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Recently the vector inflation has been proposed as the alternative toinflationary models based on scalar bosons and quintessence scalar fields. Inthe vector inflationary model, the vector field non-minimally couples togravity. We should, however, inquire if there exists a relevant fundamentaltheory which supports the inflationary scenario. We investigate the possibilitythat Weyl's gauge gravity theory could be such a fundamental theory. That isthe reason why the Weyl's gauge invariant vector and scalar fields arenaturally introduced. After rescaling the Weyl's gauge invariant Lagrangian tothe Einstein frame, we find that in four dimensions the Lagrangian isequivalent to Einstein-Proca theory and does not have the vector fieldnon-minimally coupled to gravity, but has the scalar boson with a polynomialpotential which leads to the spontaneously symmetry breakdown.

On the cosmology of Weyl's gauge invariant gravity