Spectrum of Perturbations in Anisotropic Inflationary Universe with Vector Hair

We study both the background evolution and cosmological perturbations of anisotropic inflationary models supported by coupled scalar and vector fields. The models we study preserve the U(1) gauge symmetry associated with the vector field, and therefore do not possess instabilities associated with longitudinal modes (which instead plague some recently proposed models of vector inflation and curvaton). We first intoduce a model in which the background anisotropy slowly decreases during inflation; we then confirm the stability of the background solution by studying the quadratic action for all the perturbations of the model. We then compute the spectrum of the $h_{\times}$ gravitational wave polarization. The spectrum we find breaks statistical isotropy at the largest scales and reduces to the standard nearly scale invariant form at small scales. We finally discuss the possible relevance of our results to the large scale CMB anomalies.