Bayesian analysis of multimessenger M-R data with interpolated hybrid EoS
A. Ayriyan, D. Blaschke, A. G. Grunfeld, D. Alvarez-Castillo, H. Grigorian, V. Abgaryan
EEPJ manuscript No. (will be inserted by the editor)
Bayesian analysis of multimessenger M-R data with interpolatedhybrid EoS
A. Ayriyan , , D. Blaschke , , , A. G. Grunfeld , , D. Alvarez-Castillo , , H. Grigorian , , , and V. Abgaryan , , Laboratory of Information Technologies, JINR, 6 Joliot-Curie St, Dubna 141980, Russian Federation IT and Computing Division, A. Alikhanyan National Laboratory, 2 Alikhanian Brothers Street, Yerevan, 0036, Armenia Institute of Theoretical Physics, University of Wroclaw, 9 M. Borna Sq, Wroclaw, 50-204, Poland Bogoliubov Laboratory of Theoretical Physics, JINR, 6 Joliot-Curie St, Dubna 141980, Russian Federation National Research Nuclear University (MEPhI), 31 Kashirskoe Hwy, Moscow, 115409, Russian Federation CONICET, Godoy Cruz 2290, Buenos Aires, Argentina Departamento de F´ısica, Comisi´on Nacional de Energ´ıa At´omica, Av. Libertador 8250, (1429) Buenos Aires, Argentina Henryk Niewodnicza´nski Institute of Nuclear Physics, 152 Radzikowskiego St, Cracow, 31-342, Poland Department of Physics, Yerevan State University, 1 Alex Manoogian St, Yerevan, 0025, Armenia Theoretical Physics Division, A. Alikhanyan National Laboratory, 2 Alikhanian Brothers Street, Yerevan, 0036, Armenia Peoples’ Friendship University of Russia (RUDN University), 6 Miklukho-Maklaya St, Moscow, 117198, Russian FederationReceived: date / Revised version: date
Abstract.
We introduce a family of equations of state (EoS) for hybrid neutron star (NS) matter that isobtained by a two-zone parabolic interpolation between a soft hadronic EoS at low densities and a set of stiffquark matter EoS at high densities within a finite region of chemical potentials µ H < µ < µ Q . Fixing thehadronic EoS as the APR one and chosing the color-superconductiong, nonlocal NJL model with two freeparameters for the quark phase, we perform Bayesian analyses with this two-parameter family of hybridEoS. Using three different sets of observational constraints that include the mass of PSR J0740+6620,the tidal deformability for GW170817 and the mass-radius relation for PSR J0030+0451 from NICER asobligatory (set 1), while set 2 uses the possible upper limit on the maximum mass from GW170817 asadditional constraint and set 3 instead the possibility that the lighter object in the asymmetric binarymerger GW190814 is a neutron star. We confirm that in any case the quark matter phase has to becolor superconducting with the dimensionless diquark coupling approximately fulfilling the Fierz relation η D = 0 .
75 and the most probable solutions exhibiting a proportionality between η D and η V , the couplingof the repulsive vector interaction that is required for a sufficiently large maximum mass. We anticipatethe outcome of the NICER radius measurement on PSR J0740+6220 as a fictitious constraint and findevidence for claiming that GW190814 was a binary black hole merger if the radius will be 11 km or less. PACS.
The observation of the first binary neutron star mergerGW170817 in gravitational waves [1] and the subsequentelectromagnetic signals from the gamma-ray burst to thelight curve of the kilonova [2] have opened the era of multi-messenger astronomy. This extends the available mass rangefor neutron star observations up to 2 . M (cid:12) for the com-panion star of the 23 M (cid:12) black hole in the binary mergerGW190814 [3], if that object was indeed the heaviest neu-tron star and not the lightest black hole, which is a cur-rently disputed question. The observation of gravitationalwaves from the inspiral phase of the merger GW170817did allow to extract for the first time a new constraint onthe equation of state (EoS) of dense matter, the tidal de- formability, to be in the range of 70 < Λ . <
580 [4] for aneutron star with the mass of 1 . M (cid:12) . From this measure-ment, together with other constraints, the authors of [5]could constrain the radius of a neutron star in that massrange to the rather narrow limits of R . = 11 . +0 . − . km.An open and controversially discussed question is the in-terior composition of neutron stars, in dependence of theirmass.It is very likely that the quark substructure of nucleonsmanifests itself at increasing densities first by a stiffeningof the EoS due to quark Pauli blocking in nuclear mat-ter [6] and at still higher densities by a delocalization ofthe quark wave function and the occurrence of deconfinedquark matter. For a recent discussion of soft delocalizationvs. hard deconfinement in the transition from nuclear to a r X i v : . [ a s t r o - ph . H E ] F e b A. Ayriyan et al.: Bayesian analysis of multimessenger M-R data with interpolated hybrid EoS quark matter, see [7]. A crucial open question, to whichthe present work intends to contribute, concerns the onsetmass of deconfinement and the character of the transition[8]. A standard approach to the hadron-to-quark-mattertransition would start from separate EoS models for thesetwo phases and obtain the phase transition from a Maxwellconstruction (for sufficiently large surface tension betweenthese phases) or a Glendenning construction of a homoge-neous mixed phase (for vanishing surface tension) [9]. In-between these limiting cases, the more realistic scenario ofthe first-order phase transition would consider structuresof finite size formed by the balance between Coulomb in-teractions and surface tension (pasta phases), see [10] andreferences therein. This approach has been used recentlyfor a Bayesian analysis with observational constraints formasses and radii of neutron stars [11] which reaches theconclusion that very likely the phase transition onset oc-curs in the center of neutron stars with masses around1 M (cid:12) and would then match the observed compactness[5] in this way. For this scenario to work, it is customaryto have a sufficient stiffness of nuclear matter at supersat-uration densities so that the deconfinement transition isdriven to relatively low densities. m [MeV]020406080100120140160 P [ M e V / f m ] sun sun sun sun APR m H = 991 MeV h V = 0.12 h V = 0.14 h V = 0.16 h V = 0.18 h V = 0.20 h D = 0.71, 0.79 "terra incognita" A P R a pp li c a b l e "terra incognita" n l N J L m o d e l a pp li c a b l e Fig. 1.
Pressure vs. chemical potential for the nlNJL EoS withdifferent values of η V = 0 . . .
20 and the two limitingcases of η D = 0 .
71 and η D = 0 .
79 is compared to that of theAPR EoS. The point µ H corresponds to n H = 1 . n . Thevalues of central pressure and chemical potential in neutronstars with 1 . M (cid:12) and 2 . M (cid:12) are indicated by labelled bluearrows, for orientation. In Fig. 1 we illustrate this situation. A soft hadronicEoS like that of Akmal, Pandharipande and Ravenhall(APR) [12] has either no crossing (Maxwell construction)with the color superconducting quark matter EoS (con-sidering a nonlocal version of the NJL model, nlNJL, de-scribed below) for the weak diquark coupling strength( η D = 0 .
