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Statistical Anisotropy in Inflationary Models with Many Vector Fields and/or Prolonged Anisotropic Expansion

We study the most general contributions due to scalar field perturbations, vector field perturbations, and anisotropic expansion to the generation of statistical anisotropy in the primordial curvature perturbation ζ. Such a study is done using the δN formalism where only linear terms are considered. Here, we consider two specific cases that lead to determine the power spectrum P_ζ(k) of the primordial curvature perturbation. In the first one, we consider the possibility that the n-point correlators of the field perturbations in real space are invariant under rotations in space (statistical isotropy); as a result, we obtain as many levels of statistical anisotropy as vector fields present and, therefore, several preferred directions. The second possibility arises when we consider anisotropic expansion, which leads us to obtain I+a additional contributions to the generation of statistical anisotropy of ζcompared with the former case, being I and a the number of scalar and vector fields involved respectively.