Intergalactic Magnetic Field and Arrival Direction of Ultra-High

27 Pages·2009·0.37 MB·English

TheAstrophysicalJournal,710:1422–1431,2010February20 doi:10.1088/0004-637X/710/2/1422
(cid:2)C2010.TheAmericanAstronomicalSociety.Allrightsreserved.PrintedintheU.S.A.
INTERGALACTICMAGNETICFIELDANDARRIVALDIRECTIONOFULTRA-HIGH-ENERGYPROTONS
DongsuRyu1,4,SantabrataDas2,andHyesungKang3
1DepartmentofAstronomyandSpaceScience,ChungnamNationalUniversity,Daejeon305-764,Korea;[email protected]
2DepartmentofPhysics,IndianInstituteofTechnologyGuwahati,Guwahati781039,Assam,India;[email protected]
3DepartmentofEarthSciences,PusanNationalUniversity,Pusan609-735,Korea;[email protected]
Received2009October18;accepted2010January11;published2010January29
ABSTRACT
Westudiedhowtheintergalacticmagneticfield(IGMF)affectsthepropagationofsuper-Greisen–Zatsepin–Kuz’min
(GZK) protons that originate from extragalactic sources within the local GZK sphere. To this end, we set up
hypothetical sources of ultra-high-energy cosmic rays (UHECRs), virtual observers, and the magnetized cosmic
webinamodeluniverseconstructedfromcosmologicalstructureformationsimulations.Wethenarrangedasetof
referenceobjectsmimickingactivegalacticnuclei(AGNs)inthelocaluniverse,withwhichcorrelationsofsimulated
UHECReventsareanalyzed.WithourmodelIGMF,thedeflectionanglebetweenthearrivaldirectionofsuper-GZK
protonsandtheskypositionoftheiractualsourcesisquitelargewithameanvalueof(cid:3)θ(cid:4)∼15◦andamedianvalue
ofθ˜ ∼7◦–10◦.Ontheotherhand,theseparationanglebetweenthearrivaldirectionandtheskypositionofnearest
referenceobjectsissubstantiallysmallerwith(cid:3)S(cid:4) ∼ 3◦.5–4◦,whichissimilartothemeanangulardistanceinthe
skytonearestneighborsamongthereferenceobjects.Thisisadirectconsequenceofourmodelthatthesources,
observers,referenceobjects,andtheIGMFalltracethematterdistributionoftheuniverse.Theresultimpliesthat
extragalacticobjectslyingclosesttothearrivaldirectionofUHECRsarenotnecessarilytheiractualsources.With
ourmodelforthedistributionofreferenceobjects,thefractionofsuper-GZKprotonevents,whoseclosestAGNsare
truesources,islessthan1/3.WediscussedimplicationsofourfindingsforcorrelationstudiesofrealUHECRevents.
Keywords: cosmicrays–large-scalestructureofuniverse–magneticfields–methods:numerical
Online-onlymaterial:colorfigures
2004). More recent data from the Pierre Auger Observatory
1.INTRODUCTION
(Auger)supporttheexistenceoftheGZKsuppression(Abraham
The nature and origin of ultra-high-energy cosmic rays etal.2008b;Schu¨ssleretal.2009).BelowE ,however,the
GZK
(UHECRs), especially above the so-called Greisen–Zatsepin– fourexperimentsreportedthefluxesthataredifferentfromeach
Kuz’min(GZK)energyofE ≈50EeV(1EeV=1018eV), otherbyuptoafactorofseveral,implyingthepossibleexistence
GZK
hasbeenoneofthemostperplexingpuzzlesinastrophysicsover of systematic errors in their energy calibrations (Berezinsky
fivedecadesandstillremainstobeunderstood(seeNagano& 2009).
Watson2000,forareview).ThehighestenergyCR(cosmicray) The overall sky distribution of the arrival directions of
detectedsofaristheFly’sEyeeventwithanestimatedenergyof UHECRsbelowE seemstosupporttheisotropyhypothesis
GZK
∼300EeV(Birdetal.1994).Atthesehighenergies,protonsand (see,e.g.,Nagano&Watson2000;Abbasietal.2004;Mollerach
nucleicannotbeconfinedandacceleratedeffectivelywithinour etal.2008).Thisisconsistentwiththeexpectationofuniform
Galaxy,sothesourcesofUHECRsarelikelytobeextragalactic. distribution of distant extragalactic sources; the interaction
AtenergieshigherthanE ,itisexpectedthatprotonslose length (i.e., horizon distance) of protons below E is a
GZK GZK
energy and nuclei are photo-disintegrated via the interactions few Gpc and the universe can be considered homogeneous
withthecosmicmicrowavebackground(CMB)radiationalong and isotropic on such a large scale.5 The horizon distance for
their trajectories in the intergalactic space (Greisen 1966; super-GZKevents,however,decreasessharplywithenergyand
Zatsepin & Kuz’min 1966; Puget et al. 1976). The former R ∼100MpcforE =100EeV(Berezinsky&Grigor’eva
GZK
is known as the GZK effect. So a significant suppression in 1988). The matter distribution inside the local GZK horizon
the energy spectrum above E could be regarded as an (R )isinhomogeneous.Sincepowerfulastronomicalobjects
GZK GZK
observationalevidencefortheextragalacticoriginofUHECRs are likely to form at deep gravitational potential wells, we
(see, e.g., Berezinsky et al. 2006). However, the accurate expect that the distribution of the UHECR sources would be
measurement of the UHECR spectrum is very difficult, partly inhomogeneous as well. Hence, if super-GZK proton events
becauseofextremelylowfluxofUHECRs.Butamoreserious pointtheirsources,theirarrivaldirectionsshouldbeanisotropic.
