JHEP10(2011)111

Published for SISSA by Springer

Received: September 6, 2011

Accepted: October 17, 2011

Published: October 24, 2011

Holographic equations of state and astrophysical

compact objects

Youngman Kim,a,b Chang-Hwan Lee,c Ik Jae Shina and Mew-Bing Wana

aAsia Pacific Center for Theoretical Physics, Pohang University of Science and Technology,Pohang, Gyeongbuk 790-784, Korea

bDepartment of Physics, Pohang University of Science and Technology,Pohang, Gyeongbuk 790-784, Korea

cDepartment of Physics, Pusan National University,Busan 609-735, Korea

E-mail: ykim@apctp.org, clee@pusan.ac.kr, geniean@apctp.org,mbwan@apctp.org

Abstract: We solve the Tolman-Oppenheimer-Volkoff equation using an equation ofstate (EoS) calculated in holographic QCD. The aim is to use compact astrophysical objectslike neutron stars as an indicator to test holographic equations of state. We first try anEoS from a dense D4/D8/D8 model. In this case, however, we could not find a stablecompact star, a star satisfying pressure-zero condition with a radius R, p(R) = 0, within areasonable value of the radius. This means that the EoS from the D4/D8/D8 model maynot support any stable compact stars or may support one whose radius is very large. Thismight be due to a deficit of attractive force from a scalar field or two-pion exchange in theD4/D8/D8 model. Then, we consider D4/D6 type models with different number of quarkflavors, Nf = 1, 2, 3. Though the mass and radius of a holographic star is larger than thoseof normal neutron stars, the D4/D6 type EoS renders a stable compact star.

Keywords: Gauge-gravity correspondence, Phenomenological Models

c© SISSA 2011 doi:10.1007/JHEP10(2011)111

JHEP10(2011)111

Contents

1 Introduction 1

2 D4/D6 model 2

3 Numerical results 6

4 Summary 10

1 Introduction

The approaches based on the Anti de Sitter/conformal field theory (AdS/CFT) corre-spondence [1–3] are showing many interesting possibilities to explore strongly interactingsystems such as dense baryonic matter, stable/unstable nuclei, and strongly interactingquark gluon plasma. The basic strategy is to introduce an extra space, which roughlycorresponds to the energy scale of 4D boundary field theory, and try to construct a 5Dholographic dual model that captures certain non-perturbative aspects of strongly cou-pled field theory. There are in general two different routes to modeling holographic dualof quantum chromodynamics (QCD). One way is a top-down approach based on stringyD-brane configurations. The other way is so-called a bottom-up approach, in which a 5Dholographic dual is constructed from a boundary field theory such as QCD. For reviews onholographic QCD we refer to [4–10].

Neutron stars are the most important astrophysical compact objects in which densematter physics can be explored and tested. Since one cannot obtain the information onthe inner structure of neutron star directly from observation, mainly masses and radii ofneutron stars are being considered. Due to the uncertainties in dense matter physics at highdensities, various equations of states have been suggested [11]. One of the main problemslies in the fact that most of the microscopic theories are based on the data at low densities,only up to normal nuclear matter density. Hence, in order to guide high density behavior,new approaches which are based on QCD fundamental symmetries have been suggested.This work is along this line of work even though it is still far from realistic.

Most accurate measurements on the neutron star masses has been done in doubleneutron star binaries and well measured neutron star masses in these systems are consistentwith < 1.5M [11]. However, recently, a 1.97 ± 0.04M neutron star has been observedin a neutron star - white dwarf binary J1614−2230 using Shapiro delay [12]. In addition,both masses and radii of neutron stars in low mass X-ray binaries have been estimatedand more data will come soon [13, 14]. These observations already ruled out some setsof equations of state. In the future, once the advanced LIGO is operating, gravitational

– 1 –

JHEP10(2011)111

t 1 2 3 (τ) z ψ1 ψ2 r φ

D4 • • • • •

D6 • • • • • • •

Table 1. The brane configurations: the background D4 and the probe D6

waves from neutron star binary inspirals are expected to be detected. These observationsmay be able to provide clues on the inner structure of neutron stars.

