Uncovering neutrinos from cosmic ray factories: the Multi Point Source method

0.25 MB·English

by  Yolanda Sestayo

Uncovering neutrinos from cosmic ray factories: the Multi Point Source method
YolandaSestayoa,ElisaResconib
aDepartamentd’AstronomiaiMeteorologia,UniversitatdeBarcelona(IEEC-UB),Mart´ıiFranque`s1,E-08028,Barcelona,Spain
bT.U.Munich,D-85748Garching,Germany
Abstract
Wepresentanovelmethodforthesearchofhighenergyextraterrestrialneutrinosinextendedregions.Themethodisbasedon
3
thestudyofthespatialcorrelationsbetweentheeventsrecordedbyneutrinotelescopes. Extendedregionsradiatingneutrinosmay
1
existintheGalaxyduetothehierarchicalclusteringofmassivestars,theprogenitorsofalltheGalacticacceleratorsknownsofar.
0
2Theneutrinoemissionassociatedtosuchextendedregionsmightbefaintandcomplexduetoboththeescapeofcosmicraysand
theintricatedistributionofgasintheenvironmentoftheaccelerators. Wehavesimulatedextendedneutrinoemissionoveranarea
n
of10◦ ×10◦, wheretheintensityfluctuationsacrosstheregionare modelledasa Gaussianrandomfield witha givencorrelation
a
Jstructure. We tested our proposed method over realizations of this intensity field plus a uniform random field representative of

8the spatial distribution of the atmospheric neutrino background. Our results indicate that the method proposed here can detect
significanteventpatternsthatwouldbemissedbystandardsearchmethods,mostlyfocusedinthedetectionofindividualhotspots.

]
E
Keywords:
HNeutrinos,Galacticcosmicrays,Spatialcorrelations
.
h
p
-1. Introduction [19], instruments which also have provided the first evidence
o
of particle acceleration up to TeV energies from a number of
tr Thesearchforthesourcesofcosmicraysisoneofthemost individualsources[20].
sstandingissuesinhigh-energyastrophysics.Majoradvancesin Further progress is expected in the upcoming future with
a
[thefieldduringthelastyearshavebeenrealizedbytheconsid- neutrino detectors such as IceCube [21]. In the energy range
erablenumberofinstrumentsdedicatedtoGalacticcosmicray 300 GeV < E < 10 PeV, the IceCubeexperimentat the South
1
vstudies[1,2,3,4,5,6].Theon-goingmeasurementsofthecos- Pole uses the Earth as a filter to observe neutrinos from the
9micradiationfromtheGalaxy[7,8,9]arehelpingtoconstruct wholenorthernsky. Itslargefield ofview anddynamicrange
2a comprehensivepictureaboutthe sourcesofcosmicrays, the offer a unique opportunity to unveil the sites of both produc-
5mechanismsofpropagation,andtheinteractionofcosmicrays tion and interaction of the high energy cosmic rays from the
1withthegasandradiationfieldsoftheGalaxy[10]. Currently, Galaxy.Asinthecaseofhadronicgamma-rays,neutrinoemis-
.
1gamma-rayastronomyhasa leadingrole in the explorationof sionimpliestheefficientaccelerationofcosmicraysinGalac-
0theskyattheveryhighenergies. Measurementsofthespatial tic sources, as well as their interaction with matter and radia-
3and spectral distributions of the diffuse Galactic gamma-rays tion. Surveysofatomicandmoleculargas[22,23], combined
1
:provide information about the propagation and interaction of withinfraredandopticaldata,haveprovidedlarge-scalemaps
vGalactic cosmic rays [11, 12, 13], whereas localized excesses of the Milky Way’s stellar and gas distribution with unprece-
i
Xof gamma-rayemission with respect to models of cosmic ray dented spatial resolution [24], despite the uncertainties in the
rpropagationcanbeinterpretedastheresultofcosmicrayinter- distanceestimations.Fromtheseobservationsweknowthatthe
aactionsinregionsofenhancedmatterdensity(whicharebelow progenitorsofalltheGalacticacceleratorsknownsofar,mas-
the spatial resolution of current cosmic ray propagationmod- sive stars, are not located at random within the Galactic disk,
els), or due to the presence of Galactic accelerators injecting butrathershowingahierarchicalstructure.
