Maximum mass of compact stars from gravitational wave events with finite-temperature equations of state
Sanika Khadkikar, Adriana R. Raduta, Micaela Oertel, Armen Sedrakian
aa r X i v : . [ a s t r o - ph . H E ] F e b Maximum mass of compact stars from gravitational wave events withfinite-temperature equations of state
Sanika Khadkikar, ∗ Adriana R. Raduta, † Micaela Oertel, ‡ and Armen Sedrakian
4, 5, § Birla Institute of Technology and Science, Pilani, Hyderabad Campus, India 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, 92190 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 (Dated: 01/29/2021)We conjecture and verify a set of universal relations between global parameters of hot and fast-rotating compact stars, including a relation connecting the masses of the mass-shedding (Kepler) andstatic configurations. We apply these relations to the GW170817 event by adopting the scenario inwhich a hypermassive compact star remnant formed in a merger evolves into a supramassive compactstar that collapses into a black hole once the stability line for such stars is crossed. We deduce anupper limit on the maximum mass of static, cold neutron stars 2 . +0 . − . ≤ M ⋆ TOV ≤ . +0 . − . forthe typical range of entropy per baryon 2 ≤ S/A ≤ Y e = 0 . PACS numbers:
I. INTRODUCTION
Neutron (or compact) stars, containing matter at den-sities exceeding that at the centers of atomic nuclei, rep-resent unique laboratories to probe the matter under ex-treme conditions. Considerable effort is underway to pindown the dense matter equation of state (EoS) as presentin neutron stars, which is pressed ahead by many re-cent observations and the prospects opened by the dawnof multi-messenger astrophysics. Among these are theprecise pulsar mass determinations from the pulsar tim-ing analysis [1–5], measurements of compact star massesand radii through the x-ray observations of their surfaceemission [6, 7] in particular, the results of the NICER ex-periment [8, 9], and the gravitational wave detection ofbinary neutron star (BNS) mergers by the LIGO-Virgocollaboration [10, 11]. Among the events in the last cate-gory, the GW170817 event is currently outstanding, sinceit has been possible to measure not only the neutron startidal deformability during inspiral, but also electromag-netic counterparts [12, 13]. As a result, the GW170817event has triggered a large number of works which areaimed at constraining neutron star properties and theEoS, either from the analysis of the tidal deformabilityalone (see for example [14–22]), or from a combination oftidal deformability and the electromagnetic signal [23–30]. Including the information from the electromagneticsignal requires as an input numerical modeling of themerger process, which introduces additional uncertain- ∗ [email protected] † [email protected] ‡ [email protected] § sedrakian@fias.uni-frankfurt.de ties but, at the same time, broadens the experimentalbase of the analysis.Another interesting event is GW190814, where themass of the lighter object has been determined (at 90%credible level) to be 2.50-2.67 M ⊙ [31]. In the standardinterpretation [32–38] this is either the most massive neu-tron star observed to date or is a black hole that is locatedin the so-called mass-gap. Other, more exotic models in-clude for example a strange star [39, 40] or a compact starin an alternative theory of gravity [41]. The neutron starinterpretation of the light companion in the GW190814challenges our current understanding of the EoS, even ifone assumes that this star is rotating very rapidly [32–38].An important aspect of the merger process is that be-fore the merger the two stars are well described by a one-parameter EoS of cold matter in weak ( β -)equilibrium,which typically relates pressure to (energy) density.This means that the measured tidal deformabilities andmasses of the two merging stars essentially concern thiscold EoS of dense matter in β -equilibrium. In contrast,after the merger the evolution of the post-merger rem-nant (if there is no prompt black hole formation) requiresas an input an EoS at non-zero temperature and out of(weak) β -equilibrium, i.e., the pressure becomes a func-tion of three thermodynamic parameters [42–45]. Mostcommonly, these are chosen to be baryon number density, n B , temperature T and charge fraction Y Q = n Q /n B ,where n Q is defined as the total hadronic charge den-sity [46]. The electron fraction Y e = Y Q due to electricalcharge neutrality. In the following, when referring tocold compact stars, we will assume that they are in β -equilibrium. Small deviations from β -equilibrium, whichcan lead to some kinematical effect (bulk viscosity, etc)will be neglected.Alongside full-fledged hydrodynamics simulations ofthe post-merger phase, different studies focused on sta-tionary solutions for compact star configurations, whichgive, among other things, hints on the magnitude of themaximum mass supported by a post-merger object andthus the conditions for the formation of a black hole.As evidenced by numerical simulations, post-merger ob-jects are rapidly rotating and support significant inter-nal flows. Therefore, to assess the stability of the post-merger object rapidly and differentially rotating configu-rations of compact stars should be studied.Universal relations, i.e. , relations between differentglobal quantities of the star found empirically to be inde-pendent of the EoS, have attracted much attention in thiscontext. Such relations have been established for bothuniformly [47–50] and differentially rotating stars [51, 52]in the case of cold stars, described by zero-temperatureEoS with the matter under β -equilibrium. However, forthe merger remnant the thermal effects cannot be ignoredand can influence, among other observables, the maxi-mum mass of a static or rapidly rotating star [53, 54]as well as the applicability of universal relations. InRefs. [53, 55, 56], it has been shown that thermal effectsinduce deviations from the universal relations obtainedfor β -equilibrated matter at zero temperature. Subse-quently, Ref. [57] demonstrated that universality is re-stored if finite-temperature configurations with the sameentropy per baryon and electron fraction are considered.Here we will extend the study of Ref. [57] which has fo-cussed on non-rotating or slowly rotating stars to rapidrotation.As a consequence of our findings on universality forhot stars, we revisit the inference of the maximum massof a compact star from the analysis of the GW170817event. This problem has been addressed by several au-thors, see Refs. [24, 28–30] using the scenario of the for-mation of a hypermassive compact star in the mergerevent and its delayed collapse to a black hole closeto the neutral stability line for supramassive compactstars. Some of these authors employed the universal-ity of the linear relation between the maximum gravi-tational mass for uniformly rotating stars at the Keplerlimit, M ⋆K , and the same quantity for a non-rotating star M ⋆ TOV = max ( M TOV ) [47, 50, 58] M ⋆K = C ⋆M M ⋆ TOV . (1)Here and below the superscript ⋆ refers to quantities char-acterizing the maximum mass objects. The employedvalue for C ⋆M ≈ . M ⋆K and M ⋆ TOV has, however, been determined assuming that the star ro-tating at Kepler frequency is cold and in β -equilibrium,which is not necessarily the case for the merger remnant.Therefore we will revisit this question and will determinethe impact of nonzero temperature and matter out of β -equilibrium on the value of C ⋆M .This paper is organized as follows. In Sec. II wedescribe briefly the numerical setup for modeling fast-rotating hot compact stars and our collection of EoS. In Section III we investigate different universal relations forfast-rotating stars. Section IV is devoted to the discus-sion of the maximum mass of fast-rotating compact stars.We derive a new upper limit on M ⋆ TOV using the univer-sal relations in Sec. V, Our conclusions are collected inSec. VI. Throughout this paper we use natural units with c = ~ = k B = G = 1. II. SETUP
