Search for Matter-Dependent Atmospheric Neutrino Oscillations in Super-Kamiokande

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by  The Super-Kamiokande Collaboration

Search forMatter-Dependent Atmospheric Neutrino OscillationsinSuper-Kamiokande
The Super-Kamiokande Collaboration
K.Abe1,Y.Hayato1,T.Iida1,M.Ikeda24,J.Kameda1,Y.Koshio1,A.Minamino1,M.Miura1,S.Moriyama1,M.Nakahata1,
S.Nakayama2,Y.Obayashi1,H.Ogawa1,H.Sekiya1,M.Shiozawa1,Y.Suzuki1,A.Takeda1,Y.Takeuchi1,K.Ueshima1,
H.Watanabe1,S.Yamada1,I.Higuchi2,C.Ishihara2,T.Kajita2,K.Kaneyuki2,G.Mitsuka2,H.Nishino2,K.Okumura2,
C.Saji2,Y.Takenaga2,S.Clark3,S.Desai3,†,F.Dufour3,E.Kearns3,S.Likhoded3,M.Litos3,J.L.Raaf3,J.L.Stone3,
L.R.Sulak3,W.Wang3,M.Goldhaber4,D.Casper5,J.P.Cravens5,J.Dunmore5,W.R.Kropp5,D.W.Liu5,S.Mine5,
C.Regis5,M.B.Smy5,H.W.Sobel5,M.R.Vagins5,K.S.Ganezer6,B.Hartfield6,J.Hill6,W.E.Keig6,J.S.Jang7,
I.S.Jeong7,J.Y.Kim7,I.T.Lim7,K.Scholberg8,M.Fechner8,N.Tanimoto8,C.W.Walter8,R.Wendell8,S.Tasaka10,
G.Guillian11,J.G.Learned11,S.Matsuno11,M.D.Messier12,T.Hasegawa13,T.Ishida13,T.Ishii13,T.Kobayashi13,
T.Nakadaira13,K.Nakamura13,K.Nishikawa13,Y.Oyama13,Y.Totsuka13,‡,A.T.Suzuki14,T.Nakaya15,H.Tanaka15,
8 M.Yokoyama15,T.J.Haines16,5,S.Dazeley17,R.Svoboda17,A.Habig19,Y.Fukuda20,T.Sato20,Y.Itow21,T.Koike21,
0
T.Tanaka21,C.K.Jung22,T.Kato22,K.Kobayashi22,C.McGrew22,A.Sarrat22,R.Terri22,C.Yanagisawa22,N.Tamura23,
0
2 Y.Idehara24,M.Sakuda24,M.Sugihara24,Y.Kuno25,M.Yoshida25,S.B.Kim26,B.S.Yang26,T.Ishizuka27,H.Okazawa28,
Y.Choi29,H.K.Seo29,Y.Gando30,K.Inoue30,Y.Furuse31,H.Ishii31,K.Nishijima31,Y.Watanabe32,M.Koshiba33,
n
a S.Chen34, Z.Deng34,Y.Liu34,D.Kielczewska35,5,H.Berns36,K.K.Shiraishi36,E.Thrane36,R.J.Wilkes36
J
1 KamiokaObservatory,InstituteforCosmicRayResearch,UniversityofTokyo,Kamioka,Gifu,506-1205,Japan

5 2ResearchCenterforCosmicNeutrinos,InstituteforCosmicRayResearch,UniversityofTokyo,Kashiwa,Chiba277-8582,Japan
3 Department of Physics, Boston University, Boston, MA 02215, USA

] 4 Physics Department, Brookhaven National Laboratory, Upton, NY 11973, USA
x 5 DepartmentofPhysicsandAstronomy, UniversityofCalifornia, Irvine, Irvine, CA92697-4575, USA
e 6 Department ofPhysics, California StateUniversity, DominguezHills, Carson, CA90747, USA
- 7 Department ofPhysics, Chonnam National University, Kwangju 500-757, Korea
p
8 Department of Physics, Duke University, Durham, NC 27708, USA
e
9 Department ofPhysics, George MasonUniversity, Fairfax, VA22030, USA
h
10 Department of Physics, Gifu University, Gifu, Gifu 501-1193, Japan
[
11 Department ofPhysicsandAstronomy, University ofHawaii, Honolulu, HI96822, USA
1 12 Department ofPhysics, Indiana University, Bloomington, IN47405-7105, USA
v 13 HighEnergyAccelerator ResearchOrganization(KEK),Tsukuba, Ibaraki305-0801, Japan
6 14 Department of Physics, Kobe University, Kobe, Hyogo 657-8501, Japan
15 Department of Physics, Kyoto University, Kyoto 606-8502, Japan
7
16 Physics Division, P-23,LosAlamosNational Laboratory, LosAlamos, NM87544, USA
7
17 DepartmentofPhysicsandAstronomy,LouisianaStateUniversity, BatonRouge, LA70803, USA
0
18 Department ofPhysics, University ofMaryland, College Park, MD 20742, USA
.