In this work, to see what the equation of state (EoS) from holography says aboutnature, we solve the Tolman-Oppenheimer-Volkoff (TOV) equation using the EoS derivedfrom a top-down model. We use compact astrophysical objects like neutron stars as atesting agent. We first take an EoS from a cold and dense D4/D8/D8 model. With thisEoS, we cannot find any stable compact stars with a reasonable radius. This might bedue to a weakness of attractive force in the D4/D8/D8 [15]. In nuclear physics, the longrange attractive force is mostly given by one-pion exchange, while the intermediate range∼ 1 fm attraction is dominated by a scalar field or two-pion exchange. Note that in theD4/D8/D8 [15] a scalar that couples to baryon fields is missing. Then, we consider a D4/D6type model with different number of quark flavors, Nf = 1, 2, 3. Since the D4/D6 typemodels have both vector and scalar bulk fields, we might expect that the EoS calculatedin this model could support a stable compact star. We find that the D4/D6 type EoSallows a stable compact star, though the mass and radius is a bit larger than observedneutron stars.

One cautionary remark is that a simple classical analysis in the holographic dual model,both top-down and bottom-up, captures only the leading Nc contributions, and we arebound to suffer from sub-leading corrections. For instance for low densities we may neglecta back-reaction from a bulk U(1) gauge field whose boundary value defines a chemicalpotential and corresponding density. However, at high densities, though we have no clear-cut number to tell low density from high density, the back-reaction will not be negligible.Another point is that we need to impose the charge neutrality and β-equilibrium conditionson our system to study neutron stars. For instance, in neutron stars, charge neutralityfavors neutrons rather than protons since protons should come in together with electrons.However, in our case we do not impose these two conditions, which may be partiallyresponsible for the large radii of the compact stars from our study.

2 D4/D6 model

We briefly summarize the D4/D6 system [16] and comment on its extended version indense matter with Nf ≥ 1 [17, 18]. The model contains Nc number of D4 branes and Nf

flavor D6 branes whose configurations are given in table 1. In the probe limit, the Nc D4branes are replaced by corresponding type IIA SUGRA background, while Nf D6 branes

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JHEP10(2011)111

are treated as probes. The solution of confining D4 brane reads

ds2 =(U

L

)3/2(ηµνdx

µdxν + f(U)dτ2)

+(L

U

)3/2( dU2

f(U)+ U2dΩ2

4

)=(U

L

)3/2(ηµνdx

µdxν + f(U)dτ2)

+(L

U

)3/2(Uξ

)2

(dξ2 + ξ2dΩ24) , (2.1)

eφ = gs

(U

L

)3/4

, F4 = dC3 =2πNcε4

Ω4, (2.2)

where f(U) = 1− (UKK/U)3 . The D4 branes are compactified on the τ -direction, and U isconsidered as a radial coordinate of the transverse to the world-volume of branes. ε4 andΩ4 denote the volume form and volume of the unit S4, respectively. For convenience, weintroduce the dimensionless radial coordinate ξ defined as(

U

UKK

)3/2

≡ 12

(ξ3/2 +

1ξ3/2

). (2.3)

The parameters in string theory are related to those of the gauge side as

L3 =λl2s

2MKK, UKK =

2λMKKl2s

9, gs =

λ

2πNcMKKls, (2.4)

with the ’t Hooft coupling λ ≡ g2YMNc. To describe the D6 brane configuration, the back-

ground metric (2.1) is rewritten as

ds2 =(U

L

)3/2(ηµνdx

µdxν +f(U)dτ2)

+(L

U

)3/2(Uξ

)2(dz2 +z2dΩ2

2 +dr2 +r2dφ2)

