gravitational collapse, holography, and hydrodynamics … · paul chesler 1 gravitational collapse,...

Paul Chesler

1

Gravitational collapse, holography, and

hydrodynamics in small systems

2

• Definition: Long wavelength, low frequency e↵ective

description of transport of conserved currents.

• Relativistic hydrodynamics:

Hydrodynamics

Macroscopically large system

2




Hydrodynamics


Assumptions:

• Local equilibrium.

• Local isotropy.

Constitutive relations (in local fluid rest frame):

T 00

hydro

= ✏,

T 0ihydro

= 0,

T ijhydro

= p �ij � ⌘⇥riuj +rjui � 2

3

�ijr · u⇤+ . . . .

Dynamics: @µTµ⌫hydro

= 0.}Dissipative stress ⇧µ⌫

2




Hydrodynamics


Assumptions:

• Local equilibrium.

• Local isotropy.

Two remarkable observations from LHC and RHIC

. 1 fm/c

timeliquid

nuclei QGP

?

1. Rapid equilibration in AA collisions:

2. A tiny drop of QGP in pA collisions?

[CMS: 1210.5482], [ALICE: 1212.2001],[ATLAS: 1212.5198], [PHENIX: 1303.1794],[Bzdak, Schenke, Tribedy, Venugopalan: 1304.3403]

Evolution consistent with hydrodynamics.

3

time } R ⇠ 1 fm

R

– Long lived: ⌧m.f.t. � 1/✏

quasiparticle

– Weakly interacting: �m.f.p. � �

de Broglie

– thydro

� ⌧m.f.t.,

– R � �m.f.p.,

4

• Microscopic description: Nearly all practical formulationsof real-time QFT rely on existence of quasiparticles.

• Kinetic theory: Hydrodynamics only applies overtime and length scales

What is the domain of utility of hydrodynamics?

5

Hydrodynamics in extreme conditions

• ✏quasiparticle

⇠ T , �de Broglie

⇠ 1

T .

• Experimentally accessible temperatures:

1

T ⇠ 1 fm.

) Relaxation time: thydro

⇠ 1/✏quasiparticle

.

) pA system size: R ⇠ �de Broglie

.

microscopic

scales

}

• Since �m.f.p. ⇠ 1g4T , natural to expect domain of hydro

to be maximal at strong coupling.

Interesting questions:

1. Is hydro theoretically consistent in such small systems?

2. How big are the smallest drops of liquid?

6

Strongly coupled dynamics and holographic duality

• Equivalence between certain QFTs and theories of of gravityin one higher dimension.

• Weak/strong equivalence:

– Strongly coupled QFT = classical gravity.

– All QFT dynamics — from far-from-eq dynamics to

hydrodynamics — encoded in numerical relativity problem.

• Holography = spherical cow.

– Extra symmetries.

A diverse set of holographic field theories

• Conformal theories

– N = 4 SYM

• Confining theories

– AdS soliton

• Confinement & chiral symmetry breaking

– Sakai-Sugimoto model

• Superconducting theories

– Abelian Higgs model

Very di↵erent ground states.

Universal character of many body systems

• Dual classical equations of motion

RMN � 1

2GMN (R� 2⇤) = 8⇡G

Newton

TMNmatter

.

• TMNmatter

is theory-dependent.

• States with O(N2

c

) entropy dual to black holes.

Universal character of many body systems

• Dual classical equations of motion

RMN � 1

2GMN (R� 2⇤) = 8⇡G

Newton

TMNmatter

.

• TMNmatter

is theory-dependent.

• States with O(N2

c

) entropy dual to black holes.

Universal features of black holes

()universal features of strongly coupled many body systems.