This section is devoted to a description of our strategyto solve for the structure of a hot rapidly rotating rela-tivistic star. More details on the formalism can be foundin [53, 59, 60]. Combined Einstein and equilibrium equa-tions are solved, assuming stationarity and axisymmetry.Besides, we assume the absence of meridional currentssuch that the energy-momentum tensor fulfills the cir-cularity condition, i.e. there is no convection. An EoSis needed to close the system of equations. In neutronstars older than several minutes matter is cold, neutrino-transparent, and in (approximate) β -equilibrium. ItsEoS is barotropic, i.e. it depends only on one variable,which commonly is chosen as baryon number density n B .In contrast, the merger-remnant matter is hot and notnecessarily in β -equilibrium, such that the EoS dependsin addition to n B on temperature T and electron fraction Y e = n e /n B or thermodynamically equivalent variables.Under the above-mentioned assumptions, in particularstationarity, the most general solution for the star’s struc-ture becomes again barotropic, i.e. , the electron fractionand the temperature need to be related to n B [59–61].To fulfill this requirement, we consider below stars char-acterized by constant entropy per baryon S/A and somefixed value of the electron fraction or constant electronlepton fraction Y L = ( n e + n ν ) /n B = n L /n B ( n ν and n L being the neutrino and electron lepton number densities,respectively). It should be stressed that this simplifiedsetup does not reflect realistic conditions in the mergerremnant. A variation of the values of S/A and Y e or Y L should nevertheless allow us to cover the relevant condi-tions and thus to estimate the sensitivity of the universalrelations and those observables needed to place limits on M ⋆ TOV on the thermal and out of β -equilibrium effectsand to give an uncertainty range. A. Numerical models of rapidly rotating hot stars
For computing numerical models of hot rapidly ro-tating stars, we have used the
LORENE library [62] . LORENE is a set of C++ classes developed for solv-ing problems in numerical relativity. It contains tools for https://lorene.obspm.fr computing equilibrium configurations of relativistic ro-tating bodies [63] for which combined Einstein and equi-librium equations are solved assuming stationarity, ax-isymmetry, asymptotic flatness, and circularity.Using a quasi-isotropic gauge, the line element ex-pressed in spherical-like coordinates reads [63] ds = − N dt + A (cid:0) dr + r dθ (cid:1) + B r sin θ (cid:0) dϕ + N ϕ dt (cid:1) , (2)with N, N ϕ , A , and B being functions of coordinates( r, θ ). Under the present symmetry assumptions, Ein-stein equations for the four metric potentials reduce to aset of four elliptic (Poisson-like) partial differential equa-tions, in which source terms contain both contributionsfrom the energy-momentum tensor (matter) and non-linear terms with non-compact support, involving thegravitational field itself. More details and explicit ex-pressions can be found in [63].The matter is assumed to behave as a perfect fluid suchthat the energy-momentum tensor can be written as T αβ = ( ε + p ) u α u β + p g αβ , (3)where ε is the total energy density (including rest mass), p the pressure, and u α the fluid four-velocity. The angu-lar velocity of the fluid then becomes Ω := u ϕ /u t . Equi-librium equations are derived from energy and momen-tum conservation, ∇ α T αβ = 0, and become within thepresent setup [53, 59–61] ∂ i ( H + ln N − ln Γ) = e − H m B [ T ∂ i ( S/A ) + µ L ∂ i Y L ] − u ϕ u t ∂ i Ω , (4)where Γ = N u t is the Lorentz factor of the fluid withrespect to the Eulerian observer and S/A the entropyper baryon ( k B = 1), H = ln (cid:18) ε + pm B n B (cid:19) , (5)is the pseudo-log enthalpy with m B being a constant ofthe dimension of a mass . Since in this work we con-sider only uniform rotations with Ω = const, constant S/A , and constant Y e with µ L = 0 or constant Y L , theright hand side of Eq. (4) vanishes and the equilibriumequation takes the same form as in the zero temperatureand β -equilibrium case.Upon computing models of rotating stars, at finitetemperature, an additional difficulty arises from the factthat the surface of the star is no longer well defined sincean extended dilute atmosphere can form, see for instancethe discussion in [57, 64]. For simplicity, we assume thatthe surface corresponds to the density n B = 10 − fm − We chose the value m B = 939 .
565 MeV. ) g/cm (10 ∈ ) d y n / c m P ( RG(SLy4); N ρω NL3- ; NY ρω NL3-FSU2H; NFSU2H; NYSRO(APR)HS(DD2) φΛ BHBSFHoSFHoYHS(IUF) =0 L µ T=0,
FIG. 1. Pressure of cold, β -equilibrated neutron star matteras function of its energy density according to the EoS mod-els employed in this work. The symbols indicate the centralenergy density of the maximum mass configuration for cold, β -equilibrated matter. for all EoS models and any considered combinations of S/A and Y e /Y L . We have checked that our conclusionsdo not depend on the choice of the definition of the sur-face, see Appendix A.Employing LORENE , we find global stellar parame-ters such as gravitational, M G , and baryon, M B , mass,and equatorial circumferential radius R . We additionallycompute the angular momentum, the moment of inertia,and the quadrupole moment. The corresponding expres-sions for the quadrupole moment can be found in [68] and[69]. For our setup with constant S/A , the star’s totalentropy is simply given by
S/A M B . B. Equations of state
The system of equations for solving for the star’sstructure discussed in the preceding section is closedby an EoS. To ensure that our results are not an ar-tifact of a particular choice of EoS model, we haveperformed the same calculations for a set of differentEoS models. There exists a large number of EoS mod-els obtained for cold matter in compact stars. Thenumber of EoS covering the regimes of finite temper-ature and varying electron fraction is however small.These are mostly based on density functional theory.Here we choose a set of EoS models that are based ei-ther on relativistic density functional theory with vari-ous parameterizations or a non-relativistic model basedon Skyrme functional and an empirical extension of avariational microscopic model. These models are rea-sonably compatible with existing constraints from nu-clear experiments, theory, and astrophysics, in partic-ular, they predict maximum masses above 2 M ⊙ [1–3, 70] or at least marginally consistent with this value.To be specific, we consider one non-relativistic density-functional (DFT) model, RG(SLy4) [71, 72]; five vari- Model M ⋆ TOV M ⋆B R . ˜Λ E B n s K E S L ( M ⊙ ) ( M ⊙ ) (km) (MeV) (fm − ) (MeV) (MeV) (MeV)RG(SLy4) 2.06 2.46 11.73 322-353 -15.97 0.159 230.0 32.0 46.0HS(DD2) 2.42 2.92 13.2 758-799 -16.00 0.149 242.6 31.7 55.0HS(IUF) 1.95 2.27 12.64 499-530 -16.40 0.155 231.3 31.3 47.2SFHo 2.06 2.45 11.9 366-401 -16.19 0.158 245.4 31.6 47.1NL3- ωρ φ ωρ NY 2.35 2.77 13.82 1042-1051 -16.24 0.148 271.6 31.7 55.5FSU2H NY 1.99 2.37 13.28 637-653 -16.28 0.150 238.0 30.5 44.5TABLE I. Global parameters of cold neutron stars (first four columns) for EoS considered in this work. These columns list(from left to right) the EoS model acronym, maximum gravitational and baryonic masses, radius of a 1 . M ⊙ star and the tidaldeformability ˜Λ range for the GW170817 event. The latter quantity is calculated assuming for the merger stars the masses m ∈ (1 . , . M ⊙ and m ∈ (1 . , . M ⊙ , which corresponds to the mass ratio range 0 . ≤ q = m /m ≤
1. Theremaining columns list properties of symmetric nuclear matter at saturation density according to the employed EoS model:the binding energy per nucleon E B , saturation density n s , compression modulus K , symmetry energy E S and its slope L .Presently available observational and experimental constraints on listed quantities include a lower limit on the maximumgravitational mass M ⋆ TOV ≥ . ± . M ⊙ [2], simultaneous constraint on the radius and mass of a compact star from theNICER experiment for PSR J0030+0451 R (1 . +0 . − . M ⊙ ) = 13 . +1 . − . km [9] and R (1 . +0 . − . M ⊙ ) = 2 . +1 . − . km [8], and arange for the tidal deformability obtained from the GW170817 event ˜Λ = 300 +500 − (90% credible interval) or ˜Λ = 300 +420 − (90%highest posterior density) for a low spin prior [65]. The nuclear matter properties have been determined as E B = − . ± . n s = 0 . ± .
