Quantum fluctuations in brane-world inflation without an 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 φ with m2=V''(φ). We decompose the bulk scalar field into the infinite mass spectrum of four-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 vertical bar m2 vertical bar l-bar2<<1, where l(bar sign) is the curvature radius of the effectively anti-de Sitter bulk, but it is violated if vertical bar m2 vertical bar l-bar2>>1, though the violation is very small. Then we evaluate the vacuum expectation value <δφ2> on the brane. We find the zero-mode contribution completely dominates if vertical bar m2 vertical barl-bar2<<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 vertical bar m2 vertical barl-bar2 is large