How (non-)linear is the hydrodynamics of heavy ion collisions?

Contents lists available atScienceDirect

Physics

Letters

B

www.elsevier.com/locate/physletb

How

(non-)linear is

the

hydrodynamics

of

heavy

ion

collisions?

Stefan Floerchinger

a

,

Urs

Achim Wiedemann

a

,

∗

,

Andrea Beraudo

a

,

b

,

Luca Del Zanna

c

,

d

,

e

,

Gabriele Inghirami

c

,

d

,

Valentina Rolando

f

,

g

a_{PhysicsDepartment,TheoryUnit,CERN,CH-1211Genève23,Switzerland}

b_{Dep.deFisicadeParticulas,U.deSantiagodeCompostela,E-15782SantiagodeCompostela,Galicia,Spain} c_{DipartimentodiFisicaeAstronomia,UniversitàdiFirenze,ViaG.Sansone1,I-50019SestoF.no(Firenze),Italy} d_{INFN- SezionediFirenze,ViaG.Sansone1,I-50019SestoF.no(Firenze),Italy}

e_{INAF- OsservatorioAstrofisicodiArcetri,L.goE.Fermi5,I-50125Firenze,Italy} f_{INFN- SezionediFerrara,ViaSaragat1,I-44100Ferrara,Italy}

g_{DipartimentodiFisicaeScienzedellaTerra,UniversitàdiFerrara,ViaSaragat1,I-44100Ferrara,Italy}

a

r

t

i

c

l

e

i

n

f

o

a

b

s

t

r

a

c

t

Articlehistory:

Received26January2014

Receivedinrevisedform4June2014 Accepted17June2014

Availableonline20June2014 Editor:J.-P.Blaizot

Weprovideevidencefromfullnumericalsolutionsthatthehydrodynamicalevolutionofinitialdensity fluctuations in heavy ion collisions can be understood order-by-order in a perturbative series in deviations from a smooth and azimuthally symmetric background solution. To leading linear order, modeswithdifferentazimuthalwavenumbersdonotmix.Whenquadraticandhigherordercorrections are numerically sizable,theycan beunderstoodas overtoneswith correspondingwavenumbers ina perturbativeseries. Severalfindings reportedinthe recentliterature resultnaturallyfrom thegeneral perturbativeseriesformulatedhere.

©2014TheAuthors.PublishedbyElsevierB.V.ThisisanopenaccessarticleundertheCCBYlicense (http://creativecommons.org/licenses/by/3.0/).FundedbySCOAP3_.

Inrecentyears,fluid dynamic simulationsof relativisticheavy ioncollisions haveprovidedstrongevidencefora picture accord-ing to which themomentum distributions ofsoft hadronsresult fromafluid dynamic evolutionof initialdensityfluctuations, see Refs.[1–4]forrecentreviews.Theresearchfocuses nowon under-standingindetailthemappingfromfluctuationsintheinitialstate toexperimentally accessibleobservablesin thefinal state [5–16]. While hydrodynamic evolution always shows non-linearities of somesize, weask herewhetherthehydrodynamicsof heavyion collisions is sufficiently weakly non-linear to be described by a perturbative series.We shall demonstratethat thefluid dynamic responsetoinitialperturbationsobeysageneralorderingprinciple inthat it can be organized interms of a perturbative expansion inpowersoftheamplitudesofinitialfluctuationsarounda back-ground,seeEq.(2)below.Thisperturbativeserieswillbeshownto applyalsoincaseswherenon-linearities aresizableordominant. Thisisofinterestsinceamappinginwhichnon-linearitiesare or-ganizedascorrectionstoalinearresponseprovidesaparticularly simpleandthusparticularpowerfultoolforrelatingexperimental observablestotheinitialconditionsofheavy ioncollisions andto thosepropertiesofmatter thatgoverntheir fluiddynamic evolu-tion[17].

*

Correspondingauthor.

