Event-by-event hydrodynamics and elliptic flow from fluctuating initial state

0.29 MB·English

by  Hannu Holopainen

Event-by-event hydrodynamics and elliptic flow from fluctuating initial state
H. Holopainena,b,∗ H. Niemic,† and K.J. Eskolaa,b‡
aDepartment of Physics, P.O.Box 35, FIN-40014 University of Jyva¨skyla¨, Finland
bHelsinki Institute of Physics, P.O.Box 64, FIN-00014 University of Helsinki, Finland
cFrankfurt Institute for Advanced Studies, Ruth-Moufang-Str. 1, D-60438 Frankfurt am Main, Germany
(Dated: 3 January,2011)
Wedevelop aframework for event-by-eventideal hydrodynamicsto studythedifferential elliptic
flowwhichismeasuredatdifferentcentralitiesinAu+AucollisionsatRelativisticHeavyIonCollider
(RHIC).Fluctuating initial energy density profiles, which here are the event-by-eventanalogues of
the eWN profiles, are created using a Monte Carlo Glauber model. Using the same event plane
method for obtaining v2 as in the data analysis, we can reproduce both the measured centrality
dependence and the pT shape of charged-particle elliptic flow up to pT 2 GeV. We also consider
1 therelationofellipticflowtotheinitialstateeccentricityusingdifferentre∼ferenceplanes,anddiscuss
1 the correlation between the physical event plane and the initial participant plane. Our results
0
demonstrate that event-by-event hydrodynamics with initial state fluctuations must be accounted
2
for before a meaningful lower limit for viscosity can be obtained from elliptic flow data.

n
a PACSnumbers: 25.75.-q,25.75.Dw,25.75.Ld,47.75.+f
J

6
2 I. INTRODUCTION nucleons have been shown to increase the initial eccen-
tricity, which is then suggested to translate into elliptic

] flow of final state particles [20]. Furthermore, the refer-
h Azimuthal anisotropy of final state particles produced
enceplaneplaysacrucialrole: theeccentricityislargerif
p in ultrarelativistic heavy ion collisions can be used to
onecalculatesit usingthe participantplane (determined
- measure the collective behavior of the dense particle
p by the transverse positions of the participant nucleons
system formed in such collisions [1]. The strong az-
e and the beam axis) instead of the reaction plane (deter-
h imuthalanisotropy,whichhasbeenobservedinthetrans-
mined by the impact parameter and the beam axis).
[ versemomentumspectraofhadronsinAu+Aucollisions
at the Relativistic Heavy Ion Collider (RHIC) of the Recently, in Ref. [21], hydrodynamical calculations
2
were performed using averaged initial density profiles
v Brookhaven National Laboratory, is also a signature of
which were obtained from MCG calculations. Before av-
8 theformationofstronglyinteractingpartonicmatter,the
eraging over the profiles, the transverse coordinate axes
6 Quark-Gluon Plasma (QGP).
3 wererotatedineacheventso thatthe participantplanes
Ideal hydrodynamics has been successful in predicting
0 wereontopofeachother. Inthismanneritispossibleto
andexplainingthemeasuredellipticflowinAu+Aucolli-
. getanaveragedinitialprofilethattakesintoaccountthe
7 sionsatRHIC[2–13]. Currently,alotofeffortisdevoted
0 eccentricity fluctuations in the initial state. For Au+Au
for developing a description of the QCD-matter evolu-
0 collisionsat RHIC, however,the effects ofsuch plane ro-
tion in terms of dissipative hydrodynamics. The recent
1 tations on the integrated v were small.
resultsshow thatevena smallviscositycanconsiderably 2
:
v decrease the elliptic flow [14–19]. Whiletheabovestudiesarestepstotherightdirection,
i itisobviousthatwithoutdoingevent-by-eventhydrody-
X However, all these ideal and viscous hydrodynamic
namic simulations, it is impossible to know how closely
r studies tend to underestimate the elliptic flow in most
the computational participant plane corresponds to the
a central collisions. Generally, the explanation for the
physical event plane which is determined from the ob-
deficit has been thought to be the initial state density
served final state hadron momenta.
fluctuations whichhavenotbeenaccountedfor. Inaddi-
So far, genuine event-by-event models where hydro-
tiontotakingintoaccountthedensityfluctuationsthem-
dynamics is run event by event using fluctuating initial
selves, special care should be taken in computing the el-
densityprofiles,havebeenpresentedinRefs.[22–27]. In-
liptic flow with respect to the same reference plane as in
terestingly,asimilartwo-particlecorrelationridgeasob-
the data analysis.
served in the experiments [28], is seen to form into the
The initial state fluctuations can be implemented e.g.
rapidity–azimuth-angle –plane both in NeXSpherio [24]
via a Monte Carlo Glauber (MCG) model which makes
and more recently in Ref. [25]. This suggests that the
possible to study the fluctuations of the initial matter
puzzling ridge may well be another consequence of the
eccentricity. Geometric fluctuations in the positions of
fluctuations in the initial state.