1 19 Department ofPhysics, University ofMinnesota, Duluth, MN55812-2496, USA
0 20 Department ofPhysics, Miyagi University ofEducation, Sendai, Miyagi 980-0845, Japan
8 21 SolarTerrestrialEnvironmentLaboratory, NagoyaUniversity, Nagoya, Aichi464-8602, Japan
0 22 DepartmentofPhysicsandAstronomy,StateUniversityofNewYork,StonyBrook,NY11794-3800,USA
: 23 Department of Physics, Niigata University, Niigata, Niigata 950-2181, Japan
v 24 Department ofPhysics, Okayama University, Okayama, Okayama 700-8530, Japan
i 25 Department ofPhysics, OsakaUniversity, Toyonaka, Osaka560-0043, Japan
X
26 Department ofPhysics, Seoul National University, Seoul 151-742, Korea
r 27 DepartmentofSystemsEngineering, ShizuokaUniversity, Hamamatsu,Shizuoka432-8561, Japan
a 28 DepartmentofInformaticsinSocialWelfare,ShizuokaUniversityofWelfare,Yaizu,Shizuoka,425-8611,Japan
29 Department ofPhysics, Sungkyunkwan University, Suwon 440-746, Korea
30 ResearchCenter forNeutrinoScience, TohokuUniversity, Sendai, Miyagi980-8578, Japan
31 Department ofPhysics, Tokai University, Hiratsuka, Kanagawa 259-1292, Japan
32 Department ofPhysics, TokyoInstitute forTechnology, Meguro, Tokyo152-8551, Japan
33 The University of Tokyo, Tokyo 113-0033, Japan
34 Department ofEngineering Physics, Tsinghua University, Beijing, 100084, China
35 Institute ofExperimental Physics, WarsawUniversity, 00-681 Warsaw, Poland
36 Department ofPhysics, University ofWashington, Seattle, WA98195-1560, USA
†Presentaddress: CenterforGravitationalWavePhysics,PennsylvaniaStateUniversity,UniversityPark,PA16802,USA
(Dated:February2,2008)
Weconsider n µ→n t oscillationsinthecontextoftheMassVaryingNeutrino(MaVaN)model, wherethe
neutrino mass can vary depending on the electron density along the flight path of the neutrino. Our analy-
sisassumes amechanism withdependence only upon theelectron density, hence ordinary matter density, of
themediumthroughwhichtheneutrinotravels. Fully-contained,partially-containedandupward-goingmuon
atmospheric neutrino data from the Super–Kamiokande detector, taken from the entire SK–I period of 1489
livedays, arecompared toMaVaNmodel predictions. Wefindthat, forthecaseof 2-flavoroscillations, and
forthespecificmodelstested,oscillationindependent ofelectrondensityisfavoredoverdensitydependence.
Assumingmaximalmixing,thebest-fitcaseandthedensity-independentcasedonotdiffersignificantly.
2
PACSnumbers:PACSnumbers:14.60.Pq,14.60.St,96.50.Sf
Keywords:neutrinooscillations,Super-Kamiokande,mavans,atmosphericneutrinos
I. INTRODUCTION the detector had an inner detector(ID) equippedwith 11146
50cm PMTs aimed inward, and an outer detector (OD) vol-
umeinstrumentedwith188520cmPMTs,aimedoutwardand
Neutrinooscillationsresultwhenatleast oneneutrinohas
equipped with wavelength-shifting plastic plates. The OD
massdifferentfromtheothers, so thatneutrinosmasseigen-
functionsprimarily as a veto counter, tagging chargedparti-
statesaredistinctfromtheflavoreigenstates.Foratmospheric
neutrinos, n µ →n t oscillations [1] are strongly favored over clesthatenterorexittheID.Within theID we definea cen-
n →n oscillations[2]andotherexoticmechanismsforneu- tral 22.5 kiloton fiducial volume, within which detector re-
µ s
sponseisexpectedtobeuniform. Fully-contained(FC)neu-
trino disappearance [3]. Neutrino oscillation without the in-
trinoeventsarethosewhereinteractionproductsareobserved
troductionofsterileflavorshasalsobeensufficienttoresolve
in the ID, with no significant correlated activity in the OD,
thesolarneutrinoproblem[4,5].
