Rotating neutron stars with quark cores
Ishfaq A. Rather, Usuf Rahaman, M. Imran, H. C. Das, A. A. Usmani, S. K. Patra
aa r X i v : . [ nu c l - t h ] F e b Rotating Neutron stars with Quark cores
Ishfaq A. Rather , ∗ Usuf Rahaman , M. Imran , H. C. Das , , A. A. Usmani , and S. K. Patra , Department of Physics, Aligarh Muslim University, Aligarh 202002, India Institute of Physics, Bhubaneswar 751005, India and Homi Bhabha National Institute, Training School Complex, Anushakti Nagar, Mumbai 400094, India
The rotating neutron star properties are studied with a phase transition to quark matter. Thedensity-dependent relativistic mean-field model (DD-RMF) is employed to study the hadron matter,while the Vector-Enhanced Bag model (vBag) model is used to study the quark matter. The starmatter properties like mass, radius,the moment of inertia, rotational frequency, Kerr parameter, andother important quantities are studied to see the effect on quark matter. The maximum mass ofrotating neutron star with DD-LZ1 and DD-MEX parameter sets is found to be around 3 M ⊙ forpure hadronic phase and decreases to a value around 2.6 M ⊙ with phase transition to quark matter,which satisfies the recent GW190814 constraints. For DDV, DDVT, and DDVTD parameter sets, themaximum mass decreases to satisfy the 2 M ⊙ . The moment of inertia calculated for various DD-RMFparameter sets decreases with the increasing mass satisfying constraints from various measurements.Other important quantities calculated also vary with the bag constant and hence show that thepresence of quarks inside neutron stars can also allow us to constraint these quantities to determinea proper EoS. Also, the theoretical study along with the accurate measurement of uniformly rotatingneutron star properties may offer some valuable information concerning the high-density part of theequation of state. I. INTRODUCTION
The compact objects like neutron stars in the knownuniverse are the ideal sources to study the properties andcomposition of high dense matter. The measurement ofmass and the radius for spherically symmetric and staticstars impose constraints on the properties of matter athigh density. The study of rotating neutron star prop-erties may lead to significant new constraints. From thepast decade, the successful discoveries of various gravi-tational waves by LIGO and Virgo collaborations haveallowed us to study the dense matter properties withmore constraints imposed on the neutron star Equationof State (EoS). The measurement of tidal deformabilityfor static neutron stars ruled out many EoSs with eithertoo large or too small maximum mass.The Binary Neutron star (BNS) merger eventGW170817 [1, 2] constrained the maximum mass andthe tidal deformability of neutron stars and hence on theEoS. The total mass of the GW170817 event was around2.7 M ⊙ with the heavier component mass 1.16-1.60 M ⊙ forlow spin priors. The maximum mass approached 1.9 M ⊙ for high spin priors [3]. The tidal deformability depen-dence on the NS radius Λ ∝ R provided a more strongconstraint on the high dense nuclear EoS. A new grav-itational wave event was observed recently by LVC asGW190814 [4] with a black hole merger of mass 22.2-24.3 M ⊙ and a massive secondary component of mass2.50-2.67 M ⊙ . The secondary component of GW190814gained a lot of attention about its nature whether it is ablack hole, or a neutron star, or some other exotic object[5–15].The maximum mass of a neutron star is assumedto be the most important parameter determining the ∗ [email protected] possible outcome of a BNS merger [16–24]. Also,proper knowledge of a neutron star maximum massand radius constraints the EoS at high density [25–29].The precise measurement of masses of millisecond pul-sars like PSR J1614-2230 (1.928 ± M ⊙ [30],PSRJ0348+0432(2.01 ± M ⊙ [31], and PSR J0740+6620(2.04 +0 . − . ) M ⊙ [32] show that the theoretical maximummass of a neutron star should be around 2 M ⊙ . The2.3 M ⊙ of GW170817 is interpreted as possibly an up-per limit on the NS maximum mass [33–35]. However,with the secondary component of GW190814 predictinga maximum mass around 2.5 M ⊙ , the maximum masslimit for neutron star seems to be weekly constrained.The effect of EoS on the properties of rotating neu-tron stars has been studied since the late 90s by variousgroups[35–38]. To investigate the neutron star structureand its properties, the choice of EoS becomes the startingpoint. There proper choice of EoS for neutron stars mat-ter invites theoretical discussions. Every single EoS pro-duces a neutron star with different properties. Despitepredicting several neutron star properties, the composi-tion at several times the normal nuclear density is stillnot known properly. The core of a neutron star is consid-ered to be a nuclear matter in β -equilibrium and charge-neutral conditions. Neutron, proton, electron, and muonare the basic components of the core of a neutron star.The neutron star structure with several exotic degrees offreedom like quarks, kaons, and hyperons are also stud-ied [8, 39–42]. The presence of such exotic phases affectsthe neutron star properties significantly.The neutron star matter containing only hadrons isstudied employing different model parameters at highdensities. The Density functional theories (DFT) havebeen widely used to determine the saturation propertiesof high dense nuclear matter [43–48]. At saturation den-sity, the nuclear matter EoS is well constrained. Thenuclear matter properties are determined with less uncer-tainty. These EoSs at several times the normal nucleardensity describe the neutron star properties. The rela-tivistic mean-field (RMF) model has been very successfulin describing both finite and infinite nuclear matter[49].The basic mechanism involves the interaction of nucleonsvia mesons. Different mesons like ρ , σ , ω , δ have reducedthe large uncertainties present in the nuclear matter prop-erties and constrained the nuclear matter properties towell within the limits [50–56]. The RMF Equation ofstates like NL3 [60] and BigApple [13, 61] determine neu-tron stars with a maximum mass in the range 2.6-2.7M ⊙ .The density-dependent RMF (DD-RMF) model containsthe density-dependent coupling constants replacing theself-and cross-coupling of various mesons in the basicRMF model [57]. DD-RMF parameters like DD-ME1[58], DD-ME2 [59] generate very massive neutron starswith a 2.3-2.5 M ⊙ maximum mass. Several new DD-RMFparameter sets were proposed recently like DD-LZ1[62],DD-MEX [63], DDV, DDVT, and DDVTD [64]. Theserecently proposed parameter sets are divided into twocategories. The DD-LZ1 and DD-MEX parameter setsproduce very stiff EoS and hence a large NS maximummass, belong to the stiff EoS group. Parameter sets likeDDV, DDVT, and DDVTD produce soft EoS and hencelie in the softer EoS group. Both the stiff and the softEoS groups are used in the current study to determinethe neutron star properties for the static and rotatingcase.The exotic degrees of freedom like quarks have beenstudied over the past decade. The presence of quarksin the core of neutron stars at very high densities hasbeen proposed [29]. Thus the phase transition to thequark matter inside neutron stars is possible at very highdensity [65, 66]. A neutron star with hadrons in the corefollowed by a phase transition to the quark matter atseveral times the normal nuclear density is termed as theHybrid star [46, 67–70].The MIT Bag model [66, 71, 72] was first proposed tostudy the strange and hybrid stars. The Nambu-Jona-Lasinio (NJL) model [73–77] was later introduced whichexplained the quark matter more precisely than the bagmodel. The modified NJL models have been very success-ful in explaining the stable HSs and also satisfying therecent GW170817 constraints [78, 79]. The modified Bagmodel, termed as Vector-Enhanced Bag model (vBag)[80] was introduced as an effective model to study theastrophysical processes. The vBag model is favored overthe simple bag model and NJL model as it accounts forthe repulsive vector interactions along with the DynamicChiral Symmetry Breaking (D χ SB). The repulsive vectorinteraction and the deconfinement for the construction ofa mixed-phase allowed it to describe the strange/hybridstars which attain the 2 M ⊙ limit.In the present work, we study the properties of therotating neutron stars by considering a phase transitionfrom hadron matter to quark matter. The star matter properties like mass, radius, the moment of inertia, Kerrparameter are studied along with some other importantproperties. The dependence of these quantities on theneutron star mass is discussed. Several properties of astatic star like mass, radius, and tidal deformability arealso discussed.This article is organized as follows: the DD-RMFmodel for hadron matter and vBag model for quark mat-ter and the phase transition properties are discussed insection(II. The static and rotating neutron structure andvarious properties associated with the star matter are dis-cussed in sec.III. In sec. (IV), the parameter sets for thenuclear matter and the saturation density properties aredefined. The EoS for the hadronic and hybrid star config-urations are explained. The static and rotating neutronstar properties like mass, radius, the moment of inertiaare discussed in subsection(IV.3). Finally, the summaryand concluding remarks are given in sec. (V). II. THEORY AND FORMALISM
