Bayesian inference of dense matter EOS encapsulating a first-order hadron-quark phase transition from observables of canonical neutron stars
aa r X i v : . [ nu c l - t h ] S e p Bayesian inference of dense matter EOS encapsulating a first-order hadron-quarkphase transition from observables of canonical neutron stars
Wen-Jie Xie and Bao-An Li ∗ Department of Physics, Yuncheng University, Yuncheng 044000, China Department of Physics and Astronomy, Texas A & M University-Commerce, Commerce, TX 75429, USA (Dated: September 30, 2020)
Background:
The remarkable progress in recent multimessenger observations of both isolated neutron stars (NSs) and theirmergers has provided some of the much needed data to improve our understanding about the Equation of State (EOS) ofdense neutron-rich matter. Various EOSs with or without some kinds of phase transitions from hadronic to quark matter(QM) have been widely used in many forward-modelings of NS properties. Direct comparisons of these predictionswith observational data sometimes also using χ minimizations have provided very useful constraints on the modelEOSs. However, it is normally difficult to perform uncertain quantifications and analyze correlations of the EOS modelparameters involved in forward-modelings especially when the available data are still very limited. Purpose:
We infer the posterior probability distribution functions (PDFs) and correlations of nine parameters characterizingthe EOS of dense neutron-rich matter encapsulating a first-order hadron-quark phase transition from the radius data ofcanonical NSs reported by LIGO/VIRGO, NICER and Chandra Collaborations. We also infer the QM mass fraction andits radius in a 1.4 M ⊙ NS and predict their values in more massive NSs.
Method:
Meta-modelings are used to generate both hadronic and QM EOSs in the Markov-Chain Monte Carlo samplingprocess within the Bayesian statistical framework. An explicitly isospin-dependent parametric EOS for the npeµ matterin NSs at β equilibrium is connected through the Maxwell construction to the QM EOS described by the constant speedof sound (CSS) model of Alford, Han and Prakash. Results: (1) The most probable values of the hadron-quark transition density ρ t /ρ and the relative energy density jump there∆ ε/ε t are ρ t /ρ = 1 . +1 . − . and ∆ ε/ε t = 0 . +0 . − . at 68% confidence level, respectively. The corresponding probabilitydistribution of QM fraction in a 1.4 M ⊙ NS peaks around 0.9 in a 10 km sphere. Strongly correlated to the PDFs of ρ t and ∆ ε/ε t , the PDF of the QM speed of sound squared c /c peaks at 0 . +0 . − . , and the total probability of being lessthan 1/3 is very small. (2) The correlations between PDFs of hadronic and QM EOS parameters are very weak. Whilethe most probable values of parameters describing the EOS of symmetric nuclear matter remain almost unchanged, thehigh-density symmetry energy parameters of neutron-rich matter are significant different with or without considering thehadron-quark phase transition. Conclusions:
The available astrophysical data considered together with all known EOS constraints from theories and terrestrialnuclear experiments prefer the formation of a large volume of QM even in canonical NSs.
