Etudes des proprietes des neutrinos dans les contextes ...

tel-00450051, version 1 - 25 Jan 2010
The approximations in the analytical treatment
To obtain such result we made several approximations. First, we made the approximation
that, at each transition, one of the neutrinos is decoupled so that the
calculation of probability reduces to a 2 flavours problem, which can be a good
approximation (see Fig.(1.6)). Such an approximation could be avoided if the
evolution of the three matter eigenstates were perfectly adiabatic. This would
mean that the effective matter Hamiltonian could be diagonal all along the evolution
of neutrinos from the neutrino sphere to the surface of the star. This is
of course not the case, since we can only make an instantaneous diagonalization
of the Hamiltonian. Therefore, each neutrinos interfere with the two others at
every moment.
Second, the factorization can be performed because we suppose that the two resonances
are well separated and do not influence each other. This approximation
is reasonable because, for instance in supernova, at least an order of magnitude
separate in distance the two resonances, due to the fact that there are two scales
of ∆m 2 , namely ∆m 2 13 /∆m2 12
∼ 40.
The third approximation made was to consider that the matter eigenstates are
equal to the flavour eigenstates initially, which is true only in the approximation
of an initial infinite density. Fourth and last approximation, we consider that
the fluxes of νµ and ντ were equal at the neutrino sphere. This approximation is
quite good depending on the studied problem, and was made here for a question
of simplicity.
The analytical treatment just described, with the factorization approximations,
has been extensively used in the literature since it gives quite accurate
physical results. However the first approximation has actually an important
drawback, it only considers factorized transition probabilities and neglects all
possible phase effects between the eigenstates. It also neglects the CP-violating
phase. As we are going to see from our results (chapter 4, 6 and 7), such approximation
can therefore miss relevant physics. Only a complete numerical code
which solves with a good accuracy the system of coupled differential equations
describing exactly the neutrino evolution in media can encode all the interesting
and relevant physical phenomena, as we will discuss in the following.
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