hurdle is the uncertainties in the energy calibration inherent The anisotropy of super-GZK events, hence, has been re-
in detecting and modeling extensive air shower events (e.g., gardedtoprovideanimportantcluethatunveilsthesourcesof
Nagano&Watson2000;Watson2006).Nevertheless,boththe UHECRs.Sofar,however,theclaimsderivedfromanalysesof
Yakutsk Extensive Air Shower Array (Yakutsk) and the High differentexperimentsareoftentantalizingandsometimescon-
Resolution Fly’s Eyes (HiRes) reported observations of the flicting.Forinstance,withanexcessivenumberofpairsandone
GZK suppression (Egorova et al. 2004; Abbasi et al. 2008a), tripletinthearrivaldirectionofCRsabove40EeV,theAGASA
while the Akeno Giant Air Shower Array (AGASA) claimed
a conflicting finding of no suppression (Shinozaki & Teshima 5 Wenotethatinprinciple,iflocalinhomogeneoussourcesdominateover
distantbackgroundsources,onemightgetanisotropysignalsevenbelow
EGZK,whichhavenotbeenobservedinreality(see,e.g.,Lemoine&Waxman
4 Authortowhomanycorrespondenceshouldbeaddressed. 2009).
1422
No.2,2010 IGMFANDARRIVALDIRECTIONOFUHEPROTONS 1423
datasupporttheexistenceofsmallscaleclustering(Hayashida intheIGMF(seeKotera&Lemoine2009,foradescriptionof
et al. 1996; Takeda et al. 1999). On the other hand, the HiRes thepicture).Inthepicture,whateverthedeflectionangleis,the
stereodataareconsistentwiththehypothesisofnullclustering arrival direction of UHECRs is expected to be correlated with
(Abbasietal.2004,2009).Theautocorrelationanalysisofthe thedistributionoftheIGMFwhichtracesthematterdistribution
Auger data reported a weak excess of pairs for E > 57 EeV inthecosmicweb.
(Abraham et al. 2008a). In addition, the Auger Collaboration In this contribution, as a follow-up work of Paper I, we
foundacorrelationbetweenhighestenergyeventsandthelarge- investigate the effects of the IGMF on the arrival direction
scalestructure(LSS)oftheuniverseusingnearbyactivegalactic of super-GZK protons above 60 EeV coming from sources
nuclei(AGNs)intheVe´ron-Cetty&Ve´ron(2006)catalog(The within75Mpc.Thelimitingparametersforenergyandsource
PierreAugerCollaboration2007;Abrahametal.2008a;Hague distance are chosen to match the recent analysis of the Auger
etal.2009)aswellasusingnearbyobjectsindifferentcatalogs collaboration. Without knowing the true sources of UHECRs,
(Aublin et al. 2009). A correlation between highest AGASA thestatisticsthatcanbeobtainedwithobservationaldatafrom
eventswithnearbygalaxiesfromSloanDigitalSkySurveywas experiments are limited; some statistics that are essential to
reported (Takami et al. 2009). The HiRes data, however, do revealthenatureofsourcesaredifficultorevenimpossibletobe
notshowsuchcorrelationofhighestenergyeventswithnearby constructed.Ontheotherhand,withdatafromsimulations,any
AGNs(Abbasietal.2008b),butinsteadindicateapossible(yet statisticscanbeexplored.Inthatsense,simulationscomplement
controversial)correlationwithdistantBLLacobjects(see,e.g., experiments. Here, with the IGMF suggested by Ryu et al.
Tinyakov&Tkachev2001;Abbasietal.2006). (2008), we argue that the large deflection angle of super-GZK
The interpretation of anisotropy and correlation analyses protonsduetotheIGMFisnotinconsistentwiththeanisotropy
is, however, complicated owing to the intervening galactic and correlation recently reported by the Auger collaboration.
magneticfield(GMF)andintergalacticmagneticfield(IGMF); However, the large deflection angle implies that the nearest
thetrajectoriesofUHECRsaredeflectedbythemagneticfields objecttoaUHECReventintheskyisnotnecessarilyitsactual
astheypropagatethroughthespacebetweensourcesandus,and source.InSection2,wedescribeourmodelsfortheLSSofthe
hence,theirarrivaldirectionsarealtered.Despiteconsiderable universe,IGMF,observers,andsourcesofUHECRs,reference
observationalandtheoreticalefforts,thenatureoftheGMFand objectsforcorrelationstudy,andsimulations.InSection3,we
theIGMFisstillpoorlyconstrained.Yet,modelsfortheGMF present the results, followed by a summary and discussion in
generally assume a strength of ∼ a few μG and a coherence Section4.
length of ∼1 kpc for the field in the Galactic halo (see, e.g.,
Stanev 1997) and predict the deflection of ultra-high-energy 2.MODELSANDSIMULATIONS
(UHE) protons due to the GMF to be θ ∼ a few degrees (see,
e.g., Takami & Sato 2008). The situation for the IGMF in the In our study, the following elements are necessary: (1) a
LSShasbeenconfusing.AdoptingamodelfortheIGMFwith model for the IGMF on the LSS, (2) a set of virtual observers
an average strength of (cid:3)B(cid:4) ∼ 100 nG in filaments, Sigl et al. thatrepresent“us,”anobserverattheEarth,inastatisticalway,
(2003)showedthatthedeflectionofUHECRsduetotheIGMF (3)asetofhypotheticalsourcesofUHEprotonswithaspecified
could be very large, e.g.,θ > 20◦ for protons above 100 EeV. injectionspectrum,and(4)asetofreferenceobjectswithwhich
On the other hand, Dolag et al. (2005) adopted a model with weperformedacorrelationstudyofsimulatedevents.InPaperI,
(cid:3)B(cid:4)∼0.1nGinfilamentsandshowedthatthedeflectionshould wedescribedindetailhowwesetup(1),(2),and(3)byusing
benegligible,e.g.,θ (cid:8)1◦forprotonswith100EeV. datafromcosmologicalstructureformationsimulations.Below,
Recently,Ryuetal.(2008)proposedaphysicallymotivated webrieflysummarizemodelsfor(1),(2),and(3)andexplainin
modelfortheIGMF,inwhichapartofthegravitationalenergy detailthereasontointroduce“referenceobjects”inthisstudy.
releasedduringstructureformationistransferredtothemagnetic
2.1.Large-scaleStructureoftheUniverse
fieldenergyasaresultoftheturbulentdynamoamplificationof
weak seed fields in the LSS of the universe. In the model, the WeassumedaconcordanceΛCDMmodelwiththefollowing
IaGndMtFhefosltlroewngstlharigseplyrethdeicmteadttteorbdeist(cid:3)rBib(cid:4)ut∼ion10innthGeicnosfimlaimcwenetbs,. hpa≡ramHet/e(r1s0:0ΩkBmMs=−1 0M.0p4c3−,1)Ω=DM0.7=,a0n.d22σ7,=an0d.8Ω.TΛh=em0o.7d3el,
0 8
Cho&Ryu(2009)studiedvariouscharacteristiclengthscales
universe for the LSS was generated through simulations in a
of turbulent magnetic field and predicted that those are in the cubicregionofcomovingsize100 h−1(≡143)Mpcwith5123
rangeof∼0.1–0.5MpcfortheIGMinfilaments.Suchafield gridzonesforgasandgravityand2563particlesfordarkmatter,
in filaments is expected to induce the Faraday rotation (Cho
usingaPM/Eulerianhydrodynamiccosmologycodedescribed
& Ryu 2009), which is consistent with observation (Xu et al.
