ICRR Seminar, Kashiwa
16 May 2016

Temporal Instability ofSupernova Neutrinos

Basudeb Dasgupta
TIFR, Mumbai

Sanduleak -69 202Supernova 1987A
Sanduleak -69 202SN 1987A

23 February 1987

29 years Ago

In a Galaxy Near US
GeorgRaffelt,
MPI Physics,
MunichBasudeb
Dasgupta,
TIFR Mumbai

16 May, ICRR

Massive Neutrinos, NTU, Singapore, 9–13 Feb 20152

Aimof– theThree
PhasesThree Game
Opportunities

Standard Candle (?)
Burst• SN theory
SN standard candle?
• DistanceSN•theory
Flavor conversions• Multi-messenger
Timingtime of flight
Mass hierarchy…
Georg Raffelt, MPI Physics, Munich

Basudeb Dasgupta, TIFR Mumbai

Strong variations
EoS & mass dependence
Accretion
Cooling(progenitor, 3D effects,• Testing nuclear physics
Astrophysics
Nuclear physics
black hole formation, …) • Nucleosynthesis inCollective• Testingeffects?
astrophysics of Nucleosynthesis
neutrino-driven windcorecollapse• Particle bounds fromShockrevival?
Exotics/Axions• Flavor conversion has
cooling speed (axions …)
Mass hierarchy?
…
strong impact on signal… (Collective & MSW)
Massive Neutrinos, NTU, Singapore, 9–13 Feb 2015

13 May, Koshiba Hall of the Tokyo Univ.

3

MSW Effects

NHIH

Dighe and Smirnov

L resonance : ν1 Spectral
ν2Splits(Δmsol2 , θ12) at 101 – 102 in
g/ccInverted
Always in neutrinos.
Hierarchy
Always adiabatic.
Basudeb Dasgupta, TIFR Mumbai

16 May, ICRR

H resonance : νSpectralν31/ν24 g/cc
(Δmatm2 , θ13) at 103 – 10Splitsin
In neutrinos for NH and in
antineutrinos for IH.
Adiabaticity ≈ sin2θ134

Neutrino-Neutrino Interactions

Pantaleone (1992)

Basudeb Dasgupta, TIFR Mumbai

16 May, ICRR5Collective Effects

Duan, Fuller, Carlson, Qian (2005, 2006,…)
Basudeb Dasgupta, TIFR Mumbai

16 May, ICRR

6Spectral Splits
Nontrivial Evolution only for Inverted Hierarchy

Fogli, Lisi, Marrone and Mirizzi
Basudeb Dasgupta, TIFR Mumbai

16 May, ICRR

7

Multiple Splits
Spectral Splits
in Inverted
Hierarchy

Spectral Splits
in Normal
Hierarchy

Dasgupta, Dighe, Raffelt and Smirnov (PRL 2009)
Basudeb Dasgupta, TIFR Mumbai

16 May, ICRR

8Multi-Angle Effects
SmearedSpectralSplits

Many works, e.g., Esteban-Pretel et al, Friedland, Duan and Shalgar

Basudeb Dasgupta, TIFR Mumbai

16 May, ICRR

9Multi-Angle Matter Effect

Esteban-Pretel, Mirizzi, Pastor, Tomas, Raffelt, Serpico, Sigl
Basudeb Dasgupta, TIFR Mumbai

16 May, ICRR

10

Symmetry Breaking

Basudeb Dasgupta, TIFR Mumbai

16 May, ICRR

11

Symmetry Breaking
Mirizzi, Mangano, Saviano

Basudeb Dasgupta, TIFR Mumbai

16 May, ICRR

12Instability Footprint

Chakraborty, Hansen, Izagguire, Raffelt
Basudeb Dasgupta, TIFR Mumbai

16 May, ICRR

13

Temporal Instability
Dasgupta and Mirizzi

Basudeb Dasgupta, TIFR Mumbai

16 May, ICRR

14k≠ 0p=0Increasing k102

k=0p≠ 0
Increasing p106

104
102

Shock Front

100

Matter potential (1/km)k≠0p≠0

Matterdensity
profile

k=0
p=0

103
Distance from the center of SN (km)

we shall omit showing these dependencies wherever t
ream,
butwith
forwardscatterings
with
other
neutrie.g., the background electron density ne (t, x). For n
Neutrino
Flavor
Evolutionroundmatter.
Theequationofmotion there
for suchastnianshowing
thesedependencies
whereverisnofor flavor evolution in the collisionless limit isi(@inv ·collisionlessrxH)%E,v= limit[HE,v, %E,v],
t+or evolution
isHthe=+H+H,E,vvacmat⌫⌫

