Hydrodynamic Tests of Fluctuating Initial Conditions

George Moschelli&Hannu Holopainen

Transport Meeting24 January 2012

Motivation

MC-KLN: Drescher, Nara, nucl-th/0611017

IP-Glasma: Schenke, Tribedy, Venugopalan,arXiv:1206.6805, Phys. Rev. Lett. 108 (2012) 252301

Fluctuating Initial Conditions and Event-by-Event Studies

• Local Correlations

• Global Correlations

• Geometry Fluctuations

Local CorrelationsInitial State Configuration Final State Momentum

Final state momenta are correlated to initial position. • Reaction / event plane• Common origin

Influence of fluctuating ICs• Arbitrary event shapes.• Random number of sources

and source sizes.

Goal: Determine hydro response to “common origin” correlations and dependence on choice of IC.

Global Correlations

E-by-E Hydro Evolution• Ideal Hydro• Lattice EoS• Gaussian Energy Density

lumps at mixture of MC Glauber Nbin and Npart positions

• Gaussian width: 0.4 fm

Goal: Trace the evolution of fluid element correlations to freeze out.

Global Correlations

E-by-E Hydro Evolution• Ideal Hydro• Lattice EoS• Gaussian Energy Density

lumps at mixture of MC Glauber Nbin and Npart positions

• Gaussian width: 0.4 fm

Goal: Trace the evolution of fluid element correlations to freeze out.

Flow LinesSpace Velocity

• Dots at initial positions of binary collisions• Movement indicates fluid cell position and velocity • Black line: const*e2

• Blue line: const*e3

• Green dots: randomly chosen group within 0.4 fm radius

• 20-30% centrality• Nbin = 464• Npart = 176• Freeze out: T = 120 MeV

Flow LinesSpace Velocity

• 20-30% centrality• Nbin = 464• Npart = 176• Freeze out: T = 120 MeV

• Dots at initial positions of binary collisions• Movement indicates fluid cell position and velocity • Black line: const*e2

• Blue line: const*e3

• Green dots: randomly chosen group within 0.4 fm radius

Fluid-Fluid Correlations

1-p

yx

yx

,Cov

• “Emission” angle corresponds to initial spatial angle. Expectation: central (circular) collisions agree, peripheral (elliptical) collisions should deviate

• Faster dots have larger displacement

• Final velocity depends on initial position. → Angular correlations!

• Faster dots freeze out first

• Need mixed events

Average Displacement

r0,min

r0,max

• Larger average displacement in central collisions

• central collisions live longer • greater effect on common origin

correlations than vn

• Linear correlation between r0, Dr, and vFO

• Flow lines starting at different radial positions get different transverse push.

• Enhances common source correlations

• Changes <en>time

Goal: Determine a source “resolution”.

Freeze Out Time• Faster dots freeze out first• Blue: Event average 20-30% centrality• Red: single event with 464 Flow Lines

• Average flow line lifetime longest in most central collisions

Freeze Out Time

• Freeze out histograms indicate the flux of flow lines through the freeze out surface at different times.

Freeze out and Event Planes

rwnrw n

n

e

cos

nnrw

nrwnn

cossin

arctan1

Alvioli, Holopainen, Eskola, Strikman arXiv:1112.5306

Space Velocity

n = 1 w(r) = r3

n = 2 w(r) = r2

n = 3 w(r) = r3

e2

• Difference in initial eccentricities due to Glauber mixture IC vs. Nbin Flow Lines

• Freeze out changes initial and final eccentricity

• Freeze out velocity eccentricity represent a “time averaged” freeze out surface

• Final eccentricity agrees with freeze out velocity eccentricity

Goal: Study IC structure impact on time averaged velocity eccentricity.

e3

• Difference in initial eccentricities due to Glauber mixture IC vs. Nbin Flow Lines

• Freeze out changes initial and final eccentricity

• Freeze out velocity eccentricity represent a “time averaged” freeze out surface

• Final eccentricity agrees with freeze out velocity eccentricity

Goal: Study IC structure impact on time averaged velocity eccentricity.

