Analyzing the Power Spectrum of the Little Bangs

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by  Agnes Mocsy

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NuclearPhysicsA00(2011)1–4
Analyzing the Power Spectrum of the Little Bangs
A´gnesMo´csya,PaulSorensenb
aPrattInstitute,DepartmentofMathandScience,Brooklyn,NY11205USA
bPhysicsDepartment,BrookhavenNationalLaboratory,UptonNY11973USA
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Abstract
n
a In this talk we discuss the analogy between data from heavy-ion collisions and the Cosmic Microwave Back-
J
ground. Weidentify pT correlationsdataastheheavy-ionanalogytotheCMBandextractapower-spectrumfromthe
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heavy-iondata. Wedefinetheratioofthefinalstatepower-spectrumtotheinitialcoordinate-spaceeccentricityasthe
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transfer-function.Fromthetransfer-functionwefindthathigherntermsaresuppressedandwearguethatthesuppres-

] sionprovidesinformationonlengthscaleslikethemean-free-path. Wemakearoughestimateofthemean-free-path
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andfindthatitislargerthanestimatesbasedonthecentralitydependenceofv .
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- Keywords: Quark-gluonplasma,correlations,fluctuations
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[ Acommonlyquotedgoaloftheheavy-ionprogramsatBrookhavenNationalLaboratoryandCERNistorecreate

conditionssimilartothoseshortlyaftertheBigBang[1]. Learningabouttheearlystagesofthematterproducedin
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heavy-ioncollisionsfromtheobservationofhadronsinthefinalstateisinsomesensesimilartounderstandingthe
v
6 stagesoftheearlyuniversefromtheobservationoftheCosmicMicrowaveBackground(CMB)[2]. Inthistalkwe
2 exploretheanalogybetweenheavy-ioncollisionsandBigBangcosmology[3]. Inparticular,wepresenttheheavy-
9 ionequivalentoftheCMBandfromthatwedeterminethepower-spectrumforthelittlebangs. Wethenestimatethe
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transfer-functionnecessarytoproducethespectrumfromtheinitialconditionsofthecollisions.
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1 Quantum fluctuations from the early universe show up as hotspots at the surface of last scattering, creating the
0 CMB measured for example by WMAP. Measurements of the CMB reveal temperature fluctuations corresponding
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to over- or under-densities present at the surface of last scattering at about 400,000 years after the Big Bang [2].
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Thesedensityfluctuationsultimatelyexplainthestructureinouruniverse. MapsofthetemperatureoftheCMBare
:
v determined from measurements of the blackbody spectra at 2M points in the sky. The temperature is found to be
Xi verysmoothwithrelativevariationsonlyshowingupatapproximately10−5. Thefluctuationstendtoonlybeatshort
distances;smalllumpsnotlargeones. Thisresulthasbeenexplainedbyaninflationaryperiodwhenthelargerscale
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a fluctuationswerepushedoutsideofthehorizon,causallyseparatingthem.
The scale of the correlations can be most easily studied by extracting a power-spectrum from the CMB. The
power-spectrumshowsthatmostpowerisforlargewave-numbers(meaningsmallwavelength). Thelackofpowerat
smallwave-numberisstrongevidenceofinflation. Peaksinthepower-spectrumarecausedbyacousticphenomenain
theearlyuniverseasdensityperturbationsintheuniversepropagateassoundwaves. Thisgivesrisetothestructurein
theanisotropiesofthemicrowavebackground,notablyacharacteristicangularscaleandthefamousacousticpeaks.
Fromthisspectrumtheacousticpeaksarefitwithmodelstoextractcosmologicalparameters. Inthistalkweexplore
an analogous measurement in heavy-ion collisions. We determine the power-spectrum for the little bangs and we
estimatethetransfer-functionnecessarytoproducethespectrumfromtheinitialconditions.