(2.5)

where ξ2 = z2 + r2. The location of D6 brane is described by r(z) with φ = 0 and τ = τ0.The distance between D4 and D6 branes is interpreted as a quark mass; r(∞) ∼ mq . Thenthe induced metric on the D6 brane is

ds2D6 =

(U

L

)3/2

ηµνdxµdxν +

(L

U

)3/2(Uξ

)2[(1 + r′

2)dz2 + z2dΩ22

], (2.6)

where the prime denotes the derivative with respect to z. The embedding configuration ofprobe brane is governed by Dirac-Born-Infeld (DBI) action. The DBI action for D6 brane,in the case of Nf = 1, becomes

SD6 =−µ6

∫d7σ e−φ

√−det(P[g] + 2πα′F )

=−τ6

∫dtdz z2(1 + 1/ξ3)4/3

√(1 + 1/ξ3)4/3(1 + r′2)− F 2 (2.7)

where µ−16 = (2π)6l7s , τ6 = µ6g

−1s Ω2V3(U3

KK/4) and F = (22/3/UKK)2πα′Fzt = A′t . Here,for later convenience, more general DBI action including a gauge field At is considered.

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JHEP10(2011)111

t 1 2 3 (τ) ξ θ ϕ1 ϕ2 ϕ3

D4 • • • • •

Table 2. The brane configuration: the compact D4

The source of the gauge field is the end points of fundamental strings. We define thedimensionless quantity Q,

∂LD6

∂F=

z2(1 + 1/ξ3)4/3F√(1 + 1/ξ3)4/3(1 + r′2)− F 2

≡ Q , (2.8)

and Q is related to the number of fundamental strings Q by Q = Q/(2πα′τ6)(UKK/22/3) .Here Q is identified with the total quark number of a system. The Hamiltonian of the D6brane is obtained through the Legendre transformation.

HD6 = τ6

∫dz

√(1 + 1/ξ3)4/3

[z4(1 + 1/ξ3)8/3 + Q2

](1 + r′2) (2.9)

This D4/D6 model is extended to study dense matter in confined phase [17] and to in-vestigate a model for transition from nuclear matter to strange matter [18]. Here we brieflyreview the D4/D6 model in dense matter and refer to [17–19] for details. In holographicQCD, a compact D4 brane wrapping on the 4-sphere S4 transverse to R1,3 is introduced as abaryon [19]. To describe the compact D4 brane, the S4 part of the background metric (2.1)is rewritten as dΩ2

4 = dθ2 + sin2θdΩ23 . The world-volume coordinate for the compact D4 is

( t, θ, ϕα) where θ is the polar angle from the south pole. To study dense matter, we turnon the time component of the gauge field At. The SO(4) symmetry gives the ansatz thatall fields depend on θ, that is, ξ(θ) and At(θ). The induced metric on this D4 brane is

ds2D4 = −

(U

L

)3/2

dt2 + (L3U)1/2[(1 +

ξ2

ξ2)dθ2 + sin2θdΩ2

3

](2.10)

where ξ = ∂ξ/∂θ. The DBI action of the compact D4 brane is

SD4 =−µ4

∫d5σ e−φ

√−det(P[g] + 2πα′F )

=−τ4

∫dtdθ sin3θ

√(1 + 1/ξ3)4/3(ξ2 + ξ2)− F 2 (2.11)

where µ−14 = (2π)4l5s , τ4 = µ4g

−1s Ω3L

3(UKK/22/3) and F = (22/3/UKK)2πα′Fθt = ˙At .The dimensionless displacement is defined as

∂LD4

∂F=

sin3θ F√(1 + 1/ξ3)4/3(ξ2 + ξ2)− F 2

≡ D , (2.12)

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JHEP10(2011)111

which is given by Chern-Simons (CS) term.

S(CS)D4 = µ4

∫A1 ∧G4 = τ4

∫dtdθ 3 sin3θ At (2.13)

From the equation of motion of the gauge field, D is obtained explicitly.