005 fm − [66], K = 230 ±
40 MeV [67], E s = 31 . ± . L = 58 . ± . ants of relativistic DFT, one with density-dependent cou-plings, HS(DD2) [73, 74], and four with non-linear cou-plings, HS(IUF) [75, 76], SFHo [77], NL3 ωρ [78, 79] andFSU2H [80, 81]; as well as the SRO(APR) model [82, 83].The latter is based on the APR EoS [84], which itselfis partly adjusted to the variational calculation of [85].If available, we compare the above purely nucleonic EoSmodels with the corresponding EoS allowing for the pres-ence of hyperons. These are BHBΛΦ [86] , the extensionof HS(DD2); SFHoY [87], extension of SFHo; NL3 ωρ Y,an extension of NL3 ωρ ; and FSU2HY, an extension ofFSU2H. For NL3 ωρ Y and FSU2HY we adopt the pa-rameterizations in [88] but disregard the σ ∗ -meson field.Except for FSU2H(Y) and NL3 ωρ (Y), EoS data are pub-licly available on the Compose data base [89] . Keyproperties of our collection of the EoS are summarized inTable I together with present constraints and in Fig. 1we show the pressure as a function of energy density forcold, β -equilibrated matter. III. UNIVERSAL RELATIONS FOR FASTROTATING STARS AT FINITE TEMPERATURE
Although the properties of static and rotating starsdepend strongly on the EoS, a series of “universal” re- The EoS model BHBΛ φ contains only Λ-hyperons and not thefull baryon octet. There exists a version, DD2Y [53], based onthe same nucleonic HS(DD2) EoS, which contains the full baryonoctet. For the present purpose, both give very similar results. https://compose.obspm.fr lations have been found between global parameters ofstatic stars which are almost EoS independent (see fora review [90]). These were later extended to slowly andmaximally fast-rotating stars [58, 91–96]. The practicalimportance of such relations resides in their potential toprovide constraints on quantities that are difficult to ac-cess experimentally.It was previously shown that most of the universal rela-tions for slowly rotating stars remain valid at finite tem-perature if the same thermodynamic conditions are main-tained (for example by fixing S/A and Y L ) [57]. Here weextend this investigation to rapidly rotating hot stars. InSec. III A we first address the universal relations betweenglobal properties of non-rotating and Keplerian configu-rations for stars with constant S/A and Y e . In the subse-quent Sec. III B we address the universal relations amongthe normalized moment of inertia, quadrupole moment,and the compactness for the maximum mass configura-tion of a compact star at the Kepler limit. A. Relations between global properties ofnon-rotating and Keplerian configurations
In this subsection, we are interested in a particu-lar class of universal relations, among the parametersof non-rotating and maximally rotating (at the mass-shedding limit) stars. The original motivation for study-ing these relations was to constrain on the stellar radiiusing the measurements of masses and frequencies of sub-millisecond pulsars [91–93]. The non-observation of arapidly rotating pulsar in the remnant of SN1987A led to
Thermo. cond. C ⋆M C ⋆R C ⋆f C ′ ⋆f T = 0, β -eq. 1.2187 (0.0064) 1.3587 (0.0104) 1259.63 (9.72) 1795.30 (4.35) S/A = 2, Y e = 0 . S/A = 2, Y e = 0 . S/A = 3, Y e = 0 . S/A = 3, Y e = 0 . a declining interest in these relations, although searchesof sub-millisecond pulsars continued [97]. The fastest ro-tating pulsar observed to date [98] rotates at 716 Hz,which is still far from Kepler frequencies predicted bythe various EoS of dense matter. The gravitational waveevent GW170817 and the attempt to deduce a maxi-mum mass constraint for a non-rotating cold neutronstar stimulated several recent studies of rigidly [96] anddifferentially rotating stars [51, 52]. Furthermore, theGW190814 event rekindled the interest in the subjectwithin the scenario in which the light component of thismerger event is a maximally rotating compact star [32–38, 40].Equation (1) which expresses the maximum gravita-tional mass of the Keplerian configuration as a functionof the maximum mass of a non-rotating star is an exam-ple of such relations. It was initially proposed in [47, 58]and later on confirmed by extensive computations in [50].Other examples are a relation between the circumferen-tial equatorial radius of the maximum mass configurationat the Kepler limit and the circumferential radius of themaximum mass static configuration [47, 58], R ⋆K = C ⋆R R ⋆ TOV , (6)and the dependence of the rotation frequency of this max-imum mass configuration at the Kepler limit on massand radius of the non-rotating maximum mass configu-ration [91–94], f ⋆K = C ⋆f x ⋆ TOV , (7)where x ⋆ TOV = ( M ⋆ TOV /M ⊙ ) / · (10 km /R ⋆ TOV ) / . Thisfunctional form is actually identical to the Newtonianexpression for the mass shedding frequency of a rotatingsphere, see also the discussion in [93] about its justifica-tion in the relativistic case.Motivated by the findings of Ref. [57] we reinterpretEqs. (1), (6) and (7) as relations between properties ofmaximum mass Keplerian and static configurations withidentical thermodynamic conditions M ⋆K ( S/A, Y e ) = C ⋆M ( S/A, Y e ) M ⋆S ( S/A, Y e ) , (8) R ⋆K ( S/A, Y e ) = C ⋆R ( S/A, Y e ) R ⋆S ( S/A, Y e ) , (9)and f ⋆K ( S/A, Y e ) = C ⋆f ( S/A, Y e ) x ⋆S ( S/A, Y e ) , (10) which implies that the coefficients C ⋆i , i ∈ M, R, f depend on two additional thermodynamic parameters,which are chosen here to be
