Non-radial oscillation modes in hybrid stars: consequences of a mixed phase

Deepak Kumar1,2,∗ Hiranmaya Mishra1,† and Tuhin Malik3‡1Theory Division, Physical Research Laboratory, Navarangpura, Ahmedabad 380 009, India2Indian Institute of Technology Gandhinagar, Gandhinagar 382 355, Gujarat, India and

3CFisUC, Department of Physics, University of Coimbra, P-3004 - 516 Coimbra, Portugal(Dated: October 4, 2021)

We study the possible existence of deconfine quark matter in the core of neutron stars. A rela-tivistic mean field model is used to describe the nuclear matter at low densities while Nambu–Jona-Lasinio model is used to describe the quark matter at high densities. A Gibbs construct is used todescribe the quark-hadron phase transition at large densities and at zero temperature. Within themodel, as the density is increased, a mixed phase appears at about 2.5 times the nuclear mattersaturation density (ρ0) and ends at about 5 ρ0 beyond which pure quark matter phase appear. Itturns out that a stable hybrid star of maximum mass, M = 2.27 M with radius R = 14 km, canexist with a quark matter core in a mixed phase only. As the density decreases from the center tothe surface, there is a sharp increase of the velocity of sound at the point where the quark-hadronphase transition occurs. This leads to the possibility of low frequency non-radial oscillations, the gmodes, in hybrid stars in contrast to canonical neutron stars which do not have such quark mattercore. The higher f modes frequencies of neutron stars get enhanced in the presence of quark mattercore in hybrid stars. The magnitude of g mode frequencies decrease with increase the repulsiveinteraction in quark matter.

I. INTRODUCTION

Neutron stars (NS) are exciting cosmic laboratories tostudy the behavior of matter at extreme densities. Theproperties of neutron stars not only open up many pos-sibilities related to composition, structure and dynamicsof cold matter in the observable universe but also throwslight on the interaction of matter at fundamental level[1]. Such compact stars, observed as pulsars, are believedto contain matter of densities few times nuclear satura-tion density (ρ0 ' 0.158 fm−3) in its core. To explainand understand the properties of such stars, one needsto connect different branches of physics like low energynuclear physics, QCD under extreme conditions, generaltheory of relativity (GR) etc [2–6].

The macroscopic properties of such a compact star likeits mass, radius, moment of inertia, tidal deformabilityin a binary merging system and different modes of os-cillations etc. depend crucially on its composition thataffect the variation of pressure with energy density orthe equation of state (EOS). Indeed, recent radio, x-ray and gravitational wave observations of neutron starshave provided valuable insights in to the EOS of densematter [7–9]. The observation of high mass pulsars likePSR J1614 − 2230 (M = 1.928 ± 0.04M) [10], PSRJ0348 − 0432 (M = 2.01 ± 0.04 M) [11] and PSRJ0740+6620 (M = 2.08± 0.07 M [12] and very recentlyPSR J1810+1714 with a mass (M = 2.13± 0.04 M [13]has already drawn attention on nuclear interactions athigh density with questions regarding the possible pres-ence of exotic matter in them. To constrain the nature

∗ deepakk@prl.res.in† hm@prl.res.in‡ tuhin.malik@uc.pt

of EOS more stringently, simultaneous measurements ofneutron star mass and radius are essential. The precisedeterminations of NS radii is difficult due to inaccuratemodeling the x-ray spectra emitted by the atmosphere ofa neutron star. The high-precision x-ray space missions,such as the NICER (Neutron star Interior CompositionExploreR) has already shed some light in this direction.Of late, The NICER has come up with one measurementof the radius 12.71+1.14

−1.19 km, for NS with mass 1.34+0.15−0.16

M [14], and other independent analyses show that theradius is 13.02+1.24

−1.06 km for an NS with mass 1.44+0.15−0.14 M

[15]. Further, the recent measurement of the equatorialcircumferential radius of the highest mass (2.072+0.067

−0.066

M) pulsar PSR J0740 + 6620 is 12.39+1.30−0.98 km (68 %)

[16, 17] by NICER group will play a important role in thisdomain. The empirical estimates of radius of a canonicalNS (M = 1.4M) should be R1.4 = (11.9 ± 1.22) km[18].

The core of the neutron star can, in principle, sup-ports various possible exotic phases of quantum chro-modynamics (QCD). While perturbative QCD predictsdeconfined quark matter at large densities, their appli-cability is rather limited in the sense that these conclu-sions are applicable only to very large baryon densitiesi.e.ρB ≥ 40ρ0, ρ0 = 0.158 fm−3 being the nuclear mattersaturation density [19]. The most challenging region tostudy theoretically is, however, at intermediate densitiesi.e. few times nuclear matter density which is actuallyrelevant for the matter in the core of neutron stars. Thefirst principle lattice QCD calculation in this connectionis also difficult due to sign problem in lattice simulationsat finite density. At present such calculations are lim-ited to low baryon density only i.e. µB/T ≤ 3.5 [20].On the other hand, many effective models predict possi-bilities of various exotic phases of quark matter at suchintermediate densities regions. These include pion su-

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perfluidity [21–23], various color superconducting phaseslike 2-flavor color superconductivity [24–26], color fla-vor locked phase (CFL) [27], Larkin-Ovchinkov-Fulde-Ferrel (LOFF) [28, 29] phase, crystalline superconduc-tivity phase etc for the quark matter phase. However,the signature of such phases in quark matter from thestudy of neutron stars have been rather challenging. TheGW170817 [9] event explored the constraints on the EOSusing tidal deformability extracted from the phase of thegravitational waveforms during the late stage of inspiralmerger [30–35]. Though not conclusive, it is quite possi-ble that one or both the merging neutron stars could behybrid stars i.e. the stars with a core of quark matter ora mixed phase core of quark and hadronic matter and acrust of hadronic matter [36, 37]. Within the current ob-servational status, it is difficult to distinguish between acanonical neutron star without a quark matter core froma hybrid star with a core of pure quark matter or a coreof quark matter in a mixed phase with hadronic matter.This calls for exploring other observational signature tosolve this “masquerade” problem [38, 39].