in Ryu et al. (1993). The simulations have a uniform spatial
2006).WiththismodelIGMF,Dasetal.(2008,PaperIhereafter) resolution of 195.3 h−1 kpc. The standard set of gasdynamic
calculated thetrajectories of UHE protons (E > 10EeV) that
variables, the gas density, ρ , temperature, T, and the flow
g
wereinjectedatextragalacticsourcesassociatedwiththeLSSin
velocity, v, were used to calculate the quantities required in
asimulatedmodeluniverse.Wethenestimatedthatonly∼35%
our model such as the X-ray emission weighted temperature,
ofUHEprotonsabove60EeVwouldarriveatuswithθ (cid:2) 5◦
T , the vorticity, ω, and the turbulent energy density, ε , at
andtheaveragevalueofdeflectionanglewouldbe(cid:3)θ(cid:4) ∼ 15◦. X turb
eachgrid.
Notethatthedeflectionangleof(cid:3)θ(cid:4)∼15◦ ismuchlargerthan
the angular window of 3◦.1 used by the Auger collaboration 2.2.IntergalacticMagneticField
inthestudyofthecorrelationbetweenhighestenergyUHECR
eventsandnearbyAGNs(ThePierreAugerCollaboration2007; WeadoptedtheIGMFfromthemodelbyRyuetal.(2008);
Abrahametal.2008a;Hagueetal.2009).Wepointoutthatina themodelproposesthatturbulent-flowmotionsareinducedvia
simplifiedpicture,thedeflectionoftrajectoriesofUHECRsmay the cascade of the vorticity generated at cosmological shocks
bedescribedasanaccumulationofscatteringsbyirregularities duringtheformationoftheLSSoftheuniverse,andtheIGMF
1424 RYU,DAS,&KANG Vol.710
isproducedasaconsequenceoftheamplificationofweakseed
fieldsofanyoriginbytheturbulence.Then,theenergydensity
(or the strength) of the IGMF can be estimated with the eddy
turnovernumberandtheturbulentenergydensityasfollows:
(cid:2) (cid:3)
t
ε =φ ε . (1)
B turb
t
eddy
Here,theeddyturnovertimeisdefinedasthereciprocalofthe
vorticityatdrivingscales,t ≡ 1/ω (ω ≡ ∇ ×v),and
eddy driving
φ is the conversion factor from turbulent to magnetic energy
that depends on the eddy turnover number t/t . The eddy
eddy
turnover number was estimated as the age of universe times
the magnitude of the local vorticity, that is, t ω. The local
age
vorticity and turbulent energy density were calculated from
cosmological simulations for structure formation described
above.Afunctionalformfortheconversionfactorwasderived
fromaseparate,incompressible,magnetohydrodynamic(MHD)
simulationofturbulencedynamo.
For the direction of the IGMF, we used that of the pas-
sive fields from cosmological simulations, in which magnetic
fieldsweregeneratedthroughtheBiermannbatterymechanism
(Biermann1950)atcosmologicalshocksandevolvedpassively
alongwithflowmotions(Kulsrudetal.1997;Ryuetal.1998). Figure1.DistributionoftheIGMFinatwo-dimensionalsliceof(143Mpc)2
In principle, if we had performed full MHD simulations, we inthesimulateduniverse.Locationsofvirtualobservers(circles)andmodel
couldhavefollowedtheamplificationoftheIGMFthroughtur- AGNs(stars)areschematicallymarkedatclustersandgroupsofgalaxies.Paths
ofUHECRsfromsourcestoobserversarealsoschematicallydrawn.
bulence dynamo. In practice, however, the currently available
(Acolorversionofthisfigureisavailableintheonlinejournal.)
computational resources do not allow a numerical resolution
highenoughtoreproducethefulldevelopmentofMHDturbu-
lence.Sincethenumericalresistivityislargerthanthephysical 2.3.ObserverLocations
resistivitybymanyordersofmagnitude,thegrowthofmagnetic
In the study of the arrival direction of UHECRs, the IGMF
fields is saturated before dynamo action becomes fully opera-
around us, that is, in the Local Group, is important, too. It
tive(see,e.g.,Kulsrudetal.1997).Thisisthereasonwhywe
wouldhavebeenidealtoplace“theobserver”wheretheIGMF
adoptedthemodelofRyuetal.(2008)toestimatethestrengthof
is similar to that in the Local Group. Unfortunately, however,
theIGMF,butwestillusedthepassivefieldsfromcosmological
little is known about the IGMF in the Local Group. Hence,
simulationstomodelthefielddirection.
instead, we placed “virtual” observers based on the X-ray
Figure 1 shows the distribution of magnetic field strength
in a slice of (143 Mpc)2 in our model universe. It shows that emissionweightedtemperatureTX.Thegroupsofgalaxiesthat
have the halo temperature similar to that of the Local Group,
the IGMF is structured like the matter in the cosmic web. As
0.05 keV < kT < 0.5 keV (Rasmussen & Pedersen 2001),
a matter of fact, the distribution of the IGMF is very well X
were identified. About 1400 observer locations were chosen
correlated with that of matter. The strongest magnetic field of
B (cid:3) 0.1 μG is found in and around clusters, while the field bythetemperaturecriterion.Inreality,thereshouldbeonlyone
observerontheEarth,butinourmodeling,wechoseanumberof
is weaker in filaments, sheets, and voids. In filaments which
observerlocationsinordertorepresentourpositionwithoutloss
are mostly composed of the warm–hot intergalactic medium
with T = 105–107 K, the IGMF has (cid:3)B(cid:4) ∼ 10 nG and ofgenerality,sincethesimulateduniverseisonlyonestatistical
(cid:3)B2(cid:4)1/2 ∼ a few ×10 nG (Ryu et al. 2008). Note that the representationoftherealuniverse.Then,wemodeledobservers
asasphereofradius0.5 h−1 Mpclocatedatthecenterofhost
deflectionofUHECRsarisesmostlyduetothefieldinfilaments
groups,inordertoreducethecomputingtimetoapracticallevel.