71) or, at slightly increased dimensionless diquarkcoupling ( η D = 0 .
79) an unrealistically early transition that is followed by a ”reconfinement” or there is again notransition, depending on the value of the dimensionlessvector meson coupling η V . m [MeV]020406080100120140160 P [ M e V / f m ] "terra incognita" n l N J L m o d e l a pp li c a b l e A P R a pp li c a b l e Quark EoS: nlNJL( h =0.12)Quark EoS: nlNJL( h =0.16)Quark EoS: nlNJL( h =0.20)B( m ), set 2Hadronic EoS: APRAPR with excluded volumenlNJL( h =0.16) - B( m )mixed phase = interpolation m H = 991 MeV h D =0.75 Fig. 2.
Pressure vs. chemical potential for the nlNJL EoS(orange lines) with η D = 0 .
75 and the three values η V =0 . . .
20 compared to that of the APR EoS (dashed blueline) shows that no reasonable Maxwell construction is pos-sible. When a nucleonic excluded volume is applied to APR(solid blue line) and a density-dependent bag pressure B ( µ )according to set 2 of Ref. [13] (black dotted line) to the quarkmatter EoS, a Maxwell transition point is obtained and amixed phase construction (green solid line) can be performedwhich would correspond to an interpolation between APR andnlNJL( η V = 0 . In Fig. 2 we show how such pathologies of the phasetransition construction (or its inapplicability) with toosimple EoS which are not suitable for such a construction,could be cured. A stiffening of the hadronic EoS, here re-alized by an excluded nucleon volume, leads already toa reasonable transition at not too low densities and tocircumvention of the reconfinement problem of a second(unphysical) crossing of hadronic and quark matter EoSat higher densities . The situation would still be improvedtowards a more realistic description when confining effectswould be included to the quark matter description, e.g., bya (density-dependent) bag pressure that resembles the ef-fect of a nonperturbative QCD vacuum surrounding colorcharges (quarks) and leads to their confinement in colorsinglet multiquark states (hadrons). We note that withoutsuch a negative pressure (and/or a confining force), by thelarger number of quark and gluon degrees of freedom onthe one hand and the larger masses of hadrons in the spec-trum on the other, the quark-gluon plasma phase wouldbe favorable over the hadronic matter phase at low tem-peratures and densities, see Fig. 1 and [17]. As we already For a discussion of the reconfinement problem see, e.g.,[14], for the related masquerade problem, see [15] and for theirsolution see, e.g., [16].. Ayriyan et al.: Bayesian analysis of multimessenger M-R data with interpolated hybrid EoS 3 noted above, within a first-order phase transition, the for-mation of structures such a bubbles, droplets, rods andplates (pasta phases) is likely, with their sizes defined byan interplay of surface tension and Coulomb interactioneffects.The resulting pressure (green curve in Fig. 2 looks as ifa direct interpolation between the hadronic and the quarkmatter EoS would have been performed and the under-lying three main microphysical ingredients (quark Pauliblocking, quark confinement and pasta structures in themixed phase) could be circumvented by a direct shortcutfrom the nuclear matter phase just above saturation den-sity to the quark matter phase which would then appearas a crossover-like EoS.Such crossovers have been invoked on physical groundsby symmetry arguments as a quark-hadron continuity inRefs. [18,19,20] and by the combined effects of chiral sym-metry breaking and diquark condensation intertwined bythe axial anomaly so that they result in a crossover atlow temperatures which eventually entails a second criti-cal endpoint in the QCD phase diagram [21,22,23]. Thecrossover behaviour has subsequently been realized in ef-fective interpolating constructions following [24,25,26,27]and further literature in this direction.In this paper, we will perform a Bayesian analysisstudy with modern mass and radius constraints, as wellas fictitious radius measurements, on the basis of a new,two-zone interpolation construction for obtaining hybridEoS that is described in the next section.
The idea is to interpolate between hadron and quark EoSmodels from trustable region of hadron EoS n H = 1 . ∼ n of nuclear saturation density up to trustable region ofquark model from n Q ∼ n (see fig. 1). Before we outlinein detail the new interpolation method in subsection 2.3,we specify in the following two subsections the hadronicand the quark matter EoS that we employ in the presentstudy to describe the pure phases outside the region of”terry incognita” indicated in Figs. 1 and 2. Our choice of hadronic equation of state for this work isthe well known APR model [12]. It is a non-relativisticmodel derived by means of variational chain summationmethods which included Urbana potentials of two andthree nucleon interactions and features a pion condensate.Moreover, it exhibits a causality breach in neutron starmatter for massive stars, a problem that shall not appearfor the hybrid star models build in this work. The APREoS version we have chosen is A18 + δ v+UIX ∗ which isnot extremely stiff, reaching the maximum neutron starmass right below 2M (cid:12) .In addition, in order to complete the description of theneutron star matter EoS, we adjoin a low density region EoS corresponding to the crust of neutron stars, namelythe SLy4 model [28]. For the description of the quark matter phase we considera nonlocal chiral quark model, as in Ref. [13], which in-cludes scalar quark-antiquark interaction, anti-triplet scalardiquark interactions and vector quark-antiquark interac-tions. The grand canonical thermodynamic potential perunit volume at zero temperature and finite density in themean field approximation (MFA) reads Ω MFA = ¯ σ G S + ¯ ∆ H − ¯ ω G V − (cid:90) d p (2 π ) ln det (cid:2) S − (¯ σ, ¯ ∆, ¯ ω, µ fc ) (cid:3) , (1)see Ref. [29] for details of the calculation. The input pa-rameters of the model are determined as to reproducemeson properties in the vacuum, at vanishing tempera-ture and densities, then, m c , Λ (the cutoff) and G S canbe determined under that conditions. The remaining cou-pling constants G D and G V are driving the terms that,after bosonization, give rise to the superconducting gapfield and the vector field. Then, the ratios η D = G D /G S and η V = G V /G S are input parameters. For OGE in-teractions in the vacuum, Fierz transformation leads to η D = 3 / η V = 1 /
2. As the microscopic interactionis not derived directly from QCD then, the above cou-pling ratios have in principle no strong phenomenologi-cal constraint except for the fact that η D values largerthan η ∗ D = (3 / m/ ( m − m c ) may lead to color symme-try breaking in the vacuum [30] (where m stands for thedressed mass and m c for the current quark mass). In thepresent work we consider η D and η V as free parameters tobe varied in reasonable limits, as it has been done for theNJL model case in Ref. [31]. The mean field values ¯ σ , ¯ ∆ and ¯ ω satisfy the coupled equations dΩ MFA d ¯ ∆ = 0 , dΩ MFA d ¯ σ = 0 , dΩ MFA d ¯ ω = 0 . (2)As we are focused on describing the behaviour of quarkmatter in the core of NSs, we have to impose: equilib-rium under weak interactions, chemical equilibrium, andcolor and electric charge neutrality. Then, the six differentchemical potentials µ fc in Eqn. (1) (depending on the toquark flavors u and d and quark colors r, g and b ), can bewritten in terms of three independent quantities: the bary-onic chemical potential µ , the electron chemical potential µ e and a color chemical potential µ . So basically, for eachvalue of µ we self-consistently solve the gap equations (2),complemented with the conditions for β -equilibrium andelectric charge and color charge neutrality (details of thecalculation can be found in the Appendix of Ref. [29]).In the present work, we consider a Gaussian form fac-tor g ( p ) = exp (cid:0) − p /p (cid:1) in Euclidean 4-momentum space.The fixed input parameters of the quark model are m c = A. Ayriyan et al.: Bayesian analysis of multimessenger M-R data with interpolated hybrid EoS . p = 782 .
