General Properties of the Self-tuning Domain Wall Approach to the Cosmological Constant Problem

We study the dynamics of brane worlds coupled to a scalar field and gravity, and find that self-tuning of the cosmological constant is generic in theories with at most two branes or a single brane with orbifold boundary conditions. We demonstrate that singularities are generic in the self-tuned solutions compatible with localized gravity on the brane: we show that localized gravity with an infinitely large extra dimension is only consistent with particular fine-tuned values of the brane tension. The number of allowed brane tension values is related to the number of negative stationary points of the scalar bulk potential and, in the case of an oscillatory potential, the brane tension for which gravity is localized without singularities is quantized. We also examine a resolution of the singularities, and find that fine-tuning is generically re-introduced at the singularities in order to retain a static solution. However, we speculate that the presence of additional fields may restore self-tuning.