The Relativistic Mean-Field (RMF) Lagrangian in-volves the interaction between the nucleons through var-ious mesons defined as Dirac particles. The most ba-sic and simplest RMF Lagrangian involves the scalar-isoscalar sigma σ and vector-isoscalar ω mesons withoutany interaction among themselves [81], which results inlarge nuclear matter incompressibility K [82]. Bogutaand Bodmer included a nonlinear self-coupling of the σ field which lowered the value of nuclear matter incom-pressibility to reasonable values. and vector-isovector ρ meson[83]. Apart from σ , ω , and ρ mesons, the additionof scalar-isoscalar δ meson is included to study the isovec-tor effect on the scalar potential of the nucleon. Bothnuclear matter and neutron star matter properties areobtained which lie well within the limits [54, 73]. TheEffective field theory motivated RMF (E-RMF) is the ex-tended RMF model which includes all possible self- andcross-couplings between the mesons [84–86]. The RMFmodel has gained a lot of success in investigating bothfinite and infinite nuclear matter properties. The vari-ous nonlinear meson coupling terms can be replaced bythe density-dependent nucleon-meson coupling constantsin the Density-Dependent Relativistic Hartree-Fock (DD-RHF) [87–89] and density-dependent RMF model (DD-RMF) [57]. The density-dependent models take into ac-count the nuclear medium effect originated by the rel-ativistic Brueckner-Hartree-Fock mode [57]. Unlike theRMF model, the coupling constants in the DD-RMF aredensity-dependent i.e, they vary with the density. TheDD-RMF coupling constants depend either on the scalardensity ρ s or the vector density ρ B , but the vector densityparameterizations are considered usually which doesn’tinfluence the total energy of the system.The DD-RMF Lagrangian density is given as: L = X α = n,p ¯ ψ α ( γ µ i∂ µ − g ω ( ρ B ) ω µ − g ρ ( ρ B ) γ µ ρ µ τ ! − M − g σ ( ρ B ) σ − g δ ( ρ B ) δτ !) ψ α + 12 ∂ µ σ∂ µ σ − m σ σ ! + 12 ∂ µ δ∂ µ δ − m δ δ ! − W µν W µν + 12 m ω ω µ ω µ − R µν R µν + 12 m ρ ρ µ ρ µ , (1)where ψ α , ( α = n, p ) denotes the neutron and protonwave-function. g σ , g ω , g ρ ,and g δ are the meson couplingconstants which are density-dependent, and m σ , m ω , m ρ and m δ are the masses for σ, ω, ρ and δ mesons respec-tively. The tensor fields W µν and R µν are defined as W µν = ∂ µ W ν − ∂ ν W µ ,R µν = ∂ µ R ν − ∂ ν R µ (2)The coupling constants of σ and ω mesons for DD-MEX,DDV, DDVT, and DDVTD parameter sets are expressedas the fraction function of the vector density. The density-dependent coupling constants for various parameteriza-tions are given as g i ( ρ B ) = g i ( ρ ) f i ( x ) , (3)where the function f i ( x ) is given by f i ( x ) = a i b i ( x + d i ) c i ( x + d i ) , i = σ, ω (4)as a function of x = ρ B /ρ , where ρ is the nuclear mattersaturation density.For the function f i ( x ), the number of constraint con-ditions defined as f i (1) = 1, f ′′ σ (1) = f ′′ ω (1), f ′′ i (0) = 0reduce the number of free parameters from eight to threein the eq.4. Out of them, the first two constraints are a i = 1 + c i (1 + d i ) b i (1 + d i ) , c i d i = 1 (5)For the isovector ρ and δ mesons, the coupling constantsare given by an exponential dependence as g i ( ρ B ) = g i ( ρ ) exp [ − a i ( x − g i is fixedat ρ B =0 for i = σ, ω ; g i ( ρ B ) = g i (0) f i ( x ) . (7)There are only four constraint conditions for σ and ω in the DD-LZ1 parameter set. The constraint f ′′ σ (1) = f ′′ ω (1) is removed, which changes the coupling constantof ρ meson as g ρ ( ρ B ) = g ρ (0) exp ( − a i x ) . (8)Following the Euler-Lagrange equation, we obtain theequation of motion for nucleons and mesons.The scalar density ρ s , baryon density ρ B , and theisovector densities ρ s , and ρ are defined as ρ s = X α = n,p ¯ ψψ = ρ sp + ρ sn = X α π ) Z k α d k M ∗ α E ∗ α , (9) ρ B = X α = n,p ψ † ψ = ρ p + ρ n = X α π ) Z k α d k, (10) ρ s = X α ¯ ψτ ψ = ρ sp − ρ sn , (11) ρ = X α ψ † τ ψ = ρ p − ρ n . (12)The effective masses of nucleons are given as M ∗ p = M − g σ ( ρ B ) σ − g δ ( ρ B ) δ, (13), and M ∗ n = M − g σ ( ρ B ) σ + g δ ( ρ B ) δ (14)Also, E ∗ α = p k α + M ∗ α , (15)is the effective energy of nucleon with nucleon momentum k α . The energy-momentum tensordetermines the totalenergy density and the pressure for the nuclear matteras E DD = E H + E kin ,P DD = P H + P kin (16)where, E H and P H are the energy density and the pressureof hadronic matter given as E H = 12 m σ σ − m ω ω − m ρ ρ + 12 m δ δ + g ω ( ρ B ) ωρ B + g ρ ( ρ B )2 ρρ ,P H = − m σ σ + 12 m ω ω + 12 m ρ ρ − m δ δ − ρ B X R ( ρ B ) , (17)and E kin and P kin are the energy density and pressurefrom the kinetic part, E kin = 18 π h k α E ∗ α (cid:0) k α + M α (cid:1) + M α ln M α k α + E ∗ α i ,P kin = 124 π h k α E ∗ α (cid:0) k α − M α (cid:1) + 3 M α ln k α + E ∗ α M ∗ α i . (18)For the neutron star matter, the β -equilibrium condi-tion is given as µ e = µ µ = µ n − µ p . (19)where, µ α = n,p = p k α + M ∗ α + h g ω ( ρ B ) ω + g ρ ( ρ B )2 ρτ + X R ( ρ B ) i ,µ l = µ,e = q k l + m l . (20)The charge neutrality condition implies q total = X i = n,p q i k i / (3 π ) + X l q l k l / (3 π ) = 0 . (21)To study the phase transition from hadron matter toquark matter, the Vector-Enhanced Bag model (vBag)[80] is employed which is an extension of the simple bagmodel [66, 71, 72]. The vBag model accounts for theDynamic Chiral Symmetry Breaking (D χ SB) and alsothe additional repulsive vector interactions which allowthe strange stars to achieve 2 M ⊙ limit on the maximummass and hence satisfy the constraints from recentlymeasured masses of pulsars like PSR J1614-2230 [30],PSR 0348+0432 [31], and PSR J0740+6620 [32].The energy density and pressure in the vBag modelfollow as [90] E Q = X f = u,d,s E vBag,f − B dc , (22) P Q = X f = u,d,s P vBag,f + B dc , (23)where, B dc represents the deconfined bag constant intro-duced which lowers the energy per particle and thus fa-voring stable strange matter. The energy density andpressure of a single quark flavor are defined as: E vBag,f ( µ f ) = E F G,f ( µ ∗ f ) + 12 K ν n F G,f ( µ ∗ f ) + B χ,f , (24) P vBag,f ( µ f ) = P F G,f ( µ ∗ f )+ 12 K ν n F G,f ( µ ∗ f ) − B χ,f , (25)where FG denotes the zero temperature Fermi gas. Thecoupling constant parameter K ν results from the vectorinteractions and controls the stiffness of the star mattercurve [91]. The bag constant for a single quark flavoris denoted by B χ,f . The chemical potential µ ∗ f of thesystem is defined as µ f = µ ∗ f + K ν n F G,f ( µ ∗ F ) . (26)An effective bag constant is defined in the vBag model sothat the phase transition to quark matter occurs at thesame chemical potential B eff = X f = u,d,s B χ,f − B dc . (27)The effective bag constant B eff is an extension to thedeconfined bag constant to allow the phase transitionto occur at the same chemical potential. This alsoillustrates how the B eff can be used in a two and threeflavor quark matter.The charge neutrality and β -equilibrium conditions forthe quark matter are23 ρ u −
12 ( ρ d + ρ s ) − ρ e − ρ u = 0 , (28) µ s = µ d = µ u + µ e ; µ u = µ e . (29)The density range over which a phase transition ex-ists between hadron matter and quark matter is deter-mined by beta-equilibrium and charge-neutral conditions[67, 92–95]. The phase transition can be either by alocal charge condition (Maxwell Construction) [96] orglobal charge neutrality condition (Gibbs Construction)[67]. The global charge neutrality condition allows thehadron matter and the quark matter to be separatelycharged, unlike the local charge-neutrality condition. Inthis study, I used the Gibbs method to construct thehadron-quark phase transition. The global charge neu-trality condition follows as χρ Q + (1 − χ ) ρ H + ρ l = 0 , (30)where the quark volume fraction in the mixed-phase givenby χ = V Q / ( V T ) which varies from χ = 0 to χ = 1in the pure hadron and pure quark phases respectively.The charge densities of quarks, hadrons, and leptons arerepresentes by ρ Q , ρ H , and ρ l respectively.The equations governing the mixed-phase chemical po-tential, pressure, energy, and the baryon density are de-fined as: µ B,H = µ B,Q ; µ e,H = µ e,Q , (31)and P H ( µ B , µ e ) = P Q ( µ B , µ e ) = P MP . (32) ε MP = χε Q + (1 − χ ) ε H + ε l , (33)and ρ MP = χρ Q + (1 − χ ) ρ H . (34)The above equations determine the properties of themixed-phase and combined with the hadron equationsgenerate overall the properties of the star. III. NEUTRON STAR STRUCTURE ANDPROPERTIESIII.1. Static neutron star
For a spherically symmetric, static neutron star, themetric element has the Schwarzschild form as ( G = c =1) ds = − e φ ( r ) dt + e r ) dr + r ( dθ + sin θdφ ) , (35)where the metric functions e − φ ( r ) and e r ) are definedas e − φ ( r ) = (1 − γ ( r )) − , (36) e r ) = (1 − γ ( r )) , (37)with γ ( r ) = 2 M ( r ) /r (38)The energy-momentum tensor reduces the Einstein Fieldequations to well known Tolman Oppenheimer Volkoffcoupled differential equations given by[97, 98] dP ( r ) dr = − [ E ( r ) + P ( r )][ M ( r ) + 4 πr P ( r )] r (1 − M ( r ) /r ) (39)and dM ( r ) dr = 4 πr E ( r ) (40)where M ( r ) represents the gravitational mass. Theboundary conditions P (0) = P c , M (0) = 0 allows one to solve the above differential equations and determinethe properties of a neutron star.The tidal deformability λ is defined as the ratio of theinduced quadrupole mass Q ij to the external tidal field E ij as [99, 100] λ = − Q ij E ij = 23 k R (41)The dimensionless tidal deformability Λis defined asΛ = λM = 2 k C (42)where C = M/R is the compactness parameter and k is the second love number. The expression for the lovenumber is written as [99] k = 85 (1 − C ) [2 C ( y − n C (4( y + 1) C + (6 y − C +(26 − y ) C + 3(5 y − C − y + 6) − − C ) (2 C ( y − − y + 2) log (cid:16) − C (cid:17)o − . (43)The function y = y ( R ) can be computed by solving thedifferential equation [100, 101] r dy ( r ) dr + y ( r ) + y ( r ) F ( r ) + r Q ( r ) = 0 , (44)where, F ( r ) = r − πr [ E ( r ) − P ( r )] r − M ( r ) , (45) Q ( r ) = 4 πr (cid:16) E ( r ) + 9 P ( r ) + E ( r )+ P ( r ) ∂P ( r ) /∂ E ( r ) − πr (cid:17) r − M ( r ) − " M ( r ) + 4 πr P ( r ) r (1 − M ( r ) /r ) . (46)The above equations are solved for spherically symmetricand static neutron stars to determine the properties likemass, radii, and tidal deformability. III.2. Rotating neutron stars
