Supernova turbulence and neutrino flavor transformations

Transcription

1 Supernova turbulence and neutrino flavor transformations Alexander Friedland Los Alamos work with Andrei Gruzinov (NYU)

2 Explosion and MSW transformations Schirato & Fuller (2002) astro-ph/ Density changes caused by the explosion do affect the MSW flavor transformations of the neutrinos H-res L-res Very important! An opportunity to observe the explosion in real time in neutrinos

3 Explosion and MSW: what will we see? Quite interesting: Resonant flavor transformations in many places What we ll see crucially depends on the density profile behind the shock H-res L-res The region behind the shock is unstable to convection -- A. Gruzinov a turbulent mess rather than an orderly series of resonances

4 Details of the density profile matter Thomas, Kachelrieß, Raffelt, Dighe, Janka & Scheck, JCAP09, 015 (2004) (I) Observed neutrino signal depends on the profile behind the front shock (II) Multi-dim. simulations indeed show turbulent mess rather than an orderly profile

5 3D simulations 3d simulations of the accretion shock instability Blondin, Mezzacappa, & DeMarino (2002) See pages/simulations.html Extensive, well-developed turbulence behind the shock More simulations scattered online (google is your friend!)

6 Core-collapse supernova and convection Convection behind the shock front is not just a curiosity: essential for the explosion mechanism! (Herant, Benz, Hix, Fryer, Colgate Ap. J. 435, 339 (1994)) Convection brings energy from the dense region near the proto-neutron star to the region behind the shock Carnot cycle Observing it would confirm the basic ingredient in the current paradigm of the SN explosion Scheck, Plewa, Janka, Kifonidis, and Muller, Phys. Rev. Lett. 92, (2004), t=1 s

7 Convective density fluctuations and neutrinos Convective density fluctuations persist for the duration of the neutrino burst (and beyond) Can modify 1-3 and solar MSW flavor transformations starting at t ~ 3-5 s (8-10 s) Fig. from K. Kifonidis, T. Plewa, L. Scheck, H.-T. Janka, E. Mueller, astro-ph/

8 How do we decide if the density fluctuations seen in the simulations matter for the neutrinos?

9 Find and apply an existing solution? First idea: find the solution in the existing literature and apply Well-known that density fluctuations could be important for neutrinos: A. Schafer, S. Koonin, Phys. Lett. B 185, 417 (1986) W. Haxton, W-M. Zhang, Phys. Rev. D 43, 2484 (1991) Analytical treatments of delta-correlated noise <δn(x) δn(y)>= n0 2 L0δ(x-y) Nicolaidis, Phys. Lett. B 262, 303 (1991) Loreti & Balantekin, Phys. Rev. D 50, 4762 (1994) Loreti, Qian, Fuller, Balantekin, Phys. Rev. D (1995) Balantekin, Fetter & Loreti, Phys. Rev. D 54, 3941 (1996) Burgess & Michaud, Annals Phys. 256, 1 (1997)...Yet turbulent fluctuations look nothing like delta-correlated noise. Taken literally, delta-corr. noise is unphysical. Could be a prototype of very short wavelength ( λosc) noise, but such fluctuations don t describe turbulence. Spin precession in turbulent magnetic field treated nicely in Miranda, Rashba, Rez, Valle, Phys.Rev.D70: (2004)

10 Kolmogorov turbulence In 3-d turbulence, energy pumped on large scales, dissipated on small scales Between these two scales (in the inertial range ), a turbulent cascade is formed, carrying energy from large to small scales. Scales relevant for neutrinos lie in the inertial range. For simplicity, consider incompressible fluid first (Kolmogorov) An eddy of a given size l fragments to smaller ones on the time scale of one turn, t~ l/v Energy is transported without piling up at any scale in the inertial range v l2 /(l/v l ) = const Velocities behave as a power law, v l ~ l 1/3. Density (temperatures) fluctuations in the realistic case should scale in a similar power law way. Vary power law to check robustness. Cascade forms on time scales of turn of large eddies.

11 Spectrum from 2d simulations by Kifonidis et al (Janka group) Fourier analysis done by Timur Rashba -- expected power law! numerical noise, resolution limited cascade

12 A toy problem Two-level neutrino system that goes through a noisy level-crossing H flavor 2ν = ( cos 2θ + A(x) sin 2θ sin 2θ cos 2θ A(x) ), where the matter term has a smooth linear component and a fluctuating component with variable normalization F and random phases A(x) = x + A noise (x) = x + F to model Kolmogorov, take β = 5/6 600 k=1 k β cos[kx + φ(k)] Throw on the computer and repeat a few times

13 Numerical calculations repeated 66 times with random phases for each fluctuation amplitude Three regimes are clearly seen Fluctuations negligible 1. Perturbative regime P F 2 Complete depolarization P 1/2 P Ν e Ν e Adiabatic in the absence of fluctuations noise amplitude F

14 Turbulence -> random walk on the flavor sphere Fluctuations negligible Perturbative regime Complete depolarization Instantaneous mass basis of the smooth component. In this basis, density fluctuations lead to a random walk on the flavor sphere. Don t need exact solutions: details vary between simulations and also in actual explosions! Instead, need to find the diffusion rate as a function of the amplitude and spectrum of the fluctuations analyze perturbative regime!

