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We study curvature perturbations in the anisotropic inflationary model with a complex scalar field charged under a U(1) gauge field in Bianchi I universe. Due to Abelian Higgs mechanism, the gauge field acquires a dynamical mass with an additional longitudinal mode. We verify that the dominant contributions into statistical anisotropies come from matter fields perturbations and one can neglect the contributions from the metric perturbations. The contribution of the longitudinal mode in statistical anisotropies power spectrum, though exponentially small, has an opposite sign compared to the corresponding contribution form the transverse modes. Since the inflaton field and the transverse modes of the gauge field perturbations have dynamical masses induced from the symmetry breaking the anisotropy mechanism becomes less efficient. As a result, the observed constraints on statistical anisotropies can be satisfied with a relaxed fine-tuning on the model parameters such as the time-dependent gauge kinetic coupling.
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