• Fluid/gravity duality: [Blattacharya et al: 0712.2456], [Baier et al: 0712.2451]

– IR behavior of horizons , hydro in QFT:

• Shear viscosity

⌘s = 1

4⇡ quantum & stringycorrections:

– Same in every holographic QFT.[Kovtun, Son & Starinets: hep-th/0405231]

• Non-hydrodynamic relaxation rates:

– Insensitive to choice of QFT.[Fuini & Ya↵e: 1503.07148][Buchel, Heller & Myers: 1503.07114][Janik et al: 1503.07149][Buchel & Day: 1505.05012]

A few universal features of strongly coupled dynamics

z

T 00

The simplest holographic models of collisions

• Simplest theory to study: N = 4 SYM.

• Gravitational equations of motion:

RMN � 1

2

GMN(R� 2⇤) = 0.

• Collide gravitational shockwaves and make black hole.

• Dual SYM energy density: T 00(t,x?, z) = F (x?, z ⌥ t)

SYM energy

10

Holographic description of QFT (I)Connecting 5d physics to 4d physics

State |�⇥ in QFT � classical field configuration {�i} in gravitational theory.

10


State |�⇧ in QFT ⇥ classical field configuration {�i} in gravitational theory.

Vacuum

4d QFT

• ⌅0|Tµ� |0⇧ = 0, ⌅0|Jµ|0⇧ = 0, . . .

5d Gravitational description

• Geometry: AdS5 ds2 = r2[�dt2 + dx2] + dr2

r2 .

• Properties:

i. Each slice of constant r is copy of Minkowski space.ii. 4d boundary at r = ⇤.iii. Throat

Deviations in geometry from AdS5 ⇥ excited states in QFT

4d Minkowski Space

19

The ground state and AdS5

QFT

• ⇥0|Tµ� |0⇤ = 0, ⇥0|Jµ|0⇤ = 0, . . . ,

Gravitational description

• Einstein: RMN � 12gMN (R + 2�) = 0.

• Most symmetric solution: AdS5,

ds2 =L2

u2

��dt2 + dx2 + du2

⇥. (2)

19


QFT

• ⇥0|Tµ� |0⇤ = 0, ⇥0|Jµ|0⇤ = 0, . . . ,


• Einstein: RMN � 12GMN (R + 2�) = 0.


ds2 =L2

u2

��dt2 + dx2 + du2

⇥. (2)r

ds2 = r2[�dt2 + dx2] +dr2

r2

11

10


State |�⇥ in QFT � classical field configuration {�i} in gravitational theory.

10



Vacuum

4d QFT

• ⌅0|Tµ� |0⇧ = 0, ⌅0|Jµ|0⇧ = 0, . . .


• Geometry: AdS5 ds2 = r2[�dt2 + dx2] + dr2

r2 .

• Properties:



10



Vacuum

4d QFT

• ⌅0|Tµ� |0⇧ = 0, ⌅0|Jµ|0⇧ = 0, . . .


• Geometry: AdS5: ds2 = r2[�dt2 + dx2] + dr2

r2 .

• Properties:



4d Minkowski Space

19


QFT

• ⇥0|Tµ� |0⇤ = 0, ⇥0|Jµ|0⇤ = 0, . . . ,


• Einstein: RMN � 12gMN (R + 2�) = 0.


ds2 =L2

u2

��dt2 + dx2 + du2

⇥. (2)

19


QFT

• ⇥0|Tµ� |0⇤ = 0, ⇥0|Jµ|0⇤ = 0, . . . ,


• Einstein: RMN � 12GMN (R + 2�) = 0.


ds2 =L2

u2

��dt2 + dx2 + du2

⇥. (2)r

ds2 = r2[�dt2 + dx2] +dr2

r2

11

}

7

Connecting 5d dynamics to 4d dynamics

Classical field theory:Near boundary deformation in 5dgeometry induces 4d stress tensor Tµ� .(Brown & York: gr-qc/9209012)

Object deforms 5d geometry.

