Proto-neutron stars with heavy baryons and universal relations
aa r X i v : . [ nu c l - t h ] A ug MNRAS , 1–20 (0000) Preprint 4 August 2020 Compiled using MNRAS L A TEX style file v3.0
Proto-neutron stars with heavy baryons and universalrelations
Adriana R. Raduta ⋆ , Micaela Oertel † and Armen Sedrakian , ‡ National Institute for Physics and Nuclear Engineering (IFIN-HH), RO-077125 Bucharest, Romania LUTH, Observatoire de Paris, Universit´e PSL, CNRS, Universit´e de Paris, 92195 Meudon, France Frankfurt Institute for Advanced Studies, D-60438 Frankfurt-Main, Germany Institute of Theoretical Physics, University of Wroc law, 50-204 Wroc law, Poland
ABSTRACT
We use covariant density functional theory to obtain the equation of state (EoS) ofmatter in compact stars at non-zero temperature, including the full baryon octet aswell as the ∆(1232) resonance states. Global properties of hot ∆-admixed hypernuclearstars are computed for fixed values of entropy per baryon (
S/A ) and lepton fraction( Y L ). Universal relations between the moment of inertia, quadrupole moment, tidaldeformability, and compactness of compact stars are established for fixed values of S/A and Y L that are analogous to those known for cold catalyzed compact stars. Wealso verify that the I -Love- Q relations hold at finite temperature for constant valuesof S/A and Y L . Cold, mature compact stars are well described by aone-parameter equation of state (EoS) relating pressureto (energy) density. In contrast, the studies of the dy-namics of core-collapse supernovae (CCSN) (Janka et al.2007; Mezzacappa et al. 2015; O’Connor & Couch 2018;Burrows et al. 2020), proto-neutron star (PNS) evolu-tion (Pons et al. 1999), stellar black-hole (BH) formation(Sumiyoshi et al. 2007; Fischer et al. 2009; O’Connor & Ott2011; Hempel et al. 2012) and binary neutron star(BNS) mergers (Shibata & Taniguchi 2011; Rosswog 2015;Baiotti & Rezzolla 2017; Ruiz et al. 2020) require as an in-put an EoS at non-zero temperature and out of (weak) β -equilibrium, i.e. , the pressure becomes a function of threethermodynamic parameters. For describing all of the abovementioned astrophysical scenarios one needs to considerbaryon number densities, n B , ranging from sub-saturationdensities up to several times the nuclear saturation density n s ≃ .
16 fm − , temperatures up to 100 MeV and chargefractions 0 ≤ Y Q = n Q /n B ≤ .
6, where n Q is defined asthe total hadronic charge density (Oertel et al. 2017).In recent years the EoS of cold compact stars hasbeen considerably constraint due to several new astrophysi-cal observations (compact star masses, radii and tidal de-formability), information coming from experimental nu-clear physics and the progress in ab-initio calculationsof pure neutron matter. The new data has, in particu-lar, narrowed the possible parameter space of EoS derivedfrom density functional theory (DFT). There exist a largenumber of DFT based EoS which are applicable to cold β -equilibrated neutron stars, see e.g. (Weissenborn et al.2012b,a; Colucci & Sedrakian 2013; van Dalen et al. 2014;Oertel et al. 2015; Chatterjee & Vidana 2016; Fortin et al. 2016; Chen et al. 2007; Drago et al. 2014; Cai et al. 2015;Zhu et al. 2016; Sahoo et al. 2018; Kolomeitsev et al. 2017;Li et al. 2018; Li & Sedrakian 2019b; Ribes et al. 2019;Li et al. 2020). The latter works address, in particular, theproblem of hyperonization of dense matter ( the hyperon puz-zle ) and the possible emergence of ∆-degrees of freedom indense matter at zero temperature. Models of finite temper-ature dense matter with heavy baryon degrees of freedom,applicable to CCSN and BNS mergers, which are consistentwith the constraints imposed by modern data are less nu-merous, see (Oertel et al. 2012; Colucci & Sedrakian 2013;Oertel et al. 2016; Marques et al. 2017; Dexheimer 2017;Fortin et al. 2018; Malfatti et al. 2019; Stone et al. 2019).The first aim of this work is to obtain an EoS of non-zero temperature matter, which is well-constrained by as-trophysics and laboratory data, on the basis of the DDME2parameterisation (Lalazissis et al. 2005) extended to hyper-nuclear matter by Fortin et al. (2016) . We include addi-tionally the ∆ degrees of freedom as they add an impor-tant feature: they soften the EoS in an intermediate den-sity range, reduce the compact star radii for intermediatemasses (Drago et al. 2014; Li et al. 2018) which improvesthe agreement between theoretical models and observations.The resulting finite-temperature EoS are relevant for the va-riety of above mentioned astrophysical scenarios, which areknown to depend sensitively on the EoS (Pons et al. 1999;Sekiguchi et al. 2011; Bauswein et al. 2012; Fischer et al.2014; Perego et al. 2019; Peres et al. 2013; Schneider et al. Similar extensions of the DDME2 parametrisation to the hyper-onic sector, which differs in the way the parameters in the scalar-meson sector are fixed, can be found in Colucci & Sedrakian(2013); van Dalen et al. (2014); Li et al. (2018).c (cid:13)
A. R. Raduta et al.
S/A (0 ≤ S/A ≤
4) either assum-ing constant lepton fraction Y L (0 . ≤ Y L ≤ .
4) in theneutrino-trapped regime or freely streaming neutrinos andmatter in β -equilibrium. Some of the considered thermody-namic conditions are relevant for different stages in the evo-lution from a PNS to a compact star (Burrows & Lattimer1986; Prakash et al. 1997; Pons et al. 1999). Broad coverageof all the astrophysics relevant parameter space will be givenelsewhere. Under the conditions of PNS, the neutrinos aretrapped and the lepton fraction is thus given by the sum ofcharged leptons and neutrinos fractions. For each set of ther-modynamic conditions, we will discuss properties of the EoSand matter composition as well as maximum mass and radiiof hot stars. Possible instability windows of non-accretingPNS with respect to collapse to a BH is investigated follow-ing Bombaci (1996). The maximum gravitational mass atconstant S/A , assuming neutrino-transparent β -equilibratedmatter, is related to the collapse to a BH during in failedCCSNe (Schneider et al. 2020). We study here the influenceof heavy baryons on this maximum mass.The second motivation of this study is to test the uni-versal relations among the global properties of stationary,slowly and rigidly rotating compact stars at finite temper-ature. In essence, universality means that these relations,well established at zero temperature, are independent of theunderlying EoS (Yagi & Yunes 2013a; Maselli et al. 2013;Breu & Rezzolla 2016; Yagi & Yunes 2017; Paschalidis et al.2018) and are thus very helpful for the interpretation of ob-servational data since they allow to mitigate the uncertain-ties related to the EoS. Given the importance of such rela-tions, it is interesting to investigate them under new con-ditions, such as finite temperatures and out-of-equilibriumwith respect to weak interactions.The possibility of some of these relations beinguniversal at finite temperature has been studied be-fore (Martinon et al. 2014; Marques et al. 2017; Lenka et al.2019). The moment of inertia as a function of compact-ness was studied by Lenka et al. (2019), who concluded thatthermal effects lead to deviations from the universal rela-tions obtained for β -equilibrated matter at zero temper-ature. Similar conclusions have been reached for the so-called I − Love − Q relations, where again it was claimed thatat finite temperature deviations from the zero-temperatureuniversality are found (Martinon et al. 2014; Marques et al.2017). Below we show that, if one considers fixed values of( S/A, Y
L,e ), where Y L,e is the electron fraction, the ¯ I − ¯ λ and ¯ I − ¯ Q relations are universal to accuracy comparable tothat obtained for cold compact stars. The same is true forseveral global properties of compact stars, such as the mo-ment of inertia, quadrupole moment, tidal deformability asfunction of compactness. Only the binding energy per unitof the gravitational mass shows deviations from universality.This paper is organized as follows. In Sec. 2 we describethe details of the parameterisation that has been used inmodeling the EoS from relativistic density functional the-ory. Section 3 is devoted to the discussion of the results forthe mass and radius of hot compact stars. In Section 4.1we investigate the behavior of different (normalised) global properties as a function of the star’s compactness. Sec-tion 4.2 focuses on I − Love − Q relations. We conclude inSec. 5. Throughout this paper we use the natural units with c = ~ = k B = G = 1. We consider matter at non-zero temperature as it oc-curs in CCSN, PNS and BNS mergers. Leptons (electrons,muons and neutrinos) and photons are considered as non-interacting gases, whereas under the relevant conditions, thetau lepton can safely be neglected. The partition function ofthe system thus factorizes into a product of baryonic (index B ), leptonic (index L ) and photonic ( γ ) partition functions: Z = Z B Z L Z γ . In thermodynamic equilibrium, within thegrand-canonical ensemble,1 V ln Z L = X j g j Z d k (2 π ) k ε j ( k )[ f FD ( ε j ( k ) − µ j ) + f FD ( ε j ( k ) + µ j )] , (1)1 V ln Z γ = g γ Z d k (2 π ) k ǫ γ ( k ) f BE ( ǫ γ ( k )) . (2)where f FD and f BE are the Fermi-Dirac and Bose-Einsteindistribution functions at temperature T , respectively, g j =(2 s j + 1) is the spin degeneracy factor with s j being thelepton’s spin [ g j = 2 for charged leptons and g j = 1 for (left-handed) neutrinos] and ε j ( k ) and µ j are the single particleenergy and the chemical potential of the lepton j . In Eq. (2) g γ = 2 and ǫ γ ( k ) are the photon spin degeneracy and energy.The partition function of the strongly interacting baryonsrequires a model of the strong nuclear interaction. Withinthe relativistic density functional theory it is given by a sumof kinetic part (which has a form analogous to that of anideal gas) and a potential term. We write it as1 V ln Z B = 1 V ln Z m + X i ∈ baryons J i + 13 Z d k (2 π ) k E i ( k ) · [ f FD ( E i ( k ) − µ ∗ i ) + f FD ( E i ( k ) + µ ∗ i )] , (3)where (2 J i + 1) denotes the spin degeneracy factor, E i = p k + M ∗ i is the single-particle energy. Interactions entervia the effective masses M ∗ i and effective chemical potentials µ ∗ i . The potential term is given by TV ln Z m = − m σ ¯ σ + 12 m ω ¯ ω + 12 m ρ ¯ ρ + 12 m φ ¯ φ . (4)The values of the mean-fields having the quantum numbersof the corresponding mesons, are determined by m σ ¯ σ = X i g σi n si , (5a) m ω ¯ ω = X i g ωi n i , (5b) m φ ¯ φ = X i g φi n i , (5c) m ρ ¯ ρ = X i g ρi t i n i , (5d)where t i is the third component of isospin of baryon i , n si and n i are the scalar and the number density. These are MNRAS , 1–20 (0000) caling in hot stellar matter given by n si = 1 π Z k M ∗ i E i ( k ) [ f FD ( E i ( k ) − µ ∗ i ) + f FD ( E i ( k ) + µ ∗ i )] dk, (6) n i = 1 π Z k [ f FD ( E i ( k ) − µ ∗ i ) − f FD ( E i ( k ) + µ ∗ i )] dk. (7)Note that Eq. (4) does not contain mesonic self-interactions,as is the case of many DFT models. Here, we employ in-stead a model belonging to the subclass of DFTs which allowfor density-dependent couplings, g m , B ( n B ) = g m , B ( n s ) h m ( x )with x = n B /n s , where n s is the nuclear saturation density.We will further assume that the couplings of mesons to hy-perons and ∆-resonances have the same density dependenceas those to nucleons. The effective chemical potentials aregiven by µ ∗ i = µ i − g ωi ¯ ω − g ρi t i ¯ ρ − g φi ¯ φ − Σ R , (8)whereΣ R = X i (cid:18) ∂g ωi ∂n i ¯ ωn i + t i ∂g ρi ∂n i ¯ ρn i + ∂g φi ∂n i ¯ φn i − ∂g σi ∂n i ¯ σn si (cid:19) , (9)is the rearrangement term. The effective baryon masses de-pend on the scalar mean field according to M ∗ i = M i − g σi ¯ σ. (10)From the partition function one obtains pressure, en-tropy density, energy density, and particle number densitiesin a standard fashion P = TV ln Z , (11) s = 1 V ∂ ( T ln Z ) ∂T (cid:12)(cid:12)(cid:12) V, { µ i } , (12) e = − TV ln Z + X i µ i n i + T s, (13) n i = 1 V ∂ ( T ln Z ) ∂µ i (cid:12)(cid:12)(cid:12) V,T . (14)Note that each component contributes additively to the netthermodynamic quantity of interest, which is easy to see bysubstituting Eqs. (1)-(3) in Eqs. (11)-(13).In this work, we use the density-dependent DDME2parameterisation for the nucleonic sector (Lalazissis et al.2005) and its extension to the hyperonic sector byFortin et al. (2016), which fixes the hyperonic couplingsin the vector meson sector to the values implied by the SU (6) symmetric quark model and adjusts the scalar cou-plings to the depth of the hyperon potentials in nuclearmatter at saturation. Alternative extensions of the DDME2model to the baryonic octet have been carried out and ap-plied to compact stars elsewhere (Colucci & Sedrakian 2013;van Dalen et al. 2014; Fortin et al. 2017; Li et al. 2018). Thedifferences between the models reside in the choice of the dif-ferent hyperonic couplings, which are not well constrained.In addition to hyperons, we include the ∆(1232) resonancestates of the baryon J / -decouplet, which were brought re-cently into focus in the context of compact stars by sev-eral groups (Chen et al. 2007; Drago et al. 2014; Cai et al.2015; Zhu et al. 2016; Sahoo et al. 2018; Kolomeitsev et al.2017; Li et al. 2018; Li et al. 2020; Li & Sedrakian 2019b;Ribes et al. 2019). Model n s E s K inf J L K sym [fm − ] [MeV] [MeV] [MeV] [MeV] [MeV]DDME2 0.152 -16.1 250.9 32.3 51.2 -87.1 Table 1.