(see Paper I). The energy density of the IGMF in filaments is
ε ∼10−16ergcm−3,whichisafewtimessmallerthanthegas The distribution of handful observers is shown schematically
B
in Figure 1. One can see that the observers (groups) are not
thermalenergydensityandanorderofmagnitudesmallerthan
the gas kinetic energy density there.6 The IGMF in filaments distributeduniformly,butinsteadtheyarelocatedmostlyalong
filaments.
inducestheFaradayrotation;thermsvalueofrotationmeasure
(RM)ispredictedtobeafewradm−1(Cho&Ryu2009).This
2.4.AGNsasReferenceObjects
is consistent with the values of RM toward the Hercules and
Perseus–PiscessuperclustersreportedinXuetal.(2006).7
As noted in Section 1, the Auger Collaboration recently
reported a correlation between the direction of their highest
6 Wenotethatourmodeldoesnotincludeapossiblecontributiontothe energy events and the sky position of nearby AGNs from the
IGMFfromgalacticblackholes,AGNfeedback(see,e.g.,Kronbergetal.
12theditionoftheVe´ron-Cetty&Ve´ron(2006,VCV)catalog;
2001);soourmodelmayberegardedtoprovideabaselinefortheIGMF.With
suchacontribution,therealIGMFmightbeevenstronger,resultingineven thecorrelationhasthemaximumsignificanceforUHECRswith
largerdeflection(seeSection3.1). E (cid:3) 60 EeV and AGNs with distance D (cid:4) 75 Mpc (The
7 Thevaluesof|RM|inXuetal.(2006)areanorderofmagnitudelargerthan PierreAugerCollaboration2007;Abrahametal.2008a;Hague
thevalueabove,afewradm−1.Butthepathlengthtowardthesuperclusters, et al. 2009). There are about 450 AGNs with D (cid:4) 75 Mpc
throughwhichRMisinduced,isabout2ordersofmagnitudelargerthanthe
thicknessoftypicalfilaments. (moreprecisely,442AGNswithredshiftz (cid:2) 0.018,forwhich
No.2,2010 IGMFANDARRIVALDIRECTIONOFUHEPROTONS 1425
themaximumsignificanceofthecorrelationwasfound)inthe Table1
VCV catalog. In that study, it is not known which subclass of ModelsofDifferentNumberDensitiesofSources
thoseAGNsorwhatfractionofthemarereallytruesourcesof Model Nsrca HostClusters (cid:3)θ(cid:4)b θ˜ (cid:3)Ssim(cid:4)b S˜sim
UHECRS.Here,weregardthoseAGNsas“referenceobjects,”
A ∼500 kTX(cid:2)0.1keV 13.98 7.01 3.58 2.80
withwhichcorrelationstudiesareperformed. B ∼60 kTX(cid:2)0.55keV 15.33 8.80 3.97 3.19
In order to compare our correlation study with that of the C ∼28 kTX(cid:2)0.8keV 17.76 10.45 4.23 3.43
Auger collaboration, we specified the following condition to
determine“model”referenceobjectsinthesimulateduniverse: Notes.
(1)thenumberoftheobjectswithin75Mpcfromeachobserver aAveragenumberofsourceswithinsphereofradius75Mpc.
should be on average ∼500 and (2) their spatial distribution bDeflectionangleθandseparationangleSaredefinedinSection3.
shouldtracetheLSSinawaysimilartotheAGNdistribution
that the spatial clustering of our model AGNs is on average
in the real universe. To set up the location of such reference
objects,weidentified“clusters”ofgalaxieswithkT (cid:3)0.1keV comparable to that of the AGNs from the VCV catalog. This
X
providesajustificationforourselectioncriteriaformodelAGNs
inthesimulateduniverse.Ofcoursesomeoftheseclusterswith
kT (cid:4)afewkeVshouldbeclassifiedasgroupsofgalaxies.But inthesimulateduniverse.WenotethatQisanintrinsicproperty
X
of the distribution of the reference objects in the sky and has
for simplicity we call all of them clusters. The reason behind
nothingtodowithUHECRs.
this selection condition is that the gas temperature is directly
relatedwiththedepthofgravitationalpotentialwell;thehottest
2.5.SourcesofUHECRs
gas resides in the densest, most nonlinear regions of the LSS
where the most luminous and energetic objects (e.g., AGNs) Although AGN is one of the viable candidates that would
form through frequent mergers of galaxies. We then assumed produce UHECRs (see, e.g., Nagano & Watson 2000, for the
that each cluster hosts one reference object at its center. For listofviablecandidates),thereisnocompellingreasonthatall
eachobserver,wegeneratedalistofreferenceobjectsinsidea the nearby AGNs are the sources of UHECRs. In this paper,
sphereofradius75Mpc,whosenumberisonaverage∼500;the weconsideredthreemodelswithdifferentnumbersofsources
exactnumberofreferenceobjectsvariessomewhatfordifferent (see Table 1) to represent different subsets of AGNs. (1) In
observers. Then, each observer has its own sky distribution ModelA,weregardedallthemodelAGNs(referenceobjects)
of reference objects, with which we studied the correlation as true sources of UHECRs. (2) In Model B, we regarded on
of simulated events. Although our reference objects could be average 60 model AGNs located at 60 hottest host clusters
anyastronomicalobjectsthattracetheLSS,hereafterwerefer (kT (cid:3) 0.55 keV) within a sphere of radius 75 Mpc as true
x
to them as “model AGNs,” because the selection criteria were sources of UHECRs. Based on the ratio of singlet to doublet
chosentomatchthenumberofAGNswiththatfromtheVCV events, Abraham et al. (2008a) argued that the lower limit on
catalog.Figure1showstheschematicdistributionofahandful the number of sources of UHECRs would be around 61. (3)
ofmodelAGNsatthecenterofhostclusters.Obviouslythehost In Model C, we regarded on average 28 model AGNs located
clusters(andtheAGNs)arenotuniformlydistributedeither. at 28 hottest host clusters (kT (cid:3) 0.8 keV) within a sphere
x
In our setup, the distance to the model AGNs, D, can be of radius 75 Mpc as true sources of UHECRs. Among AGNs,
arbitrarily small. In reality, however, the closest AGN in the radiogalaxiesareconsideredtobethemostpromisingsourcesof
VCV catalog is NGC 404 at D ∼ 3 Mpc in the constellation UHECRs (see, e.g., Biermann & Strittmatter 1987), and there
Andromeda (Karachentsev et al. 2004). So the model AGNs are 28 known radio galaxies within D (cid:2) 75 Mpc. We note,
withthedistancefromeachobserverD <D ≡3Mpcwere however,thatnotallradiogalaxieswithin75Mpcresideinside
min
excluded. clusters,andinfactFRIIgalaxiesarefoundmainlyinfilaments.