7




Analogous classical EM problem: image charges

conductor

7





conductor

Jµinduced

Image problem from electrodynamics

Excited states and expectation values

“Image” Tµ⌫

12

}

7




7





conductor

7





conductor

Jµinduced

Image problem from electrodynamics

Excited states and expectation values

“Image” Tµ⌫

[de Haro, Solodukhin & Skenderis: hep-th/0002230]

Reinterpret Tµ⌫ ! hTµ⌫i

12

5

What should the geometry correspondingto a liquid look like?

Local fluid rest frame:

• T

µ⌫(x) = diag[✏(x), p(x), p(x), p(x)] + gradients.

• Local temperature T (x) ⇠ ✏(x)

1/4.

• Local entropy s(x) ⇠ T (x)

3.

• Dynamics: @µTµ⌫

= 0

5

What should the geometry correspondingto a liquid look like?

Local fluid rest frame:

• T

µ⌫(x) = diag[✏(x), p(x), p(x), p(x)] + gradients.

• Local temperature T (x) ⇠ ✏(x)

1/4.

• Local entropy s(x) ⇠ T (x)

3.

• Dynamics: @µTµ⌫

= 0

Gravitational description:

• Dynamics: R

MN

� 12GMN

(R+ 2⇤).

• Local geometry: AdS-Schwarzschild black brane

ds

2= r

2[�f(x, r)dt

2+dx

2]+

dr

2

r

2f(x, r)

+ gradients, f = 1�⇣

rh(x)r

⌘4.

[Bhattacharyya, Hubeny,Minwalla, Rangamani: 0712.2456]

timespace

Black hole

“Holographic image”energy density

Colliding particles

Gravitational models of heavy ion collisions

14

timespace

Black hole


Colliding particles

liquidtime


14

timespace

Black hole


Colliding particles

liquidtime


5D gravity: a challenging numerical problem

14

Numerical relativity in a box

6

Gau

ge/g

ravi

tydual

ity

Mal

dac

ena

conje

cture

(Mal

dac

ena:

hep

-th/9

7112

00)

•C

onje

ctur

edeq

uiva

lenc

ebe

twee

nst

ring

sin

asym

ptot

ical

lyA

dS5�

S5an

d4d

N=

4SY

M.

•D

omai

nof

utili

ty:

Larg

eN

c,�

SYM⇥⇤

clas

sica

lapp

roxi

mat

ions

.

Bas

icid

ea:

•4d

field

theo

ryliv

eson

the

boun

dary

ofhi

gher

dim

ensi

onal

curv

edsp

ace.

•O

bjec

tsex

isti

ngin

high

erdi

men

sion

s:st

ring

s,bl

ack

hole

s,E

Mfie

lds

...

•O

bjec

tsex

isti

ngin

QFT

:qua

rks,

plas

mas

,con

serv

edcu

rren

ts..

.

5th

dim

ensi

on

bla

ckbra

ne

Dyn

amic

sin

5den

codes

dyn

amic

sin

fiel

dth

eory

.

{ �GMN

Asymptotically AdS:

• ds

2 ! r

2⌘µ⌫dx

µdx

⌫+

dr2

r2 .

• Time-like boundary at r = 1.

) must impose BCs at r = 1.

• Saving grace: infinite redshift.

15

�GMN


6

Gau

ge/g

ravi

tydual

ity

Mal

dac

ena

conje

cture

(Mal

dac

ena:

hep

-th/9

7112

00)

•C

onje

ctur

edeq

uiva

lenc

ebe

twee

nst

ring

sin

asym

ptot

ical

lyA

dS5�

S5an

d4d

N=

4SY

M.

•D

omai

nof

utili

ty:

Larg

eN

c,�

SYM⇥⇤

clas

sica

lapp

roxi

mat

ions

.

Bas

icid

ea:

•4d

field

theo

ryliv

eson

the

boun

dary

ofhi

gher

dim

ensi

onal

curv

edsp

ace.

•O

bjec

tsex

isti

ngin

high

erdi

men

sion

s:st

ring

s,bl

ack

hole

s,E

Mfie

lds

...

•O

bjec

tsex

isti

ngin

QFT

:qua

rks,

plas

mas

,con

serv

edcu

rren

ts..

