Δ-admixed neutron stars: spinodal instabilities and dUrca processes
aa r X i v : . [ nu c l - t h ] J a n ∆ -admixed neutron stars: spinodal instabilities and dUrca processes Adriana R. Raduta National Institute for Physics and Nuclear Engineering (IFIN-HH), RO-077125 Bucharest, Romania
Within the covariant density functional theory of nuclear matter we build equations of state of ∆ -admixed compact stars. Uncertainties in the interaction of ∆(1232) resonance states with nuclearmatter, due to lack of experimental data, are accounted for by varying the coupling constants toscalar and vector mesonic fields. We find that, over a wide range of the parameter space allowedby nuclear physics experiments and astrophysical observations, cold catalyzed star matter exhibitsa first order phase transition which persists also at finite temperature and out of β -equilibriumin the neutrino-transparent matter. Compact stars featuring such a phase transition in the outercore have small radii and, implicitly, tidal deformabilities. The parameter space is identified wheresimultaneously ∆ -admixed compact stars obey the astrophysical constraint on maximum mass andallow for dUrca processes, which is otherwise forbidden. INTRODUCTION
With densities exceeding several times the nuclear sat-uration density, n s ≈ .
16 fm − , the core of neutron stars(NS) has been acknowledged since long ago to representa unique environment for studies of compressed baryonicmatter. In the innermost shells of massive stars severalnon-nucleonic degrees of freedom (d.o.f.) - such as hyper-ons, kaon and pion condensates, ∆ -resonances and quarkgluon plasma - have been conjectured [1] to exist in ad-dition or instead of the nucleonic ones.The high-precision mass measurements, during thisdecade, of several massive pulsars in binary systems withwhite dwarfs [2–6] re-opened the issue of dense matter hy-peronisation. A large variety of theoretical frameworkshas been employed, such as relativistic density functionaltheory (DFT), quark-meson coupling model, auxiliaryfield diffusion Monte Carlo approach, cluster variationalmethod and Brueckner-Hartree-Fock theory. Within themost frequently employed DFT class of models reconcili-ation among two solar mass NS and hyperonic d.o.f. hasbeen possible either employing a sufficiently stiff nucle-onic equation of state (EoS) or going beyond the SU(6)symmetry ansatz to fix the vector meson couplings, for adetailed discussion see [7]. The relativistic quark modeland DFT have been also used to address nucleation of ∆(1232) -resonances in stellar matter, see [8–16]. Withmasses lying between those of Σ and Ξ hyperons and anattractive potential in nuclear matter, ∆ -resonances areexpected to be populated based on the same generic ener-getic arguments with which hyperonisation is advocated.Similarly to the appearance of any other new hadronicspecies above its production threshold ∆ s will soften theEoS and, thus, modify the mass-radius diagram. Refs.[10, 11] have shown that nucleation of ∆ in purely nucle-onic NS decreases the maximum mass by up to ≈ . M ⊙ and the radii of intermediate mass NS by up to ≈ km.The magnitude of these effects is nevertheless very sen-sitive to the underlying nucleonic EoS, ∆ s effective massand strength of ∆ − N interaction. Ref. [9, 12–16] have shown that in hypernuclear stars the only significant ef-fect induced by the nucleation of ∆ s consists in the reduc-tion of radii. High precision determination of NS radii,expected from the currently operating NICER mission[17] and also from future x-ray observatories like Athenax-ray telescope [18] and eXTP [19], could thus contributeto shed light on the ∆ d.o.f. in NS and, implicitly, on ∆ − N interaction. Complementary information