Expanding AdS5 Branes: Time Dependent Eigenvalue Problem and Production of Particles

An analysis is first given of the situation where a scalar field is contained between two fixed, spatially flat, branes. The usual fine-tuning (RS) condition is relaxed, and the branes are allowed to possess a positive effective cosmological constant \lambda. We first analyze the eigenvalue problem for the Kaluza-Klein masses when the metric is time dependent, and consider in detail the case when \lambda is small. Thereafter we consider, in the case of one single brane, the opposite limit in which \lambda is large, acting in a brief period of time T, and present a "sudden approximation" calculation of the energy produced on the brane by the rapidly expanding de Sitter space during this period.