Also higher flow coefficients have been measured [29–
31] and recent studies [32] show that the initial state
densityfluctuationsmayplayanimportantroleinunder-
∗Electronicaddress: hannu.l.holopainen@jyu.fi
†Electronicaddress: [e-mail] standing the centralitydependence of the ratio v4/(v2)2.
‡Electronicaddress: [e-mail].fi Triangular flow arising from event-by-event fluctuations
2
[33] is also one of the things that should be studied fur- dius and d = 0.54 fm for the surface thickness. In the
ther with event-by-eventhydrodynamics. transverse(x,y)planethetwonucleiareseparatedbyan
Inthispaper,weintroduceanevent-by-eventidealhy- impact parameter b between the centers of mass of the
drodynamicsframeworktostudythefollowingv -related nuclei, which is determined by sampling the distribution
2
problems: Withidealhydrodynamicsusing averagedini- dN/db b in the region 0 b b = 20 fm > 2r .
max 0
∝ ≤ ≤
tial states, (i) there is a v deficit in central collisions, Thelongitudinalz coordinateistakenintoaccountwhen
2
as discussed above; (ii) the shape and centrality depen- samplingtheinitialnucleonpositionsbutinwhatfollows
denceofv (p )areunsatisfactoryinthatthep slopesof it does not play any role.
2 T T
v2 easily become too steepand elliptic flow increasestoo Nucleonsiandjfromdifferentnucleiarethenassumed
much towards noncentral collisions; (iii) elliptic flow is to collide if their transverse distance is small enough,
computed relative to the initial reaction plane or in the
best case to the participant plane [21] but not relative (x x )2+(y y )2 σNN, (1)
to the event plane, which is commonly used in the ex- i− j i− j ≤ π
periments; (iv) one does not know how closely the event
planeandtheinitialparticipantplanecorrespondtoeach whereσ isthe inelasticnucleon-nucleoncrosssection.
NN
other. Aconcreteillustrationoftheproblems(i)–(ii)can We apply here σ = 42 mb for Au+Au collisions at
NN
be found in Fig. 7.5. of Ref. [34], and also in Fig. 5 of √sNN =200 GeV.
our previous elliptic flow study [12]. WenotethatoursimpleMCGmodelfailstoreproduce
Wewillshowhowevent-by-eventidealhydrodynamics, the correlations between the nucleons, since we use the
initiatedwithafluctuatinginitialdensityprofileobtained WS distribution for determining the nucleon positions
from a MC Glauber model, and especially the determi- independently from each other. In [35] it was observed
nationofv2 withrespecttotheeventplane,conveniently that a realistic model, which accounts for nucleon corre-
solves the problem of the v2 deficit in the most central lations[36],canbewellapproximatedusinganexclusion
Au+Au collisions at RHIC. Simultaneously, we can sig- radius which prevents nucleon overlap. Using such ra-
nificantly improve the agreement with the data for v2 dius, or giving a finite size for the nucleons [21], causes
at all centrality classes up to 30-40% most central colli- deviations from the WS distribution which should then
sions in the typical applicability region of hydrodynam- be compensated by tuning of the parameters in the ini-
ics,pT <2GeV.Thisinturnhastheveryimportantim- tially sampled WS distribution.
plicationthat viscouseffects canin factbe allowedto be To keep our modeling as transparent as possible we,
smaller than previously thought. Finally, we also show however,choosenottoapplyanexclusionradiusoranu-
the correspondence between the event and participant cleonsizeinourMCGmodelsinceaccordingto Ref.[21]
planes and study the relation between the elliptic flow only a 10% uncertainty in the initial eccentricity can be
and initial eccentricity using different reference planes. expected, which is a much smaller effect than e.g. the
The rest of the paper is constructed as follows: First, overall uncertainties related to the choice of the initial
in Sec. II we introduce our frameworkfor event-by-event density profiles.
hydrodynamics. Details discussed there are our MCG
Next, we define the centrality classes using the num-
model,computationofthefluctuatinginitialenergyden-
ber of participant nucleons, N , for simplicity. We
part
sity profiles, MC modeling of thermal spectra of final
have plotted the distribution of events as a function of
state hadrons, and MC modeling of the resonance de-
N in Fig. 1. As indicated there, we slice our total
part
cays. We alsotry to discuss the points whereour model-
event distribution in N so that each N interval
part part
ingcouldbeimproved. SectionIIIisdevotedfordefining
correspondstoacentralityclasswhichcontainsacertain
the event plane and elliptic flow. Also eccentricity issues
percent of total events. The impact parameter may thus
are discussed there. Our results are presented in Sec. IV
freely fluctuate within each centrality class.
and conclusions are given in Sec. V.