Weconsiderinitialconditionsofheavy ioncollisions, specified intermsoffluctuating fluiddynamic fieldshi on ahypersurface

atfixedinitialtime

τ

0.Here,theindexi runsoverallindependent fields, hi

(

τ

,

r,

ϕ

,

η

)

=



w,ur

,u

φ

,u

η

,

π

bulk

,

π

ηη

, . . .



,

(1)

includinge.g.theenthalpydensityh1

=

w,threeindependentfluid velocity components, the bulk viscous tensor, the independent components ofthe shearviscous tensor, etc. Inthe followingwe assumeBjorkenboostinvarianceanddroptherapidity-argument

η

inthehydrodynamicalfields.FollowingRefs.[17,18],weexpresshi

intermsofabackgroundcomponenthBG_i andanappropriately nor-malizedperturbationh

˜

i.The backgroundistakentobe asolution

of the non-linear hydrodynamic equations initialized at

τ

0 with anazimuthally symmetricaverageovermanyevents.Itisevolved withthefluid dynamic solverECHO-QGP

[19]

.Foranysample of events,thisbackgroundneedstobedeterminedonlyonce.Weask towhatextentthetimeevolutionoftheh

˜

i,obtainedfromthefull

numericalECHO-QGPsolutionswithoutanylinearized approxima-tion,canbeunderstoodintermsofaperturbativeseriesontopof thebackgroundfields,

˜

hi

(

τ

,

r,

ϕ

)

=



r,ϕ

G

i j



τ

,

τ

0

,

r,r

,

ϕ

−

ϕ



˜

hj



τ

0

,r



,

ϕ





http://dx.doi.org/10.1016/j.physletb.2014.06.049

0370-2693/©2014TheAuthors.PublishedbyElsevierB.V.ThisisanopenaccessarticleundertheCCBYlicense(http://creativecommons.org/licenses/by/3.0/).Fundedby SCOAP3_.

306 S. Floerchinger et al. / Physics Letters B 735 (2014) 305–310

+

1 2



r,r,ϕ,ϕ

H

i jk



τ

,

τ

0

,

r,r

,

r

,

ϕ

−

ϕ



,

ϕ

−

ϕ





× ˜

hj



τ

0

,

r

,

ϕ



˜

hk



τ

0

,r



,

ϕ





+

O

˜

h3



,

(2)

where



_r

=



₀∞dr r,



_ϕ

=



₀2πd

ϕ

etc. The kernels

G

i j,

H

i jk (and

correspondingtermsforhigherordersinh

˜

i)dependonthe

time-evolvedbackgroundhBG_i only. Duetotheazimuthalrotation sym-metryofthebackground,

G

i j dependsontheangles

ϕ

and

ϕ

only

viathe difference

ϕ

−

ϕ

 andsimilarlyfor

H

i jk.Thequestion we

raiseinthetitlecannowbemademoreprecise:Weaskwhether theexpansion

(2)

ispossibleforasuitablychosenbackground,1 _in

whichrangeitisdominatedbythefirstlinearterm,andwhether non-linearities even iflarge can be understood perturbatively on thebasisof

(2)

.

Fortheinitialconditions,wemakeassumptionsthatarewidely spread in the phenomenological literature. The initial transverse velocity components vanish, the longitudinal velocity is Bjorken boostinvariant,theshearstresstensorisinitializedbyitsNavier– Stokes value, and the bulk viscous pressure is neglected. Initial fluctuationsresidethenonlyintheinitialenthalpydensityw

(

τ

,



r

)

, that we parametrize in terms of an azimuthally averaged back-groundwBG

(

τ

,

r

)

andtheweights w

˜

(_lm) oftheazimuthal (m)and

radial (l) wave numbersof adiscrete orthonormal Bessel–Fourier decomposition[17] w(

τ

0

,

r,

ϕ

)

=

wBG

(

τ

0

,

r)



1

+

∞



m=−∞

˜

w(m)

(

τ

0

,

r)eimϕ



,

˜

w(m)

(

τ

0

,r)

=

∞



l=1

˜

w(_lm)Jm



k(_lm)r



.

