neutron stars and speed limits in ads/cft · sound and equation of state the equation of state xes...

Neutron stars and speed limits in AdS/CFT

Carlos HoyosUniversidad de Oviedo

w/.David Rodriguez, N. Jokela, A. Vuorinen,

arXiv:1603.02943

String at DunesJuly 13, 2016

C. Hoyos (Oviedo U.) Neutron stars Natal 2016 1 / 25

Outline

Introduction

Speed limits in holographic models

Quark matter in neutron stars


Introduction


QCD phase diagram


Possible neutron star structures

(From [Weber astro-ph/0407155])C. Hoyos (Oviedo U.) Neutron stars Natal 2016 3 / 25

Matter in neutron stars

(From [Kurkela, Fraga, Schaffner-Bielich, Vuorinen 1402.6618])


Sound and equation of state

The equation of state fixes p vs ε

p = p(ε)

In a CFT

0 =⟨T µµ

⟩= −ε+ 3p ⇒ v2

s =13.

Causality vs ≤ 1 restricts the equation of state

p ≤ ε


Stiffness

Compress the fluid ⇒ increase energy density

Pressure also increases ⇒ opposes compression

The larger ∂p∂ε is, the less compressible is the fluid

The EoS is ‘stiffer’ or ‘softer’ for larger or smaller speeds of sounds


Neutron stars EoS


Speed limits in holographic models


Holographic models (RGflows)

First computed in N = 4 SYM [Policastro, Son, Starinets ’02]

vs =1√3

Masses for fermions (mf ) and scalars (mb) (N = 2∗)[Benincasa, Buchel, Starinets ’05]

vs =1√3

(1−

[Γ(

34

)]43π4

(mf

T

)2− 1

18π4

(mb

T

)4+ · · ·

)

Klebanov-Strassler (N = 1): logarithmic running with scale Λ[Aharony, Buchel, Yarom ’05]

v2s =

1

3− 2

9

1

log TΛ

+ · · ·


Holographic models (RGflows)

For a class of models:relevant deformation (∆ < 4) of a CFT4 = minimally coupled scalar field

S =1

2κ2

∫d5x

(R− 1

2(∂φ)2 − V (φ)

)At high temperatures the speed of sound is below the conformal value[Cherman, Cohen, Nellore ’09; Hohler, Stephanov ’09]

v2s =

1

3− C(∆)(LT )∆−4 + · · ·

Holds for several scalars [Cherman, Nellore ’09]

Conjecture: universal bound

Further evidence: D-brane intersections


D-brane intersections

D3/D7: flavor mass m, condensate c [Mateos, Myers, Thomson ’07]

v2s −

1

3' λYM

24π2

Nf

Nc

(mc+

1

3mT

∂c

∂T

)< 0.

D3/D7 (flavors) at T = 0, µ 6= 0Massless [Karch, Son, Starinets ’08] and massive [Kulaxizi, Parnachev ’08]

v2s =

µ2 −m2

3µ2 −m2≤ 1

3(v2s < 1)


The bound in neutron stars

Neutron stars: largest mass depends on equation of state

Observations find up to ∼ 2M

Needs stiff equation of state

Bound on the speed of sound strongly disfavored [Bedaque, Steiner ’14]

Can the bound be violated in holographic models?


‘Trivial’ breaking of the bound

= the UV is not a CFT


D-brane intersections

Dp/Dq: (n|p ⊥ q) n = common spatial directions (q ≥ p)[Karch, Kulaxizi, Parnachev ’09; Goykham, Parnachev, Zaanen ’12;

DiNunno, Ihl, Jokela, Pedraza ’14; Jokela, Ramallo ’15]

v2s =

2

2n+ p−32 (p+ q − 2n− 8)

Dp/Dq n SUSY v2s > 1/n

Dp/D(p+ 4) p X p > 3Dp/D(p+ 2) p− 1 X p > 3Dp/Dp p− 2 X p > 3

D4/D8/D8 3 7 XD3/D7′ 2 7 7

D2/D8′ 2 7 X


Non-relativistic theories

Non-relativistic scaling

t→ λzt, xi → λxi, i = 1, . . . , n

Equation of state

p =z

nε

Stiffer than a CFT for z > 1

Sound dispersion relation[Hoyos, O’Bannon, Wu ’10; Lee,Pang, Park’10; Dey, Roy ’13]

ω2 =z

nCz,nρ

2(z−1)/nk2

Valid for 2 > z ≥ 1. For z > 2 there is no sound mode.