Just as quantum fluctuations stretched to cosmic sizes by inflation show up in the CMB, we expect fluctuations
from the beginning of the little bangs to show up in heavy-ion data. Mishra et. al first have explored the analogy
/NuclearPhysicsA00(2011)1–4 2
Figure1: Top:thelengthscalesintheCMB.Bottom:thelengthscalesprobedwithhigherninheavy-ioncollisions.
betweentheexpansionofheavy-ioncollisionsstartingfromalumpyinitialenergydensityandtheexpansionofthe
universestartingwithquantumfluctuationsstretchedtocosmologicalsizes,andbasedonthatanalogysuggestedmea-
suringRMSvaluesoftheFouriercoefficientsv (knownasflowcoefficients)asthepower-spectrum[3]. Theydidn’t,
n
however,makeaconnectionbetweentheRMSofv andthealreadyexistingtwo-particlecorrelationmeasurements.
n
Sorensenlaterpointedout[4]thatmanyofthesurprisingcorrelationstructuresseeninRHICdata(theridgeandmach
cones[5])maybeunderstoodintermsoffluctuationsofhigherFouriercomponentsofv whichMishraet. al. had
n
arguedwouldarisefromanisotropiesintheinitialenergydensityconvertedintomomentum-spaceduringtheexpan-
sion. Severalotherauthorshavealsopreviouslyexploredtheconnectionbetweeninitialconditionsandtwo-particle
correlations[6,7,8,9,10,11].
While quantum fluctuations stretched to cosmic sizes by inflation explain the temperature fluctuations in the
CMB, lets see how density fluctuations at the beginning of heavy-ion collisions might show up in the correlations
data. Wehavesomeexpectationsabouttheinitialstate: giventhatthecollisionsinvolvenucleiwithafinitenumber
of nucleons, we expect that the initial density is inhomogeneous, it has a lumpiness. Every event is unique and no
event is homogeneous. We do not dwell for the moment on the exact theory to describe these initial fluctuations,
butonlypointoutthatthefinitenumberofchargesintheapproachingnucleiimpliesthattheinitial(energy)density
is lumpy (even if we consider a larger number of gluons in the nucleus, those gluons are still confined in the radial
direction within the typical size of the nucleons). The lumpy initial density may survive through the evolution and
show up at the end as hotspots in the surface of last scattering. We expect this because the rapid expansion boosts
theselumpsoutwardandthedecayofthehotspotsincoordinate-spacemanifestsincorrelationsinmomentum-space
duetoflow[6]. Thelumpyinitialconditionsshowupinthe2-particlecorrelationstructures. Theplausibilityofthis
conjecturehasbeenverifiedwithmultiplemodelsforheavy-ioncollisions[9,10,11,12].
For the analogy with the CMB we are interested in temperature-correlations. Thus we need to consider how
muchoftheinitialinhomogeneitywillbetransferredintothefinalstate. Simulationsbasedonhydrodynamicmodels
indicate that collisions of Au nuclei (12 fm across) may contain hotspots of size 1.5 fm and that remnants of those
hotspots persists during the collisions evolution [12]. How much of this initial inhomogeneity is being transferred
into the final state depends on the mean-free-path l and it’s relation to the size of the hotspots. The higher the
mfp
valueofn,thesmallerthelengthscaleweareprobing. Ann = 2asymmetry(thinkofv )canbetranslatedbecause
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the l is small compared to the length probed by n = 2. That’s why ideal hydro works well for v . The fact that
mfp 2
hydrodynamic models do a reasonable job of predicting the value of v suggests that l can be considered small