15 To find the diffusion rate Rotate to the instantaneous mass basis of the smooth density component [( ) ( )] iψ m idθ = m /dx cos 2θm sin 2θ + δa m ψ. idθ m /dx m sin 2θ m cos 2θ m Assume that without the fluctuations the evolution would be adiabatic. Then transitions between the basis states are driven by the off-diagonal fluctuating matter term δa sin 2θm. Perturbatively, P (ν 1 ν 2 ) G2 F 2 dk 2π C(k) xf x i x dx sin 2θ m (x) exp[i dx (2 m (x) k)] C(k) dx δn(0)δn(x) e ikx = C 0 k α. 2. For the linear smooth profile, can be evaluated by the saddle point method Θ(p 1) P G ( ) F k dkc(k)g, G(p) p p n 0 2 sin 2θ

16 General properties of the solution G(p) Θ(p 1) p p 2 1. The spectral function G(2Ek/Δm 2 sin2θ13) is peaked at k Δm 2 sin 2θ13/2E, which up to a factor equals to the inverse neutrino oscillation length λ H 47 km (E ν /15 MeV)(0.3/ sin 2θ 13 ) For fluctuations on longer distance scales, the response is approximately zero (exp. suppressed) these fluctuations are followed adiabatically Contributions of fluctuations on shorter scales are power-law suppressed ( k -2 ) Previously known analytical result for δ-correlated noise <δn(0)δn(x)>=n0 2 L0δ(x) is correctly reproduced (in the region of applicability P<<1)

17 Solution and Kolmogorov spectrum The integral over the spectrum of fluctuations for power-law spectrum can be done analytically πγ (1/2 α/2) In particular, for Kolmogorov turbulence α = -5/3 C(k) dx δn(0)δn(x) e ikx = C 0 k 5/3. we get P power law G F C 0 2 n 0 (2 sin 2θ 13) α+1 2Γ (1 α/2) P Kolm 0.84G F C 0 (2 sin 2θ 13 ) 2/3 / 2 n 0. We can get the overall normalization of the fluctuations C0 from the simulations, by examining large scales that are well resolved Remember this is in the perturbative regime. If it turns out that Pperturb > 1/2, the actual value is P =1/2 -- complete flavor depolarization P { Pperturb, P perturb 1/2 1/2, P perturb 1/2

18 K. Kifonidis et al. Observable effect To achieve complete depolarization, density fluctuations on large scales need to satisfy Details in A.F., A. Gruzinov, δρr 1/3 astro-ph/ ! 0.1θ13 ρr Simulations show order one fluctuations criterion satisfied and by a large margin P!Νe"Νe" We are here noise amplitude F

19 Some comments

20 The result is quite robust to variations in the turbulent spectrum For the general noise exponent α in C(k) dx δn(0)δn(x) e ikx = C 0 k α. the depolarization criterion is δn r /n r > fθ (α+1)/2 13 where the coefficient f varies from 0.04 to 0.25 as α varies from -1.5 to -2

21 Adiabaticity assumption? What if the evolution without fluctuations is non-adiabatic? If the neutrino oscillation length, λosc (Δm 2 /(2E) sin2θ) -1, is comparable or greater than than the outer scale of the turbulence (the radius of the shock) -> the effect of the turbulence comes in the power-law tail of G(p). Statistical averaging not guaranteed. If λosc is much greater than the radius of the shock the evolution nonadiabatic with or without turbulent fluctuations. For the H resonance, this would be the case for small sin2θ13. For the L resonance, the adiabaticity condition is satisfied -- the solar parameters are known.

22 Off-resonance depolarization Since on resonance the effect is strongly oversaturated, by continuity depolarization must become important before the density in the turbulence is diluted down to the resonance value -> The depolarization effect on the H-resonance starts setting in earlier, possibly at 3 seconds Turns on gradually (more so than the shock effect) See astro-ph/ in prep. for details

23 Turbulent shadow Turbulence produces 50/50 incoherent mixture of the two states ρ = ( 1/2 0 ) 0 1/2 This density matrix commutes with any Hamiltonian -> any other features neutrino encounters, before or after turbulence, have no effect for transitions between the states involved in the H-resonance Sensitivity to front shock in the range of densities covered by the turbulence is lost, replaced by the distinct signal from turbulence For δ-corr. noise this is noted by Fogli, Lisi, Mirizzi, hep-ph/ Remember this assumes adiabaticity the absence of the turbulent fluctuations, i.e. for H-resonance sin 2 2θ ) ) )

24 L-resonance The discussion concentrated on the H-resonance As the shock expands and the density behind drops, the L-resonance also becomes depolarized. Given the solar oscillation parameters, the depolarization is guaranteed in this case. Because the solar mixing angle is large, the L-resonance depolarization occurs both in neutrinos and antineutrinos. In contrast, the depolarization of the H-resonance occurs only in one channel -- which one depends on the sign of mass hierarchy When the L-resonance is depolarized, any matter effect in the Earth disappears -- turbulence again casts a shadow.

25 Summary and conclusions Turbulence is ubiquitous in all modern multi-dimensional simulations. The density fluctuations in the turbulence are large enough to cause flavor depolarization of both H and L resonances as the shock expands This leads to a time-dependent signature with certain characteristic features. For H-resonance, either in neutrinos or antineutrinos, for L-resonance in both. Can be used to test the key ingredient in the explosion paradigm and also learn about θ13 and the sign of the hierarchy. A standard model of the supernova neutrino signal is incomplete without the effects of turbulence. What is needed is a global calculation that puts together everything we know at the moment: best initial spectra, collective effects, turbulence, front shock, earth effect. Fun continues!