Key nuclear matter properties of the DDME2 model(Lalazissis et al. 2005): the binding energy per nucleon ( E s ) andcompression modulus ( K inf ) of symmetric nuclear matter at sat-uration density ( n s ) together with the symmetry energy ( J ), itsslope ( L ) and curvature ( K sym ). Our choice of the nucleonic DDME2 EoS is motivatedby the following factors: i) the parameters of isospin symmet-ric ( i.e. equal numbers of protons and neutrons) nuclear mat-ter around saturation density are in good agreement withpresent experimental constraints (Lalazissis et al. 2005) (seeTable 1); ii) the properties of atomic nuclei, such as bind-ing energies, rms radii of charge distribution, neutron skinthickness, quadrupole and hexadecupole moments of heavyand superheavy nuclei, excitation energies of the iso-scalargiant monopole- and iso-vector giant dipole-resonances inspherical nuclei, are in good agreement with experimentalvalues (Lalazissis et al. 2005); iii) the energy per baryon oflow-density neutron matter predicted by the DDME2 func-tional is in good agreement with that from ab initio calcula-tions (Gandolfi et al. 2012; Hebeler et al. 2013) [see Fig. 12in (Fortin et al. 2016) and Fig. 1 in (Li & Sedrakian 2019a)].Point (iii) implies that the model predicts a relatively lowvalue for the slope of the symmetry energy L , which lieswithin the domains 40 . L .
62 MeV (Lattimer & Steiner2014) or 30 . L .
86 MeV (Oertel et al. 2017) deduced fromexperiments [see Fig. 13 in Fortin et al. (2016)]. The curva-ture of the symmetry energy falls in the intervals K sym = − . ± . K sym = − +82 − MeV (Baillot d’Etivaux et al. 2019) and K sym = − ± σ -meson to hyperons to reproduce the empirical depths ofthe hyperon potentials in symmetric nuclear matter at sat-uration. The potential for particle j in k -particle matter isthereby defined via the effective masses and chemical poten-tials as U ( k ) j ( n k ) = M ∗ j − M j + µ j − µ ∗ j . (15)The DDME2Y parameterisation uses the values U ( N )Λ ≈ − U ( N )Ξ ≈ −
14 MeV, U ( N )Σ ≈
30 MeV in isospin sym-metric nuclear matter (Gal et al. 2016). Coupling constantsof the hyperons Y to the meson fields are customarily ex-pressed in terms of the coupling constants of the nucleons N to the meson fields, x m,Y = g m,Y /g m,N , where m ∈ σ, ρ, ω ,etc. labels the meson. Adopting this convention, DDME2Yparameterisation is defined by the following coupling con-stants: x σ Λ = 0 . x σ Ξ =0.3225, x σ Σ = 0 . x ω Λ = 2 / x ω Ξ = 1 / x ω Σ = 2 / x ρ Λ = 0, x ρ Ξ = 1, x ρ Σ = 2. Thecoupling constants of the hidden strangeness meson φ to the MNRAS , 1–20 (0000)
A. R. Raduta et al. baryons are g φN =0, g φ Λ = −√ / g ωN , g φ Ξ = − √ / g ωN , g φ Σ = −√ / g ωN .The DDME2Y parameterisation above fulfills not onlythe existing constraints from terrestrial experiments and abinitio calculations, but it is in agreement with existing com-pact star observations as well (see Section 3 and Table 2).Specifically, the predicted value of the compact star max-imum mass exceeds the observational lower bound on themaximum mass of a compact star 2 M ⊙ (Demorest et al.2010; Antoniadis et al. 2013; Arzoumanian et al. 2018). Fur-thermore, the predicted value of the radius of the canon-ical 1 . M ⊙ compact star is in agreement with the recentinferences R = 13 . +1 . − . km (Miller et al. 2019) and R =12 . +1 . − . km (Riley et al. 2019) from the data obtained bythe NICER mission. The tidal deformability for a 1 . M ⊙ compact star is Λ . = 712. This value lies outside therange Λ . = 190 +390 − (Abbott et al. 2018) and at the up-per limit of the interval 300 +420 − (Abbott et al. 2019) ex-tracted, at a 90% confidence level, from the analysis of theGW170817 event. Note that the values of tidal deformabil-ity in Abbott et al. (2018) have been obtained assuming thatboth compact objects are NSs obeying a common equation ofstate; on the other hand, the values in Abbott et al. (2019)have been obtained, as in the initial analysis of GW170817event (Abbott et al. 2017), by making minimal assumptionsabout the nature of the compact objects and allowing thetidal deformability of each object to vary independently.The masses of ∆-resonances, which form an isospinquadruplet, lie between those of Σ and Ξ hyperons, there-fore they are expected to nucleate in dense stellar matteraccording to the same energetic arguments employed for thenucleation of hyperons (Glendenning 1985). While in thevacuum ∆s are broad resonances which decay into nucleonswith emission of a pion, in stellar matter they are thoughtto be stabilised by the Pauli-blocking of the final nucleonstates. Apart from the narrowing the quasi-particle widthof the ∆s, matter effects may shift the quasi-particle energyto larger values (Sawyer 1972; Ouellette 2011), which wouldsuppress the ∆ degrees of freedom. Below, we will assumethat the ∆s retain their vacuum masses and have negligiblewidth, as has been done in the recent literature (Chen et al.2007; Drago et al. 2014; Cai et al. 2015; Zhu et al. 2016;Sahoo et al. 2018; Kolomeitsev et al. 2017; Li et al. 2018;Li et al. 2020; Li & Sedrakian 2019b; Ribes et al. 2019).The information about nucleon-∆ interaction is ex-tracted from pion-nucleus scattering and pion photo-production (Nakamura et al. 2010), electron scattering onnuclei (Koch & Ohtsuka 1985) and electromagnetic exci-tations of the ∆-baryons (Wehrberger et al. 1989). As re-viewed by Drago et al. (2014) and Kolomeitsev et al. (2017)(a) the potential of the ∆ in the nuclear medium is slightlymore attractive than the nucleon potential −
30 MeV + U ( N ) N . U ( N )∆ . U ( N ) N , which translates in values of x σ ∆ slightly larger than 1, (b) 0 . x σ ∆ − x ω ∆ . . x ρ ∆ .Since there remain large uncertainties on the valuesof the ∆ couplings, they are commonly varied in a cer-tain plausible range. Previous works employed the ranges0 . . x σ ∆ . .
15, 0 . . x ω ∆ . .
2, and 0 . . x ρ ∆ . • small values of x σ ∆ (which lead to small values U ( N )∆ )result in larger values of compact star radii (Zhu et al. 2016;Kolomeitsev et al. 2017; Spinella 2017; Li et al. 2018); theeffect on the maximum mass is small and dependent on thehigh density part of the nucleonic EoS, • small values of x ω ∆ and x ρ ∆ imply lower values ofcompact star radii and maximum masses (Drago et al.2014; Cai et al. 2015; Sahoo et al. 2018; Li et al. 2020;Li & Sedrakian 2019b; Ribes et al. 2019), • the threshold density for the onset of ∆ is correlatedwith x σ ∆ (Zhu et al. 2016; Kolomeitsev et al. 2017; Spinella2017; Li et al. 2018), x ρ ∆ (Cai et al. 2015; Zhu et al. 2016;Sahoo et al. 2018), x ω ∆ (Ribes et al. 2019) and the ∆-effective mass (Cai et al. 2015; Sahoo et al. 2018), • The effective mass of ∆s significantly impacts com-pact star radii and maximum masses (Cai et al. 2015;Sahoo et al. 2018), • the threshold density for the onset of ∆ strongly de-pends on the slope of the symmetry energy at saturation(Drago et al. 2014; Cai et al. 2015). Cai et al. (2015) alsoshowed that it is less sensitive to the other parameters ofsymmetric saturated nuclear matter.The appearance of ∆ resonances in hot stellar matterwith fixed lepton fraction was investigated only recently byMalfatti et al. (2019). It was found that: (i) the lower thelepton fraction, the higher the ∆ abundances; (ii) at highenough temperatures and densities the four isobars are pop-ulated in addition to all hyperonic degrees of freedom; (iii)the most abundant of the ∆-isobars is ∆ − .In this work, we use the following values of the cou-plings of mesons to ∆s: x σ ∆ = 1 .