We note that Cuoco et al. (2008) showed that local AGNs ThemainreasontoconsiderModelsBandChereistoexplore
with z (cid:4) 0.02 in the VCV catalog exhibit, similarly to the howthenumberdensityofsourcesaffectsouranalysis.
brightestgalaxies,strongersmall-scaleclusteringthanaverage
2.6.SimulationsofPropagationofUHEProtons
galaxiesinPSCzcatalog.ThedistributionofourmodelAGNs
insidehostclustersshouldbebiasedagainsttheaveragematter At sources, UHE protons were injected with the power-law
distribution too, in the sense that they represent statistically energy spectrum; N (E ) ∝ E−γ for 6×1019 eV (cid:2) E (cid:2)
the highest density peaks of baryonic matter in the simulated inj inj inj inj
1021eV,whereγ istheinjectionspectralindex.Weconsidered
universe.
thetwocasesofγ =2.7and2.4.Ateachsource,protonswere
WecheckedtheangulardistanceQbetweenagivenreference
randomlydistributedoverasphereofradius0.5 h−1 Mpcand
objecttoitsnearestneighboringobject.Forasetof442objects
(thenumberoftheAGNswithz(cid:2)0.018intheVCVcatalog), thenlaunchedinrandomdirections.
We then followed the trajectories of UHE protons in our
iftheyaredistributedisotropicallyovertheskyof4πradian,the
averagevalueofQwouldbe(cid:3)Q (cid:4)≈11◦.Withthe442AGNs modeluniversewiththeIGMF,bynumericallyintegratingthe
from the VCV catalog, on theisoother hand, (cid:3)Q (cid:4) = 3◦.55. equationsofmotion,
VCV
The fact that (cid:3)Q (cid:4) < (cid:3)Q (cid:4) means that the distribution of
VCV iso dr dp
the AGNs from the VCV catalog is not isotropic, but highly =v, =e(v×B), (2)
dt dt
clustered, following the matter distribution in the LSS of the
universe.Wenotethat(cid:3)QVCV(cid:4)=3◦.55issimilartotheangular where r, v, and p are the position, velocity, and momentum,
windowof3◦.1usedintheAugerstudy.Clearlythisagreement respectively.DuringpropagationUHEprotonsinteractwiththe
isnotaccidental,butratherconsequential.Forthesetsof∼500 CMB,andthedominantprocessesforenergylossarethepion
modelAGNsinoursimulations,theaverageangulardistanceis and pair productions. The energy loss was treated with the
(cid:3)QAGN(cid:4) = 3◦.68±1◦.66.Theerrorwasestimatedwith(cid:3)QAGN(cid:4) continuous-loss approximation (Berezinsky et al. 2006). The
for∼1400observers.Thefactthat(cid:3)QAGN(cid:4)∼(cid:3)QVCV(cid:4)indicates adiabaticlossduetothecosmicexpansionwasignored.
1426 RYU,DAS,&KANG Vol.710
Figure 2. Geometry of the deflection angle, θ, the separation angle, S, the
distancetothetruesource,Dθ,andthedistancetothenearestobject,DS.The
path of the UHECR event from the source to the observer is schematically
drawn.
(Acolorversionofthisfigureisavailableintheonlinejournal.)
We let UHE protons continue the journey, visiting several
observersduringflight,untiltheenergyfallsto60EeV.Atthe
observerlocations,theeventswithE (cid:5)60EeVwererecorded
andanalyzed.
3.RESULTS
3.1.DeflectionAngle
In Paper I, we considered the deflection angle, θ, between
the arrival direction of UHECR events and the sky position
of their sources (see Figure 2). Obviously this angle can be
calculatedonlywhenthetruesourcesareknown,whichisnot
the case in experiments. In the simulated universe, our model
AGNsandobserversarelocatedinstronglymagnetizedregions.
As illustrated in Figure 1, UHECRs first have to escape from
magnetic halos surrounding sources, then travel through more
orlessvoidregions(path1)orthroughfilaments(path2),and
finallypenetrateintomagnetichalosaroundobservers,toreach
observers.Sothedegreeofdeflectiondependsnotonlyonthe
magnetic field along trajectories but also on the fields at host
clusters and groups of sources and observers. Since the gas
temperature, the depth of gravitational potential well, and the
magnetic field energy density are related as kT ∝ Φ ∝ ε
X B
in our model, hotter clusters and groups would have stronger
fields.Soweexpectthatifsourcesandobserversarelocatedat
hotterhosts,θ wouldbelargeronaverage.
Figure3showsθ versusD forUHEprotoneventsrecorded
θ
at the observer locations in our simulations. Here, D denotes
θ
thedistancetotheactualsourcesofevents.Thetop,middle,and
bottompanelsareforModelsA,B,andC,respectively.Onlythe
caseofinjectionspectralindexγ = 2.7ispresented.Thecase
ofγ =2.4issimilar.Eachdotrepresentsonesimulatedevent,
andthereareabout105eventsineachModel.Theuppercircles
connected with dotted lines represent the mean values of θ in
thedistancebinsof[D ,D +ΔD ].Themeandeflectionangle
θ θ θ
averaged over all the simulated events is (cid:3)θ(cid:4) = 13◦.98, 15◦.33,
and17◦.76forγ =2.7inModelsA,B,andC,respectively.The
lower circles connected with solid lines represent the median Figure3.Distributionofdeflectionangles(θ)asafunctionofdistancetotrue
values of θ. The median value for all the simulated events is sources(Dθ).DotsrepresentUHEprotoneventsrecordedinoursimulations.
θ˜ = 7◦.01, 8◦.80, and 10◦.45 for γ = 2.7 in Models A, B, Circlesconnectedwithdottedlines(upper)andsolidlines(lower)showthe
and C, respectively. The values of (cid:3)θ(cid:4) and θ˜ for γ = 2.4 mfiresatnanadndthmireddqiaunarvtialleusesin,rgeisvpeenctDivθelby.inVse.rTtiocpa,lmlinidedslceo,nannedcbtothtteommapraknseolsftahree
are similar. The marks connected with vertical solid lines on forModelsA,B,andC,respectively.Thecasesofγ =2.7areshown.
both sides of median values are the first and third quartiles in (Acolorversionofthisfigureisavailableintheonlinejournal.)
No.2,2010 IGMFANDARRIVALDIRECTIONOFUHEPROTONS 1427
the distance bins, which provide a measure of the dispersion
ofθ.
Wenotethefollowingpoints.(1)WithourIGMF,themean
deflectionangleofUHEprotonsduetotheIGMFisquitelarge,
muchlargerthantheangularwindowof3◦.1usedintheAuger
correlationstudy.Itisalsomuchlargerthanthemeandeflection
anglethatisexpectedtoresultfromtheGalacticmagneticfield,
which is a few degrees (Takami & Sato 2008). (2) The mean
deflectionangleislargestinModelCandsmallestinModelA.