In this context, it has been suggested that the studyof the non-radial oscillation modes of neutron stars canhave the possibility of providing the compositional infor-mation regarding the matter in the interior of the neutronstars. This includes the neutron stars with a hyperon core[40–42], a quark core or a mixed phase with quark andhadronic matter components [39, 43–48]. This is becausethe non-radial oscillations not only depend upon the EOSi.e. P (ε) but also on the velocity of sound c2s = dP/dε.Since the appearance of hyperons does not involve anyphase transition, their effects on the non-radial oscillationmodes can be milder compared to a hadron quark phasetransition at finite densities whose effect can be more pro-nounced. The non-radial oscillation modes can be stud-ied within the frame work of general relativity [49, 50].Here the fluid perturbation equations can be decomposedinto spherical harmonics leading to two classes of oscilla-tions depending upon the parity of the harmonics. Theeven parity oscillations produce the spheroidal(polar) de-formation while the odd parity produce toroidal one. Thepolar quasi-normal modes can further be classified intodifferent kinds of modes depending upon the restoringforce that acts on the fluid element when it gets dis-placed from its equilibrium position [51]. These oscilla-tions couple to the gravitational wave and can be diag-nostic tools for studying the phase structure of the matterinside the neutron stars. The important modes for thisare the pressure (p) modes, fundamental (f) modes andgravity (g) modes. The frequency of the g-modes is lowerthan that of p-modes while the frequency of f-modes lie inbetween. These are the fluid modes to be distinguishedfrom w-modes which are associated with the perturba-tion of space-time metric itself. In the present work, wefocus of g and f modes oscillations arising from quark-hadron mixed phase. The origin of the g modes here isdue to the discontinuity in the energy density as a resultof the first order hadron-quark phase transition at finite

density.It may be noted that g modes oscillations have been

studied earlier in the context of quark-hadron phase tran-sitions [39, 43–48]. In most of the these investigationsthe hadronic matter description is through a relativis-tic mean field model while the quark matter descriptionis through a bag model or an improved version of thesame. In the present investigation, for the descriptionof quark matter we use two flavor Nambu–Jona-Lasinio(NJL) model where the parameters of the model are fixedfrom the physical variables like the pion mass, the piondecay constant and the light quark condensates that en-codes the physics of the chiral symmetry breaking. Forthe nuclear matter sector we use a relativistic mean field(RMF) theory involving nucleons interacting with scalarand vector mean fields along with self-interactions of themesons leading to reasonable saturation properties of nu-clear matter. The phase transition from hadronic mat-ter to quark matter can be considered either througha Maxwell construct or a Gibbs construct leading to amixed phase [52]. It ought to be noted that the kind ofphase transition depends crucially on the surface tension[53–59] of the quark matter which, however, is poorlyknown. Gibbs construct is relevant for smaller value ofsurface tension while Maxwell construct becomes relevantfor large values of surface tension [60, 61].

We organize this paper as follows. In section II A wediscuss salient features of RMF model describing the nu-clear matter and in section II B we discuss NJL modeland write down the equation of state for the quark mat-ter. Section II C details the hadron-quark transition andmixed phase structure using Gibbs construct of phasetransition with multiple chemical potentials. In sectionIII, we discuss the stellar structure equations as well asthe non-radial fluid oscillations for compact stars. In sec-tion IV we discuss the results of the present investigationregarding thermodynamics of the dense matter, mixedphase structure, hybrid star structure and the non-radialmode oscillations. Finally in section V, we summarize theresults and give an outlook for the further investigation.We use natural units here where ~ = c = G = 1.

II. FORMALISM

A. Equation of state for nuclear matter

We discuss briefly the relativistic mean field frameworkto construct EOS for nucleonic matter. In this frame-work, the force between two nucleons is realized by thethe exchange of mesons, when the nucleons are close to-gether of the order of femtometer. The relevant mesonsfor this purpose are the σ, ω and ρ mesons [62–65]. Whilethe scalar σ mesons create a strong attractive centralforce, the vector ω mesons on the other hand are respon-sible for the repulsive short range force. The protons andneutrons only differ in terms of their isospin projections,the isovector ρ mesons are included to distinguish be-

3

tween these baryons. The Lagrangian including nucleonfield, the σ, ω and ρ mesons and their interactions canbe written as [66],

L = ψ

[γµ (i∂µ − gvVµ − gρτ · bµ)− (m+ gsφ)

]ψ

+ 12∂µφ∂

µφ− 12m

2sφ

2 − 13g2φ

3 − 14g3φ

4 − 14V

µνVµν

+ 12m

2vV

µVµ − 14b

µν · bµν +12 m2

ρbµbµ. (1)

In the above equation the ψ is nucleon doublet. The σ,ω, ρ meson fields are denoted by φ, Vµ and bµ, respec-tively and their masses are ms, mv and mρ, respectively.The m denotes the nucleon masses. The gs, gv and gρare the scalar, vector and isovector coupling constants,respectively. In the mean-field procedure, one assumesthe mesonic fields to be classical and uniform mean fieldswhile retaining the quantum nature of the baryonic fieldi.e. 〈φ〉 = φ0, 〈Vµ〉 = V0δµ0, 〈baµ〉 =δµ0δ

a3b

03.

The baryon density, ρB and scalar density, ρs are

ρB =〈ψ†ψ〉

=∑i=n,p

γkiF3

6π2

(2)

ρs =〈ψψ〉

=γ

(2π)3

∑i=n,p

∫ kiF

0

m∗√m∗2 + k2

d3k

=∑i=n,p

γm∗kiF2

2π2G(m∗/kiF )

(3)

where

G(x) =1

2

[√1 + x2 − x2 ln

(1 +√

1 + x2

x

)](4)

where kiF is the Fermi momentum for the nucleonic mat-ter and the γ is the spin degeneracy factor of nucleon.Further, γ = 2 for neutron and proton individually andm∗ = m + gsφ0 is the effective nucleon mass. The totalenergy density εHP and the pressure PHP are

εHP =1

π2

∑i=n,p

H(m∗/kiF )

+1

2m2sφ

20 +

1

3g2φ

30 +

1

4g3φ

40 +

1

2m2vV

20 +

1

2m2ρb

032

(5)

and

PHP =1

3π2

∑i=n,p

I(m∗/kiF )

− 1

2m2sφ

20 −

1

3g2φ

30 −

1

4g3φ

40 +

1

2m2vV

20 +

1

2m2ρb

032

(6)

TABLE I. The nucleon masses (m), σ meson mass (ms), ωmeson mass (mv), ρ meson mass (mρ) and couplings gs, gv,gρ, g2, g3 of NL3 parameterization [66].