.

5th

dim

ensi

on

bla

ckbra

ne

Dyn

amic

sin

5den

codes

dyn

amic

sin

fiel

dth

eory

.

{ �GMN

Asymptotically AdS:

• ds

2 ! r

2⌘µ⌫dx

µdx

⌫+

dr2

r2 .




15

�GMN


6

Gau

ge/g

ravi

tydual

ity

Mal

dac

ena

conje

cture

(Mal

dac

ena:

hep

-th/9

7112

00)

•C

onje

ctur

edeq

uiva

lenc

ebe

twee

nst

ring

sin

asym

ptot

ical

lyA

dS5�

S5an

d4d

N=

4SY

M.

•D

omai

nof

utili

ty:

Larg

eN

c,�

SYM⇥⇤

clas

sica

lapp

roxi

mat

ions

.

Bas

icid

ea:

•4d

field

theo

ryliv

eson

the

boun

dary

ofhi

gher

dim

ensi

onal

curv

edsp

ace.

•O

bjec

tsex

isti

ngin

high

erdi

men

sion

s:st

ring

s,bl

ack

hole

s,E

Mfie

lds

...

•O

bjec

tsex

isti

ngin

QFT

:qua

rks,

plas

mas

,con

serv

edcu

rren

ts..

.

5th

dim

ensi

on

bla

ckbra

ne

Dyn

amic

sin

5den

codes

dyn

amic

sin

fiel

dth

eory

.

{ �GMN

Challenges:

• Must impose BCs at r = 1.

– Imposing BCs at r = finite

yields unstable evolution.

• Einstein singular at r = 1.

Asymptotically AdS:

• ds

2 ! r

2⌘µ⌫dx

µdx

⌫+

dr2

r2 .




15

Characteristic formulation of PDEs

Example: 1 + 1D wave equation

• Equation of motion:

gµ⌫@µ@⌫� = (�@2⌧ + @2

r )� = f(�).

• Null coordinate: t = ⌧ � r


(2@r@t + @2r )� = f(�).

• Integral form:

@t� =

1

2

Z r

0dr0[f(�)� @2

r�] + C(v).

16

Characteristic formulation of PDEs

Example: 1 + 1D wave equation


gµ⌫@µ@⌫� = (�@2⌧ + @2

r )� = f(�).

• Null coordinate: t = ⌧ � r


(2@r@t + @2r )� = f(�).

• Integral form:

@t� =

1

2

Z r

0dr0[f(�)� @2

r�] + C(v).

Benefits:

1. Easier implementation

of BCs

2. Easier to deal with

singularities.

16

Einstein’s equations in characteristic form

black brane

radialdirection

time-like boundary

17

Metric ansatz:

ds

2= �Adt

2+ 2Fidx

idt+ 2drdt+ ⌃

2gijdx

idx

j, det gij = 1.

Infalling null geodesics: t = const., x = const.

Schematic form of Einstein’s equations:

�@

2r +Q⌃[g]

�⌃ = 0,

�@

2r + PF [g,⌃]@r +QF [g,⌃]

�F = SF [g,⌃],

(@r +Q⌃[g,⌃])˙

⌃ = S⌃[g,⌃, F ],

⇣@r +Q ˙g[g,⌃]

⌘˙

g = S ˙g[g,⌃, F,˙

⌃],

@

2rA = SA[g,⌃, F,

˙

⌃,

˙

g],

with

˙

h ⌘ @th+

12A@rh.

Einstein’s equations in characteristic form

black brane

radialdirection

time-like boundary

Never solve more than linear ODE!17

Metric ansatz:

ds

2= �Adt

2+ 2Fidx

idt+ 2drdt+ ⌃

2gijdx

idx

j, det gij = 1.

Infalling null geodesics: t = const., x = const.

