collective supernova neutrino oscillations georg raffelt, mpi physik, munich, germany

Georg Raffelt, Max-Planck-Institut für Physik, München, Germany JIGSAW 07, 12-23 Feb 2007, TIFR, Mumbai, India

Collective Supernova Neutrino OscillationsCollective Supernova Neutrino OscillationsGeorgGeorg Raffelt,Raffelt, MPIMPI Physik,Physik, Munich,Munich, GermanyGermany

JIGSAW 07, 12JIGSAW 07, 1223 Feb 2007, TIFR, Mumbai, India23 Feb 2007, TIFR, Mumbai, India


Sanduleak Sanduleak 69 69 202202

Large Magellanic Cloud Large Magellanic Cloud Distance 50 kpcDistance 50 kpc (160.000 light years)(160.000 light years)

Tarantula NebulaTarantula Nebula

Supernova 1987ASupernova 1987A 23 February 198723 February 1987



Neutrino Signal of Supernova 1987ANeutrino Signal of Supernova 1987A

Within clock uncertainties,Within clock uncertainties,signals are contemporaneoussignals are contemporaneous

Kamiokande-II (Japan)Kamiokande-II (Japan)Water Cherenkov detectorWater Cherenkov detector2140 tons2140 tonsClock uncertainty Clock uncertainty 1 min1 min

Irvine-Michigan-Brookhaven (US)Irvine-Michigan-Brookhaven (US)Water Cherenkov detectorWater Cherenkov detector6800 tons6800 tonsClock uncertainty Clock uncertainty 50 ms50 ms

Baksan Scintillator TelescopeBaksan Scintillator Telescope(Soviet Union), 200 tons(Soviet Union), 200 tonsRandom event cluster ~ 0.7/dayRandom event cluster ~ 0.7/dayClock uncertainty +2/-54 sClock uncertainty +2/-54 s


Flavor-Dependent Fluxes and SpectraFlavor-Dependent Fluxes and Spectra

Broad characteristicsBroad characteristics• Duration a few secondsDuration a few seconds

• EE ~ ~ 101020 MeV20 MeV

• EE increases with timeincreases with time

• Hierarchy of energiesHierarchy of energies

• Approximate equipartitionApproximate equipartition

of energy between flavorsof energy between flavors

xeeEEE xeeEEE

Livermore numerical modelLivermore numerical modelApJ 496 (1998) 216ApJ 496 (1998) 216

Prompt Prompt ee

deleptonizationdeleptonizationburstburst

ee

ee

xx__ However, in traditionalHowever, in traditional

simulations transport simulations transport

of of and and schematic schematic

• Incomplete microphysicsIncomplete microphysics

• Crude numerics to coupleCrude numerics to couple neutrino transport withneutrino transport with hydro codehydro code


Flavor-Dependent Neutrino Fluxes vs. Equation Flavor-Dependent Neutrino Fluxes vs. Equation of Stateof State

Kitaura, Janka & Hillebrandt, “Explosions of O-Ne-Mg cores, the CrabKitaura, Janka & Hillebrandt, “Explosions of O-Ne-Mg cores, the Crabsupernova, and subluminous Type II-P supernovae”, astro-ph/0512065supernova, and subluminous Type II-P supernovae”, astro-ph/0512065

Wolff & Hillebrandt nuclear EoS (stiff)

ee

,, ,

Lattimer & Swesty nuclear EoS (soft)

ee

,, ,


Level-Crossing Diagram in a SN EnvelopeLevel-Crossing Diagram in a SN Envelope

Dighe & Smirnov, Identifying the neutrino mass spectrum from a supernovaDighe & Smirnov, Identifying the neutrino mass spectrum from a supernovaneutrino burst, astro-ph/9907423neutrino burst, astro-ph/9907423

Normal mass hierarchyNormal mass hierarchy Inverted mass hierarchyInverted mass hierarchy


Self-Induced Flavor Oscillations of SN Self-Induced Flavor Oscillations of SN NeutrinosNeutrinos

Survival probability Survival probability eeSurvival probability Survival probability ee

NormalNormalHierarchyHierarchy

atm atm mm22

1313closeclose

to Choozto Choozlimitlimit

InvertedInvertedHierarchyHierarchy

NoNonu-nu effectnu-nu effect

NoNonu-nu effectnu-nu effect

MSWMSWeffecteffect

MSWMSWeffecteffect

RealisticRealisticnu-nu effectnu-nu effect

BipolarBipolarcollectivecollectiveoscillationsoscillations(single-angle(single-angle approximation)approximation)

