Inflationary models including vector fields have attracted a great deal ofattention over the past decade. Such an interest owes to the fact that theymight contribute to, or even be fully responsible for, the curvatureperturbation imprinted in the CMB. However, the necessary breaking of thevector field's conformal invariance during inflation is not without problems.In recent years it has been realized that a number of instabilities endangeringthe consistency of the theory arise when the conformal invariance is broken bymeans of a non-minimal coupling to gravity. In this paper we consider a massivevector field non-minimally coupled to gravity through the Gauss-Bonnetinvariant, and investigate whether the vector can obtain a nearlyscale-invariant perturbation spectrum while evading the emergence ofperturbative instabilities. We find that the strength of the coupling must beextremely small if the vector field is to have a chance to contribute to thetotal curvature perturbation.
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