Schematic form of Einstein’s equations:

�@

2r +Q⌃[g]

�⌃ = 0,

�@

2r + PF [g,⌃]@r +QF [g,⌃]

�F = SF [g,⌃],

(@r +Q⌃[g,⌃])˙

⌃ = S⌃[g,⌃, F ],

⇣@r +Q ˙g[g,⌃]

⌘˙

g = S ˙g[g,⌃, F,˙

⌃],

@

2rA = SA[g,⌃, F,

˙

⌃,

˙

g],

with

˙

h ⌘ @th+

12A@rh.

A few (of many) technical details

18

• Horizon excision & residual di↵eomorphism invariance.

• Field redefinitions to ameliorate boundary singularities.

• Discretize using pseudo-spectral methods.

• Employ domain decomposition in radial direction.

• Filtering.

• Parallelization.

• Choice of units: max(initial energy density) = 1.

• Total runtime: 14 days on 6-core desktop computer.

Results animated

Energy density T 00

“proton” “nucleus”

[PC: 1506.02209]

z

x

t = �1.125 t = 0 t = 1.125 t = 2.25

Energy

Momentum

x?

Results illustrated

z

x?

A tiny drop of liquid

Energy at t = 1.5

Temperature

x?

z

20%

15%

Stress at x? = z = 0

How small of a droplet?

• E↵ective temperature

T�1

e↵

⌘ @seq

@✏eq

��✏eq=✏

.

• Result: RTe↵

⇡ 1.

Rapid equilibration?

• Result: thydro

Te↵

⇡ 0.3.

Rel. Mag. of 1

st

order corrections

Rel. Mag. of 2

nd

order corrections

z

x?


Hydrodynamics in extreme conditions (II)

Rel. Mag. of 1

st

order corrections

Rel. Mag. of 2

nd

order corrections

z

x?


} thydro

� ⌧m.f.t. �

1

T,

R � �m.f.p. �

1

T,

Common lore:

Lesson:


Rel. Mag. of 1

st

order corrections

Rel. Mag. of 2

nd

order corrections

z

x?


} thydro

� ⌧m.f.t. �

1

T,

R � �m.f.p. �

1

T,

Common lore:

Lesson:

No theoretical inconstancy with hydro in pA.


Universality and critical gravitational collapse

�

Critical gravitational collapse: [Choptuik: 1993]

• Vary dimensionless parameter p ⌘ �Eprobe

.

• Must exists critical p = pc which for p < pcno black hole forms.

) Hydrodynamic evolution for p > pc.

) No hydrodynamic evolution for p < pc.


�



.




Universal gravitational dynamics as p ! pc (w.r.t. initial conditions)

• Self-similar geometry, scalings

– Entropy production: �S ⇠ (p� pc)3� , � ⇠ 0.4.

• What are universal dynamics in dual QFT?

• Interesting to look at low energy dynamics where

signs of hydrodynamic behavior turn o↵.


�



.




Universal gravitational dynamics as p ! pc (w.r.t. initial conditions)

• Self-similar geometry, scalings

– Entropy production: �S ⇠ (p� pc)3� , � ⇠ 0.4.

• What are universal dynamics in dual QFT?

• Interesting to look at low energy dynamics where

signs of hydrodynamic behavior turn o↵.

theory-dependent

Thank you

t

T 00

z

T 00

t z

Planar shock collisions

Key observations:

1. Sensitivity to shock profilenear lightcone.

[Casalderrrey-Solana et al: 1305.4919]

2. Hydrodynamic flow insidethe lightcone.

[PC & Yaffe: 1011.3562]

3. Hydrodynamic flow isinsensitive to shock profile.

[PC, van der Schee & Kilbertus]

thick width w

narrow width w

Universal hydrodynamic flow

ξ/ξFWHM

-1 -0.5 0 0.5 1

ϵ/ϵ(ξ=

0)

0

0.2

0.4

0.6

0.8

1

w = wo

to 6wo

Key points:

• Initial hydro data well described by

boost invariant uµand

✏(⇠, w)|⌧=const. = A(w)f(⇠

⇠FWHM(w)).