Parameters Values

m (MeV) 939ms (MeV) 508.194mv (MeV) 782.501mρ (MeV) 763.000

gs 10.217gv 12.868gρ 4.474

g2(fm−1) -10.431g3 -28.885

where we have defined the functions H(x) and I(x) asfollows,

H(x) =1

8

(√1 + x2(2 + x2)− x4 ln

(x+√

1 + x2

x

)), (7)

I(x) =1

8

(√1 + x2(2− 3x2) + 3x4 ln

(x+√

1 + x2

x

)).

(8)

We have considered the NL3 parameterization [66] forthe nucleonic EOS. The parameters are listed in Table I.

B. Equation of state for quark matter

We note down here, for the sake of completeness, thesalient features of the thermodynamics of NJL modelwith two flavors that we use to describe the equation ofstate of the quark matter. The Lagrangian of the modelwith four point interactions is given by

L = q(iγµ∂µ −m)q + gs((qq)2 + (qiγ5τ q)2

)+gv

((qγµq)2 + (qiγµγ5τ q)2

). (9)

Here, q is the doublet of u and d quarks. We have alsotaken here a current quark mass, m which is same for uand d quarks. The second term describes the four pointinteractions in the scalar and pseudo-scalar channel. Thethird term is a phenomenological vector interaction giv-ing rise to repulsive interaction for gv > 0 which can makethe equation of state stiffer. Except for the explicit sym-metry breaking term proportional to current quark mass,the Lagrangian is chirally symmetric. Using the standardmethod of thermal field theory one can write down thethe thermodynamic potential Ω(β, µ) within a mean fieldapproximation as a given temperature, (T = β−1) andquark chemical potential (µ = µB/3) [67] as

4

Ω(M,β, µ) = −2NfNc

∫dp

(2π)3

Ep

+1

βlog(1 + exp

(− β(Ep − µ)

))+

1

βlog(1 + exp

(− β(Ep + µ)

))

+(M −m)2

4gs− (µ− µ)2

4gv.

(10)

In the above, Ep(p) =√

p2 +M2 is the on shell singleparticle energy of the quark with “constituent quark massM” and µ being an effective quark chemical potential inthe presence of the vector interaction.

The constituent quark mass, M satisfies the mass gapequation

M = m− 2gsρs, (11)

where we have introduced the scalar density, ρs which isgiven as

ρs =〈qq〉

=− 2NfNc

∫dp

(2π)3

M

Ep

(1− f−(p, β, µ)− f+(p, β, µ)

)(12)

with f±(p, β, µ) =[1 + exp

(− β(Ep ± µ)

)]−1are the

fermi-distribution functions of the quark and anti-quarkwith the constituent quark mass M and an effectivechemical potential µ. The effective chemical potentialis given by

µ = µ− 2gvρv (13)

where the vector density, ρv is given as

ρv =〈q†q〉

=2NfNc

∫dp

(2π)3

(f−(p, β, µ)− f+(p, β, µ)

) (14)

In Eq.(10), the first term in the curly bracket correspondsto the Dirac vacuum contribution to the thermodynamicpotential with a fermion massM which is divergent and isregulated by a cut-off in the momentum. The other termscorrespond to the medium contribution to the thermo-dynamic potential. The difference of the vacuum energydensity between the non-perturbative vacuum (charac-terized by the constituent quark mass, M) and energydensity of the perturbative vacuum (characterized by cur-rent quark mass, m) is the bag constant, B.

B = Ω(M,β =∞, µ = 0)− Ω(m,β =∞, µ = 0) (15)

This bag constant is to be subtracted from Eq.(10) sothat the thermodynamic potential vanishes at vanishing

temperature and density. Here we focus our attentionto T = 0 which is applicable to the cold neutron stars.Using the relation limβ→∞

1β log

(e−βx + 1

)= −xΘ(−x),

the pressure i.e. the negative of the thermodynamic po-tential and the energy density of the quark matter in NJLmodel are given as

PNJL = Pvac + Pmed +B, (16)

and

εNJL =∑i=u,d

niµi − PNJL. (17)

In Eq.(16), the vacuum contribution to the pressure,Pvac is given by

Pvac =2NcNf(2π)3

∫|p|≤Λ

dp√

p2 +M2

=NcNfπ2

Λ4 H(M/Λ)

(18)

and the medium contribution to the pressure, Pmed isgiven by

Pmed =∑i=u,d

2Nc(2π)3

∫ piF

0

dp√p2 +M2

=∑i=u,d

Ncπ2

(piF )4 H(M/piF )

(19)

where piF = Θ(µi − M)√µ2i −M2. In the above, the

function H(x) is already defined in Eq.(7).The mass gap equation, Eq.(11) reduces in the zero

temperature limit as

M =m+ 4Ncgs∑i=u,d

∫|p|≤Λ

dp

(2π)3

M

Ep

(1−Θ(µi − Ep)

)=m+ 4Ncgs

∑i=u,d

[M(

Λ2G(M/Λ)− piF2G(M/piF )

)](20)

where the function G(x) is defined in Eq.(4) and the ef-fective chemical potential in the T = 0 limit is given asfollows

µi = µi − 4Ncgv∑i=u,d

piF3

6π2(21)

where µi = µB/3 + qiµE , qi is the electric charge of uand d quarks and µE is the chemical potential associatedwith electric charge.

Further we use β-equilibrium condition i.e. the quarkmatter in the interior of the star is in equilibrium underweak interaction, d→ u+ e+ νe. So that,

µd = µu + µe. (22)

5

we have taken here the chemical potential for neutrino tobe zero as they can defuse out of the star. This, howevercan be relevant for proto-neutron star. Thus there aretwo independent chemical potentials to describe the mat-ter in the interior of the neutron stars which we take to bethe quark chemical potential µq and the electric chargechemical potential µE . In terms of these, the chemicalpotentials are given as µu = µq + 2

3µE , µd = µq − 13µE

and µe = −µE . In addition, for the description of thecharge neutral matter, there is a further constraint ofcharge neutrality given as

2

3ρu −

1

3ρd − ρe = 0 (23)

where ρi =γ(piF )3

6π2 , i = u, d, e. With the degeneracyfactor γ = 6 for quarks and γ = 2 for electron. Sincethe typical electric charge chemical potential is of theorder of MeV, one can neglect the electron mass so thatpeF = |µe|.