Recall that in Model C sources are located only at 28 hottest
clusters, while in Model A all 500 clusters include sources
(see Table 1). The UHECR events from hotter clusters tend
toexperiencemoredeflection,asnotedabove.Sothemeanof
deflectionanglesinthemodelwithhotterhostclustersislarger.
(3)ThemeanvalueofθhasaminimumatD ∼20–30Mpc,
θ,min
whichcomparestoatypicallengthoffilaments.Aswepointed
outinPaperI,thisisaconsequenceofthestructuredmagnetic
fieldsthatareconcentratedalongfilamentsandatclusters.Inan
eventwithD <D ,thesourceandobserveraremorelikely
θ θ,min
tobelongtothesamefilament,andsotheparticleismorelikely
to travel through strongly magnetized regions and suffer large
deflection(seepath2inFigure1).Intheoppositeregime,the
sourceandobserverarelikelytobelongtodifferentfilaments,
sotheparticlewouldtravelthroughvoidregions(seepath1in
Figure 1). (4) For the events with D > D , the mean and
θ θ,min
dispersionofθincreasewithD .Suchatrendisexpected,since
θ
inthediffusivetransportmodelofthepropagationofUHE√CRs,
the deflection angle increases with distance as θ ∝ D
rms θ
(see,e.g.,Waxman&Miralda-Escude´1996).(5)Therearemore
eventsfromnearbysourcesthanfromdistantsources,although
allthesourcesinjectthesamenumberofUHECRsinourmodel.
The smaller number of events for larger D should be mostly
θ
a consequence of energy loss due to the interaction with the
CMB.
3.2.SeparationAngles
Inthisstudy,wealsoconsidertheseparationangle,S,between
the arrival direction of UHECR events and the sky position of
nearestreferenceobjects(seeFigure2).Theanglecanbecal-
culatedwithobservationdata,onceaclassofreferenceobjects
(e.g.,AGNs,galaxiesgamma-raybursts,etc.)isspecified.For
example,ThePierreAugerCollaboration(2007)tooktheAGNs
within75MpcintheVCVcatalogasthereferenceobjectsfor
theircorrelationstudy.However,foragivenUHECRevent,the
nearestAGNintheskymaynotbetheactualsource;hence,the
separationanglebetweenaUHECReventanditsnearestAGN
isnotnecessarilythesameasthedeflectionangleoftheevent
(Hillas2009;Ryuetal.2009).
WeobtainedSforsimulatedeventswithourmodelreference
objects(AGNs).Figure4showsSversusD .Here,D denotes
S S
thedistancetonearestAGNs.Againonlythecaseofγ = 2.7
ispresented,andthecaseofγ = 2.4issimilar(seeFigure5).
Thecirclesconnectedwithsolidlinerepresentthemeanvalues
of S for the events with nearest AGNs in the distance bins of
[D ,D +ΔD ].Themeanseparationangleaveragedoverallthe
S S S
simulatedeventsis(cid:3)Ssim(cid:4) = 3◦.58,3◦.97,and4◦.23forγ = 2.7 Figure 4. Distribution of separation angles between the directions of UHE
inModelsA,B,andC,respectively. protonsandnearestmodelAGNs(S)asafunctionofdistancetonearestmodel
Wenotethefollowingpoints.(1)Themeanseparationangleis AGNs(DS).DotsrepresentUHEprotoneventsrecordedinoursimulations.
muchsmallerthanthemeandeflectionangle,(cid:3)S (cid:4)∼(1/4)(cid:3)θ(cid:4) Circles connected with solid lines show the mean values. Top, middle, and
sim bottom panels are for Models A, B, and C, respectively. The cases of the
(see the next section for further discussion). (2) The mean injectionspectralindexγ =2.7areshown.
separationangleislargestinModelCandsmallestinModelA,
(Acolorversionofthisfigureisavailableintheonlinejournal.)
althoughthedifferenceof(cid:3)S (cid:4)amongthemodelsislessthan
sim
1428 RYU,DAS,&KANG Vol.710
that of (cid:3)θ(cid:4). With larger deflection angles in Model C, there is
ahigherprobabilityforaeventtobefoundfurtherawayfrom
theregionwheremodelAGNsareclustered,soSisonaverage
largeraswell.(3)SimilarlyasintheθversusD distribution,the
θ
distributionof(cid:3)S (cid:4)hastheminimumataroundD ∼35Mpc.
sim S
This is again a signature of the filamentary structures of the
LSS.(4)ContrarytoD ,therearemoreeventswithlargerD
θ S
thanwithsmallerD .ItissimplybecausetherearemoreAGNs
S
withlargerD .
S
3.3.ComparisonwiththeAugerData
We also obtained S for the 27 Auger events of highest
energies,publishedinAbrahametal.(2008a),with442nearby
AGNs from the VCV catalog. In Figure 5, we compare the
S versus D distribution for the Auger data with that of our
s
simulations. The upper circles connected with dashed/dot-
dashed lines represent the mean values of S of simulated
sim
events as in Figure 4, but this time both cases of γ = 2.7
and2.4arepresented.Thelowercirclesconnectedwithsolid/
dottedlinesrepresentthemedianvaluesofS .Thedifference
sim
between the cases of γ = 2.7 and 2.4 is indeed small. The
medianvalueofS forallthesimulatedeventsisS˜ =2◦.80,
sim sim
3◦.19,and3◦.43forγ =2.7inModelsA,B,andC,respectively.
AsterisksdenotetheAugerevents.Themeanseparationangle
fortheAugereventsis(cid:3)S (cid:4)=3◦.23for26events,excluding
Auger
oneeventwithlargeS(≈27◦),while(cid:3)S (cid:4)=4◦.13forallthe
Auger
27events.Wenotethat(cid:3)S (cid:4)∼(cid:3)S (cid:4),eventhoughthemean
Auger sim
deflectionangleismuchlargerthanthemeanseparationangle
inoursimulations,thatis,(cid:3)θ(cid:4)(cid:13)(cid:3)S (cid:4).Inallthemodels,about
sim
half of the Auger events lie within the quartile marks: 15, 13,
and13eventsforModelsA,B,andC,respectively.
