Damping of neutrino flavor conversion in the wake of the supernova shock wave by G.L. Fogli, E. Lisi, D. Montanino, A. Mirizzi Based on hep-ph/ : Damping of supernova neutrino transitions in stochastic shock-wave density profiles Based on hep-ph/ : Damping of supernova neutrino transitions in stochastic shock-wave density profiles

Core collapse SN’s is one of the most energetic event in nature. It corresponds to the terminal phase of a massive star which becomes instable at the end of its life. It collapses and ejects its outer mantle in a shock wave driven explosion. ENERGY SCALES: 99% of the released energy available in the core collapse (~10 53 erg) is emitted by (anti)neutrinos of all flavors with energies of order of a tenth of MeVs. TIME SCALE: The duration of the burst lasts ~10s EXPECTED RATE: 1-3 SN/century in our galaxy (d  O(10) kpc).

cooling “Hot” PN star Shock wave Stellar core Schirato & Fuller astro-ph/ T. Totani et al., Astrophys. J. 496, 216 (1998). R c  10 4 km R PN  10 2 km

transitions The flavor evolution in matter is described by the MSW equation: where: not relevant for SNe not known butfrom solar & KamLAND  m 12 2 Normal Hierarchy (NH)  m 13 2 k H > Inverted Hierarchy (IH)  m 12 2 k H <0 from solar & KamLAND from atmospheric & K2K i i Neutrino potential in matter

Behind this phenomenology and neglecting Earth matter crossing, the (relevant) survival probability P ee  P( e  e ) can be decomposed into two effective “high” (H) and “low” (L) 2 subsystems, up to small terms of the order of O(  m 12 2 /  m 13 2, sin 2  13 ) : [see, e.g., G.L. Fogli et al., PRD68 (2003) , Dighe and Smirnov, PRD62, (2000) ] where the only dependence on the matter effect is encoded in terms of P 2,H ee. Clearly, a time dependent potential V=V(t) induced by the passage of the shock modulates the survival probability P 2,H ee and thus leaves an “imprint” on the time spectrum of the neutrino burst.

Livermore group (Schirato & Fuller), astro-ph/ Forward shock Models of shock Forward shock Reverse shock Garching group (Tomàs et al.), astro-ph/ Forward shock No reverse shock Tokio group (Kawagoe et al.), unpublished Our parameterization of the shock profile

Kifonidis (PhD thesis) X 10 9 cm t=4s t=10s t=20s

Stochastic scale density fluctuation of various magnitudes and correlation lengths may reasonably arise in the wake of a shock front (i.e., for r<r shock ). A SN neutrino “beam” traveling to the Earth might thus experience stochastic matter effects while traversing the stellar envelope. We shall assume L 0 =10km. With this hypothesis the density fluctuations can be considered “  -correlated”, i.e.: We will consider only “small” scale fluctuations, i.e., fluctuations whose correlation length is smaller than the typical oscillation wavelength in matter at resonance: with

 represents the “r.m.s” of the amplitude of the stochastic fluctuations and, in principle,  =  (r). Unfortunately, there is not an ab initio theory of small scale fluctuation, so we make the simplifying assumption that fluctuation arise only after the passage of the shock wave: We conservatively assume that  4%.

Evolution in fluctuating potentials Suppose that a system is described by an Hamiltonian which is composed by one “deterministic” and one “stochastic” component: where  (t) is random fluctuating  -correlated function: In this case Schrödinger equation is no longer adequate to describe the evolution of the system: [see, e.g., Balantekin et al., Burgess et al.]

“damping” term (perturbation for  up to ~10%) In our case Q=| e  e |. The previous modified “Liouville” equation can be written as a “Bloch” equation in term of the polarization vector Application to SN ’s standard term (leading) with, and the probability of observing an electron neutrino at distance r can be calculated as

We suppose that the fluctuations are sufficiently small to affect only the “High” subsector. After some calculations, the probability P 2,H ee in presence of random noise can be recast as with whereis the effective “13” mixing angle in matter. The effect of noise is thus to suppress the MSW effect into the stellar medium. In the limit of large fluctuations, one gets P 2,H ee  1/2, which correspond to a sort of complete “flavor depolarization” for the effective states in the H subsystem.

Conclusions The observation of a modulation in the survival probability caused by the passage of the shock wave inside the exploding supernova can give us valuable information on the unknown oscillation parameters (mass hierarchy,  13 ) as well as on the internal structure of the exploding star But small-scale fluctuations can partially hide this effect and cause a dangerous confusion scenario (no prompt shock? “wrong” hierarchy? too small  13 ?) For this reason, a better theoretical understanding of stochastic density fluctuations behind the shock front would be of great benefit for future interpretation of SN neutrino events

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Bonus slides

( ) in inverse (direct) hierarchy

Analytical vs Runge Kutta

X 10 8 cm Kifonidis et al., Astrophys. J. Lett., 531, L123 (2000) Evolution of entropy in SN explosion