II. EVENT-BY-EVENT HYDRODYNAMICS B. Initial density profiles
FRAMEWORK
In orderto utilize the MCG-giveninitial state to start
A. MC Glauber model and centrality classes hydrodynamics, we must next somehow transform the
positions of the wounded nucleons or binary collisions
We use here a MCG model to define the centrality into energy density or entropy density. These would
classesandtoforminitialstateswithfluctuating density be the fluctuating event-by-eventMCG analogues of the
profiles. First,wedistributethe nucleonsinthecolliding conventional eWN, eBC and sWN, sBC [5] average ini-
nucleirandomlyusingthestandard,sphericallysymmet- tial densities. For simplicity, we consider here just the
ric, two-parameter Woods-Saxon (WS) nuclear density eWN-type of profile and leave the profile fine tuning for
profile as the probability distribution. Our WS param- future work. The energy density is now distributed in
eters for the gold nucleus are R 6.37 fm for the ra- the(x,y)planearoundthewoundednucleonsusinga2D
A
≈
3
densityprofilesineachevent,wehaveamoredirectcon-
trolontheinputenergydensity(pressure)gradientsthat
-1
10 drive the evolution of the transverse flow and its asym-
metries. Inourcase,thetotalenergyperrapidityunitin
Au+Au each event, dxdyǫ(x,y), thus remains independent of
σ, while theRtotal entropy per rapidity unit and thereby
√s = 200 GeV
10-2 NN also the final state multiplicity depend on σ.
For Au+Au collisions at √sNN = 200 GeV, we use
the value K = 37.8 GeV/ fm. With this, we reproduce
ot the initial total entropy of Ref. [12] when averagingover
Nt
10-3 many initial states in central (b = 0) collisions when
/
N σ =0.4 fm. MotivatedbytheEKRTminijet(finalstate)
% % %
% saturationmodel[37]andRef.[12],wefixtheinitialtime
0 0 0 %
4 3 2 0 to τ =0.17 fm for all events.
- - - 1 5 0
-4 30 20 10 5- 0-
10
C. Hydrodynamics, freeze out and resonance
decays
-5
10
0 100 200 300 400 For obtaining the ideal-fluid hydrodynamic evolution
N of the system, we solve the standard equations
part
∂ Tµν =0 (3)
µ
FIG. 1: Our definition of centrality classes for Au+Au col- together with an Equation of State (EoS) which re-
lisions at √sNN = 200 GeV. Distribution of the number of lates pressure with the energy density and net-baryon
participantsiscalculated fromaMonteCarloGlaubermodel number density, P = P(ǫ,n ). As we are interested
B
without a nucleon exclusion radius.
in particle production at mid-rapidity, we assume the
net-baryon density to be negligible. Since the rapid-
ity distributions of hadrons are approximately flat at
Gaussian as a smearing function,
mid-rapidities we can safely simplify our hydrodynami-
calequations by assuming longitudinalboost-invariance.
K Npart (x xi)2+(y yi)2 We solve this (2+1)-dimensional numerical problem us-
ǫ(x,y)= exp − − , (2)
2πσ2 X (cid:16)− 2σ2 (cid:17) ing the SHASTA algorithm [38, 39] which is also able to
i=1
handle shock waves.
where K is a fixed overall normalization constant and σ As the Equation of State (EoS), we choose the EoS
is afree smearingparametercontrollingthe widthofour fromLaine andSchr¨oder[40]. Athightemperaturesthis
Gaussian. In each event, the impact parameter defines EoShasbeenmatchedwiththe lattice-QCDdataandat
the direction of the x axis and the origin of the (x,y) lowtemperatureswithahadronresonancegascontaining
plane is determined so that the energy-density weighted particles of mass m < 2 GeV. This EoS has a ”cross-
coordinate averages become x = y =0 fm. over” transition from the QGP to the hadron gas.