16 MeV and G S p = 19 . η D and η V as free parameters. The idea of interpolation is between n h and n q is to usetwo parabolic functions: (cid:40) P η ( µ ) = a η ( µ − µ H ) + b η ( µ − µ H ) + c η µ ≤ µ c P ρ ( µ ) = a ρ ( µ − µ Q ) + b ρ ( µ − µ Q ) + c ρ µ ≥ µ c (3)where µ H and µ Q correspond to n H and n Q respectively,and µ c is free parameter taking value between them: µ H <µ c < µ Q .Four parameters b η , b ρ , c η and c ρ can be immediatelydefined from the following conditions: P η ( µ H ) = P H ( µ H ) ⇒ c η = P H ( µ H ) n η ( µ H ) = n H ( µ H ) ⇒ b η = n H ( µ H ) P ρ ( µ Q ) = P Q ( µ Q ) ⇒ c ρ = P Q ( µ Q ) n ρ ( µ Q ) = n Q ( µ Q ) ⇒ b ρ = n Q ( µ Q ) (4)The parameters a η , a ρ will be defined by the idea thatat µ c both functions should be sewed (cid:40) P η ( µ c ) = P ρ ( µ c ) n η ( µ c ) = n ρ ( µ c ) (5)These conditions produce the following System of LinearAlgebraic Equations (SLAE): a η ( µ c − µ H ) − a ρ ( µ c − µ Q ) = κ a η ( µ c − µ H ) − a ρ ( µ c − µ Q ) = κ (6)where κ = n Q ( µ c − µ Q ) − n H ( µ c − µ H ) + P Q − P H ,κ = n Q − n H (7)The determinant of this SLAE ∆ = 2( µ c − µ Q )( µ c − µ H )( µ H − µ Q ) (8)shows that the system has solution always when µ c (cid:54) = µ Q , µ c (cid:54) = µ H and µ H (cid:54) = µ Q .The solution to the SLAE is a η = − κ + κ ( µ c − µ Q )2( µ c − µ H )( µ H − µ Q ) a ρ = − κ + κ ( µ c − µ H )2( µ c − µ Q )( µ H − µ Q ) (9)We note, that this two-zone interpolation allows for ageneralization to describe a first-order phase transition.This could be achieved by adding a jump in the density ∆n ( µ c ) as additional parameter to the second equation ofthe system (5). The continuous interpolation discussed inthis work is then the limiting case for ∆n ( µ c ) → m H m * m Q chemical potential m p r e ss u r e P Hadrons (trustable)Hadrons ( not trustable)Quarks (trustable)Quarks ( not trustable)Interpolation (hadron-like)Interpolation (quark-like) m H m * m Q chemical potential m p r e ss u r e P Hadrons (trustable part)Hadrons ( not trustable part)Quarks (trustable part)Quarks ( not trustable part)Interpolation (hadron-like)Interpolation (quark-like)
Fig. 3.
A hybrid equation of state that joins a nuclear equationof state with a quark matter equation of state by interpolationin the intermediate region between µ H and µ Q for n ( µ H ) = n H ∼ n and n ( µ Q ) = n Q ∼ (4 . . . n . The dotted curvesindicate where the extrapolations of the nuclear and quarkmatter equations of state become unreliable. We can treat thecase where these extrapolations cross each other (upper panel)as well as the case when they do not cross (lower panel). Due to a backbending of the given quark curves on P - ε plot a causality violation at high energy densities appears.In order to avoid such a problem, we employ an extrap-olation with a constant-speed-of-sound (CSS) EoS. Theformalism of the extrapolation is following: P ( µ ) = P + P (cid:18) µµ x (cid:19) β for µ > µ x ε ( µ ) = − P + P ( β − (cid:18) µµ x (cid:19) β for µ > µ x n B ( µ ) = P βµ x (cid:18) µµ x (cid:19) β − for µ > µ x (10) . Ayriyan et al.: Bayesian analysis of multimessenger M-R data with interpolated hybrid EoS 5 Here µ x is the chemical potential where the matching tothe CSS extrapolation starts. The parameter β is directlyrelated to the squared speed of sound c s = ∂P/∂µ∂ε/∂µ = 1 β − β = 1 + 1 c s . (12)It is obvious that fulfilling the causality constraint c s ≤ β ≥
2. The coefficients P and P in Eq. (10)are defined as P = [( β − P x − ε x ] /β (13) P = ( P x + ε x ) /β , (14)where P x = P ( µ x ) and ε x = ε ( µ x ).The results below have been obtained for the choice β = 2, i.e. c s = 1, with µ x being the chemical potentialat which for first time the squared speed of sound of thequark matter EoS reached the causality limit, c s ( µ x ) = 1.For those EoS which have no causality violation the CSSextrapolation has been performed with c s correspondingto the value at end of the given table. The interpolation method has been implemented for APRand NJL models with different η D = 0 . . .
79 and η V = 0 . . .
20 (see Figs. 4–8). The onset densityfor the interpolation has been set to n H = 2 n , and theand the density where the interpolation matches the quarkmatter EoS has been varied depending on η V as n Q =5 . . . n while simultaneously η V was incremented.The value of µ c has been fixed as µ c = µ H +0 . µ Q − µ H ). m [MeV] P [ M e V / f m ] APR h V = 0.06 h V = 0.08 h V = 0.10 h V = 0.12 h V = 0.14 h V = 0.16 h V = 0.18 h V = 0.20 h D = 0.71, 0.73, 0.75, 0.77, 0.79 Fig. 4.
Two-zone interpolation construction between APR andnlNJL on P - µ plot. e [MeV/fm ] P [ M e V / f m ] Fig. 5.
Two-zone interpolation construction between APR andnlNJL on P - ε plot. n B / n e [ M e V / f m ] Fig. 6.
Energy density dependence on the baryon density forthe EoS curves under consideration.
Neutron stars are computed within the framework of gen-eral relativity by using the corresponding EoS in the formof p ( ε ) to solve the Tolman-Oppenheimer-Volkoff (TOV)equations [32,33]d p ( r )d r = − ( ε ( r ) + p ( r )) (cid:0) m ( r ) + 4 πr p ( r ) (cid:1) r ( r − m ( r )) , (15)d m ( r )d r = 4 πr ε ( r ) . (16)which describe a static, spherical star. Radial mass m ( r ),energy density ε ( r ) and pressure p ( r ) stellar internal pro-files help to determined the mass M and radius R ofa star with central density ε c with boundary conditions m ( r = 0) = 0 and p ( r = R ) = 0. In Fig. 9 and Fig. 10 we A. Ayriyan et al.: Bayesian analysis of multimessenger M-R data with interpolated hybrid EoS e [MeV/fm ] c S Fig. 7.
