Etudes des proprietes des neutrinos dans les contextes ...

tel-00450051, version 1 - 25 Jan 2010
with the corresponding diagonal term ̺ββ(pi)), and M12→34 is the process amplitude.
We approximate collisions with a simple damping prescription of the form
C[ραβ(p, t)] = −Dpραβ(p, t) . These damping factors for the off-diagonal elements
of ρ(p, t) in the weak interaction basis, mimicks the destruction of phase coherence
by flavor-sensitive collisions. Here we have decided to use the simplified
assumption following [48]. To sum up about the evolution equation, the diagonal
elements change by collisions and by oscillations, whereas the off-diagonal
elements change by oscillations and damping. Finally we can write
⎛
iDe−µ,τ(f(p, ξe) − ρνeνe) −iDe−µ,τρνeνµ −iDe−µ,τρνeντ
C[ρ(p, t)] ≡ ⎝
−iDe−µ,τρνµνe iDe−µ,τ(f(p, ξµ) − ρνµνµ) −iDµτρνµντ
−iDe−µ,τρντνe −iDµ−τρντνµ iDe−µ,τ(f(p, ξτ) − ρντντ)
(8.27)
with De−µ,τ = 2 × (4sin 4 θW − 2sin 2 θW + 2)F0 and Dµ−τ = 2 × (2sin 4 θW + 6)F0
[48].
8.2.2 The comoving variables
It is more convenient since the universe is expanding to use comoving variable.
We therefore define the dimensionless expansion rate by:
x ≡ mR , y ≡ pR , (8.28)
where R is the universe scale factor and m an arbitrary mass scale that we choose
to be 1 MeV. With such new variable we rewrite Eq.(8.19). We first consider the
l.h.s. of Eq.(8.19) by expressing the differential of ρ(x, y) over dt:
dρ(t, p)
dt
∂ρ dx
=
∂x y dt +
∂ρ dy
∂y x dt
= m ˙
∂ρ
R + m
∂x y
˙
∂ρ
R
∂y
∂ρ ∂ρ
= Hx + pH
∂x ∂p
y
x
x
(8.29)
which finally yields :
∂ρ ∂ρ
− pH = Hx
∂t ∂p
∂ρ
(8.30)
∂x
Consequently, the equation of motion for the period of expansion we are interested
in, is:
UM
iHx(∂x)ρ(x, y) =
2U †
2y − 8√
2GFy
E + √
2GF(ρ − ρ), ρ(x, y) +C[ρ(x, y)],
3m 2 W
142
(8.31)
⎞
⎠