For the hydrodynamicaldehsciripthioni to be meaningful, Thermal spectra for hadrons are calculated using the
the initialstateshouldnothavetoosharpdensitypeaks. conventional Cooper-Frye method [41], where particle
InourMCGmodelwehavegivenaneffectiveinteraction emission from a constant-temperature surface σ is cal-
radius σ /π/2 0.6 fm for the colliding nucleons, culated according to
NN
≈
which spets a natural order of magnitude for σ. To probe
dN
the sensitivity of our results to the initial state smear- d2p dy =Z f(x,p)pµdσµ, (4)
ing, we will consider two values, σ = 0.4 fm and 0.8 fm. T σ
With the current setup, we cannot reduce σ further, as where f(x,p) is the particle number-distribution func-
thiswouldrequireasmallerstepsizeinourhydrodynam- tion in momentum at a certain space-time location. The
ical code, and consequently much more CPU time. One freeze-out temperature T = 160 MeV is fixed so that
dec
should then also develop a way to handle multiple sep- we reproduce the measured p spectrum of pions [42]
T
arate freeze-out surfaces, see the discussion in Sec. IIC. when averagedinitial states are considered.
These developments we leave as future improvements. Our surface finding algorithm operates in the (r,τ)-
Thereasontochoosetheenergydensitytobesmeared plane for all spatial azimuthal angles. Currently, we can
rather than the entropy density, is mostly technical and find only surfaces which go through r = 0. Due to the
due to the fact that our focus here is on understanding initialstatefluctuationstheremightsimultaneouslyexist
the transverse flow phenomena. Since we now avoid us- alsoother,disconnected,freeze-outsurfaceswhichoural-
ing the Equation of state in forming the initial energy gorithmdoesnotrecognize. Wehavecheckedthatforthe
4
centrality classes and smearing parameters σ considered 20000initial states generatedby ourMCG model. Such
here, only a a few percent of the events actually con- largenumberofeventsisrequiredsincefluctuationsnear
tain such a surface. In any case, since these additional the edges of the system easily affect the final value of
surfacestypicallyoriginatefromafew-nucleoncollisions, ellipticflowifthedensityprofileisotherwisesmooth. We
they contribute negligibly to particle production in not then do one hydro runwith the averagedinitialstate for
too peripheral Au+Au collisions. Making σ smaller can eachcentralityclass. Tomakeafarecomparisonwiththe
also increase the number of disconnected freeze-out sur- event-by-eventhydroresults,wedotheresonancedecays
faces. To ensure the applicability of our framework, we andanalysis using the same code for the averagedinitial
prefer not to consider centrality classes more peripheral state case as for the event-by-event hydro case, making
than 30–40%or σ <0.4 fm in the present study. 10 000 final state events from this one hydro run.
For the flow analysis, we need individual final state
particles. In generating these using the computed ther-
mal spectrum as the probability distribution, we assume III. ELLIPTIC FLOW ANALYSIS
thetotalnumberofthermalparticlesinarapidityunitto
be fixed individually in each event. The transverse mo- A. Elliptic flow and event plane
mentum (p ,p ) for each particle is thus sampled from
x y
the distribution dN/d2pTdy calculated in Eq. (4). Due The transverse momentum spectra of hadrons can be
to the assumed boost-symmetry, we are not equipped to written as a Fourier series,
consider rapidity distributions, thus y is sampled from a
∞
flat distribution in the interval y 0.5. dN 1 dN
Note that above we have negl|ec|t≤ed the fluctuations in d2pTdy = πdp2Tdy(cid:16)1+2nX=1vncos(nφ)(cid:17), (5)
the number of emitted thermal particles. In principle
one could derive these fluctuations separately from the whereφisthehadronmomentum’sazimuthalanglewith
thermaldistributionsforeachfreeze-outsurfaceelement. respect to the reaction plane defined by the impact pa-
However, it is not so clear how to treat the space-like rameter. The flow coefficients vn can then be computed
parts of the surface in this case. Since in the collisions from
considered here there are of the order of 1000 particles dφcos(nφ) dN(b)
pexepreucnteitdrtaopibdeitny,egtlhigeisbelefluinctcuoamtiponarsiscoannwinithantyhecainseitibael vn(pT)= R dφ dN(dbp)2Tdφdy. (6)
state fluctuations. R dp2Tdφdy
Once we have generated all the thermal hadrons, we Whenwehavefluctuationsintheinitialstate,calcula-
still need take into account the strong and electromag- tionofvn isnotsostraightforward. Inthehydrodynamic
netic decays. We let the thermal resonances decay one runs, where we always know the direction of our impact
by one using PYTHIA 6.4 [43]. Some decay products parameter,wecancalculatetheellipticflowwithrespect
can fall outside our rapidity interval y < 0.5. On the to the reaction plane. If we want to compare with ex-
| |
other hand, there would also be decay products arriving periments,weshouldusethe sameanalysismethodsand
from y > 0.5 which we do not consider here. We have definitions as in the data analysis. In this work we use
| |
checked that instead of increasing the width of our ther- the event plane method [44, 45] which is a common way
mal particle rapidity window, to speed up the analysis, to calculate v2. Since it is not (yet) typically used in hy-
we cansimply countalldecay products into our rapidity drodynamicalcalculations,let us briefly recapitulate the
acceptance regardless of their actual rapidity. main points (see Ref. [44] for details).
