1. DAMPING THE FLAVOR PENDULUM BY BREAKING HOMOGENEITY (Alessandro MIRIZZI, Hamburg U.) NOW 2014 Neutrino oscillation workshop Conca Specchiulla, 07-14 September 2014 (Based on work in collaboration with G. Mangano & N. Saviano, 1403.1892)

2. DENSITY MATRIX FOR THE NEUTRINO ENSEMBLE Diagonal elements related to flavor content Off-diagonal elements responsible for flavor conversions In 2  scenario. Decompose density matrix over Pauli matrices to get the “polarization” (Bloch) vector P. Survival probability Pee =1/2(1+P z ). P z = -1 -> Pee =0 ; P z = 0 -> Pee =1/2 (flavor decoherence) Alessandro Mirizzi NOW 2014 Conca Specchiulla, 8 September 2014

3. EQUATIONS OF MOTION FOR A DENSE NEUTRINO GAS (Sigl & Raffelt, 1992) Liouville operator Hamiltonian Explicit time evolution Drift term due to space inhomogeneities 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. Alessandro Mirizzi NOW 2014 Conca Specchiulla, 8 September 2014

4. SPACE/TIME HOMOGENEITIY Space Homogeneity: Pure temporal evolution (Neutrinos in Early Universe) Time Homogeneity: Stationary space evolution (SN neutrinos) However, small deviations from these symmetries have to be expected. Can these act as seed for new instabilities? Alessandro Mirizzi NOW 2014 Conca Specchiulla, 8 September 2014

5. TOY MODEL: PENDULUM IN FLAVOR SPACE Two-flavor polarization vectors Vacuum oscillation frequency Matter potential. Large HOMOGENEOUS can be rotated away !  potential Alessandro Mirizzi NOW 2014 Conca Specchiulla, 8 September 2014

6. FLAVOR OSCILLATIONS AS SPIN PRECESSION Alessandro Mirizzi NOW 2014 Conca Specchiulla, 8 September 2014 Slide from G. Raffelt

7. HOMOGENEOUS PENDULUM Periodic pair conversions in IH (( For homogeneous   [Hannestad et al, astro-ph/0608695] Alessandro Mirizzi NOW 2014 Conca Specchiulla, 8 September 2014 

8. NON-HOMOGENEOUS BACKGROUNDS The partial differential equation can be transformed into a tower of ordinary differential equations for the Fourier modes (1D spatial motion)

9. MONOCHROMATIC MATTER INHOMOGENEITY FT Alessandro Mirizzi NOW 2014 Conca Specchiulla, 8 September 2014

10. EOMs for the n=0,1 modes Starting from homogeneous initial condition: only, n ≥ 1 modes are excited in sequence Alessandro Mirizzi NOW 2014 Conca Specchiulla, 8 September 2014

11. DAMPING THE FLAVOR PENDULUM A small seed of inhomogeneity is enough to produce a run-away from the stable pendulum behavior. The average P 0 tends towards the flavor equilibrium. pendulum oscillation frequency 1403.1892 Alessandro Mirizzi NOW 2014 Conca Specchiulla, 8 September 2014

12. TRAJECTORIES OF THE FLAVOR PENDULUM (( Stable pendulum Unstable pendulum Alessandro Mirizzi NOW 2014 Conca Specchiulla, 8 September 2014

13. EVOLUTION OF DIFFERENT FOURIER MODES n=1 n=2 n=3 n=4 After P 1 starts rising, the higher Fourier modes are also rapidly excited in sequence reaching | P n |~0.1 Alessandro Mirizzi NOW 2014 Conca Specchiulla, 8 September 2014

14. DEPENDENCE ON PERTURBATION WAVE-NUMBER The flavor decoherence is approached earlier Longer perturbation wave-length. The system needs more cycles to feel inhomogeneities Perturbations are averaged during an oscillation cycle. The effect is shifted al later times Alessandro Mirizzi NOW 2014 Conca Specchiulla, 8 September 2014

15. NON-TRIVIAL SPACE BEHAVIOR homogeneous solution inhomogeneous solutions Alessandro Mirizzi NOW 2014 Conca Specchiulla, 8 September 2014

16. DECLINING NEUTRINO DENSITY 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 Alessandro Mirizzi NOW 2014 Conca Specchiulla, 8 September 2014

17. 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. Alessandro Mirizzi NOW 2014 Conca Specchiulla, 8 September 2014