Etudes des proprietes des neutrinos dans les contextes ...

tel-00450051, version 1 - 25 Jan 2010
Figure 5.6: Evolution of Pz and ¯ Pz for the system of equations Eq. (5.27) with
˜θ0 = 0.01 (i.e., inverted hierarchy), ω = 1, and strong neutrino-neutrino interaction
µ = 10. Taken from [72]
at their initial value which is 1. In order to estimate the time it takes for the
polarisation vectors to flip, we write the equation of motion for this case,
¨ϕ = κ 2 (ϕ − 2 ˜ θV ) . (5.43)
still in the approximation where ˜ θV ≪ 1 and ϕ is small. Using the initial conditions
ϕ(0) = −(ω/µQ) 2˜ θV and ˙ϕ(0) = 0, this equation is solved by
ϕ(t) = 2˜
θV 1 − 1 + ω
cosh(κt) . (5.44)
µQ
Therefore, the time it takes for ϕ to reach −π is 8
τbipolar ≈ −κ −1 ln[( ˜ θV
π
) (1 + ω/µQ)] . (5.45)
Therefore, the time for Pz and ¯ Pz to show a noticeable flavour conversion scales
logarithmically with ˜ θV . On Fig.(5.6), we show what are the respective evolution
of Pz and ¯ Pz in the inverted hierarchy case.
The matter background and the rotating frame
We now include the matter potential λL × P in Eqs.(5.27) to observe the consequence
on the pendulum behaviour. Using a change of coordinates in another
8 We considered here that at t of order κ −1 e κt dominates.
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