(3)

Here k_l(m)

=

z_l(m)

/

R, where z(_lm) is the l-th zero of the modified Bessel function Jm and R

=

8 fm throughout this work. Since

˜

w

(

τ

,

r

,

ϕ

)

is real, we have w

˜

(m)

₍

_τ

_,

_r

₎

_{= ˜}

_w(−m)∗

₍

_τ

_,

_r

₎

_{. In the} fol-lowing,we taketheweights withm

≥

0 astheindependentones andwrite

˜

w(_lm)

=

˜

w_l(m)

e−imψl(m)

_.

₍₄₎

The corresponding modeswith m

<

0 are then not independent andaredefinedbythecondition

| ˜

w_l(m)

|

= | ˜

w(_l−m)

|

withazimuthal angle

ψ

_l(−m)

= ψ

_l(m)

±

π

.

We next discuss the physically relevant range of

| ˜

w(_lm)

|

. For centralheavyioncollisions,theeventaveragedweights

| ˜

w_l(m)

|

O

(

0

.

1

)

andthetailsofeventdistributionssatisfy

| ˜

w(_lm)

|



0

.

5,see e.g.Fig.13ofRef.[18].Inperipheralcollisions,eventdistributions shift to larger

| ˜

w_l(m)

|

with increasing impact parameter b, since theexpansion

(3)

isaroundanazimuthallysymmetricbackground. Alsointhesenon-centralcollisions,

| ˜

w(_lm)

|

ismuchsmallerthan unity(e.g.

| ˜

w_l(m)

|

0

.

5 atb

=

6 fm forthe modelinRef. [18]). Foran intuitive understanding ofthesevalues,one mayconsider thecaseforwhichonesinglefluctuatingbasismode,saythemode withtheweight w

˜

(₁2),isembeddedontopofwBG

(

τ

0

,

r

)

w(

τ

0

,



r)

=

wBG

(

τ

0

,

r)

1

+

2

˜

w(₁2)

J2



k(₁2)r



cos



2



ϕ

− ψ

(2) 1



.

(5)

1 _{Hydrodynamicevolutionisgovernedbynon-linearpartialdifferentialequations} anditmaybechaoticoritmaycontaintermsthatarenon-analyticintheinitial fluidfieldsh˜j.Hence,thevalidityoftheexpansion(2)isnotguaranteed.Also,it

willdependonthechoiceofthebackgroundhBG_{andonthestrengthofthe}

pertur-bationsh.˜

For one single mode, we can set without loss of generality

ψ

₁(2)

=

0. The Bessel function J2 takes a maximal value max

[

J2

(

r

)

]

=

0

.

4865, and the enthalpy density (5) is therefore positive definite at all transverse positions only for

| ˜

w(₁2)

|

<

1

/(

2max

[

J2

(

r

)

])

=

1

.

028. Larger values for

| ˜

w(12)

|

can arise only ifthe presence ofadditionalmodes

| ˜

w_l(m)

|

ensures that the neg-ativecontributionfrom

| ˜

w(₁2)

|

to w

(

τ

0

,



r

)

iscanceledeverywhere. Thelargerthecancellation needed,thesmallertheprobabilitythat itarisesinaneventsample.Thismayprovidean intuitive under-standingforwhyeventhetailsofeventdistributionsofperipheral collisions are confinedtovalues

| ˜

w₁(2)

|

smallerthan 1.5,andwhy the mostlikely initial conditionsshow valuesthat are O

(

0

.

5

)

or smaller.Numericallysimilarconstraintsareobtainedforother ba-sismodes.

In

Fig. 1

,wetestthefluiddynamicresponsetoasingle fluctu-ating basis mode (5) fortheentire physicallyrelevant parameter range

| ˜

w(_lm)

|

<

1

.