I. INTRODUCTION
Probing the Equation of State (EOS) of dense neutron-rich matter has been a long standing and shared goal ofboth astrophysics and nuclear physics. Much progresshas been made in realizing this goal using various mes-sengers from both isolated neutron stars (NSs) and theirmergers especially since LIGO/VIRGO’s observation ofGW170817. For recent reviews, see, e.g., Refs. [1–7].Among the many interesting questions studied in the lit-erature, significant efforts have been devoted for a longtime to investigating whether quark matter exists in NSs,the nature as well as where and when the hadron-quarkphase transition may happen, what its size and EOSmay be if quark matter does exist in NSs or can be cre-ated during their mergers, see, e.g., Refs. [8–18]. Sincethe earlier debate [19, 20] on whether the mass and ra-dius of EXO 0748-676 can rule out the existence of itsquark core, despite of the great progresses made usingvarious astrophysical data including the latest ones from ∗ Corresponding author: [email protected]
LIGO/VIRGO, NICER and Chandra observations, state-of-the-art theories and models as well as updated nuclearphysics constraints, see, e.g., Refs. [5, 21–24] and refer-ences therein, no consensus has been reached on mostof the issues regarding the nature and EOS of dense NSmatter.Most of the studies about the nature of hadron-quarkphase transition and the size of possible quark mattercore in NSs have been carried out by using the tradi-tional forward-modeling approach based on various the-ories for both the hadronic and quark phases, perhapsexcept very few recent studies using Bayesian analyses,see, e.g., Refs. [5, 24]. Often, various forms of poly-tropes or spectrum functions are used to interpolate theNS EOS starting slightly above the saturation density ρ of nuclear matter (below which reliable theoretical pre-dictions and some experimental constraints exist) to veryhigh densities where predictions of perturbative QCD ex-ist. Comparisons of model predictions with observationaldata have provided very useful constraints on the modelEOSs considered. Although χ minimizations are some-times used, often conclusions are strongly model depen-dent. Moreover, it is normally difficult to perform uncer-tain quantifications and analyze correlations of the EOSmodel parameters involved in forward-modelings espe-cially when the available data are still very limited.In this work, meta-modelings are used for bothhadronic and quark phases to construct very generallythe EOSs of NS matter. An explicitly isospin-dependentEOS [25] for the npeµ matter in NS at β equilibriumis connected through the Maxwell construction to theconstant speed of sound (CSS) quark matter EOS [26].With totally 9 parameters in their prior ranges allowedby general physical principles and available constraints,the constructed NS EOS is so generic that it can es-sentially mimic any NS EOS available in the literature.Without restrictions and possible biases of underlyingenergy density functionals of specific theories, we in-fer the probability distribution functions (PDFs) of thenine EOS parameters using the available NS radius datafrom LIGO/VIRGO, NICER and Chandra, satisfyingthe causality and dynamical stability condition withinthe Bayesian statistical framework. We found that theavailable astrophysical data considered together with allknown EOS constraints from theories and terrestrial nu-clear experiments prefer the formation of a large volumeof QM even in canonical NSs. II. THEORETICAL APPROACH
Here we summarize the major features of our approach.In the CSS model of Alford, Han and Prakash [26] forhybrid NSs, the pressure in NSs is parameterized as ε ( p ) = (cid:26) ε HM ( p ) p < p t ε HM ( p t ) + ∆ ε + c − ( p − p t ) p > p t , (1)where ε HM ( p ) is the hadronic matter (HM) EOS belowthe hadron-quark transition pressure p t , ∆ ε is the dis-continuity in energy density ε at the transition, and c QM is the QM speed of sound. Once the HM EOS ε HM ( p ) isspecified, the transition pressure p t and energy density ε t are uniquely related to the hadron-quark transition den-sity ρ t . In our Bayesian analyses using the CSS model,we use the ρ t /ρ , ∆ ε/ε t and c /c as three indepen-dent parameters to be generated randomly with uniformprior PDFs in the range of 1 to 6 (or 10 for comparison),0 . − −