The total pressure and the energy density for thecharge neutral quark matter are then given by

PQP = PNJL + Pe (24)

εQP = εNJL + εe, (25)

where εe ' µ4e

4π2 and Pe ' εe/3.We may note that NJL model has four parameters −

namely, the current quark mass m, the three momen-tum cutoff Λ, and the two couplings gs and gv. Thevalues of the parameters are usually chosen by fitting thepion decay constant fπ = 92.4 MeV, the chiral conden-sate 〈−uu〉 = 〈−dd〉 = (240.8 MeV)3 and the pion massmπ = 135.0 MeV. This fixes m = 5.6 MeV, gsΛ

2 = 2.44and Λ = 587.9 MeV. As mentioned gv is not fitted fromany other physical constraint and we take it as a free pa-rameter. We shall show our results for the two values ofgv namely gv = 0 and gv = 0.2 gs. With this param-eterization, the constituent quark mass M is 400 MeV,the critical chemical potential µc for the chiral transitionturns out to be µc = 1168 MeV for the vector couplingconstant gv = 0 in NJL model.

C. Hadronic-quark phase transition and mixedphase

The number density or the quark chemical potential atwhich the hadronic-quark phase transition occurs is notknown precisely from the first principle calculations inQCD but it is expected from various model calculationsto occur at a density which is few times the nuclear mat-ter saturation density. In the context of neutron stars,two types of this phase transition can be possible depend-ing upon the surface tension [53–59] of the quark matter.If the surface tension is large then there will be sharpinterface and one can have a Maxwell construct for thephase transition. On the other hand if the surface tension

is small we can have a Gibbs construct for the phase tran-sition, where there is a mixed phase of nuclear and quarkmatter. It ought to be mentioned however, the estimatedvalues of the surface tension for quark matter vary overa wide range and is very much model dependent. As thevalue of the surface tension is not precisely known yetboth the scenarios, (Maxwell and Gibbs) are plausible.We adopt here the Gibbs construct for the hadron-quarkphase transition as nicely outlined in Ref. [68]. In thiscase, one can achieve charge neutrality in two ways. Onecan have a mixed phase of charge neutral hadronic matterand a charge neutral quark matter. In this situation, theelectric chemical potential µE is determined as a functionof µB demanding local charge neutrality a condition ineach phase. This makes the pressure a function of onechemical potential, µB only. On the other hand, one canachieve the charge neutrality with a positively chargedhadronic matter mixed with a negatively charged quarkmatter in necessary amounts leading to a global chargeneutrality condition. In the later case, the pressures ofboth hadronic matter and quark matter are functions oftwo independent chemical potentials µB and µE . TheGibbs condition for the equilibrium at the zero temper-ature between the two phases for such a two componentsystem is given Ref. [52] by

PHP(µB , µE) = PQP(µB , µE) = PMP(µB , µE), (26)

where the pressure for the hadronic phase PHP is given inEq.(6) and the pressure for the quark phase PQP is writ-ten down in Eq.(24) without imposing the condition ofcharge neutrality. In Fig. 1 we illustrate this calculation,where the pressure is plotted as a function of neutronchemical potential µn(µB) and the electron chemical po-tential µe(= −µE). The green color surface denotes thepressure in the hadronic phase estimated from the RMFmodel using the NL3 parameters. The purple color sur-face denotes the pressure in the quark matter phase es-timated in NJL model. The two surfaces intersect alongthe curve AB satisfying the global charge neutrality con-dition

χ ρQPc + (1− χ) ρHPc = 0, (27)

where ρHPc and ρQPc denote the total charge densitiesin HP and QP respectively and χ defines the volumefraction of the quark matter in the mixed phase definedas

χ =VQP

VQP + VNP. (28)

Explicitly, for a given µB we calculate the electric chargechemical potential µE such that the pressure in boththe phases are equal satisfying the Gibbs conditionEq.(26). This gives the intersection line of the twosurfaces as shown in Fig. 1. Further imposing the

6

global charge neutrality condition Eq.(27) one obtainsthe volume fraction χ occupied by the quark matter inthe mixed phase. Thus along the line AB in Fig. 1,the volume fraction occupied by quark matter increasesmonotonically from χ = 0 to χ = 1. This gives thepressure for the charge neutral matter in the mixedphase. Below χ < 0, the EOS corresponds to thehadronic charge neutral matter shown as the red dashcurve while for χ > 1 the EOS corresponds to thecharge neutral quark matter shown as the purple dashcurve. With the present parametrization of the RMFmodel for hadronic matter and NJL model for the quarkmatter, the mixed phase starts at (µB , µE , P ) =

(1423 MeV, 289.26 MeV, 144.56 MeV/fm3)

and ends at (µB , µE , P ) =

(1597 MeV, 102.40 MeV, 266.23 MeV/fm3). This

corresponds to the starting of the mixed phase at baryondensity ρB = 2.75 ρ0 and ending of the mixed phaseat baryon density ρB = 5.72 ρ0. For NJL model wehave taken here gv = 0.2 gs. For gv = 0, the mixedphase starts little earlier i.e. ρB = 2.36 ρ0 and ends atρB = 5.22 ρ0. After the mixed phase, as baryon densityincreases the matter is in pure charge neutral quarkmatter phase.

We can find the energy density in the MP as follows

εMP = χ εQP + (1− χ) εHP . (29)

We can also see the fraction of particles normalized withrespect to baryon density in different phases which wehave plotted in the Fig. 2 for gv = 0.2 gs. Similar toEq.(29), the baryon number density in the mixed phase

ρBMP = χρBQP + (1− χ)ρBHP . (30)

In the mixed phase region, nuclear matter density de-creases while quark matter density increases as ρB in-creases. As ρB is increased further the nuclear mattermelts completely to quark matter which occurs for den-sities beyond ρB = 5.72 ρ0.