Speed of sound vs energy density for the two-zoneinterpolation construction between APR and nlNJL. Legendas in Fig. 4. n B [n ] c S Fig. 8.
Speed of sound vs. density for the two-zone interpo-lation construction between APR and the set of nlNJL quarkmatter EoS from Fig. 5. Legend as in Fig. 4. show the compact star mass as a function of the centralenergy density and baryon density, respectively, for thetwo-zone interpolation construction. The dashed blue linecorresponds to the hadronic APR EoS and the remain-ing curves show the hybrid EoS for a range of input pa-rameters for the quark matter phase (same patters/colorsas in previous figures). The compact star mass-radius se-quence is a benchmark for every EoS model commonlypresented in a mass-radius diagram that includes pulsarmeasurement regions as well as excluded ones by other as-trophysical observations, see for instance Fig. 11. The EoSsequences displayed in there are obtained by a systematicintegration of the the TOV equations for increasing p c foreach single star up to the value of the maximum mass forwhich the condition ∂M/∂ε c > e [MeV/fm ] M [ M O . ] Fig. 9.
Mass vs. central energy density for the two-zone in-terpolation construction between APR and the set of nlNJLquark matter EoS. n B [n ] M [ M O . ] Fig. 10.
Mass vs. central baryon density for the two-zone in-terpolation construction between APR and the set of nlNJLquark matter EoS.
In addition, deformations of the compact star is a fea-ture of each EoS closed related to the physic process ofcompact stars mergers. They are quantified by computingthe tidal deformability Λ , for which estimated regions werederived by the observation of the GW170817 event [1,4],also displayed in Fig. 11. The corresponding equations arederived from perturbations of the spherical metric of thecompact star supplemented with the stellar internal pro-files for physical quantities derived from the TOV equa-tions. The equation Λ = 23 R M k (17)relates the dimensionless tidal deformability Λ with theLove number k and the total mass and radius of the . Ayriyan et al.: Bayesian analysis of multimessenger M-R data with interpolated hybrid EoS 7 Radius [km] M a ss [ M O . ] APR h V = 0.06 h V = 0.08 h V = 0.10 h V = 0.12 h V = 0.14 h V = 0.16 h V = 0.18 h V = 0.20 h D = 0.71, 0.73, 0.75, 0.77, 0.79 Fig. 11.
Mass-radius relations for the two-zone interpolationconstruction between APR and nlNJL. star. Details on the derivation of the above formula canbe found in [34,35,36,37,38].
In this section we introduce the Bayesian methodologyand its application to the set of considered EoS character-ized by the parameters η D and η V in order to find theirbest values that fulfill observational data.The a posteriori probability P ( π q | E ) is a conditionalprobability of the given vector of parameters π q (intro-duced below), where q denotes the indexes the values ofparameters for each alliterative representation of the modelof EoS. The condition E is the set of the observational data(events), its likelihood for the given model is representedas a product P ( E | π q ) = (cid:89) α P ( E α | π q ) , (18)which is the the conjunction of all events E α (where α is an index of an event). The a posteriori probabilities,likelihoods and a priori probabilities are connected to eachother via the Bayes formula P ( π q | E ) = P ( E | π q ) P ( π q ) N − (cid:80) p =0 P ( E | π p ) P ( π p ) , (19)where the factor P ( π q ) is the prior of a given model.First, we define the set of values of the parameters η D and η V as H D = { . , . , . , . , . } and H V = { . , . , . , . , . , . , . , . } . M [ M O . ] P S R J + M i l l e r e t a l . A p J L GW190814GW170817M M GW170817 PSR J0740+6620
GW170817 excluded(Bauswein et al.)
GW170817 excluded (Rezzolla et al.)
GW170817 excluded (Annala et al.) h D = 0.71, 0.79 h V = APR0.060.080.100.120.140.160.180.20
Fig. 12.
Mass-radius relations for EoS models featuring aninterpolation scheme [8] between the low density APR modelfor hadronic matter and high density nlNJL quark matter. Afew compact star sequences are displayed for two fixed quarkmatter parameter values of η D with varying η V together withthe hadronic APR EoS. Different color regions correspond toeither pulsar measurements or forbidden regions that serveas constraints for the compact star EoS. The green band re-gion above 2M (cid:12) corresponds to the mass measurement of PSRJ0740+6620 [39], the blue ellipse corresponds to the massand radius measurement of PSR J0030+0451 by NICER [40]whereas the grey and light green regions correspond to the es-timates of the components of the binary system labeled as M and M of the GW170817 merger [4]. Red bands correspondto excluded regions derived from GW170817 observations byBauswein et al. [41], Annala et al. [42] and Rezzolla et al. [43].The black dashed horizontal lines are the upper and lower limitfor the mass 2 . +0 . − . M (cid:12) of the lighter component in the bi-nary merger event GW190814 [3]. Then, the vector of parameters π q will be an elementof the set H D × H V , π q ∈ { η D ( i ) , η V ( j ) | i = 0 ..N D − , j = 0 ..N V − } , (20)where q = iN V + j and N V = 8, N D = 5. Therefore, q = 0 ..N − N = 40 is the full number of modelrepresentations. For the choice of the uniform distributionof the prior we have P ( π q ) = 1 /N. In the next section, we discuss the specific astrophysicalconstraints that we will employ in our Bayesian analysis(BA).
A. Ayriyan et al.: Bayesian analysis of multimessenger M-R data with interpolated hybrid EoS
Mass [M O. ] L GW170817 (Low-spin prior)
Fig. 13.
Dimensionless tidal deformabilities of hybrid com-pact stars together with the corresponding measurement fromGW170817 [4]. L L
90 %50 %
Fig. 14.