For a rapidly rotating neutron star with a nonax-isymmetric configuration, they would emit gravitationalwaves until they achieve axisymmetric configuration.The rotation deforms the neutron stars. Here we studythe rapidly rotating neutron stars assuming a stationary,axisymmetric space-time. The energy-momentum tensorfor such a perfect fluid describing the matter is given by T µν = ( E + P ) u µ u ν + P g µν , (47)where the first term represents the contribution from mat-ter. u µ denotes the fluid-four velocity, E is the energy den-sity, and P is the pressure. For rotating neutron stars,the metric tensor is given by [102–104] ds = − e ν ( r,θ ) dt + e ψ ( r,θ ) ( dφ − ω ( r ) dt ) + e µ ( r,θ ) dθ + e λ ( r,θ ) dr , (48)where the gravitational potentials ν , µ , ψ , and λ are thefunctions of r and θ only. The Einstein’s field equationsare solved for the given potential to determine the phys-ical properties that govern the structure of the rotatingneutron stars. Global properties like gravitational mass,equitorial radius, the moment of inertia, angular momen-tum and quadrupole moment are calculated.For rotating neutron stars, the angular momentum J iseasy to calculate. By defining the angular velocity of thefluid relative to a local inertial frame, ¯ ω ( r ) = Ω − ω ( r ),the ¯ ω satisfies the following differential equation1 r ddr r j d ¯ ωdr ! + 4 r djr ¯ ω = 0 , (49)where j = j ( r ) = e − ( ν + λ ) / .The angular momentum of the star is then given bythe relation J = 16 R d ¯ ωdr ! r = R , (50)which relates the angular velocity asΩ = ¯ ω ( R ) + 2 JR . (51)The moment of inertia defined by I = J/ Ω, is given by[105, 106] I ≈ π Z R ( E + P ) e − φ ( r ) h − m ( r ) r i − ¯ ω Ω r dr, (52)The properties of a rotating neutron star are calculatedby using the RNS code [46, 107–109]. IV. RESULTS AND DISCUSSIONSIV.1. Parameter sets
To determine the properties of static and rotating neu-tron stars, we have used several recent DD-RMF pa-rameterizations like DD-MEX [63], DD-LZ1 [62], andDDV, DDVT, DDVTD [64]. Apart from the basic DD-MEX and DD-LZ1 parameter sets, the DDV, DDVT, andDDVTD sets include the necessary tensor couplings ofthe vector mesons to nucleons.Table(I) shows the necessary nucleon masses, mesonmasses, and the coupling constants of the used parame-ter sets. The meson coupling constants in the DD-LZ1
TABLE I. Nucleon and meson masses and different couplingconstants for various DD-RMF parameter sets.DD-LZ1 DD-MEX DDV DDVT DDVTD m n m p m σ m ω m ρ g σ ( ρ ) 12.0014 10.7067 10.1369 8.3829 8.3793 g ω ( ρ ) 14.2925 13.3388 12.7704 10.9871 10.9804 g ρ ( ρ ) 15.1509 7.2380 7.8483 7.6971 8.0604 a σ b σ c σ d σ a ω b ω c ω d ω a ρ E/A ),incompressibility ( K ), symmetry energy ( J ), slope parame-ter ( L ) at saturation density for various DD-RMF parametersets. DD-LZ1 DD-MEX DDV DDVT DDVTD ρ ( fm − ) 0.158 0.152 0.151 0.154 0.154 E/A (MeV) -16.126 -16.140 -16.097 -16.924 -16.915 K (MeV) 231.237 267.059 239.499 239.999 239.914 J (MeV) 32.016 32.269 33.589 31.558 31.817 L (MeV) 42.467 49.692 69.646 42.348 42.583 M ∗ n /M M ∗ p /M parameter set are the values at zero density while forthe other parameter sets, these coupling constants areobtained at the nuclear matter saturation density ρ .Table (II) displays the nuclear matter properties likeSymmetry energy, incompressibility, slope parameter atsaturation density for various DD-RMF parameter sets.The binding energy E/A for all the parameter sets lieswell around -16 MeV. The symmetry energy value J liesin the range 31-33 MeV which is compatible with themeasurement from various astrophysical observations, J = 31 . ± .
66 MeV [110]. The slope parameter L lies outside the constraints L = 59 . ± .
100 1000 ε (
MeV/fm )1101001000 P ( M e V / f m ) DD-LZ1DD-MEXDDVDDVTDDVTD GW170817+GW190814
FIG. 1. (color online) Equation of State (EoS) profile forDD-LZ1, DD-MEX, DDV, DDVT, and DDVTD parametersets. The recently combined constraints from GW170817[2]and GW190814 [4] are also shown.
IV.2. Equation of State
Fig.1 displays the various EoS for various DD-RMFparameter sets for a neutron star in beta-equilibriumand charge-neutrality conditions. The DDVTD param-eter set produces the stiffest EoS at low densities andsoftest EoS at high density as compared to other param-eter sets. DDV and DDVT sets produce soft EoS at highdensities which represent a neutron star with low maxi-mum mass. The DD-LZ1 and DD-MEX parameter setsproduce stiff EoSs at high densities and hence large NSmaximum mass. The recently combined constraints fromthe gravitational wave data GW170817 and GW190814are also shown in the green shaded region. For a unifiedEoS, Baym-Pethick-Sutherland (BPS) EoS [113] is usedfor the outer crust part which lies in the density region10 -10 g/cm . The outer crust EoS does not effect theneutron star maximum and the radius, so we have chosenthe outer crust BPS EoS for all parameter sets. The in-ner crust EoS has an high imapct on the NS radius at thecanonical mass, R . M ⊙ , while a small change is seen inthe maximum mass and the maximum radius [114]. Forthe parameter sets used in this work, the inner crust EoSis not available. Thus, we have employed the DD-ME2inner crust EoS [59] for all the parameter sets but withmatching symmetry energy slope parameter [115, 116].For the mixed-phase hadron matter and quark matter,the Gibbs Construction method, which corresponds tothe global charge neutrality between two different phases,has been employed. The effective bag model with aneffective bag constant B / is used to study the quarkmatter. The coupling constant parameter K ν is fixed at6 GeV − for three flavor configuration. Three differentvalues of effective bag constant are used B / eff =130, 145& 160 MeV.Fig. (2) shows the hadron-quark phase transition withDD-RMF parameter sets for hadronic matter and vBag
500 1000 1500 2000 2500 3000 ε (MeV/fm )100200300400500600700800900 P ( M e V / f m ) DD-LZ1DD-MEXDDVDDVTDDVTDB=130 MeVB=145 MeVB=160 MeV
FIG. 2. (color online) Equation of state for the hadron-quarkphase transition for DDV, DDVT, DDVTD, DD-LZ1, andDD-MEX hadronic parameter sets and vBag quark matter atdifferent effective bag constants. The solid lines represent thehybrid EoS at B / eff =130 MeV while dashed and dot-dashedlines represent the hybrid EoS at B / eff =145 & 160 MeV, re-spectively. model for quark matter using the Gibbs method for con-structing mixed-phase which ensures a smooth transitionbetween the two different phases. With the increasing ef-fective bag constant B / eff , the phase transition densityincreases, and the mixed-phase region also expands. Forbag constant B / eff =130 MeV, the mixed-phase regionstarts from ρ = 2 . ρ and extends upto 4.03 ρ . For B / eff =145 & 160 MeV, the mixed-phase region lies inthe density range (3.03-4.82) ρ and (3.69-5.31) ρ , respec-tively. DD-LZ1 and DD-MEX parameter sets producestiff EoS and thus the mixed-phase region lies in higherpressure region than the DDV, DDVT, and DDVTD pa-rameter sets. The mixed-phase region in the DD-LZ1parameter sets lies in the density range (2.56-4.23) ρ for130 MeV, (2.73-4.95) ρ for 145 MeV ,and (3.04-5.43) ρ for 160 bag constants. Thus, DD-LZ1 and DD-MEX setspredict a large mixed-phase region as compared to theother parameter sets. IV.3. Neutron star properties
Figure (3) displays the hadronic mass vs radius curvesfor DD-LZ1, DD-MEX, DDV, DDVT, and DDVTD pa-rameter sets. The DD-LZ1 set produces an NS witha maximum mass of 2.55 M ⊙ with a radius of 12.30km. DD-MEX set produces a 2.57 M ⊙ neutron starwith a 12.46 km radius. Both these parameter sets sat-isfy the constraints from recent gravitational wave dataGW190814 and recently measured mass and radius ofPSR J0030+0451, M = 1 . +0 . − . M ⊙ and R = 12 . +1 . − . km by NICER [28, 117]. The DDV, DDVT, and DDVTDpredict a maximum mass of 1.95, 1.93, and 1.85 M ⊙ for astatic neutron star with radius 12.11, 11.40, and 11.33 km M ( M O ) DD-LZ1DD-MEXDDVDDVTDDVTD . GW190814
PSR J0740+6620PSR J0348+0433PSR J1614-2230
FIG. 3. (color online) Mass vs Radius profiles for pure DD-LZ1, DD-MEX, DDV, DDVT, and DDVTD parameters for astatic neutron star. The recent constraints on mass from thevarious gravitational wave data and the pulsars are shown inthe shaded region [4, 30–32] and radii [28, 117] are also shown. at the canonical mass, R . M ⊙ , respectively. DDV andDDVT satisfy the mass constraint from PSR J1614-2230and radius constraint from PSR J0030+0451. DDVTDparameter set produces a slightly lower maximum massneutron star than PSR J1614-2230. The shaded regionsdisplay the constraints on the maximum mass of neu-tron star from PSR J1614-2230 (1.928 ± M ⊙ )[30],PSR J0348+0432 (2.01 ± M ⊙ ) [31], PSR J0740+6620(2.14 +0 . − . M ⊙ ) [32], and GW190814 (2.50-2.67 M ⊙ ) [4].
