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The \delta N formalism is extended to include the perturbation of the vectorfield. The latter is quantized in de Sitter space-time and it is found that ingeneral the particle production process of the vector field is anisotropic.This anisotropy is parametrized by introducing two parameters p and q, whichare determined by the conformal invariance breaking mechanism. If any of themare non-zero, generated \zeta is statistically anisotropic. Then the powerspectrum of \zeta and the non-linearity parameter fNL have an angularmodulation. This formalism is applied for two vector curvaton models and theend-of-inflation scenario. It is found that for p \ne 0, the magnitude of fNLand the direction of its angular modulation is correlated with the anisotropyin the spectrum. If p \gtrsim 1, the anisotropic part of fNL is dominant overthe isotropic one. These are distinct observational signatures; their detectionwould be a smoking gun for a vector field contribution to \zeta . In the firstcurvaton model the vector field is non-minimally coupled to gravity and in thesecond one it has a time varying kinetic function and mass. In the former, onlystatistically anisotropic \zeta can be generated, while in the latter,isotropic \zeta may be realized too. Parameter spaces for these vector curvatonscenarios are large enough for them to be realized in the particle physicsmodels. In the end-of-inflation scenario fNL have similar properties to thevector curvaton scenario with additional anisotropic term.

Quantum Fluctuations of Vector Fields and the Primordial Curvature Perturbation in the Universe