5. Assuming for simplicity a Bjorken-boost in-variantlongitudinaldependence,weevolvetheseinitialconditions with the 2

+

1 dimensional version of the hydrodynamical code ECHO-QGP [19] with a value

η

/

s

=

1

/

4

π

for the ratio of shear viscositytoentropydensity.2FollowingRef.[20],weusethe equa-tionofstates95p-PCEwhichcombineslatticeQCDresultsathigh temperatures witha hadron resonance gas atlow temperatures. The background wBG used throughout thispaperis initialized at

τ

0

=

0

.

6 fm/c with an azimuthally symmetricaverage ofGlauber modelinitialconditionsforPb

+

PbcollisionsattheLHC,described in Ref. [18]. The time evolution of wBG determined from

ECHO-QGPisshownin

Fig. 1

.Thetime-evolvedfluctuation w

˜

(2)

(

τ

,

r

)

is determinedfromthefullhydrodynamicevolutionviaFourier anal-ysis.3 The mainobservationfrom

Fig. 1

isthat the fluiddynamic response to

(5)

scalesapproximately linearlywiththeinitial am-plitude w

˜

(₁2) ofthe perturbation. Thisscaling, expected fromthe linearterminEq.(2),isverygood forvalues w

˜

(₁2)

<

0

.

6.We ob-serve thislineardependence withsimilaraccuracy alsoforother basicmodes(datanotshown).Moresizabledeviationsfromlinear scalingareobservedforw

˜

₁(2)

>

0

.

6,see

Fig. 1

.

The main message of this work is that even for the largest physically relevant amplitudes w

˜

(_lm), where non-linear

contribu-tions can dominate the hydrodynamic response, the observed

non-linearities can be understood in terms of the perturbative ansatz (2). To explain this point, we note first that the dom-inant non-linearity in the hydrodynamic response to a fluctua-tion with weights w(₁2) is in modesm

=

2. Consider the Fourier series h

˜

i

(

τ

,

r

,

ϕ

)

=

₂1_π



∞m=−∞ eimϕ h

˜

(m)

i

(

τ

,

r

)

, where h

˜

(m) i

(

τ

,

r

)

are in general complex expansion coefficients, but h

˜

_i(m)

(

τ

,

r

)

=

˜

h(_i−m)∗

(

τ

,

r

)

since h

˜

(

τ

,

r

,

ϕ

)

∈ R

. Since the kernels in (2) depend only onthe backgroundfield, they areinvariant underazimuthal rotationandtheirFourierexpansionsread

G

i j



τ

,

τ

0

,

r,r

, 

ϕ



=

1 2

π

∞



m=−∞ eimϕ

G

(m) i j



τ

,

τ

0

,

r,r



,

2 _{Forthese2+1 dimensionalsimulationsinBjorkencoordinates,weadopta} uni-formgridinx andy withaspatialresolutionof0.2fm,whereasthetime-stepisset to10−3_{fm/c,withaCourantnumberof0.2toensurestability.Spatial} reconstruc-tionisachievedbyemployingtheMPE5scheme,themostaccurateoneavailablein ECHO-QGP(fifthorderforsmoothflows).Forfurthertechnicaldetails,seeRef.[19].

3 _{Fluctuationsattimeτ}

0arecut-offintheregionofverylowbackground den-sity,seee.g.w˜(2)_(τ

0, r)inFig. 1.Also,fortheextremeweightsw˜(12)=1.2 and1.4, thedistribution(5)iscutoffintheregionsinwhichitwouldturnnegative.For values| ˜w(12)|<1/(2max[J2(r)])=1.028,thesinglemodecanbepropagated with-outsuchanadhocmodification.Wehavecheckedthatthesecutsdonotaffectour conclusions.