III. NON-RADIAL FLUID OSCILLATIONMODES OF COMPACT STARS

In this section, we outline the equations governing theoscillation modes of the fluid comprising the neutron starmatter. We shall estimate here the eigen frequencies ofthe oscillations of the spherically symmetric non-rotatingcompact objects. The static spherically symmetric met-ric is given by the line element

ds2 = −e2Φ(r)dt2 + e2Λ(r)dr2 + r2(dθ2 + sin2 θdφ2

),

(31)where Φ(r) and Λ(r) are the metric functions which areto be obtained by solving the Einstein field equations. Itis convenient to define the “mass function”, m(r) in thefavour of Λ(r) as

e2Λ(r) =

(1− 2m(r)

r

)−1

. (32)

FIG. 1. Pressure is plotted as a function of µn(µB) andµe(−µE) for the hadronic and quark phases. The green sur-face is for the hadronic phase and the purple surface is forthe quark matter phase. The two surfaces intersect along thecurve AB. The along the dashed portion on this line, the elec-trical charge neutrality is maintained. Along the red dashedline and magenta dashed line charge neutrality is maintainedin the hadronic phase and in the quark phase respectively.The quark matter fraction χ increases monotonically fromχ = 0 to χ = 1 along the curve AB.

0

0.2

0.4

0.6

0.8

1

0 1 2 3 4 5 6 7 8

ρi/ρ

B

ρB/ρ0

Hadronic Phase Mixed Phase Quark Phase

n

p

e

µ

d

u

e

gv=0.2gs

FIG. 2. The particle fractions normalized with respect tobaryon density for the charge neutral matter are plotted as afunction of the baryon number density. The plot is for gv =0.2 gs. At ρB = 2.75 ρ0 the quark matter starts appearingand at ρB = 5.72 ρ0 the hadronic matter melts completely tothe quark matter.

7

Starting from the line element Eq.(31) one can obtain theequations governing the structure of spherical compactobjects, Tolman-Oppenheimer-Volkofv (TOV) equationsas

dP (r)

dr=−

(ε(r) + P (r))(m(r) + 4πr3P (r)

)r(r − 2m(r))

(33)

dm(r)

dr=4πr2ε(r) (34)

dΦ(r)

dr=m(r) + 4πr3P (r)

r(r − 2m(r)). (35)

In the above ε(r), P (r) are the energy density and pres-sure respectively. m(r) is the total mass of the compactstar enclosed within a radius r. To solve these equations,one has to supplement these equations with an equationrelating pressure and energy density i.e. an equation ofstate (EOS). Further, one has to set the boundary con-ditions at the center and surface as

m(0) =0 (36)

P (R) =0 (37)

e2Φ(R) =1− 2m(R)

R, (38)

where the total mass of the compact object is given byM = m(R), R being it’s radius which is defined as theradial distance where the pressure vanishes while inte-grating out Eqs.(33 - 34 and 35) from the center to thesurface of the star. One can solve these equations alongwith a boundary conditions Eqs.(36 - 37 and 38) for a setof central densities εc to obtain the mass-radius, (M−R)curve.

Next, we consider the perturbation equations for thenon-radial oscillations of the spherically symmetric neu-tron star. The framework for studying the perturbationwithin general relativity was developed in Ref. [49] andfurther extended in Ref. [50]. The perturbation can bein the metric as well as in the fluid element and in gen-eral, the perturbation equations can be decomposed intospherical harmonics giving rise to two types of oscillationequations depending upon the parity of the harmonics.The even parity oscillations produce spheroidal deforma-tions while the odd parity oscillations produce toroidaldeformations. In the present study, we confine our atten-tion to spheroidal oscillations. In such a case, the linearperturbation equations for the fluid and metric pertur-bation get coupled. However, to study the oscillationmodes, a great simplification occurs when the metric per-turbations are negligible. This was explicitly shown byCowling [46, 69–71] for newtonian stars and further ex-tended to its relativistic counterpart in Ref. [72]. It turnsout that within such relativistic Cowling approximation( which we use in our analysis), the f modes differ byabout 20% and the deviation is less than few percent forthe g modes [43, 73].Thus, in this approximation, thefluid perturbation oscillates in a fixed background andone cannot study the pure gravitational w modes.

The fluid perturbations are decomposed in sphericalharmonics with a harmonic time dependence exp(iωt).Explicitly the fluid perturbations are described by theLagrangian fluid displacement vector

ξi = eiωt(e−Λ(r)W (r),

− V (r) ∂θ, − V (r) sin−2 θ ∂φ

)r−2Ylm(θ, φ).

(39)

where, ω, W (r), V (r) characterize the perturbations.The pulsation equations are given as

dW (r)

dr=

dε

dP

(ω2r2eΛ(r)−2Φ(r)V (r)

+W (r)dΦ(r)

dr

)− l(l + 1)eΛ(r)V (r)

(40)

dV (r)

dr= 2V (r)

dΦ(r)

dr− 1

r2W (r)eΛ(r) (41)

These two equations are making an eigen value problemfor ω. Here these are solved with the appropriate bound-ary conditions at center and the surface. Near the centerof the compact star the behavior of the functions W (r)and V (r) is given by [43]

W (r) = Crl+1 and V (r) = −Crl/l (42)

where C is an arbitrary constant. While other boundarycondition at the stellar surface is the vanishing of theLagrangian perturbation pressure, i.e. ∆P = 0. Thiscondition leads to [43, 74]

ω2R2eΛ(R)−2Φ(R)V (R) +W (R)dΦ(r)

dr

∣∣∣∣r=R

= 0 (43)

at the surface of the star. Furthermore, if one consid-ers stellar models with phase transition i.e. having anequation of state with a discontinuity then the additionaljunction conditions have to be satisfied which are the con-tinuous conditions for W (r) and ∆P . These continuityconditions are written in terms of functions W (r) andV (r) as

W+ = W−, (44)

V+ =e2Φ

ω2rc

ε− + P

ε+ + P

(ω2r2

ce−2ΦV−

+ e−ΛΦ′W−

)− e−ΛΦ′W+

.

(45)

here, rc is the radial distance from the center where theenergy density has a discontinuity. Minus(plus) sub-scripts correspond to the quantities before(after) thephase transition.

8

To calculate the eigen frequencies ω, we proceed asfollows. For a given central density εc, we first solve theTOV equations Eqs.(33 - 35) to get the profile of the un-perturbed metric functions Λ(r), Φ(r) and also the massM and the radius R of the spherical star. For a given ω,we solve the pulsating equations Eqs.(40 - 41) to deter-mine the fluid perturbing functions W (r) and V (r) as afunction of r. To solve these equations, we take the initialvalues for W and V consistent with Eq.(42). Specificallywe took C of the order 1. The solutions of W and V areindependent of this choice. While solving the pulsatingequations we also take into account the junction condi-tions Eqs.(44 - 45) at the transition radius rc. We thencalculate LHS of Eq.(43). The value of ω is then variedsuch that the boundary condition, Eq.(43), is satisfied.This gives the frequency, ω as function of mass and ra-dius. It may be noted that there can be multiple solutionsof ω satisfying the pulsating equations and the boundaryconditions corresponding to different initial trail valuesfor ω. These different solutions for ω correspond to fre-quencies of different modes of oscillations of the compactstar.