10 12 14 16 18 20 22 24R (km)01234 M ( M O ) Pure HadronB = 130 MeV145 MeV160 MeV
12 14 16 18 20 22 24R (km) a) DD-LZ1 b) DD-MEX . GW190814
PSR J0740+6620PSR J0348+0432PSR J1614-2230
FIG. 4. (color online) Mass-Radius profile for pure hadronicand hybrid rotating neutron stars for a) DD-LZ1 and b) DD-MEX parameter sets at bag values B / eff =130, 145 & 160 MeV.The shaded regions represent recent constraints on the massfrom various measured astronomical observables. The rotating neutron star mass-radius profile for DD-LZ1 and DD-MEX parameter sets are shown in fig.(4).The solid lines represent the pure hadronic star while
10 12 14 16 18 20 22 2400.511.522.53 M ( M O )
12 14 16 18 20 22 24R (km) 12 14 16 18 20 22 24
Pure HadronB =130 MeV145 MeV160 MeV a) DDV b) DDVT c) DDVTD . GW190814
PSR J0740+6620PSR J0348+0432PSR J1614-2230
FIG. 5. (color online) Same as fig.(4), but for a) DDV, b)DDVT, and c) DDVTD EoSs. the dashed lines represent the hybrid star at differentbag constants. The effective bag constant B / eff is writ-ten as B / for ease convenience. The DD-LZ1 EoS pro-duces a pure hadronic rotating neutron star with a max-imum mass of 3.11 M ⊙ with a radius of 18.23 km. Withthe phase transition from hadron matter to quark mat-ter, the maximum mass and the corresponding radiusdecrease with the increase in the bag constant. For DD-LZ1 set, the maximum mass decreases from 3.11 M ⊙ to2.98 M ⊙ for B / =130 MeV, and to 2.75 M ⊙ and 2.64 M ⊙ for B / =145 & 160 MeV, respectively. The radius at thecanonical mass decreases from 18.32 km for pure hadronmatter to 16.64 km for hybrid star matter at 160 MeVbag value. Similarly, for the DD-MEX parameter set, themaximum mass for pure hadronic matter is 3.15 M ⊙ atradius 16.53 km which reduces to 2.69 M ⊙ at 16.63 km forbag constant 160 MeV. Thus while the pure hadronic ro-tating neutron stars predict a large maximum mass, thephase transition to quark matter lowers the maximummass and the radius thereby satisfying the constraintfrom GW190814.Figure (5) displays the mass-radius relation forhadronic and hybrid rotating neutron stars with DDV,DDVT, and DDVTD EoSs. The maximum mass for arotating neutron star with DDV EoS is 2.37 M ⊙ with a17.41 km radius at the canonical mass. Both the maxi-mum mass and the radius decrease to 2.23, 2.13, 2.01 M ⊙ and 16.91, 16.68, 16.13 km for bag constants B / =130,145, & 160 MeV, respectively, thereby satisfying the 2 M ⊙ constraint. For DDVT, the maximum mass reduces from2.28 to 1.99 M ⊙ . The radius at the canonical mass also de-creases from 17.82 km to 16.01 km. Similarly for DDVTDEoS, the rotating neutron star maximum mass reduces to1.93 M ⊙ from 2.21 M ⊙ at B / =160 MeV. For all the pa-rameter sets, the phase transition to quark matter lowersthe maximum mass which satisfies the 2 M ⊙ limit.The measurement of neutron star moment of inertiais important as it follows a universal relation with thetidal deformability and the compactness of an NS. Themoment of inertia as a function of gravitational mass forthe rotating neutron stars is displayed in fig.(6). The con-straint on the moment of inertia obtained from the jointPSR J0030+0451, GW170817, and the nuclear data anal-ysis predicting I . = 1 . +0 . − . × kg.m is shown[118]. The predicted moment of inertia of pulsar PSRJ0737-3093A, I . = 1 . +0 . − . × g.cm is alsoshown [119]. For pure hadronic matter, DD-LZ1 andDD-MEX EoSs predicts an NS with a moment of iner-tia 2.22 & 2.35 × g.cm , respectively. The phasetransition to the quark matter reduces the moment ofinertia to a value 1.65 & 1.93 × g.cm for DD-LZ1and DD-MEX parameter sets at bag constant B / =160MeV, which satisfies the constraint from [118–120]. O )01234567 I ( g c m ) Pure HadronB =130 MeV145 MeV160 MeV O ) Jiang et al 2019Lim et al 2019Landry et al 2019 a) DD-LZ1 b) DD-MEX . .
FIG. 6. (color online) Moment of inertia variation with thegravitational mass for a) DD-LZ1 and b) DD-MEX EoSs. Theconstraints on the moment of inertia of canonical neutron starmass are also shown [120]. The constraint from joint PSRJ0030+0451, GW170817, and the nuclear data analysis areshown by green bar [118]. The predicted moment of inertia ofpulsar J0737-3039A using Bayesian analysis of nuclear EoS isshown by brown bar [119].
Fig.(7) displays the moment of inertia variation withthe gravitational mass for DDV, DDVT, and DDVTD pa-rameter sets. The solid lines represent the pure hadronicmatter, while the dashed lines represent the hadron-quark mixed phase at bag constants B / =130, 145, &160 MeV. The constraints on the moment of inertia ob-tained from millisecond pulsars (MSP) with GW170817universal relations are shown [121]. For DDV EoS, themoment of inertia of a pure hadronic star is found tobe 2.01 × g.cm while for DDVT and DDVTD EoSs,the value is found to be 1.95 & 1.88 × g.cm , respec-tively. For hybrid EoS, the moment of inertia is loweredto a value of 1.71 × g.cm for DDV set at bag con-stant 160 MeV. For DDVT and DDVTD sets, this valuereduces to 1.68 & 1.64 × g.cm respectively for 160MeV bag constant. The phase transition to quark matter I( g c m ) Pure HadronB =130 MeV145 MeV160 MeV O ) 0 0.5 1 1.5 2 2.5 3 . a) DDV b) DDVT c) DDVTD FIG. 7. (color online) Same as fig.(6) but for a) DDV, b)DDVT, and c) DDVTD parameter sets. The constraints onthe moment of inertia of MSPs obtained from universal rela-tions with GW170817 are shown [121]. produces as NS with the moment of inertia that satisfiesthe constraints from various measurements.For a static neutron star, the maximum mass is usu-ally determined as the first maximum of a M- ε c curve,i.e, ∂M/∂ε c =0, where ε c is the central energy density.For rotating neutron stars, the situation becomes compli-cated. To determine the axisymmetric instability points,several methods have been used in the literature. Fried-man et al. [122] described a method to determine thepoints at which instability is reached in rotating neutronstars [123, 124]. (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) ∂M ( ε c , J ) ∂ε c (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) J = constant = 0 , (53)where J is the angular momentum of the star. Once thesecular instability is initiated, the star evolves until itreaches a point of dynamical instability where the gravi-tational collapse starts [107].The above equation defining an upper limit on themass at a given angular momentum is sufficient but nota necessary condition for the instability. The limit on thedynamic instability is shown in ref.[125].Figure (8) shows the variation in the gravitational massof a rotating neutron star with the central density forDD-LZ1 and DD-MEX parameter sets. The maximummass of 3.11 M ⊙ for DD-LZ1 EoS is produced at density1.40 × g/cm . The phase transition to quark mat-ter at bag constant B / =160 MeV reduces the maxi-mum mass to 2.64 M ⊙ at 1.17 × g/cm energy den-sity. For the DD-MEX parameter set, the maximummass of 3.15 M ⊙ occurs at 1.47 × g/cm reduces to2.69 M ⊙ at 1.25 × g/cm . Figure (9) represents thesame variation in the gravitational mass of a NS with thecentral density for softer EoS group (DDV, DDVT, andDDVTD).0 ε c (g/cm )0123 M ( M O ) Pure HadronB =130 MeV145 MeV160 MeV ε c (g/cm ) . a) DD-LZ1 b) DD-MEX FIG. 8. (color online) Gravitational mass versus central den-sity for a) DD-LZ1 and b) DD-MEX EoSs. The solid linesrepresent pure hadronic rotating stars while dashed lines rep-resent the hybrid stars at different bag constants. M ( M O ) Pure HadronB =130 MeV145 MeV160 MeV ε c (x10 g/cm ) 0.5 1 1.5 2 2.5 3 . a) DDV b) DDVT c) DDVTD FIG. 9. (color online) Same as fig.(8) but for a) DDV, b)DDVT, and b) DDVTD EoSs.