high energy particles which interact close to their sources of All present-daystar formationappearsto take place in gi-
origin [14, 15, 16, 17]. The GeV gamma-ray diffuse emis- ant molecularclouds. Massive open star clusters do not form
sionisbeingmeasuredbytheFermisatellite[7],showingthat inisolation,buttendtobeclusteredthemselvesintheso-called
gamma-raysare producedthroughoutthe Galactic disk by the clusterscomplexes[25],associatedwithgiantmolecularclouds
interactionsofcosmicrayswiththegasandradiationfromthe followingthespiralarmsoftheMilkyWaydisk[26,27,28].De
interstellar medium, after their diffusion in the Galactic mag- laFuenteMarcos&DelaFuenteMarcos[26]foundstatistical
neticfields. TheGalacticTeVdiffuseemissioniswithinreach evidenceoftheexistenceofatleastfivedynamicalfamiliesof
ofgroundCherenkovtelescopeslikeH.E.S.S.[18]andMilagro youngandmassiveopenclustersinthesolarneighborhood(dis-
tance<2.5kpc),whichmayextendupto10degreesinGalac-
ticlongitude. Therelevantscenarioforhighenergyastronomy
Emailaddress:[email protected](YolandaSestayo)
PreprintsubmittedtoAstroparticlePhysics January9,2013
is one in which we can find groups of stellar open clusters in the simulations of the event pattern. In section 3 we present
an evolutionary stage in which the combined winds of mas- ourproposedmethodfortheanalysisofextendedregions.Sec-
sivestarsandsupernovaexplosionsdominatetheenergeticsof tion 4 shows the performance of both our correlation method
theregion. ThisestablishestheexistenceofpotentialGalactic andstandardsearchmethodsfromtheresultsobtainedwiththe
cosmic ray factoriesin regionsof massive star formation[29] simulatedeventpatterns.Section4hasthediscussion.
whichmaywellextendseveraldegreesinthesky,possiblyaf-
fectingthespatialdistributionofneutrinoeventsrecordedwith
2. Simulationsoftheeventpattern
neutrinotelescopes.
In this paper we focus on the potential of neutrino tele- An event pattern consists of a set of event locations r =
scopes such as IceCube to detect a significant neutrino event (r,r,...,r )withinaregion(R),atwhichNeventshavebeen
i j N
patternassociatedwiththeseactiveregionsoftheGalaxy. This recorded. Inthesimplestcase,thedatasetcomprisesonlythe
couldpointtothelocationanddistributionofGalacticcosmic event locations. However, in some cases we may have addi-
ray and neutrino sources during the first years of the IceCube tional information related to the events which might have a
experiment, when sensitivity mightbe still too limited for the bearing with the nature of the analysis. For instance, the ar-
detectionofindividualsteadypoint-sources. rivaltimeoftheevents,foranalysisofvariablesources[36],or
Apossibleneutrinoemissionintheseregionsmayshowan theenergyoftheeventsforahighersignaltonoisediscrimina-
intricate,andlikelyfaint,intensitypattern,duetothecomplex tion power[37]. These cases correspondto what is knownas
structure of the matterin regionsharboringmassive stars. On markedpointpattern. Althoughthisinformationcanbeincor-
onehandwehavetheclusteringofpotentialaccelerators,where porated in any analysis, in this paper we have not considered
highenergycosmicraysare originated,andonthe otherhand marksintheeventpattern,andwe focusedonlyonthespatial
theavailabletargetforcosmicrayinteractions,consistingofthe distributionofevents.
gas left from the parent molecular cloud, which has not been
used for star formation nor evacuated from the region by the 2.1. Simulationsoftheastrophysicalneutrinosignal
radiationandwindsfromthestarsandsupernovaexplosions.
Aspatialeventpatterncanbethoughtofastherealizationof
Bothobservationsandtheoryaboutthespatialstructureof
aspatialprocess. Inmanyastrophysicalscenarios,suchphysi-
themolecularandatomicgasinalargerangeofenvironments
calprocessesdependonrandomcomponents,andtheyareusu-
haveshownthatitpresentssomedegreeofspatialcorrelation,
allymodelledintermsofrandomfields[39].
usually characterized in terms of the power spectra [30, 31].
We adopteda Gaussianrandomfield with a givencorrela-
The causes of the observed spatial correlations are manifold,
tionstructurefortheintensityfluctuationsofapossibleneutrino
butlargescale turbulenceandsome localeffectslike the radi-
signalinsidearegionthatismuchlargerthantheangularreso-
ationpressureandwindsfromthemassivestarsandsupernova
lutionoftheexperiment.
explosions are the most important[32, 33, 34, 35]. These ef-
If the field is Gaussian, we can determine it completely
fects produce spatial fluctuations in the gas density, which in
given its mean and correlation function (or power spectrum),
turn, may contribute to the fluctuations of a possible neutrino
whichfacilitatesthesimulations. Thestartingpointofthesim-
signalinsideanareapopulatedwithcosmicraysources.
ulations of Gaussian random fields is the convolution of the
We offer here an alternative to the search of these poten-
powerspectrumofthefieldwithawhitenoise(i.e.,correlation
tial Galactic cosmic ray factories, in a situation in which lit-
free)randomsignal[40]:
tle knowledgeaboutthe high energyprocessestakingplace is
available. We introducehere the use of analysisof the spatial
I(r)= d3keik·r P(k)W(k) (1)
correlationsbetweenneutrinoeventsforthediscoveryofGalac- Z
p
tic cosmic ray sites. We will study the advantagesand poten-
which gives the intensity of the field at each point r from the
tialofcorrelationanalysiswithrespecttostandardsearchesfor
powerspectrumP(k),relatedtothespatialcorrelationfunction
neutrinosources. Themethodpresentedhereandthestandard C(r) by the Fourier transform: P(k) = F−1C(r). W(k) is the
searchmethodsaretestedonsimulatedeventpatterns,inwhich
white noise, which gives the random amplitude and phase to
thephysicaloriginofthepossibleneutrinosignalistranslated
thefield.
intostatisticalpropertiesoftheneutrinointensityfield.