We first define an event flow vector Qn for the nth
harmonic. The event flow vector in the transverse plane
D. Event statistics is
Qn = (pTicos(nφi),pTisin(nφi)), (7)
Our main goal is to compare the event-by-event hy- X
i
drodynamic results with the ones obtained by more con-
wherewe sumovereveryparticle inthe eventandwhere
ventional non-fluctuating hydrodynamics initiated with
φ is measured from the x axis which is here fixed by the
averagedinitial states.
impact parameter. The event plane angle ψ for each
Forevent-byeventhydrodynamics,wemake500hydro n
event is then defined to be
runs in each centrality class. This amount of hydro runs
seems enough for the hadron spectra and elliptic flow arctan(Qn,y/Qn,x)
ψ = , (8)
analysis. To increase statistics we make 20 final state n n
events fromeveryhydro run, thus we have10 000events
with arctan placed into the correct quadrant. The ”ob-
in total. To check that using each hydro run 20 times is
served” v is calculated with respect to the event planes
n
sensible, we have checkedthat doing 250 hydroruns and
obtained above,
40 events from each leads to the same flow results.
To create an averaged initial state, we sum together v obs = cos(n(φ ψ )) , (9)
n i n events
{ } hh − ii
5
where the inner angle brackets denote an average over B. Initial eccentricity and participant plane
allparticles i in one event andthe outer ones anaverage
over all events. In order to remove autocorrelations, the The reactionplane eccentricity of the hydrodynamical
particleiisexcludedfromthedeterminationoftheevent initial state can be defined as (see e.g. Ref. [20])
flow vector when correlating it with the event plane.
Sinceinourfiniterapidityintervalwehaveonlyafinite σ2 σ2
ǫ = y − x (14)
number of particles available for the event plane deter- RP σ2+σ2
y x
mination, the obtained event plane fluctuates from the
”true”event plane. (In ourevent-by-eventhydrodynam- where
ics, the true event plane in each event would correspond
to the average event plane obtained by generating in- σ2 = y2 y 2
y h i−h i
finitely many final states from one hydro run.) The ob- σ2 = x2 x 2, (15)
tained v obs is correctedusing the eventplane resolu- x h i−h i
n
{ }
tion for the harmonic n
where the averagingis done overthe energydensity pro-
Rn =hcos(n(ψn−ψntrue))i, (10) file of Eq. (2).
Sincethepositionsofwoundednucleons,however,fluc-
where ψtrue defines the true event plane and the an- tuate from one event to another, tilting the transverse
n
coordinate axes suitably we can actually get a larger ec-
gle brackets stand for an average over a large sample of
centricity than ǫ above. Thus it is not so clear what
events. Because experimentally it is not possible to find RP
the most correct reference plane should be.
the true event plane, the event plane resolution must be
estimated. The reference plane that maximizes the initial eccen-
tricitycanbe expectedtocorrelatebetter withthe event
In the two-subevents method, which also we will use,
planethanthereactionplane. Forthispurpose,onemay
each event is randomly divided into two equal subevents
define the participant eccentricity [20]
A and B. The event plane resolution for each of these
subevents is then [44]
(σ2 σ2)2+4σ2
ǫ = q y − x xy, (16)
PP σ2+σ2
sub = cos(n(ψA ψB)) . (11) y x
Rn qh n − n i
where σ = xy x y . In this case the reference
xy
h i −h ih i
IfthefluctuationsfromthetrueeventplaneareGaussian, plane is the participant plane which is defined by the z
one can analytically obtain the following result [44]
√π
= χ exp( χ2/4) I (χ2/4)+I (χ2/4) , (12) 103
Rn 2√2 n − n h 0 n 1 n i = 0.4 fm
0-5%
where I and I are modified Bessel functions and χ -2V] 102 = 0.8 fm
√N, wi0th N re1ferring to the number of particles. Sinnc∼e e = 0.4 fm ave
G 1
we can calculate sub from the subevents, we can nu- [ 10 PHENIX
merically solve χsuRbnfrom Eq. (12). Because the number y)
n
of particles in the subevents is half of those in the full 2d 100
events, χfnull = √2χsnub, and we can calculate the resolu- pT
tionRfnull forthefullevents. Finally,theflowcoefficients /d 10-1 20-30%
are obtained as N
d +
-2 π
( 10
v obs
n )
vn = R{fnull}. (13) 1/ 10-3 A√us+Au= 200 GeV
( NN
The elliptic flow results computed with this method
are denoted here as v EP . We also compute the ellip-
2 0.0 0.5 1.0 1.5 2.0 2.5 3.0
{ }
tic flow from Eq. (9) with respect to the reaction plane
p [GeV]
using both fluctuating and averaged initial states. In T
the reaction plane case we have no corrections coming
FIG.2: (Color online)Transversemomentumspectraofpos-
from statistical fluctuations. These results are denoted
itive pions for Au+Au collisions at √sNN = 200 GeV cal-
as v RP in what follows.