1, respectively. We thus use the CSSmodel as a generic meta-model for generating the QMEOS.In several recent applications of the CSS model, see,e.g., Refs. [27–30], various HM EOSs predicted by micro-scopic nuclear many-body theories and/or phenomeno-logical models have been used. These HM EOSs areoften restricted by the underlying energy density func-tionals of the theories used and are usually not flexi-ble enough in statistical analyses to explore the wholeEOS parameter space permitted by general physics prin-ciples and known constraints as pointed out already inRefs. [22, 31]. On equal footing as the generic QM EOS,we use the meta-model of Ref. [25] for generating the HM EOS. The explicitly isospin dependence of the latterbuilt into the EOS at the level of average nucleon en-ergy in neutron-rich matter is an important distinctioncompared to directly parameterizing the HM pressureas a function of energy or baryon density with piece-wise polytropes or spectrum functions. Such kinds ofparameterizations with minor variations for HM EOSshave been widely used in both nuclear physics, see, e.g.Refs. [32, 33] and astrophysics applications, see, e.g.,Refs. [2, 22, 25, 31, 34–41]. For this work, we calculatethe pressure within the npeµ model for the core of NSsusing P ( ρ, δ ) = ρ dǫ HM ( ρ, δ ) /ρdρ (2)where the HM energy density ǫ HM ( ρ, δ ) = ǫ n ( ρ, δ ) + ǫ l ( ρ, δ ) with ǫ n ( ρ, δ ) and ǫ l ( ρ, δ ) being the energy den-sities of nucleons and leptons, respectively. While the ǫ l ( ρ, δ ) is calculated using the noninteracting Fermi gasmodel [42], the ǫ n ( ρ, δ ) is from ǫ n ( ρ, δ ) = ρ [ E ( ρ, δ ) + M N ] (3)where M N is the average nucleon mass. The average en-ergy per nucleon E ( ρ, δ ) in neutron-rich matter of isospinasymmetry δ = ( ρ n − ρ p ) /ρ is parameterized in terms ofthe energy per nucleon E ( ρ ) ≡ E ( ρ, δ = 0) in symmetricnuclear matter (SNM) and the symmetry energy E sym ( ρ )as [43] E ( ρ, δ ) = E ( ρ ) + E sym ( ρ ) δ . (4)The E ( ρ ) and E sym ( ρ ) are parameterized respectivelyas E ( ρ ) = E ( ρ ) + K ρ − ρ ρ ) + J ρ − ρ ρ ) , (5) E sym ( ρ ) = E sym ( ρ ) + L ( ρ − ρ ρ ) + K sym ρ − ρ ρ ) + J sym ρ − ρ ρ ) (6)where E ( ρ ) = − . K , J , E sym ( ρ ), L, K sym and J sym in theircurrently known uncertain ranges: 220 ≤ K ≤ − ≤ J ≤
400 MeV, 28 ≤ E sym ( ρ ) ≤ ≤ L ≤
90 MeV, − ≤ K sym ≤
100 MeV,and − ≤ J sym ≤
800 MeV, respectively.The density profile of isospin asymmetry δ ( ρ ) in chargeneutral NSs at β equilibrium is uniquely determined bythe symmetry energy E sym ( ρ ). Once the δ ( ρ ) is deter-mined, both the P ( ρ, δ ) and ǫ HM ( ρ, δ ) become barotropicfunctions of density ρ . The core EOS outlined above isthen connected smoothly to the NV EOS [45] for the in-ner crust and the BPS EOS [46] for the outer crust usingthe crust-core transition density and pressure evaluatedconsistently using a thermodynamical approach from thecore side with the same parameters given above [25].In preparing the EOS for the entire NS, we explicitlyenforce the Seidov stability condition [8, 9, 11]∆ εε t ≤
12 + 32 p t ε t . (7)Consequently, only a stable hybrid star branch is ex-pected to be connected to the NS branch for a givenEOS allowing the formation of twin stars [26].We use the standard Bayesian formalism and theMarkov-Chain Monte Carlo (MCMC) technique to eval-uate the posterior PDFs of EOS parameters as in ourprevious work where no hadron-quark phase transitionwas considered but using the same HM meta-model forthe entire core of NSs [38, 39]. For easy of the follow-ing discussions, we notice the following key inputs andaspects of our Bayesian analyses: • The likelihood function P [ D |M ( p , , ··· )] measuresthe ability of the model M with 9 parameters p , , ··· to reproduce the observational data D. Weuse [38, 39] P [ D |M ( p , , ··· )] = P filter × P mass , max × P radius where the P filter is a filter selecting EOS parame-ter sets satisfying the following conditions: (i) Thecrust-core transition pressure always stays positive;(ii) At all densities, the thermaldynamical stabilitycondition (i.e., dP/dε ≥ P mass , max stands for the requirement that eachaccepted EOS has to be stiff enough to support theobserved NS maximum mass M max . We presentresults with M max =1.97 M ⊙ to be consistent withthat used by the LIGO/VIRGO Collaborations intheir extraction of the NS radius from GW170817[47]. Using 2.01 or 2.14 M ⊙ for M max has only someminor quantitative effects. • We use the following radii of canonical NSs asindependent data: 1) R . = 11 . ± . R . = 10 . +2 . − . extracted inde-pendently also from GW170817 by De et al. [48],3) R . = 11 . +1 . − . from earlier analysis of quies-cent low-mass X-ray binaries observed by Chan-dra and XMM-Newton observatories [49], and 4) R = 12 . +1 . − . km for PSR J0030+0451 of mass M = 1 . ± .