can, inprinciple, be provided also by NS cooling history, consid-ering that, because of its negative charge, ∆ − can shiftthe threshold of nucleonic dUrca to lower densities and,possibly, open new dUrca processes. The first effect isparticularly interesting as hyperonisation has been shownto not alter the threshold of nucleonic dUrca [20].A sufficiently attractive ∆ N interaction potential is ex-pected not only to modify NS global properties but alsothe thermodynamic stability of dense matter. The occur-rence of thermodynamic instabilities related to the onsetof ∆ s has been so far discussed in connection with hot anddense hadronic matter produced in high-energy heavy-ioncollisions [21] and, more recently, cold hypernuclear stars[22]. [22] shown that such instabilities manifest over abaryonic density domain which corresponds to the innercore, n s . n B . n s , for coupling constants spanningwide domains of values in agreement with available ex-perimental constraints. The limited number of EoS dis-cussed in [22] nevertheless suggests that when the twosolar mass constraint is imposed ∆ -driven instabilitiesare ruled out.The first aim of the present work is to investigatewhether ∆ -driven instabilities occur also in NS builtupon EoS which fulfill the M ⊙ constraint [3]. To thisaim a systematic investigation of the parameter space isperformed. As [22] we use the covariant density func-tional theory but rely on a different nucleonic EoS and,for the sake of clarity, disregard hyperonic d.o.f. Muonsare also disregarded. The second motivation of this studyis to identify the parameter space of the ∆ − N interac-tion where compact stars that obey the two solar masslimit allow for dUrca processes. To better address thisissue we select a density-dependent nucleonic EoS whichdoes not allow for n → p + e − + ˜ ν e and p + e → n + ν e dUrca processes in stable stars.The paper is organized as follows. In Section wepresent the EoS models. In Section the formalismfor the analysis of thermodynamic instabilities in multi-component systems is revisited. Properties of ∆ -admixedcompact star are discussed in Section . Finally, the con-clusions are drawn in Section . EQUATIONS OF STATE
The phenomenological EoS considered in our study areobtained by extending the density-dependent DDME2[23] nucleonic parameterisation such as to additionallyaccount for ∆(1232) resonance states of the baryon J / decuplet. Interaction among different baryonic species isrealized by the exchange of scalar-isoscalar ( σ ), vector-isoscalar ( ω ) and vector-isovector ( ρ ) mesons. Cou-pling constants of ∆ s to mesonic fields are expressed interms of coupling constants of nucleons to mesonic fields x m, ∆ = g m, ∆ /g m ; N , where m = σ, ω, ρ ; as common in lit-erature, the assumption on the same density dependencefor ∆ - and nucleon-meson couplings is made.Our choice of the nucleonic DDME2 is motivated bythe following factors: i) the parameters of isospin sym-metric nuclear matter around saturation density are ingood agreement with present experimental constraints;ii) the values of the symmetry energy at saturation J = 32 . MeV as well as its slope L = 51 . MeV andcurvature K sym = − . MeV lie within the domainsdeduced from nuclear data and neutron star observa-tions; for constraints on L , see [24, 25]; for constraintson K sym , see [26–28]; the relatively low value of L ex-plains the good agreement of the energy per baryon with ab initio calculations of low density pure neutron matter[29, 30] (see fig. 12 in [31]); iii) the maximum mass ofpurely nucleonic NS, M ( N )max = 2 . M ⊙ [31], built uponDDME2 fulfills the M ⊙ observational constraint on thelower limit of maximum masses [3]; iv) the radius of thecanonical . M ⊙ NS, R . = 13 . km, agrees with the re-cent measurements R (1 . +0 . − . M ⊙ ) = 13 . +1 . − . km [32]and R (1 . +0 . − . M ⊙ ) = 12 . +1 . − . km [33] of the PSRJ0030+0451 millisecond pulsar obtained by the NICERmission, v) for purely nucleonic NS, the tidal deforma-bility range ≤ ˜Λ ≤ , corresponding to the massratio range . ≤ q ≤ of the merger stars, slightlyovershoots the upper limit of the interval ˜Λ = 300 +500 − (symmetric 90% credible interval) for a low spin prior ex-tracted from the GW170817 event [34], vi) the agreementwith above cited astrophysical data is maintained uponinclusion of hyperons [13, 31, 35, 36]; vii) the nucleonicdUrca process is not allowed in stable stars; this providesus with the perfect framework to assess the impact thenucleation of ∆ s has on chemical composition.In vacuum ∆(1232) s are broad resonances which decay into nucleons with emission of a pion. In medium theyare considered to be stabilized by the Pauli blocking ofthe final nucleon states. Other effects of the mediumregard the narrowing of the quasi-particle width and shiftof quasi-particle energy to larger values [37, 38]. Withinthe DFT and relativistic quark model it was assumedthat ∆ s preserve their vacuum masses and have vanishingwidths [8–15]. In the present work we employ the samehypothesis.The information about ∆ -nucleon interaction is scarce.Data extracted from pion-nucleus scattering and pionphoto-production [39], electron scattering on nuclei [40]and electromagnetic excitations of the ∆ -baryons [41]have been reviewed by Drago et al. [9] and Kolomeit-sev et al. [12] with the aim of constraining the values ofthe coupling constants at saturation. Their conclusionsmay be summarized as follows: i) the potential of the ∆ sin the nuclear medium is slightly more attractive thanthe nucleon potential −
30 MeV + U ( N ) N . U ( N )∆ . U ( N ) N ;this translates in values of x σ ∆ slightly larger than 1, ii) . x σ ∆ − x ω ∆ . . , and iii) no experimental constraintsexist for the value of x ρ ∆ .The still remaining uncertainties are customarily ac-counted for by allowing for variation of coupling con-stants within large domains of values. Previous works[9, 12–15] have considered the ranges . ≤ x σ ∆ ≤ . , . ≤ x ω ∆ ≤ . and . ≤ x ρ ∆ ≤ . We shall here-after adopt the same strategy and the following domains . ≤ x σ ∆ ≤ . ; x σ ∆ − . ≤ x ω ∆ ≤ x σ ∆ + 0 . ; . ≤ x ρ ∆ ≤ . .The core EoS calculated within DFT is smoothlymerged with the crust EoS. For the latter we em-ploy the Negele and Vautherin [42] and Haensel-Zdunik-Dobaczewski [43] EoS. THERMODYNAMIC INSTABILITIES
Spinodal instabilities manifest as convexity anoma-lies of thermodynamic potentials expressed in terms ofextensive variables. Mathematically they are signaledby negative eigenvalues of the curvature matrix C i,j = ∂ f ( { n i } ; i = 1 , ..., N ) /∂n i ∂n j , where i, j = 1 , ..., N ; f stands for the free energy density; n i represents the num-ber density of each conserved species i , whose total num-ber is N . The eigenvectors associated to negative eigen-values correspond to the directions in the density spacealong which density fluctuations are spontaneously andexponentially amplified in order to achieve phase sepa-ration. Each negative eigenvalue corresponds to a phasetransition whose order parameter is given by the eigen-vector. The maximum number of phase transitions ina multi-component system is equal to the dimension ofthe curvature matrix which, in its turn, is equal to thenumber of conserved species. The curvature matrix anal-ysis has been often employed in mean-field studies ofbaryonic matter. For the case of liquid-gas phase tran-sition taking place at sub-saturation densities, see [44–47]; for strangeness-induced instabilities