Excitation of a Kaluza-Klein mode by parametric resonance

In this paper we investigate a parametric resonance phenomenon of a Kaluza-Klein mode in a $D$-dimensional generalized Kaluza-Klein theory. As the origin of the parametric resonance we consider a small oscillation of a scale of the compactification around a today's value of it. To make our arguments definite and for simplicity we consider two classes of models of the compactification: those by $S_{d}$ ($d=D-4$) and those by $S_{d_{1}}\times S_{d_{2}}$ ($d_1\ge d_2$, $d_{1}+d_{2}=D-4$). For these models we show that parametric resonance can occur for the Kaluza-Klein mode. After that, we give formulas of a creation rate and a number of created quanta of the Kaluza-Klein mode due to the parametric resonance, taking into account the first and the second resonance band. By using the formulas we calculate those quantities for each model of the compactification. Finally we give conditions for the parametric resonance to be efficient and discuss cosmological implications.