Anti-Gravitating Brane-World Solutions for a de Sitter Brane in Scalar-Tensor Gravity

In the context of a five-dimensional theory with a scalar field non-minimally-coupled to gravity, we look for solutions that describe novel black-string or maximally-symmetric solutions in the bulk. The brane line-element is found to describe a Schwarzschild-(Anti)-de Sitter spacetime, and, here, we choose to study solutions with a positive four-dimensional cosmological constant. We consider two different forms of the coupling function of the scalar field to the bulk scalar curvature, a linear and a quadratic one. In the linear case, we find solutions where the theory, close to our brane, mimics an ordinary gravitational theory with a minimally-coupled scalar field giving rise to an exponentially decreasing warp factor in the absence of a negative bulk cosmological constant. The solution is characterised by the presence of a normal gravity regime around our brane and an anti-gravitating regime away from it. In the quadratic case, there is no normal-gravity regime at all, however, scalar field and energy-momentum tensor components are well-defined and an exponentially decreasing warp factor emerges again. We demonstrate that, in the context of this theory, the emergence of a positive cosmological constant on our brane is always accompanied by an anti-gravitating regime in the five-dimensional bulk.