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Abstract

Due to their special properties neutrinos are valuable messengers of physical information. In particular the low interaction rate in matter allows us to receive neutrino signals from otherwise inaccessible regions, like the interior of stars. For the correct interpretation of these signals a profound comprehension of associated processes is necessary. One of them is the conversion of neutrino flavours caused by neutrino mixing. When neutrinos propagate in a medium, the interaction with the environment can influence the mixing, i.e. enhance or suppress flavour conversions. At high neutrino densities, as they occur for example in core-collapse supernovae or neutron-star mergers, neutrinos themselves are the medium and neutrino-neutrino interactions can dominate the flavour evolution, which leads to the appearance of collective flavour modes. One class of collective phenomena is called fast flavour conversion because the length scale at which the oscillation takes place is much shorter than the scale of other oscillation effects. The conversion occurs when the initially small flavour coherence becomes unstable and grows exponentially. The existence of unstable fast-flavour modes depends crucially on the angular lepton-number distribution. This thesis is dedicated to investigate the theoretical origin of fast flavour conversions. For this purpose the formalism based on the matrix of flavour densities is applied and the corresponding equation of motion is derived including refraction terms from the interaction with a medium. The result is used to calculate the dispersion relation for the flavour correlation function as long as its equation of motion can be linearised. The connection between instabilities in the correlation function and crossings in the angular lepton number distribution is proven for axially symmetric systems by deriving the full set of eigenfunctions and matching them for collective and non-collective modes. In this context the difference between the stability of symmetry preserving and breaking modes is demonstrated. Finally the assumption of axial symmetry is dropped and examples are studied for general settings, i.e. with arbitrary wave vectors and non-symmetric lepton-number distributions.