Fig. 1. Thehydrodynamicevolutionoftheinitialcondition(5),obtainedwithECHO-QGP.Upperrow:Thetimedependenceoftheenthalpydensityisshownseparately forthebackgroundwBG(τ, r)andfortheperturbationw˜(2)_{(τ, r)initializedwith w˜}(2)

1 =0.5.Middlerow:Thedependenceoftheperturbationsw˜(2)ontheinitialweight forτ=τ0+10 fm/c (left)andscaledbytheinitialweights,w˜(2)(τ, r)/w˜(12) (right).Thisscalingshowstowhatextentthefluiddynamicresponsetoperturbationsis approximatelylinear.Lowerrow:Thedeviationfromlinearscaling



linear scaling= ˜w(2)(τ, r)/w˜(12)− ˜w

(2)

(τ, r)/w˜(₁2)|w˜₁(2)=0.1.Thisquantitywouldvanishforperfectlinear scalinganditscaleswith O((w˜1(2))2)ifitisdominantedbythethirdordercorrectionin(2).Thisscalingshowsthatnon-linearitiescanbeunderstoodperturbativelyin termsofEq.(7).

H

i jk



τ

,

τ

0

,

r,r

,

r

, 

ϕ



, 

ϕ





=

1

(

2

π

)

2 ∞



m,m=−∞ ei(mϕ+mϕ)

H

(m,m) i jk



τ

,

τ

0

,r,r



,

r



,

(6) andsoon.From

G

i j

(

τ

,

τ

0

,

r

,

r

,



ϕ

)

∈ R

oneobtains

G

i j(m)

=

G

(−m)∗

i j

and similarly

H

_{i jk}(m,m)

=

H

_{i jk}(−m,−m)∗. One obtains then from Eq.(2)

˜

h(_im)

(

τ

,r)

=



r

G

(m) i j



τ

,

τ

0

,r,r



˜

h(jm)



τ

0

,

r



+

1 2



r,r 1 2

π



m,m

δ

m,m+m

H

(m _,m₎ i jk



τ

,

τ

0

,

r,r

,

r



× ˜

h(_jm)



τ

0

,r



˜

h(m ₎ k



τ

0

,r





+ . . .

(7)

For the case that initial conditions contain only fluctuations of enthalpy density, we have h

˜

(_jm)

(

τ

0

,

r

)

= δ

j1 w

˜

(m)

(

τ

0

,

r

)

. Using the orthonormalexpansion

(3)

forw

˜

(m)

₍

_τ

0

,

r

)

,onecanwriteEq.

(7)

as

˜

h(_im)

(

τ

,r)

=



l

G

(m) i1;l

(

τ

,

τ

0

,r)

w

˜

(m) l

+

1 4

π



m,m,l,l

δ

m,m+m

H

(m _,m₎ i11;ll

(

τ

,

τ

0

,r)

× ˜

w_l(m)w

˜

(m ₎ l

+ . . .

(8) with

G

(m) i1;l

(

τ

,

τ

0

,r)

=



r

G

(m) i1



τ

,

τ

0

,

r,r



Jm



k_l(m)r



,

(9)

andsimilarlyfor

H

i11;ll.

To explain how results in the recent literature are related to Eq.(8),werecall thattheinitial amplitudes w

˜

_l(m) arerelated lin-early to the n-th radial moments that define the eccentricities

m,n [18].Alinearrelationshipbetweeneccentricitiesandflow

ob-servable would thus correspond to a truncation of (8) at linear order.There isevidence thatthisisa goodapproximation for el-liptic andtriangularflow, m

=

2

,

3,see e.g. Ref.[12].Also, it has been found that corrections quadratic in eccentricities can give sizable contributions to quadrangular and pentagonal flow coef-ficients whenthey involveatleastone power of

2,n (or w

˜

l(2) in

308 S. Floerchinger et al. / Physics Letters B 735 (2014) 305–310

Fig. 2. Leftcolumn:Thezeroth,fourthandsixthharmonicperturbationsinducedbyaninitialfluctuationinthesecondharmonic,shownfordifferentvaluesoftheinitial weightw˜(12).Rightcolumn:Thesame resultsbutrescaledbythesecond(third)poweroftheweightw˜