S/A and Y e . The subscript S refers to static, hot configurations and the subscript“TOV” refers to cold static stars in β -equilibrium.The relations (8), (9), (10) are shown in Fig. 2 forvarious combinations of S/A = 2 , Y e = 0 . , . C ⋆i obtained by a fit to these results are provided inTable II for each considered thermodynamic condition.In the bottom panel of Fig. 2 the dependence of f ⋆K on x ⋆K is considered, too. As a trivial consequence of thelinear dependencies in Eqs. (8), (9), (10) one finds againa linear relation f ⋆K = C ′ ⋆f x ⋆K [96]. Our results show thatuniversality holds reasonably well for hot rapidly rotatingstars as well if the same constant S/A - and Y e -values areconsidered. Similar results were obtained and discussedfor non-rotating in Ref. [57]. Moreover, since our set ofEoS models contains purely nucleonic models as well asmodels with hyperons, we conclude that these relationsare insensitive to the baryonic composition of matter,be it purely nucleonic or with an admixture of hyper-ons. As mentioned above, the proportionality coefficientsdepend, however, on the thermodynamic conditions. Asmall residual dependence of C ⋆R on the EoS remains. Itarises, as previously discussed for cold stars [58], from aweak dependence of the maximum mass static configura-tion on the compactness Ξ S = M ⋆S /R ⋆S , see the inset inFig. 2.Refs. [93, 95, 99] suggested that relations analogousto Eqs. (6) and (7) hold for stars with the same gravita-tional mass (and not only at the maximum of a sequence).These can again be generalized to configurations with thefixed S/A and Y e to find R K ( M ) = C R R S ( M ) , (11) f K ( M ) = C f x S ( M ) , (12)where x S = h ( M/M ⊙ ) · (10 km /R S ( M )) i / . Eqs. (11)and (12) are plotted in Fig. 3, left and right panels re-spectively, for our collection of eleven EoS. The same for acold star as well as for stars with ( S/A = 2, Y e = 0 .
1) arealso plotted. It can be seen that the relation (11) holds,but the proportionality constant C R slightly depends onthe EoS for finite S/A . The relation (12) is confirmed too. M ⋆ S ( M ⊙ ) M ⋆ K ( M ⊙ ) Ξ ⋆ S C ⋆ M R ⋆ S ( km ) R ⋆ K ( k m ) eq.; N β T=0; eq.; NY β T=0; =0.1; N e S/A=2; Y =0.1; NY e S/A=2; Y =0.4; N e S/A=2; Y =0.4; NY e S/A=2; Y =0.1; N e S/A=3; Y =0.1; NY e S/A=3; Y =0.4; N e S/A=3; Y =0.4; NY e S/A=3; Y Ξ ⋆ S C ⋆ R x ⋆ S , x ⋆ K f ⋆ K ( H z ) SK FIG. 2. Top panel: maximum gravitational mass at the Ke-pler limit ( M ⋆K ) vs. maximum gravitational mass of a staticstar ( M ⋆S ), Eq. (8); middle panel: equatorial circumferentialradius of the maximum mass Keplerian configuration ( R ⋆K )vs. circumferential radius of the maximum mass static con-figuration ( R ⋆S ), Eq. (9); bottom panel: rotation frequencyof the maximum mass configuration at the Kepler limit f ⋆K as function of x ⋆S or x ⋆K , i.e., for the maximum mass static( S ) and Keplerian ( K ) configurations, see Eq. (10). The in-sets show the dependence of C ⋆M (top) and C ⋆R (middle) onthe compactness of the maximum mass static configuration.The results correspond to eleven EoS models and differentthermodynamic conditions expressed in terms of S/A and Y e .Results for cold stars are shown for comparison. The observed deviations occur only for M S & . . M ⋆S ,in agreement with previous findings [95]. We thus findagain that the different thermodynamic conditions leadto different values for the proportionality coefficients inEqs. (11) and (12), but the linear relationships remainwell fulfilled. B. Relations between global parameters of themaximum mass configuration at the Kepler limit
For cold compact stars in β -equilibrium numerousother universal relations between global properties havebeen found, notably the so-called “I-Love-Q” rela-tions [100, 101] between the moment of inertia ( I ), thetidal deformability ( λ ), and the quadrupole moment ( Q ).In this context, different relations expressing global prop-erties in terms of the star’s compactness Ξ have receivedmuch attention, too [49, 50, 102, 103].Here, we will consider as an example two such relationsand investigate whether they hold for rapidly rotating hotcompact stars. These are ¯ I = I/M and ¯ Q = QM/J ,with J standing for the angular momentum, expressed aspolynomials of Ξ − [50]¯ I = a Ξ − + a Ξ − + a Ξ − + a Ξ − , (13)¯ Q = b + b Ξ − + b Ξ − + b Ξ − , (14)which were shown to be universal for rigidly and slowlyrotating cold stars [50]. In Ref. [57] these relations wereshown to be universal also for hot stars, as long as thesame pair of constant S/A and Y e /Y L is considered.More specifically, we will investigate the behavior ofthe different quantities taken for the maximum mass Ke-plerian configuration, i.e. , we study ¯ I ⋆K and ¯ Q ⋆K as afunction of Ξ K = M ⋆K /R ⋆K . Note that because of ro-tational stretching of the star, the equatorial and polarradii are different; we recall that R ⋆K refers to the equa-torial circumferential one. Fig. 4 depicts these relation-ships. Each symbol indicates a particular EoS modeland the different colors differentiate different thermody-namic conditions among S/A = 2 , Y e = 0 . , . T = 0 , µ L,e = 0) (solid lines),(
S/A = 2 , Y e = 0 .
1) (dashed lines) and (
S/A = 3 , Y e =0 .
4) (dotted lines). These fits reproduce the exact re-sults with good accuracy; the χ -values are of the orderof 10 − − − (10 − ) for ¯ Q ⋆ vs. Ξ ⋆ ( ¯ I vs. Ξ ⋆ ) are andslightly increasing with S/A . Although some scattering isseen in Fig. 4, the functional form obtained for slowly ro-tating stars applies reasonably well also for the maximummass configuration at the Kepler limit, and universalityis again reasonably well fulfilled. However, the relativedisplacement of points corresponding to a given combi-nation of entropy and electron fraction indicates that thevalues of the parameters a i , b i entering Eqs. (13), and(14) depend on thermodynamic conditions, as expected.