while partially-contained (PC) events are those where some
Super-Kamiokandepreviouslyreportedbestfitparameters
for atmospheric two-flavor n µ → n t oscillations, providing interaction productsexit the ID. Upward-goingmuon events
arethosewhereapenetratingparticletravellingintheupward
an explanation of the atmospheric neutrino anomaly [6]. In
directionentersandeitherstopsorpassesthroughthedetector,
Super-Kamiokande,where previousanalyses always consid-
andareattributedtomuonsproducedbyneutrinointeractions
eredonlygeometricpathlengthindependentofmedium,neu-
in the surrounding rock. In general terms, FC, PC, and up-
trino disappearanceeffects are strikingly evidentfor upward
wardmuoneventsrepresentsuccessivelyhigherenergysam-
going neutrinos, which have passed through many km of
ples of neutrino interactions, ranging from 200 MeV for the
rock, in comparison with downward-goingneutrinos, which
lowestenergyFCeventstoabove1TeVforthehighestenergy
travel mainly through air. We note that other experiments
upward-going muons. Further details regarding the Super-
thatobservedneutrinooscillations, such as KAMLAND [7],
Kamiokandedetectordesign,operation,calibrations,anddata
K2K[8],andMINOS[9],detectedneutrinoswhosepathwas
reductioncanbefoundin[6,12].
almostentirelythroughrock.
Super-Kamiokande-I(SK-I)datatakingforphysicsanaly-
HereweconsiderapossibleconsequenceoftheMassVary-
sis began on May 17, 1996 and lasted until a planned shut-
ingNeutrino(MaVaN)model[10].Inthismodel,theneutrino
down for refurbishingon July 16, 2001, including a total of
masscanvarydependingonthematterdensityalongthepath
1489.2daysofeffectivelivetime(92kiloton-years)forFCand
oftheneutrino. MaVaNscouldprovideasourceforthedark
PC events, and 1645.9 days for upward-going muon events.
energy[11]. Here we will assume thatthe massvariationof
ThefullMonteCarlo(MC)eventsamplegeneratedforcom-
the neutrinos depends only upon the electron density of the
parison is equivalent to a 100 live-year period. The SK-I
environment, a possible side effect from radiative couplings
databaseusedintheanalysespresentedherehasthefollowing
of active neutrinos and electrons [10]. (In ordinary matter
statistics:forFCevents,12180dataand13676.7MCeventsin
comprisingtheEarth’satmosphereandinterior,overallmatter
alivetime-scaledsample,forPCevents,911dataand1129.6
density and baryondensity, consideredin otherMaVaN sce-
MCevents. Theupward-goingmuonstatistics are: forstop-
narios,areessentiallyproportionaltoelectrondensity.)Ifthis
pingmuontracks(thosethatenterbutdonotexitthedetector),
hypothesisis accurate, it may be possible to probe the mass
417.7dataand713.5MCevents,andforthrough-goingmuon
dependencewithSuper–Kamiokandedata.
tracks, 1841.6data and1669.5MC events, afterbackground
We used atmospheric neutrino data from the Super-
subtractionfornear-horizontaldownward-goingmuonsasde-
Kamiokande-I running period (1996–2001) to test whether
scribedinref.[6].
n µ → n t oscillations have an apparent dependence on the
Twodifferentanalysesareconsideredhere. Thefirstanal-
electron density of the material the neutrino passes through.
ysis does not take into account path length in air for neu-
For this analysis, we assumed that the mass squared differ-
trinooscillations.Instead,itassumesMaVan-typeoscillations
ence is proportional to some power of the electron density,
D m2 ∼r n . Wealsoassumedthatthemixingangleiscon- onlyoccurinhigh-densitymatter,takingintoaccountsurface,
eff crust, mantle, and Earth core densities. In other words, the
stantforallmedia.