A star rotating at a keplerian rate becomes unstabledue to the loss of mass from its surface. The massshedding limit angular velocity which is the maximumangular velocity of a rotating star is the keplerianangular velocity evaluated at the equatorial radius R e ,i.e, Ω J =0 K = Ω orb ( r = R e ).Figure (10) displays the neutron star gravitationalmass as a function of the Kepler frequency ν k for a)DD-LZ1 and b) DD-MEX EoSs. The limits imposedon the rotational frequency by various pulsars like PSRB1937+21 ( ν =633 Hz)[126], PSR J1748-2446ad ( ν =716Hz)[127], and XTE J1739-285( ν =1122 Hz)[128] are alsoshown. For DD-LZ1 EoS, the pure hadronic star rotateswith a maximum frequency of 1525 Hz. For hybrid starat bag value B / =130 MeV, the star rotates with a fre-quency of 1405 Hz. For 145 and 160 bag values, thefrequency obtained is 1431 and 1497 Hz, respectively. ν κ ( kHz)00.511.522.533.5 M ( M O ) Pure HadronB /14 =130 MeV145 MeV160 MeV ν κ ( kHz) . a) DD-LZ1 b) DD-MEX PS R B + PS R J - a d X TE J - FIG. 10. (color online) Kepler frequency v k variation with theneutron star gravitational mass for a) DD-LZ1 and b) DD-MEX EoSs. Solid lines represent pure hadronic star while thedashed lines represent hybrid star at bag constants B / =130,145 & 160 MeV. The vertical lines represent the observationallimits imposed on the frequency from rapidly rotating pul-sars like PSR B1937+21 ( ν =633 Hz)[126], PSR J1748-2446ad( ν =716 Hz)[127], and XTE J1739-285( ν =1122 Hz)[128]. Similarly for DD-MEX EoS, the maximum rotational fre-quency for a pure hadronic star is found to be 1503 Hz,which changes to 1361 Hz for the hybrid star at bag con-stant 130 MeV, 1408, and 1438 Hz for the hybrid star at145 & 160 MeV bag values. Both pure hadronic and hy-brid stars rotate at a frequency greater than ν =1122 Hz.Also, the hybrid star M- ν curves coincide with the purehadronic curves upto ν k ≈
400 Hz, which then show atransition towards higher frequency depending upon thebag constant. M ( M O ) Pure HadronB /14 =130 MeV145 MeV160 MeV ν κ ( kHz) 0 0.5 1 1.5 2 . a) DDV b) DDVT c) DDVTD PS R B + PS R J - a d X TE J - FIG. 11. (color online) Same as fig.(10) but for a) DDV, b)DDVT, and b) DDVTD EoSs.
Figure (11) displays the same gravitational mass vari-ation with the Kepler frequency for DDV, DDVT, and1DDVTD parameter sets. For the DDV set, the purehadronic star rotates with a rotational frequency of 1498Hz. The hybrid stars produced with bag constants B / =130, 145, and 160 MeV have a rotational frequencyof 1454, 1446, and 1520 Hz, respectively. SImilarly forDDVT and DDVTD EoSs, the pure hadronic star hasa rotational frequency of 1473 and 1418 Hz respectivelywhich then changes to 1503 and 1456 Hz respectively forthe hybrid star at 160 MeV bag constant. Thus it is seenthat the hybrid stars with a hadron-quark phase transi-tion initially produce a low mass neutron star with a lowrotating frequency than the pure hadronic star at lowbag constant ( B / =130 MeV). By increasing the bagconstant, the NS maximum mass further decreases butthe rotational frequency increases. Thus the hybrid starscan withstand higher rotation as the star is denser andhas low maximum mass as compared to the pure hadronicstar.A useful parameter to characterize the rotation of astar is the ratio of rotational kinetic energy T to thegravitational potential energy W , β = T /W . For a rotat-ing neutron star, if β > β d , where β d is the critical value,the star will be dynamically unstable. The critical value β d for a rigidly rotating star is found to be 0.27 [129, 130].However, for different angular momentum distributions,the value lies in the range 0.14 to 0.27 [131–133].The variation in the T /W ratio of the pure hadronand hybrid stars with the gravitational mass is shown infig.(12). The
T /W ratio for pure hadronic stars is 0.147and 0.145 for DD-LZ1 and DD-MEX parameter sets, re-spectively. The hybrid stars have large
T /W ratio andincrease with the bag constant. For DD-LZ1 set, theratio increases from 0.150 at B / =130 MeV to 0.153 at B / =160 MeV. For the DD-MEX set, the ratio increasesto 0.149 and 0.151 for bag values 130 MeV and 160 MeVrespectively. The large value of the T /W ratio in hybridstars is since the quark stars are bound by strong inter-action, unlike hadron stars which are bound by gravity.Figure(13) depicts the
T /W variation with the gravi-tational mass for DDV, DDVT, and DDVTD parametersets. For DDV EoS, the pure hadronic star predicts a
T /W ratio of 0.127, which lies below the critical value β d . For hybrid stars, this ratio increases 0.142 for bagconstant 160 MeV thereby satisfying the critical β d limitand hence becomes dynamically unstable and emits grav-itational waves. Similarly, for DDVT and DDVTD EoS,the pure hadron star produces a ratio of 0.115 and 0.108while the hybrid star at B / =160 MeV gives a value of0.127 and 0.125, respectively.The Einstein’s field equations provide Kerr space-timefor so-called Kerr black holes which can be fully de-scribed by the angular momentum J and the gravita-tional mass M of rotating black holes[134, 135]. Thecondition J ≥ GM /c must be satisfied to define a sta-ble Kerr black hole. The gravitational collapse of massiverotating neutron stars constrained to angular momentumconservation creates a black hole with mass and angularmomentum resembling that of a neutron star. Thus, it’san important quantity used in the study of black holes as O )00.020.040.060.080.10.120.140.160.180.2 T / W Pure HadronB =130 MeV145 MeV160 MeV O ) a) DD-LZ1 b) DD-MEX . . FIG. 12. (color online) Rotational kinetic energy to the grav-itational potential energy ratio
T /W variation with the grav-itational mass for a) DD-LZ1 and b) DD-MEX EoSs. Solidlines represent pure hadronic stars while the dashed lines rep-resent hybrid stars at bag constants B / =130, 145 & 160MeV. T / W Pure HadronB =130 MeV145 MeV160 MeV O ) 0 0.5 1 1.5 2 2.5 a) DDV b) DDVT c) DDVTD . FIG. 13. (color online) Same as fig(12) but for a) DDV, b)DDVT, and c) DDVTD EoSs. well as rotating neutron stars. The Kerr parameter leadsto the possible limits on the compactness of neutron starsand also can be an important criterion for determiningthe final fate of the collapse of a rotating compact star[134, 136]. The Kerr parameter is described by the rela-tion κ = cJGM (54)where, J is the angular momentum and M is the grav-itational mass of rotating neutron star. The Kerr pa-rameter for black holes is an important and fundamentalquantity with a maximum value of 0.998 [137], but it’simportant for other compact stars as well.To constrain the Kerr parameter for neutron stars, we2 O )0.40.50.60.70.8 κ Pure HadronB =130 MeV145 MeV160 MeV O ) . . a) DD-LZ1 b) DD-MEX FIG. 14. (color online) Kerr parameter κ as a function ofgravitational mass for a) DD-LZ1 and b) DD-MEX EoSs. Theplot shows both pure hadronic stars (solid lines) and hybridstars (dashed lines) at different bag constants. studied the dependence of the Kerr parameter on the neu-tron star gravitational mass as displayed in fig.(14) andfig.(15) for the given parameter sets. From fig.(14), theKerr parameter for pure hadronic DD-LZ1 and DD-MEXparameter sets is found to be 0.64 and 0.67 respectively.This parameter increases for the hybrid stars with a max-imum value of 0.73 at B / =160 MeV for the DD-LZ1set. For the DD-MEX set, the maximum value of theKerr parameter is 0.75 at 160 MeV bag constant. ForDD-LZ1 parameter sets, the Kerr parameter remains al-most unchanged once the star reaches a mass of around1.4 M ⊙ for pure hadronic matter and around 1.2 M ⊙ forhybrid configurations. For DDV, DDVT, and DDVTDparameter sets as shown in fig.(15), the Kerr parametervalue for pure hadronic stars at the maximum mass is0.64, 0.62, and 0.61 respectively. For hybrid star configu-rations, the value increases to 0.75 for all parameter setsat bag constant B / =160 MeV. The Kerr parameter forhybrid star configurations remains almost identical to thehadron star up to almost 0.4 M ⊙ .Another important quantity related to the neutronstars is the redshift which has been investigated deeply[36, 103, 138]. The measurement of redshift can imposeconstraints on the compactness, and in turn, on the neu-tron star EoS. For a rotating neutron star, if the detectoris placed in the direction of the polar plane of the star,the polar redshift, also called gravitational redshift, canbe measured. For a detector directed tangentially, theforward and backward redshifts can be measured. Theexpression for the polar redshift is given as Z P (Ω) = e − ν (Ω) − ν is the metric function. The variation of thepolar redshift with the gravitational mass is depictedin fig.(16) for DD-LZ1 and DD-MEX EoSs. For pure κ Pure HadronB =130 MeV145 MeV160 MeV a) DDV b) DDVT c) DDVTD
M(M O ) . FIG. 15. (color online) Same as fig(14) but for a) DDV, b)DDVT, and c) DDVTD EoSs. hadronic stars, the polar redshift is found to be around1.1 for both EoSs. With the quark matter present in theneutron stars, the polar redshift for DD-LZ1 decreases toa value 0.89, 0.84, and 0.64 for bag constants B / =130,145, & 160 MeV, respectively. Similarly for the DD-MEX set, the redshift decreases up to 0.68 for the 160MeV bag constant. The observational limits imposed onthe redshift from 1E 1207.4-5209 ( Z P =0.12-0.23)[139] ,RX J0720.4-3125( Z P =0.205 +0 . − . )[46], and EXO 07482-676( Z P =0.35)[140] are also shown. The redshift predic-tion of Z P =0.35 for EXO 07482-676 was based of thenarrow absorption lines in the x-ray bursts. However, itwas later seen that the spectral lines from EXO 07482-676 may be narrower than predicted[141]. Therefore theestimates of the redshift from EXO 07482-676 are uncer-tain.For the softer EoS group, the polar redshift variationwith the gravitational mass is shown in fig.(17) for bothpure hadron matter and the hybrid star configurations.For theDDV set, the polar redshift is found to be 0.75 forpure hadronic star and decreases to 0.50 for the hybridstar at bag constant 160 MeV. For DDVT and DDVTDEoSs, the redshift decreases from 0.72 and 0.70 for purehadron matter to 0.55 and 0.53 respectively for hybridstar at B / =160 MeV. The neutron star redshift pro-vided by measuring the gamma-ray burst annihilationlines has been interpreted as gravitationally redshifted511 keV electron-positron pair annihilation from the neu-tron star surface[142]. If this interpretation is correct,then it will support an NS with redshift in the range0 . ≤ Z P ≤ . M ⊙ andhence the phase transition to quark matter will decrease3 O )0.250.50.7511.25 Z p Pure HadronB =130 MeV145 MeV160 MeV O ) a) DD-LZ1 b) DD-MEX . .