11 12 13 14 15 16 17 ( k m ) K R RG(SLy4); N ρω NL3- ; NY ρω NL3-FSU2H; NFSU2H; NYSRO(APR) HS(DD2) φΛ BHBSFHoSFHoYHS(IUF) =0 L,e µ T=0;
11 12 13 14 15 16 17 (km) S R ( k m ) K R =0.1 e S/A=2; Y ( H z ) K f =0 L,e µ T=0; S x ( H z ) K f RG(SLy4); N ρω NL3- ; NY ρω NL3-FSU2H; NFSU2H; NYSRO(APR)HS(DD2) φΛ BHBSFHoSFHoYHS(IUF) =0.1 e S/A=2; Y
FIG. 3. Left panels: equatorial circumferential radius of the Keplerian configuration R K vs. radius of the non-rotatingconfiguration R S for the same mass, see Eq. (11). Solid lines correspond to the value C R = 1 .
44, obtained by [95] for cold stars.Right panels: rotation frequency at the Kepler limit f K as a function of the x -parameter corresponding to a static configurationwith the same mass, see Eq. (12). The upper panel corresponds to cold stars, the lower panel to hot stars with S/A = 2 and Y e = 0 .
1. The results are shown for a set of EoS models as indicated by the labels.
IV. MAXIMUM MASS OF RIGIDLY ROTATINGHOT STARS
As well-known, for cold compact stars the value of M ⋆K , is 20% larger than M ⋆ TOV , independent of theEoS [47, 50, 58]. As seen in the previous section, thevalue of C ⋆M ≈ . M ⋆ TOV and M ⋆K are computed for cold, β -equilibrated, stars.The assumption of a cold star fails for the merger rem-nant, as the EoS obtains significant thermal correctionsand a hot star potentially out of β -equilibrium shouldbe considered for M ⋆K , as has been argued in the caseof GW170817 event [24, 28–30]. The purpose of this sec-tion is to investigate the relation between M ⋆K for variousthermodynamic conditions and the cold M ⋆ TOV to verifyto which extent thermal and out of equilibrium effectscan change the estimated value of M ⋆ TOV .What are the effects of finite-temperature EoS on themaximum masses of a static and a rapidly rotating star,respectively? First, compact stars expand due to ther-mal effects (e.g. [57, 104]), therefore a same-mass hotstar will have a larger radius than its cold counterpart.Consequently, the larger centrifugal force acting on par-ticles on the stellar surface will be larger and, therefore,the Keplerian limit will be achieved for smaller frequen-cies, which will result in smaller masses at the Keplerlimit. Second, the thermal pressure adds to the degen- eracy pressure which means that a larger mass can besupported against the gravitational pull. Thus we seethat there is an interplay between two competing effects.Figure 5 shows the variation of C ⋆M with S/A for differ-ent purely nucleonic EoS and a constant electron frac-tion of Y e = 0.1. The value of the Keplerian maximummass M ⋆K is normalized to that of the TOV maximummass M ⋆ TOV computed for a cold star. An inspectionof Fig. 5 shows that one EoS model (RG(SLy4)) man-ifests a monotonic increase of C ⋆M over the considered S/A range while the remaining six models show a non-monotonic behavior; the position of the minimum valueof C ⋆M for the latter category of models is situated inthe domain 1 ≤ S/A ≤ .
5. This variety of behaviors isassociated with the interplay between the effects of theincrease of the pressure due to the thermal contributionand expansion of the star with temperature and the as-sociated reduction of the Keplerian frequency. The firsteffect increases C ⋆M , whereas the second one decreases it.In addition to the two factors described above, C ⋆M is ex-pected to depend on the composition of matter as well.The reason is that different compositions and electronfractions were shown to influence the maximum mass andthe star’s compactness [53, 57, 105], too.To disentangle the different effects discussed above,Fig. 6 shows M ⋆K this time normalized to the maximummass of a non-rotating configuration with the same val- ( Q ⋆ M ⋆ / J ⋆ ) K eq.; N β T=0; eq.; NY β T=0; =0.1; N e S/A=2; Y =0.1; NY e S/A=2; Y =0.4; N e S/A=2; Y =0.4; NY e S/A=2; Y =0.1; N e S/A=3; Y =0.1; NY e S/A=3; Y =0.4; N e S/A=3; Y =0.4; NY e S/A=3; Y Ξ ⋆ K ( I ⋆ / M ⋆ ) K FIG. 4. Relations between global properties of maximummass configurations at the Kepler limit: normalized momentof inertia ¯ I as function of the star’s compactness (bottom)and normalized quadrupole moment ¯ Q as function of com-pactness (top). Compactness is here defined with the equa-torial radius. The results correspond to eleven different EoSmodels and various thermodynamic conditions as indicated inthe legend. The lines correspond to Eqs. (13) and (14), re-spectively for ( S/A = 0 , µ L = 0) (solid), ( S/A = 2 , Y e = 0 . S/A = 3 , Y e = 0 .
4) (dotted).
S/A M ⋆ K / M ⋆ T O V RG(SLy4); N ρω NL3-FSU2H; NSRO(APR)HS(DD2)SFHoHS(IUF) =0.1 e Y FIG. 5. Dependence of C ⋆M [see Eq. (1)] on entropy per baryon S/A for fixed Y e = 0 . S/A M ⋆ K / M ⋆ S RG(SLy4); N ρω NL3-FSU2H; NSRO(APR)HS(DD2)SFHoHS(IUF) =0.1 e Y FIG. 6. Same as in Fig. 5 except that the normalization isdone by the maximum gravitational mass of the non-rotatingstar with the same
S/A and Y e values. S/A M ⋆K M ⋆B R ⋆K Ξ ⋆S n ( c ) B ( k B ) ( M ⊙ ) ( M ⊙ ) (km) (fm − )1 2.92 3.44 16.0 0.27 0.722 2.84 3.27 17.1 0.25 0.723 2.79 3.09 19.7 0.21 0.654 2.84 3.01 26.3 0.16 0.46TABLE III. Dependence on S/A of some global propertiesof the maximum mass configuration of stars at Kepler limitfor the HS(DD2) EoS [73, 74] and for fixed Y e = 0 .