denoteusing
HE,vlimit,=E,Hvvacdependence+ Hmat + H⌫⌫, subscripts. Of courflavorx). This space-timeis always understo✓ 2 dependence◆
1e.g., them1background0
quantities,electrondensityn†
eHvac =U✓U2✓
2E0m✓ omit◆e shallshowingthesedependencies wherever22m1 0✓◆†p 2 U ✓ ne 0U✓
E Hmat 0= m2G2 Fan forflavor
0 0 in the collisionless limit is✓◆ evolution
Zne 0p
12GF020200H⌫⌫ 0= 0 2GHFE,v = Hvac
Hmatdv+ (1H⌫⌫ ,v · v )%E 0 ,v0
3 E+ dE
Z
(2⇡)
10202
00
Basudeb Dasgupta, TIFR Mumbai

13 May, Koshiba Hall of the Tokyo Univ.

16

has ! > 0 for neutrinos and ! < 0 for antineutrinos.

2
Whatisthe“spectrum”
scillation frequency ! = | m |/(2E) is a ?
more con◆has ! > 0 forTr%
neutrinos and ! ✓< 0 for antineutrinoss!,v S!,v!,v⌫%!,v =I+g!,v,⇤S!,v s!,v2
2✓◆Tr%!,vs!,v S!,v⌫%!,v=I + no flavor
g!,v change,⇤, withs being
1 whenhasoccurre
S!,v s!,v
2
2emission spectrum, defined by
Z 0 1 when no flavor change has occur, with s beingd⌫ gdefined!,v = (by⌫¯e⌫¯x ) ,emission spectrum,1

Z 0and ⌫¯e,x are theflavor-dependentneutrino
numbed ⌫ g!,v = ( ⌫¯e⌫¯x ) ,o asymmetry1parameter ✏ is given by
Basudeb Dasgupta, TIFR Mumbai

13 May, Koshiba Hall of the Tokyo Univ.

17ntineutrino
asymmetry✏ is givenby the spectrum is given b
Thisfixes the frequency
scaleparameterof flavor evolution.
Therefore,⇥⇤ ⇥⇤g!,v =(! 1++!0✏) +(v vL)).+ (v vR ) ,0) ⇥=(1( + ✏) (! !)/(Fourier Modes⌫e⌫x

⌫¯e

⌫¯x

which implies the normalization ⌫ = ( ⌫¯eare constant along x⌫¯µ )/2.
⌫ andPrecession-likemotionforthemodes
equationsof motion
be written
as flavor composition can depe
but vary alongz. Awaycan
fromthen
the source,
the neutrino
Z P!,v will have to be specified along
as well as x and t. The boundary
conditions on
00 symmetries.
andat vz=0, andbreakthe !Bcorresponding
space-timetranslation(@tt +·r)P=+L+µd(1v·v)P!0 ,v0 ⇥ P!,vx!,vThe flavor evolution can be studied using Fourier modes of the polarizationEq.
be converted
a towerofof ODEs
ordinaryby
di↵erential
equationsthe
by decompo
canbecanconverted
topatotower
decomposing
polarizp(2.9)
polarizationconstituent
modes
by (Rep andk [23–2=2GFFourier
nFourierand modes.
µinto= their
2G. TheFourier
vectorP!,vlabeled= g!,vS!,v, Im
decomposition
evectors
F ⌫writes
ituentOnewritesd from the density matrix, X
is a so-called polarization vector that eTTi(pt+kx)p,kXi(pt+kx)
p,kcomposition,PwhileB=(cos✓,
0,sin✓)
andL=(0,0,1)
are th(t,x,z)=eP(z),!,v!,vP!,v (t, x, z) =e
P!,v (z) ,nstructed from Hvac and Hmatp,k, respectively.
In this paper we presentp,kmal neutrino mass ordering (m1 < m2 ), and for the inverted ordering
wherep, k are theandtemporaland spatial
frequency
modesofof thethe polarization
vectorhetemporal
spatialfrequencymodes
polarizationv
shis
B decomposition,! B.
Equationfor
Fouriermodes
p,k
one gets a tower of equations for the Fourier modes P!,v , p,kion,one
getsatowerof equations
thegiven
Fourierproblem boils down
to calculating
P!,vfor(t, x),
theirmodes
values Pat!,v
theX000 0p,kTypically thisp,kboundary conditionp,kpbep ,kstationarykp ,kf vtime.
istakento(!BL) ⇥ P!,v +µv P 0 0⇥ P!,v .
z @z P!,v = i(p + vx k)P!,v! ,v
0 0 0 X
0ous over the source, which may naively suggest
the solution
ough
0 0
p ,k ,!that,vp,kp,kp ,kp,k=i(p+vk)P(!BL)
⇥P+µP
⇥P,
metries as xwell. !,v
However, such a solution
unstable0 0 to small!,v is oftenv!,v0 ! ,v
Basudeb Dasgupta, TIFR Mumbai