Tidal deformabilities diagram. Λ and Λ correspondto the dimensionless tidal deformability for each of the compo-nents of the binary in GW170817. Light and dark green regionscorrespond to the 50% and 90% credibility regions for the pos-teriors used in the LIGO-Virgo analysis [4]. All the EoS in thiswork result in curves that fall inside the lighter region as wellas the APR EoS which is displayed in blue. Recently, the mass of the PSR J0740+6620 is estimatedby combining data from the North American NanohertzObservatory for Gravitational Waves (NANOGrav) anddata from the orbital-phase-specific observations using theGreen Bank Telescope. The 68.3% of the credibility inter-val is given as 2 . +0 . − . M (cid:12) in [39]. This value has beenchosen as the lower limit of maximum mass ( M lowmax ). The likelihood for the lower limit of maximum massconstraint is given by P ( E M | π q ) = Φ ( M q , µ l , σ l ) , (21)where M q is the maximum mass of the sequence of neutronstar configurations for the given π q , and Φ ( M, µ, σ ) is the cumulative distribution function (CDF) of the standardnormal distribution. And µ l and σ l are the parameters ofthe uncertainty of a low limit maximum mass estimation.Additionally, the assumption that one of the compo-nent of the binary merger GW190814 [44] is a neutronstar gives an estimation of the lower limit of the maxi-mum mass as 2 . +0 . − . M (cid:12) [3]. This value has been usedin the Bayesian Inference as an alternative scenario forthe lower limit. In Fig. 12 we display the mass-radius rela-tions for our hybrid configurations together with the APRmodel as the low density baseline for hadronic matter thatbecomes invalid at higher densities where it crosses overto the nlNJL quark matter model. The different coloredregions correspond to either mass and/or radius measure-ments or to forbidden regions following from GW170817phenomenology that serve as constraints for the compactstar EoS. The horizontal black dashed lines show the massrange for the lighter object in the binary merger GW190814,that we employ as a possible lower limit on the maximummass, in the case that this object was a neutron star. There is an estimation of the upper limit of maximummass of neutron star in the litrature [43]. It was estimatedwith combination of the observation of gravitational waves(GW170817) and drawing from basic arguments on kilo-nova modeling of GRB 170817A, together with the quasi-universal relation between the maximum masses of staticneutron stars and the fastest stable star under uniformrotation [46]. The upper limit of the maximum mass is2 . +0 . − . M (cid:12) as shown in Fig. 12.The likelihood for the upper limit of maximum massconstraint is given by P ( E M | π q ) = 1 − Φ ( M q , µ u , σ u ) , (22)where M q is the maximum mass of the sequence of neutronstar configurations for the given π q , and µ u = 2 . M (cid:12) and σ u = 0 . M (cid:12) .However, in Ref. [45] a relationship between the onsetmass of prompt collapse to a black hole, the tidal deforma-bility at half this mass and the maximum TOV mass hasbeen derived, according to which the fact that the mergerGW170817 did not promptly collapse to a black hole im-plies a lower limit on the maximum TOV mass. Therefore,we will include the disputable maximum mass constraintof Ref. [43] only in one of the sets of our Bayesian analysis. The observation of the gravitational waves from binaryNS-NS merger GW170817 allows to calculate relation be-tween tidal deformabilities of the primary and secondary . Ayriyan et al.: Bayesian analysis of multimessenger M-R data with interpolated hybrid EoS 9 components [1,4]. In order to implement the tidal de-formability constraint to Bayesian Inference the Gaussiankernel density estimation has been used to recover theprobability distribution function with use of the data onthe Λ − Λ publicly shared by LIGO collaboration .The likelihood of the gravitational wave constraint isintroduced as P ( E GW | π q ) = (cid:90) l β ( Λ ( n c ) , Λ ( n c )) d n c , (23)where l is the length of the line on the Λ – Λ diagramproduced by −→ π q , and n c is the central baryon density ofa star. β ( Λ , Λ ) is the probability distribution function (PDF) that has been reconstructed (as previously donein [47,48]). In Fig. 13 we show the dimensionless tidaldeformabilities of hybrid com-pact stars configurations.The line colors/patters are the same as in previous fig-ures. We display as well the corresponding measurementfrom GW170817.Fig. 14 shows our results for the tidal deformabilities Λ asa function of Λ . Dark and light green regions correspondto the 50% and 90% credibility for the posteriors used inthe LIGO-Virgo Collaboration analysis, respectively. Allthe hybrid EoS in this work result in curves that fall in-side the 90% region as the APR EoS which is displayed asdashed blue line. Simulations estimation of a mass and a radius of the pul-sar PSR J0030+0451 has been done with use of the datacollected from the Neutron Star Interior Composition Ex-plorer (NICER) space observatory. The results of obser-vation have been reported in a collection of publications .There were two estimates of the mass and equato-rial radius based on mutually exclusive assumptions aboutthe uniform-temperature emitting spots. The first radiusand mass estimates are M = 1 . +0 . − . M (cid:12) and R =13 . +1 . − . km [49] whereas the second estimates are M =1 . +0 . − . M (cid:12) and R = 12 . +1 . − . km [50]. A bivariateprobability distribution function α ( M, R ) has been recon-structed by the method of Gaussian the Gaussian kerneldensity estimation using the data [51]. A likelihood is for-mulated as P ( E GW | π q ) = (cid:90) l α ( M ( n c ) , R ( n c )) d n c , (24)where M ( n c ) and R ( n c ) are mass and radius of sequenceof neutron star for a given q th equation of state, and n c isthe central baryon density. The above constraints can beseen in Fig. 12. https://dcc.ligo.org/LIGO-P1800115/public Z. Arzoumanian & K. C. Gendreau.
Focus on NICER Con-straints on the Dense Matter Equation of State , ApJ 887, 2019
An essential input for the BA would be precise radiusmeasurements for pulsars with known masses. One exam-ple is the millisecond pulsar PSR J0740+6620 for whichthe mass is known from Ref. [39] and for which the analy-sis of data taken by the NICER experiment shall result ina radius measurement that ideally would have the designprecision of 0 . R = 11, 12 and 13 kmwith the the same σ = 0 . We suggest three sets of constraints: the mass measurement for PSR J0740+6620 [39] asthe lower limit for the maximum mass, the tidal de-formability from GW170817 [4] and the mass-radiusconstraint from PSR J0030+0451 [49] (set 1); in addition to the constraints of set 1, the constrainton the upper limit of the maximum mass from Ref. [43]is included; as for set 1, but assuming that the lower mass compan-ion of the black hole in the asymmetric binary mergerGW190814 [3] was a neutron star, the lower limit forthe maximum mass is replaced by the lower limit onits mass M = 2 . +0 . − . M (cid:12) .Besides these ”pure” sets of constraints, we investigatefor each of them the possibility of an additional mass-radius constraint for the pulsar PSR J0740+6620, for whichthe mass 2 . +0 . − . M (cid:12) is rather precisely measured anda radius measurement by the NICER experiment is underway. We anticipate for this to be measured radius the val-ues of 11, 12 and 13 km and denote these subsets with thenumbers 1, 2 and 3, respectively. The entirety of Bayesianconstraint sets in this work is synoptically summarized inTab. 1. In this section we show and discuss our results for the BAof the hybrid EoS for compact stars under the constraintsdefined in the previous section and summarized in Tab.1. The details of the considered EoS, the interpolationmethod, the constant speed of sound extrapolation, theBA and astrophysical constraints were already presentedin previous sections.Figure 15 shows the results of our BA for the η V and η D values of our EoS models under consideration. Thedifference between the three LEGO is that the for theirderivation the values on the maximum mass constraint hasbeen changed. These three different cases comprise 1) themass measurement of the object PSR J0740+6620, [39], Set Maximum mass [ M (cid:12) ] M - R Λ - Λ Fictitious radius measurement Fig.Lower limit Upper limitPSR J0740+6620 GW190814 GW170817 J0030+0451 [49] GW170817 [4] on PSR J0740+6620 [km]2 . ± .
09 [39] 2 . ± .