1. Listedare gravitational and baryonic masses, equatorial circumfer-ential radius, compactness of the non-rotating configurationand central baryonic number density. ues of
S/A and Y e (instead of the non-rotating TOV massof a cold star). In this way, we eliminate the thermal and Y e -dependence and we observe the change in C ⋆M due en-tirely to the expansion of the star. Indeed, the masses inFig. 6 are observed to almost linearly decrease with S/A and increasing radii as expected. For completeness, wereproduce in Table III as an example the results in thecase of the HS(DD2) EoS. The compactness is given herefor the non-rotating configuration as an indication for theexpansion of the star with increasing entropy. We thusconfirm the earlier expectation born out from the analy-sis of Fig. 5, namely that for low entropies the variationin the mass is controlled predominantly by the expan-sion. As the entropy increases, however, the thermal ef-fects lead to a substantial increase in mass and outweighthe effect due to the growth in radius and thus reducedcompactness.Up to now, we have investigated configurations with aparticular value of constant electron fraction, Y e = 0 . Y L changes our find-ings. To examine the dependence on Y e and Y L , we showin Fig. 7 the maximum masses at Kepler frequency nor-malized to the non-rotating maximum mass as functionof S/A for different values of constant Y e and Y L . TheSFHoY EoS model [87] has been chosen for that purpose,we have checked that other EoS models behave qualita-tively similarly.First, since neutrinos themselves only contributeweakly to the EoS at high density and therefore onlyhave a very small impact on maximum masses, we ob-serve that the main difference between choosing Y e or Y L arises from the fact that the electron fraction is equal tothe hadronic charge fraction Y Q , whereas due to the pres-ence of neutrinos Y L = Y Q . This shift in Y Q induces adifferent behavior of the hadronic part of the EoS whichis well visible in the maximum masses. This implies, too,that for our study it is sufficient to vary either Y e or Y L ifthe range is chosen large enough. Second, since a higherelectron/lepton fraction increases the star’s radius, theKepler frequency is lower and the supported mass, too.Thus the ratio of the Kepler maximum mass M ⋆K and thestatic one M ⋆S decreases with increasing Y e /Y L with themost pronounced reduction observed at low entropies perbaryon, where thermal effects are small. A related ques-tion is whether the presence of muons would change ourresults. It is obvious that in equilibrium, for the ther-modynamic conditions considered here, charged muonswill be abundant. In contrast to core-collapse super-novae, where there are no muons in the progenitor starand complete equilibrium has to be reached by dynam-ical reactions (see e.g. [107]), the two neutron stars be-fore merger contain already muons, such that the mergerremnant should indeed contain a non-negligible fractionof muons. The EoS itself is, however, still dominated bythe hadronic part, such that again the influence of muonson our results would manifest itself only by a potentialshift in the hadronic charge fraction since in the pres-ence of charged muons we have Y Q = Y e + Y µ . In thefollowing discussion we choose Y e = 0 .
1, which should beclose to the conditions in the central part of the mergerremnant, see e.g. Ref.[108], keeping in mind that, if theelectron fraction in the merger remnant is higher, then C ⋆M is reduced. A. Comparison between nucleonic and hyperonicequations of state
So far, when selecting the EoS of dense matter, we as-sumed that neutron star matter contains nucleons andleptons. At densities exceeding several times the nu-clear saturation density, non-nucleonic degrees of free-dom, such as hyperons, meson-condensates, and evenquark matter may appear [109]. Below we explore theeffect of different compositions on the observables dis-cussed by comparing the results for purely nucleonic EoSwith those obtained in the models allowing for the pres-ence of hyperons. In the present context, the focus willbe on the changes in the composition of matter at finitetemperature favoring the onset of hyperons [110, 111],
S/A M ⋆ K / M ⋆ S =0.1 L Y =0.2 L Y =0.4 L Y =0.1 e Y =0.2 e Y =0.4 e Y SFHoY
FIG. 7. The same dependence as in Fig. 6, for three cases ofconstant electron fraction Y e and constant lepton fraction Y L and one specific EoS model, SFHoY [87]. which is expected to change the value of C ⋆M at high en-tropies.Fig. 8 depicts this comparison in detail. The bottompanels display C ⋆M vs S/A for four different EoS mod-els and their hyperonic counterparts. Although qualita-tively the behavior for all the EoS models is the same, aquantitative difference exists between the purely nucle-onic models and those with an admixture of hyperons.More precisely, for low
S/A -values the hyperonic modelsstart with higher values of the ratio C ⋆K /C ⋆ TOV and man-ifest a much stronger decrease of C ⋆M with S/A than thenucleonic models. To understand this, different effectshave to be considered. First, M ⋆ TOV for hyperonic modelsis much smaller than fore purely nucleonic models, sincethe presence of hyperons softens the EoS. Second, thissoftening reduces the radius, thus leading to a compara-tively higher rotation frequency and supported mass atKepler limit, see the upper panels in Fig. 8. The increas-ing abundance of hyperons with increasing
S/A leads toa less pronounced increase in the supported mass dueto thermal effects, which explains the more pronounceddecrease in M ⋆K /M ⋆ TOV with
S/A . V. MAXIMUM TOV MASS FROM GW170817
The event GW170817 and its electromagnetic coun-terpart have been used by several authors to place anupper limit on the value of the maximum mass of staticcold compact star configurations, M ⋆ TOV [24, 28–30]. InRef. [24] a selection of microscopic zero-temperature, EoSwere approximated by piecewise polytropes and a maxi-mum mass M ⋆ TOV ≤ . M ⊙ was inferred from conserva-tive estimates of energy deposited into the short-gamma-ray burst and kilonova ejecta. Ref. [28] used the uni-versal relation between the mass of Keplerian configura-tions and static ones, derived for cold compact stars, seeEq. (1), to place a limit M ⋆ TOV ≤ . +0 . − . M ⊙ consistentwith the one derived in Ref. [24]. A weaker constraint M ⋆ TOV ≤ . M ⊙ was found in Ref. [29], who used EoS0 S/A M ⋆ K / M ⋆ T O V ; N ρω NL3-FSU2H; NHS(DD2)SFHo =0.1 e Y M ⋆ K ( M ⊙ ) S/A =0.1 e Y ; NY ρω NL3-FSU2H; NY φΛ BHBSFHoY