D(θ) = 2− (3 cos θ − cos3 θ) (2.14)

The Hamiltonian of the compact D4 brane is also given by the Legendre transformation.

HD4 = τ4

∫dθ

√(1 + 1/ξ3)4/3(ξ2 + ξ2)

√D2 + sin6 θ (2.15)

We consider the system in which the number of fundamental strings is Q. Since Nc

fundamental strings are attached to each compact D4 brane, there exist NB(= Q/Nc)number of D4 branes. The other end of the fundamental string is located on the D6 braneas a source of the gauge field. Since the tension of the string is always larger than that ofthe brane, the fundamental strings pull both D4 and D6 branes. Eventually the length ofthe string vanishes and the compact D4 and D6 branes meet at a point. To be a stablesystem, the force at the cusp should be balanced: (Q/Nc)fD4 = fD6[Q] . Force due to thebrane tension at the point is given by the variation of its Hamiltonian with respect to theposition of contact point.

f ≡ ∂H

∂ξc

∣∣∣∣fixed other values

(2.16)

At zero temperature, energy density ε and pressure p are calculated as

ε =1V3

(Q

NcΩD4 + Ωreg.

D6 [Q])

and p =1V3

Ωreg.D6 [Q] (2.17)

where Ω and Ω denote the grand canonical potential and its Legendre transformed one,respectively. These quantities can be obtained as on-shell action and Hamiltonian.

Ω ≡ S|on-shell , Ω ≡ H|on-shell (2.18)

When we calculate the on-shell values of probe D6 branes, the integration result with theinfinite range of z diverges. Therefore, we should subtract out the zero-density contribution.

Ωreg.[Q] ≡ Ω[Q]− Ω[Q = 0] (2.19)

The baryon number density is also given by ρ = Q/(NcV3) .If we want to extend the single flavor description to a multi-flavor case (Nf > 1), it is

convenient to introduce the quantity q ≡ Q/Nf . In this work we consider all Nf D6 probebranes have identical asymptotic heights, i.e., Nf quark flavors with the same masses. Inthis case we can trivially extend the single flavor case to the multi-flavor one. For a fixedQ, Q/Nf number of fundamental strings are attached on each D6 probe brane. Then theforce balancing condition is modified as (Q/Nc)fD4 = NffD6[q] . The energy density andthe pressure are

ε =1V3

(Q

NcΩD4 +Nf Ωreg.

D6 [q])

, p =Nf

V3Ωreg.

D6 [q] . (2.20)

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JHEP10(2011)111

3 Numerical results

Now, we use the EoS obtained in holographic QCD models to calculate the mass-radiusrelation of a compact star. To this end, we solve the general-relativistic TOV equationusing the holographic EoS:

dp

dr= −1

2(ε+ p)

2m+ 8πr3p

r(r − 2m), (3.1)

where r is the circumferential radius, and geometric units, c = G = 1 are used. Theenergy density, ε, is given by ε ≡ (E/A + mb)ρ, where E/A is the energy per baryon,mb = 938 MeV is the mass of the baryon, and ρ is the baryonic number density. Theintegrated mass of the compact star is given by m(r) ≡ 4π

∫ r0 ε(r

′)r′2dr′. We employ afourth-order Runge-Kutta method together with a tabulated equation of state obtainedfrom each of the holographic models.

We first consider an EoS calculated in the D4/D8/D8 model. Physics of dense matterin the Sakai-Sugimoto model [15] has been developed with/without the source term forbaryon charge [20–24]. One thing that might need to be improved with the Sakai-Sugimotomodel for nuclear matter might be the absence of the scalar field which is responsible forthe intermediate attraction of the nuclear force. In this work we adopt a simple EoS atzero temperature with a point-like baryon source given in [24]. Hereafter, we adopt thenatural units. The regularized Helmholtz free energy is given by

Freg(nB)V3

= a

∫ ∞−∞

dZK2/3

(√1 +

(NcnB)2

4a2bK−5/3 − 1

), (3.2)

where nB is the baryon number density and

K = 1 + Z2 , a = 3.76× 109 MeV4 , b = 7.16× 10−6 MeV−2 .