in dense strangehadronic matter, see [48–52].Spinodal instabilities of multi-component systemswhose direction of phase separation is dominated by oneof the conserved densities, n j , are revealed by convex in-truders in the free energy density Legendre conjugatedwith respect to the remaining ( N − ) chemical poten-tials, ¯ f ( n j , { µ i } i =1 ,..,N ; i = j ) = f − P i =1 ,..,N ; i = j µ i n i or,alternatively, back-bending behavior of the chemical po-tential µ j = (cid:0) ∂ ¯ f /∂n j (cid:1) n i ; i =1 ,...,N ; i = j as a function of theconjugate species density n j . For a detailed discussion,see [53].In the most simple case, that we employ here, in whichmuons and hyperons are disregarded ∆ -admixed stel-lar matter is made of neutrons, protons, the four ∆ -isobars ( ∆ − , ∆ , ∆ + , ∆ ++ ) and electrons. The netcharge neutrality condition links the baryon charge den-sity, n Q = n p + n ∆ + + 2 n ∆ ++ − n ∆ − , to the electron den-sity, n e = n Q . In the neutrino-transparent matter regime- characterized by vanishing lepton chemical potential, µ L = 0 , - the electron density equals the lepton density, n e = n L . This means that the thermodynamic potentialsmay be expressed in terms of the total baryon numberdensity, n B , and charge (or lepton) density, n Q ( L ) .In the following spinodal instabilities will be iden-tified via the negative eigenvalues of C B,L = ∂ e ( n B , n L ) /∂n B ∂n L , where e stands for the total en-ergy density, the relevant thermodynamic potential atzero temperature. Despite the fact that, being interestedin neutron stars, we impose β -equilibrium, the stabilityanalyses are performed in two dimensions, which meansthat we allow density fluctuations to drive matter out of µ L = 0 . Phase transitions will be discussed in the hybridensemble ( n B , µ L ) [53] for µ L = 0 , which corresponds to β -equilibrated matter. Extension to finite temperaturesand non-vanishing values of µ L will allow us to check thepersistence of ∆ -driven instabilities in hot matter out of β -equilibrium. RESULTS
We now turn to investigate chemical composition andthermodynamic instabilities of cold catalyzed ∆ -admixedstellar matter, for different values of the coupling con-stants of ∆ to mesonic fields. The onset of ∆ isobars isdiscussed in connection with density thresholds of dUrcaprocesses involving nucleons, n → p + e − + ˜ ν e , as wellas the two most abundant ∆ s: ∆ − → n + e − + ˜ ν e , ∆ → p + e − + ˜ ν e . Then we address the compatibil-ity of ∆ -driven phase transitions and dUrca reactionswith compact stars EoS that obey the M ⊙ constraint[3]. Finally the relevance of thermodynamic instabilitiesin hot stellar matter beyond β -equilibrium is studied by ) -3 (fm B n − − − − − ) ( M e V ) ∆ U ( SNM ) -3 (fm B n SNM ) -3 (fm B n SNM − − − − − ) ( M e V ) ∆ U ( - ∆ ∆ + ∆ ++ ∆ PNM =1 ∆ρ =0.9, x ∆ω =0.9, x ∆σ x PNM =1 ∆ρ =1.2, x ∆ω =1.2, x ∆σ x PNM =1 ∆ρ =1, x ∆ω =1.2, x ∆σ x FIG. 1: Single-particle potentials of ∆ isobars as a function ofbaryonic particle number density in symmetric nuclear mat-ter (SNM) (bottom) and pure neutron matter (PNM) (toppanels) for different values of x σ ∆ and x ω ∆ , as indicated ineach panel. In all cases x ρ ∆ = 1 . plotting the phase diagram. U ( N )∆ potentials Nucleation of ∆ -isobars obviously depends on their in-teraction potential in nuclear matter, U ( N )∆ = − g σ ∆ ¯ σ + g ω ∆ ¯ ω + g ρ ∆ t ¯ ρ, (1)where ¯ σ , ¯ ω , ¯ ρ stand for the mean field expectation valuesof the mesonic field and t represents the third compo-nent of the isospin, with the convention t ++ = 3 / ;the dependence on baryonic particle number densitieshas been omitted for the potential, coupling constantsand mesonic fields. Eq. (1) suggests that, for a given