MSWMSW

RealisticRealisticnu-nu effectnu-nu effect

MSWMSWeffecteffect


Matrices of Density in Flavor SpaceMatrices of Density in Flavor Space

Neutrino quantum fieldNeutrino quantum field

Spinors in flavor spaceSpinors in flavor space

Quantum states (amplitudes)Quantum states (amplitudes)

Variables for discussing neutrino flavor oscillationsVariables for discussing neutrino flavor oscillations

““Matrices of densities” Matrices of densities” (analogous to occupation numbers)(analogous to occupation numbers)

““Quadratic” quantities, required forQuadratic” quantities, required fordealing with decoherence, collisions,dealing with decoherence, collisions,Pauli-blocking, nu-nu-refraction, etc.Pauli-blocking, nu-nu-refraction, etc.

Sufficient for “beam experiments,”Sufficient for “beam experiments,”but confusing “wave packet debates”but confusing “wave packet debates”for quantifying decoherence effectsfor quantifying decoherence effects

xpi

p†

p3

3evp,tbup,ta

2

pd)x,t(

xpi

p†

p3

3evp,tbup,ta

2

pd)x,t(

3

2

1

3

2

1

3

2

1

a

a

a

a

3

2

1

a

a

a

a

3

2

1

b

b

b

b

3

2

1

b

b

b

bDestructionDestructionoperators foroperators for(anti)neutrinos(anti)neutrinos

0

a

p,ta

p,ta

p,t

p,t

p,t†

p,t3

2

1

3

2

1

0

a

p,ta

p,ta

p,t

p,t

p,t†

p,t3

2

1

3

2

1

NeutrinosNeutrinos

Anti-Anti-neutrinosneutrinos

p,tap,tap,t i†jij

p,tap,tap,t i

†jij

p,tbp,tbp,t j†iij

p,tbp,tbp,t j

†iij


General Equations of MotionGeneral Equations of Motion

pqqqp3

3

FpFp

2

pt ),)(cos1(2

qdG2],L[G2,

p2M

i

pqqqp3

3

FpFp

2

pt ),)(cos1(2

qdG2],L[G2,

p2M

i

pqqqp3

3

FpFp

2

pt ),)(cos1(2

qdG2],L[G2,

p2M

i

pqqqp3

3

FpFp

2

pt ),)(cos1(2

qdG2],L[G2,

p2M

i

Usual matter effect withUsual matter effect with

nn00

0nn0

00nn

Lee

nn00

0nn0

00nn

Lee

• Vacuum oscillationsVacuum oscillations M is neutrino mass M is neutrino mass matrixmatrix

• Note opposite sign Note opposite sign betweenbetween neutrinos and neutrinos and antineutrinosantineutrinosNonlinear nu-nu effects are importantNonlinear nu-nu effects are importantwhen nu-nu interaction energy exceedswhen nu-nu interaction energy exceedstypical vacuum oscillation frequencytypical vacuum oscillation frequency(Do not compare with matter effect!)(Do not compare with matter effect!)

cos1nG2E2

mF

2

osc

cos1nG2E2

mF

2

osc


Two-Flavor Neutrino Oscillations in VacuumTwo-Flavor Neutrino Oscillations in Vacuum

““Magnetic field”Magnetic field”in flavor spacein flavor space 2

Bp2

mm

p4

mmp

21

22

21

220

p

2B

p2

mm

p4

mmp

21

22

21

220

p

2cos

0

2sin

B

2cos

0

2sin

B

PolarizationPolarization vectorvector 2

Pf ppp

2

Pf ppp

2

P1f ppp

2

P1f ppp

or differentor differentnormalizationnormalization

ii Pauli Pauli

matricesmatrices

Neutrino flavor oscillation as a spin precessionNeutrino flavor oscillation as a spin precession

NeutrinosNeutrinos

p

2

pt PBp2

mP

p

2

pt PBp2

mP

SpinSpin 1/21/2

MagneticMagneticmomentmoment

++mm22/2p/2p

Anti-neutrinosAnti-neutrinos

p

2

pt PBp2

mP

p

2

pt PBp2

mP

SpinSpin 1/21/2

MagneticMagneticmomentmoment

--mm22/2p/2p

BB

pPpP

22

xxyy

zz


Synchronizing Oscillations by Neutrino Synchronizing Oscillations by Neutrino InteractionsInteractions