Here they took λ ' 16.71 and MKK ' 950 MeV [24]. We adopt the EoS in [24] to solvethe TOV equation. In this case, however, we could not find a stable compact star, i.e., astar satisfying pressure-zero condition with a radius R, p(R) = 0, within a reasonable valueof the radius. This means that the EoS from the D4/D8/D8 model may not support anystable compact stars or may support one whose radius is very large. This might be dueto a deficit of attractive force in the D4/D8/D8 since the gravity alone may not balanceagainst the pressure of the D4/D8/D8 EoS. In nuclear physics, the long range attractiveforce is governed by one-pion exchange, while the intermediate range ∼ 1 fm attractionis dominated by a scalar field or two-pion exchange. In the D4/D8/D8 [15] a scalar thatcouples to baryon fields is missing. We may find a stable compact star if we try some otherEoS based on the Sakai-Sugimoto model other than the one given in [24]. However, theradius of the resulting star will be much larger than the one in nature and also the oneobtained in the D4/D6 models given below.

– 6 –

JHEP10(2011)111

0.0⋅100

5.0⋅109

1.0⋅1010

1.5⋅1010

2.0⋅1010

2.5⋅1010

3.0⋅1010

3.5⋅1010

0 1 2 3 4 5 6 7 8 9

ener

gy d

ensi

ty, ε

(MeV4)

ρ/ρ0

D4/D6 (λ=6)D4/D6/D6 (λ=6)

D4/D6/D6/D6 (λ=6)

(a)

0.0⋅100

1.0⋅109

2.0⋅109

3.0⋅109

4.0⋅109

5.0⋅109

6.0⋅109

7.0⋅109

8.0⋅109

9.0⋅109

0 1 2 3 4 5 6 7 8 9

pres

sure

, p (

MeV

4)

ρ/ρ0

D4/D6 (λ=6)D4/D6/D6 (λ=6)

D4/D6/D6/D6 (λ=6)

(b)

Figure 1. (a): Relation between energy density and baryon number density, and (b): relationbetween pressure and baryon number density for the three holographic equations of state considered.Here we take λ = 6 and MKK = 1.04 GeV.

Now, we move on to the D4/D6 models. The EoS is given in eq. (2.20) as

ε(ρ) = a

(Q

4τ4ΩD4 +

Nf

τ6Ωreg.

D6 [q])

p(ρ) = a

(Nf

τ6Ωreg.

D6 [q])

(3.3)

where ρ = b Q is the baryon number density and

a ' 4.45× 108 MeV4 , b ' 1.07× 106 MeV3 . (3.4)

Here we take λ = 6 and MKK = 1.04 GeV in D4/D6 model [25, 26]. Since the value ofMKK is fixed uniquely by a meson mass, MKK ∼ 1 GeV is rather robust compared to thevalue of the ’t Hooft coupling. With this choice of our model parameters, we plot theenergy density and pressure with respect to the baryon number density in figure 1. For agiven central baryon number density, a configuration is obtained by integrating eq. (3.1)from the center, r = 0, to the point where the pressure vanishes, i.e., p(r = R) = 0,with R being the radius of the configuration. The mass of this configuration is given bym(r = R) = M . For low baryon number densities, i.e., at ρ < 3 × 10−4 · ρ0, the Baym-Pethick-Sutherland equation of state [27] is used. Here ρ0 is the normal nuclear density, andwe use ρ0 = 1.3×106 MeV3. This equation of state is patched smoothly to the holographicequation of state at ρ ≈ 3× 10−4 · ρ0 MeV3.