With(cid:3)θ(cid:4)∼15◦inoursimulations,onemightnaivelyexpect
that such large deflection would erase the anisotropy in the
arrival direction and the correlation between UHECR events
and AGNs (or the LSS of the universe). However, we argue
thatthelargedeflectiondoesnotnecessarilyleadtothegeneral
isotropyofUHECRarrivaldirection,iftheagentofdeflection,
the IGMF, traces the local LSS. Suppose that UHECRs are
ejected from sources inside the Local Supercluster. Some of
them will fly along the supergalactic plane and arrive at the
Earth;theirtrajectorieswouldbedeflectedbythemagneticfield
between sources and us, but the arrival directions still point
toward the supergalactic plane. Others may be deflected into
voidregions,andthentheywillhavelesschancetogetreflected
back to the direction toward us due to lack of the turbulent
IGMF there. As pointed out in Section 1, we may regard the
irregularitiesintheIGMFasthe“scatters”ofUHECRs(Kotera
&Lemoine2009);inthepicture,thelastscatteringpointwould
be the arrival direction of UHECRs. As a result, even with
large deflection, we still see more UHECRs from the LSS of
clusters,groups,andfilaments,andfewerUHECRsfromvoid
regions where both sources and scatters are underpopulated.
Figure 5. Mean (upper circles) and median (lower circles) values of S as Consequently,theanisotropyinthearrivaldirectionofUHECRs
functionsofDSforUHEprotoneventsinoursimulations.Verticallinesconnect canbemaintainedandthearrivaldirectionstillfollowstheLSS
dthaeshmeadrlkisneosf(tuhpepfierrs)taannddstohliirddlqinueasrt(illoeswienr)gaivreenfoDrSγb=ins2..C7,iracnledsthcoonseneccotnendewctiethd oftheuniverse.
withdot-dashedlines(upper)anddottedlines(lower)areforγ =2.4.Themean Below the GZK energy, the proton horizon reaches out to
Sforγ =2.7arethesameasthoseinFigure2.Themedianandquartilesfor afewGpc,sothesourcedistributionshouldlookmoreorless
γ =2.4arehorizontallyshiftedforbettervisibility.AsterisksdenoteSforthe isotropicandthearrivaldirectionsshouldnotshowacorrelation
27Augereventsofhighestenergies,publishedinAbrahametal.(2008a),with
with nearby AGNs. Thus, we do not expect to see anisotropy
nearbyAGNsfromtheVCVcatalog.Top,middle,andbottompanelsarefor
andcorrelationforUHECRswithsuchenergy.
ModelsA,B,andC,respectively.
InSection2.4,weshowedthatthedegreeofclusteringofour
(Acolorversionofthisfigureisavailableintheonlinejournal.)
modelAGNsissimilartothatofAGNsfromtheVCVcatalog;
No.2,2010 IGMFANDARRIVALDIRECTIONOFUHEPROTONS 1429
the mean of the angular distance Q between a given AGN to
itsnearestneighboringAGNissimilar,(cid:3)Q (cid:4) ∼ (cid:3)Q (cid:4).In
AGN VCV
both cases, AGNs follow the matter distribution in the LSS,
highlystructuredandclustered.Wepointoutthatifalongwith
thereferenceobjects,theCRsourcesandtheIGMFalsofollow
thematterdistribution,with(cid:3)θ(cid:4)(cid:13)(cid:3)S(cid:4),(cid:3)S(cid:4)∼(cid:3)Q(cid:4)isexpected.
Theresultthat(cid:3)S (cid:4)∼(cid:3)S (cid:4)∼(cid:3)Q (cid:4)∼(cid:3)Q (cid:4)isindeed
Auger sim AGN VCV
consistentwithsuchexpectation.Thismeans,however,thatthe
statistics of S reflect mainly on the distribution of reference
objectsratherthanthedeflectionangle.
TofurthercomparetheAugerdatawithoursimulations,we
plotthecumulativefractionofevents,F((cid:2)logS),versuslogS
for the simulated events (lines) and the Auger events (open
circles)inFigure6.Thesolidanddottedlinesareforthecases
of γ = 2.7 and 2.4, respectively, and the difference between
thetwocasesisagainsmall.TheKolmogorov–Smirnov(K–S)
test yields the maximum difference of D = 0.17, 0.23, and
0.26betweentheAugerdataandthesimulationdata(γ =2.7)
in Models A, B, and C, respectively; the significance level of
the null hypothesis that the two distributions are statistically
identicalisP ∼ 0.37,0.09,and0.04forModelsA,B,andC,
respectively. So the null hypothesis that the two distributions
for our simulated events and the Auger events are statistically
identical cannot be rejected, especially for Model A. Also we
seethatModelAwithmoresourcesispreferredoverModelsB
andCwithfewersources.Butthisdoesnotnecessarymeanthat
alltheAGNswouldbetheactualsourcesofUHECRs.Wenote
thatthenumberoftheAugereventsused,27,isstilllimited.In
addition, we consider only UHE protons in this paper. Hence,
before we argue the above statements for sure, we will need
more observational events and need to know the composition
of UHECRs (see summary and discussion section for further
discussiononcomposition).
3.4.ProbabilityofFindingTrueSources
With (cid:3)θ(cid:4) (cid:13) (cid:3)S (cid:4), there is a good chance that the AGNs
sim
found closest to the direction of UHECRs are not the actual
sources of UHECRs. To illustrate this point, we first show the
distributionofD versusD inFigure7.Forsomeevents,the
S θ
closest AGNs are the actual sources. They are represented by
thediagonalline.Aroundthediagonalline,anoticeablefraction
ofeventsarefound.Thosearetheeventsforwhichtheclosest
AGNsarefoundaroundthetruesources;bothsourcesandclose-
byAGNsareclusteredasapartoftheLSSoftheuniverse.For
the events away from the diagonal line, it is more likely that
D > D . It is because there are more AGNs with larger D ;
S θ S
awayfromtruesources,observedeventsaremorelikelytopick
upclosestAGNswithlargerD .
S
Toquantifytheconsequenceof(cid:3)θ(cid:4) (cid:13) (cid:3)S (cid:4),wecalculated
sim
thefractionoftrueidentification,f,astheratioofthenumberof
eventsforwhichnearestAGNsaretheirtruesourcestothetotal
numberofsimulatedevents.Thisisameasureoftheprobability
tofindthetruesourcesofUHECRs,whennearestcandidatesare
blindlychosen(whichisthebestwecandowithobserveddata).