(2)

1 .Thisscalingestablishesthatw˜(0)(τ, r)andw˜(4)(τ, r)(w˜(6)(τ, r)) canbeunderstoodasovertonesthatareinduced bytheinitialsecondharmonicperturbationasaperturbativesecond(third)ordercorrectionto(2).Theshort-range fluctuationsintherescaledw˜(6)_{(τ, r)resultfromamplifyingthenumericaluncertaintiesofverysmallnumberbyalargescalingfactor}

₍

_1/w˜(2)

1 ) 3_=1000.

ourformalism)[5,10–12,21]andthatacombinationoflinearplus quadratictermsin eccentricitiesis agood“predictor” forthe hy-drodynamic response in a large set of realistic collisions, in the sensethat anappropriately definedPearsoncoefficientiscloseto unity[12].

We further note that the perturbative series (8)contains also informationabouttheazimuthalorientation ofnon-linear correc-tions.This isinparticularrelevant forreactionplane correlations such astheonesstudied inRefs. [7,21–23].The findings recalled hereweredemonstratedmainlyonthelevelofparticlespectra af-terthefluidhasfrozenout.Thereareonlyfewcommentsthat at-tributespecific non-linearcorrectionseitherto thehydrodynamic evolution[5]ortothehadronicfreeze-out[24].

We now turn to a quantitative test of the perturbative se-ries (7). According to this equation, fluctuations initialized asin

Fig. 1 with a single mode of weight w

˜

_l(m) receive corrections to secondorderinh

˜

i

(

τ

0

)

thatdonotappearinthetime-evolved har-monicsh

˜

(_im),butintheharmonicsh

˜

_i(2m)andh

˜

(_i0).Alsothethird or-dercorrectionentersthefluctuatingfieldsinh

˜

(_i3m)(anditentersin

˜

h(_im) asacorrectionthatissubleadingbytwoorderscomparedto theleadinglinearresponse).

Fig. 2

,showsthatthenon-linearities observedintheovertonesh

˜

(_i2m),h

˜

_i(0)(h

˜

(_i3m)), scaleindeedtohigh accuracywiththesecond(third)powerof

| ˜

w₁(2)

|

,asexpectedfrom

(7).Also, thelast rowof Fig. 1 illustrates that the sizeable non-linearities in h

˜

(_im) observed for extreme values of w

˜

_l(m) scale as expectedfrom

(7)

.Thissuggeststhatfortheentirephysically

rele-vantrangeofvalues

| ˜

w_l(m)

|

,theperturbativeseries

(2)

providesan ordering principlethatexplainshowlinearandnon-linear contri-butionsfeedintodifferentazimuthalmodesm,andhowtheyscale characteristicallydifferentwiththeinitialamplitudesw

˜

(_lm).

Eq.(2)explainsalsothestructureandsymmetriesofthe hydro-dynamic interactions betweeninitial perturbations withdifferent wave numbers. Fig. 3 illustrates thispoint witha caseforwhich two perturbations w

˜

(₂2), w

˜

₁(3) are embedded ontop of the initial background fields.We havechecked that the second (third) har-monics w

˜

(2)

₍

_τ

_,

_r

₎

_(w

_˜

(3)

₍

_τ

_,

_r

₎

_{) ofthe fluiddynamic responsescale}

almost linearly withthe initial weight w

˜

₂(2) (w

˜

(₁3)) andthat they agree to highaccuracy withthe responseto an initial configura-tion inwhichonly onemode w

˜

(₂2) (w

˜

(₁3)) isembedded ontop of

wBG (data not shown). Also, w

˜

(4)

(

τ

,

r

)

scales withthe square of

˜

w(₂2) (data not shown),similarly tothe caseshownin

Fig. 2

.For studyinginteractions betweendifferentmodes, weshow in

Fig. 3

thefirstandfifthharmonicsthataccordingtoEq.(8)aretheonly harmonicsthatreceiveleadingsecondordercontributions propor-tionalto w