neutrino flight path length is in effect taken to be only that
portion of the geometric path that lies in high density mat-
ter. Because the path length systematic error is not relevant
II. DATAANALYSIS inthistest,itisremovedfromthelistof39systematicerrors
that was used in previousSuper-Kamiokandeanalyses. The
Super-Kamiokande is a water Cherenkov experiment lo- secondanalysisalsotakesintoaccountpathlengthinairand
cated within Mt. Ikeno-yama in central Japan, under 2700 its density in considering neutrino oscillations effects in the
meters water equivalent rock overburden. It has a cylindri- contextoftheMaVaNmodel.
cal design, holds 50 kilotons of water, and is divided into A detailed description of the SK-I atmospheric neutrino
two optically separated sections by a structural framework chi-squaredzenithangleanalysisusingthe“pull”method[13]
that supports the photomultiplier tubes (PMTs). During the canbefoundin[6],andthesamemethodsareappliedhere.A
SK-Irunningperiodfromwhichthedataanalyzedherecame, profile of the mountain surroundingSuper-Kamiokandewas
3
takenfromtopologicalmapstodeterminethedownwardneu- forthecaseofmatter-dependentoscillationsassumingmaxi-
trinopathlengthinrock.ThemapsusedaretheUnitedStates mal mixing (sin2(2q )=1) in all densities (exceptair). This
GeologicalSurveyagency’sDigitalElevationMaps[14],with analysisusesboth n andD m2 as the variedparameters. D m2
datapointsat7.5minute(about30meters)spacing. valuesfrom10−4to100eV2wereusedover81bins,andden-
Intheconventional2-flavorn µ→n t oscillationframework, sitydependencepowersfromn=−3to+3wereconsidered,
theoscillationprobabilitycanbewrittenas over 61 bins. The results produce a c 2 = 187.3/178 d.o.f.
min
for(n,D m2)=(−0.30,3.16×10−3eV2). Thus,themodelin
P(n µ→n t )=sin22q sin2(cid:18)1.27ED nm2LkmG·eeVV2(cid:19), iwnhriocchka,iirspdaitshfalevnogrethdiastnaebgoluectttehde,3i..e5.,soslecvilelaltrieolnastiovcectuorsotannly-
dardoscillations.
whereListhedistancetravelledbytheneutrinobetweenpro-
ductionanddetection,andEn istheneutrinoenergy,whilethe
oscillation parameters are the mixing angle q , and the mass
B. AnalysisIncludingAirpathlength
differencesquared, D m2 =|m2−m2| in the usualnomencla-
2 3
ture. To test for density dependence, we replaced D m2 in
Werepeatedtheanalysis,thistimetakingintoaccountthe
the equation above with an effective mass difference that is
proportional to the electron density of the medium, D m2 → pathlengthinair. Nowthen=0case, wherethereisnode-
D m2× r e n, where r is the electron density of the matter pendenceontheelectrondensity,correspondstotheconven-
r e tionaloscillationsmodel,sincethefullgeometricpathlength
(cid:16) o(cid:17)
inneutrinotrajectory,andnparameterizesthedensitydepen- isused.
dence. For neutrinos passing through layers of matter with Theairmassdensitydependsgreatlyonthealtitudeofthe
different density, the path length in each layer is taken into air above sea level. For simplicity, a constantvalue of 10−3
account. r 0 issetat6.02×1023e/cm3. Severaldifferentval- g/cm3 (corresponding to actual air density at mountain alti-
uesofnaretestedhere,asdescribedbelow. Toapproximate tude)wasusedforthemassdensity.
themassdensityoftheEarthforupwardtravelingneutrinos, The same set of fixed n values are tested again, to check
thePreliminaryEarthReferenceModel(PREM)[15]isused. variousdensitydependences,thistime includingtheair path
Theelectrondensityistakentobethemassdensitymultiplied length. The results are listed in Table II. All models
bythecharge-to-massratio[16]. testedareexcludedby≥3.8s whencomparedtothedensity-
Atmospheric neutrinos are produced at altitudes around independentcase.
15 km above sea level. A neutrino travelling nearly verti- Finally, we again treat density dependencen as a free pa-
callydownwardthuspassesthroughabout15kmofair(low- rameter, this time taking air path length into account, once
density matter) followed by 1-2 km of rock (high density more assuming maximal mixing. The results are shown in
matter)beforeinteractingwithinthedetector,whileanearly- Figure 1. A minimum c 2 = 174.3/178 d.o.f. is found at
horizontaldownward-goingneutrinotravelsthroughover200 (n,D m2)=(−0.04,1.95×10−3eV2). Thisresultandthestan-
km of air and about10 km of rock. Thus, a downwardneu- dard2-flavoroscillationresult(i.e. withn≡0)do notdiffer
trino has 8-20 times as much path length in air as in rock. significantly.
