DAMPING THE FLAVOR PENDULUM BY BREAKING HOMOGENEITY (Alessandro MIRIZZI, Hamburg U.)

DAMPING THE FLAVOR PENDULUMDAMPING THE FLAVOR PENDULUMBY BREAKING HOMOGENEITYBY BREAKING HOMOGENEITY

(Alessandro MIRIZZI, Hamburg U.)(Alessandro MIRIZZI, Hamburg U.)

NOW 2014NOW 2014Neutrino oscillation workshopNeutrino oscillation workshop

Conca Specchiulla, 07-14 September 2014Conca Specchiulla, 07-14 September 2014

(Based on work in collaboration with G. Mangano & N. Saviano, (Based on work in collaboration with G. Mangano & N. Saviano, 1403.1892)1403.1892)

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*

e

e

eeee

DENSITY MATRIX FOR THE NEUTRINO ENSEMBLE

Diagonal elements related to flavor content

Off-diagonal elements responsible for flavor conversions

),(

),(

rEF

rEF

In 2scenario. Decompose density matrix over Pauli matrices to get the “polarization” (Bloch) vector P. Survival probability Pee =1/2(1+Pz) . Pz = -1 -> Pee =0 ; Pz = 0 -> Pee =1/2 (flavor decoherence)

Alessandro Mirizzi NOW 2014 Conca Specchiulla, 8 September 2014

EQUATIONS OF MOTION FOR A DENSE NEUTRINO GAS

],[ ,,,,, xpxpxppxpxpxpt ipv

(Sigl & Raffelt, 1992)

Liouville operator Hamiltonian

xpt , Explicit time evolution

xpxpv , Drift term due to space inhomogeneities

xppp , Force term acting on neutrinos (negligible)

7-dimensional problem. Never solved in its complete form. Symmetries have been used to reduce the complexity of the problem.


mattvacxp,

SPACE/TIME HOMOGENEITIY

Space Homogeneity:

],[ ,,,, xpxpxpxpxpt iv

Pure temporal evolution (Neutrinos in Early Universe)

Time Homogeneity:

],[ ,,,, xpxpxpxpxpt iv

Stationary space evolution (SN neutrinos)

However, small deviations from these symmetries have to be expected. Can these act as seed for new instabilities?


TOY MODEL: PENDULUM IN FLAVOR SPACE

PDLBP

PDLBP

t

t

][

][

PPD

)1(2

1 Pp Two-flavor polarization vectors

E

m

2

2 Vacuum oscillation frequency

eFnG2 Matter potential. Large HOMOGENEOUS can be rotated away !

nGF2 potential


FLAVOR OSCILLATIONS AS SPIN PRECESSION


Slide from G. Raffelt

HOMOGENEOUS PENDULUM

Periodic pair conversions in IHxxee

((

For homogeneous

[Hannestad et al, astro-ph/0608695]


,e e

,

NON-HOMOGENEOUS BACKGROUNDS

),()],(),(),([),()( txPtxDtxLtxBtxPtx

The partial differential equation can be transformed into a tower of ordinary differential equations for the Fourier modes

ikxk etxdxPtP ),()(

)],([)(

)],([)(

txFTt

txFTt

k

k

(1D spatial motion)

MONOCHROMATIC MATTER INHOMOGENEITY

)cos( 0xk )]()([ 00 kkkkk FT

const )(2 kk

0nkkn

nkn PP


EOMs for the n=0,1 modes

Starting from homogeneous initial condition: only 00 P , n ≥ 1 modes are excited in sequence


DAMPING THE FLAVOR PENDULUM

310

0

710

A small seed of inhomogeneity is enough to produce a run-away from the stable pendulum behavior. The average P0 tends towards the flavor equilibrium.

20 k pendulum oscillation frequency

1403.1892


TRAJECTORIES OF THE FLAVOR PENDULUM

((

0 310

Stable pendulum Unstable pendulum


EVOLUTION OF DIFFERENT FOURIER MODES

310

n=1

n=2

n=3 n=4

After P1 starts rising, the higher Fourier modes are also rapidly excited in sequence reaching | Pn|~0.1


DEPENDENCE ON PERTURBATION WAVE-NUMBER

ck 0

1c

210c

210c

1c The flavor decoherence is approached earlierkk 0

210c Longer perturbation wave-length. The system needs more cycles to feel inhomogeneities210c Perturbations are averaged during an oscillation cycle. The effect is shifted al later timesAlessandro Mirizzi NOW 2014 Conca Specchiulla, 8 September 2014

NON-TRIVIAL SPACE BEHAVIOR

ikxk etdkPtxP )(),(

homogeneous solution

inhomogeneous solutions


DECLINING NEUTRINO DENSITY

)/exp(0 t

Quick decoherence. Similar to the case of constant

Lowering the system requires more time to decohere. Decoherece is not complete.

For a too fast decline, the system has not enough time to decohere


CONCLUSIONS

We studied the effects of small inhomogeneities on the self-induced evolution of a dense neutrino gas, by Fourier transforming the EOMs

We found that the neutrino flavor pendulum is not stable under the effects of small inhomogeneities

However, a declining neutrino potential can suppress the effect of the inhomogeneities

The effect on the flavor evolutions of neutrinos in SN or in the Early Universe needs further investigations with more realistic models.


DAMPING THE FLAVOR PENDULUM BY BREAKING HOMOGENEITY (Alessandro MIRIZZI, Hamburg U.)

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