09 [3] 2 . ± .
17 [43] 11 ± . ± . ± . (cid:88) (cid:88) (cid:88) (cid:88) (cid:88) (cid:88) (cid:88) (cid:88) (cid:88) (cid:88) (cid:88) (cid:88) (cid:88) (cid:88) (cid:88) (cid:88) (cid:88) (cid:88) (cid:88) (cid:88) (cid:88) (cid:88) (cid:88) (cid:88) (cid:88) (cid:88) (cid:88) (cid:88) (cid:88) (cid:88) (cid:88) (cid:88) (cid:88) (cid:88) (cid:88) (cid:88) (cid:88) (cid:88) (cid:88) (cid:88) (cid:88) (cid:88) (cid:88) (cid:88) (cid:88) (cid:88) (cid:88) (cid:88) (cid:88) Table 1.
Overview on Bayesian analysis constraints employed in the present work. The rightmost column refers to the setassociated figure number in the text and posterior probability LEGO plot position in it according to the following convention:columns are indicated by left (l), center (c), and right (r), while rows are labelled upper (u), middle (m), and bottom (b).
Fig. 15.
Bayesian analysis using the mass measurement for PSR J0740+6620 [39] as the lower limit for the maximum mass,the tidal deformability from GW170817 [4] and the mass-radius constraint from PSR J0030+0451 for the class of hybrid EoSobtained with a two-zone interpolation between APR and nlNJL in the two-dimensional EoS parameter plane spanned by η V and η D (left panel; set 1). In the middle panel the constraint on the upper limit of the maximum mass from Ref. [43] has beenadded (set 2) and in the right panel this limit has been lifted again in favor of the new lower limit on the maximum mass fromlower-mass companion of the black hole in the asymmetric binary merger GW190814 [3] that replaces the one from [39] (set 3),if the former object would not be a black hole.
2) the previous measurement plus the upper limit of themaximum mass from Ref. [43], 3) the first measurementplus the mass estimation for GW190814 [3] assuming thethere was a compact star in the binary system. We cansee that whereas for the first to cases the posterior prob-ability distributions peak at intermediate values of the η V parameter of quark matter, the third case favours itshighest values. η D values remain somehow uniform for thethree cases.The situation becomes more interesting when we con-sider a new, fictitious mass radius measurement whichmay well correspond to a future NICER observation ofthe heavy pulsar PSR J0740+6620. In figure 16 each col-umn corresponds to the results for such a radius mea-surement with a value of R = 11 km, R = 12 km or R = 13 km where as the rows represent the three samethree aforementioned cases for the constraints. Just likebefore, case 3) displays a more selective effect on the η V parameter, however the inclusion of this new radius ob-servation favours the highest η D values for most of theplots.In addition, figures 17 and 18 present the equivalentcontour plots constraints for set 1 and set 2 respectively, aswell the Bayesian most probably EoS together with theirmass-radius curves and tidal deformabilities values for theGW170817 components. . Ayriyan et al.: Bayesian analysis of multimessenger M-R data with interpolated hybrid EoS 11 R →
11 km 12 km 13 km ↓ Constraints set 1 :inf { M max } [39], Λ . [4],( M, R ) J0030+0451 [49] set 2 :inf { M max } [39], Λ . [4],( M, R ) J0030+0451 [49]sup { M max } [43] set 3 :inf { M max } [3], Λ . [4],( M, R ) J0030+0451 [49]
Fig. 16.
Probabilities when an additional, yet fictitious, measurement of the radius R of PSR J0740+6220 as expected fromNICER is taken into account with R = 11 km (second column), 12 km (third column) or 13 km (fourth column) with a standarddeviation of σ R = 0 . R = 11 km set 1 + R = 12 km set 1 + R = 13 km η D η V η D η V η D η V η D η V Radius [km] M a ss [ M O . ] h D = 0.75, h V = 0.14 h D = 0.79, h V = 0.14 h D = 0.77, h V = 0.14 h D = 0.77, h V = 0.16 h D = 0.73, h V = 0.14 h D = 0.73, h V = 0.12 h D = 0.71, h V = 0.12 h D = 0.79, h V = 0.16 h D = 0.75, h V = 0.12 h D = 0.71, h V = 0.14 h D = 0.79, h V = 0.18 h D = 0.75, h V = 0.16 h D = 0.73, h V = 0.16 h D = 0.77, h V = 0.12 h D = 0.75, h V = 0.18APR Radius [km] M a ss [ M O . ] h D = 0.79, h V = 0.12 h D = 0.77, h V = 0.12APR Radius [km] M a ss [ M O . ] h D = 0.71, h V = 0.10 h D = 0.75, h V = 0.12 h D = 0.73, h V = 0.12 h D = 0.77, h V = 0.12 h D = 0.73, h V = 0.10APR Radius [km] M a ss [ M O . ] h D = 0.77, h V = 0.16 h D = 0.79, h V = 0.18 h D = 0.75, h V = 0.14 h D = 0.71, h V = 0.14 h D = 0.73, h V = 0.14 h D = 0.73, h V = 0.16 h D = 0.75, h V = 0.16 h D = 0.79, h V = 0.16 h D = 0.75, h V = 0.18 h D = 0.71, h V = 0.12 h D = 0.77, h V = 0.14 h D = 0.73, h V = 0.12 h D = 0.71, h V = 0.16 h D = 0.79, h V = 0.14 h D = 0.77, h V = 0.18APR e [MeV/fm ] P [ M e V / f m ] h D = 0.75, h V = 0.14 h D = 0.79, h V = 0.14 h D = 0.77, h V = 0.14 h D = 0.77, h V = 0.16 h D = 0.73, h V = 0.14 h D = 0.73, h V = 0.12 h D = 0.71, h V = 0.12 h D = 0.79, h V = 0.16 h D = 0.75, h V = 0.12 h D = 0.71, h V = 0.14 h D = 0.79, h V = 0.18 h D = 0.75, h V = 0.16 h D = 0.73, h V = 0.16 h D = 0.77, h V = 0.12 h D = 0.75, h V = 0.18APR e [MeV/fm ] P [ M e V / f m ] h D = 0.79, h V = 0.12 h D = 0.77, h V = 0.12APR e [MeV/fm ] P [ M e V / f m ] h D = 0.71, h V = 0.10 h D = 0.75, h V = 0.12 h D = 0.73, h V = 0.12 h D = 0.77, h V = 0.12 h D = 0.73, h V = 0.10APR e [MeV/fm ] P [ M e V / f m ] h D = 0.77, h V = 0.16 h D = 0.79, h V = 0.18 h D = 0.75, h V = 0.14 h D = 0.71, h V = 0.14 h D = 0.73, h V = 0.14 h D = 0.73, h V = 0.16 h D = 0.75, h V = 0.16 h D = 0.79, h V = 0.16 h D = 0.75, h V = 0.18 h D = 0.71, h V = 0.12 h D = 0.77, h V = 0.14 h D = 0.73, h V = 0.12 h D = 0.71, h V = 0.16 h D = 0.79, h V = 0.14 h D = 0.77, h V = 0.18APR L L h D = 0.75, h V = 0.14 h D = 0.79, h V = 0.14 h D = 0.77, h V = 0.14 h D = 0.77, h V = 0.16 h D = 0.73, h V = 0.14 h D = 0.73, h V = 0.12 h D = 0.71, h V = 0.12 h D = 0.79, h V = 0.16 h D = 0.75, h V = 0.12 h D = 0.71, h V = 0.14 h D = 0.79, h V = 0.18 h D = 0.75, h V = 0.16 h D = 0.73, h V = 0.16 h D = 0.77, h V = 0.12 h D = 0.75, h V = 0.18APR L L h D = 0.79, h V = 0.12 h D = 0.77, h V = 0.12APR L L h D = 0.71, h V = 0.10 h D = 0.75, h V = 0.12 h D = 0.73, h V = 0.12 h D = 0.77, h V = 0.12 h D = 0.73, h V = 0.10APR L L h D = 0.77, h V = 0.16 h D = 0.79, h V = 0.18 h D = 0.75, h V = 0.14 h D = 0.71, h V = 0.14 h D = 0.73, h V = 0.14 h D = 0.73, h V = 0.16 h D = 0.75, h V = 0.16 h D = 0.79, h V = 0.16 h D = 0.75, h V = 0.18 h D = 0.71, h V = 0.12 h D = 0.77, h V = 0.14 h D = 0.73, h V = 0.12 h D = 0.71, h V = 0.16 h D = 0.79, h V = 0.14 h D = 0.77, h V = 0.18APR Fig. 17.
