Does anisotropic inflation produce a small statistical anisotropy?

Anisotropic inflation is an interesting model with an U(1) gauge field and it predicts the statistical anisotropy of the curvature perturbation characterized by a parameter g(*). However, we find that the background gauge field does not follow the classical attractor solution due to the stochastic effect. We develop the stochastic formalism of a vector field and solve Langevin and Fokker-Planck equations. It is shown that this model is excluded by the CMB constraint g(*) <= 10(-2) with a high probability about 99.999%.