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Braneworlds with field derivative coupling to the Einstein tensor

In this paper, we investigate the Randall-Sundrum type braneworld models in the scalar-tensor theory with field derivative coupling to the Einstein tensor. We first formulate the generalized junction conditions of the metric and the scalar field on the timelike codimension-one hypersurface (=brane). With the use of these junction conditions, we then derive the Minkowski and de Sitter brane solutions embedded into the $Z_2$-symmetric five-dimensional anti-de Sitter (AdS) bulk spacetime. The configuration of the scalar field depends on the slice of the AdS spacetime. These branes are supported by the tension and not coupled to the scalar field. The Minkowski brane solution can be obtained when the brane tension is tuned to the bulk contributions. Once this tuning relation is broken, the de Sitter brane solutions are obtained. The de Sitter brane solutions have two branches. One has the smooth limit to the case where the scalar field becomes trivial, while the other branch does not. The latter branch has the upper bound on the brane tension, where the expansion rate vanishes. Finally, we investigate the low energy effective gravitational theory realized on the brane, which is given by the four-dimensional Einstein-scalar theory with the corrections from the bulk. In the case that there is no generation of the dark radiation from the bulk scalar field, we recover the the four-dimensional Einstein-scalar theory.