By solving the TOV equation, we obtain a mass-radius relation for each equation ofstate. In figure 2, we compare the mass-radius relations for three holographic equationsof state with Nf = 1, 2, 3. As we increase the number of quark flavors we find thatthe maximum mass and corresponding radius becomes smaller. This is consistent with thesimple picture that larger Nf favors a soft EoS, thereby smaller maximum mass and radius.

– 7 –

JHEP10(2011)111

0.5

1.0

1.5

2.0

2.5

3.0

3.5

4.0

4.5

5.0

5.5

0 50 100 150 200 250 300 350 400 450 500 550 600 650 700

mas

s, M

(M

sun

)

radius, R (km)

D4/D6 (λ=6)D4/D6/D6 (λ=6)

D4/D6/D6/D6 (λ=6)

Figure 2. Comparison of mass-radius relations for three holographic equations of state with λ = 6and MKK = 1.04 GeV.

0⋅100

1⋅1010

2⋅1010

3⋅1010

4⋅1010

5⋅1010

6⋅1010

7⋅1010

0 1 2 3 4 5 6 7 8 9

ener

gy d

ensi

ty, ε

(MeV4)

ρ/ρ0

D4/D6 (λ=6)D4/D6/D6 (λ=6)

D4/D6/D6/D6 (λ=6)D4/D6 (λ=17)

D4/D6/D6 (λ=17)D4/D6/D6/D6 (λ=17)

(a)

0.0⋅100

2.0⋅109

4.0⋅109

6.0⋅109

8.0⋅109

1.0⋅1010

1.2⋅1010

1.4⋅1010

1.6⋅1010

0 1 2 3 4 5 6 7 8 9

pres

sure

, p (

MeV

4)

ρ/ρ0

D4/D6 (λ=6)D4/D6/D6 (λ=6)

D4/D6/D6/D6 (λ=6)D4/D6 (λ=17)

D4/D6/D6 (λ=17)D4/D6/D6/D6 (λ=17)

(b)

Figure 3. (a): Relation between energy density and baryon number density, and (b): relationbetween pressure and baryon number density for the three holographic equations of state considered.

Roughly speaking, the observed neutron stars have masses around 1.5M and theirradii are around 10 km. Compared to these observed neutron stars, what we obtained aremore massive and larger.

To see the λ-dependence of our results, we take λ = 17 and MKK = 1.04 GeV. Weshow the energy density and pressure for each model with different values of λ in figure 3.

In figure 4, we collect mass-radius relations with λ = 6, 17 and MKK = 1.04 GeV. Wefind that with a bit larger value of the ’t Hooft coupling, the mass and radius of our compactstars becomes smaller. For instance the maximum mass for Nf = 2 with λ = 6 was about3.86M and its radius was 45 km, which are now M = 1.36M and R = 17 km respectively.

So far we have focused on the mass-radius relation from holographic EoSs. Now welook into the holographic EoS in detail to see why our compact stars are different fromobserved neutron stars and to check if it satisfies constraints from thermodynamics or

– 8 –

JHEP10(2011)111

0.5

1.0

1.5

2.0

2.5

3.0

3.5

4.0

4.5

5.0

5.5

0 50 100 150 200 250 300 350 400 450 500 550 600 650 700

mas

s, M

(M

sun

)

radius, R (km)

D4/D6 (λ=6)D4/D6/D6 (λ=6)

D4/D6/D6/D6 (λ=6)D4/D6 (λ=17)

D4/D6/D6 (λ=17)D4/D6/D6/D6 (λ=17)

Figure 4. Comparison of mass-radius relations for three holographic equations of state.

0

2

4

6

8

10

12

-5 -4 -3 -2 -1 0 1 2

log(p(M

eV4))

log(ρ/ρ0)

D4/D6 (λ=6)D4/D6/D6 (λ=6)

D4/D6/D6/D6 (λ=6)D4/D6 (λ=17)

D4/D6/D6 (λ=17)D4/D6/D6/D6 (λ=17)

APRFPSSly4

Figure 5. Comparison of the pressure from D4/D6 models with those from a few typical EoSs.

causality. In figure 5, we compare the pressure from the D4/D6 models with those froma few conventional EoSs: APR [28], FPS [29], Sly4 [30]. Note that pressures from D4/D6models are quite larger than those from the conventional EoSs in most of the density regionthat are relevant to neutron stars. This might be the primary reason why our compactstars are quite large in radius compared to other conventional results.

