Damping of neutrino flavor Damping of neutrino flavor conversion in the wake of conversion in the wake of the supernova shock wavethe supernova shock wave
by G.L. Fogli, E. Lisi, D. Montanino, A. Mirizziby G.L. Fogli, E. Lisi, D. Montanino, A. Mirizzi
Based on hep-ph/0603033:Damping of supernova neutrino transitions in
stochastic shock-wave density profiles
Based on hep-ph/0603033: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 (~1053 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 (dO(10)
kpc).
cooling
“Hot” PN star
Shock wave
Stellar core
Schirato & Fullerastro-ph/0205390
T. Totani et al., Astrophys. J. 496, 216 (1998).
Rc104km RPN102km
transitions The flavor evolution in matter is described by the MSW equation:
where:
not relevant for SNenot known but from solar & KamLAND
12
3
m122
Normal Hierarchy (NH)
m132
kH>0
12
3
Inverted Hierarchy (IH)
m122
kH<0
from solar & KamLAND
from atmospheric & K2K
ii
Neutrino potential in matter
Behind this phenomenology and neglecting Earth matter crossing, the (relevant) survival probability PeeP(ee) can be decomposed into two effective “high” (H) and “low” (L) 2 subsystems, up to small terms of the order of O(m12
2/m132, sin213):
[see, e.g., G.L. Fogli et al., PRD68 (2003) 033005, Dighe and Smirnov, PRD62, (2000) 033007] where the only dependence on the matter effect is encoded in terms of P2,H
ee.
Clearly, a time dependent potential V=V(t) induced by the passage of the shock modulates the survival probability P2,H
ee and thus leaves an “imprint” on the time spectrum of the neutrino burst.
Livermore group (Schirato & Fuller), astro-ph/0205390
Forward shock
Models of shock
Forward shock
Reverse shock
Garching group (Tomàs et al.), astro-ph/0205390
Forward shockNo reverse shock
Tokio group (Kawagoe et al.), unpublished
Our parameterization of the shock profile
Kifonidis (PhD thesis)
0 5 10 15X 109cm
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<rshock). A SN neutrino “beam” traveling to the Earth might thus experience stochastic matter effects while traversing the stellar envelope.
We shall assume L0=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 potentialsSuppose 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=|ee|. 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 P2,Hee in
presence of random noise can be recast as
with
where is 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 P2,H
ee1/2, 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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() in inverse (direct) hierarchy
Analytical vs
Runge Kutta
0 1 2 33 2 1 X 108cm
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Kifonidis et al., Astrophys. J. Lett., 531, L123 (2000)
Evolution of entropy in SN explosion