The statistically anisotropic curvature perturbation generated by f2(φ)FμνFμν

The inflaton might be coupled to a gauge field through a term f(2)(phi)F mu nu F mu nu. If f proportional to 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.