13 May, Koshiba Hall of the Tokyo Univ.

18teflavor evolution
in Fourier
this rather
simplified model is quite compstability
analysisevenusingmodes
to flavor
linearize
the equations.
linearorderin S, themodel
EOMsis simplifytteevolution
even inTo
thisrather
simplified
quite comor evolution even in this rather simplified model isZquite complicated,o linearize the equations. To linear order in S, the EOMs simplify
0rizei(@the+equations.
To=linearorderin S,the EOMs
simplifyv·r)S(!
++✏µ)Sµdg!0 ,v0toS!0 ,v0 .
Ztx !,vv !,vv
Zi(@t + v · rx )S!,v = ( ! + + ✏µv )S!,v 0 µv d 0 g!0 ,v0 S!0 ,v0 .
+ vexpress· rx )S!,v= (using! +its+Fourier✏µv )S!,vµv d g!0 ,v0 S!0 ,v0 .
(2ain
S!,v
transform,X transform,ainexpress
S
usingits
Fourier
Use
Fourier
decomposition
p,k!,v
ress S!,v using its
Fouriertransform,
S!,v(t, x, z)=e i(pt+kx) S!,v(z) ,X i(pt+kx) p,kX i(pt+kx)
p,kp,kS!,veS!,vS!,v (t, x,z) (t,= x, z)
e=(z) , S!,v (z) ,(2,kp,kiz⌦p,kp,k=Qe. The eigenvalue
equation for the Fourier amplitudes,v!,v
an eigenvalue equation
kp,k Getiz⌦
p,kiz⌦
p,k=Qe.
Theeigenvalue
equation
fortheFourier
amplitudee.
Theeigenvalue
equation
fortheFourier
amplitudes
Qv!,v!,v!,v is t✓◆Z! kvx p + + ✏µvµvp,k0p,kp,k p,k00Qd!
gQ=⌦0!,v!,v Q!,v!
,v
◆✓◆
Z!,vZ
v+z ✏µvz
kv!
p+µkvx p +v +p,k✏µ0µvp,kp,kx
0p,k p,kp,k p,0 0Q
Q!,v v Qvp,kd!
g=
⌦Q.
00d!
g
Q
=
⌦Q(20!,v 0!,v!,v!,v!
!
,v!
,v!,v!,vvz In principle
el we consider,neutrino
momentummodeslabeledvzthere arev
thisonlyis avfoursolvableeigenvalue
problemzz
and vR , and the above equation becomes an eigenvalue equation
Lnsider,
there areonlyare
fourneutrino
momentummodes labeled
bylabele!=lweconsider,
there
only
four
neutrino
momentummodes
h 4 eigenvalues ⌦i=1,2,3,4 , as a function of p and k. These eigenvalues
vRand, andtheabove
equation
becomes becomesan eigenvalue
equation for
thev
,
and
theaboveequationaneigenvalue
equatioR
13 May, Koshiba Hall of the Tokyo Univ.
nBasudeb Dasgupta, TIFR Mumbai
19

Linearization

A simple example
zvLvR
θRθL

z=0x

Neutrinos emitted from a surface, along two directions L and RThe initial condition along x are noisy and not steadyHow does the flavor content evolve along z?
How does it break the x and t translation symmetry?

