Isotropisation of flat homogeneous universes with scalar fields

Starting from an anisotropic flat cosmological model(Bianchi type $I$), we show that conditions leading to isotropisation fall into 3 classes, respectively 1, 2, 3. We look for necessary conditions such that a Bianchi type $I$ model reaches a stable isotropic state due to the presence of several massive scalar fields minimally coupled to the metric with a perfect fluid for class 1 isotropisation. The conditions are written in terms of some functions $\ell$ of the scalar fields. Two types of theories are studied. The first one deals with scalar tensor theories resulting from extra-dimensions compactification, where the Brans-Dicke coupling functions only depend on their associated scalar fields. The second one is related to the presence of complex scalar fields. We give the metric and potential asymptotical behaviours originating from class 1 isotropisation. The results depend on the domination of the scalar field potential compared to the perfect fluid energy density. We give explicit examples showing that some hybrid inflation theories do not lead to isotropy contrary to some high-order theories, whereas the most common forms of complex scalar fields undergo a class 3 isotropisation, characterised by strong oscillations of the $\ell$ functions.