Various types of five-dimensional warp factor and effective Planck scale

Based on the assumption that the warp factor of four-dimensional spacetime and the one of fifth dimension are tied through a parameter α, we consider five-dimensional gravity with a 3-brane coupled to a bulk scalar field. For arbitrary value of α, the form of the warp factor is implicitly determined by hypergeometric function. Concretely, we show that the warp factor becomes explicit form for appropriate value of α, and study the relation between four-dimensional effective Planck scale and the brane tension. This setup allows the possibility of extending the diversity of brane world.