1, which corresponds to a∆ potential at the saturation U ( N )∆ ≈ −
83 MeV, x ω ∆ = 1 . x ρ ∆ = 1 . x φ ∆ = 0. In the following this model willbe referred to as DDME2Y∆.We assume strangeness changing weak equilibrium lead-ing to the following equilibrium conditions, µ Λ = µ Σ = µ Ξ = µ ∆ = µ n = µ B ; (16) µ Σ − = µ Ξ − = µ ∆ − = µ B − µ Q ; (17) µ Σ + = µ ∆ + = µ B + µ Q ; (18) µ ∆ ++ = µ B + 2 µ Q , (19)where µ B is the baryon number chemical potential and µ Q = µ p − µ n is the charge chemical potential. The equilibriumconditions, Eqs. (16)-(19), together with the total baryoniccharge n p + n Σ + + 2 n ∆ ++ + n ∆ + − ( n Σ − + n Ξ − + n ∆ − ) = n Q determine the composition of baryonic matter for a given( n B , T, Y Q = n Q /n B ). The requirement of global electricalcharge neutrality of stellar matter then fixes the chargedlepton density Y Q = Y e + Y µ where Y e = ( n e − − n e + ) /n B and Y µ = ( n µ − − n µ + ) /n B . For free-streaming neutrinos µ e = µ µ = − µ Q , whereas for trapped neutrinos µ e/µ = µ L,e/µ − µ Q , where µ L,e/µ denotes the (electron/muon) lep-ton number chemical potential. The lepton Y L,e/µ fractions,which are conserved separately, are then defined via the totallepton number density divided by n B . Throughout this pa-per “ β -equilibrium” refers to β -equilibrated matter which,in addition, is transparent to neutrinos . This means that MNRAS , 1–20 (0000) caling in hot stellar matter the corresponding lepton number chemical potentials vanish µ L,e = µ L,µ = 0.At densities below nuclear saturation density and nottoo high temperature, the matter becomes unstable to-wards density fluctuations, because of the competition be-tween nuclear and Coulomb interactions. As a consequence,a large variety of clusters are formed, all of which arein chemical and thermal equilibrium with the unboundbaryons. If the temperature is high enough, hyperons and∆s are in principle expected to nucleate within the clus-ters and unbound components as well, but for simplic-ity, we neglect this possibility here. The theoretical frame-work suitable under these conditions is the Nuclear Statis-tical Equilibrium (NSE) (Hempel & Schaffner-Bielich 2010;Raduta & Gulminelli 2010; Gulminelli & Raduta 2015). In-teractions among unbound particles and clusters are usu-ally accounted for in the excluded volume approximation,while those in the homogeneous matter within a cho-sen mean-field approach. The transition between the in-homogeneous and homogeneous phases is in principle re-alized by minimizing, for equal values of the intensivethermodynamic observables, the relevant thermodynamicpotential. The EoS used in our work are obtained bysmoothly merging the uniform matter EoS to the NSE modelHS(DD2) (Hempel & Schaffner-Bielich 2010) for inhomoge-neous matter. The latter is publicly available on the Com-pose database (Typel et al. 2015). In principle, the transi-tion density depends on the EoS, temperature and chargefraction, see e.g. the discussion in (Ducoin et al. 2008, 2007;Pais et al. 2010), but for simplicity, the matching is per-formed here at a fixed transition density n t = n s /
2. Inho-mogeneous and homogeneous matter are considered at thesame
S/A and Y L,e /µ L,e . The similarity between the effec-tive interactions of DDME2 (Lalazissis et al. 2005) and DD2(Typel et al. 2010) leads to a coherent treatment of the EoSover the whole density regime and the fixed transition den-sity only induces very small thermodynamic inconsistencieswith little impact on the EoS and the global star propertiesstudied here.
A PNS is born in the aftermath of a successful supernova ex-plosion, when the stellar remnant and the expanding ejectaget gravitationally decoupled. The evolutionary epoch dur-ing which the remnant changes from a hot and lepton-richPNS to a cold and deleptonized compact star lasts for sev-eral tens of seconds and consists of two major evolution-ary stages (Prakash et al. 1997; Pons et al. 1999): the delep-tonization stage and the cooling stage. The deleptonizationstage is characterized by a gradual decrease of the net lep-ton and proton fractions and the heat-up of the core, dueto the diffusion of trapped electron neutrinos from the cen-tral region outward. The cooling stage is characterized bya simultaneous decrease of both entropy and lepton con-tent. The structure and composition of the PNS during thisepoch will be investigated here in a schematic way, assum-ing entropy per baryon and lepton fraction with typical https://compose.obspm.fr/ ) -3 (fm B n − − −
10 1 i X n p Λ - Σ - Ξ e µ DDME2Y − − −
10 1 i X n p Λ - Ξ - ∆ e µ ∆ DDME2Y
Figure 1.
Relative abundances in β -equilibrated, neutrino-transparent, cold compact star matter as predicted by DDME2Y(bottom panel) and DDME2Y∆ (top panel) models as function ofbaryon number density. Note that the nucleation of ∆ − resonanceleads to a suppression of Σ − abundance. Thin vertical lines markthe central baryon number densities corresponding to a 1 . M ⊙ star (dot-dashed) and the maximum mass star (dotted), respec-tively. values (Prakash et al. 1997; Pons et al. 1999): ( S/A = 1, Y L,e = 0 . S/A = 2, Y L,e = 0 . S/A = 1, µ L,e/µ = 0)and (
S/A = 0, µ L,e/µ = 0).
Fig. 1 shows relative particle abundances as a function ofbaryon number density in cold β -equilibrated compact starmatter. Results corresponding to baryonic matter composedof NY ∆ (top panel) are compared with those correspondingto NY (bottom panel). In both cases, the net charge neutral-ity is guaranteed by electrons and muons. In the case of NY matter, the only non-nucleonic baryonic degrees of freedomthat nucleate in stable stars are Λ, Ξ − and Σ − , as alreadydiscussed elsewhere (Fortin et al. 2016; Raduta et al. 2018).This can be explained by the large negative charge chemicalpotential in matter featuring low charge fractions favoringnegatively charged particles. Under the considered condi-tions, the Λ-hyperons remain nevertheless the most abun-dant non-nucleonic species. Their lower mass compared withthose of Σ − - and Ξ − -hyperons and more attractive poten-tial in nuclear matter compensate the effect of the chargechemical potential.In the case of NY ∆ matter, ∆ − is the first heavy baryonto appear. Its onset strongly affects the hyperonic abun-dances: the threshold densities for the appearance of Λ and MNRAS , 1–20 (0000)
A. R. Raduta et al.
Model M G, max n c, max Y n Y M Y Y n Y M Y Y n Y M Y R . M ⊙ Λ . ( M ⊙ ) (fm − ) (fm − ) ( M ⊙ ) (fm − ) ( M ⊙ ) (fm − ) ( M ⊙ ) (km)DDME2Y 2.113 0.93 Λ 0.34 1.39 Ξ − − − − Table 2.
Properties of non-rotating spherically symmetric cold β -equilibrated, neutrino-transparent, compact stars based on the EoSmodels considered in this work. n c, max stands for the central baryon number density of the maximum gravitational mass ( M G, max )configuration. Columns 4, 7 and 10 specify the heavy baryon species that nucleate in stable stars. n i represents the threshold densityat which the species i is produced, while M i gives the corresponding gravitational mass of the star. R . M ⊙ indicates the radius of afiducial 1 . M ⊙ star; Λ . represents the tidal deformability of a canonical 1 . M ⊙ star. ) -3 (fm B n − − − i X n =0.4 L,e
S/A=1, Y p =0.4 L,e
S/A=1, Y Λ =0.4 L,e
S/A=1, Y - Σ =0.4 L,e
S/A=1, Y Σ =0.4 L,e
S/A=1, Y + Σ =0.4 L,e
S/A=1, Y - Ξ =0.4 L,e
S/A=1, Y Ξ =0.4 L,e
S/A=1, Y ++ ∆ =0.4 L,e
S/A=1, Y + ∆ =0.4 L,e
S/A=1, Y ∆ =0.4 L,e
S/A=1, Y - ∆ =0.4 L,e
S/A=1, Y e =0.4 L,e
S/A=1, Y =0.4
L,e
S/A=1, Y e ν =0.4 L,e
S/A=1, Y =0.4
L,e
S/A=1, Y − − − i X n =0.2 L,e
S/A=2, Y p =0.2 L,e
S/A=2, Y Λ =0.2 L,e
S/A=2, Y - Σ =0.2 L,e
S/A=2, Y Σ =0.2 L,e
S/A=2, Y + Σ =0.2 L,e
S/A=2, Y - Ξ =0.2 L,e
S/A=2, Y Ξ =0.2 L,e
S/A=2, Y ++ ∆ =0.2 L,e
S/A=2, Y + ∆ =0.2 L,e
S/A=2, Y ∆ =0.2 L,e
S/A=2, Y - ∆ =0.2 L,e
S/A=2, Y e =0.2 L,e
S/A=2, Y =0.2
L,e
S/A=2, Y e ν =0.2 L,e
S/A=2, Y =0.2
L,e
S/A=2, Y − − − i X n =0 L,e µ S/A=1, p =0 L,e µ S/A=1, Λ =0 L,e µ S/A=1, - Σ =0 L,e µ S/A=1, Σ =0 L,e µ S/A=1, =0
L,e µ S/A=1, - Ξ =0 L,e µ S/A=1, Ξ =0 L,e µ S/A=1, =0
L,e µ S/A=1, + ∆ =0 L,e µ S/A=1, ∆ =0 L,e µ S/A=1, - ∆ =0 L,e µ S/A=1, e =0 L,e µ S/A=1, =0
L,e µ S/A=1, =0
L,e µ S/A=1, =0
L,e µ S/A=1,
Figure 2.
Relative particle abundances in hot NY ∆ matter with( S/A = 1, Y L,e = 0 .
4) (bottom), (
S/A = 2, Y L,e = 0 .
2) (middle)and (
S/A = 1, µ L,e = 0) (top), as predicted by the DDME2Y∆model. Muons are not considered. Thin vertical lines mark thecentral baryon number densities corresponding to a 1 . M ⊙ star(dot-dashed) and a maximum mass star (dotted), respectively. Ξ − hyperons are shifted to higher densities and the Σ − hy-peron is completely suppressed. Nucleation of ∆ − s also mod-ifies the neutron and proton abundances: neutron (proton)abundance in NY ∆ matter is smaller (larger) than in NY matter. The proton fraction remains below the threshold forthe nucleonic dUrca process. Recall that, due to the rel- Note that in several models, e.g.