2 mfp
comparedtothesizeofthesystem. Butwhataboutn = 3orevenlarger? AsillustratedinFig.1,asweincreasen,
wereducethelengthscaleprobed. Ifthemean-free-pathisgreaterthanthelengthscale,thereisnotranslationfrom
coordinate-spaceintomomentum-space. Weonlyexpectanefficientconversionofcoordinate-spaceanisotropiesinto
/NuclearPhysicsA00(2011)1–4 3
momentum-space when l < 2π(cid:104)R(cid:105)/n where (cid:104)R(cid:105) is the average radial position of the systems constituents. This
mfp
relationwascheckedinasimulationbycalculatingtheinitialparticipanteccentricityforallvaluesofn(ε2 )[10]
n,part
fromaMonteCarloGlaubermodel[13]. Forcentralcollisions,whentheparticipantsaretreatedaspointlike,ε2
n,part
isnearlyindependentofn. Whentheparticipantsaresmearedoveraregionofsizer ,ε2 athighernbecomes
part n,part
quenched. Wefindthatε2 isreducedbyhalfwhenr <2π(cid:104)R(cid:105)/nandarguethatl willhaveasimilareffectas
n,part part mfp
r . Thepower-spectrumtellsusthepowertransferredintoeachharmonicnasafunctionofn. Clearly,thebehavior
part
of the power-spectrum can be used to extract information about length scales like l . By examining the power-
mfp
spectrumofheavy-ioncollisionswhichincludesinformationforallvaluesofn(beyondjustn = 2or3),wehopeto
betterconstrainl [14].Wenotethatotherlengthscalesthatcouldbeimportanthavebeenidentified:thermalscale;
mfp
timeoffreestreaming;sizeofhadrons;etc[15].
Todeterminethepower-spectrumwetaketheSTARdataof p correlationsvstherelativeazimuthalanglesand
T
the relative rapidity between the particles [16]. We use this correlation measurement since this is a good proxy for
point-to-pointtemperature-fluctuations:justliketheblackbodyspectrum,thep spectracanreflectthetemperaturein
T
heavy-ioncollisions.Anarrowpeakpositionedaroundsmallangleseparationisobservedinthedata.Thistellsusthat
ifaparticlecomesoutwithaboveaverage p ,thenthenearbyparticlesalsotendtohavelarge p . Thisisconsistent
T T
withexpectationsfromhotspotsonthesurfaceoflastscattering. Thecorrelationofthesefastparticlessuggeststhat
theyarebornoutofthesamehigh-density,high-temperaturelump,i.e.thesamehotspot.Therelevanceoftemperature
fluctuationstoheavy-iondataissupportedbytheimprovementto p spectrafitsusingaTsallisdistribution[17].
T
We considered a slice of the p correlations at zero relative rapidity vs relative azimuthal angle. We Fourier-
T
transform this correlation to generate a power-spectrum vs harmonic number n. The coefficients of the Fourier de-
composition a give us the power-spectrum. This is shown in the left plot of Figure 2. The a are similar to v2 but
n n n
extractedfrom p correlationsinsteadofnumbercorrelations[18]. a isthepower-spectrumforheavy-ioncollisions
T n
anditistheanalogousmeasurementtothatoftheCMB.Wefindthatmostofthepowerisinthelowestmodesanda
n
fallsoffrapidlywithn. Modesotherthann=2howeverstillshowasignificantcontribution.
FortheCMB,thelowestmodesarestronglysuppressedandsignificantacousticoscillationsshowupatl = 200.
Thisisduetothefactthatintheearlyuniverseinflationstretchesthefluctuationsby30ordersofmagnitudesothat
thesmallwavenumbersareoutsidetheacoustichorizon. Mishraet. alhadproposedlookingforsuppressionofsmall
nmodesduetoanacoustichorizoninheavy-ioncollisions[3],butweseeisclearlyasuppressionoflargenmodes.
Wearguethatthisisbecauseviscosityisfarmoreimportantinheavy-ioncollisionsthanintheearlyuniverse[19].