Vacuum oscillation frequency Vacuum oscillation frequency of mode with momentum p ~ Eof mode with momentum p ~ E p2

m2

osc

p2

m2

osc

Modified in a medium by theModified in a medium by theusual weak-interaction potentialusual weak-interaction potential

In an ensemble withIn an ensemble witha broad momentuma broad momentumdistribution, thedistribution, thep-dependent oscillationp-dependent oscillationfrequency quickly leadsfrequency quickly leadsto kinematicalto kinematicalflavor decoherenceflavor decoherence

In a dense neutrino gas,In a dense neutrino gas,all modes go with theall modes go with thesame frequency:same frequency:““SynchronizedSynchronized flavor oscillations” orflavor oscillations” or““self-maintainedself-maintained coherence”coherence”


Bipolar Oscillations of Neutrinos in a BoxBipolar Oscillations of Neutrinos in a Box

Dense gas of and Dense gas of and

Take equal densities withTake equal densities with

ee ee

eenG2nG2 FF eenG2nG2 FF

Assume only one energy withAssume only one energy with

1012 E2m 1012 E2m

Small mixing angle, inverted hierarchySmall mixing angle, inverted hierarchy

01.0 01.0

Survival probability of both and Survival probability of both and ee ee

)1P()(prob z21

ee )1P()(prob z21

ee

• Time scale set byTime scale set by

• “ “Plateau phases” mean exponentialPlateau phases” mean exponential growth, increase by growth, increase by log(log())

• Reduced mixing angle by ordinaryReduced mixing angle by ordinary matter unimportantmatter unimportant

2 2 Note:Note:This is no real “flavor conversion”,This is no real “flavor conversion”,Rather a “collective pair conversion” Rather a “collective pair conversion”

ee ee

Time scale Time scale proportional to proportional to FGFG


Synchronized vs. Bipolar OscillationsSynchronized vs. Bipolar Oscillations

Asymmetric system, Asymmetric system, initially consisting of initially consisting of unequalunequal numbersnumbers with equal energies (equal oscillation frequency with equal energies (equal oscillation frequency ) and ) and

eenn eenn

enG2 F enG2 F

PP

PP

BB

Free oscillationsFree oscillations

≪≪

PP

PP

)1( )1(

BB

Bipolar oscillationsBipolar oscillations

2)1(

1

2)1(

1≪≪ ≪≪

PP

PP

11

synch

11

synch

BB

Synchronized oscillationsSynchronized oscillations

2)1(

1

2)1(

1≪≪



PP

PP

BB


≪≪

PP

PP

)1( )1(

BB


2)1(

1

2)1(

1≪≪ ≪≪

PP

PP

11

synch

11

synch

BB


2)1(

1

2)1(

1≪≪

Energy of the system: Energy of the system: kinpot2

12

1EE)PP()PP(BE

kinpot2

12

1EE)PP()PP(BE

EEpotpot always dominates always dominatesEEkinkin always dominates always dominates



PP

PP

BB


≪≪

PP

PP

)1( )1(

BB


2)1(

1

2)1(

1≪≪ ≪≪

PP

PP

11

synch

11

synch

BB


2)1(

1

2)1(

1≪≪

SupernovaSupernovaCoreCore R = 40R = 4060 km60 km R R 200 km 200 km


Equations of Motion for Two-Flavor CaseEquations of Motion for Two-Flavor Case

pqqqp3

3

FpeFp

2

pt P)PP)(cos1(2

qdG2PznG2PB

p2m

P

pqqqp3

3

FpeFp

2

pt P)PP)(cos1(2

qdG2PznG2PB

p2m

P

pqqqp3

3

FpeFp

2

t P)PP)(cos1(2

qdG2PznG2PB

p2m

P

pqqqp3

3

FpeFp

2

t P)PP)(cos1(2

qdG2PznG2PB

p2m

P

Isotropic neutrinos,Isotropic neutrinos,ignore matter, onlyignore matter, onlyone neutrino energy one neutrino energy

p2m2

p2

m2P)PP(PBPt

P)PP(PBPt

P)PP(PBPt

P)PP(PBPt

nG2 F nG2 F

SDDBSt

SDDBSt

SumSum

Diff.Diff. SBDt

SBDt

Neglect first term for Neglect first term for ≪≪

S)SB(S2t

S)SB(S2

t

Length S = 2 approximately conservedLength S = 2 approximately conserved

Tilt angle of S Tilt angle of S relative torelative toB-directionB-direction

)sin(2 )sin(2 Pendulum in flavour spacePendulum in flavour space• Inverted (unstable) for inverted hierarchyInverted (unstable) for inverted hierarchy• Stable (harmonic oscillator) otherwiseStable (harmonic oscillator) otherwise