IftheangularextentofRislarge,asphericalharmonicde-
Modelling the exact characteristics of a possible neutrino
compositionofthefieldwouldbeneeded.Inthiscase,thefield
intensityfieldinaparticularregionoftheGalaxyisoutofthe
wouldhavethespectralrepresentation:
scope of this paper, andthe examplesconsideredaim to illus-
tratetheperformanceofsearchmethods.Withintheregionun- ∞ l
derstudy,weassumethatthefluctuationsintheintensityofthe I(r)= al,mYl,m(r) (2)
X X
neutrinosignalvaryspatially accordingto a Gaussian random l=0 m=−l
fieldwithagivencorrelationstructure. Thelastsourceofneu- where Y is a basis of spherical harmonics, and the coeffi-
l,m
trinos we consider is the isotropic foreground of atmospheric cientsa arecomplexuncorrelatedrandomvariablesthatde-
l,m
neutrinos, the statistics of which is consistent with a uniform pendonlyonthecovarianceorpowerspectrumofthefield P.
l
Poissondistribution. Inourexamples,RissmallenoughastoassumeEuclidean
Thepaperisorganizedasfollows. Insection2wedescribe space.WegeneratedtheGaussianrandomfieldsthroughFourier
2
phase randomization by discretizing equation 1 with a Carte- map EMEMMRRMRRnnMMMMeeeettaaaaSSSSrrnnnnii eexyxy sxsxyy mmaapp 22 66 322322 22 0..0..337979115555242444....5325324422
sianmeshdefinedoverR.Forsimplicity,weassumedthatRis 40
80
a square region of length L = 10◦, and used a mesh of 512 × 39
60
512nodes,givinga0.02◦samplingintervalofR,muchsmaller 38
40
thantheangularresolutionofneutrinotelescopes. 37 20
ThesamplingofRisdoneinFourierspacedirectlyassum- g) 36 0
ing a power spectrum for the intensity field. The Fourier do- c (de 35 -20
mainiscommonlyusedinthesimulationsofrandomfieldsbe- de 34 -40
cause the random variables at different points are statistically -60
33
independent, whereas in the spatial domain they have long- -80
32
rangecorrelationsthataredifficulttosample. -100
31 -120
Forsimplicity,weassumedanisotropic,homogeneousran-
30
domfield;thatis,thecorrelationbetweentwopoints r and r 300 301 302 303 304 305 306 307 308 309 310
1 2 R. A. (deg)
depends on the length τ = |τ| of the vector τ = r − r , but
1 2
notonits direction. We adopteda power-lawpowerspectrum Figure1:RealizationofaGaussianrandomfieldwithpowerspectrumP(|k|)=
P(|k|) = (α+|k|2)−β,whichhasanassociatedcovariancefunc- (1+|k|2)−2overthe10◦×10◦regionusedinourexamples.Inthisspecificre-
tion describedalso bya power-lawplusan exponentialdecay, alizationunitsarearbitrary,theintensityofthefieldisnormalizedwithrespect
toareferencevalue.Inthefollowingexamplesthesignalwillbenormalizedto
with α,β as the parameters that regulate the decay rate of the
agivennumberofevents.
longrangecorrelationswithdistance: C(τ)∝τβ−1e−ατ.
Figure1showsarealizationofanisotropic,homogeneous
Gaussianrandomfieldwiththepowerspectrumdescribedabove 3. Analysisofthespatialcorrelations:theMultiPointSource
withα=1,β=2. Infigure1thecolorscalehasarbitraryunits, method(MPS)
buttheobtainedspatialdistributionwasusedfortherepresen-
Methodsto accountfor the spatial correlationsin the data
tation of the astrophysicalneutrinosignal, where the intensity
areextensivelyusedinastronomy[43],startingfromthework
will be normalizedto a certain numberof signaleventsin the
ofPeeblesandco-workerstostudythelarge-scalematterdistri-
final data sample of neutrino telescopes. The test region over
butionin the Universethroughthe two-pointcorrelationfunc-
whichweinjectthesignalisdefinedintherangeinequatorial
tionofgalaxies[44]. Inhighenergyastrophysics,thesearches
coordinates30◦ <dec<40◦,300◦ <R.A.<310◦.
forcorrelationsin thedatahavebeenappliedmainlyto detect
Theinstrumentationintroducesanadditionalcorrelationin
anisotropiesin the spatial distribution of cosmic-rays[45, 46]
the intensity pattern due to the errors in the reconstruction of
andneutrinos[47].
the event directions. Therefore, for the analysis of the event
Themostcommonlyusedestimatorsforthetwo-pointcor-
patterninsectionweconvolvedthesignalfieldoffigure1with
relationfunctioninastronomyarebasedonpaircounting(see
thePointSpreadFunction(PSF)typicalofneutrinotelescopes.