SWL (Spinella 2017) and atively low L -value, neither purely nucleonic nor hyperonadmixed compact stars based on DDME2(Y) EoS allow fora nucleonic dUrca process (Fortin et al. 2016). Finally, weobserve that the onset of ∆ − leads to a fast drop of lep-ton densities, as they compensate for the charge of leptonsand are more energetically favorable than the electrons andmuons.The properties of non-rotating spherically-symmetriccold β -equilibrated compact stars based on DDME2Y(∆)models are summarized in Table 2. Listed are the maxi-mum gravitational mass M G, max , the corresponding centralbaryon number density n c, max , the threshold densities fornucleation of heavy baryons and the corresponding minimalcompact star masses, the radius of a compact star with acanonical mass of 1 . M ⊙ . We note that (i) with M G, max =2 . M ⊙ , DDME2Y and DDME2Y∆ models both fulfil the2 M ⊙ constraint on the lower limit of maximum mass of acompact star (Demorest et al. 2010; Antoniadis et al. 2013;Arzoumanian et al. 2018); (ii) the radii, R ( NY )1 . M ⊙ = 13 . R ( NY ∆)1 . M ⊙ = 13 .
09 km, are in agreement with re-cent NICER results giving 13 . +1 . − . km (Miller et al. 2019)and, respectively, 12 . +1 . − . km (Riley et al. 2019) for PSRJ0030+0451 with a mass of M G = 1 . +0 . − . (Miller et al.2019) and M G = 1 . +0 . − . M ⊙ (Riley et al. 2019); (iii)Λ . = 653 is in better agreement with Abbott et al. (2018);(iv) the 1 . M ⊙ stars will have a tiny fraction of Λ-hyperons,and, within the DDME2Y∆ model, a core which is rich with∆ − resonances.The composition of matter at non-zero temperatureis modified because of the thermal excitation of new de-grees of freedom. Possible neutrino trapping additionallymodifies the composition. Fig. 2 illustrates, as a functionof baryon number density, the relative particle abundancesof hot NY ∆ matter under thermodynamic conditions rele-vant for different stages in the evolution of PNS (Pons et al.1999). Each panel corresponds to a set of constant values oftotal entropy per baryon and electron lepton fraction. In allcases, a vanishing muon lepton fraction Y L,µ = 0 is assumed.The bottom panel corresponds to a moment shortly aftercore bounce, when the star is hot and lepton rich (
S/A = 1, Y L,e = 0 . S/A = 2, Y L,e = 0 . S/A = 1, µ L,e = 0).At high enough temperatures and low- Y L,e values heavybaryons can nucleate already at sub-saturation densities.
FSU2H (Tolos et al. 2016, 2017), the nucleation of ∆ − opens upnucleonic dUrca process or shifts its threshold to much lower den-sities. MNRAS000
FSU2H (Tolos et al. 2016, 2017), the nucleation of ∆ − opens upnucleonic dUrca process or shifts its threshold to much lower den-sities. MNRAS000 , 1–20 (0000) caling in hot stellar matter Moreover, the heavy baryon fractions increase with temper-ature, and eventually, all allowed particle degrees of free-dom can be populated. As visible from Fig. 2, the depen-dence on n B is not always monotonic, due to the competi-tion among different species. Still, due to the large negativecharge chemical potential in matter with low charge frac-tions, negatively charged baryons nucleate at lower densi-ties and are more abundant than their neutral and positivecounterparts. The charge chemical potential decreases withthe charge fraction, such that the effect becomes more pro-nounced at low values of Y L,e . Thus, Λ-hyperons are themost abundant non-nucleonic species, except for low valuesof Y L,e , where the ∆ − abundance can exceed the Λ one. Ourresults qualitatively agree with the findings by Oertel et al.(2012); Oertel et al. (2016), where a large number of EoSmodels with hyperons was considered, and by Malfatti et al.(2019) concerning the ∆s. Fig. 3 shows the temperature as function of baryon num-ber density for matter composed of nucleons (N), nucleonsand hyperons ( NY ), and nucleons, hyperons and ∆ ( NY ∆),as predicted by the DDME2, DDME2Y, and DDME2Y∆model, respectively. The top panel compares different val-ues of S/A for the same Y L,e = 0 . Y L,e for the same
S/A = 1. For purely nucleonic matter, a strong increaseof temperature with density is observed over the entire den-sity range, whereas the increase is much less steep as soonas additional particles appear. Temperature can even de-crease with the density over a certain range, see e.g. thecurves for
S/A = 1 for the DDME2Y and DDME2Y∆ mod-els. The change of slope is due to the sequential onset ofheavy baryons (see Fig. 2). A comparison between the re-sults obtained for N , NY and, respectively, NY ∆ matterproves that for fixed values of ( n B , Y L,e , S/A ), lower valuesof temperature are reached in systems with more particle de-grees of freedom. This can be explained by the fact that, ata given temperature, the entropy of a system increases withthe number of constituent particles. The effect was alreadydiscussed by Oertel et al. (2016), who confronted the behav-ior of matter composed of nucleons and hyperons to that ofpurely nucleonic matter and, respectively, nucleons and Λ-hyperons. They also showed that, for fixed n B and S/A ,the uncertainties in the modeling of the nucleonic sector,especially in the isovector channel, induce larger variationsin temperature than those arising from modeling hyperonicmatter, see the results corresponding to the purely nucle-onic EoS of Lattimer & Swesty (1991), labelled “LS220”, theIUFSU (Fattoyev et al. 2010) version of Fischer et al. (2014)labeled “HS(IUF)” as well as the hyperonic EoS “SFHoY”(Fortin et al. 2018) at
S/A = 2 and Y L,e = 0 . , plotted inthe bottom panel.The middle panel of Fig. 3 demonstrates that, for givenbaryon content of matter ( i.e. N vs. NY vs. NY ∆) andfixed values of S/A and n B , the temperature depends on Y L,e . In all cases, for a given
S/A , the temperature decreaseswith increasing Y L,e . The amplitude of this effect depends These EoS are available on the Compose database. ) -3 (fm B n T ( M e V ) N, 1, 0.2NY, 1, 0.2, 1, 0.2 ∆ NY N, 2, 0.2NY, 2, 0.2, 2, 0.2 ∆ NY N, 4, 0.2NY, 4, 0.2, 4, 0.2 ∆ NY ) -3 (fm B n T ( M e V ) eq. β N, 1, eq. β NY, 1, eq. β , 1, ∆ NY N, 1, 0.2NY, 1, 0.2, 1, 0.2 ∆ NY N, 1, 0.4NY, 1, 0.4, 1, 0.4 ∆ NY ) -3 (fm B n T ( M e V ) eq. β SFHoY; NY, 1, SFHoY; NY, 1, 0.2SFHoY; NY, 1, 0.4 eq. β HS(IUF); N, 1, HS(IUF); N, 1, 0.2HS(IUF); N, 1, 0.4 LS220; N, 2, 0.2HS(IUF); N, 2, 0.2SFHoY; NY, 2, 0.2
Figure 3.
Temperature versus baryon number density for hotstar matter whose baryonic sector allows for nucleons ( N ), NY and NY ∆. Top panel: predictions of DDME2(Y∆) model for dif-ferent values of S/A and Y L,e = 0 .
2, as given in the legend inthe format
S/A, Y
L,e . Middle panel: the same for
S/A = 1 andvarying Y L,e as well as µ L,e = 0. Bottom panel: predictions ofpurely nucleonic EoS models LS220 (Lattimer & Swesty 1991)and HS(IUF) (Fischer et al. 2014) as well as hyperonic SFHoYEoS (Fortin et al. 2018), for different values of
S/A, Y
L,e . In allcases Y L,µ = 0.MNRAS , 1–20 (0000)
A. R. Raduta et al. ) -3 (fm B n − − − − − i s o s p i n a sy m . eq. β N, 1, eq. β NY, 1, eq. β , 1, ∆ NYN, 1, 0.2NY, 1, 0.2, 1, 0.2 ∆ NY N, 1, 0.4NY, 1, 0.4, 1, 0.4 ∆ NY Figure 4.
Isospin asymmetry P i t ,i n i as a function of particlenumber density for β -equilibrated, neutrino-transparent, matterand matter with constant (electron) lepton fractions at S/A = 1,as predicted by DDME2, DDME2Y and DDME2Y∆ models. on the available particle degrees of freedom via the overallisospin asymmetry of matter, shown in Fig. 4. The smallerthis asymmetry, the smaller the temperature variation as afunction of Y L,e . In particular, in the presence of hyperons,isospin asymmetry becomes reduced for a given Y L,e , reflect-ing the fact that some neutrons are replaced by Λ-hyperons,such that the temperature variation is weaker.Fig. 5 shows the dependence of total pressure on thetotal energy density. Predictions for purely nucleonic mat-ter (top) are confronted with those corresponding to mat-ter composed of nucleons and hyperons (middle panel) for
S/A = 1 and 4, at µ L,e = 0 and Y L,e = 0 .
4. The lower panelshows the impact of the onset of ∆s at µ L,e = 0 and
S/A = 0and 4. We find that i) for low energy densities e . , the pressure increases with ( S/A ), whereas theopposite is is true at higher energy densities; ii) the depen-dence of the pressure on Y L,e is complex; for purely nucleonicmatter at high e , pressure is higher for lower Y L,e -values,whereas the opposite is observed for matter containing hy-perons; for intermediate energy-densities lower Y L,e -valueslead to lower values of pressure, no difference between nu-cleonic and hyperonic EoS are observed here since hyperonshave not yet nucleated; iv) the nucleation of ∆s softens P ( e )over intermediate energy-densities and stiffens it for high e ;the magnitude of the modification increases with ∆ abun-dances and, thus, with ( S/A ). These results are in agreementwith (Li et al. 2018), who discussed in detail the effect of ∆son the EoS of cold β -equilibrated matter. Now we turn to the discussion of global properties of hotcompact stars using as input the EoS models presented inthe previous sections. Before discussing our detailed findingsfor hot stars, let us recall some general relations observed inolder, β -equilibrated stars. • The maximum mass of a compact star is sensitiveto the interactions in the high-density domain and servesas a useful diagnostics of the composition of matter,in particular, nucleation of heavy baryons and/or quark ) P ( M e V /f m =0 L,e µ N; S/A=1; =0
L,e µ N; S/A=4; =0.4
L,e
N; S/A=1; Y =0.4
L,e
N; S/A=4; Y − − − log(e) − − − − − − − − l og ( P ) ) P ( M e V /f m =0 L,e µ NY; S/A=1; =0
L,e µ NY; S/A=4; =0.4
L,e
NY; S/A=1; Y =0.4
L,e
NY; S/A=4; Y ) e (MeV/fm ) P ( M e V /f m =0 L,e µ NY; S/A=4; =0
L,e µ ; S/A=4; ∆ NY =0
L,e µ NY; T=0; =0
L,e µ ; T=0; ∆ NY Figure 5.