• Insensitive to functional form of shockwave.

properenergy✏

fluidvelocityu⌧

[PC,van der Schee, Kilbertus, to appear shortly]

ξ-1 0 1

ϵ/µ4

0

0.05

0.1

0.15

0.2

0.25

ξ-1 0 1

uτ

0.999

1

1.001

1.002

1.003

1.004w = 7w

o

w = wo

w = 7wo

w = wo

ξ-1 0 1

ϵ/µ4

0

0.05

0.1

0.15

0.2

0.25

ξ-1 0 1

uτ

0.999

1

1.001

1.002

1.003

1.004w = 7w

o

w = wo

w = 7wo

w = wo

rapidity ⇠ rapidity ⇠

Hydro variables at fixed proper time ⌧ = const.

Decreasing probe width

probe

target

How to make a small droplet of liquid

Viscous flow at LHC C. Gale, S. Jeon, B.Schenke, P.Tribedy, R.Venugopalan, PRL110, 012302 (2013)

10-5

10-4

10-3

0 500 1000 1500 2000 2500

P(d

Ng/d

y)

dNg/dy

Glasma centrality selection

0-5

%

5-1

0%

10-2

0%

20-3

0%

30-4

0%

40-5

0%

50-6

0%

0

0.2

0.4

0.6

0.8

1

0 2 4 6 8 10 12

P(b

)

b [fm]

Distribution of b in 20-30% central bin

0

0.05

0.1

0.15

0.2

0 0.5 1 1.5 2

⟨vn2⟩1

/2

pT [GeV]

ATLAS 20-30%, EP

η/s =0.2

v2 v3 v4 v5

Hydro evolution

MUSIC

0

0.02

0.04

0.06

0.08

0.1

0.12

0.14

0 10 20 30 40 50

⟨vn

2⟩1

/2

centrality percentile

η/s = 0.2ALICE data vn{2}, pT>0.2 GeV v2

v3 v4 v5

Experimental data:ATLAS collaboration, Phys. Rev. C 86, 014907 (2012)ALICE collaboration, Phys. Rev. Lett. 107, 032301 (2011)

Björn Schenke (BNL) BNL, March 2013 32/45

Viscous flow at RHIC and LHC C. Gale, S. Jeon, B.Schenke,P.Tribedy, R.Venugopalan, PRL110, 012302 (2013)

RHIC �/s = 0.12

0

0.05

0.1

0.15

0.2

0.25

0 0.5 1 1.5 2

⟨vn2⟩1

/2

pT [GeV]

RHIC 200GeV, 30-40%

open: PHENIX

filled: STAR prelim.

v2 v3 v4 v5

LHC �/s = 0.2

0

0.05

0.1

0.15

0.2

0.25

0 0.5 1 1.5 2

⟨vn2⟩1

/2

pT [GeV]

ATLAS 30-40%, EP v2 v3 v4 v5

Experimental data:A. Adare et al. (PHENIX Collaboration), Phys.Rev.Lett. 107, 252301 (2011)Y. Pandit (STAR Collaboration), Quark Matter 2012, (2012)ATLAS collaboration, Phys. Rev. C 86, 014907 (2012)

Lower effective ⌘/s at RHIC than at LHC needed to describe dataHints at increasing ⌘/s with increasing temperatureAnalysis at more energies can be used to gain information on (⌘/s)(T )

Björn Schenke (BNL) RHIC AGS Users’ Meeting 2013, BNL 19/28


RHIC �/s = 0.12

0

0.05

0.1

0.15

0.2

0.25

0 0.5 1 1.5 2

⟨vn2⟩1

/2

pT [GeV]

RHIC 200GeV, 30-40%

open: PHENIX


v2 v3 v4 v5

LHC �/s = 0.2

0

0.05

0.1

0.15

0.2

0.25

0 0.5 1 1.5 2

⟨vn2⟩1

/2

pT [GeV]

