Excitation of the Kaluza-Klein gravitational mode

We investigate excitation of Kaluza-Klein modes due to the parametric resonance caused by oscillation of the radius of compactification. We consider a gravitational perturbation around a D-dimensional spacetime, which we compactify on a $\left(D-4\right)$ sphere to obtain a four-dimensional theory. The perturbation includes the so-called Kaluza-Klein modes, which are massive in four dimensions, as well as zero modes, which are massless in four dimensions. These modes appear as scalar, vector, and second-rank symmetric tensor fields in the four-dimensional theory. Since Kaluza-Klein modes are troublesome in cosmology, quanta of these Kaluza-Klein modes should not be excited abundantly. However, if the radius of compactification oscillates, then the masses of Kaluza-Klein modes also oscillate and, thus, parametric resonance of Kaluza-Klein modes may occur to excite their quanta. In this paper we consider parts of the Kaluza-Klein modes that correspond to massive scalar fields in four dimensions and investigate whether or not quanta of these modes are excited in the so-called narrow resonance regime of the parametric resonance. We conclude that at least in the narrow resonance regime quanta of these modes are not excited so catastrophically.