EoS for different values of
S/A and Y L,e (or µ L,e = 0),as given in the legend according to the DDME2( Y ∆). The insetin the top panel illustrates the behavior of the nucleonic matterEoS in the intermediate energy density range. The modificationsof the EoS due to the onset of ∆s are illustrated in the bottompanel for the case of β -equilibrated, neutrino-transparent matter.The onset of heavy baryons is marked, in each case, by a symbol. matter (Weissenborn et al. 2012a,b; Bonanno & Sedrakian2012; Colucci & Sedrakian 2013; Miyatsu et al. 2013;van Dalen et al. 2014; Gusakov et al. 2014; Oertel et al.2015; Fortin et al. 2016, 2017), • The radius of a canonical mass M G ≃ . M ⊙ compactstar significantly constrain the intermediate-density domain,where the value of the symmetry energy and its slope playan important role (Steiner et al. 2010; Lattimer & Steiner2014). • The tidal deformabilities of compact stars are stronglycorrelated with the radius of the star (Postnikov et al. 2010)and put constraint(s) on the intermediate-density range ofthe EoS (Raaijmakers et al. 2019). Their measurement inthe GW170817 event (Abbott et al. 2017) has already ruledout stiff EoS (Most et al. 2018; Paschalidis et al. 2018). • The compact star moment of inertia depends sen-sitively to the EoS in the intermediate- to low-densityregime (Ravenhall & Pethick 1994; Lattimer & Prakash
MNRAS000
MNRAS000 , 1–20 (0000) caling in hot stellar matter • The thermal evolution of compact stars is a sensi-tive probe of their interior physics and depends on theEoS mostly via the composition of matter in the denseinteriors and occurrence of various direct Urca processesat high densities. Other uncertainties include the com-position of the atmosphere and pairing gaps of vari-ous baryonic species that experience attractive interac-tions (Sedrakian & Clark 2019). Conservative models basedon “minimal cooling paradigm” which assume nucleonic starsand absence of dUrca processes are so far consistent withthe data (Page et al. 2004, 2009). Recent studies of thethermal evolution of hypernuclear compact stars with mod-ern EoS (Raduta et al. 2018, 2019; Negreiros et al. 2018;Grigorian et al. 2018) show that agreement with the datacan be achieved in this case as well due to a combination ofaccelerated cooling via various dUrca processes and suppres-sion of their rates by superfluidity of heavy baryons. Still,any evidence for accelerated cooling cannot be attributed toheavy baryonic cores of compact stars, as models which fea-ture nucleonic dUrca (see e.g.
Beznogov & Yakovlev (2015);Wei et al. (2019)) or quark matter (Hess & Sedrakian 2011;de Carvalho et al. 2015; Sedrakian 2016a; Wei et al. 2020)in the centers of compact stars predict accelerated cool-ing as well. An accelerated cooling can also result from theemission of particles beyond the standard model, such asQCD axions or axion-like particles (Sedrakian 2016b, 2019;Beznogov et al. 2018; Leinson 2019).Fig. 6 illustrates the dependence of the star’s gravi-tational mass on the circumferential radius. β -equilibratedstars and stars with constant electron lepton fraction areconsidered for different values of entropy per baryon. Theresults for baryonic matter composed of N , NY , NY ∆ arecompared assuming Y L,µ = 0.Let us start the discussion with the results at zero tem-perature, depicted in the top panel of Fig. 6. Naturally, thepopulation of additional degrees of freedom such as hyper-ons or ∆s modifies both compact star masses and radii. Nu-cleation of hyperons entails a significant reduction of themaximum mass; if in addition ∆s are accounted for, themaximum mass is only slightly modified; see Table 2, too.Both the population of hyperons and ∆s reduce thestar’s radius. Since the critical density for the onset of∆s is lower than that for hyperons, the effect on the ra-dius is visible for the DDME2Y∆ parameterisation at lowermasses than for the DDME2Y parameterisation, see Ta-ble 2 for the respective onset masses. Since for a canoni-cal mass star 1 . M ⊙ , hyperons are present only in a verysmall amount, the impact on the radius is small, whereas,within DDME2Y∆, the reduction is noticeable (0.16 km),see Table 2. This finding is in agreement with the re-sults by Spinella (2017); Li et al. (2018); Li et al. (2020);Li & Sedrakian (2019b); Ribes et al. (2019), where differentunderlying nucleonic EoS models have been employed.Let us now turn to the case of hot compact stars. Topand bottom panels respectively depict results correspondingto β -equilibrated stars and stars with constant (electron)
11 12 13 14 15 16 17 18 19 20 ) ( M G M N, 0N, 1N, 2NY, 0NY, 1NY, 2, 0 ∆ NY , 1 ∆ NY , 2 ∆ NY - transparent e ν
11 12 13 14 15 16 17 18 19 20
R (km) ) ( M G M N, 1, 0.4N, 2, 0.2N, 1, 0.2NY, 1, 0.4NY, 2, 0.2NY, 1, 0.2NY, 2, 0.4 , 1, 0.4 ∆ NY , 2, 0.2 ∆ NY , 1, 0.2 ∆ NY =const. L,e Y Figure 6.
Gravitational mass M G versus radius for non-rotatingspherically-symmetric stars for the DDME2( Y ∆) EoS models.Top panel: β -equilibrated, neutrino-transparent stars for differentvalues of S/A . Bottom panel: Stars with constant electron leptonfraction for different values of
S/A as indicated in the legend inthe format
S/A, Y
L,e . lepton fraction. A word of caution is necessary here whichconcerns, in particular, the shown radii. For cold compactstars it is well known that radii are sensitive to the crustEoS and the matching of core and crust (Fortin et al. 2016).Good experimental constraints on the compact star outercrust composed of stable nuclei combined with a thermody-namically consistent treatment employing the same interac-tion over the whole density range, limit these uncertaintiesto ≈
5% (Fortin et al. 2016). Finite-temperature EoS areexpected to be affected by consistency issues related to thetransition to a clustered matter close to the surface, too.An additional problem arises since in general the tabulatedfinite-temperature EoS contain entries for T ≥ . S/A can be found in the tables liesin the range n ll ≈ − − − fm − , the exact values de-pending on the EoS and the S/A and Y L,e values. To definethe surface in a coherent way for all models and (
S/A , Y L,e )conditions, we choose a common n min ≈ − fm − andextrapolate all EoS for n min ≤ n B < n ll with linear depen-dencies log( n B ) − log( e ) and, respectively, log( n B ) − log( P ).This extrapolation together with the arbitrarily chosen tran-sition density n t = n s / e.g. compactness MNRAS , 1–20 (0000) A. R. Raduta et al.
Model M B, max /M ⊙ Y L,e instab. domain Y L,e instab. domain Y L,e instab. domainHS(IUF) 2.26 0.4
S/A ≥ .
28 0.2
S/A ≥ . S/A ≤ . S/A ≤ .
82 0.2
S/A ≤ .
67 0.1
S/A ≤ . S/A ≤ .
64 0.2
S/A ≤ .
60 0.1
S/A ≤ . Table 3.
Maximum baryonic masses of cold catalyzed neutrino-transparent compact stars based on different EoS models and domainsof instability with respect to collapse to BH (Bombaci 1996), for Y L,e = 0 . , . , . C = M G /R , the moment of inertia, quadrupole moment andtidal deformability. Masses are not affected. For S/A . M G, max on the star’s entropy per baryon is shown in Fig.7. Note that the maximum masses in Fig. 7 have been de-termined at constant total entropy ( S/A ) M B , where M B denotes the star’s baryonic mass following the turning pointcriterion for a configuration to be secularly stable; see(Sorkin 1982; Goussard et al. 1998; Marques et al. 2017) fora detailed discussion. The following features are observed inFigs. 6 and 7:i) for purely nucleonic stars thermal effects increase thegravitational mass and, thus, M G, max , whereas stars withan admixture of heavy baryons manifest a non-monotonicdependence of M G, max on S/A . The reason is that as longas thermal effects favor nucleation of new species, the max-imum mass decreases with
S/A ; as soon as all available de-grees of freedom are populated, we recover the behavior ob-served for purely nucleonic stars, i.e. M G, max increases with S/A .ii) for purely nucleonic stars at fixed
S/A M G, max ( µ L,e =0) > M G, max ( Y L,e = 0 . > M G, max ( Y L,e = 0 .
4) with theexception of LS220 at
S/A > .
5. Compact stars with anadmixture of heavy baryons most frequently show the oppo-site effect, i.e. M G, max increases with Y L,e ; the reason lies infact that a lepton rich environment with large Y L,e disfavorsheavy baryons, such that they become less populated andthe maximum mass can thus increase with Y L,e for given
S/A ; out of the considered cases the only exception to thisrule is the case of SFHoY EoS with
S/A ≤ M G, max ( Y L,e = 0 . < M G, max ( Y L,e = 0 . Y L,e reduce the star’s compactness, seeFig. 6;iv) for a given mass and composition, radii increase with
S/A , i.e. thermal effects reduce the star’s compactness, seeFig. 6;v) the magnitude of thermal effects depends on the EoS;for DDME2 thermal effects are smaller than those due to Y L,e and the number of allowed degrees of freedom; forLS220 thermal effects dominate over those related to Y L,e .vi) nucleation of ∆s reduces the gravitational mass, includ-ing M G, max . Since the effect scales with ∆ abundance, itincreases with S/A and decreases with Y L,e . The maximum gravitational mass of isentropic compactstars in β -equilibrium is relevant for BH formation in a failedCCSN. By performing many simulations, Schneider et al.(2020) have recently shown that BH formation occurs soonafter the PNS’s gravitational mass overcomes M G, max corre-sponding to its most common entropy value. The trajectoryis thereby essentially determined by the progenitor compact-ness, such that the EoS dependence enters mainly via thebehavior of M G, max as function of S/A .The maximum baryon mass is an interesting quantityin the context of stability against collapse to a black holeduring PNS and BNS merger evolution. In the absence of ac-cretion, M B is a conserved quantity during evolution, suchthat if it exceeds the maximum baryon mass of the cold β -equilibrated configuration, the star necessarily becomesunstable against collapse to a black hole at some point in-dependently of the mechanism stabilising it temporarily,be it strong differential rotation (Baumgarte et al. 2000;Morrison et al. 2004; Kastaun & Galeazzi 2015), the leptonrich environment in PNS (Prakash et al. 1997) or thermaleffects (Prakash et al. 1997; Kaplan et al. 2014). Followingthis reasoning, Bombaci (1996); Prakash et al. (1997) con-jectured that thermally populated non-nucleonic degrees offreedom lead to the existence of meta-stable objects, i.e.stars which during the PNS evolution have a larger max-imum baryonic mass than their cold, β -equilibrated coun-terparts and which upon deleptonization and cooling neces-sarily collapse to a black hole.Our results for the maximum baryonic mass, shownas a function of entropy per baryon S/A for constant(electron) lepton fractions are plotted in Fig. 8 togetherwith results corresponding to LS220, HS(IUF), SFHoY andSFHo (Steiner et al. 2013) EoS models. The chosen val-ues of the (electron) lepton fraction, Y L,e = 0 . S/A ) M B , see Goussard et al. (1998); Marques et al.(2017). Table 3 gives, for all models, the values of M B, max (0 , µ L,e = 0); also we give, for the considered Y L,e values, the instability domains defined according to Bombaci(1996). The values of
S/A where M B, max ( S/A, Y
L,e ) = M B, max (0 , µ L,e = 0) are marked with symbols in Fig. 8.DDME2Y(∆) models show an instability for relatively low
S/A , whereas the nucleonic version DDME2 stays stable overthe entire range of considered values for
S/A . In simulations,during the Kelvin-Helmholtz phase,
S/A stays around 1-2,such that within DDME2 the population of hyperons and/or∆ might indeed lead to the formation of a meta-stable ob-ject. However, this does not seem to be possible exclusively
MNRAS000
MNRAS000 , 1–20 (0000) caling in hot stellar matter S/A / M G , m a x M DDME2; NDDME2; NY ∆ DDME2; NY =0 L,e µ / M G , m a x M SFHo; NSFHoY; NY =0.2
L,e Y / M G , m a x M IUF(HS); NLS220; N =0.4
L,e Y Figure 7.