Kerscheretal.(2000)[48]foracomparisonofdifferentestima-
TheeffectofthePSFistosmearthesignaloversomearea,ac-
tors). FollowingthenotationofSzapudi&Szalay(1998)[49],
cordingtoitsshape.WeassumethePSFhasaGaussianprofile
thenumberofeventpairswithinadistancerisdefinedas
withσ = 1,whichcorrespondstoamedianof0.7◦,similarto
whatisexpectedfortheIceCubeneutrinodetector[41].
P (r)= Ψ(x,y) (x,y) (3)
DD r
XX
2.2. Simulationofthebackgroundofatmosphericneutrinos x∈Dy∈D
Asimulatedsampleconsistingonlyofatmosphericneutrino P (r)= Ψ(x,y) (x,y) (4)
RR r
events was constructed injecting uniform random events on a Xx∈RXy∈R
sphere.Inreality,atmosphericneutrinosreachthedetectorwith
where the summations run over event coordinates in the data
a zenith dependent component [42], but within a declination
sample D, and in a sample R of randomly distributed events.
bandof∼10◦scaleslikeintheexamplesweconsiderhere,the Ψ (x,y)= 1onlyifacertainpairselectioncriteriaaresatisfied
r
spatial distribution of atmospheric neutrino events in the final
inthereal(random)sample,andisequalto0otherwise. After
sample is uniform random, where the intensity at each point
thenormalizationofthenumberofpaircounts:
followsaPoissonprocess.
Atotalof50000backgroundeventswereinjectedrandomly DD(r)= P (r)N(N−1) (5)
DD
through the northern sky, a number which could be represen- X
tative of the total atmospheric neutrino events with E > 500 RR(r)= P (r)N (N −1) (6)
RR R R
GeVexpectedfortheIceCubedetectorinitsfinalconfiguration. X
withNandN beingthetotalnumberofeventsintherealand
Thisproducesandexpectationof207backgroundeventsinthe R
10◦×10◦regionwithinthedeclinationband30◦ <δ<40◦that random sample, the so-call natural estimator of the two-point
correlationfunction,η ,is:
wehaveconsideredinourexamples. n
DD
η (r)= −1 (7)
n
RR
3
which representsthe excess probability, with respect to a ran- detection. To obtain the p-value of an observation, we must
domdistribution,offindinganeventatadistancer ofanother therefore know the distribution of our test statistics under the
event. background-onlyhypothesis.IntheMPSmethod,theteststatis-
Inthissection,weintroducethisformalismforthedetection tics is a function of the angular scale Θ. For each Θ exam-
ofhighenergyneutrinosfromanextendedregion.Theclassical ined, we must construct the probability distribution of Φ(Θ).
two-pointcorrelationfunctionmethod,brieflydescribedabove, The number of background events in a given area A = πΘ2
was adapted to our specific case and optimized for discovery. isproducedbya Poissonprocess; asaconsequence,thenum-
The aim of our method is to determine whether the observed berofpairsofbackgroundeventswithinadistanceΘfollowsa
spatialpropertiesofneutrinoeventsinsidetheareaunderinves- Gammadistribution.However,withthedefinitionofeventpairs
tigationarecompatiblewiththoseexpectedfromadistribution inMPS,theparametersoftheGammadistributionaredifficult
ofbackground-onlyevents(atmosphericneutrinoevents),orif toestimatefromobservables,andweobtainedthedistributions
anextraterrestrialcomponenthastobeinvokedinordertoex- withMonteCarlosimulations.
plaintheobservations,andatwhichlevelthebackground-only
hypothesisisrejected.