nu-cleonic EoS, more attractive potentials are provided bylarge values of x σ ∆ and low values of x ω ∆ . It also showsthat in neutron rich matter, as it is the case of NS, thepotential felt by positively charged isobars is larger inabsolute values than that felt by negative isobars since ¯ ρ < . At variance with this, in symmetric nuclear mat-ter (SNM), all isobars experience the same potential.Quantitative information is provided in Fig. 1 for theextreme cases of pure neutron matter (PNM) and SNM.Different combinations of ( x σ ∆ , x ω ∆ ) are considered; forall cases we assume x ρ ∆ = 1 . It comes out that, depend-ing on the coupling constants, U ( N )∆ in SNM is attractivefor < n B . . − n s . For the PNM case a strongerdependence on the values of the coupling constants isobtained along with a significant dispersion among thepredictions corresponding to different isobars. ∆ s in cold catalyzed matter The different attractive potentials felt by the differentisobars in neutron rich matter together with the chemical ∆σ x ) - ( f m B n - ∆ ∆ + ∆ ++ ∆ =1.0 onset ∆ρ =1.0; x ∆ω x ∆σ x (n,p),n) - ∆ ( ,p) ∆ ( =1.0 dUrca ∆ρ =1.0; x ∆ω x ∆ω x ) - ( f m B n =1.0 onset ∆ρ =1.1; x ∆σ x ∆ω x =1.0 dUrca ∆ρ =1.1; x ∆σ x ∆ρ x ) - ( f m B n =1.0 onset ∆ω =1.1; x ∆σ x ∆ρ x =1.0 dUrca ∆ω =1.1; x ∆σ x FIG. 2: Onset densities of ∆ isobars (left) and threshold den-sities of various dUrca processes (right) in cold catalyzed neu-tron star (NS) matter as a function of x σ ∆ (top), x ω ∆ (middle)and x ρ ∆ (bottom). equilibrium conditions µ ∆ − = µ B − µ Q ; µ ∆ = µ B ; µ ∆ + = µ B + µ Q ; µ ∆ ++ = µ B +2 µ Q , (2)determine the onset densities of each species. Here µ B and µ Q stand for baryon and charge chemical poten-tials and can be expressed in terms of chemical poten-tials of nucleons, leptons and electrons as µ B = µ n and µ Q = µ p − µ n = µ L − µ e . The left panels of Fig. 2 il-lustrate the role played by each coupling constant whenthe other two are kept constant. For all considered setsof ( x σ ∆ , x ω ∆ , x ρ ∆ ) the first particle that nucleates is ∆ − . Despite the low attractive potential its populationis favored by charge conservation. The other three parti-cles nucleate only for certain values of the coupling con-stants; the second most favored particle is ∆ , while theless favored one is ∆ ++ . We also note that: i) the on-set densities decrease (increase) with the increase of x σ ∆ ( x ω ∆ ), reflecting thus the evolution of U ( N )∆ with the twocoupling constants; ii) x ρ ∆ has almost no influence onthe onset densities; iii) the onset density of ∆ − is sig-nificantly lower than those of the other three particles,which do not differ much one from the other.By partially replacing the electrons, which regulate therelative abundances of neutrons and protons, ∆ − willimplicitly modify the neutron and proton abundances.If protons will become sufficiently abundant such as thetriangle inequalities of Fermi momenta are satisfied [54],nucleonic dUrca - forbidden in DDME2 - will become ∆σ x − − − − ∆ ω - x ∆ σ x FIG. 3: Range of coupling constants ( x σ ∆ , x ω ∆ ) for whichcold β -stable matter features one (red stars) or two (blue dots)spinodal instability domains. Green squares: EoS which al-low for at least one dUrca process at n B < . fm − . Solid(dashed) line: contours of coupling constants corresponding to M max = 2 . M ⊙ ( . M ⊙ ). Results corresponding to x ρ ∆ = 1 . energetically allowed. If, additionally, ∆ s are abundantenough to satisfy, along with electrons and nucleons, theappropriate triangle inequalities, also dUrca processes in-volving these particles will open up.The right panels of Fig. 2 illustrate the threshold den-sities of nucleonic dUrca as well as those of two dUrcaprocesses