Quantum fluctuations on a thick de Sitter brane

We investigate quantum fluctuations on a de Sitter (dS) brane, which has its own thickness, in order to examine whether or not the finite thickness of the brane can act as a natural cut-off for the Kaluza-Klein (KK) spectrum. We calculate the amplitude of the KK modes and the bound state by using the zeta function method after a dimensional reduction. We show that the KK amplitude is finite for a given brane thickness and in the thin wall limit the standard surface divergent behavior is recovered. The strength of the divergence in the thin wall limit depends on the number of dimensions, e.g., logarithmic on a two-dimensional brane and quadratic on a four-dimensional brane. We also find that the amplitude of the bound state mode and KK modes depends on the choice of renormalization scale; and for fixed renormalization scales the bound state mode is insensitive to the brane thickness both for two- and four-dimensional dS branes.