4. Analysisoftheeventpattern
Duetothedifferencesourapproachposestothenaturaldef-
initionofthetwo-pointcorrelationfunction,wedefinedourtest
In this section, we present our results about the ability of
statistics as a clusteringfunction, and in what followswe will
both a stardard search method and the MPS method to find a
refertoourproposedmethodastheMultiPointSource(MPS).
significantneutrinoeventspatternoverextendedregions. The
HerewedescribehowtheMPSmethodisdefined.
testswereperformedoversimulatedskymaps,inwhicharegion
Let R be the region under study, in which a total number Rof10◦×10◦exhibitsneutrinoemissionwithsomecorrelation
of N events have been registered in the final data sample
inside structure(section2.1).
atthelocations(r ,r ,...r ),definedbythecoordinatesr =
1 2 Ninside i Standardsearchmethodsdealwiththeidentificationofhot
(θ,φ)onthesky. MPSmakesatwo-pointsamplingofRwith
i i spots, definedas a spatialconcentrationofevents. Thesearch
circularbinsofvariableareaA =πΘ2centeredatthelocations forhotspots is done by mappingthe sky using differenttech-
of each of the events inside the region, r. In the sampling of
i niques; the ones commonly used in neutrino astronomy are
the region, the angular distance, Θij, from an event i located basedoneither”classical”binnedmethods,orlikelihood-based
insideRtoaneventj(atanylocation)ismeasuredforeachof methods with different degrees of sophistication in the mod-
theN eventspresentinR(i=1,...,N ;j=1,...,N ).
inside inside total ellingofthedata[37,38].Regardlessofthemethodusedinthe
Thenumberofpairsijismeasuredasafunctionoftheangular
analysis,thesearcheshavebeenusuallyoptimizedforthedis-
separationandthehistogramofeventpairsasafunctionofthe
coveryofpoint-likeemission,andhencecorrelationsofevents
angulardistanceΘisconstructedfromthesemeasurements.
beyondthesinglepoint-sourcescaledonotenterintheanalysis.
Theanalysisthenmakesuseofascale dependentcumula-
Theexplorationofthesky,seekingforsourcesofhighenergy
tive clustering function, Φ(Θ), defined as the excess, with re-
neutrinos,is doneby superimposinga gridoverthe eventdis-
specttothenullhypothesis,inthenumberofeventpairswithin tribution.Classicalbinnedmethodscountthenumberofevents
acertainangulardistanceΘ: within a circular fixed-area bin centered at each of the nodes
in the grid, resulting in estimates of the intensity of the pro-
Θ
DD(Θ)dΘ
cessesthatgeneratethe observedeventpatternateachsample
Φ(Θ)= R0 (8)
Θ point.Thegridstepismuchsmallerthantheangularresolution
RR(Θ)dΘ
R0 achievedin the analysis, and the bin size is usually optimized
whereDD(Θ)= DD ,RR(Θ)= RR ,andthesumruns tohavethebestsignaltonoiseratioforpoint-sources. Likeli-
ij ij
Xij Xij hoodmethodsusethesametypeofscananalysis,buttheygive
over all non-repeated pairs in the real data sample and in the a probabilistic assignment of an event to a component(signal
backgroundcase,respectively.Inourcase,DD (RR )=1only
ij ij orbackground)accordingto its distance to the pointat which
ifeithertheeventiortheeventj,orboth,arewithintheregion
the intensityisbeingestimated. Ateachgridpoint, the likeli-
under study, and it is equal to zero otherwise. That is, MPS
hoodofamixturemodelofsignalplusbackgroundiscompared
does not correlate events with specific locations, but rather it
toapurebackgroundhypothesis,andtheregionunderstudyis
considerseacheventinsidetheregionasapointsourceinorder
imagedasaprobabilitydensitymap.
todetermineitsdegreeofcorrelationwiththerestoftheevents.
The performanceof both classical and likelihoodmethods
Thisfeatureispreciselywhatnamesthemethodas”MultiPoint
aresimilarintermsofdiscovery,aslongastheyuse thesame
Source”. Withthisdefinitionwemeasureboththeintensityof eventinformation[37]. Giventhattheeventdensitymapsob-
the process thatgenerated the observedneutrinoeventpattern tainedfromabinnedanalysisofferaclearvisualinspectionof
aswellasitscorrelationstructure.
thespatialeventpattern,aswellasamorestraightforwardcom-
Theultimateresultoftheanalysisinneutrinotelescopesis
parisonwiththeresultsfromMPS,weusedaclassicalbinned
the p-value of the observation, that is, the probabilitythat the
methodasthestandardscananalysisofoursimulatedskymaps.
observed event pattern is just a realization of the atmospheric Thescanof thetest regionof10◦ ×10◦ was donein steps
neutrino background field. A discovery is claimed when this of 0.25◦ × 0.25◦. At each point, the local event density was
probability is below 2.8 × 10−7, which corresponds to a 5σ
4
measured and compared to the one expected under the back- eachangularscale. Themeasuredspatialcorrelationsbetween
groundhypothesis,wherebylocalwemeanwithinafixed-area events below 5◦ follow a power-law with exponential decay,
binaroundthesamplepoints. consistentwiththesignalfieldthatwasusedinthesimulations,
Both the MPS and the binned scan of R were applied to and the discoveryof the simulated astrophysicalneutrino sig-
background+signalsamplesandthecorrespondingsignificances nalgoesuptoangularscalesof∼2.5◦. Therefore,MPSresults
werecalculatedwiththep-valueoftheoutcomesofeachmethod. havethepotentialtodiscoveramorecomplicatedstructureofa
The probability distributions were computed after the analy- possibleastrophysicalneutrinosignalthanwhatwas observed
sis of n randombackground-onlysamples with both the MPS withthestandardscanoftheregion.