Apparently, the damping due to various length-scales in heavy-ion collisions (l , etc.) is the dominant effect.
mfp
Nextwewanttocomparethefinalpower-spectrumtotheinitialcoordinate-spaceanisotropy. Theazimuthaldistribu-
tionofmatterintheinitialoverlapzoneiscalculatedusingtheparticipanteccentricityforallharmonicsnε2 [10].
n,part
WestartwithaGlaubermodel[13]fortheinitialspatialdistributionofparticipantsforperfectlycentralcollision(b=0
fm)inordertocomparetodataalsoforthe5%mostcentralcollisions. Theresultsareshowninthemiddleplotof
Figure2. Thelargen=2termpersistsevenforcentralcollisionsbecauseweincludetheintrinsicdeformationofthe
AunucleusinourMonteCarlo. Then=1termissmallbecausetheparticipantsarere-centeredtothecenterofmass.
For central collisions for n ¿ 2 the eccentricity is nearly independent of n (the higher harmonics are important too).
Thisisbecauseforsymmetriccollisionsandpoint-likeparticipants,ε2 dependsonlyonthenumberofparticipants,
n,part
independentofn. Weexpect,however,thattheconversionofhigherharmoniceccentricitywillbedampedduetothe
existenceofthelengthscalel . Givingtheparticipantsasizecausestheeccentricitytofallofffasterwithn. The
mfp
drop occurs where the size of the participant is larger than the lengthscale corresponding to n the ε2 is smeared
n,part
out[20].
Wedefinetheratioa /ε2 asthetransfer-function[14]andshowitontherightplotofFigure2. Thetransfer-
n n,part
functiongivesusinformationabouthowefficientlythecoordinate-spaceanisotropyhasbeenconvertedintomomentum-
space. The transfer-function derived from data and the Glauber model shows that as n increases, the efficiency for
converting coordinate-space anisotropy into momentum-space quickly drops off. This can be expected from the
condition that the transfer-function should go to zero when l is larger than the length-scale associated with n:
mfp
n > 2π(cid:104)R(cid:105)/l . Wecanmakeacrudeestimateforl basedonthetransfer-function. Ifwetake(cid:104)R(cid:105) = 3fmforthe
mfp mfp
averageradialpositionofaparticipantandn=6astheharmonicbeyondwhichtheconversionisinefficient,thenwe
getl = 3fm. Thisestimatecanbeimprovedwithamorecompletemodelofheavy-ioncollisions. Thisestimate
mfp
correspondstoaviscosityseveraltimeslargerthanestimatesbasedonthecentralitydependenceofv [19]orfrom
2
/NuclearPhysicsA00(2011)1–4 4
Figure2:Left:thepower-spectrumanextractedfrompT correlationsinheavy-ioncollisions.Middle:theparticipanteccentricitycalculatedfrom
aMonte-CarloGlaubermodel. Right: thetransfer-functionforheavy-ioncollisionsshowingthesuppressionathighervaluesofninthefinal
momentum-spacecomparedtotheinitialcoordinate-space.
the longitudinal width of p correlations [21]. Our crude estimate, however, is geometry based, not accounting for
T
thevariousphasesoftheexpansion.
We’ve used p correlations from heavy-ion collisions to extract a power-spectrum analogous to the power-
T
spectrum extracted from the Cosmic Microwave Background. We calculated the initial state eccentricity and then
find the transfer-function which we define as the ratio of the power-spectrum to the initial eccentricity. We argue
that the transfer-function required to describe the RHIC data may potentially be understood in terms of an ineffi-
ciency in conversion of coordinate-space anisotropy into momentum-space when l > 2π(cid:104)R(cid:105)/n. We used this
mfp
transfer-function to make a rough estimate of the mean-free-path of the system’s constituents. The introduction of
theharmonicaxisnprovidesmoredetailedinformation. Thisapproachrepresentsanewmethodfordeterminingthe
characteristics of heavy-ion collisions and the QGP and should be further investigated. It stands as a challenge for
models of heavy-ion collisions to correctly describe this transfer-function: in other words, to correctly describe the
conversionofcoordinate-spaceanisotropiesatallscalesintomomentum-spaceanisotropiesatallscales.
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