2 culated with averaged and fluctuating initial states varying
{ }
thewidthofGaussian smearing. DataarefromthePHENIX
collaboration [42].
6
0.2 0.2
v2{EP} PHENIX
(a) (b)
v2{obs} STAR
0-5% v2{RP} 5-10%
charged v2{RP,ave} charged
v2 0.1 v2{PP} 0.1 v2
0.0 0.0
(c) (d)
10-20% 20-30%
charged charged
v2 0.1 0.1 v2
0.0 0.0
0.0 0.5 1.0 1.5 0.0 0.5 1.0 1.5 2.0
p [GeV] p [GeV]
T T
FIG. 3: (Color online) Elliptic flow of charged particles as a function of pT at different centralities for Au+Au collisions at
√sNN =200 GeV. Hydrodynamical calculations with fluctuating and averaged initial states are shown for σ =0.4 fm. Data
are from the PHENIX [31, 48] and STAR [49] collaborations. The statistical errors in the experimental data are smaller than
thesymbol size.
axis(beamdirection)andthexaxiswhichisfirstrotated IV. RESULTS
around the z axis by the angle
Below, we present the results for pion spectra, elliptic
2σ flow, eccentricities and the correlation of the event and
xy
ψPP =arctan − . (17) participant planes. The genuine event-by-event calcula-
σ2 σ2+ (σ2 σ2)2+4σ2
y − x q y − x xy tions using smearings σ = 0.4 and 0.8 fm, are compared
with the results obtained using an averagedinitial state.
In Fig. 2 we show the p spectra of positive pions
T
In what follows, we will compute the elliptic flow also from these three different hydro calculations and from
with respect to the participant plane, thePHENIXcollaboration[42]. AsexplainedinSec.IIB,
ourmultiplicitydependsontheGaussiansmearingwidth
σ, hence the (small) difference between the points with
v PP = cos(2(φ ψ )) (18)
2{ } hh i− PP iievents σ =0.4 fm and σ =0.8 fm at low pT.
Wecanalsoseethatathigherp wegetmoreparticles
T
and consider the relation of elliptic flow to the initial withthe fluctuatinginitialstatesthanwiththe averaged
eccentricity using both the reaction plane and the par- initial state case. This follows from the fact that in the
ticipant plane as the reference. fluctuating initial states there are larger pressure gradi-
7
ents present. For the same reason, the high-p spectra ciallythatincentralcollisionsthereisasignificantdeficit
T
are quite sensitive to the value of σ: with a larger σ, the ofv relativetothedata. Second,weseethatourv EP
2 2
{ }
pressuregradientsaresmallerandthep spectrasteeper. agrees very well (within the estimated errors) with the
T
Thisisinfactaninterestingobservation,suggestingthat data up to p 2 GeV in all centrality classes. No-
T
∼
with fluctuating initial states the applicability region of tice also the difference between the uncorrected v obs
2
{ }
(event-by-event)hydrodynamicsmayextendtohigherp and the corrected, final, v EP ; especially for central
T 2
{ }
than previously (see e.g. Refs. [46, 47]) thought. In any collisions, the corrections are quite large. Thus, fluc-
2
R
case, the obtained p spectra agree with the data suffi- tuationsalonearenotsufficientinexplainingthev data
T 2
ciently well, so that we can meaningfully next study the but that – in addition to taking into account the fluctu-
elliptic flow. ations – the computed v must be defined in the same
2
In Fig. 3 we plot the elliptic flow of charged particles way as in the experimental analysis.
as a function of pT atdifferent centralities. We show the Third, we notice that the relative increase from
event-by-event results for v2{EP}, v2{RP} and v2{PP}, v2{RP} to v2{EP} decreases from central to peripheral
as well as v2{RP,ave} which is obtained from averaged collisions: v2{EP}/v2{RP} = O(10) in panel (a) and
initial states. (1.2) in panel (d). Fourth, contrary to our original ex-
O
First, we observe, that v2{RP} and v2{RP,ave} are pectation,v2{RP,ave}forsemi-peripheralcollisionsisac-
quite close to eachother (althoughin the panel(c) some tually below (and not above) the data at p 1.5 GeV.