involving ∆ s as a function coupling constants.The same cases as in the left panels are considered. Withthe exception of x σ ∆ = 1 . and x ω ∆ ≥ . , all consideredsets of parameters allow for nucleonic dUrca, otherwiseforbidden in NS built upon DDME2. ∆ − → n + e − + ˜ ν e is energetically allowed roughly over the same parame-ter ranges which allow for nucleonic dUrca; its thresholddensity is only slightly higher than the one of nucleonicdUrca. ∆ → p + e − + ˜ ν e opens up at higher densitiesand operates over a reduced range of parameters. Simi-larly to the onset densities, density thresholds of dUrcaprocesses show strong (poor) sensitivity to x σ ∆ and x ω ∆ ( x ρ ∆ ). Spinodal instabilities in cold catalyzed matter
We now turn to investigate the occurrence of ∆ -driveninstabilities in cold catalyzed matter. With the purposeto limit the parameter space and the motivation that x ρ ∆ does not impact the onset of ∆ s, we fix x ρ ∆ = 1 .We generate EoS of ∆ -admixed cold β -equilibratedmatter in the neutrino-transparent regime for . ≤ x σ ∆ ≤ . and x σ ∆ − . ≤ x ω ∆ ≤ x σ ∆ + 0 . . Ther-modynamic instabilities are sought after by calculat-ing the eigenvalues of the curvature matrix C B ; L = ∂ e ( n B , n L ) /∂n B ∂n L . Three situations are encountered:a) no instability, b) one domain of instabilities, c) two do-mains of instabilities. The sets of ( x σ ∆ , x ω ∆ ) correspond-ing to the latter two situations are illustrated in Fig. 3 ( M e V ) B µ T=0T=10 MeV =1.0 ∆ρ =0.85; x ∆ω =1.05; x ∆σ x =0 L µ ) P (MeV/fm ( M e V ) B µ ) -3 (fm B n − − −
10 1 i X np ++ ∆ + ∆ ∆ - ∆ e =0; T=0 L µ ( M e V ) B µ =1.0 ∆ρ =1.0; x ∆ω =1.2; x ∆σ x =0; T=0 L µ ) P (MeV/fm ( M e V ) B µ ) -3 (fm B n − − −
10 1 i X np ++ ∆ + ∆ ∆ - ∆ e =0; T=0 L µ FIG. 4: Baryonic chemical potential (top) and relative particle densities (bottom) as a function of baryonic particle numberdensity for ∆ -admixed β -equilibrated neutrino-transparent matter at T = 0 . The insets in the top panels depict the baryonicchemical potential as a function of total pressure over a density domain which contains the threshold for the onset of ∆ − s. Redopen circles and horizontal segments indicate the Gibbs constructions. Stable (unstable) solutions of mean-field calculationsare illustrated with solid (dotted) lines. Results corresponding to x σ ∆ = 1 . , x ω ∆ = 0 . (left) and x σ ∆ = 1 . , x ω ∆ = 1 . (right) and x ρ ∆ = 1 . . On the top left panel also shown is µ B ( n B ) for T =10 MeV. with red stars and, respectively, blue solid dots. Instabili-ties occur for a narrow domain of . ≤ x σ ∆ − x ω ∆ ≤ . and, for the lowest values of x σ ∆ , . ≤ x σ ∆ ≤ . , twoinstability domains are present. For all matter states fea-turing spinodal instability only one negative eigenvaluewas found for C B ; L .Further insight into the thermodynamic behavior isprovided in Fig. 4 in terms of µ B ( n B ) (top panels); thechemical composition as a function of baryonic particlenumber density is represented as well (bottom panels).Two sets ( x σ ∆ , x ω ∆ ) are considered, as mentioned in leftand right panels. For the lower value of x σ ∆ two insta-bility domains are obtained, signaled by backbendings of µ B ( n B ) . Inspection of the bottom panel indicates thatthe first instability is due to the onset of ∆ − and thesecond one to the onset of the other three isobars. Forthe higher value of x σ ∆ only the low density instabilityrelated to nucleation of ∆ − is present. The fact that -despite their earlier onset - ∆ , ∆ + and ∆ ++ do not leadto instabilities is due to the less steep increase of theirabundances, caused by the higher value of x ω ∆ . µ B ( n B , µ L = 0) considered in Fig. 4 corresponds tohybrid ensembles which allow multi-component systemsbe