On Einstein equations with cosmological constant in braneworld models

In this paper, we investigate the Einstein equations with cosmological constant for Randall–Sundrum (RS) and Dvali–Gabadadze–Porrati (DGP) models to determine the warp functions in the context of warp product spacetimes. In RS model, it is shown that Einstein’s equation in the bulk is reduced into the brane as a vacuum equation, having vacuum solution, which is not affected by the cosmological constant in the bulk. In DGP model, it is shown that the Einstein’s equation in the bulk is reduced into the brane and along the extra dimension, where both equations are affected by the cosmological constant in the bulk. We have solved these equations in DGP model, subject to vanishing cosmological constants on the brane and along extra dimension, and obtained exact solutions for the warp functions. The solutions depend on the typical values of cosmological constant in the bulk as well as the dimension of the brane. So, corresponding to the typical values, some solutions have exponential behaviors which may be set to represent warp inflation on the brane, and some other solutions have oscillating behaviors which may be set to represent warp waves or branes waves along the extra dimension.