Statistical anisotropy of the curvature perturbation from vector field perturbations

The δ N formula for the primordial curvature perturbation ζ is extended to include vector as well as scalar fields. Formulas for the tree-level contributions to the spectrum and bispectrum of ζ are given, exhibiting statistical anisotropy. The one-loop contribution to the spectrum of ζ is also worked out. We then consider the generation of vector field perturbations from the vacuum, including the longitudinal component that will be present if there is no gauge invariance. Finally, the δ N formula is applied to the vector curvaton and vector inflation models with the tensor perturbation also evaluated in the latter case.