treated as uni-component ones and have the Gibbsconstruction reduced to the Maxwell construction [53].Coexisting phases, characterized by equal values of inten-sive quantities, i.e. pressure ( P ) and the two chemical potentials ( µ B and µ L ), are represented with red opendots. Complementary information is provided in the in-sets in terms of P ( µ B ) .The instabilities induced by ∆ − occur over n s . n B . n s , which corresponds to the outer core of NS. As suchit is expected to impact the radii of all mass NS and havelittle or no effect on the maximum mass. The instabilitiesinduced by (cid:0) ∆ , ∆ + , ∆ ++ (cid:1) occur over . n s . n B . n s ,which corresponds to more central shells in the core. Inaddition to radii also the maximum NS mass may beaffected. However some of the coupling constant setsmight not be compatible with the M ⊙ constraint [3]. Tocheck the issue we plot in Fig. 3 the sets of ( x σ ∆ , x ω ∆ ) forwhich the maximum NS mass equals M ⊙ and . M ⊙ .Only the coupling constants sets lying at the r.h.s. ofthese curves provide maximum NS masses larger than thevalue to which the contour corresponds. This means thatthe occurrence of thermodynamic instabilities associatedto (cid:0) ∆ , ∆ + , ∆ ++ (cid:1) is ruled out by the astrophysical limiton pulsar masses. This is the case of EoS discussed by [22]which, indeed, provide low mass NS. The explanation ofthe low maximum NS masses obtained for this domain ofthe parameter space consists in the fact that, because ofvanishing nucleon effective masses, the baryonic densitycan not exceed − n s . In conclusion the only phasetransition compatible with M ⊙ is the low density one.Also shown in Fig. 3 is the domain of parameter sets ) -3 (fm B n ) - ( f m Q n T=0T=5 MeVT=10 MeV =1.0 ∆ρ =1.0; x ∆ω =1.2; x ∆σ x FIG. 5: Phase coexistence domains (shaded regions) betweenpure ( n, p, e ) -matter and ∆ admixed matter in n B − n Q co-ordinates for T = 0 , 5 and 10 MeV. Open circles correspondto coexisting phases at µ L = 0 . Results corresponding to x σ ∆ = 1 . , x ω ∆ = 1 . , x ρ ∆ = 1 . . for which the density threshold of the first energeticallyallowed dUrca process is smaller than 0.8 fm − . Thisarbitrarily chosen value roughly coincide with the cen-tral baryonic density of the maximum mass configura-tion provided by DDME2 when only nucleonic d.o.f. areaccounted for. The phase diagram of ( n, p, ∆ , e ) matter The phase diagram of ( n, p, ∆ , e ) matter correspondingto the coupling constants set considered in the right pan-els of Fig. 4 is plotted in Fig. 5. In addition to T = 0 alsothe domains of phase coexistence between pure ( n, p, e ) -matter and ∆ -resonance admixed matter for T = 5 and10 MeV are represented. As in Fig. 4 the open dots mark,for each temperature and µ L = 0 , the coexisting phases.It comes out that, contrary to the liquid-gas phase transi-tion taking place in sub-saturated nuclear matter, the or-der parameter of the ∆ -driven phase transition has onlya tiny component along n Q . Other features are never-theless similar to those of the liquid-gas phase transition:the coexistence domain gets quenched as the tempera-ture increases and second order phase transitions occuras well. First order phase transitions in hot and densematter have been shown to impact star stability with re-spect to collapse to a black hole during the core collapse[55]. Second order phase transitions have been proven todrastically reduce the neutrino mean free path and, thus,slow down the proto-NS cooling [49]. Though addressedin connection with a strangeness driven phase transitionassociated to hyperonisation the two effects are expectedto hold also in the case of the presently discussed phasetransition and, thus, lead to observable effects.