method and with a binnedscan of R. Each of these simulated
datasetsyieldsneventpatternswithadifferentnumberN(n) 4.2. LowS/Ncases
inside
ofbackgroundeventsinsideR.Theaverageclusteringfunction In this example a total of 40 signal events were injected
in the background case and its dispersion at different angular in the region. With a background expectation of 207 events
scalesisobtainedafterapplyingMPSovern=104background- withinR,thissituationcorrespondstoabackgrounddominated
onlydatasets. Theobtainedprobabilitydistributionswerefitted sample.Thiscaseexemplifiesamorerealisticsituation,sincea
to a Gamma distribution. The binnedscan was applied to the totalof40signaleventspresentatthefinalsampleofneutrino
samebackground-onlydatasetsandthedistributionofthenum- telescopesfromaregionof10◦×10◦constitutesamorerealistic
berofeventswithinthesearchbinforeachofthenodesinthe situationgiventhecurrenteffectiveareasofneutrinodetectors
gridwasfittedtoaPoissondistribution. andsourceswithmoderateneutrinofluxes[41].
Inthenextsections,wepresentourresultsundertwodiffer- We illustrate here three simulated examples of such situa-
entscenarios:asituationofhighsignaltonoiseratio(S/N),and tion. Fig.4showstheresultofa1◦ binnedscanforthreereal-
asituationinwhichtheS/Nisverylow.ThecaseofahighS/N izationsofthebackgroundfieldplus40signaleventsfollowing
underlinesthedifferencesintheinformationextractedfromthe thespatialdistributionoffigure1. Thethreecasesarecompat-
sameeventpatternwiththetwodifferentmethodsusedinthis iblewithbackgroundfluctuations,reachinga3σlevelatmost.
paper. Thestudyofalow-signalcaseillustrateshowtheMPS Fig. 5 shows the same scan, but in which signal events have
and a standardscan ofR dealwith the backgrounddominated been removed. The comparisonof figs. 4 and 5 manifests the
samplescharacteristicofneutrinotelescopes. random appearance of events even in the signal+background
maps,wherethemostsignificantspotsoccurredalwaysatran-
4.1. HighS/Ncase dom locations, as expected in background-onlyskymaps, but
In this example a total of 100 signal events were injected with a slightly higher significance due to the additional pres-
randomlyfollowingthedistributionoffigure1convolvedwith ence of signal events in the region. If we combine the three
a Gaussian with σ = 1, representative of the PSF of current simulatedsamples(figures6(a),6(b)),correspondingtoalarger
neutrino telescopes. On top of the signal, background events periodofdatatakingintherealsituation,theS/Nisstilltoolow
wererandomlygeneratedasexplainedinsection2.2. toproduceasignificantdetectionfromthebinnedscan,andthe
Fig. 2 shows a realization of the signal+background field minimump-valueachievedonlyreaches3σat0.5◦ scales(fig-
imaged as a significance event density map from the results ure 6(a)). If, in addition, we correctfromthe trials associated
of the binned scan of R with circular bins of radius 0.5◦, 1◦, to the scan of an extended region, this significance is further
1.5◦, 2◦. In this simulated data set, a total of N = 294 reduced. However, althoughthe significance of these individ-
inside
signal+backgroundeventshavefallen withinR, yieldinga 6σ ualspotsisnotenoughforadetection,theirdistributionisnot
excessofeventsinsidetheentireR,clearlyrepresentingahigh commonofarandomdistributionofbackgroundevents.Thisis
S/Ncase. illustratedwhenwe useMPSto quantifythedeparturesofthe
Theappearanceoftheresultingimagesisacombinationof wholeeventpatternwithinRfromadistributionofbackground
diverseeffects:thespatialstructureofthesignalintensityfield, events. WithMPS,theminimump-valueoftheobservedevent
thesmearingofthissignalbecauseoftheinstrumentalerrorin pattern reaches 4.3σ observation (figure 6(c)). The trial fac-
theeventlocations,andthesizeofthesearchbinarea.Thesig- torsfromscanningseveralclusteringscaleswithMPSarevery
nificances obtained then reflect the balance between the clus- small, and the MPS post-trial significance remains practically
tering of the signal at the scales of the search bin radius, and unchanged, from the pre-trial p-value of 10−5.5 to a post-trial
thescalesatwhichrandomaccumulationofbackgroundevents p-valueof10−5.3,yieldingalsoa4.3σfinalresult.