1E 1207.4-5209RX J0720.4-3125EXO 07482-676
FIG. 16. (color online) Polar redshift vs gravitational mass forpure hadron stars and hybrid star configurations for a) DD-LZ1 and b) DD-MEX EoSs. The observational limits imposedon the polar redshift from 1E 1207.4-5209 (grey band)[139],RX J0720.4-3125 (brown band)[46], and EXO 07482-676 (or-ange horizontal line)[140] are shown. O )00.10.20.30.40.50.60.70.80.9 Z p Pure HadronB =130 MeV145 MeV160 MeV O ) 0 0.5 1 1.5 2 2.5M(M O ) a) DDV b) DDVT c) DDVTD . . . FIG. 17. (color online) Same as fig.(16) but for a) DDV, b)DDVT, and c) DDVTD EoSs. the maximum mass to a value not satisfying any recentconstraints on the mass and other neutron star proper-ties. However, to study the properties of pure hadronicEoS, the mass-radius profile for static stars is explainedabove (fig.3). In addition to this, we study the tidal de-formability of the given parameter sets. The equationsdescribing the tidal deformation and its dependence onthe star matter properties are described above.The dimensionless tidal deformability Λ as a func-tion of NS mass for the hadronic EoSs is shown infig.(18). The constraints from the joint PSR J0030+0451,GW170817, and the nuclear data analysis at the NScanonical mass,Λ . = 370 +360 − [118],and the tidal de-formability of MSP obtained from GW170817 with uni-versal relations[121] are shown. The non-parametric con-straints on the tidal deformability, Λ = 451 +241 − is alsoshown[119]. The tidal deformability depends upon the O )040080012001600 Λ DD-LZ1DD-MEXDDVDDVTDDVTV . Landry et al (2020)Jiang et al (2019)MSP Kumar-Landry (2019)
FIG. 18. (color online) Dimensionless tidal deformabilityas a function of NS mass for DD-LZ1, DD-MEX, DDV,DDVT, and DDVTD EoSs. The non-parametric constraintson the tidal deformability of canonical neutron star massare shown[119]. The constraint from joint PSR J0030+0451,GW170817, and the nuclear data analysis[118],and the tidaldeformability of MSP obtained from GW170817 with univer-sal relations[121] are also shown.
NS mass and the radius. The value decreases with theincreasing mass and becomes very small at the NS max-imum mass. The dimensionless tidal deformability forDD-LZ1, DD-MEX, DDv, DDVT, and DDVTD EoSs atthe canonical mass is found to be 727.17, 791.60, 391.23,337.51, and 281.05, respectively. All these values lie wellwithin the constraints defined. The DD-MEX set pro-duces a little higher value of the tidal deformability. Thevalue of Λ . for softer group EoS (DDV, DDVT, andDDVTD) is significantly lower than the stiffer group (DD-LZ1 and DD-MEX) because of the small maximum massand the corresponding radius. However, the stiffer groupEoSs cannot be neglected in comparison to the softergroup. The precise measurement of the tidal deformabil-ity for the BNS mergers with a maximum mass around2 M ⊙ by future gravitational wave detectors will lowerthe uncertainties in these values thereby constraining theEoSs. V. SUMMARY AND CONCLUSION
The properties of static and rotating neutron starsare studied with a hadron-quark phase transition. Thehadronic matter is studied by employing the density-dependent Relativistic Mean-Field(DD-RMF) model.Recent parameter sets like DDV, DDVT, and DDVTDalong with the DD-LZ1 and DD-MEX are used to studythe hadronic EoS. The quark matter is studied using amodified version of the Bag model, the Vector-EnhancedBag model (vBag). The vBag model includes the nec-essary repulsive vector interactions and Dynamic Chi-ral Symmetry Breaking (D χ SB). The vBag model cou-pling parameter K ν controlling the stiffness of the EoS4curve is held constant at 6 GeV − . The effective bagconstant B / eff is varied taking the values 130, 145, and160 MeV. The Gibbs technique is used to construct themixed-phase between hadrons and quarks which accountsfor the global charge neutrality of the system. The prop-erties like mass, radius, and the tidal deformability ofstatic NS are studied. For rotating neutron stars, thevariation in the NS properties like maximum mass, ra-dius, the moment of inertia, rotational frequency, Kerrparameter, etc are studied in the presence of quark mat-ter.For static NSs, the maximum mass for the DD-LZ1and DD-MEX is found to be 2.55 and 2.57 M ⊙ respec-tively, forming a stiffer EoS group. For DDV, DDVT,and DDVTD EoSs, the maximum is found to be around1.9 M ⊙ , thus lying in the softer EoS group. The phasetransition properties for static NSs are not studied forthe softer EoS group as it would result in a very lowmaximum mass not satisfying any mass constraints.For rotating neutron stars, the maximum mass isfound to be 3.11 M ⊙ for the DD-LZ1 set which in pres-ence of quark matter reduces to 2.64 M ⊙ satisfying therecent GW190814 mass constraint. The DD-MEX setalso predicts a maximum mass of 3.15 M ⊙ decreasing to2.69 M ⊙ for B / eff =160 MeV bag constant. For the softerEoS group, the rotating neutron star mass lies in therange 2.2-2.3 M ⊙ which then reduces with the increasingbag constant to satisfy the 2 M ⊙ limit. The radiusalso decreases with the increase in the bag constant.The moment of inertia for the stiffer group lies in therange (2.2-2.3) × g.cm for pure hadron EoSs. Thephase transition to quark matter reduces the value to1.7 × g.cm satisfying the recent constraints. For thesofter group of EoSs, The moment of inertia is loweredin presence of quark matter to satisfy the constraintsfrom GW170817 with universal relations.The variation in the rotational frequency of an NS withthe gravitational mass is also studied. The pure hadronicEoSs produce NSs with high rotational frequencies. ForDD-LZ1 and DD-MEX, the rotational frequency at themaximum mass is 1525 and 1503 Hz, respectively. ForDDV, DDVT, and DDVTD EoSs, the frequency obtainedis in the range 1400-1500 Hz. The quarks produce the hy-brid star configurations with larger rotational frequenciesas the quark star are more compact than hadron stars.Initially, for hybrid star configuration at B / =130 MeV,the rotating with frequency smaller than a pure hadronicstar is formed. As the bag constant increases, the max-imum mass decreases, and the corresponding frequencyincreases. All the pure hadronic and hybrid star config-urations produce NS with a frequency higher than thehighest measured frequency of ν =1122 Hz. The ratio of rotational kinetic energy to the gravita-tional potential energy β = T /W is studied to determinethe dynamical stability of the rotating neutron stars. For β > β d (= 0 . − . T /W ratio for rotating pure hadronic stars is foundto be 0.147 and 0.145 for DD-LZ1 and DD-MEX EoSs.The quark matter phase transition tends to increase the
