSmax ]. The reason for assumingthat GW170817 1 is a QS is phenomenological: the presence of strong electromagnetic counterparts associated withGW170817 (GRB170817A and AT2017gfo) implies that the merger event did not produce a prompt collapse. Since thethreshold mass for a prompt collapse for a HS-HS binary has been estimated in [22] to be m thr = 2 . M (cid:12) , that mergerevent should be classified as HS-QS merger. Thus, the heaviest component, GW170817 1, within the two-familiesscenario, is most probably a QS. III. RESULTS AND DISCUSSIONS
Let us present the results of our Bayesian analysis on the sources explained in II. We have used the CSS parameter-ized EoS described before, with two free parameters: e and c s . We select the priors on e to be in the range between160 and 232 MeV fm − and on c s in the interval [0 . , . M (cid:12) on theEoS and this has consequences on the ranges of the parameters. We have checked the prior range for e if c s is set to1 / e >
220 MeV fm − are ruled out. This is no longer true in the two parameterscase, since an increase in c s allows also the previously excluded configurations to fulfill the constraint, but restricts c s above 0 .
26. Details of this analysis alongside a comparison with the neural network based prediction techniques canbe found in [49].Next, we construct the joint posterior and investigate the conformal limit on c s given the present data. We use thepublicly available posterior samples of X-ray measurements from ¨Ozel et al. [43] . The mass and tidal deformabilitysamples of the binary merger components are provided by LVC , as well. In figure 2, we display the marginalizedPDFs for e and c s along with their most probable values and 1 σ errors and the 2 D distribution with 1 σ (39 . c s with correspondingnot too big values of e . Indeed a positive correlation among these two parameters is found. The most probable pointof the joint PDF is represented as a blue circle in figure 2 and it is located at e = 183 .
48 MeV fm − , c s = 0 . c s are far from the causal limit (which is instead adoptedin the twin-stars scenario) but even the violations of the conformal bound seems to be unnecessary.The M − R and M − Λ curves corresponding to the 68% CI are shown in figure 3. The higher likelihood region ofthe 2 D posterior is associated to EoSs which are characterized by maximum masses within ∼ . − . M (cid:12) and radiiin the interval R . ∼ . − . M max = 2 . M (cid:12) and R . = 12 . The mass-radius distributions of the sources of [43] can be found at http://xtreme.as.arizona.edu/neutronstars/. The data from GW170817 and GW190425 are available respectively at https://dcc.ligo.org/LIGO-P1800115/public and,https://dcc.ligo.org/LIGO-P2000026/public. distribution at the 68% CI. However at 90% CI the agreement is met also with GW170817 1. We believe that adensity dependence of c s could probably solve this (mild) tension. FIG. 2: Posterior for e and c s resulting from the Bayesian analysis. The blue circle shows the most probable point of the jointdistribution. The green and red lines in the marginalized PDFs plots correspond to the mode and 1 σ errors respectively.FIG. 3: M − R (left) and M − Λ (right) curves for the 68% CI of the posterior distribution. The black dashed lines representthe most probable EoS.
IV. WHAT IF GW190814 HAS A . M (cid:12) COMPACT STAR COMPONENT?
The detection of the gravitational waves signal GW190814 by the LVC collaboration [31] led to the discovery ofa binary system made of a 23 M (cid:12) BH and a CS companion of ∼ . M (cid:12) . The nature of the companion is rather
300 400 500 600 700 800 900 1000 µ [MeV]0.310.320.330.340.350.36 c s UnpairedCFL
FIG. 4: Speed of sound as a function of the quark chemical potential for unpaired quark matter and CFL matter. uncertain since its mass falls within the BH lower mass gap but, on the other hand, such a massive neutron starchallenges previous astronomical constraints and nuclear physics constraints [50]. In the recent Ref. [51], it has beenproposed that the companion could actually be a QS.From the discussion presented in Sec.I it is clear that to explain the existence of a QS with M = 2 . M (cid:12) theconformal limit must be violated also in dense quark matter. Namely, it is necessary that for a certain range ofdensities c s is larger than 1 /
3. In [51] this requirement is achieved by assuming that quark matter could be in a colorsuperconducting state, namely the CFL phase. The superconducting gap modifies the density dependence of c s asshown in Fig.4 and acts also as an effective reduction of e (encoded in the bag constant). Notice that, with respectto the CSS EoS, in presence of massive quarks (the strange quark with mass m s ) and of a superconducting gap (∆)there is an additional term in the gran-potential which is proportional to : m s − , see [27]. One can show that,if ∆ > m s / c s will approach the conformal limit from above as the chemical potential increases. In the exampledisplayed in Fig.4, m s = 100 and ∆ = 80 MeV and the pQCD correction parameter a = 0 .
7. In this way it is possibleto reach values of m max larger than 2 . M (cid:12) as shown in [51] and to pin down a physical mechanism for explaining theviolation of the conformal bound at low density. Notice that the deviations from the conformal limit are rather smalland that actually the most important effect for increasing the value of the maximum mass derives from the value of e which is reduced by the presence of a large superconducting gap. On the other hand, the twin-star scenario seemsto be ruled out by the existence of very massive CSs, as recently found in [52]. V. CONCLUSIONS
We have shown that the existence of massive compact stars, with masses up to 2 . M (cid:12) does not necessarily implythe violation of the conformal limit in dense strongly interacting matter. One has however to abandon the assumptionthat only one family of compact stars exists and assume that two separated branches in the mass-radius plot arepossible: the HS branch and the QS branch. Within the two-families scenario, some astrophysical sources can be(with a certain degree of uncertainty) identified as QSs. A Bayesian analysis on such selected sample of sources hasallowed to estimate the posterior distributions of the two parameters of the model for the EoS, namely e and c s .Interestingly, the presently available data suggest that the distribution of c s is actually peaked at a value close to 1 / c s has a distribution peaked at 0.95 [53], thus totally different withrespect to our findings.If compact stars with masses larger than 2 . M (cid:12) will be discovered, then even in the two-families scenario theviolation of the conformal limit is mandatory. A possible mechanism has been suggested which is based on theformation of a (sizable) superconducting gap.Finally, while in this paper the Bayesian analysis has been performed by using the simple constant speed ofsound model, it would be important in future investigations to use more sophysticated models allowing for a densitydependence of c s such as e.g. the color dielectric model of Ref.[29] or a simple parametrization with piecewise constantspeed of sound models. Acknowledgments
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