ismoreprobable. Adiscovery(i.e. ap-value< 2.8×10−7 )is Therefore,inthetypicalsituationsofneutrinotelescopesof
achievedwhensamplingtheregionwithbinsof1.5◦radius(fig- lowS/N,theonlyhintofapossibleneutrinosignalwouldbethe
ure2(c)),butthesignificancedropsconsiderablywhenobserv- accumulationofrelativelyweakspotswithintheregionwhich
ing at different angular scales, and any information regarding is believedto bea factoryofcosmic rays. Aslongas theS/N
thespatialstructureofthesignalwithintheregionunderexam- is low, these most significant spots do not necessarily have to
inationislost. However,thisinformationisrecoveredwhenwe appearat the same location in the analysisof the same region
studythespatialcorrelationsbetweeneventswithMPS.There- in different epochs, since they have an important background
sultsoftheanalysisoftheregionwithMPSareshowninfigure component,whichfluctuatesrandomlywithintheregion. The
3. Figure 3(a) shows the clustering function of the events up signal would be missed by standard scan analysis, but could
to5◦scales,andfigure3(b)showsthesignificancesobtainedat bedetectedbycorrelationanalysislikeMPS.Eventually,after
5
map EMMRREMMRRnnMMMMeeeettaaaaSSSSrrnnnnii eexyxy sxysxymm aapp 33223322 1104..04..4747665.5.63963988..4664466422 map EMMRREMMRRnnMMMMeeeettaaaaSSSSrrnnnnii eexyxy sxysxymm aapp 3333 2222110404....6616165.5.55881414..6666222121
40 4.5 40
39 39
4
5
38 38
3.5
37 37 4
dec (deg) 333456 223.5(p_value)og10 dec (deg) 333456 3 (p_value)og10
1.5-l 2 -l
33 33
32 1 32
1
31 0.5 31
30 30
300 301 302 303 304 305 306 307 308 309 310 300 301 302 303 304 305 306 307 308 309 310
R.A. (deg) R.A. (deg)
(a) Significancemapofeventdensitiesat0.5◦scales (b) Significancemapofeventdensitiesat1◦scales
map EMMRREMMRRnnMMMMeeeettaaaaSSSSrrnnnnii eexyxy sxysxymm aapp 3333 1212110404....6684845.5.66889595..7676221111 map EMMRREMMRRnnMMMMeeeettaaaaSSSSrrnnnnii eexyxy sxysxymm aapp 33123312 1104..04..5252665.5.77877888..8733873322
40 40
8
39 39 3
38 7 38
2.5
37 6 37
dec (deg) 333456 45 (p_value)og10 dec (deg) 333456 12.5(p_value)og10
3 -l -l
33 33 1
2
32 32
0.5
31 1 31
30 0 30 0
300 301 302 303 304 305 306 307 308 309 310 300 301 302 303 304 305 306 307 308 309 310
R.A. (deg) R.A. (deg)
(c) Significancemapofeventdensitiesat1.5◦scales (d) Significancemapofeventdensitiesat2◦scales
Figure2:Resultsfromthebinnedscan.Thecolorscaleisinunitsofstandarddeviation.
Ex2 100 signal events Ex2 100 signal events
2.2 10
8
2 e)
u
al
Θ) _v 6
Φ(1.8 (pg10
o
-l 4
1.6
2
1.4
0
0.5 1 1.5 2 2.5 3 3.5 4 4.5 5 0.5 1 1.5 2 2.5 3 3.5 4 4.5 5
Θ (deg) Θ (deg)
(a) Clusteringfunction (b) Significance oftheclustering function ateachoftheangularscales
tested.
Figure3:ResultsfromtheMPS
someyearsofintegrationtime,theS/Nwouldbehighenough background events, provided there is some clustering of neu-
to overcome the backgroundfluctuations, providingan image trinosourceswithintheregionunderexamination.