Large pressure implies large speed of sound. In order to check if the EoSs from D4/D6models do not violate causality, we calculate the square of the sound velocity, c2

s = dp/dε

on the left panel of figure 6. Our result shows that the speed of sound is less than thatfor the fully relativistic Fermi gas limit, c2

s = 1/3. We also calculated the adiabatic indexγ defined by γ = d ln p/(d ln ε) on the right panel of figure 6. Note that the results withλ = 17 shows lower speed of sound but higher adiabatic index compared to those withλ = 6. This result can be understood from the relation γ = c2

s(ε/p) and the results infigure 3.

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JHEP10(2011)111

0.00

0.05

0.10

0.15

0.20

0.25

0.30

0.35

0 1 2 3 4 5 6 7 8 9

cs

2

ρ/ρ0

D4/D6 (λ=6)D4/D6/D6 (λ=6)

D4/D6/D6/D6 (λ=6)D4/D6 (λ=17)

D4/D6/D6 (λ=17)D4/D6/D6/D6 (λ=17)

(a)

1.10

1.15

1.20

1.25

1.30

1.35

1.40

1.45

1.50

1.55

0 1 2 3 4 5 6 7 8 9

γ

ρ/ρ0

D4/D6 (λ=6)D4/D6/D6 (λ=6)

D4/D6/D6/D6 (λ=6)D4/D6 (λ=17)

D4/D6/D6 (λ=17)D4/D6/D6/D6 (λ=17)

(b)

Figure 6. (a) Sound velocity squared c2s, and (b) the adiabatic index γ, for the holographic EoSswith respect to the normalized baryonic number density.

4 Summary

We solved the TOV equation using the EoS calculated in holographic QCD. We first usedthe EoS from a dense D4/D8/D8 model. In this case, however, we could not find a stablecompact star that satisfies pressure-zero condition p(R) = 0 at a reasonable value of theradius R. This means that the EoS from the D4/D8/D8 model may not support any stablecompact stars or may support one whose radius is very large. This might be due to the lackof attractive force mediated by a scalar field or two-pion exchange in the model. Next, weconsidered the D4/D6 models with different number of quark flavors, Nf = 1, 2, 3. Thoughthe mass and radius of a holographic star are larger than those of normal neutron stars,we found stable compact stars in this case. To see why the compact stellar object obtainedfrom the D4/D6 model is quite different from observed compact stars, we compared thepressure from our model with a few typical ones. We found that the pressure in ourwork is an order of magnitude larger than that of conventional works. We also confirmedthat the EoSs considered in our work do not violate causality by considering the soundvelocity square c2

s and the adiabatic index γ. Since holographic QCD is based on the QCDsymmetries which are believed to be restored at very high densities, extending the modelto low density region may not be valid. In order to have a more realistic compact star, onemay have to find a way of introducing proper attraction in holographic QCD or have tomake a hybrid star in which low density EoS is replaced by the conventional one.

Acknowledgments

Y.K. thanks Keun-Young Kim and Hyun Kyu Lee for useful discussions. M.B. Wan thanksGordon Baym, Feryal Ozel and Chris Pethick for useful discussions. Y.K. and I.J.Shin ac-knowledge the Max Planck Society(MPG), the Korea Ministry of Education, Science andTechnology(MEST), Gyeongsangbuk-Do and Pohang City for the support of the Indepen-dent Junior Research Group at APCTP. C.H.L. was supported by the BAERI Nuclear R& D program (M20808740002) of MEST/KOSEF and the Mid-career Researcher Programthrough NRG grant funded by the MEST (No. 2009-0083826).

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