Ifoscillationonlyoccursinhighdensitymatter,theeffective
pathlengthwillbeshorterthanthegeometricpathlengthtrav-
elledbytheneutrinofrompointofproductiontodetection.In
III. SUMMARYANDCONCLUSIONS
thatcase, the standardoscillation model, which uses the full
geometricpathlength,assumesmoreoscillationcyclesoccur
for a given D m2, and oscillation effects are more significant We used data from Super-Kamiokande-I to test mass-
varying neutrino models, by considering oscillation effects
for neutrinos at low energies. Analyses both including, and
where the effective mass of a neutrino is dependent on the
neglecting,theairpathlengthareconsideredhere.
matter environment of the neutrino flight path. Our analy-
sis assumed varying neutrino mass was due entirely to the
electron density of the material in a neutrino’s path, a pos-
A. AnalysisNeglectingAirpathlength
sibleconsequenceoftheMaVaNtheory.Assumingaconstant
mixing angle in all environments, we substituted D m2 with
gleDcteinnsgittyhedeapiernpdaetnhtloesncgiltlhataionndmusoidneglsspweecriefiefidrsvtatleusetesdonfen- D m2×(cid:16)rr oe(cid:17)n (wherer o =6.02×1023e/cm3) for specific val-
suggestedbytheorists[17]. TheresultsareshowninTableI ues of n suggested by theorists, as well as with n as a fitted
for two cases: fits constrained to lie in the physical region, parameter. Twodifferentmodelswereconsidered. Inthefirst
wheremaximalmixingisassumedforalldensities,andwith analysis, we neglectedthe portionof the neutrinoflightpath
thisconstraintrelaxed,wheremixingangleisallowedtovary. inlowdensitymatter(air),andassumedmass-varyingeffects
All models tested are ruled out, with confidence level better occuronlyinhighdensitymatter(rock).Forthesecondanal-
than 99.9%, whencomparedto fits assumingoscillationsin- ysis,weallowedoscillationeffectsinallportionsoftheneu-
dependentofdensityofthemedium(includingair). trinopath,but(incontrasttotheconventionaloscillationanal-
Next,wetreatthedensitydependenceasafreeparameter, ysis,whereonlythegeometricpathlengthisconsidered)took
and find the best fit electron density dependenceexponentn intoaccountmassdensities.Theprimarydifferenceinthetwo
4
TABLEI:Comparisonsofc 2Valuesfordifferentmodelswithoutairpath.Thereareatotalof178degreesoffreedominthec 2(seeref.[6]).
Notethatn=0isnotequivalenttooscillationsindependentofmedium,becauseherewedonotallowforanyoscillationsinair.
Model PhysicalRegion UnphysicalRegion
D m2×(r /r o)n c 2min D m2(eV2) sin2(2q ) Ds c 2min D m2(eV2) sin2(2q )
n=2 203.4 4.84×10−4 1.00 5.3 203.1 4.67×10−4 1.02
n=1 194.7 1.12×10−3 1.00 4.4 194.1 1.19×10−3 1.04
n= 2 192.5 1.46×10−3 1.00 4.2 191.7 1.46×10−3 1.04
3
n= 1 190.5 1.60×10−3 1.00 3.9 189.5 1.60×10−3 1.04
3
n=0 189.8 2.66×10−3 1.00 3.8 188.9 2.66×10−3 1.04
n=−1 188.5 3.16×10−3 1.00 3.7 187.5 2.92×10−3 1.04
3
n=−2 187.9 3.12×10−3 1.00 3.6 187.0 3.12×10−3 1.04
3
n=−1 188.5 3.80×10−3 1.00 3.7 187.6 3.80×10−3 1.04
n=−2 190.1 4.73×10−3 1.00 3.9 188.7 4.73×10−3 1.05
TABLEII:Comparisonsofc 2ValuesforDifferentModelsincludingairpath. Notethathere,n=0isequivalenttooscillationsindependent
ofmedium.