Basudeb Dasgupta, TIFR Mumbai

13 May, Koshiba Hall of the Tokyo Univ.

20

vz

vz

Explicit Computation

we consider, there are only four neutrino momentum modes labele
and vR , and the above equation becomes an eigenvalue equatio
4 eigenvalues ⌦i=1,2,3,4 , as a function of p and k. These eigenvalue
01!+L + ⌦0(1 + ✏)µLµLBC0!
+⌦(1+✏)µµLLLC = 0,
Det B@(1 + ✏)µR
A
µR !+R + ⌦0
(1 + ✏)µRµR0! R+⌦!±(L,R) = (±!0 + kvx,(L,R) + p✏µv )/vz,(L,R)µ(L,R) = µv /vz,(L,R) .

Basudeb Dasgupta, TIFR Mumbai

13 May, Koshiba Hall of the Tokyo Univ.

21

are however artifacts of explicitly breaking the L $ R symme
matter e↵ect. In a truly multi-angle scenario with a L $ R
with many modes, these artifacts are avoided. Note, howev
real and will only give real or complex-conjugate eigenvaluesWithout NoiseOut[142]=Out[143]=

Capozzi, Dasgupta, Mirizzi (2016, JCAP)
Basudeb Dasgupta, TIFR Mumbai

13 May, Koshiba Hall of the Tokyo Univ.

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With Noise2

Out[92]=Out[93]=Out[94]=2Out[93]=Out[94]=Out[95]=

Capozzi, Dasgupta, Mirizzi (2016, JCAP)

Figure 2. Footprints of instability for various values of spatial and temporal frequency modes, k andp respectively, on the plane of the matter potential and the neutrino potential µ, all in units of!0 . The warmer colors represent larger magnitudes of the imaginary part of the eigenvalues given byEq. (2.19). The dark blue color signifies that all eigenvalues are real. The dashed white line represents
Koshiba
Hall areof theTokyo
Univ.
Basudeb Dasgupta, TIFR Mumbai
a correlation between µ and , e.g., 13in May,a model
where bothvaryingwithz (see Sec. 3.2). Note23

Fully Nonlinear Evolution

Figure 3. Amplitudes of the o↵-diagonal components of the di↵erent np modes, Aeµ , as a functionof the distance from the neutrino source, z, and of the Fourier mode with index np , for ⌧ = 1 (leftpanel), 30 (middle panel) and 10 (right panel) respectively. The closer is Aeµ to 0 (red color in theplot), the stronger are the flavor conversions.
Basudeb Dasgupta, TIFR Mumbai

13 May, Koshiba Hall of the Tokyo Univ.

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Linear vs. Nonlinear

Figure 3. Amplitudes of the o↵-diagonal components of the di↵erent np modes, Aeµ , as a functionof the distance from the neutrino source, z, and of the Fourier mode with index np , for ⌧ = 1 (leftpanel), 30 (middle panel) and 10 (right panel) respectively. The closer is Aeµ to 0 (red color in theplot), the stronger are the flavor conversions.
τλ=∞00τλ=300−55

τλ=10055eµA

np = 0np = 50−1010

np = 901010linear
0510z

15200510z

152005101520z

Figure 4. Comparison linear vs nonlinear growth of Aeµ for di↵erent Fourier modes for ⌧ = 1 (leftpanel), 30 (middle panel) and 10 (right panel) respectively.

The instability develops only at z & 10 for np ⇠ 50. Modes with np . 50 grow till
Aeµ ' 1, when nonlinearity takes over, while larger np modes are suppressed by thelarge p that mimics the matter suppression.
13 May, Koshiba Hall of the Tokyo Univ.
25
Basudeb Dasgupta, TIFR Mumbai

More work in progress

Basudeb Dasgupta, TIFR Mumbai

16 May, ICRR

26

Moving Footprints

Matter Density

Dasgupta and Mirizzi

Temporalfluctuations

Usual
Instability
spatialfluctuations

Distance from the center of SN
Basudeb Dasgupta, TIFR Mumbai

16 May, ICRR

27