Impact of Ordinary MatterImpact of Ordinary Matter

p2m2

p2

m2P)PP(PLPBPt

P)PP(PLPBPt

P)PP(PLPBPt

P)PP(PLPBPt

2 2

eFnG2 eFnG2• Matter has identical effectMatter has identical effect on nus and anti-nuson nus and anti-nus• In rotating frame (frequency In rotating frame (frequency )) no matter effect, rotating B-no matter effect, rotating B-fieldfield• For inverted hierarchy:For inverted hierarchy: “ “driven” inverted harmonicdriven” inverted harmonic oscillator, exponentially oscillator, exponentially growinggrowing amplitudeamplitude

Exponentially growing phase,Exponentially growing phase,scales withscales with

nG2 F nG2 F

22vac1

bipol ln

22vac1

bipol ln

Analytic solution in:Analytic solution in:Hannestad, Raffelt, Sigl & Wong,Hannestad, Raffelt, Sigl & Wong,Self-induced conversion in denseSelf-induced conversion in denseneutrino gases:neutrino gases:Pendulum in flavour spacePendulum in flavour spaceastro-ph/0608695astro-ph/0608695


Asymmetric System as a Pendulum with SpinAsymmetric System as a Pendulum with Spin

P)PP(PBPt

P)PP(PBPt

P)PP(PBPt

P)PP(PBPt

Neutrinos and anti-neutrinosNeutrinos and anti-neutrinos

New variablesNew variables

BPPQ

BPPQ

PPD

PPD

Essentially particleEssentially particlenumber ( )number ( ) nn nnLepton number Lepton number ( )( ) nn nn

QDQ QDQ QBD QBD

Equations of motionEquations of motion

Conserved quantitiesConserved quantities

QQ

QQ

Spherical pendulumSpherical pendulum

BD

BD

QD

QD

22DQBE

22DQBE

EnergyEnergy

Angular momentumAngular momentumalong force directionalong force direction

Angular momentumAngular momentumalong pendulumalong pendulum

Interpretation as pendulum with spinInterpretation as pendulum with spin

QQq

QQq

qqq

D

qqq

D

Pendulum orientationPendulum orientation

Angular momentumAngular momentum

1 1 Moment of inertiaMoment of inertia

1

|PP|

)PP()PP(

1|PP|

)PP()PP(

SpinSpin

Equations of motion once moreEquations of motion once more

qBQqqq

qBQqqq

qB)1(q)1(qq

qB)1(q)1(qq

Precession (synchronized oscillations)Precession (synchronized oscillations)

Pendulum (bipolar oscillations)Pendulum (bipolar oscillations)

NO FREE LUNCHNO FREE LUNCH• Net flavor lepton number along massNet flavor lepton number along mass direction is conserveddirection is conserved• “ “Bipolar conversions” are PAIR Bipolar conversions” are PAIR conversions,conversions, no net flavor-lepton number violationno net flavor-lepton number violation• True flavor conversion only caused byTrue flavor conversion only caused by vacuum flavor oscillation effectvacuum flavor oscillation effect• No unusual enhancementNo unusual enhancement (unlike MSW effect)(unlike MSW effect)


Pendulum in Flavor SpacePendulum in Flavor Space

Mass directionMass directionin flavor spacein flavor space

PrecessionPrecession(synchronized oscillation)(synchronized oscillation)

NutationNutation(bipolar(bipolar oscillation)oscillation)

SpinSpin(Lepton Asymmetry)(Lepton Asymmetry)• Very asymmetric systemVery asymmetric system

- Large spin - Large spin - Almost pure precession - Almost pure precession - Fully synchronized oscillations- Fully synchronized oscillations

• Perfectly symmetric systemPerfectly symmetric system - No spin- No spin - Simple spherical pendulum- Simple spherical pendulum - Fully bipolar oscillation- Fully bipolar oscillation

nn nn

Polarization vectorPolarization vectorfor neutrinos plusfor neutrinos plusantineutrinos antineutrinos

nn nn

≫≫


Toy Supernova in “Single-Angle” Toy Supernova in “Single-Angle” ApproximationApproximation

BipolarOscillations

• Assume 80% anti-neutrinosAssume 80% anti-neutrinos• Vacuum oscillation frequencyVacuum oscillation frequency

= 0.3 km= 0.3 km11

• Neutrino-neutrino interaction Neutrino-neutrino interaction energy at nu sphere (r = 10 km)energy at nu sphere (r = 10 km)