FIG. 8. Dependence of M ⋆K (upper panels) and M ⋆K /M ⋆ TOV (lower panels) on entropy per baryon (see also Fig. 5). The lefttwo panels correspond to nucleonic EoS, the right two panels to EoS which allow for hyperons. based on ad-hoc piecewise polytropic parameterizationin combination with the angular momentum conserva-tion and numerical simulation to show that the mergerremnant at the onset of collapse to a black hole needs notto rotate rapidly.The physical picture of the GW170817 event that un-derlies the argumentation for placing the upper limit on M ⋆ TOV is as follows. Initially, the merger leaves behinda hypermassive neutron star (HMNS) which is differen-tially rotating. The HMNS star spins-down by losses toradiation, whereas the internal dissipation leads to van-ishing internal shears and eventually to the uniform ro-tation. At this stage, the star is in the region of stabilityof supramassive neutron stars, which support themselvesagainst gravitational collapse due to uniform rotation.Subsequently, the star crosses the stability line beyondwhich it is unstable to collapse. While in principle thestar may cross this line (which connects M TOV and M K )at any point, it has been argued that the dynamics of themerger suggest that this crossing occurs in the vicinity of M ⋆K (see, however, Ref. [29], where this assumption hasbeen questioned and the resulting corrections to the lim-its have been explored.) Since the slower rotation impliesa larger maximum mass limit, one should keep in mindthat our estimate below may be relaxed somewhat.The extraction of the upper limit circumvents the fulldynamical study and uses the baryon mass conservationbetween the instances of creation of HMNS in the merger (hereafter t = 0) and the moment of collapse to a black-hole ( t = t c ), which reads M B ( t c , S/A, Y e ) = M B (0) − M out − M ej , (15)where M out refers to the baryon mass of the torus formedaround the black-hole, after the merger and M ej refers tothe baryon mass of the ejecta. The left-hand-side of (15)refers here to a hot supramassive compact star at theinstance of collapse, M B (0) is the baryonic mass of theHMNS formed in the merger at the initial time t = 0.As already mentioned in the introduction, the previ-ous estimates of the M ⋆ TOV were based on EoS of coldbaryonic matter, i.e. they do not account for the ther-mal pressure in the BNS merger remnant and consider inparticular the cold mass on the left-hand side of Eq. (15).Numerical simulations, however, show evidence that theBNS merger remnant is heated up to temperatures ofthe order of tens of MeV. Thus, it is necessary to carryout the analysis of the post-merger remnant taking intoaccount the finite-temperature EoS of baryonic matter.In the left-hand side of Eq. (15) we now substitute M B ( t c , S/A, Y e ) = η ( S/A, Y e ) M ( t c , S/A, Y e )= η ( S/A, Y e ) M ⋆K ( S/A, Y e ) , (16)where the second equality assumes that at the instanceof collapse the star is rotating at the maximum of itsrotational speed, consistent with Ref. [28], but see also1Ref. [29]. The coefficient η ( S/A, Y e ) relates the bary-onic and gravitational masses of the hot compact starat the instance of collapse and is an EoS-dependentquantity. On the right-hand side of Eq. (15) we intro-duce the same quantity for the newly formed object via M B (0) = η (0) M (0), where M (0) = 2 . M ⊙ [14] is thegravitational mass of the merger as measured during in-spiral for the GW170817 event, i.e. for cold stars. Thus,the mass conservation equation (15) can be rewritten as M ⋆K ( S/A, Y e ) = 1 η ( S/A, Y e ) [ η (0) M (0) − M out − M ej ] . (17)It has been estimated from the analysis of GW170817that M ej ≃ . − . M ⊙ [112] and 0 . ≤ M out ≤ . M ⊙ [29]. Taking M out = 0 . ± . M ⊙ and M ej =0 . ± . M ⊙ we have M out + M ej = 0 . ± . η -coefficientsallows one to estimate the Keplerian maximum mass ofa hot supramassive compact star on the left-hand side ofEq. (17).As illustrated in Fig. 9, for cold compact stars basedon our collection of EoS we have η (0) ≃ . ± .
002 for M = 1 . M ⊙ and η (0) ≃ . ± .
001 for M = 1 . M ⊙ .The chosen values of gravitational masses bracket therange 1 . ≤ M ⊙ ≤ . . +0 . − . M ⊙ of the merger remnant at t = 0 [14]. Forour estimates we adopt the value η (0) ≃ . +0 . − . leading to M B (0) = 3 . +0 . − . M ⊙ . We extract values of η ( S/A, Y e ) for two values of entropy as given in Fig. 9assuming that the star is rotating at the Keplerian fre-quency. We then find that η (2 , . ≃ . ± . η (3 , . ≃ . ± . M out + M ej ) /η ( S/A, Y e ) we obtain 0 . ± .
036 and 0 . ± . S/A = 2 and 3 and Y e = 0 .
1, respectively. Substitut-ing the numerical values we find M ⋆K (2 , .
1) = 2 . +0 . − . , M ⋆K (3 , .
1) = 2 . +0 . − . . (18)It was shown recently that several universal relations holdfor hot, isentropic stars out of β -equilibrium [57], if ther-modynamic conditions in terms of entropy per baryonand electron/lepton fraction are fixed. In Section IIIwe have extended these findings to relations betweenstars rotating at Kepler frequency and non-rotating ones.The above limits can thus be used to set a limit on the maximum mass of non-rotating hot compact stars , usingEq. (8) and fitting parameters in Table II. We find M ⋆S (2 , .
1) = 2 . +0 . − . , M ⋆S (3 , .
1) = 2 . +0 . − . . (19)We can also use the limits (18) in combination with theresults shown in Fig. (5) to deduce an upper limit on the maximum mass of cold compact stars . Let us stress η =0; non-rot. L µ T=0; η =0.1; Kepler e S/A=2; Y M ( M ⊙ ) η RG(SLy4); N ρω NL3- ; NY ρω NL3-FSU2H; NFSU2H; NYSRO(APR) HS(DD2) φΛ BHBSFHoSFHoYHS(IUF) =0.1; Kepler e S/A=3; Y
FIG. 9. Dependence of the η parameter on the gravitationalmass for spherically symmetric (non-rotating) stars at T = 0and in β -equilibrium (top), for hot stars rotating at the Keplerlimit for S/A = 2 (middle) and
S/A = 3 (bottom panel) forfixed Y e = 0 . that in this case universality is lost, and C ⋆M assumesvalues in a range 1 . < C ⋆M < .
23 (
S/A = 2) and1 . < C ⋆M < .
29 (
S/A = 3) for the eleven EoS modelsconsidered here. We adopt, for simplicity, a mean value C ⋆M ≃ .
18 for the range 2 ≤ S/A ≤ M ⋆ TOV = 2 . +0 . − . , M ⋆ TOV = 2 . +0 . − . . (20)In this last relation, the errors correspond to 2 σ stan-dard deviation. Here and in the formulas for the massesabove the error propagation for the upper and lower lim-its was computed independently. When comparing the2limits (20) with those of previous works [28, 29], oneshould keep in mind that we used a (recent) value forthe mass M (0), which is slightly lower than the valueof 2 . M ⊙ [113] used in these studies. Our limits on M ⋆ TOV would have been higher had we adopted the largervalue of M (0). It is seen that, if just before collapsethe supermassive neutron star has average entropy perbaryon S/A = 3, then the estimate of the TOV massis significantly relaxed compared to the bound placed inRefs. [24, 28, 30]. According to the discussion in Sec. IV,a higher electron fraction in the merger remnant wouldfurther relax the bound on the TOV mass. The limit wefound is similar to the one in Ref. [29] but for a physi-cally different reason. The last fact indicates that liftingthe assumption that the star rotates at the Keplerianfrequency would further loosen the bound on the TOVmass. Let us, however, stress the fact that universality islost when extracting the cold TOV mass limits (20) fromthe information on the hot merger remnant, independentof the assumption about rotation at collapse, i.e. , thesefinal limits become EoS dependent.
