The statistically anisotropic curvature perturbation generated by f(φ)² F²

The inflaton might be coupled to a gauge field through a term f^2(\phi) F_\mu\nu F^\mu\nu. If f \propto a^{-2} where a(t) is the scale factor, the perturbation \delta W of the gauge field generates a potentially observable statistically anisotropic contribution to the primordial curvature perturbation during slow-roll inflation. The spectrum and bispectrum of this contribution have been calculated using the in-in formalism of quantum field theory. We give a simpler and more complete calculation using only the classical perturbations. The results suggest that either the entire curvature perturbation \zeta (both the statistically isotropic and anisotropic parts) is generated during slow-roll inflation, or else it is generated afterwards.