Maximum gravitational mass M G, max versus entropyper baryon S/A for non-rotating spherically-symmetric compactstars based on the DDME2( Y ∆), HS(IUF) (Fischer et al. 2014),LS220 (Lattimer & Swesty 1991), SFHo (Steiner et al. 2013) andSFHoY (Fortin et al. 2018) models for µ L,e = 0 (bottom) as wellas Y L,e = 0 . in models with non-nucleonic degrees of freedom. The purelynucleonic HS(IUF) and LS220 show instabilities too. Al-though it is not clear whether they will be experienced inthe stellar evolution without performing simulations, it can-not be excluded that meta-stable nucleonic stars could form.These results show the importance of the nuclear interactionin understanding the properties of the high-density EoS. Although NS properties depend sensitively on the EoS, sev-eral “universal” relations have been found between globalquantities. The term “universality” refers here to veryweak dependence on the EoS which holds well for cold β -equilibrated stars. Although so far the reason for this uni-versal behavior is not well understood, it may be exploited to S/A / M B , m a x M DDME2; NDDME2; NY ∆ DDME2; NY =0.1
L,e Y / M B , m a x M SFHo; NSFHoY; NY =0.2
L,e Y / M B , m a x M IUF(HS); NLS220; N =0.4
L,e Y Figure 8.
Maximum baryonic mass M B, max versus entropyper baryon S/A for non-rotating spherically-symmetric compactstars based on the DDME2( Y ∆), HS(IUF) (Fischer et al. 2014),LS220 (Lattimer & Swesty 1991), SFHo (Steiner et al. 2013) andSFHoY (Fortin et al. 2018) EoS models for Y L,e = 0 . S/A where the maximum baryonic mass at Y L,e = const . equals the maximum baryonic mass of the corre-sponding cold β -equilibrated, neutrino-transparent star is markedby solid square (DDME2(NY)), solid triangle (DDME2(NY∆),open triangle (IUF(HS)), and diamond (LS220). constrain quantities difficult to access observationally, elim-inate the uncertainties related to the EoS in the analysis ofthe data, or break degeneracies between integral quantities,( e.g. , the quadrupole moment and the neutron-star spins inbinary in-spiral waveforms).In this section, we shall investigate to what extent thisuniversality remains valid for hot and lepton rich stars, al-lowing for various particle degrees of freedom in the EoS.Section 4.1 will address relations between the normalisedmoment of inertia, quadrupole moment, tidal deformabilityand binding energy and the star’s compactness, and Sec-tion 4.2 will address the I -Love- Q relations. Binding en- MNRAS , 1–20 (0000) A. R. Raduta et al.
Thermo. cond. p p p p p Refs. c c c c c T = 0, β -eq. 0.244 0.638 0 0 3.202 (Breu & Rezzolla 2016) S/A = 2, Y L,e = 0 . . · − a a a a T = 0, β -eq. 8 . · − . · − . · − − . · − (Breu & Rezzolla 2016) S/A = 2, Y L,e = 0 . . · − . · − − . · − . · − this work b b b T = 0, β -eq. 3 . · − − . · − . · − (Maselli et al. 2013) S/A = 2, Y L,e = 0 . . · − − . · − . · − this work e e e e T = 0, β -eq. -2.7157 0.7017 0.1611 − . · − this work S/A = 2, Y L,e = 0 . − . · − this work d d T = 0, β -eq. 0.6213 0.1941 (Breu & Rezzolla 2016) S/A = 2, Y L,e = 0 . Table 4.
Fitting parameters entering Eqs. (21)-(25) under different indicated thermodynamic conditions. ergies and tidal deformabilities will be calculated for non-rotating spherically-symmetric stars; moments of inertia andquadrupole deformations will be calculated for rigidly andslowly rotating stars.
For cold, β -equilibrated stars in the slow-rotation approxi-mation, several authors have established relations betweenthe compactness of a star C = M G /R and normalised mo-ment of inertia, quadrupole moment, tidal deformability andbinding energy, which show universal character, i.e. are al-most EoS independent.First, by considering different EoS for cold β -equilibrated neutron star matter Ravenhall & Pethick(1994) noted that, except for very low mass stars, the nor-malised moment of inertia ˜ I = I/ (cid:0) M G R (cid:1) behaves as auniversal function of the star’s mass and radius,˜ I ≈ .
21 11 − M G /R . (20)Later, Lattimer & Schutz (2005) proved that ˜ I can be ex-pressed as a polynomial in compactness:˜ I = c + c C + c C + c C + c C . (21)The issue was recently reconsidered by Breu & Rezzolla(2016) who showed that the dispersion between differentEoS in Eq. (21) is reduced if only models which fulfill the2 M ⊙ maximum mass constraint are considered. Further-more, they found another universal relation, relating alter-natively normalised moment of inertia ¯ I = I/M G [note thedifferent normalisation with respect to Eq. (21)] to compact-ness ¯ I = a C − + a C − + a C − + a C − . (22)Maselli et al. (2013) introduced a universal expression re-lating compactness to the normalised tidal deformability¯ λ = λ/M G C = b + b ln ¯ λ + b (cid:0) ln ¯ λ (cid:1) . (23)Earlier, by considering a collection of different EoSmodels, Yagi & Yunes (2013a) have shown that the scaledquadrupole moment ¯ Q = QM G /J , where J stands for the angular momentum, as function of compactness is onlyweakly dependent on the EoS. Our results suggest that ¯ Q can be expressed as a polynomial of C − ¯ Q = e + e C − + e C − + e C − . (24)which is analogous to Eq (22) for ¯ I . The neutron star bind-ing energy, defined as the difference between baryonic andgravitational masses E B = M B − M G , shows little sensitiv-ity on the underlying EoS model if normalised by the grav-itational mass (Lattimer & Prakash 2001; Breu & Rezzolla2016). According to Lattimer & Prakash (2001) it behavesas E B M G = d C − d C . (25)Again, as in the case of ˜ I ( C ), limiting the considered EoSmodels to those which are consistent with the 2 M ⊙ massconstraint improves the quality of the fit (Breu & Rezzolla2016). The values of the different fitting parameters, a i , b i , c i , d i , e i , entering Eqs. (21)-(25) are provided in Table 4.Let us now turn to the discussion of universality atnonzero entropy. The top panels of Figs. 9, 10, 11 and 12 il-lustrate, respectively, the behavior of ˜ I , ¯ I , ¯ Q , ¯ λ , and E B /M G as a function of compactness for different combinations ofconstant S/A and Y L,e , as obtained from the DDME2(Y∆)model. The moment of inertia is thereby calculated to lead-ing order in the slow, rigid rotation approximation (Hartle1967) and the tidal deformability λ is computed followingHinderer (2008) and Hinderer et al. (2010). The quadrupolemoment is computed for a rotation frequency of 100 Hz usingthe public domain code Lorene (Gourgoulhon et al. 2016).For comparison, for cold, β -equilibrated matter the resultsobtained from the DDME2(Y∆) model as well as the pre-dictions of Eqs. (21)-(25) are shown, see Table 4 for theparameter values.For all these quantities, a significant scatter of the re-sults for different values of S/A and Y L,e is observed. Thedeviation from the results corresponding to cold catalyzedmatter increases with
S/A and/or Y L,e . This indicates thatuniversality does not hold when stars with different entropies https://lorene.obspm.fr MNRAS000
S/A and/or Y L,e . This indicates thatuniversality does not hold when stars with different entropies https://lorene.obspm.fr MNRAS000 , 1–20 (0000) caling in hot stellar matter I ~ NY, 1, 0.2NY, 2, 0.2NY, 4, 0.2NY, 1, 0.4NY, 2, 0.4NY, 4, 0.4, 1, 0.2 ∆ NY , 2, 0.2 ∆ NY , 4, 0.2 ∆ NY -equil. β NY, 0, -equil. β , 0, ∆ NY I ~ DDME2; NDDME2; NY ∆ DDME2; NYLS220; NHS(IUF); NSFHoY; NY =0.2
L,e
S/A=2; Y C − ( % ) f i t I ~ | /I ~ ∆ | =0.2 L,e
S/A=2; Y I NY, 1, 0.2NY, 2, 0.2NY, 4, 0.2NY, 1, 0.4NY, 2, 0.4NY, 4, 0.4, 1, 0.2 ∆ NY , 2, 0.2 ∆ NY , 4, 0.2 ∆ NY -equil. β NY, 0, -equil. β , 0, ∆ NY I DDME2; NDDME2; NY ∆ DDME2; NYLS220; NHS(IUF); NSFHoY; NY =0.2
L,e
S/A=2; Y − − C − ( % ) f i t I | /I ∆ | =0.2 L,e
S/A=2; Y
Figure 9.
Normalised moments of inertia ˜ I = I/ (cid:0) M G R (cid:1) (left panels) and ¯ I = I/M G (right panels) as function of compactness C = M G /R . Top panels: results corresponding to different thermodynamic conditions and matter compositions, as obtained from theDDME2(Y∆) model. At finite temperatures the thermodynamic conditions are indicated in terms of constant S/A and Y L,e ; the label“0, β -equil.” corresponds to cold catalyzed neutrino-transparent matter. Middle panels: results corresponding to ( S/A = 2, Y L,e = 0 . and lepton contents are compared. For ˜ I and ¯ I this conclu-sion has recently been reached by Lenka et al. (2019). How-ever, one needs in fact to compare quantities under identicalthermodynamic conditions. To illustrate this point, we dis-play in the middle panels of Figs. 9, 10 and 12 and bottompanel of Fig. 11 the results for different EoS models and com-positions, for the case ( S/A = 2, Y L,e = 0 . N , NY or NY ∆ based on the DDME2(Y∆)models, we show results corresponding to the purely nucle-onic LS220 and HS(IUF) EoS models (Lattimer & Swesty1991; Fischer et al. 2014) as well as the hyperonic SFHoYEoS (Fortin et al. 2018). The agreement between the dif-ferent EoS models is very good, except for ˜ I , ¯ Q , and E B at large values of C & .