Results of the BA for set 1 which includes the constraints (inf { M max } [39], Λ . [4], ( M, R ) J0030+0451 and [49]) inthe leftmost column and with an additional (yet fictitious) NICER radius measurement for PSR J0740+6620 of R = 11, 12 or13 km with an estimated standard deviation of σ R = 0 . Λ − Λ (4th row) relationships correspond to the parameter sets with at least 75% ofthe maximum probability as shown in the first row. In the present work we have applied Bayesian methodsto investigate the most likely quark-hadron hybrid EoSamong a family of models that are agnostic about the de-tailed microphysical scenario of the hadron-to-quark mat-ter transition. The model uses the APR EoS for densitiesbelow 1 . n and above n = 3 . n a set of nonlocal,color superconducting chiral quark model parametriza-tions of the NJL type with a covariant Gaussian form-factor, whereby the vector meson coupling η V and thescalar diquark coupling η D are varied as free parametersthat determine the stiffness of high-density quark mat-ter. The transition between both regimes is constructedby a new two-zone interpolation that realizes a smooth crossover behaviour, due to assumption that the nature ofthe transition is a mixing of phases. Thus, in performingthis interpolation we exclude the possibility of a first-ordertransition associated with a jump in the density. However,the construction itself allows such a possibility. This ex-tension of the new two-zone interpolation construction weplan to systematically investigate in future work.In our Bayesian study we apply standard constraintsfor mass and radius measurements in set 1 and demon-strate their effect of narrowing the viable range of param-eter values. We obtain the result that the most probableEoS lie along a line of proportionality between η V and η D ,whereas the higher values of η V are favorable for obtain-ing larger maximum masses of hybrid neutron stars. Thisfinding confirms similar results of earlier studies in [52, . Ayriyan et al.: Bayesian analysis of multimessenger M-R data with interpolated hybrid EoS 13set 2 set 2 + R = 11 km set 2 + R = 12 km set 2 + R = 13 km η D η V η D η V η D η V η D η V Radius [km] M a ss [ M O . ] h D = 0.79, h V = 0.14 h D = 0.73, h V = 0.12 h D = 0.75, h V = 0.12 h D = 0.71, h V = 0.12 h D = 0.77, h V = 0.12 h D = 0.77, h V = 0.14 h D = 0.75, h V = 0.14 h D = 0.79, h V = 0.12APR Radius [km] M a ss [ M O . ] h D = 0.79, h V = 0.12 h D = 0.77, h V = 0.12APR Radius [km] M a ss [ M O . ] h D = 0.71, h V = 0.10 h D = 0.73, h V = 0.10 h D = 0.75, h V = 0.12 h D = 0.77, h V = 0.12 h D = 0.73, h V = 0.12APR Radius [km] M a ss [ M O . ] h D = 0.71, h V = 0.12 h D = 0.73, h V = 0.12 h D = 0.75, h V = 0.14 h D = 0.75, h V = 0.12 h D = 0.77, h V = 0.14 h D = 0.79, h V = 0.14 h D = 0.73, h V = 0.14 h D = 0.71, h V = 0.10APR e [MeV/fm ] P [ M e V / f m ] h D = 0.79, h V = 0.14 h D = 0.73, h V = 0.12 h D = 0.75, h V = 0.12 h D = 0.71, h V = 0.12 h D = 0.77, h V = 0.12 h D = 0.77, h V = 0.14 h D = 0.75, h V = 0.14 h D = 0.79, h V = 0.12APR e [MeV/fm ] P [ M e V / f m ] h D = 0.79, h V = 0.12 h D = 0.77, h V = 0.12APR e [MeV/fm ] P [ M e V / f m ] h D = 0.71, h V = 0.10 h D = 0.73, h V = 0.10 h D = 0.75, h V = 0.12 h D = 0.77, h V = 0.12 h D = 0.73, h V = 0.12APR e [MeV/fm ] P [ M e V / f m ] h D = 0.71, h V = 0.12 h D = 0.73, h V = 0.12 h D = 0.75, h V = 0.14 h D = 0.75, h V = 0.12 h D = 0.77, h V = 0.14 h D = 0.79, h V = 0.14 h D = 0.73, h V = 0.14 h D = 0.71, h V = 0.10APR L L h D = 0.79, h V = 0.14 h D = 0.73, h V = 0.12 h D = 0.75, h V = 0.12 h D = 0.71, h V = 0.12 h D = 0.77, h V = 0.12 h D = 0.77, h V = 0.14 h D = 0.75, h V = 0.14 h D = 0.79, h V = 0.12APR L L h D = 0.79, h V = 0.12 h D = 0.77, h V = 0.12APR
100 200 300 400 500 600 700 L L h D = 0.71, h V = 0.10 h D = 0.73, h V = 0.10 h D = 0.75, h V = 0.12 h D = 0.77, h V = 0.12 h D = 0.73, h V = 0.12APR L L h D = 0.71, h V = 0.12 h D = 0.73, h V = 0.12 h D = 0.75, h V = 0.14 h D = 0.75, h V = 0.12 h D = 0.77, h V = 0.14 h D = 0.79, h V = 0.14 h D = 0.73, h V = 0.14 h D = 0.71, h V = 0.10APR Fig. 18.
Results of the BA for set 2 which includes the constraints (inf { M max } [39], Λ . [4], ( M, R ) J0030+0451 [49], sup { M max } [43]) in the leftmost column and with an additional (yet fictitious) NICER radius measurement for PSR J0740+6620 of R = 11,12 or 13 km with an estimated standard deviation of σ R = 0 . Λ − Λ (4th row) relationships correspond to the parameter sets with at least75% of the maximum probability as shown in the first row.