T
∼
statisticalfluctuationsseemtobestillpresent),andespe- This is due to the fact that with our MCG model and
smearing,the actual energy density profiles become flat-
terandlesseccentricthantheconventionaleWNprofiles
obtainedfromanopticalGlaubermodel. Asa result,we
getasmallerv RP thane.g. inRef.[12],andthusalso
0.2 2
{ }
v2{EP} =0.4fm in the 20-30% centrality class there is room for an in-
(a) creasefromv RP to v EP . Fromthese observations
v2{EP} =0.8fm 2 2
{ } { }
we can conclude that we have answered the questions
v2{RP} =0.4fm
(i)–(iii) presented in Sec. I.
v2{RP} =0.8fm
Fourth, Fig. 3 indicates that v PP is very close to
2
{ }
v EP inallcentralityclasses. Thisresultsuggeststhat
0-5% 2{ }
the participant plane indeed is quite a good approxima-
v2 0.1 charged tion for the event plane.
In Fig. 4 we show the effects of varying our Gaussian
smearing parameter σ. We see that our elliptic flow re-
sults are quite insensitive to σ: Doubling the value of σ
causes only of the order of 10% changes in our v (p ).
2 T
We remind, however, that our p -spectra and multiplic-
T
ity of pions were not as stable against σ but we expect
0.0 thatdoingmoreproperfittingtothepionspectrabyfine-
PHENIX
tuningT andtheinitialoverallnormalizationconstant
dec
STAR K, would not affect our v results significantly.
2
InFig.5weplottheintegratedellipticflowforthefour
20-30%
different cases considered above and the data from the
charged PHOBOS collaboration [50]. As expected on the basis
of Figs. 3 and 2, our results v EP and v PP now
2 2
{ } { }
v2 0.1 agree with the data very well, while the v2 RP results
{ }
fall significantly below the data.
Next, Fig. 6 shows the computational quantity v /ǫ
2
which is often discussed. In the PHOBOS result [50], v
2
isdeterminedrelativeto the eventplanewhile the initial
(b)
state eccentricity is computed relative to the participant
plane. We reproduce the PHOBOS v /ǫ if we do the
2
same, i.e. use ǫ from Eq. (16). Interestingly, if we
0.0 PP
0.0 0.5 1.0 1.5 2.0 replaceboththeellipticflowandtheeccentricitybytheir
p [GeV] reaction plane analogues, we can still get a scaling law
T
thatagreeswithourv EP /ǫ andwiththedata. This
2 PP
{ }
figureillustratesagainthe importanceofthe consistency
FIG. 4: (Color online) Elliptic flow of charged particles as in the reference plane definition.
a function of pT at different centralities with two different Finally,weanswerthequestion(iv)presentedinSec.I.
values for Gaussian smearing parameter σ. Figure 7 shows the correlation between the event plane
8
and the participant plane as well as the correlation be-
tween the event plane and the reaction plane. We plot
the distribution of events as a function of the angle dif- 0.4
v {EP}
ferences ψ2 ψPP and ψ2 ψRP. For this figure, we 2
have used e−ach hydro run −only once. We notice that 0.35 v {RP}
2
in central collisions, the planes are more weakly cor-
PHOBOS
related than in semi-peripheral collisions where clearer 0.3
peaksaroundψ =ψ ,ψ arise. Asexpectedbasedon
2 PP RP
Fig.3,theparticipantplaneisindeedfoundtobequitea 0.25
good approximation for the event plane, in all centrality
/
classes. However, fluctuations of the event plane around 0.2
2
the ”true” event plane are much larger in central colli- v
sions and thus the correlation between the event plane 0.15
and the participant plane in Fig. 7 looks weakerfor cen-
tral collisions than for the more peripheral ones. 0.1
Au+Au
0.05 √s = 200 GeV
NN
V. CONCLUSIONS
0.0
0 50 100 150 200 250 300 350 400
The main result of this paper is that using event-by-
event ideal hydrodynamics with MCG-generated fluctu- Npart
ating initial density profiles we can simultaneously re-
produce the measured centrality dependence and the p
T
shape ofcharged-particleelliptic flowup to p 2 GeV. FIG.6: (Coloronline)Integratedellipticflowofchargedpar-
T
Also the measured pion spectra are quite we∼ll repro- ticles for Au+Au collisions at √sNN = 200 GeV divided by
initial eccentricity. Theoretical calculations correspond to
duced, although we have not made an effort to fine-
fluctuating initial states with σ = 0.4 fm. Data from the
tune the model parameters. In particular, in addition
PHOBOS collaboration [50] are shown with statistical and
to accountingforthe fluctuations inthe system,we have
systematic errors added in quadrature.