2, where some deviations can beseen. This confirms that indeed universality holds well forall these relations if the same thermodynamic conditions areconsidered.To quantify this universality, we have performed fits to the nucleonic DDME2 results following Eqs. (21)-(25), in-dicated in the figures by the green open circles. The corre-sponding parameter values are listed in Table 4. Overall, thegood description of the results indicates that the functionalrelations proposed for cold β -equilibrated matter hold forhot matter with trapped neutrinos, too. Only for ˜ I ( C ), thedata are better described by a third-order polynomial, seeTable 4. For ˜ I ( C ), ¯ I ( C ), ¯ Q and C (¯ λ ) the deviations from thefits are of the order of a few percents, as illustrated in thebottom panels of Figs. 9 and 10 and, respectively, the insetin the bottom panel of Fig. 11, where the relative residualerrors with respect to the fit functions are depicted. We con-clude that for these quantities universality holds with a pre-cision only slightly inferior to that for the cold β -equilibratedcase. For E B /M G ( C ) the deviations from the fit are larger,up to 20%, see the bottom panel of Fig. 12. MNRAS , 1–20 (0000) A. R. Raduta et al. Q NY, 1, 0.2NY, 2, 0.2NY, 4, 0.2NY, 1, 0.4NY, 2, 0.4NY, 4, 0.4, 1, 0.2 ∆ NY , 2, 0.2 ∆ NY , 4, 0.2 ∆ NY -eq. β NY, 0, -eq. β , 0, ∆ NY Q DDME2; NDDME2; NY ∆ DDME2; NYLS220; NHS(IUF); NSFHoY; NY =0.2
L,e
S/A=2; Y − C ( % ) f i t Q | / Q ∆ | =0.2 L,e
S/A=2; Y
Figure 10.
Normalised quadrupole moment ¯ Q of slowly andrigidly rotating stars as function of compactness, C . Top panel: re-sults corresponding to different thermodynamic conditions (men-tioned in the legend) and matter compositions derived fromthe DDME2(Y∆) model. At finite temperatures the thermo-dynamic conditions are specified in terms of constant S/A and Y L,e ; the label “0, β -equil.” corresponds to cold catalyzedneutrino-transparent matter. Middle panel: results correspondingto ( S/A = 2, Y L,e = 0 .
2) for different EoS models and for differ-ent matter compositions. The red dot-dashed curves correspondto fits in Eq. (24) with parameters values from Table 4. Bottompanel: relative residual errors with respect to the fit in Eq. (24)for the cases considered in the middle panels. I -Love- Q universal relations Yagi & Yunes (2013b,a) identified universal relations amongthe pairs of quantities ¯ I - ¯ Q , ¯ I - ¯ λ and ¯ Q - ¯ λ . Numerically,these relations can be cast in a polynomial form on a log-logscale,ln Y i = a i + b i ln X i + c i (ln X i ) + d i (ln X i ) + e i (ln X i ) , (26)with the pairs ( Y i , X i ) corresponding to ( ¯ I, ¯ λ ); ( ¯ I, ¯ Q ); ( ¯ Q, ¯ λ ).Originally established in the slow rotation limit, these rela-tions remain EoS independent for fast rotating stars withrotation frequency dependent fit parameters (Doneva et al.2013). λ NY, 1, 0.2NY, 2, 0.2NY, 4, 0.2NY, 1, 0.4NY, 2, 0.4NY, 4, 0.4, 1, 0.2 ∆ NY , 2, 0.2 ∆ NY , 4, 0.2 ∆ NY -equil. β NY, 0, -equil. β , 0, ∆ NY − − C λ DDME2; NDDME2; NY ∆ DDME2; NYLS220; NHS(IUF); NSFHoY; NY =0.2
L,e
S/A=2; Y ) λ log( ( % ) f i t C | / C ∆ | Graph
Figure 11.
Normalised tidal deformability ¯ λ of non-rotatingspherically-symmetric stars as function of compactness C . Toppanels: results corresponding to different thermodynamic condi-tions (mentioned in the legend) and matter compositions, as ob-tained from the DDME2(Y∆) model. At finite temperatures thethermodynamic conditions are specified in terms of constant S/A and Y L,e ; the label “0, β -equil.” corresponds to cold catalyzedneutrino-transparent matter. Bottom panels: results correspond-ing to ( S/A = 2, Y L,e = 0 .
2) for different EoS models for differentmatter compositions. The red dot-dashed curves correspond to fitin Eq. (23) with parameters values from Table 4. Relative resid-ual errors with respect to the the fit in Eq. (23) are shown in theinset.
Let us now consider I − Love − Q universality at nonzerotemperature. We will focus here on ¯ I − ¯ λ and ¯ I − ¯ Q . If uni-versality holds for these two pairs of quantities, it is verylikely that it will hold for the third pair as well. In Fig. 13we plot ¯ I as function of ¯ λ (left) and ¯ Q (right). Again, in thetop panels, we compare results for different values of S/A and Y L,e employing the DDME2(Y∆) models with those forcold β -equilibrated stars. The dot-dashed line indicates theresult of Eq. (26) with fitting parameters obtained for cold β -equilibrated stars according to the DDME2Y model, seeTable 5. As can be seen from the middle panels, the resid-ual errors become an order of magnitude larger than thoseobtained when considering only cold β -equilibrated stars(Yagi & Yunes 2013a). This deviation of the I − Love − Q -relations from universality due to thermal effects have al-ready been noted by Martinon et al. (2014); Marques et al.(2017).The discussion in Sec. 4.1 suggests that universality atnonzero entropy can be recovered under identical thermody-namic conditions. The bottom panels of Fig. 13 show the rel-ative error of the results with respect to Eq. (26) with refit-ted parameters at S/A = 2 and Y L,e = 0 .
2, see Table 5. Re-sults corresponding to DDME2(Y∆) model are confrontedwith those of LS220 (Lattimer & Swesty 1991), HS(IUF)
MNRAS000
MNRAS000 , 1–20 (0000) caling in hot stellar matter Y X
Thermo. cond. a b c d e
Refs.¯ I ¯ λ T = 0, β -eq. 1.47 7 . × − . × − − . × − . × − this work¯ I ¯ Q T = 0, β -eq. 1.50 4 . × − . × − . × − . × − this work¯ I ¯ λ S/A = 2, Y L,e = 0 . . × − . × − − . × − − . × − this work¯ I ¯ Q S/A = 2, Y L,e = 0 . . × − . × − − . × − . × − this work Table 5.
Fitting parameters of eq. (26) for different thermodynamic conditions. / M B E NY, 1, 0.2NY, 2, 0.2NY, 4, 0.2NY, 1, 0.4NY, 2, 0.4NY, 4, 0.4, 1, 0.2 ∆ NY , 2, 0.2 ∆ NY , 4, 0.2 ∆ NY -equil. β NY, 0, -equil. β , 0, ∆ NY / M B E DDME2; NDDME2; NY ∆ DDME2; NYLS220; NHS(IUF); NSFHoY; NY =0.2
L,e
S/A=2; Y C r e l . r e s i dua l e rr o r s ( % ) DDME2; NDDME2; NY ∆ DDME2; NYLS220; NHS(IUF); NSFHoY; NY =0.2
L,e
S/A=2; Y
Figure 12.
Binding energy per unit of gravitational mass E B /M G of non-rotating spherically-symmetric stars as a func-tion of compactness C . Top panel: results corresponding to dif-ferent thermodynamic conditions (indicated in the legend) andmatter compositions, as obtained from the DDME2(Y∆) model.At finite temperatures the thermodynamic conditions are spec-ified in terms of constant S/A and Y L,e ; the label “0, β -equil.”corresponds to cold catalyzed neutrino-transparent matter. Mid-dle panel: results corresponding to ( S/A = 2, Y L,e = 0 .
2) fordifferent EoS models and for different matter compositions. Thered dot-dashed curves correspond to fits in Eq. (25) with param-eters values from Table 4. Bottom panel: relative residual errorswith respect to the fit for the cases considered in the middle panel. (Fischer et al. 2014) and SFHoY (Fortin et al. 2018). Thequality of the fit by Eq. (26) is considerably improved reach-ing the accuracy of zero temperature, β -equilibrated case.Thus, universality again holds under the same thermody-namic conditions.This phenomenologically observed universality is notyet understood. Analytical solutions in the Newtonianlimit (Yagi & Yunes 2013a) as well as those obtained us-ing an expansion around this limit (Jiang & Yagi 2020) cor-roborate the EoS independence, but without providing adefinitive insight into their origin. Yagi & Yunes (2013a) ad-vanced two possible explanations. First, these dimension-less quantities mainly depend on the outermost shells of thecore and on the crust, where, by construction, ”realistic”EoS, i.e. nuclear EoS with parameters fitted to nuclear dataand/or astrophysical observations, agree with each other.As in (Yagi & Yunes 2013a) the term ”realistic” is meanthere to distinguish nuclear EoS from polytropic EoS. If thiswas the case, finite-temperature EoS should necessarily vi-olate universality as their low-density behaviors differ fromeach other, and from that of cold β -equilibrated matter, seethe discussion in Sec. 3. Second, Yagi & Yunes (2013a) sug-gested that this universality could be a reminiscence of no-hair theorems as the neutron star’s compactness approachesthe black hole limit. In this case, too, since hot stars are lesscompact, see Sec. 3, universality should be less well satisfiedfor hot stars.Yagi & Yunes (2013a) corroborated their guess aboutthe origin of universality by investigating the radial depen-dence of the integrands entering the calculation of momentof inertia, quadrupole moment and tidal deformability in theNewtonian limit, normalised by their values at the star’s cen-ter, showing that they are peaked around 0 . . r/R . . r is the radial distance from the star’s center. Themain contribution to these quantities thus indeed comesfrom the outer core and crust. In Fig. 14 we show for starswith compactness C = 0 .
17 the radial profiles of the quan-tities (cid:0) er /e c R (cid:1) and (cid:2) ep c ( de/dp ) /e c (cid:3) , where (cid:0) er (cid:1) cor-responds to the integrand of the moment of inertia andquadrupole moment in the Newtonian limit (top panel), and( ede/dr ) to the EoS-dependent contribution to the tidal de-formability in the Newtonian limit (bottom panel); here in-dex c indicates corresponding values at the star’s center.Comparing results for different S/A and Y L,e (left) we notethat with increasing temperature and, slightly more pro-nounced, increasing Y L,e , the maximum of the integrandsmigrate to lower r/R values and smear out. For the high-est considered value
S/A = 4, (cid:2) er /e c R (cid:3) manifests a widepeak centered at r/R ≈ .