Model PhysicalRegion UnphysicalRegion
D m2×(r /r o)n c 2min D m2(eV2) sin2(2q ) Ds c 2min D m2(eV2) sin2(2q )
Equalmediumoscillations(n=0) 175.0 2.11×10−3 1.00 - 174.7 2.11×10−3 1.02
n=2 203.4 4.84×10−4 1.02 5.3 203.1 4.67×10−4 1.00
n=1 194.7 1.12×10−3 1.00 4.4 194.1 1.12×10−3 1.04
n= 2 192.4 1.46×10−3 1.00 4.2 191.7 1.46×10−3 1.04
3
n= 1 189.6 1.60×10−3 1.00 3.8 188.7 1.60×10−3 1.04
3
n=−1 228.5 7.50×10−4 0.91 7.9 228.5 7.50×10−4 0.91
3
n=−2 392.1 1.50×10−4 0.63 14.7 392.1 1.50×10−4 0.70
3
n=−1 447.8 3.63×10−2 0.60 16.5 447.8 3.63×10−2 0.60
n=−2 447.9 3.73×10−3 0.60 16.5 447.9 3.73×10−3 0.60
hypothesisliesintheeffectivepathlengthsfornear-horizontal squaredvalueofc 2 =174.3/178d.o.f.at(n=−0.04,D m2=
min
downward-goingneutrinos. 1.95×10−3eV2),consistent(within1.4s )withthen=0(no
densitydependence)case.
Inthe“pull”fittingprocedureused,someofthesystematic
The results of our analyses show no evidence that the en-
errorsare allowedto varyin the fit. Whenfitted systematics
vironmentalelectrondensityinfluencestheeffectiveD m2 de-
are forcedsignificantlyoutsidea reasonablerangeof values,
terminedusingSuper-Kamiokandeatmosphericneutrinodata.
it is an indication that the model used is disfavored by the
We have not explicitly considered models where the mixing
data. For the high-density-only oscillation model, some of
the systematics, particularly the n /n ratio for energies be- angleisnotconstantinalldensities,orwhichassume3flavor
µ e
oscillations,nordoweexcludeallvariationsofMaVaNmod-
low5GeV, adjustedtovaluesbeyondareasonabledeviation
from the expected value. This does not happen when fitting els. We find that conventionaldensity independentn µ →n t
oscillationsaresufficienttoexplaintheatmosphericneutrino
to the conventionaloscillation model, and results in a larger
data.
chi-squaredvalue for the mass-dependentmodels. Based on
relativechi-squaredvaluesin ourresults, the hypothesisthat
neutrinos oscillate only in high density matter is disfavored
relativetoconventionaloscillationsbyatleast3.5s ,without IV. ACKNOWLEDGEMENTS
requiringadditionaleffects.
For the second analysis, where oscillations in low density WegratefullyacknowledgethecooperationoftheKamioka
matter were included, a mass difference with electron den- MiningandSmeltingCompany. TheSuper-Kamiokandeex-
sity dependence is disfavored at more than the 3.8-s level perimentwas builtandhasbeenoperatedwith fundingfrom
when compared with neutrino oscillations with a fixed mass theJapaneseMinistryofEducation,Science,SportsandCul-
difference, for all fixed values of n tested. In addition, a ture,andtheUnitedStatesDepartmentofEnergy. We grate-
freely varying density dependence analysis was performed, fully acknowledge individual support by the National Sci-
assuming maximal mixing. It produced a minimum chi- enceFoundation,andthePolishCommitteeforScientificRe-
5
search.
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6
)
2
V
e
(

2
m
D
99% C.L.
-3
90% C.L.
10
68% C.L.
-0.5-0.4-0.3-0.2-0.1 0 0.1 0.2 0.3 0.4 0.5
n
20
18 99% C.L.
16 90% C.L.
68% C.L.
14
min 12
2
2c - 10
8
c
6
4
2
0
-3
10
D m2 (eV2)
20
18 99% C.L.
16 90% C.L.
68% C.L.
14
min 12
2
2c - 10
8
c
6
4
2
0
-0.5-0.4-0.3-0.2-0.1 0 0.1 0.2 0.3 0.4 0.5
n
FIG. 1: D m2 →D m2× rr e n (including air path length). Upper
(cid:16) o(cid:17)
plotshowsrelative-c 2confidencelevelcontoursontheD m2versusn
plane,obtainedwhentakingintoaccountbothhighandlowdensity
matterpathlengths. Thelowerplotsdisplaythec 2−c 2 contours,
min
withconfidencelevelsshown,atthebest-fitparametervalues.