en Distributions

Cartesian Space

Velocity Space#

Even

ts#

Even

ts

Fluctuations can differentiate initial conditions

Multiplicity Fluctuations

Fluctuations per source

Fluctuations in the number of sources

For K sources that fluctuate per event

KK

KK

K11

2

22

2

2

R

Negative binomial distribution 1 NBDkR

Schenke, Tribedy, Venugopalan, arXiv:1206.6805, Phys. Rev. Lett. 108 (2012) 252301

Gelis, Lappi, McLerran Nucl.Phys. A828, 149 (2009)

Gavin, Moschelli Phys.Rev. C79, 051902 (2009)

Negative Binomial Distribution

Fluctuations per source

Fluctuations in the number of sources

For K sources that fluctuate per event

KK

KK

K11

2

22

2

2

R

Negative binomial distribution 1 NBDkR

Schenke, Tribedy, Venugopalan, arXiv:1206.6805, Phys. Rev. Lett. 108 (2012) 252301

Gelis, Lappi, McLerran Nucl.Phys. A828, 149 (2009)

Gavin, Moschelli Phys.Rev. C79, 051902 (2009)

NBD put in by hand

Fluctuations and Correlationscorrelations = pairs - singles2

211121221 ,, pppppp r

R222121 1, NNNNddr pppp

ttt ppp

2121

2121 1, pppp dd

NNrpppp tttt

D

21

2122

2 cos12

,2

42 pppp ddnNN

rvv nnn

Multiplicity Fluctuations

Momentum Fluctuations

“Flow Fluctuations”

Gavin, Moschellinucl-th/1107.3317nucl-th/1205.1218

The next stepIC lumps from K random sources• Poisson flow line multiplicity per source

• Compare large <K> and small source size to small <K> and large source size

• Compare to “smooth” hydro

Angular Correlations

• Compare en and vn with different IC

• Radial cuts

• Momentum, vn (eccentricity) and vn{2}2-vn{4}2 fluctuations

Mixed Events• With and without aligned reaction / event planes

Summary

Can we use hydro select the right IC?

• Determine hydro response to “common origin” correlations and dependence on choice of IC.

• Trace the evolution of fluid element correlations to freeze out.

• Determine a source “resolution”.

• Study IC structure impact on time averaged velocity eccentricity.

Freeze out effects• Eccentricity fluctuations

• Event plane angle determination

Cumulant Expansion

212111212 ,, pppppp r

222 22 nnn vv

Pair Distribution:

Two-particle coefficient:

Correlated Part:

Borghini, Dinh, Ollitrault

vn factorization is a signature of flow if n = 0

• <vn>2 = reaction plane correlations

• 2n = other correlations

• vn{4} <vn>

Borghini, Dinh, Ollitrault;Voloshin, Poskanzer, Tang, Wang

D

21

2122

2 cos12

,2

42 pppp ddnNN

rvv nnn

The Soft Ridge

• Only cos D and cos 2D terms subtracted

•These terms also contain fluctuations

•Glasma energy dependence•R scale factor set in

Au-Au 200 GeV•Blast wave f (p,x)•Difference in peripheral

STAR→ALICE

refn

nref

rdydNnv

dydN

D

D

DD

21cos

22

1

2

Flow subtracted ridge

D

DD

ndydN

refn cos2 2

Four-Particle Coefficients

44321

44 cos224 nnn vnvv

4321

432111

4131211143214

,,

,

,,,

pppp

pppp

pppppppp

rr

r

42244321 24cos nnnn vvn

Voloshin, Poskanzer, Tang, WangBorghini, Dinh, and Ollitrault

Four-particle coefficient:

Four-Particle Distribution: keep only two-particle correlations

222244 Re24 nnnnn vvv

22224224

4321

Re224

cos

nnnnnnn vvv

n

vn{4} corrections

21

212 2cos1

,Re pppp ddnNN

rRPn

221

Four-particle coefficient:

Will cancel with vn{2} terms

Corrections of order ~1.2%

R• K flux tubes, assume • K varies event-by-event

Fluctuations per source

Fluctuations in the number of sources

For K sources that fluctuate per event

KK

KK

K11

2

22

2

2

R

KNK

KNNKK

222

KN

222222 KKKNN