6, while (cid:0) ep c ( de/dp ) /e c (cid:1) showsa plateau over r/R . . MNRAS , 1–20 (0000) A. R. Raduta et al. I NY, 1, 0.2NY, 2, 0.2NY, 4, 0.2NY, 1, 0.4NY, 2, 0.4NY, 4, 0.4, 1, 0.2 ∆ NY , 2, 0.2 ∆ NY , 4, 0.2 ∆ NY -equil. β NY, 0, -equil. β , 0, ∆ NY ∆ DDME2Y, DDME2Y ( % ) f i t I | / I ∆ | λ ( % ) f i t I | / I ∆ | DDME2; NDDME2; NY ∆ DDME2; NYLS220; NHS(IUF); NSFHoY; NY =0.2
L,e
S/A=2; Y I NY, 1, 0.2NY, 2, 0.2NY, 4, 0.2NY, 1, 0.4NY, 2, 0.4NY, 4, 0.4, 1, 0.2 ∆ NY , 2, 0.2 ∆ NY , 4, 0.2 ∆ NY -equil. β NY, 0, -equil. β , 0, ∆ NY ( % ) f i t I | / I ∆ | Q ( % ) f i t I | / I ∆ | DDME2; NDDME2; NY ∆ DDME2; NYLS220; NHS(IUF); NSFHoY; NY =0.2
L,e
S/A=2; Y
Figure 13.
Left: ¯ I vs. ¯ λ (top) and relative residual errors | ∆¯ I | / ¯ I fit (¯ λ ) with respect to the fit by eq. (26) (middle, bottom). Right:¯ I vs. ¯ Q (top) and relative residual errors | ∆¯ I | / ¯ I fit ( ¯ Q ) with respect to the fit by eq. (26) (middle, bottom). Top: results for differentthermodynamic conditions and matter compositions (indicated in the legend), as obtained from the DDME2(Y∆) model. The dot-dashed red line indicates the fit to the cold β -equilibrated results, see Table 5. Middle: relative error with respect to the fit at zerotemperature. Bottom panels: results corresponding to S/A = 2 and Y L,e = 0 .
2. The relative error with respect to refitted values underthese thermodynamic conditions, see Table 5, is shown for different matter compositions and EoS models. tions are compared. Considering again fixed
S/A = 2 and Y L,e = 0 . I − Love − Q universalitydoes not lie in the similar behavior of “realistic” EoS in theouter core and the crust but rather in the similar behavior ofEoS over the density domains which, under the consideredthermodynamic condition, play the most important role. In this work, we constructed EoS of dense matter with heavybaryons (hyperons and ∆-resonances) at non-zero tempera-ture and for different lepton fractions within the covariantdensity functional theory. This extends models of EoS forcold β -equilibrated matter (as it occurs in older compactstars) to finite temperatures and matter out of β -equilibriumas needed for the description of CCSN, PNS evolution, and BNS mergers. In particular, our finite-temperature EoSmodels include heavy baryon degrees freedom - the fullbaryon octet and ∆-resonances. Our EoS model is consis-tent with available constraints from nuclear physics experi-ments, ab initio calculations of low-density neutron matter,and observations of compact stars, specifically, massive neu-tron stars, radius and mass inferences by NICER experimentand tidal deformability derived from GW170817 event. Weplan to make tables of the EoS publicly available on theCompose database.As discussed previously for cold compact stars(Drago et al. 2014; Li et al. 2018), the population of ∆s atintermediate densities, before the onset of most hyperons,leads to smaller radii for intermediate-mass stars. The im-pact on the maximum mass compared with hypernuclearmodels is nevertheless negligible since at high densities any-way many different states are populated and the additional∆ degree of freedom only leads to a rearrangement of par- MNRAS000
S/A = 2 and Y L,e = 0 . I − Love − Q universalitydoes not lie in the similar behavior of “realistic” EoS in theouter core and the crust but rather in the similar behavior ofEoS over the density domains which, under the consideredthermodynamic condition, play the most important role. In this work, we constructed EoS of dense matter with heavybaryons (hyperons and ∆-resonances) at non-zero tempera-ture and for different lepton fractions within the covariantdensity functional theory. This extends models of EoS forcold β -equilibrated matter (as it occurs in older compactstars) to finite temperatures and matter out of β -equilibriumas needed for the description of CCSN, PNS evolution, and BNS mergers. In particular, our finite-temperature EoSmodels include heavy baryon degrees freedom - the fullbaryon octet and ∆-resonances. Our EoS model is consis-tent with available constraints from nuclear physics experi-ments, ab initio calculations of low-density neutron matter,and observations of compact stars, specifically, massive neu-tron stars, radius and mass inferences by NICER experimentand tidal deformability derived from GW170817 event. Weplan to make tables of the EoS publicly available on theCompose database.As discussed previously for cold compact stars(Drago et al. 2014; Li et al. 2018), the population of ∆s atintermediate densities, before the onset of most hyperons,leads to smaller radii for intermediate-mass stars. The im-pact on the maximum mass compared with hypernuclearmodels is nevertheless negligible since at high densities any-way many different states are populated and the additional∆ degree of freedom only leads to a rearrangement of par- MNRAS000 , 1–20 (0000) caling in hot stellar matter − − − )(r / R ) c ( e / e NY, 1, 0.2NY, 2, 0.2NY, 4, 0.2NY, 1, 0.4NY, 2, 0.4NY, 4, 0.4 r/R −
10 1 ) c / p c e ( de / dp ) / ( e , 1, 0.2 ∆ NY , 2, 0.2 ∆ NY , 4, 0.2 ∆ NY -equil. β NY, 0, -equil. β , 0, ∆ NY − − − )(r / R ) c ( e / e DDME2; NDDME2; NY ∆ DDME2; NYLS220; NHS(IUF); NSFHoY; NY r/R −
10 1 ) c / p c e ( de / dp ) / ( e =0.2 L,e
S/A=2; Y =0.2
L,e
S/A=2; Y =0.2
L,e
S/A=2; Y =0.2
L,e
S/A=2; Y =0.2
L,e
S/A=2; Y =0.2
L,e
S/A=2; Y
Figure 14.
Normalised radial profiles of ( e/e c ) ( r/R ) (top panels) and e ( de/dp ) (cid:0) e c /p c (cid:1) (bottom panels) for stars with compactness C =0 .
17, as predicted by various EoS models. Left panels illustrate results corresponding to DDME2(Y∆) model for various thermodynamicconditions specified in the legend in terms of (
S/A, Y
L,e ); the label “0, β -equil.”corresponds to cold catalyzed neutrino-transparent matter.Right panels illustrate results corresponding to different EoS models and matter compositions ( N , NY , NY ∆) for ( S/A = 2 , Y
L,e = 0 . ticle abundances. As long as no additional degrees of free-dom are populated, thermal effects lead to an increase of themaximum gravitational mass with S/A . This is particularlythe case of purely nucleonic stars and hypernuclear starsat high temperatures. The population of additional parti-cle degrees of freedom by thermal excitation can, on theother hand, reduce the maximum gravitational mass withincreasing
S/A . The lepton fraction modifies the star’s max-imum gravitational mass, too. Depending on the EoS,
S/A ,and particle degrees of freedom, compact star’s maximumgravitational mass may increase or decrease with the lep-ton fraction. Most frequently the gravitational mass of acompact star with an admixture of heavy baryons increaseswith the lepton fraction, while the opposite effect is obtainedfor purely nucleonic stars. The reason is that an increasingcharge fraction decreases the charge chemical potential andthus disfavors the appearance of, in particular negativelycharged, hyperons and ∆s. Because of an extended surface,stars at finite temperature and/or high lepton fractions areless compact and less bound than their counterparts at zerotemperature. The maximum baryonic mass of a hot star de-termines the stability against collapse to a black hole. Wehave shown that, depending on the nuclear EoS, both purelynucleonic stars, and hypernuclear stars may be stable (un-stable) within the considered domain of entropy per baryon.Several authors (Martinon et al. 2014; Marques et al.2017; Lenka et al. 2019) have argued that thermal effects in-duce deviations from the universal relations. These findingswere confirmed by comparing the relations between variousglobal properties of compact stars at finite
S/A with thoseat zero temperature. As a byproduct, we have shown that the ∆ degrees of freedom do not alter universal relationsfor cold compact stars. Finally, we have demonstrated that when the universal relations are studied at the same entropyper baryon and the same lepton fraction, universality is re-covered . We have illustrated this by establishing universalrelations between compact star’s compactness and severalother global properties as well as by testing the validity ofuniversality for the I − Love − Q relations. This EoS indepen-dence could be helpful for the analysis of observational datafrom hot, transient states of compact stars, in full analogyto the zero temperature case discussed extensively in theliterature. Our findings may also give new hints for the un-derstanding of the origin(s) of universality. ACKNOWLEDGMENTS
This work has been partially funded by the European COSTAction CA16214 PHAROS “The multi-messenger physicsand astrophysics of neutron stars”. A. R. R. acknowledgesthe hospitality of the Frankfurt Institute for AdvancedStudies. A. S. acknowledges the support by the DeutscheForschungsgemeinschaft (Grant No. SE 1836/5-1) and thehospitality of the Observatoire de Paris, Meudon.
APPENDIX: UNCERTAINTIES IN RADII OFCOMPACT STARS
To quantify the uncertainties related to the surface defini-tion and the crust-core transition we consider four different
MNRAS , 1–20 (0000) A. R. Raduta et al. case n min n t n B fit domain(1) n ll n s / n ll n s / − [fm − ] n s / n ll , n t )(4) 10 − [fm − ] n s / n ll , n LIN ) Table 1.
Definition of NS surface and matching between crustand core. n min defines the baryon number density at the star’ssurface, n t the transition density from core to crust, n ll the lowestdensity in the EoS data tables for the given conditions and n LIN the maximum density for which log( e ) − log( n B ) and log( P ) − log( n B ) show a linear beahvior to a very good precision between n ll and n LIN ; n s stands for the saturation density of symmetricnuclear matter. scenarios, see Table 1. We thereby vary on the one hand the(fixed) transition density from the core to the crust and onthe other hand the density at which we define the surfaceof the star, implying an EoS extrapolation for cases (3) and(4). The top panels of Fig. 1 illustrate, for each scenario,the total pressure as a function of the total energy densitytogether with the EoS data from the Compose database.For this example, the DD2Y EoS (Marques et al. 2017) hasbeen chosen. The bottom panels depict the mass-radius re-lation, where the cases ( S/A = 2 , Y
L,e = 0 .
2) (left) and(
S/A = 4 , Y
L,e = 0 .
2) (right) have been considered.As indicated in the main text, we find that: i) the uncer-tainties related to the arbitrarily chosen value of the crust-core transition density n t are negligible, ii) the uncertaintiesrelated to the surface definition are sizeable only for starswith high values of S/A &
3. For the shown example, at
S/A = 4 , Y
L,e = 0 .
2, the uncertainty in the radius of a 1.4 M ⊙ star amounts to 20%. This indicates an upper limit onthe uncertainty; for all other studied star properties, the un-certainty is smaller. Similar uncertainties are found for otherEoS models, showing that our results are robust. REFERENCES
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