11 11.5 12 12.5 13
R (km) M ( M ⊙ ) npe(1.1, 1.1)(1.2, 1.1)(1.2, 1.0)(1.2, 1.0) M(1.3, 1.1)(1.3, 1.1) M PSR J0348+0432
FIG. 6: Gravitational mass - radius diagram of cold cat-alyzed NS built upon DDME2. Results corresponding to ∆ -admixed matter are confronted with those corresponding topurely nucleonic matter for different values of the couplingconstants of mesons to ∆ -isobars, specified as ( x σ ∆ , x ω ∆ ). Inall cases x ρ ∆ = 1 . EoS featuring a phase transition where theMaxwell construction was performed are marked with "M".The shaded horizontal stripe shows the observed mass of pul-sars PSR J0348+0432, M = 2 . ± . M ⊙ [3]. M ( M ⊙ ) Λ npe(1.1, 1.1)(1.2, 1.1)(1.2, 1.0)(1.2, 1.0) M(1.3, 1.1)(1.3, 1.1) M -equil. β T=0,
FIG. 7: Tidal deformabilities as a function of gravitationalmass for the same EoS as in Fig. 6. The vertical arrowmarks the limits obtained for a . M ⊙ NS, < Λ . < ,as derived from the observation of GW170817 by the LVCcollaboration [56]. Global properties of NS featuring a ∆ -driven phasetransition In the following we discuss some properties of ∆ -admixed NS with focus on modifications induced bythe phase transition. Equilibrium configurations ofspherically-symmetric non-rotating NS are obtained bysolving the Tolman-Oppenheimer-Volkoff set of differen-tial equations. The tidal deformabilities are calculatedfollowing [57, 58].Figs. 6 and 7 illustrate the mass-radius diagram and,respectively, the tidal deformabilities as a function ofgravitational mass. Results corresponding to several ∆ -admixed NS are confronted with those of purely nucleonicstars. As previously discussed, nucleation of ∆ s entailsa reduction of the maximum mass of up to a few tens ofsolar mass [10, 11] and a diminish of NS radii by up to1-2 km [9, 12–15]. Related to the latter effect a signif-icant reduction of tidal deformabilities is also obtained.The amplitude of these modifications increases with theamount of heavy baryons, that is with the increase of x σ ∆ and the decrease of x ω ∆ . For the two considered sets ofcoupling constants for which matter manifests instabili-ties the Maxwell construction causes an extra reductionof the order of several percent of both radii and tidaldeformabilities. CONCLUSIONS
Inspired by previous findings of Lavagno and Pigato[22] regarding spinodal instabilities in cold catalyzed NSmatter with admixture of ∆(1232) -resonances we have in-vestigated in a systematic way the parameter space of the ∆ − N interaction and identified the domains where theonset of ∆ generates thermodynamic instabilities. EoSwhich fulfill the two solar mass astrophysics constraintmanifest spinodal instabilities over n s . n B . n s ,which corresponds to the outer shells of the core. Whenmean-field anomalous behavior is cured by Maxwell con-struction an extra diminish, of the order of several per-cent, is obtained for both NS radii and tidal deforma-bilities. ∆ − -driven instabilities persist also out of β -equilibrium in the neutrino-transparent matter and attemperatures of a few tens MeV, which suggests that -if present - the thermodynamic instabilities may affectalso other astrophysical phenomena where dense matteris populated.Finally we have shown that, for a large domain of theparameter space, nucleation of ∆ s opens-up the nucle-onic dUrca process which is otherwise forbidden. Theconsequences on thermal evolution of isolated NS and X-ray transients are beyond the aim of this work and willaddressed elsewhere. Nevertheless it is straightforwardto anticipate that ∆ -admixed massive NS, whose innercore accommodates unpaired nucleons, might meet theconditions required to describe the data of the faintest X-ray transients, XRT SAX J1808.4-3658 and 1H 1905+000[59]. The situation is interesting the more as to date noneof the covariant density functional models with couplingconstants fitted on experimental and observational datahas been able to simultaneously describe the whole set ofthermal data from isolated NS and accreting NS in quies-cence, even if the pairing gaps in different channels wereallowed to vary over wide ranges and hyperonic degreesof freedom have been accounted for [60]. Acknowledgements
This work was, in part, supported by the EuropeanCOST Action "PHAROS" (CA16214). The author ac-knowledges support from UEFISCDI via grant nr. PN-III-P4-ID-PCE-2020-0293.
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