InFigure8,weshowthefractionasafunctionoftheseparation
angle,S.ThefractionislargestinModelAwithlargestN and
src
smallestinModelCwithsmallestN .AtS ∼2◦thefractionis
src
about50%forModelA,closeto40%forModelB,andalittle
Figure 6. Cumulative fraction of UHECR events with the separation angle
above30%forModelC,butonly20%–30%atS = 3◦–4◦.As smaller than S. Solid and dotted lines denote the results calculated with the
theseparationangleincreases,thefractiondecreasesgradually UHEprotoneventsinoursimulationsforγ =2.7and2.4,respectively.Top,
to∼10%,indicatinglower probabilitiestofindtruesources at middle,andbottompanelsareforModelsA,B,andC,respectively.Opencircles
denotetheresultforthe27Augerevents(Abrahametal.2008a)withnearby
larger separation angles. On average, we should expect that in
AGNsfromtheVCVcatalog.
lessthanoneoutofthreeevents,thetruesourcesofUHECRscan
(Acolorversionofthisfigureisavailableintheonlinejournal.)
1430 RYU,DAS,&KANG Vol.710
Figure8.Fraction(f)ofUHEprotoneventsrecordedinoursimulations,for
whichtheirtruesourcesareidentifiedasclosestAGNsinthesky,asafunction
Figure7.DistancetonearestmodelAGNs,DS,vs.distancetotruesources,Dθ, ofS.Dashed,dotted,andsolidlinesareforModelsA,B,andC,respectively.
fortheUHECReventsinModelA.Thecaseofγ =2.7isshown.Colorcodes Darkandlightlinesdenotethefractionsforγ =2.7and2.4,respectively.
thenumberofeventsinthelog10scaleinbinsofΔDθ×ΔDS.Themaximum (Acolorversionofthisfigureisavailableintheonlinejournal.)
distanceis75MpcforbothDSandDθ.
(Acolorversionofthisfigureisavailableintheonlinejournal.)
the simulated universe is (cid:3)Q (cid:4) = 3◦.68±1◦.66, while that
AGN
beidentified,ifourmodelfortheIGMFisvalidandUHECRs for 442 AGNs from the VCV catalog is (cid:3)QVCV(cid:4) = 3◦.55. This
areprotons. demonstrates that the two samples have a similar degree of
clusteringandarehighlystructured(e.g.,(cid:3)Q (cid:4) ≈ 11◦ forthe
iso
isotropicdistribution).
4.SUMMARYANDDISCUSSION
With our model IGMF, the deflection angle, θ, between the
In the search for the nature and origin of UHECRs, under- arrival direction of UHE protons and the sky position of their
standingthepropagationofchargedparticlesthroughthemag- actualsourcesisquitelargewiththemean(cid:3)θ(cid:4)∼14◦–17◦.5and
netizedLSSoftheuniverseisimportant.Atpresent,thedetails the median θ˜ ∼ 7◦–10◦, depending on models with different
of the IGMF are still uncertain, mainly due to limited avail- numbers of sources (see Table 1). On the other hand, the
ableinformationfromobservation.Here,weadoptedarealistic separation angle between the arrival direction and the sky
modeluniversethatwasdescribedbysimulationsofcosmolog- position of nearest reference objects is substantially smaller
ical structure formation; our simulated universe represents the withthemean(cid:3)S (cid:4)∼3◦.5–4◦andthemedianS˜ =2◦.8–3◦.5.
sim sim
LSS,whichisdominatedbythecosmicweboffilamentsinter- Thatis,wefoundthatwhile(cid:3)θ(cid:4) ∼ 4(cid:3)S (cid:4),(cid:3)S (cid:4)issimilarto
sim sim
connectingclustersandgroups.ThedistributionoftheIGMFin (cid:3)Q (cid:4).FortheAugereventsofhighestenergiesinAbraham
AGN
theLSSoftheuniversewasobtainedwithaphysicallymotivated etal.(2008a),with442nearbyAGNsfromtheVCVcatalogas
modelbasedonturbulencedynamo(Ryuetal.2008). the reference objects, the mean separation angle is (cid:3)S (cid:4) =
Auger
ToinvestigatetheeffectsoftheIGMFonthearrivaldirection 3◦.23 for the 26 events, excluding one event with large S
ofUHECRs,wefurtheradoptedthefollowingmodels.Virtual (≈27◦), while (cid:3)S (cid:4) = 4◦.13 for all the 27 events. Hence,
Auger
observersofabout1400wereplacedatgroupsofgalaxies,which (cid:3)S (cid:4) ∼ (cid:3)Q (cid:4)∼ (cid:3)S (cid:4) ∼ (cid:3)Q (cid:4). This implies that the
Auger VCV sim AGN
represent statistically the Local Group in the simulated model separation angle from the Auger data would be determined
universe.Then,wesetupasetofabout500AGN-like“reference primarilybythe distributionof reference objects (AGNs),and
objects” within 75 Mpc from each observer, at clusters of maynotrepresentthetruedeflectionangle.
galaxies (deep gravitational potential wells) along the LSS. Wefurthertestedwhetherthedistributionsoftheseparation
They represent a class of astronomical objects with which we angle, S, for our simulated events and for the Auger events
performedacorrelationanalysisforsimulatedUHECRevents. are statistically comparable to each other. According the K–S
Weconsideredthreemodels,inwhichsubsetsofthereference test for the cumulative fraction of events, F((cid:2) logS), versus
objects were selected as AGN-like sources of UHECRs (see logS,thesignificancelevelofthenullhypothesisthatthetwo
Table 1). UHE protons of E (cid:5) 60 EeV with the power- distributionsaredrawnfromtheidenticalpopulationisaslarge
low energy spectrum were injected at those sources, and the as P ∼ 0.37 for Model A (see Table 1). Thus, we argued that
trajectoriesofUHEprotonsinthemagnetizedcosmicwebwere oursimulationdata,especiallyinModelA,areinfairagreement
followed.Attheobserverlocations,theeventswithE (cid:5)60EeV withtheAugerdata.Thistestalsoshowedthatthemodelwith
from sources within a sphere of radius 75 Mpc were recorded moresources(ModelA)ispreferredoverthemodelswithfewer
andanalyzed. sources(ModelsBandC).
To characterize the clustering of the reference objects, we The fact that (cid:3)θ(cid:4) (cid:13) (cid:3)S (cid:4) implies that the AGNs found
sim
calculatedtheangulardistance,Q,fromagivenreferenceobject closest to the direction of UHECRs may not be the true
toitsnearestneighbor.ThemeanvalueforourmodelAGNsin sources of UHECRs. We estimated the probability of finding
No.2,2010 IGMFANDARRIVALDIRECTIONOFUHEPROTONS 1431
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source identifications to the total number of simulated events.
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arXiv:0906.4850
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