Field theory counterexample

QCD at non-zero isospin chemical potential µI

For µI > mπ ⇒ Pion condensate

Speed of sound [Son, Stephanov ’00]

v2s =

µ2I −m2

π

µ2I + 3m2

π

v2s → 1 when mπ/µI → 0


‘Non-trivial’ breaking of the bound

Only known to happen for some unstable phases [Buchel, Pagnutti ’09]

However, the question has not been explored in a systematic way

Charged/dilatonic black holes?

Holographic nuclear matter models?

To explore...


Quark matter in neutron stars


Conditions on quark matter

Baryon chemical potential µB = µu + µd + µs

Chemical equilibrium under weak processes

u→ d+ e+ + νe d, s→ u+ e− + νe

µu = µd − µe, µd = µs = µu + µe

Thenµu = (µB − 2µe)/3, µd = µs = (µB + µe)/3

We take µe = 0µu = µd = µs = µB/3 ≡ µq

T ∼ 0.1− 100 keV µq ∼ 100 MeV


Holographic model

N = 4 SU(Nc) SYM plus Nf Nc fundamental hypers

Dual: AdS5 × S5 with Nf probe D7 branes

Pressure (γ ' 1.4):

P =NcNf

4γ3λYM(µq −m)3 +O(µ3

qT, T4).

[Karch, O’Bannon ’07]

EoS:

ε = µq∂P

∂µq− P = 3P +

√2m2

2π

√P

∂P

∂ε<

1

3


Extrapolation to QCD

Nc = Nf = 3

Stefan-Boltzman value as µq →∞

P ∼NcNf

12π2µ4q

This fixes

λYM =3π2

γ3' 10.74

P (µq = m) = 0 ⇒ m ∼ mp/3

m ' 308.55 MeV


Comparison to nuclear matter models

350 400 450 500 550 600 6500

100

200

300

400

500

600

700

Quark chemical potential [MeV]

Pressure

[MeV/fm3]

50 100 500 1000 5000 104

1

10

100

1000

104

Energy density [MeV/fm3]

Pressure

[MeV/fm3]


Neutron star

Self-gravitating spherically symmetric object in hydrostatic equilibrium

Metric

ds2 = −eν(r)dt2 +dr2

1− 2M(r)r

+ r2dΩ22

Structure determined by Tolman-Oppenheimer-Volkoff equation

P ′(r) = − 1

r2

ε(r) + P (r)

1− 2M(r)r

(M(r) + 4πr3P (r)

)and

ν ′(r) = − 2P ′(r)

ε(r) + P (r), M ′(r) = 4πr2ε(r)

Boundary condition on the surface P (r = R) = 0


Mass versus Radius

8 10 12 14 160.0

0.5

1.0

1.5

2.0

2.5

Radius(km)

Mass(M

⊙)

No stable star with pure quark matter at the core


Outlook

Quark matter at the core?

Mixed phases

Are there holographic models with stiffer EoS?

Strongly coupled nuclear matter

Large-Nc makes baryonic matter difficult to study

Solitons in Sakai-Sugimoto [Kim, Sin, Zahed ’06; Bergman, Lifschytz,

Lippert ’07; Rozali, Shieh, Van Raamsdonk, Wu ’07; Kim, Sin, Zahed ’07

Kaplunovsky, Melnikov, Sonnenschein ’12; Li, Schmitt, Wang ’15; Rho, Sin,

Zahed ’09] and D3/D7 [Ammon, Jensen, Kim, Laia, O’Bannon ’12]

Unrealistic neutron stars in Sakai-Sugimoto and D4/D6 [Burikham,

Hirunsirisawat, Pinkanjanarod ’10; Kim, Lee, Shin, Wan ’11; Ghoroku,

Kubo, Tachibana, Toyoda ’13,Kim, Shin, Lee, Wan ’14]

New approach needed?


neutron stars and speed limits in ads/cft · sound and equation of state the equation of state xes...

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