˜

(₂2)w

˜

(₁3).If

ψ

(2)

= ψ

(3),thentheresponseswBGw

˜

(1)and

wBGw

˜

(5) have both a real andan imaginary part.Both parts

ex-hibit the expectedscaling with w

˜

(₂2)w

˜

(₁3),as seen inFig. 3. Also, according to (8), the phases of the first and fifth harmonics are determined by the orientations of the initial perturbations. The comparisonwiththefullnumericalresultsinthemiddlepanelof

Fig. 3showsthat thisperturbative expectationisrealized approx-imately (strongdeviationsareseenonlyforvaluesoftheradiusr

Fig. 3. ResultsfromECHO-QGPforevolvinguptoτ=τ0+10 fm/c ontopofthebackgroundofFig. 1aninitialconditioncomposedoftwobasismodeswithweightsw˜(22), ˜

w(13)andangles

ψ

(2)=0 and

ψ

(3)= −0.2.Upperrow:Realandimaginaryparts ofthefirst(wBGw˜(1))andfifth(wBGw˜(5))harmonicsoftheenthalpy.Thecurvesshownare

forthefourcombinationsofw˜(₂2)=0.1, 0.25 andw˜(₁3)=0.1, 0.25 andillustratescalingbehavior.Middlerow:ThephaseArg[ ˜w(m)_{(τ, r)]ofthem-thharmonicmode(solid)}

comparedtotheperturbativeexpectation(dashedline)basedonEq.(8).

forwhichRe

[ ˜

w(m)

]

andIm

[ ˜

w(m)

]

approachzeroandforwhichthe orientationisthusnotwelldefined).WehavecheckedthatEq.(8)

accountsalsoquantitativelyforthemorecomplicateddependence ofhigherharmonicsthatreceivenon-linearcorrectionsfrommore thanone termin

(7)

.Here,thisis thecasee.g. for wBGw

˜

(6) that

receivescorrectionsofsecondorderin w

˜

₁(3) andofthirdorderin

˜

w(₂2).

Realistic initial conditions for the fluid dynamic evolution of heavyioncollisions areexpectedto involvefluctuationsonmany differentlengthscales andwithlarge amplitude. Coulditbe that theperturbative series(2) provides forthe case ofsimple initial conditionswithfewfluctuating modesanunderstanding forhow linearresponse andnon-linearities arise,butthat itisinadequate fordealing withthecomplexity ofarealistic heavy ioncollision? Tolaysuchconcernsatrest,wehaveembeddedsinglebasismodes in realistic initial conditions with many and large fluctuations.

Fig. 4showssuchan initialdistribution.It isconstructedby sub-tractingfroman arbitraryinitialconditiongeneratedbyaGlauber modelthecontributionleading toa secondharmonicandadding thenthe perturbationof

(5)

.Inthisway,we havean analytically controlled initial perturbationon top ofa realistically fluctuating background,and we can extract thisinitial perturbation andthe timedependenceof itsfluid dynamic responsevia Fourier analy-sis.Thelower panelof

Fig. 4

showsthat thisdynamical response inaneventwithrealisticfluctuationsisdescribedtohighaccuracy bytheresponseestablishedin

Fig. 1

.

Insummary,theevolutionofinitialanisotropicdensity pertur-bationsasdeterminednumericallywiththefluiddynamicssolver ECHO-QGP seems to follow a simple pattern that can be under-stoodorder-by-orderin aperturbative expansion forsmall devia-tionsfroman azimuthally symmetricevent-averagedbackground. The leading order is linear and modes with different azimuthal

wave numbers do not mix. Quadratic and higher orders can be

seen as next-to-leading order corrections. They influence modes withazimuthalwavenumbersthatcanbewrittenassums(or dif-ferences) of the seed wave numbers. If non-linear couplings are numericallysmall, the higher harmonics generatedby two-mode

or three-mode interactions will often be small in comparison to initially present and linearly evolving perturbations. But also for initial conditions for which non-linear contributions maybe siz-able, we have shown that the perturbative series expressed by Eqs.(2)or

