Instabilities and Speeded-up Flavor Equilibration in Neutrino Clouds

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

2. Overview 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 n flavors 2. Consequent hardening of ne spectrum ……softening of nm,t spectra. Mechanics: Instabilities in (mean-field) non-linear evolution equations. Bonus: Beyond the mean-field…….. comparison of stable and unstablecases

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

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

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

6. Equations of motion: oscillationterm Mean fieldapproximation: Pastor and Raffelt (2002) Take flavor diagonal initial conditions (two flavors): and solve

8. Momentum distribution nx , ne near n -sphere

9. We can delete nm,newhen paired in angle. So, in effect,

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

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

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 L • A. Inverted hierarchy: • Get a complex eigenvalue quite generally. • Normal hierarchy: • Get a complex eigenvalue only in the case of very small neexcess. In complete simulations of the evolution–with all of L: This complex e.v . completescrambling of the angles and energies on the medium-fast time scale

13. This mixing rate and restriction onne excess is roughly in accord with results of Raffelt and Sigl (suitably scaled to their region above n-surface) But wait: A. Take our initial assignments of n 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 L =0 (no n mass term at all) In these cases, complete simulation gives total scrambling for both normal and inverted hierarchies

14. Fourteen beams Inverted hierarchy Upper hemisphere ne- nx dens. Diff. Units (nn GF)-1 Normal hierarchy 50 = .5 cm 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.

15. Implications for supernova There will be rapid spectral mixing just under the n 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 ?? • If no: we may be done – • except for the possible application to R process nucleosynthesis and the neutrino pulse from SN 2047h. 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 .

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 In a box of vol.= V N=# of n’s nn =N/V Or, we can just solve the system Case A: z=1 --- stable (neutrinos) (again) Friedland & Lunardini (2003) Friedland , McKellar, Okuniewicz (2006) Case B: z=0 ---- unstable (photons) Conjecture: General connection between complex ev. in MF result and log N behavior in complete solution..

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

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

19. Conclusion Irregularities in flow Instability in mean field linearized eqns. Interesting beyond-mean-field results ?? Rapid flavor scrambling Should do beyond MF including n 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. Steps= factors of 4 N=8192 N=8 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: 2.35 eV, 100 MeVg laser Both beams linearly polarized. Angle between polarizations not = np/2

23. Mean distance for scattering of the photon –from cross-section and laser beam density -- 109 cm. Question: What is distance for polarization exchange? Answer: 3 cm. (Kotkin and Serbo) Colliding photon clouds 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