oftheneutrinoemissionintheregion.Butbeforethissituation
isachieved,weexpecttofindadistributionofneutrinoevents
whichdeparturessignificantlyfromtheaveragedistributionof
6
map EMMRREMMRRnnMMMMeeeettaaaaSSSSrrnnnnii eexyxy sxysxymm aapp 222233 11 ....44 66666633..21214488002424992255 map EMMRREMMRRnnMMMMeeeettaaaaSSSSrrnnnnii eexyxy sxysxymm aapp 3232 3322110.0.55..55664455..11884488..3232221111 map EMMRREMMRRnnMMMMeeeettaaaaSSSSrrnnnnii eexyxy sxysxymm aapp 332332 221105.05...6666665.5.00008822..7257252211
40 40 40
39 3 39 3 39 2
3378 2.5 3378 2.5 3378 11..68
dec (deg) 333456 12.5(p_value)-log10 dec (deg) 333456 12.5(p_value)-log10 dec (deg) 333456 0111...824(p_value)-log10
33 1 33 1 33 0.6
32 0.5 32 0.5 32 0.4
31 31 31 0.2
33000 301 302 303 304 305 306 307 308 309 310 33000 301 302 303 304 305 306 307 308 309 310 33000 301 302 303 304 305 306 307 308 309 310
R.A. (deg) R.A. (deg) R.A. (deg)
(a) Significance map of event densities at 1◦ (b) Significance map of event densities at 1◦ (c) Significance mapofeventdensities at1◦ in
scalesinsample1 scalesinsample2 sample3
Figure4:Resultsfromthebinnedscanoversignal+backgroundsamples
map EMMRREMMRRnnMMMMeeeettaaaaSSSSrrnnnnii eexyxy sxysxymm aapp 222233 11 ....00337676664444060688....4747882266 map EMMRREMMRRnnMMMMeeeettaaaaSSSSrrnnnnii eexyxy sxysxymm aapp 322322 11 5..5.. 65656633..54454488009869862255 map EMMRREMMRRnnMMMMeeeettaaaaSSSSrrnnnnii eexyxy sxysxymm aapp 3333 2222110505....6676765.5.33889191..5656221111
334890 122..82 334890 12.8 334890 11..46
1.6 1.2
37 1.6 37 37
dec (deg) 333456 0111...824(p_value)-log10 dec (deg) 333456 0111...824(p_value)-log10 dec (deg) 333456 001..68(p_value)-log10
33 0.6 33 0.6 33 0.4
32 0.4 32 0.4 32
31 0.2 31 0.2 31 0.2
33000 301 302 303 304 305 306 307 308 309 310 33000 301 302 303 304 305 306 307 308 309 310 33000 301 302 303 304 305 306 307 308 309 310
R.A. (deg) R.A. (deg) R.A. (deg)
(a) Significance map of event densities at 1◦ (b) Significance map of event densities at 1◦ (c) Significance mapofeventdensities at1◦ in
scalesinsample1 scalesinsample2 sample3
Figure5:Resultsfromthebinnedscanoverbackground-onlysamples
map EMMRREMMRRnnMMMMeeeettaaaaSSSSrrnnnnii eexyxy sxysxymm aapp 22223333 11....05057676665.5.19190088..8787262622 map EMMRREMMRRnnMMMMeeeettaaaaSSSSrrnnnnii eexyxy sxysxymm aapp 3232 3322110.0.55..44664455..44880066..3737221111 Combined Ex1+Ex2+Ex3
40 40 6
3.5
39 39 3
38 3 38 5
2.5
dec (deg) 3333334567 1122..55(p_value)-log10 dec (deg) 3333334567 112.5(p_value)-log10 (p_value)-log10 234
32 0.5 32 0.5 1
31 31
33000 301 302 303 304 305 306 307 308 309 310 33000 301 302 303 304 305 306 307 308 309 310 0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 5
R.A. (deg) R.A. (deg) Θ (deg)
(a) Significance map of event densities at 0.5◦ (b) Significance map of event densities at 1◦ (c) Significanceoftheclusteringfunctionateach
scalesinthecombinedsample scalesinthecombinedsample oftheangularscalestestedinthecombinedsam-
ple.
Figure6:ResultsfromthebinnedscanandfromMPSoverthecombinedsample
5. Discussion Galaxy. Massive andyoungstellar populations,the birth sites
ofGalacticcore-collapsesupernova,areassociatedtothespiral
Thediscoveryofhighenergyneutrinoemissionfromapar- armsoftheMilkyWay,followingahierarchicalschemewhere
ticularregionoftheGalaxywouldsuggestalocalinjection of
starsaregroupedinclusters,andthestellarclustersthemselves
high energy protons or heavier nuclei; as well as their inter-
formpartofopenclustercomplexes.
action not very far away from their sources of origin, before
Under the assumption that the explosion of a supernova
they lose theirenergyby diffusion in Galactic magnetic fields
is the most plausible energy source for cosmic ray accelera-
[50,51]. Whereasthediscoveryofindividualpointsourcesof
tionintheGalaxy,weproposeinthispapertosearchforneu-
highenergyneutrinosseemschallengingbecauseofthelimited
trino emission in the regions where massive stars die, that is,
sensitivityofneutrinodetectors,theagglomerationofneutrino
where the cosmic ray acceleration process takes place in the
sourcesinsidethesameregionoftheskycouldbedetectedasa supernovaremnantshocksthatformaroundtheexplosionsite.
significantdeparturefromthedistributionofatmospheric neu-
GiventhedistributionofmassivestarclustersintheGalaxy,we
trino events. Such clustering of neutrino sources would be a
can find such potential cosmic ray factories at angular scales
naturalconsequenceof the distribution of stellar matterin the whichcouldevenreacharound∼10◦ormore,dependingifwe
7
also want to consider clustering of potential accelerators due [9] Adriani,O.,Barbarino,G.C.,Bazilevskaya, G.A.,etal.2009,Nature,
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