Quantum fluctuations in brane-world inflation without inflaton on the brane

A Randall-Sundrum type brane-cosmological model in which slow-roll inflation on the brane is driven solely by a bulk scalar field was recently proposed by Himemoto and Sasaki. We analyze their model in detail and calculate the quantum fluctuations of the bulk scalar field $\varphi$ with $m^2=V^{\mathrm{\prime }\mathrm{\prime }}(\varphi )$. We decompose the bulk scalar field into the infinite mass spectrum of 4-dimensional fields; the field with the smallest mass-square, called the zero-mode, and the Kaluza-Klein modes above it with a mass gap. We find the zero-mode dominance of the classical solution holds if $\mathrm{\mid }{m}^2\mathrm{\mid }{\bar{\mathrm{\ell }}}^2\ll 1$, where $\bar{\mathrm{\ell }}$ is the curvature radius of the effectively anti-de Sitter bulk, but it is violated if $\mathrm{\mid }{m}^2\mathrm{\mid }{\bar{\mathrm{\ell }}}^2\gg 1$, though the violation is very small. Then we evaluate the vacuum expectation value $<\delta {\varphi }^2>$ on the brane. We find the zero-mode contribution completely dominates if $\mathrm{\mid }{m}^2\mathrm{\mid }{\bar{\mathrm{\ell }}}^2\ll 1$ similar to the case of classical background. In contrast, we find the Kaluza-Klein contribution is small but non-negligible if the value of $\mathrm{\mid }{m}^2\mathrm{\mid }{\bar{\mathrm{\ell }}}^2$ is large.