IV. RESULTS AND DISCUSSION

In the present section, we study the structural prop-erties and non-radial oscillations of neutron stars andhybrid stars. We consider the RMF model with NL3[66] parameters for nucleonic matter EOS (see sec. II A)and the two flavor Nambu–Jona-Lasinio model for thequark matter EOS (see sec. II B) with parameters,(gsΛ

2,Λ,m) = (2.24, 587.6 MeV , 5.6 MeV) [67] . Themixed-phase is calculated using Gibbs construction, asoutlined details in sec. II C.

A. Equation of state and properties ofneutron/hybrid star

We display the particle content as a function of densityfor the charge neutral matter for gv = 0.2 gs in Fig. 2.In the hadronic matter phase, the neutron density domi-nates with a small fraction of proton to make charge neu-tral nuclear matter. At ρB ∼ 2.36 ρ0, the mixed phasestarts and the nucleon density decreases while quark frac-tion starts increasing. Finally, at densities ρB ∼ 5.22 ρ0

and above, the quark matter phase takes over with downquark densities roughly becoming twice that of the upquarks to maintain the charge neutrality. The resultingmixed phase EOS is shown in the Fig. 3 for the two dif-ferent values of the vector coupling in the NJL model.Increasing gv results stiffer equation of state. Further,the higher gv corresponds to a larger critical energy den-sity from where the mixed phase starts.

In Fig. 4, we have plotted the quark matter fraction χas a function of density for gv = 0. Increasing gv resultsin a higher threshold for the onset of mixed phase. For

0

100

200

300

400

500

600

0 200 400 600 800 1000 1200 1400 1600 1800

P (M

eV/f

m3 )

ε (MeV/fm3)

gv=0.0gsgv=0.2gs

Hadronic Phase

Mixed Phase

Quark Phase

FIG. 3. The equation of state for the charge neutral matterincluding a mixed phase. The upper curve corresponds to thevector coupling gv = 0.2 gs while the lower curve correspondsto gv = 0. In each curve the red curve refers to hadronic phaseand blue curve refers to the mixed phase while magenta curvecorresponds to the quark matter phase. The open squarecorresponds to the central energy density of a neutron star ofmass 1.4M The triangles denote the starting of the mixedphase and correspond to neutron stars mass of 2.17 M (gv =0) and 2.50 M (gv = 0.2 gs). The circles represent themaximum masses 2.27M (gv = 0) and 2.55M (gv = 0.2 gs)hybrid stars.

0

0.2

0.4

0.6

0.8

1

0 1 2 3 4 5

χ

ρB/ρ0

gv=0

FIG. 4. The quark fraction as a function of baryon densityfor EOS given in Fig. 3. The open circle indicates the cen-tral density of the maximum mass star i.e. ρB,max ' 3.5 ρ0corresponding to Mmax = 2.27 M.

the EOS plotted in the Fig. 3, the central density of themaximum mass star indicated by the filled circle on thecurve lies in the mixed phase and hence the pure quarkmatter core is not realized.

The variation of the square of the velocity of sounddP/dε as a function of baryon number density using theequation of state for the charge neutral matter is dis-played in Fig. 5. As density increases in the hadronicmatter the velocity of sound increases monotonically up

9

0

0.2

0.4

0.6

0.8

1

0 1 2 3 4 5 6 7 8

Cs2

ρB/ρ0

gv=0.2gs

Had

roni

c Ph

ase

Mixed Phase

Quark Phase

FIG. 5. The variation of the square of speed of sound (c2s =dP/dε) as a function of baryon number density for the chargeneutral matter. The red curve corresponds to sound speedin hadronic phase, the blue curve corresponds to the soundspeed in mixed phase while the magenta curve refers to thesound speed in the quark phase.

to the critical density with a value c2s ∼ 0.68, after which,it reduces discontinuously to a value c2s = 0.1 at the be-ginning of the mixed phase. As the mixed phase endsthe sound velocity jumps up again discontinuously to thequark matter phase with a value c2s ' 0.35 for gv = 0 andc2s ' 0.42 for gv = 0.2 gs. This velocity of sound actuallycorrespond to the equilibrium sound speed as discussedin Ref. [39, 75] which is smaller than the adiabatic soundspeed and was shown to be crucial for the existence ofthe g mode. Apart from the g modes, this sudden rise ofsound velocity has also important consequences regard-ing the mass and radius relation in neutron star. Oneactually needs a rise in velocity of sound in a narrow re-gion of densities, for an explanation of the compact starsto have large mass and small radius [76]. To achieve thispossibility, a quarkyonic phase [76] or a vector conden-sate phase along with pion superfluidity [77] have beenproposed recently. On the other hand, such a steep rise inthe velocity of sound can also arise in a mixed phase con-struct within the models for hadronic matter and quarkmatter as used here.

In Fig. 6, we show the mass-radius relation. For purenucleonic matter the maximum mass turns out to be2.77 M and radius turns out to be 13.26 km. If oneuses mixed phase equation of state the maximum massreduces to 2.27 M for gv = 0 with the correspondingradius R = 14.39 km and to 2.55 M for gv = 0.2 gswith the radius being R = 14.17 km. This is essen-tially due to the fact that the quark matter EOS is softercompared to the nuclear matter EOS. The central en-ergy densities for the maximum mass hybrid stars areεmaxc = 656 MeV/fm3 (gv = 0) and εmaxc = 738 MeV/fm3

(gv = 0.2 gs). As central energy density is increased fur-ther, the hybrid stars become unstable i.e. dM/dεc < 0.Thus, within the present model, we do not find stable

1

1.5

2

2.5

3

12 12.5 13 13.5 14 14.5 15 15.5 16

M (

M⊙

)