31] and the recent work [23] that do not employ Bayesianmethods.In the sets 2 and 3 we explore the nonstandard con-straints of an upper limit on the maximum mass and thehigh mass of the lighter companion object of GW190814as a lower limit on the maximum mass, respectively. Whilefor the set 2 the narrowing of the parameter range due tothe upper limit on the maximum mass leads to an exclu-sion of the higher values of the vector meson coupling,the high value of the lower limit on the maximum massinstead allows only the highest possible vector couplingsresulting in stiffer EoS and thus larger maximum massesand radii. It is a remarkable fact that within the presentinterpolation approach the 2 . M (cid:12) companion object inGW190814 could be a hybrid star with quark matter core. Finally we have used the Bayesian approach to explorethe consequences of likely results for the ongoing radiusmeasurements on the 2 M (cid:12) pulsar PSR J0740+6620 bythe NICER experiment can have for neutron star phe-nomenology. An important observation is that in order toreach very massive compact stars the equation of stateshould feature a steep rise of the constant speed of soundreaching high values near the causality limit [53,54]. Thepossibility that GW190814 could have been a neutron star- black hole merger with the 2 . M (cid:12) object being a hybridstar and not a black hole has been pointed out in severalworks, among them [54,55].Should the NICER radius measurement on J0740+6620yield a value as small as 11 km or less, the hybrid star sce-nario for the 2 . M (cid:12) object in GW190814 is excluded and R = 11 km set 3 + R = 12 km set 3 + R = 13 km η D η V η D η V η D η V η D η V Radius [km] M a ss [ M O . ] h D = 0.71, h V = 0.2 h D = 0.73, h V = 0.2 h D = 0.77, h V = 0.2 h D = 0.75, h V = 0.2APR Radius [km] M a ss [ M O . ] h D = 0.79, h V = 0.18APR Radius [km] M a ss [ M O . ] h D = 0.77, h V = 0.2 h D = 0.79, h V = 0.2APR Radius [km] M a ss [ M O . ] h D = 0.77, h V = 0.2 h D = 0.73, h V = 0.2 h D = 0.71, h V = 0.2 h D = 0.75, h V = 0.2APR e [MeV/fm ] P [ M e V / f m ] h D = 0.71, h V = 0.2 h D = 0.73, h V = 0.2 h D = 0.77, h V = 0.2 h D = 0.79, h V = 0.2APR e [MeV/fm ] P [ M e V / f m ] h D = 0.79, h V = 0.18APR e [MeV/fm ] P [ M e V / f m ] h D = 0.77, h V = 0.2 h D = 0.79, h V = 0.2APR e [MeV/fm ] P [ M e V / f m ] h D = 0.77, h V = 0.2 h D = 0.73, h V = 0.2 h D = 0.71, h V = 0.2 h D = 0.75, h V = 0.2APR L L h D = 0.71, h V = 0.2 h D = 0.73, h V = 0.2 h D = 0.77, h V = 0.2 h D = 0.75, h V = 0.2APR L L h D = 0.79, h V = 0.18APR L L h D = 0.77, h V = 0.2 h D = 0.79, h V = 0.2APR L L h D = 0.77, h V = 0.2 h D = 0.73, h V = 0.2 h D = 0.71, h V = 0.2 h D = 0.75, h V = 0.2APR Fig. 19.
Results of the BA for set 3 of constraints (inf { M max } [3], Λ . [4], ( M, R ) J0030+0451 [49]) in the leftmost column andwith an additional (yet fictitious) NICER radius measurement for PSR J0740+6620 of R = 11, 12 or 13 km with an estimatedstandard deviation of σ R = 0 . Λ − Λ (4th row) relationships correspond to the parameter sets with at least 75% of the maximum probabilityas shown in the first row. this event was a binary black hole merger. In that case thepresent two-zone interpolation approach with continuouscrossover is not suitable since it does not produce stablehybrid stars with small enough radii above 2 M (cid:12) .As discussed, e.g., in [8] such a two-zone interpolationcould be used to parametrize a first-order phase transi-tion resulting in a density jump in the inner core andsufficiently compact high-mass hybrid star configurations.It has been demonstrated recently in Ref. [54] that fora NICER radius measurement on J0740+6620 resulting in10 km or less, the only possibility to explain the star struc-ture is a hybrid star scenario with a large quark mattercore, since there is no realistic hadronic EoS model thatcould explain the smallness of neutron stars with a massexceeding 2 M (cid:12) .Should the NICER radius measurement on J0740+6620yield a large value such as 13 km or more, the likely inter- pretation of GW190814 is that of a neutron star - blackhole merger where the neutron star possibly had a quarkmatter core. However, a purely hadronic interior could notbe excluded in that case.In future extensions of the Bayesian approach to neu-tron star phenomenology it is desirable to widen the classof EoS either by combining the results for the presentsmooth interpolation approach with those of a first-orderphase transition scenario [11] or to extend the presenttwo-zone interpolation method to capture also the strongphase transition case.In order to compare the results of two Bayesian ana-lyzes obtained on the same astrophysical data, it is nec-essary to combine the factors of the normalization of theBayesian formula (19). This allows to introduce the rela-tive posterior probability of each analysis. The set of EoSmodels, whose analysis gives a greater value of the rel- . Ayriyan et al.: Bayesian analysis of multimessenger M-R data with interpolated hybrid EoS 15 ative posterior probability can be considered to be moresuccessful in describing the observational data. A compar-ison of the result presented in this paper with the previousresult from Ref. [11], where constraints corresponding toset 1 have been employed, shows that the likelihood formodels considered in [11] exceeds that of the present workby more than 5 times. Such a direct comparison is possibleonly for the case of set 1, because the other two sets havenot been considered in [11].Furthermore, it is customary to base the Bayesian studynot only on one hadronic EoS like in the present work andin [11], but to systematically vary the high-density behav-ior of the hadronic phase, e.g., within an excluded nucleonvolume approach (see Ref. [56]) without a entering a de-confined quark matter phase in the neutron star interior.Only such a more complete EoS basis for the Bayesianstudy will allow to draw conclusions for the key questionwhether neutron stars in the observable mass range canharbor deconfined quark matter in their cores. Acknowledgements
We acknowledge the partial support by the COST Ac-tion CA16214 ”PHAROS” for our international network-ing activities in preparing this article. This work receivedsupport from the Russian Fund for Basic Research un-der grant no. 18-02-40137. The work by D.B. on the newclass of quark-hadron hybrid EoS in Section 4 was sup-ported by the Polish National Science Centre under grantnumber UMO 2019/33/B/ST9/03059. D.E.A-C. and H.G.are grateful for support from the programme for exchangebetween JINR Dubna and Polish Institutes (Bogoliubov-Infeld programme). A. G. G. would like to acknowledge toCONICET for financial support under Grant No. PIP17-700.
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