24 M ⊙ from NICER [50]. The errorsquoted are at 90% confidence level. Correspond-ingly, the P radius is a product of four Gaussian func-tions, i.e., P radius = Y j =1 √ πσ obs ,j exp[ − ( R th ,j − R obs ,j ) σ ,j ] where σ obs ,j represents the 1 σ error bar of the ra-dius from the observation j while R th ,j is the corre-sponding theoretical prediction. More details canbe found in our previous work in Refs. [38, 39]. • In the MCMC process of sampling the posteriorPDFs of EOS parameters, we throw away the initial100,000 burn-in steps/EOSs before the stationarystate is reached. Afterwards, we generate 1600,000steps/EOSs to calculate the posterior PDFs andcorrelations of EOS parameters. The acceptancerate is about 15%.We emphasize that a fundamental assumption made inthe CSS model is that once the energy density reachedin the core of NS is higher than a critical value ε c = ε HM ( p t ) + ∆ ε , QM will be formed through the first-orderhadron-quark phase transition. All results presented herethus have to be understood with this assumption in mind. III. RESULTS AND DISCUSSIONSA. Quark matter fraction and size in hybrid stars
Shown in the inner window of Fig. 1 are the mass-radius sequences in selected samples with the hadron-quark transition density ρ t /ρ = 1 . , . c /c = 1.) As expected, the stable hybrid branchesare all connected to the NS branches.In the study of hybrid stars, a key question has beenwhether the densities reached inside NSs are high enoughto form a sizeable QM core. To answer this question, weshow in the outer and middle windows the normalizedprobability distribution of the QM fraction f massQM (de-fined as the ratio of QM mass over the total NS mass)and the corresponding QM radius R QM , in regions wherethe energy density is higher than the QM critical energydensity ε c in NSs of mass 1.4, 1.6, 1.8 and 2.0 M ⊙ , re-spectively, in the default Bayesian analysis with the nineEOS parameters. It is interesting to see the two peaksindicating the formation of twin stars. The major peaksat f massQM = 0 correspond to pure hadronic NSs in caseswhere the ε c is always higher than the maximum energydensity at the core of the NSs considered. The secondpeaks around f massQM = 0 . ∼ .
95 and R QM = 10 ∼ f massQM higher than 0.1 is 77.6%,81.8%, 85.2% and 88.7% for M=1.4, 1.6, 1.8 and 2.0 M ⊙ ,respectively.By changing the prior range of hadron-quark transitiondensity ρ t /ρ from the default 1-6 to 1-10, we found verylittle effect. We also found that correlations between theHM and QM EOS parameters are very weak, thus in the P r obab ili t y d i s t r i bu t i on R QM (km)
10 11 12 13 140.00.51.01.52.02.53.03.5
L = 51 MeVE sym ( )=34.5 MeV/ t =0.4c =1 t / =1.5 t / =3.5 t / =5.5 M / M s un R(km) J =-180 MeVK = 222 MeVJ sym =800 MeVK sym =-130 MeV M/M sun =1.4 M/M sun =1.6 M/M sun =1.8 M/M sun =2.0 P r obab ili t y d i s t r i bu t i on f massQM FIG. 1: (Color online) The inner window: mass-radius sequences of selected samples with the three different hadron-quarktransition densities but all other parameters fixed at the values specified. The middle and outer windows are the normalizedprobability distribution of the quark matter radius and fraction, respectively, from all accepted EOSs in the Bayesian analysis. following we present the PDFs and correlations of quarkmatter and hadronic matter EOSs separately.