demonstrated the importance of using the same v defi-
2
0.12 nition as in the data analysis.
v {EP}
2 Wehaveperformedallhydrodynamicsimulationswith
0.1 v2{RP} zeroviscosity. Thus,ourresultssuggestthatextractinga
v {RP,ave} non-zero lower limit for the viscous coefficients from the
2
measuredv (p )ofchargedhadronsispracticallyimpos-
v {PP} 2 T
0.08 2 siblewithout furtherconstraintstothe model, especially
PHOBOS hit to the initial state. We would like to emphasize, that
PHOBOS track wehaveforsimplicityconsideredonlytheevent-by-event
v2 0.06 analogues of the eWN initial profiles whose eccentrici-
ties are typically smaller than e.g. those of the eBC- or
CGC-type [51–53] of profiles. Whether the data are still
0.04
consistent with non-zero viscosity with these initial con-
Au+Au ditions, is left as a future exploration. Nevertheless, our
0.02 √s = 200 GeV results demonstrate that event-by-event hydrodynamics
NN with initial state fluctuations must be accounted for be-
fore a more reliable lower limit for viscosity can be ob-
0.0 tained from elliptic flow data.
0 50 100 150 200 250 300 350 400
We have shown that the definition of the reference
N plane with respect to which one determines v , plays an
part 2
importantroleespeciallyincentralcollisions. Ontheone
hand,ifv iscomputedrelativetothereactionplane(de-
2
FIG.5: (Coloronline)IntegratedellipticflowforAu+Aucol- termined by the impact parameter), the fluctuating and
lisions at √sNN = 200 GeV calculated with fluctuating and averagedinitialstatesleadpracticallytothesameresults.
averaged initial states are shown for σ = 0.4 fm. Data from In this sense, the previous conventional ideal hydrody-
thePHOBOScollaboration[50]areshownwithstatisticaland namicalresultsforthe systemevolutionarestillrelevant
systematic errors added in quadrature. in central enough collisions but one should not compare
9
decoupling temperature may vary from event to event.
Instead of a fixed T applied here, one could imple-
dec
1.6 ment a dynamical freeze-out criterion as was done e.g.
0-5% PP
inRef.[54]. However,inordertoimproveuponthe well-
1.4 0-5% RP known problem of the proton p spectra when partial
T
chemicalequilibriumisnotapplied,onecouldcoupleour
30-40% PP
) 1.2 hydroto ahadroncascadeafterburnerwhichwouldhan-
ψ 30-40% RP
dle also the resonance decays [25–27]. Related to the
∆
1.0 initial state, one should more closely inspect the uncer-
d
/ taintiesdue the assumedenergydensitysmearing,which
N Au+Au
0.8 is an avoidable issue with event-by-event hydrodynam-
d
)( √sNN = 200 GeV ics. Herewefoundoutthatv2 remainedfairlyinsensitive
N 0.6 to Gaussian smearing width while pion pT spectra were
/ more sensitive to it towards larger p . Also other pos-
1 T
( 0.4 sible smearing functions should be studied. One should
also consider a dynamical QCD-based model for the ini-
0.2 tial fluctuations, in which case also the absolute initial
density profiles should be computable. These tasks, and
considering the effects of fluctuations on other observ-
0 /4 /2 ables, we leave as future developments.
Acknowledgments
ψ ψ
2 PP/RP
−
We gratefully acknowledge the financial support from
FIG. 7: (Color online) Correlation of the event plane with
the Academy of Finland, KJE’s projects 115262 and
theparticipantplane,andwiththereactionplaneatdifferent
centralities and with σ = 0.4 fm. The lines are to guide the 133005, from the national Graduate School of Particle
eye. and Nuclear Physics (HH), and from the University of
Jyva¨skyla¨(HH’s grant). The work of HN was supported
by the Extreme Matter Institute (EMMI). HH thanks
thereaction-planev totheevent-planev quotedbythe the [Department of Energy’s] Institute for Nuclear The-
2 2
experiments. On the other hand, according to our re- ory at the University of Washington for its hospitality
sults, the initial participant plane seems to be quite a and the Department of Energy for partial support dur-
good approximation for the event plane in the presence ingthecompletionofthiswork. WethankD.H.Rischke,
of hydrodynamically evolving density fluctuations. P.V.Ruuskanen,M.LuzumandJ.-Y.Ollitraultforhelp-
The present work can obviously be improved in many ful discussions. All the supercomputing was done at the
ways. Especially in event-by-event hydrodynamics the CSC – IT Center for Science in Espoo, Finland.
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