Are Kaluza-Klein modes enhanced by parametric resonance?

We study parametric amplification of Kaluza-Klein (KK) modes in a higher $D$-dimensional generalized Kaluza-Klein theory, which was originally considered by Mukohyama in the narrow resonance case. It was suggested that KK modes can be enhanced by an oscillation of a scale of compactification by the $d$-dimensional sphere $S^d~(d=D-4)$ and by the direct product $S^{d_1}\times S^{d_2}~(d_1+d_2=D-4)$. We extend this past work to the more general case where initial values of the scale of compactification and the quantum number of the angular momentum $l$ of KK modes are not small. We perform analytic approaches based on the Mathieu equation as well as numerical calculations, and find that the expansion of the universe rapidly makes the KK field deviate from instability bands. As a result, KK modes are not enhanced sufficiently in an expanding universe in these two classes of models.