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Light vector fields during inflation obtain a superhorizon perturbationspectrum when their conformal invariance is appropriately broken. Suchperturbations, by means of some suitable mechanism (e.g. the vector curvatonmechanism), can contribute to the curvature perturbation in the Universe andproduce characteristic signals, such as statistical anisotropy, on themicrowave sky, most recently surveyed by the Planck satellite mission. Themagnitude of such characteristic features crucially depends on the magnitude ofthe vector condensate generated during inflation. However, the expectationvalue of this condensate has so-far been taken as a free parameter, lacking adefinite prediction or a physically motivated estimate. In this paper, we studythe stochastic evolution of the vector condensate and obtain an estimate forits magnitude. Our study is mainly focused in the supergravity inspired casewhen the kinetic function and mass of the vector boson is time-varying duringinflation, but other cases are also explored such as a parity violating axialtheory or a non-minimal coupling between the vector field and gravity. As anexample, we apply our findings in the context of the vector curvaton mechanismand contrast our results with current observations.

Inflationary buildup of a vector field condensate and its cosmological consequences