The role of temporal instabilities in supernova neutrino flavour conversions

The role of temporal instabilities

in supernova neutrino flavour conversions

based on JCAP 1604 (2016) no.04, 043 with B. Dasgupta and A. Mirizzi

FRANCESCO CAPOZZI

(Padua University - INFN Padua)

1

Outlook

• Supernova neutrino oscillations: current status and the role of symmetries

• The line model and non-stationary neutrino emission

• Linear stability analysis of the flavour evolution

• Non linear calculation of the flavour evolution

• Conclusions

Supernova neutrinos

Core-collapse supernovae are the final explosion of stars with M>8M⦿. Most of the gravitational binding energy is released through neutrinos νe,νe,νx (~10⁵³ erg).

Neutrino emission is time dependent and lasts ~10 s.

Neutronization burst

Accretion phase

Cooling phase

Figure adapted from

Fischer et al. (Basel group), arXiv: 0908.1871

(spherically symmetric

with Boltzmann transport) 

Supernova neutrinos: oscillations

4

vacuum oscillations, ω=Δm²/2E

In such an environment νν interactions cannot be neglected when studying oscillations.

The main ingredients for the EoM are:

H = H

+ H

+ H

H

=

✓p 2G

N

0

0 0

◆

H

=

U

✓ 0 0 0 m

◆

U

2E

H

= p

2G

Z 1

(2⇡)

E

dE

dv

(1 v · v

)%

λ=√2GFNe

μ=√2GFNν

i(@

+ v · r

)%

= [H

, %

]

%

=

✓ h ⌫

| ⌫

i h ⌫

| ⌫

i h ⌫

| ⌫

i h ⌫

| ⌫

i

◆

Neutrino emission from the neutrinosphere is assumed to be uniform, half isotropical, azimuthal symmetric and stationary.

Physical conditions of the star depend only on r

Supernova neutrinos: bulb model

%(t, x) ! %(r) ı(@

+ v · r

)%(r ) = v

d

dr %(r )

Supernova neutrinos: oscillations

Mirizzi et al.

Riv.Nuovo Cim. 39 (2016) no.1-2, 1

For r<10³ km we have μ>>λ and collective oscillation phenomena take place For r>10³ km we have μ<<λ and we have only standard MSW effect

Collective oscillation phenomena

Synchronized oscillations Bipolar oscillations

Pastor et al. hep-ph/0109035v3 Hannestad et al. astro-ph/0608695

Neutrinos with different energies oscillate with same frequency

Almost complete flavour conversion νeνe—>νxνx

even with a small mixing angle

Recent developments

8

Recently it was shown that:

There is not a complete knowledge of oscillations in SN. Current understanding is that neutrinos cannot change their flavour too close to the SN core

We study the impact of non stationary emission on flavour conversion near the supernova core

1) Neutrinos can spontaneously break, during propagation, the space time symmetries assumed in the initial conditions.

[G. Raffelt et al. Phys. Rev. Lett. 111, no. 9 - Mirizzi et al. Phys. Rev. D 92, no. 2]

2) tiny space inhomogeneities in the initial conditions may lead to new flavour instabilities which can develop even at small distances from the SN core.

[H. Duan et al. Phys. Lett. B 747]

3) A large matter potential (λ>>μ) can suppress bipolar flavor conversions.

[S. Chakraborty et. al. arXiv:1507.07569]

Line model

A beam of monochromatic (E=E0) νe, νe is emitted by an infinite boundary (x-axis) and propagate in the plane (x,z). The emission occurs only in two directions (L,R) and we relax the hypothesis of stationarity and homogeneity.

! = !

= m

2E

= 1

✓

= 5⇡/18 ✓

= 7⇡/9

" = 1

1 + ✏ = ( _{⌫e ⌫x})/( _{⌫¯e ⌫¯x})

%

(z ) = Z

dxdt%

e

Linear stability analysis.