ATLAS 30-40%, EP v2 v3 v4 v5





RHIC �/s = 0.12

0

0.05

0.1

0.15

0.2

0.25

0 0.5 1 1.5 2

⟨vn2⟩1

/2

pT [GeV]

RHIC 200GeV, 30-40%

open: PHENIX


v2 v3 v4 v5

LHC �/s = 0.2

0

0.05

0.1

0.15

0.2

0.25

0 0.5 1 1.5 2

⟨vn2⟩1

/2

pT [GeV]

ATLAS 30-40%, EP v2 v3 v4 v5




η/s = 0.12η/s = 0.2

Figure 1: Model calculations compared to measurements of the harmonic decomposition of azimuthalcorrelations produced in heavy-ion collisions. The left panel shows model calculations and data for v

n

vs.collision centrality in Pb+Pb collisions at

psNN = 2.76 TeV. The right panel shows similar studies for

the p

T

dependence of v

n

in 200 GeV Au+Au collisions. The comparison of the two energies providesinsight on the temperature dependence of ⌘/s.

determination of these two unknowns is aided by measurements of multiple flow observables sensitiveto medium properties in di↵erent stages of the evolution [15, 35, 36]. Due to the large event-by-eventfluctuations in the initial state collision geometry, in each collision the created matter follows a di↵erentcollective expansion with its own set of flow harmonics (magnitude v

n

and phases �n

). Experimentalobservables describing harmonic flow can be generally given by the joint probability distribution of themagnitudes v

n

and phases �n

of flow harmonics:

p(vn

, v

m

, ..., �n

, �m

, ...) =1

Nevts

dNevts

dv

n

dv

m

. . . d�n

d�m

. (1)

Specific examples include the probability distribution of individual harmonics p(vn

), flow de-correlationin transverse and longitudinal directions, and correlations of amplitudes or phases between di↵erentharmonics (p(v

n

, v

m

) or p(�n

, �m

)). The latter are best accessed through measurements of correlationswith three or more particles. The joint probability distribution (1) can be fully characterized experimentallyby measuring the complete set of moments recently identified in Ref. [37]. With the added detail providedby these measurements, hydrodynamic models can be fine-tuned and over-constrained, thus refiningour understanding of the space-time picture and medium properties of the heavy-ion collisions. Initialmeasurements of some of these observables [38–40] and comparison to hydrodynamic models [25, 41–43]already provided unprecedented insights on the nature of the initial density fluctuations and dynamicsof the collective evolution. However, at this point none of the state-of-the-art hydrodynamic modelsproperly accounts for the dynamical fluctuations generated during the evolution by thermal noise [44–47]– future quantitative work will need to address these, too.

Precision determination of key QGP parameters

The agreement between the model and the data shown in Figure 1 suggests that the essential featuresof the dynamic evolution of heavy-ion collisions are well described by current models. These modelcalculations depend on a significant number of parameters that are presently poorly constrained byfundamental theory, and a reliable determination of the QGP properties requires a systematic explorationof the full parameter space. An example of such an exploration [48,49] is shown in Figure 2 where theshape of the QCD EOS is treated as a free parameter. The left panel shows a random sample of thethousands of possible Equations of State, constrained only by results on the velocity of sound obtained byperturbative QCD at asymptotically high temperature and by lattice QCD at the crossover transition

13

[Gale et al: 1209.6330]

[Kovtun, Son, Starinets: hep-th/0309213],

[Buchel, Liu: hep-th/0311175],

[Benincasa, Buchel, Naryshkin: hep-th/0610145].

RHICLHC

[Arnold, Moore, Ya↵e:hep-ph/0010177]

Applying holographic duality to heavy ion collisions

28

All holographic theories have

⌘

s=

1

4⇡+ finite coupling corrections.

Weak coupling:

⌘s ⇠ ⌧

m.f.t.✏quasiparticle =5.12

g(T )

4log 2.42/g(T )

gravitational collapse, holography, and hydrodynamics … · paul chesler 1 gravitational collapse,...

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