= 0.3= 0.3101055 km km11

• Falls off approximately as Falls off approximately as rr44

(geometric flux dilution and nus(geometric flux dilution and nus become more co-linear)become more co-linear)

Decline of oscillation amplitudeDecline of oscillation amplitudeexplained in pendulum analogyexplained in pendulum analogyby inreasing moment of inertiaby inreasing moment of inertia(Hannestad, Raffelt, Sigl & Wong(Hannestad, Raffelt, Sigl & Wong astro-ph/0608695)astro-ph/0608695)


Nonlinear Neutrino Conversion in SupernovaeNonlinear Neutrino Conversion in Supernovae



atm atm mm22

1313closeclose



Duan, Fuller, Carlson, Qian: “Simulation of Coherent Non-Linear NeutrinoDuan, Fuller, Carlson, Qian: “Simulation of Coherent Non-Linear Neutrino Flavor Transformation in the Supernova Environment. 1. CorrelatedFlavor Transformation in the Supernova Environment. 1. Correlated Neutrino Trajectories”, astro-ph/0606616. See also: astro-ph/0608050Neutrino Trajectories”, astro-ph/0606616. See also: astro-ph/0608050


Flux Term as a Source of DecoherenceFlux Term as a Source of Decoherence

pqqqp3

3

Fp

2

pt P)PP)(cos1(2

qdG2PB

p2m

P

pqqqp3

3

Fp

2

pt P)PP)(cos1(2

qdG2PB

p2m

P

ppqqq3

3

pqq3

3

Fp

2

pt Pcos)PP(cos2

qdP)PP(

2

qdG2PB

p2m

P

ppqqq3

3

pqq3

3

Fp

2

pt Pcos)PP(cos2

qdP)PP(

2

qdG2PB

p2m

P

General two-flavor expressionGeneral two-flavor expression

Axial symmetry around some direction, e.g., supernova radial directionAxial symmetry around some direction, e.g., supernova radial direction

p0 PD

p0 PD

pp1 cosPD

pp1 cosPD

Density (isotropic) termDensity (isotropic) termResponsible for usualResponsible for usualcollective phenomenacollective phenomena(“self-maintained(“self-maintained coherence”)coherence”)

Flux termFlux termResponsible forResponsible forkinematical decoherencekinematical decoherence(“self-induced(“self-induced decoherence”) decoherence”)

Raffelt & Sigl,Raffelt & Sigl,Self-induced decoherence in dense neutrino gasesSelf-induced decoherence in dense neutrino gaseshep-ph/0701182hep-ph/0701182


Kinematical Decoherence - Symmetric CaseKinematical Decoherence - Symmetric Case


Isotropic (single angle)Isotropic (single angle) Large flux (“half isotropic”)Large flux (“half isotropic”)



Inverted Hierarchy - Asymmetric Case (Inverted Hierarchy - Asymmetric Case ( = 0.8) = 0.8)

Single angleSingle angle Multi angleMulti angle

PolarizationPolarizationvectorsvectors• LengthLength• z-componentz-component

EnergyEnergycomponentscomponents

zPzPzPzP

potEpotE

EEkin EEkin

potEpotE

EEkin EEkin


Inverted Hierarchy - Asymmetric Case (Inverted Hierarchy - Asymmetric Case ( = 0.6) = 0.6)




zPzPzPzP

potEpotE

EEkin EEkin


Normal Hierarchy - Asymmetric Case (Normal Hierarchy - Asymmetric Case ( = 0.8) = 0.8)




potEpotE

EEkin EEkin

potEpotE

EEkin EEkin


Normal Hierarchy - Asymmetric Case (Normal Hierarchy - Asymmetric Case ( = 0.9) = 0.9)




potEpotE

EEkin EEkin

potEpotEEEkin EEkin


Nonlinear Neutrino Conversion in SupernovaeNonlinear Neutrino Conversion in Supernovae



atm atm mm22

1313closeclose



Duan, Fuller, Carlson, Qian: “Simulation of Coherent Non-Linear NeutrinoDuan, Fuller, Carlson, Qian: “Simulation of Coherent Non-Linear Neutrino Flavor Transformation in the Supernova Environment. 1. CorrelatedFlavor Transformation in the Supernova Environment. 1. Correlated Neutrino Trajectories”, astro-ph/0606616. See also: astro-ph/0608050Neutrino Trajectories”, astro-ph/0606616. See also: astro-ph/0608050