Let SE,v be the off-diagonal component of the density matrix. The linear stability analysis [A. Banerjee et al. Phys. Rev. D 84, 053013 (2011)] is performed by expanding the EoM at linear order in S, working in the Fourier space

and taking

Det 0 B B

@

!

+ ⌦ 0 (1 + ✏)µ

µ

0 !

+ ⌦ (1 + ✏)µ

µ

(1 + ✏)µ

µ

!

+ ⌦ 0

(1 + ✏)µ

µ

0 !

+ ⌦

1 C C

A = 0 ,

!

= ( ± !

+ kv

+ p ✏µ

)/v

µ

= µ

/v

.

where

S

= Q

e

We are looking for exponentially growing solution, i.e. those with a non vanishing imaginary part of Ω

Out[94]=

Out[95]=

2

Linear stability analysis

Out[92]=

Out[93]=

The eigenvalues depend on λ only through the combination λ+εμ-p. There is always a p compensating λ and thus eliminating the matter suppression, leading to growing instabilities.

This interplay remains true at a nonlinear level

Non linear evolution

For the mode k=0 and p=0 we take the off diagonal components of the density matrix to be 10^-7, while a numerical seed of 10^-12 for all the other modes.

Our notation:

p = n

¯(z = 0)

¯ = + "µ 100

We are interested in the evolution of off diagonal terms

A

(n

, z ) ⌘ log

| %

(z) |

The cancellation of λ occurs for np~100, which is the most unstable mode. Instability

then propagates to lower and higher scales

k = 150!

µ = 40 km

= 4 ⇥ 10

exp( z/⌧ ) km

Non linear evolution

10 5 0

10 5 0

p = 0 n

= 50 np

= 90 np

linear

10 5 0

∞

λ=

τ τ_λ=30 τ_λ=10

0 5 10 15 20 0 5 10 15 20 0 5 10 15 20

10

− 5

− 0

z z z

µ

Ae

• linear solution is a good approximation of the non-linear evolution only until nonlinearities remain small

• in the case of variable λ the adiabaticity of the flavor evolution plays a significant role.

When the numerical solution shows non-adiabatic features resulting from resonances the deviations with respect to the linear approximation become significant.

µ = 40 km

= 4 ⇥ 10

exp( z/⌧ ) km

Non linear evolution

• For the model A, we realize that the instability starts at z~2 from modes at np~100

• For model B, due to the reduced adiabaticity in the flavor evolution the instability never grows significantly.

µ = 60e ^z/30 km ¹ = 7 ⇥ 10⁴e ^z/15 km ¹ µ = 60e ^z/15 km ¹ = 10⁵e ^z/7.5 km ¹

Non linear evolution

• For the model A, we realize that the instability starts at z~2 from modes at np~100

• For model B, due to the reduced adiabaticity in the flavor evolution the instability never grows significantly.

µ = 60e

= 7 ⇥ 10

e

µ = 60e

= 10

e

10 5 0

10 5 0

p = 0 n

= 50 np

= 90 np

linear

model A model B

0 5 10 15 20 0 5 10 15 20

−10 5

− 0

z z

eµ

A

Conclusions

16

• Time inhomogeneities can compensate the phase dispersion associated with a large matter term that would otherwise suppress the flavour conversions, producing a cascade in the Fourier modes

• The development of this cascade crucially depends on the adiabaticity of the matter and neutrino potential.

• The linearized solution of the equations of motion can be a valuable tool to find the growing modes and to predict the flavor evolution till non-linearities and non-adiabatic features become important

We have studied the development of the temporal instability in self-induced flavour evolution in the line model. We have shown that:

Our model of course is too simple to gain a definite answer on the development of such effects in the flavour conversions on SN neutrinos, but still points out that such details must be taken into account

Thank you

17

18

Equation of motion for the line model

ı d

dz %

(z ) = 1 v

 U M

U

2E , %

+ 1

v

(kv

+ p)%

+

v

[L, %

] + 1

v

Z

dx dt e

[H

, %

]