T /W ratio with decreasing mass. For bag constant 160MeV, the ratio is found to be 0.153 and 0.151 for DD-LZ1and DD-MEX EoSs, respectively. For softer EoS group,this ratio lies below the critical limit for pure hadronicstars, but increases to a value well within the criticallimit.The Kerr parameter is calculated for the rotating neu-tron stars whose measurement allows to constrain thecompactness of a star and hence EoS. The precise valueof the Kerr parameter for neutron stars is not knownyet, but a maximum value of 0.75 is seen in most of thetheoretical works. For the given parameterization sets,the Kerr parameter value lies around 0.65 for the stiffergroup and 0.6 for the softer group. Following the in-verse relationship with the gravitational mass, the Kerrparameter increases in the presence of quarks. For bothstiffer and softer EoS groups, the value attains a maxi-mum value of 0.75, which remains almost unchanged asthe mass increases beyond 1 M ⊙ . The dependence of po-lar redshift on the NS mass is also calculated. It is seenthat the polar redshift decreases in presence of quarks.The redshift parameter measured for all hybrid star con-figurations lies well above the predicted value from EXO07482-676, Z P =0.35.For static, spherically symmetric stars, we have alsocalculated the dimensionless tidal deformability. It isseen that all the parameter sets predict a value of tidaldeformability satisfying the constraints from various mea-surements.Thus, it’s clear that the presence of quarks inside theneutron stars affects both static and rotating star mat-ter properties. Eliminating the uncertainties present inthe values of these quantities will allow us to rule outvery stiff and very soft EoSs. The measurement of tidaldeformability for rotating neutron stars will help us toconstraint its properties and hence determine a properEoS in the near future. Additional gravitational-waveobservations of binary neutron star mergers and more ac-curate measurements of other neutron star properties likemass, radius, tidal deformability will allow the universalrelation-based bounds on canonical deformability to befurther refined. The theoretical study of uniformly rotat-ing neutron stars, along with the accurate measurements,may offer new information about the equation of state inhigh denstiy regime. Besides, neutron stars through theirevolution may provide us with a criterion to determinethe final fate of a rotating compact star. [1] B. P. Abbott and R. Abbott et al. , . [2] B. P. Abbott and R. Abbott et al. (The LIGO Sci- entific Collaboration and the Virgo Collaboration),Phys. Rev. Lett. , 161101 (2018).[3] B. P. Abbott and R. Abbott et al. (LIGO Sci-entific Collaboration and Virgo Collaboration),Phys. Rev. X , 011001 (2019).[4] R. Abbott and T. D. A. et al. ,The Astrophys. Jour. , L44 (2020).[5] V. Dexheimer, R. O. Gomes, T. Kl¨ahn, S. Han, andM. Salinas, “Gw190814 as a massive rapidly-rotatingneutron star with exotic degrees of freedom,” (2020),arXiv:2007.08493 [astro-ph.HE].[6] H. Tan, J. Noronha-Hostler, and N. Yunes, “Neutronstar equation of state in light of gw190814,” (2020),arXiv:2006.16296 [astro-ph.HE].[7] M. Fishbach, R. Essick, and D. E. Holz,Astrophys. J. , L8 (2020).[8] I. A. Rather, A. A. Usmani, and S. K. Patra, “Hadron-quark phase transition in the context of gw190814,”(2020), arXiv:2011.14077 [nucl-th].[9] D. A. Godzieba, D. Radice, and S. Bernuzzi, (2020),arXiv:2007.10999 [astro-ph.HE].[10] E. R. Most, L. J. Papenfort, L. R. Weih, and L. Rez-zolla, MNRAS , L82–L86 (2020).[11] N.-B. Zhang and B.-A. Li,The Astrophys. Jour. , 38 (2020).[12] A. Tsokaros, M. Ruiz, and S. L. Shapiro, (2020),arXiv:2007.05526 [astro-ph.HE].[13] F. J. Fattoyev, C. J. Horowitz, J. Piekarewicz, andB. Reed, (2020), arXiv:2007.03799 [nucl-th].[14] Y. Lim, A. Bhattacharya, J. W. Holt, and D. Pati,(2020), arXiv:2007.06526 [nucl-th].[15] I. Tews, P. T. H. Pang, T. Dietrich, M. W. Coughlin,S. Antier, M. Bulla, J. Heinzel, and L. Issa, (2020),arXiv:2007.06057 [astro-ph.HE].[16] M. Shibata and K. Taniguchi,Phys. Rev. D , 064027 (2006).[17] Y. Sekiguchi, K. Kiuchi, K. Kyutoku, and M. Shibata,Phys. Rev. Lett. , 051102 (2011).[18] K. Hotokezaka, K. Kiuchi, K. Kyutoku, T. Mu-ranushi, Y.-i. Sekiguchi, M. Shibata, and K. Taniguchi,Phys. Rev. D , 044026 (2013).[19] A. Bauswein, T. W. Baumgarte, and H.-T. Janka,Phys. Rev. Lett. , 131101 (2013).[20] C. Palenzuela, S. L. Liebling, D. Neilsen, L. Lehner,O. L. Caballero, E. O’Connor, and M. Anderson,Phys. Rev. D , 044045 (2015).[21] S. Bernuzzi, D. Radice, C. D. Ott, L. F.Roberts, P. M¨osta, and F. Galeazzi,Phys. Rev. D , 024023 (2016).[22] L. Lehner, S. L. Liebling, C. Palenzuela, O. L. Ca-ballero, E. O’Connor, M. Anderson, and D. Neilsen,Class. and Quan. Grav. , 184002 (2016).[23] D. Radice, A. Perego, F. Zappa, and S. Bernuzzi,The Astrophys. Jour. , L29 (2018).[24] S. K¨oppel, L. Bovard, and L. Rezzolla,The Astrophys. Jour. , L16 (2019).[25] K. Hebeler, J. M. Lattimer, C. J. Pethick, andA. Schwenk, Phys. Rev. Lett. , 161102 (2010).[26] K. Hebeler, J. M. Lattimer, C. J. Pethick, andA. Schwenk, The Astrophys. Jour. , 11 (2013).[27] J. M. Lattimer, Annual Review of Nuclear and Particle Science , 485–515 (2012).[28] M. C. Miller, C. Chirenti, and F. K. Lamb,The Astrophys. Jour. , 12 (2019).[29] E. Annala, T. Gorda, A. Kurkela, J. N¨attil¨a, andA. Vuorinen, Nature Phys. , 907–910 (2020). [30] P. B. Demorest, T. Pennucci, S. M. Ransom,M. S. E. Roberts, and J. W. T. Hessels,Nature , 1081–1083 (2010).[31] J. Antoniadis and P. C. C. Freire et al. ,Science (2013), 10.1126/science.1233232.[32] H. T. Cromartie and E. Fonseca et al. ,Nature Astronomy , 72–76 (2019).[33] M. Shibata, E. Zhou, K. Kiuchi, and S. Fujibayashi,Phys. Rev. D , 023015 (2019).[34] L. Rezzolla, E. R. Most, and L. R. Weih,The Astrophys. Jour. , L25 (2018).[35] B. Margalit and B. D. Metzger,The Astrophys. Jour. , L19 (2017).[36] G. B. Cook, S. L. Shapiro, and S. A. Teukolsky,Astrophys. J. , 823 (1994).[37] N. Stergioulas, L. Rev. Relativ. , L19 (1998).[38] V. Paschalidis and N. Stergioulas,L. Rev. Relativ. , L19 (2017).[39] I. A. Rather, A. A. Usmani, and S. K. Patra,J. Phys. G: Nucl. and Part. Phys. , 105104 (2020).[40] I. A. Rather, A. Kumar, H. C. Das, M. Im-ran, A. A. Usmani, and S. K. Patra,Int. J. Mod. Phys. E , 2050044 (2020).[41] D. D. Ofengeim, M. E. Gusakov, P. Haensel, andM. Fortin, Phys. Rev. D , 103017 (2019).[42] A. Sulaksono, International Journal of Modern Physics E , 1550007 (2015),https://doi.org/10.1142/S021830131550007X.[43] D. Vautherin and D. M. Brink,Phys. Rev. C , 626–647 (1972).[44] H. Shen, H. Toki, K. Oyamatsu, and K. Sumiyoshi,Nuclear Physics A , 435 – 450 (1998).[45] H. Shen, Phys. Rev. C , 035802 (2002).[46] Douchin, F. and Haensel, P.,A&A , 151–167 (2001).[47] S. S. Bao and H. Shen, Phys. Rev. C , 045807 (2014).[48] S. S. Bao, J. N. Hu, Z. W. Zhang, and H. Shen,Phys. Rev. C , 045802 (2014).[49] J. D. Walecka, Ann. Phys. , 491 (1974).[50] C. J. Horowitz and J. Piekarewicz,Phys. Rev. Lett. , 5647–5650 (2001).[51] Y. Sugahara and H. Toki,Nuclear Physics A , 557 – 572 (1994).[52] J. Boguta and A. Bodmer,Nuclear Physics A , 413 – 428 (1977).