1. Instabilities and speeded-up flavor equilibration in neutrino clouds Ray Sawyer UCSB

2. Domain of Application: Neutrino clouds with # densities of about (7 MeV) 3 (but with non-thermal distributions) just under “neutrino-surface” in SN. Phenomena: 1. Rapid exchange of  flavors 2. Consequent hardening of e spectrum ……softening of  spectra. Mechanics: Instabilities in (mean-field) non-linear evolution equations. Bonus: Beyond the mean-field…….. comparison of stable and unstable cases Overview

3. Time scales: Very fast: Medium fast: Kostelecki & Samuel 1995 Raffelt, Sigl 2007 Esteban-Pretel et al 2007 G Fuller group RFS 2004 2005 Hannestad et al 2006 (“bimodal”) Oscillation:

4. Time scales: Very fast: Medium fast: Oscillation: this talk Faster flavor scrambling of spectra —in a disorganized medium. Also: relatively independent of electron neutrino excess, and normal vs. Inverted hierarchy

5. This term doesn’t contribute anything. Density operators: Forward Hamiltonian : Commutation rules: Angle dependence is key  Interactions: Sum is over states, p, q that are occupied by ’s of some flavor.

6. Equations of motion : oscillation term Mean field approximation : Take flavor diagonal initial conditions (two flavors): and solve Pastor and Raffelt (2002)

8. Momentum distribution  x  e near -sphere

9. We can delete    e when paired in angle. So, in effect,

10. Two beams------N up, N down (and no  osc. terms) For the up-moving states define, For the down-moving states similarly, Collective coordinates Hamiltonian E of M Initials:  =1 neutrinos  =0 photons

11. With these initial conditions (and in mean field): Nothing happens. But when  =0 there is an instability for rapid growth of a perturbation. Eigenvalues of linearized problem: Growth So - in the photon problem  we would get mixing time scale, if no matter how tiny In  problem  =1, ., marginally stable. Including flavor-diagonal parts of mass terms (inverted hierarchy) get =complex. and “medium fast” (Kotkin & Serbo)

12. Back to Now do 14 beam case but in 3-space, & interpolating the up-down asymmetry of 2-beam simulation Linearized stability analysis-- taking only the flavor-diagonal part of  A. Inverted hierarchy : Get a complex eigenvalue quite generally. B.Normal hierarchy: Get a complex eigenvalue only in the case of very small e excess. In complete simulations of the evolution –with all of  : This complex e.v. complete scrambling of the angles and energies on the medium-fast time scale

13. This mixing rate and restriction on  excess is roughly in accord with results of Raffelt and Sigl (suitably scaled to their region above  surface) But wait: A. Take our initial assignments of densities in the various rays — B. Introduce random variations of order 10% in these assignments Then-- 60% of the time there is a complex ev even when  (no  mass term at all) In these cases, complete simulation gives total scrambling for both normal and inverted hierarchies

14. Normal hierarchy Inverted hierarchy Results of solving 56 coupled nonlinear equations over 1000’s of oscillation times. We have found that complete mixing ensues whenever the 28x28 matrix for the linear response has a complex eigenvalue. Fourteen beams Units (n G F ) -1 50 =.5 cm Upper hemisphere e  -  x  dens. Diff.

15. Implications for supernova There will be rapid spectral mixing just under the  surface, given the chaotic flows produced by SN simulations The most interesting question is: If in SN simulations the spectra are artificially scrambled at each time step, ……. would it affect the gross features of the hydrodynamics ?? A.If no: we may be done – except for the possible application to R process nucleosynthesis B. If yes, there is more to do.-- Take some data from a simulation with detailed angular distributions. Fit to a bunch of rays, and test the eigenvalues of the linearized system. and the neutrino pulse from SN 2047h.

16. Beyond the mean field (in two beam case) With no oscillation term or no Initial tilt------- the MF equations say that nothing happens. But we can estimate in PT a flavor mixing time Or, we can just solve the system Case A:  1 --- stable (neutrinos) ( again) Case B:  0 ---- unstable (photons) Friedland & Lunardini (2003) Friedland, McKellar, Okuniewicz (2006) In a box of vol.= V N=# of ’s n =N/V Conjecture: General connection between complex ev. in MF result and log N behavior in complete solution..

17. under the influence of Spin system: How long for this : to go into this? or this?

18. time in units (GN) -1 N=512 N=8 Comments: A logarithm is never large. Coherence lengths?

19. Conclusion Irregularities in flow Instability in mean field linearized eqns. Rapid flavor scrambling Interesting beyond-mean-field results ?? Should do beyond MF including mass terms

20. We defined MF approximation by taking operators, O: taking commutators, [ H,O], to get E of M, and then <>’s to get MF eqns. Now, instead, take the operators commute with H and take <>’s, getting closed set of 6 eqns. Scaling time and density:

21. N=8192 N=8 Steps= factors of 4 Solid curves --- solutions to Dashed curves --- solutions

22. Photon-photon scat: (polarizations now take the place of flavors and Heisenberg-Euler replaces Z-exchange.) laser Laser: 2.35 eV, 100 MeV  Both beams linearly polarized. Angle between polarizations not = n 

23. Mean distance for scattering of the photon –from cross-section and laser beam density -- 10 9 cm. Question: What is distance for polarization exchange? Answer: 3 cm. (Kotkin and Serbo) laser photon cloud Now with one cloud unpolarized and the other polarized: The polarized cloud loses polarization in distance 3 log[N] cm. RFS Phys.Rev.Lett. 93 (2004) 133601 Colliding photon clouds