B. Posterior probability distribution functions ofquark matter EOS parameters and their correlations
Shown in Fig. 2 are the posterior PDFs and correla-tions of QM EOS parameters ρ t /ρ , ∆ ε/ε t and c /c ,as well as the f massQM and R QM for canonical NSs in thedefault calculation. Several interesting features deserveemphasizing: • The most probable values of the QM EOS param-eters are ρ t /ρ = 1 . +1 . − . , ∆ ε/ε t = 0 . +0 . − . and c /c = 0 . +0 . − . at 68% confidence level. Be-cause the transition density peaks at a rather lowdensity, and the energy jump at the transition isalso relatively low, the QM stiffness represented byits c value is rather high to provide the necessarypressure in QM. Since the average baryon densityof a canonical NS with a 12 km radius is about 2 ρ ,it is thus not surprising that for canonical NSs thePDFs of QM fraction and its radius peak around f massQM ≈ . R QM ≈
10 km, respectively. • The total probability for c /c ≤ / c /c in QM is likelyvery high while the strength of the phase transitionmeasured with the energy density jump ∆ ε/ε t ismodest (around 0.4). • The f massQM , R QM and ∆ ε/ε t are all anti-correlatedwith ρ t /ρ as one expects. When the transitiondensity is low and the energy jump is weak, therequired c /c has an approximately equally highprobability to be between 0.5 to 1. C. The role of the speed of sound in quark matter
Motivated by perturbative QCD predictions at ex-tremely high densities or the casual limit, often inforward-model predictions one sets c /c =1/3 or 1among other constants examined. In fact, much ef-forts have been devoted to finding signatures/imprintsof c /c from/on astrophysical observables especially / t c Q M L (GeV) f m a ss Q M R Q M t / / t c f massQM R QM FIG. 2: (Color online) The posterior probability distribution functions and correlations of the three quark matter EOS param-eters as well as the fraction and radius of quark matter in hybrid neutron stars. since LIGO/VIRGO Collaborations’ recent discoverythat GW190814’s secondary component has a mass of(2.50-2.67) M ⊙ , see, e.g., Ref. [51] and references therein.While in our default Bayesian analysis we have gener-ated c /c randomly with a uniform prior PDF in therange of 0 to 1, it is interesting to compare the defaultresults with calculations setting c /c to certain con-stants. Shown in Fig. 3 are the posterior PDFs of thetransition density (upper) and the jump of energy den-sity there (lower) with c /c =1/3 and 1, respectively.While the results with c /c = 1 are very close to thedefault ones, setting c /c =1/3 requires a much highertransition density and a larger energy density jump. Thisis simply because the resulting very soft QM EOS can’tsupport the NSs considered if the hadron-quark transi-tion happens at too low densities. Consequently, onlyvery small QM fractions are allowed in the hybrid NSs.Quantitatively, we find that with c /c =1/3 the f massQM has a value of only 2.3%, 2.3%, 2.3% and 2.8% for1.4, 1.6, 1.8 and 2.0 M ⊙ NS, respectively. While with c /c =1 the f massQM almost remains the same as in the default calculation where the PDF of c /c peaks at c /c = 0 . +0 . − . at 68% confidence level as shown inFig. 2. D. Posterior probability distribution functions ofnuclear matter EOS parameters extracted with andwithout considering the hadron-quark phasetransition in neutron stars
Properties of NSs have been studied extensively usingvarious models with or without considering the hadron-quark phase transition in the literature for many years.Within the framework of the present work, it is thus in-teresting to study effects of considering the hadron-quarkphase transitions in NSs on extracting nuclear matterEOSs using astrophysical observables. Shown in Fig. 4are our results. Some interesting observations can bemade: • The incompressibility K of symmetric nuclearmatter and the symmetry energy E sym ( ρ ) at sat- P r obab ili t y d i s t r i bu t i on t / =1/3 c =1 P r obab ili t y d i s t r i bu t i on / t FIG. 3: (Color online) The posterior