R (km)

gv=0gv=0.2gs

Neutron Star

2.50 M⊙

2.55 M⊙

1.40 M⊙

2.17 M⊙

2.27 M⊙

FIG. 6. The compact stars masses are plotted as a function ofcorresponding radius. The circles denote the maximum masshybrid stars having quark matter inside its core for differentvalues of vector interaction in NJL model. While the trianglesrepresents the maximum mass neutron stars having hadronicmatter inside the core. The maximum mass of hybrid star is2.27 M (gv = 0) and 2.55 M (gv = 0.2 gs).

hybrid stars with pure quark matter core. The quarkmatter, if present in the core, is always in the mixedphase. As gv increases, the mixed phase starts at higherenergy density and hence larger fraction of hadronic mat-ter contributes to the total mass of the star. This leadsto an increase of the maximum mass of the hybrid star.With increasing gv further we might expect the neutronstars without any quark matter in the core. The radiusR1.4 for the canonical mass of 1.4M neutron stars turnsout to be 14.52 km. It may be noted that the x-ray pulseanalysis of NICER data from PSR J0030+0451 by Milleret.al. found R = 13.02+1.14

−1.19 km for M = 1.44± 0.15 M[15]. Such a star will not have a quark core within thepresent model for the EOS of dense matter.

In Fig. 7 we have plotted the energy density profile i.e.energy density as a function of the radial distance fromthe center. This we have plotted corresponding to thehybrid star of maximum mass. As mentioned earlier thecore of a such star is in a mixed phase with about the 50percent of quark matter and rest being nuclear matter.The radius of the mixed phase core is about 3.8km withthe total radius of 14.17 km. We have taken here thevector coupling gv/gs = 0.2 in NJL model. For gv = 0the core radius slightly larger i.e. 4.2 km while the radiusbeing about 14.39 km. At r = rc, the critical radial dis-tance, where matter goes from a mixed phase to hadronicphase, the energy density changes discontinuously whilepressure shows smooth behavior as may be observed infigure.

The behavior of sound speed c2s = dP/dε is plottedin Fig. 8 as a function of radial distance from the cen-ter of the star for both hybrid star as well as a neu-tron star without a quark matter core. Both the plotscorrespond to stable maximum mass stars which are

10

0

100

200

300

400

500

600

700

800

0 0.2 0.4 0.6 0.8 1

ε an

d P

(MeV

/fm

3 )

r/R

Energy DensityPressure

FIG. 7. The energy density and pressure as a function ofradial distance. This is the star’s profile of the maximum masshybrid star in terms of energy density and pressure (for thevector coupling gv/gs = 0.2 in NJL model). The transitionfrom mixed phase to nuclear matter phase occurs at ρB =2.75 ρ0 corresponds to rc = 0.27 R. This is reflected in theenergy density profile at the onset of nuclear matter whereit shows a discontinuous behavior. On the other hand, thepressure varies smoothly as it may be expected from the Gibbsconditions of phase transition.

0

0.2

0.4

0.6

0.8

1

0 0.2 0.4 0.6 0.8 1

c s2

r/R

Nuclear StarHybrid Star

FIG. 8. The velocity of sound as function of radial distance.The plot is for the maximum mass hybrid star. The abruptchange in velocity of sound onset of the quark-hadron phasetransition r/R = 0.27. The red curve indicates velocity ofsound for a neutron star with nuclear matter only which cor-respond to a maximum mass neutron star of M/M = 2.77.

M = 2.55 M (gv/gs = 0.2) for the hybrid star andM = 2.77 M for the neutron star. The quark hadronphase transition in the hybrid stars is reflected in the ve-locity of sound which changes abruptly from c2s = 0.15 toc2s = 0.65 at the critical radius rc. This plays an impor-tant role in the determination of non-radial oscillationfrequencies which we discuss in the next subsection.

1.9

1.95

2

2.05

2.1

2.15

2.2

2.2 2.3 2.4 2.5 2.6 2.7 2.8

f (k

Hz)

M (M⊙)

Hadronic Mattergv=0 gv=0.2gs

FIG. 9. The frequencies of the f mode f = ω/2π in kHz asa function of star mass in units of M. The magenta curvecorresponds to neutron stars i.e. without any quark mattercore. Blue(Red) curve corresponds to the hybrid stars forgv = 0 (gv/gs = 0.2).

B. Oscillation modes in hybrid stars

We next show here the results for f and g modes forthe neutron and hybrid stars. We shall consider thequadrapole mode (l = 2). It may be expected from thecoupled Eqs.(40 - 41) for the fluid perturbation functionsW (r) and V (r) that the (inverse of) velocity of soundplays an important role in the determination of the dif-ferent solutions for this function and hence on the fre-quencies of oscillation modes. The typical frequency of gmodes are from few 100 Hz up to 1 kHz while that of fmodes lie in the range 1 − 3 kHz. As mentioned in sec.III, we solve Eqs.(40 - 41) in a variational method to de-termine the oscillation frequencies. As this is computedusing a variational method, the final solutions dependupon the initial guess for the frequencies. To get a so-lutions of the f mode, we give the initial guess for thefrequency (f = ω/2π) in the range of kHz. On the otherhand to look for a g mode we give the initial guess for thesame in the range of few hundred Hz. In Fig. 9, we showthe f mode frequencies as a function of mass of com-pact stars for the both hybrid and neutron stars. Herethe blue curve refers to the frequencies for hybrid starswith gv = 0 and the red curve refers to the frequenciesfor hybrid stars with gv/gs = 0.2 and the magenta curverefers the frequencies for the neutron stars. We may ob-serve here there is a mild rise in the frequencies for thef modes as the mixed matter is considered in the core ofthe neutron star. This rise of frequencies at the onset ofmixed phase was observed also in Ref. [39, 75]. However,the corresponding frequencies where the few hundred Hzi.e. for g mode unlike the case of f mode here. The rea-son for this difference could be the two different modelstaken for hadronic and quark matter considered in Ref.[39, 75]. Infact, the hadronic matter considered in thisRef. [39] is rather soft compared to the EOS for the RMFmodel considered in the present investigation.

11

0

0.05

0.1

0.15

0.2

0.25

0.3

0.35

0.4

2.2 2.25 2.3 2.35 2.4 2.45 2.5 2.55 2.6

f (k

Hz)

M (M⊙)

gv=0 gv=0.2gs

FIG. 10. The g mode frequencies for hybrid stars as a functionof stellar mass are shown for two different vector coupling forNJL model. The banding portion of the curve correspond tounstable dM/dεc > 0 stars.