Different Oscillation Modes in SupernovaeDifferent Oscillation Modes in Supernovae

CenterCenter

NeutrinoNeutrinospheresphere~15~15

00

R [km]R [km]

~10~1044

~200~200

Bipolar oscillations forBipolar oscillations forinverted hierarchyinverted hierarchy

0 0 MeV100

e MeV100

e

FluxesFluxes

FFFF FFFF

H-Resonance (atm)H-Resonance (atm)

FreeFreestreamingstreaming

nnne nnne

nnne nnne

≫≫≪≪

FF,FFee FF,FFee

M

att

er

eff

ect

im

port

an

tM

att

er

eff

ect

im

port

an

t

Usu

ally

su

pp

ress

es

mix

ing

an

gle

Usu

ally

su

pp

ress

es

mix

ing

an

gle

MSWMSW

Neu

trin

o-n

eu

trin

oN

eu

trin

o-n

eu

trin

oco

llect

ive

colle

ctiv

e e

ffect

seff

ect

s st

ron

gst

ron

g

~10~1055L-Resonance (sol)L-Resonance (sol) MSWMSW

FF,FFee FF,FFee

SynchronisedSynchronisedoscillationsoscillations

Little effect because ofLittle effect because ofmatter-suppressedmatter-suppressedmixing anglemixing angle

~80~80

Importance of bipolarImportance of bipolaroscillations in this SNoscillations in this SNregion first noted byregion first noted byDuan, Fuller & QianDuan, Fuller & Qianastro-ph/0511275 astro-ph/0511275


Papers on collective neutrino oscillationsPapers on collective neutrino oscillations

1992 Flavor off-diagonal1992 Flavor off-diagonalrefractive indexrefractive index

1992-19981992-1998NumericalNumerical && analytic studies,analytic studies,but not much impactbut not much impact(nobody really understood)(nobody really understood)

2001-20022001-2002Flavor equilibration ofFlavor equilibration ofcosmological neutrinos withcosmological neutrinos withchemical potential beforechemical potential beforeBBN epochBBN epoch

1994-20041994-2004SN neutrino oscillations and SN neutrino oscillations and r-process nucleosynthesisr-process nucleosynthesis(everybody missed the main(everybody missed the main point)point)

2006-20072006-2007““Bipolar” oscillationsBipolar” oscillationscrucial for SN neutrinos crucial for SN neutrinos

Pantaleone, PLB 287(1992) 128Pantaleone, PLB 287(1992) 128

Samuel, Kostolecký & PantaleoneSamuel, Kostolecký & Pantaleonein various combinations:in various combinations:PLB 315:46 & 318:127 (1993), 385:159 (1996)PLB 315:46 & 318:127 (1993), 385:159 (1996)PRD 48:1462 (1993), 49:1740 (1994), 52:621 & 3184PRD 48:1462 (1993), 49:1740 (1994), 52:621 & 3184(1995), 53:5382 (1996), 58:073002 (1998)(1995), 53:5382 (1996), 58:073002 (1998)

Pastor, Raffelt & Semikoz, hep-ph/0109035Pastor, Raffelt & Semikoz, hep-ph/0109035Lunardini & Smirnov, hep-ph/0012056Lunardini & Smirnov, hep-ph/0012056Dolgov et al., hep-ph/0201287Dolgov et al., hep-ph/0201287Wong, hep-ph/0203180Wong, hep-ph/0203180Abazajian, Beacom & Bell, astro-ph/0203442Abazajian, Beacom & Bell, astro-ph/0203442

Pantaleone, astro-ph/9405008Pantaleone, astro-ph/9405008Qian & Fuller, astro-ph/9406073Qian & Fuller, astro-ph/9406073Sigl, astro-ph/9410094Sigl, astro-ph/9410094Pastor & Raffelt, astro-ph/0207281Pastor & Raffelt, astro-ph/0207281Balantekin & Yüksel, astro-ph/0411159Balantekin & Yüksel, astro-ph/0411159

Duan, Fuller & Qian, astro-ph/0511275Duan, Fuller & Qian, astro-ph/0511275Duan, Fuller, Carlson & Qian, astro-ph/0606616Duan, Fuller, Carlson & Qian, astro-ph/0606616Hannestad, Raffelt, Sigl & Wong, astro-ph/0608695Hannestad, Raffelt, Sigl & Wong, astro-ph/0608695Raffelt & Sigl, hep-ph/0701182Raffelt & Sigl, hep-ph/0701182

collective supernova neutrino oscillations georg raffelt, mpi physik, munich, germany

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