probability distributionfunctions of the hadron-quark matter transition density (up-per) and the energy density jump at the transition (lower)from Bayesian analyses by setting the quark matter speed ofsound squared c /c to 1/3 and 1, respectively. uration density ρ are not affected at all. In fact,their posterior PDFs are not much different fromtheir uniform prior PDFs. These are not surpris-ing and consistent with earlier findings. Whilethe most probable value of J characterizing thestiffness of symmetric nuclear matter at supra-saturation densities does not change, the hadron-quark phase transition requires more contributionsfrom larger J values as it generally softens the EOSunless the c /c is close to 1. • The L and K sym parameters together characterizethe density dependence of nuclear symmetry energyaround (1 − ρ . They are known to have signif-icant effects on the radii of canonical NSs in bothforward-modelings and Bayesian inferences, see,e.g., Ref [2], for a recent review. It is seen that theirposterior PDFs shift significantly to higher valuesespecially for L when the hadron-quark phase isconsidered. This can be well understood as thehadron-quark phase transition reduces significantlythe pressure above ρ t compared to the extension ofthe hadronic pressure into higher density regions. -800-600-400-200 0 200 400 J (MeV)
220 240 260 K (MeV) -200 0 200 400 600 800 with phase transition w/o phase transition prior J sym (MeV) -400 -300 -200 -100 0 100 K sym (MeV)
40 60 80
L(MeV)
28 30 32 34 36 E sym ( )(MeV) FIG. 4: (Color online)The posterior probability distributionfunctions of nuclear matter EOS parameters inferred fromBayesian analyses with and without considering the hadron-quark phase transition in neutron stars in comparison withtheir uniform priors.
To reproduce the same radius data, the contribu-tion to pressure from the symmetry energy in the(1 − ρ density region has to increase. Thus, theL and K sym parameters have to be higher. Sincethe J sym characterizes the symmetry energy at den-sities above about (2 − ρ [35], with the PDF of ρ t /ρ peaks at 1 . +1 . − . and all the QM EOS pa-rameters are isospin-independent, the analysis con-sidering the hadron-quark phase transition is thusnot sensitive to what one uses for the J sym . Conse-quently, the posterior PDF of J sym is almost identi-cal to its prior PDF. Therefore, the high-density be-havior of nuclear symmetry energy extracted fromNS properties does depend on whether one con-siders the hadron-quark phase transition or not.Moreover, the nuclear symmetry energy loses itsphysical meaning above the hadron-quark transi-tion density. • While the most probable values of L and K sym ex-tracted from the astrophysical data with and with-out considering the hadron-quark phase transitionare significantly different, they are unfortunately allconsistent with currently known theoretical predic-tions and findings from terrestrial nuclear experi-ments [2, 52]. Moreover, to our best knowledge,there is currently no terrestrial experimental con-straint on the J sym at all. Thus, the available con-straints on the nuclear EOS from terrestrial nuclearlaboratory experiments do not provide any addi-tional preference on whether QM exists or not inNSs. IV. SUMMARY AND CONCLUSIONS
In summary, within the Bayesian statistical frame-work using generic EOS parameterizations for boththe hadronic and quark matter connected through theMaxwell construction we inferred the PDFs of EOS pa-rameters as well as QM fraction and its size from NSradius data from several recent observations. We foundthat the available astrophysical data and all known EOSconstraints prefer the formation of a large volume of QMeven in canonical NSs. Future Bayesian inferences using unified EOS models describing both NSs and heavy-ionreactions with possible phase transitions from combinedmultimessenger data from both fields will significantlyimprove our knowledge about the EOS of super-denseneutron-rich matter.
V. ACKNOWLEDGMENTS
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