Thick braneworlds generated by a non-minimally coupled scalar field and a Gauss–Bonnet term: conditions for the localization of gravity

Abstract

We consider warped five-dimensional thick braneworlds with four-dimensional Poincaré invariance originated from bulk scalar matter non-minimally coupled to gravity plus a Gauss–Bonnet term. The background field equations as well as the perturbed equations are investigated. A relationship between 4D and 5D Planck masses is studied in general terms. By imposing finiteness of the 4D Planck mass and regularity of the geometry, the localization properties of the tensor modes of the perturbed geometry are analysed to first order, for a wide class of solutions. In order to explore the gravity localization properties for this model, the normalizability condition for the lowest level of the tensor fluctuations is analysed. It is found that for the examined class of solutions, gravity in four dimensions is recovered if the curvature invariants are regular and the 4D Planck mass is finite. It turns out that both the addition of the Gauss–Bonnet term and the non-minimal coupling between the scalar field and gravity reduce the value of the 4D Planck mass compared to its value when the scalar field and gravity are minimally coupled and the Gauss–Bonnet term is absent. The above discussed analysis depends on the explicit form of the scalar field (through its non-minimal coupling to gravity), making necessary the construction of explicit solutions in order to obtain results in closed form, and is illustrated with some examples which constitute smooth generalizations of the so-called Randall–Sundrum braneworld model. These solutions were obtained by making use of a detailed singular perturbation theory procedure with respect to the non-minimal coupling parameter between the scalar field and gravity, a difficult task that we managed to perform in such a way that all the physically meaningful conditions for the localization of gravity are fully satisfied. From the obtained explicit solutions, we found an interesting effect: when we consider a non-minimally coupled scalar–tensor theory, there arise solutions for which the symmetries of the background geometry are not preserved by the scalar matter energy density distribution. In particular, the value of the ‘5D cosmological constant’ of the asymptotically AdS5 spacetime (which is even with respect to the extra coordinate) gets different contributions at −∞ and +∞ from the asymptotic values of the self-interaction potential of the scalar field. Thus, an asymmetric energy density distribution of scalar matter gives rise to a spacetime which is completely even with respect to the fifth coordinate, in contrast to braneworld models derived from minimally coupled scalar–tensor theories, where both entities possess the same symmetry.