In Fig. 10, we plot the g mode frequencies as a func-tion of mass of hybrid stars. The back bending part ofthe curves is for the unstable stars. We do not observeany g modes for the pure neutron stars. The g modearises here because of discontinuity in density [43, 48].As gv increases, the size of mixed phase core decreasesand the magnitude of g mode also decreases. For gv = 0the masses of stable hybrid stars range from 2.22 M to2.27M beyond which the stars become unstable. Theg mode frequency in this range vary from 221 to 244Hz. For gv/gs = 0.2 the hybrid star masses increaseand range from 2.5 M to 2.55 M beyond which starsbecome unstable. The corresponding g modes frequen-cies decrease and lie in the range 141 to 253 Hz. As gvincreased further, we expect the mixed phase core de-crease and finally vanishing for sufficiently large value ofgv. This also means the decrease in the value of g modefrequency with increasing the repulsive quark interactionstrength gv and finally vanishing as there will not be anymixed phase core for large gv.

To understand the behavior of g modes associated withthe phase transition, it is useful to examine the Eqs.(40- 41) and the boundary conditions Eqs.(44 - 45) in somedetail.

Let us first note that near r = 0, W (r) and V (r) areof opposite sign where W (r) being positive. Looking atEq.(43) we can have a solution for ω with V (r) and W (r)remain of opposite sign so that they can vanish for a finiteω. The other possibility is that both are of the same signand vanish at the surface. The former case correspondsto the solution for the f mode while the later correspondsto the g mode as we discuss below.

Let us next focus our attention to the boundary con-dition Eqs.(44 - 45). We note that the pre-factor for theparenthesis is approximately 1 and infact little greaterthan 1 as may be inferred from the EOS Fig. 7. Whenω is large corresponding to f mode then V+ ' V− andthe sign of V (r) and remains negative. On the other

-0.04

-0.02

0

0.02

0.04

0 2 4 6 8 10 12 14

W(r

)/V

(r)

r

f-mode V W

g-mode V W

FIG. 11. The solutions for the fluid perturbation functions Wand V are plotted as a function of radial distance. The blackand red curves correspond to the function V (r) for the f andg modes, respectively. The blue and green curves correspondto the function W (r) for the f and g modes, respectively.

hand, W remains positive and is an increasing functionof r as may be inferred from Eq.(40) as W ′ remains posi-tive because the second term on the RHS dominates overthe first term. This also makes V (r) positive as V ′(r) re-mains positive as may be inferred from the Eq.(41). Thusthis corresponds to the first case and we have a solutionfor f mode. In Fig. 11, we have plotted W (r) and V (r)as a function of r for the maximum mass star.

On the other hand for g modes where the ω is small thefirst term in the parenthesis in Eq.(45) become negligiblecompared to the second term and V+ become positivei.e. V changes sign from negative to positive at rc. Thismakes both W ′(r) and V ′(r) negative beyond rc as maybe inferred from Eqs.(40 - 41). This correspond to thesecond case and we have a solution for the g mode. Thissolution is shown in Fig. 11. Therefore, it is critical tohave a discontinuity in the energy density associated withthe phase transition to have a g mode solution.

V. SUMMARY AND CONCLUSION

Let us summarize the salient features of the presentinvestigation.We have looked into possible distinct features of neutronstars with a quark matter core and a neutron star with-out a quark matter core. This is investigated by lookinginto non-radial oscillations of compact stars. The neu-tron stars are constructed using the equation of state i.e.constructed using a relativistic mean field theory (RMF)for nuclear matter and NJL model for the quark matter.Gibbs criterion for the mixed phase is used to constructthe mixed phase with two chemical potentials (µB andµE) imposing global global charge neutrality condition.It is observed that the cores of hybrid stars can accom-modate a mixture of nucleonic and quark matter, thepure quark matter phase being never achieved. Inclu-

12

sion of a mixed phase of matter softens the equation ofstate which results in a smaller values for the maximummass and a larger corresponding radius as compared toa neutron star without a quark core.

Determining the composition of neutron star throughobservables it is necessary to break the degeneracy be-tween normal and hybrid star. To this end, we lookedinto non-radial oscillating modes of such compact ob-jects. Unlike M-R curves for which equation of state issufficient, analysis of these modes require the velocity ofsound of the charge neutral matter. Using a mixed phasestructure it is observed that the velocity of sound shootsup at the transition between the mixed phase and nu-clear matter phase in such a construct. It may be notedthat such a steep rise in the velocity of sound in a narrowregion of density as one comes from the core towards thesurface was also seen in a quarkyonic to hadronic mat-ter transition [76] as well as in an equation of state withω0 condensate and fluctuations in pion condensate [77].Such a steep rise in velocity in sound speed is generatednaturally here through the mixed phase construct. ThisEOS is used to determine the frequencies of non-radialoscillations in neutron star within a relativistic Cowlingapproximation that neglects the fluctuation of the spacetime metric and results in a much simpler equation tosolve and analyze. While this is not strictly consistentwith the fully relativistic treatment, the impact of suchsimplifying approximation is not severe, typically effect-ing the g modes at the 5− 10% level while f modes aremore sensitive to Cowling approximation [73]. This has

its effects in having lower frequencies (∼ few hundred Hz)modes the g modes, which is a distinct feature of hybridstars. On the other hand, a mixed phase core enhancesthe higher frequencies (∼ few kHz) f modes slightly.

The novel feature of the present investigation has beenused in nucleonic EOS modeled through RMF with itsparameters determined from the nuclear matter proper-ties at saturation density with the NL3 parameterizationand a model for the quark matter that incorporates theimportant feature of chiral symmetry breaking of stronginteraction with its parameters fixed from the low en-ergy hadronic properties. On the other hand, it oughtto be mentioned that the such models do not includethe confinement properties of strong interactions. In fu-ture work, we would like to include the strange quarks inquark matter sector and correspondingly hyperons in thehadronic sector. It will also be interesting and importantto include the effect of strong magnetic field in the struc-ture of neutron stars [78] and its effect on the non-radialoscillation modes. We have focused out attention for neu-tron star matter which is at zero temperature and van-ishing a neutrino chemical potential. However, to studythe proto-neutron stars we should take into account thethermal effect on the oscillations and the effect of tem-perature and neutrino trapping on the phase structure ofmatter.

ACKNOWLEDGMENTS

The authors gratefully acknowledge discussions with P.Jaikumar, Bharat Kumar and their useful suggestions.

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