VI. SUMMARY AND CONCLUSIONS
In this work, we have addressed two interrelated top-ics that rely on the knowledge of finite temperature EoSof dense matter. First, we have extended the universalrelations, previously found for hot slowly rotating com-pact stars, to rapidly rotating stars. In particular weconsidered in detail the mass-shedding (Keplerian) limit.Secondly, we discussed an improvement of the previousmaximum mass limits for non-rotating compact stars ob-tained from the GW170817 event in the scenario wherethe merger remnant is a hypermassive compact star thatcollapses to a black hole upon crossing the neutral stabil-ity line as a supramassive (uniformly rotating) compactstar.Our analysis was carried out using a variety of finite-temperature EoS. The collection used includes relativisticdensity functional theory based EoS with nucleonic de-grees of freedom as well as EoS models allowing for thepresence of hyperons. These EoS satisfy the astrophys-ical constraints on neutron stars and nuclear data (nu-clear binding energies, rms radii, etc). As an alternativeto the covariant description, we used a non-relativisticmodel based on a Skyrme-type functional and a param-eterization of a microscopic model. In this way, we wereable to bracket the range of possible predictions for theobservables stemming from various EoS with different un-derlying methods of modeling.When considering universal relations, we followed thestrategy of Ref. [57] to search universality under the samethermodynamical conditions, meaning that we compareobservables of the same star or various rotating and non-rotating configurations at the same fixed entropy perbaryon
S/A and electron fraction Y e . Specifically, weconsidered a class of relations which connect the Ke- plerian configurations with their non-rotating counter-parts given by Eqs. (8)-(10) generalizing the earlier zero-temperature studies to the finite-temperature case. Wefind that these relations are universal (in the sense ofindependence on the EoS) to good accuracy. Similarly,finite-temperature universality propagates beyond zero-temperature results for the relations connecting radii andfrequencies of the same mass Keplerian and non-rotatingstars, see Eqs. (11) and (12). Finally, we have veri-fied (partially) the validity of the I -Love- Q relations bycomputing the first and the last quantity of the triple,specifically, ¯ I = I/M and ¯ Q = QM/J for maximum-mass Keplerian configurations. We find that the univer-sal functional dependence of these quantities on the com-pactness of the star obtained for slowly rotating stars isvalid for the maximum mass configurations at the Keplerlimit as well.The relation between the maximum masses of non-rotating and Keplerian sequences is an important linkneeded for placing limits on the maximum mass of acold, non-rotating star from studies of the millisecondpulsars or gravitational wave analysis of binary neutronstar mergers. We have explored this relation for finite-temperature stars finding that there are two competingeffects: one is the thermal expansion of the star, whichreduces the Kepler frequency and, implicitly, the star’smass at this limit and the additional thermal pressurewhich makes a star of a given mass more stable againstcollapse. If the static and maximally rotating configura-tions are taken at the same values of S/A and Y e , then wefind universality of the coefficient relating their masses,see Fig. 6.The second important application of our analysis con-cerns the upper limit on the maximum mass of a non-rotating cold compact star. Several works, using vari-ous methods and scenarios, claimed that this maximummass can be tightly constrained using the GW170817event [24, 28–30] to the range M ⋆ TOV ≤ . − . M ⊙ ,where the upper range in this limit arises when consider-ing below-Keplerian rotations, instead of Keplerian ones.We have improved on the previous analysis by extractingthe ratio of the baryonic to gravitational masses for hotcompact stars of given S/A and Y e and applying this tothe same scenario. Our central finding is that the upperlimit on the maximum mass of static, cold neutron starsis 2 . +0 . − . ≤ M ⋆ TOV ≤ . +0 . − . for a typical parameter range 2 ≤ S/A ≤ Y e = 0 . ACKNOWLEDGMENTS
We thank N. Stergioulas for useful comments on themanuscript. This work has been partially funded by theEuropean COST Action CA16214 PHAROS “The multi-messenger physics and astrophysics of neutron stars”. A.R. R. acknowledges support from UEFISCDI (Grant No.PN-III-P4-ID-PCE-2020-0293). The work of M. O. hasbeen supported by the Observatoire de Paris throughthe action f´ed´eratrice “PhyFog”. A. S. acknowledges thesupport by the Deutsche Forschungsgemeinschaft (GrantNo. SE 1836/5-1). The authors gratefully acknowledgethe Italian Istituto Nazionale de Fisica Nucleare (INFN),the French Centre National de la Recherche Scientifique(CNRS) and the Netherlands Organization for ScientificResearch for the construction and operation of the Virgodetector and the creation and support of the EGO con-sortium.
Appendix A: Influence of the surface definition onresults
In this appendix, we discuss the sensitivity of our re-sults on the density at which the surface is located. Thisis essential for establishing the validity of our resultsand conclusions. The available data for most finite-temperature EoS models are limited to temperaturesabove T = 0 . S/A considered, many of the EoS models used did not havesolutions for densities below roughly ( 10 − -10 − fm − ).This calls for an extrapolation of the required thermody-namic quantities from the densities where solutions were available to lower densities. Extrapolation of thermody-namic quantities introduces an error in the EoS. To avoidthe above-stated extrapolation we define the surface ofthe star at n B = 10 − fm − uniformly in our modelling.This surface definition allows us to use the data providedfor every EoS model in the parameter range used in ourcalculations.To gauge the amount by which the value of the max-imum mass changes with a variation of the locationof the surface, we refer to the results for M ⋆K in Sec-tion IV. We verified that changing the surface densityfrom n B = 10 − fm − to 10 − fm − resulted in a changeof the value of M ⋆K only in the third decimal. The ex-trapolation has thereby been performed assuming lineardependencies of log ε and log p on log n B with parameterscalculated over the densities covering the lowest availabledata, 10 − ≤ n B ≤ − fm − . The small change in M ⋆K can be understood from the fact that the maximum massis sensitive only to the high-density physics.To quantify the uncertainties on the results in Sec-tion III, we consider again two different values of thedensity at which we define the surface of the star. Thistime, in addition to the value of n B = 10 − fm − for thesurface density, we take a surface at n B = 10 − fm − ,implying again an extrapolation of EoS data over the do-main for which data are not available. We find that theextension of the surface by locating it at a lower densitydiminishes the maximum rotation frequency and that thehigher the entropy per baryon the larger the induced dif-ferences in all studied quantities. However, neither theKepler frequency, nor the quadrupole moment, the mo-ment of inertia or the values of the gravitational mass inthe ranges discussed in Section III are modified by morethan a few per mille upon varying the location of the sur-face. We, therefore, conclude that we can safely definethe surface at n B = 10 − fm − . [1] P. Demorest, T. Pennucci, S. Ransom, M. Roberts,and J. Hessels, Nature , 1081 (2010),arXiv:1010.5788 [astro-ph.HE].[2] J. Antoniadis et al. , Science , 6131 (2013),arXiv:1304.6875 [astro-ph.HE].[3] H. Cromartie et al. (NANOGrav),Nature Astron. , 72 (2019),arXiv:1904.06759 [astro-ph.HE].[4] F. ¨Ozel and P. Freire,Ann. Rev. Astron. Astrophys. , 401 (2016),arXiv:1603.02698 [astro-ph.HE].[5] A. Watts et al. , Proceedings, Advancing As-trophysics with the Square Kilometre Ar-ray (AASKA14): Giardini Naxos, Italy,June 9-13, 2014 , PoS
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