[53] B. D. Serot, Physics Letters B , 146 – 150 (1979).[54] S. K. Singh, S. K. Biswal, M. Bhuyan, and S. K. Patra,Phys. Rev. C , 044001 (2014).[55] B. Kumar, S. K. Singh, B. K. Agrawal, and S. K. Patra,Nucl. Phys. A , 197–207 (2017).[56] B. Kumar, S. K. Patra, and B. K. Agrawal,Phys. Rev. C , 045806 (2018).[57] R. Brockmann and H. Toki,Phys. Rev. Lett. , 3408–3411 (1992).[58] T. Nikˇsi´c, D. Vretenar, P. Finelli, and P. Ring,Phys. Rev. C , 024306 (2002).[59] G. A. Lalazissis, T. Nikˇsi´c, D. Vretenar, and P. Ring,Phys. Rev. C , 024312 (2005).[60] G. A. Lalazissis, J. K¨onig, and P. Ring,Phys. Rev. C , 540–543 (1997).[61] H. C. Das, A. Kumar, B. Kumar, S. K. Biswal, andS. K. Patra, (2020), arXiv:2009.10690 [nucl-th].[62] B. Wei, Q. Zhao, Z.-H. Wang, J. Geng, B.-Y. Sun, Y.-F.Niu, and W.-H. Long, Ch. Phys. C , 074107 (2020).[63] A. Taninah, S. Agbemava, A. Afanasjev, and P. Ring,Phys. Lett. B , 135065 (2020). [64] S. Typel and D. Alvear Terrero,Eur. Phys. Jour. A , 160 (2020).[65] E. Witten, Phys. Rev. D , 272 (1984).[66] E. Farhi and R. L. Jaffe, Phys. Rev. D , 2379 (1984).[67] N. K. Glendenning, Phys. Rev. D , 1274 (1992).[68] F. ¨Ozel, D. Psaltis, S. Ransom, P. Demorest, and M. Al-ford, The Astrophys. Jour. , L199–L202 (2010).[69] T. Kl¨ahn, R. Lastowiecki, and D. Blaschke,Phys. Rev. D , 085001 (2013).[70] I. Bombaci, D. Logoteta, I. Vida˜na, and C. Providˆencia,Eur. Phys. Jour. A , 58 (2016).[71] A. Chodos, R. L. Jaffe, K. Johnson, C. B. Thorn, andV. F. Weisskopf, Phys. Rev. D , 3471 (1974).[72] B. Freedman and L. McLerran,Phys. Rev. D , 1109 (1978).[73] S. Kubis and M. Kutschera,Phys. Lett. B , 191 (1997).[74] Y. Nambu and G. Jona-Lasinio,Phys. Rev. , 345–358 (1961).[75] Y. Nambu and G. Jona-Lasinio,Phys. Rev. , 246–254 (1961).[76] S. P. Klevansky, Rev. Mod. Phys. , 649–708 (1992).[77] M. Buballa, Physics Reports , 205 – 376 (2005).[78] C.-M. Li, J.-L. Zhang, T. Zhao, Y.-P. Zhao, and H.-S.Zong, Phys. Rev. D , 056018 (2017).[79] C.-M. Li, J.-L. Zhang, Y. Yan, Y.-F. Huang, and H.-S.Zong, Phys. Rev. D , 103013 (2018).[80] T. Kl¨ahn and T. Fischer,The Astrophys. J. , 134 (2015).[81] C. Horowitz and B. D. Serot,Nucl. Phys. A , 503 – 528 (1981).[82] J. Walecka, Ann. Phys. , 491–529 (1974).[83] J. Boguta and A. Bodmer,Nucl. Phys. A , 413–428 (1977).[84] R. Furnstahl, B. D. Serot, and H. B. Tang,Nucl. Phys. A , 539 (1996).[85] R. Furnstahl, B. D. Serot, and H.-B. Tang,Nucl. Phys. A , 441 – 482 (1997).[86] B. Kumar, S. Singh, B. Agrawal, and S. Patra,Nucl. Phys. A , 197 – 207 (2017).[87] A. Bouyssy, J.-F. Mathiot, N. Van Giai, and S. Marcos,Phys. Rev. C , 380–401 (1987).[88] R. Brockmann, Phys. Rev. C , 1510–1524 (1978).[89] W.-H. Long, N. Van Giai, and J. Meng,Physics Letters B , 150 – 154 (2006).[90] T. F. T. B. N. Cierniak, M.and Kl¨ahn,Universe , 30 (2018).[91] W. Wei, B. Irving, M. Salinas, T. Kl¨ahn, and P. Jaiku-mar, The Astrophys. J. , 151 (2019).[92] K. Schertler, S. Leupold, and J. Schaffner-Bielich,Phys. Rev. C , 025801 (1999).[93] B. K. Sharma, P. K. Panda, and S. K. Patra,Phys. Rev. C , 035808 (2007).[94] G. F. Burgio, M. Baldo, P. K. Sahu, and H.-J. Schulze,Phys. Rev. C , 025802 (2002).[95] M. Orsaria, H. Rodrigues, F. Weber, and G. A. Contr-era, Phys. Rev. C , 015806 (2014).[96] D. Logoteta and I. Bombaci,Phys. Rev. D , 063001 (2013).[97] R. C. Tolman, Phys. Rev. , 364–373 (1939).[98] J. R. Oppenheimer and G. M. Volkoff,Phys. Rev. , 374–381 (1939).[99] T. Hinderer, B. D. Lackey, R. N. Lang, and J. S. Read,Phys. Rev. D , 123016 (2010).[100] B. Kumar, S. K. Biswal, and S. K. Patra, Phys. Rev. C , 015801 (2017).[101] T. Hinderer, The Astrophys. J. , 1216–1220 (2008).[102] E. M. Butterworth and J. R. Ipser,Astrophys. Jour. , 200–223 (1976).[103] J. L. Friedman, J. R. Ipser, and L. Parker,Astrophys. Jour. , 115 (1986).[104] J. L. Friedman, J. R. Ipser, and L. Parker,Phys. Rev. Lett. , 3015–3019 (1989).[105] J. M. Lattimer and M. Prakash,Physics Reports , 121 – 146 (2000).[106] A. Worley, P. G. Krastev, and B.-A. Li,The Astrophys. Jour. , 390–399 (2008).[107] N. Stergioulas, Living Rev. Relativ. , 3 (2003).[108] N. Stergioulas and J. L. Friedman,‘Astrophys. Jour. , 306 (1995).[109] N. Stergioulas, RNS: Rapidly rotating neutron stars (1996).[110] B.-A. Li and X. Han,Phys. Lett. B , 276 – 281 (2013).[111] Y. Zhang, M. Liu, C.-J. Xia, Z. Li, and S. K. Biswal,Phys. Rev. C , 034303 (2020).[112] P. Danielewicz and J. Lee,Nucl. Phys. A , 1 – 70 (2014).[113] G. Baym, C. Pethick, and P. Sutherland,Astrophys. J. , 299–317 (1971).[114] I. A. Rather, A. A. Usmani, and S. K. Patra, (2020),arXiv:2009.12613 [nucl-th].[115] H. Pais and C. m. c. Providˆencia,Phys. Rev. C , 015808 (2016).[116] F. Grill, H. Pais, C. m. c. Providˆencia, I. Vida˜na, andS. S. Avancini, Phys. Rev. C , 045803 (2014).[117] T. E. Riley, A. L. Watts, S. Bogdanov, P. S. Ray, R. M.Ludlam, S. Guillot, Z. Arzoumanian, C. L. Baker, A. V.Bilous, D. Chakrabarty, K. C. Gendreau, A. K. Harding,W. C. G. Ho, J. M. Lattimer, S. M. Morsink, and T. E.Strohmayer, Astrophys. Jour. , L21 (2019).[118] J.-L. Jiang, S.-P. Tang, Y.-Z. Wang, Y.-Z. Fan, andD.-M. Wei, Astrophys. Jour. , 55 (2020).[119] P. Landry, R. Essick, and K. Chatziioannou,Phys. Rev. D , 123007 (2020).[120] Y. Lim, J. W. Holt, and R. J. Stahulak,Phys. Rev. C , 035802 (2019).[121] B. Kumar and P. Landry,Phys. Rev. D , 123026 (2019).[122] J. L. Friedman, J. R. Ipser, and R. D. Sorkin,Astrophys. Jour. , 722 (1988).[123] R. Sorkin, Astrophys. J. , 254–257 (1981).[124] R. D. Sorkin, Astrophys. J. , 847–854 (1982).[125] K. Takami, L. Rezzolla, and S. Yoshida,MNRAS , L1–L5 (2011).[126] D. C. Backer, S. R. Kulkarni, C. Heiles, M. M. Davis,and W. M. Goss, , 615 (1982).[127] J. W. T. Hessels, S. M. Ransom, I. H. Stairs,P. C. C. Freire, V. M. Kaspi, and F. Camilo,Science , 1901–1904 (2006).[128] P. Kaaret, Z. Prieskorn, J. J. M. in ' t Zand, S. Brandt,N. Lund, S. Mereghetti, D. G¨otz, E. Kuulkers, and J. A.Tomsick, Astrophys. Jour. , L97–L100 (2007).[129] D. McNall, Geophysical Journal International , 103–104 (1970).[130] J. E. Tohline, R. H. Durisen, and M. McCollough, As-trophys. J..[131] B. K. Pickett, R. H. Durisen, and G. A. Davis,Astrophys. J. , 714 (1996).[132] J. N. Imamura, J. Toman, R. H. Durisen, B. K. Pickett,and S. Yang, Astrophys. J. , 363 (1995).[133] J. M. Centrella, K. C. B. New, L. L. Lowe, and J. D. Brown, Astrophys. J. Lett. , L193–L196 (2001).[134] K.-W. Lo and L.-M. Lin, Astrophys. J. , 12 (2011).[135] F. Cipolletta, C. Cherubini, S. Filippi, J. A. Rueda, andR. Ruffini, Phys. Rev. D , 023007 (2015).[136] P. S. Koliogiannia and C. Moustakidis,Phys. Rev. C , 015808 (2020).[137] K. S. Thorne, Astrophys. J. , 507 (1974).[138] C. Xia and W. Yong-Jiu,Ch. Phys. Lett. , 070402 (2009). [139] D. Sanwal, G. G. Pavlov, V. E. Zavlin, and M. A. Teter,Astrophys. J. , L61–L64 (2002).[140] J. Cottam, F. Paerels, and M. Mendez,Nature , 51 (2002).[141] M. Baub¨ock, D. Psaltis, and F. ¨Ozel,Astrophys. J.766