(7)

providesausefulorderingschemeforunderstanding thedependenceofnon-linearitiesontheamplitudesofinitial fluc-tuations.Thismotivatesamoreformalandthoroughdevelopment of this kind of perturbation theory. In thiscontext, it would be also interesting to consider initial fluctuations invorticity where one may expect on general physics grounds that non-linearities play a more important role [25]. We also note that the relative importanceoflinearandnon-lineartermsmaydependonthe dis-sipativepropertiesofthemedium[11,15].Weintendtostudythe dependenceon

η

/

s insubsequentwork.

While our studies clearly demonstrate that the hydrodynamic responseto initialfluctuationsispredominantlylinearforawide range of physically relevant amplitudes

| ˜

w_l(m)

|

<

0

.

5, we caution that this doesnot imply that experimental observables must be dominatedbylinearcontributions.Considerforinstancean exper-imentally observable quantity in the fifth harmonic, such as v5. Based on (2)and Fig. 3, we expect that itreceives linear contri-butions

∝ | ˜

w(_l5)

|

andquadraticcontributions

∝ | ˜

w_l(2)w

˜

(_l3)

|

.Evenif thehydrodynamicresponsetoanyofthemodesm

=

2,m

=

3 and

m

=

5 ispredominantly linear,the initial conditionsforan event sample maybe such that

| ˜

w_l(5)

|

is small against

| ˜

w(_l2)w

˜

(_l3)

|

– in thiscase, v5 wouldbe dominatedby thenon-linear contribution of thehydrodynamic response,but thedependence ofthis dom-inantnon-linear responseon the initialweights

| ˜

w_l(m)

|

wouldbe describedbytheperturbativeseries

(2)

.Thus,ourfindingthatthe hydrodynamicresponsecanbebasedonaperturbativeserieswith aleadinglinearresponsetermdoesnotdisagreewiththeresultof simulations [5] that show that observables canbe dominatedby non-linearities inthe hydrodynamic response. Rather,the pertur-bativeseries

(2)

showshowsuchnon-linearitiesarisedynamically andweintendtoexploitthisfurtherinsubsequentwork.

We note that non-linear corrections to linear response can

310 S. Floerchinger et al. / Physics Letters B 735 (2014) 305–310

Fig. 4. Upperpanels:Left:Exampleofaninitialconditionwithmanyfluctuatingmodesw˜_l(m),m=2,andthemodew˜(₁2)=0.5 ontopofthebackgroundwBG.Right:The samedistribution,evolveduptoτ=τ0+5 fm/c.Lowerpanel:Thesecondharmonicsw˜(2)(τ, r)extractedfordifferenttimesτ.Resultsextractedfromthefluctuatingevent shownintheupperpanelarecomparedtothecaseshowninFig. 1inwhichw˜1(2)=0.5 istheonlymodeembeddedontopofasmoothbackground.Thisillustratesthat theassumptionofapredominantlylinearresponseontopofasuitablychosenbackgroundisapplicableforrealisticinitialconditionsthatdisplaystrongfluctuations. fromnon-linearitiesinthehadronizationprocess.There aresome

first observations that disentangle between both mechanisms

[5,24],but giventhesignificantly different physicstested inboth stages, more detailed studies would be desirable. In particular, thepresentwork couldbe extendedtoacharacterizationof non-linearitiesinthefreeze-outprocess.

Acknowledgements

AB is supported by European Research Council grant HotLHC ERC-2011-StG-279579.ECHO-QGP was developed within the Ital-ianPRIN2009MIURproject“Il Quark-GluonPlasmaelecollisioni nuclearidialta energia”, grantNo. 2009WA4R8W.We alsothank F. Becattiniforhelpingtomakethiscollaborationpossible.

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