Gravitational waves in cosmological models of Hořava-Witten theory

We study the behavior of gravitational waves and their back reaction on the background in cosmological solutions of the five-dimensional Hořava-Witten theory. As a dynamical background, we consider two cosmological solutions with spatially flat expanding FRW branes, called $\left(\uparrow \right)$ and $\left(\downarrow \right)$ solutions, in which the orbifold size increases and decreases in time, respectively. For these background solutions, the wave equation for the tensor perturbation can be solved by the method of separation of variables, and the mode functions are classified by a separation constant which can be regarded as a graviton mass. We show that the spatial behavior of the mode functions is the same for both background solutions but the temporal behavior is significantly different. We further show that for the $\left(\uparrow \right)$ solution the background bulk geometry is unstable against the back reaction of the perturbation, while for the $\left(\downarrow \right)$ solution the back reaction on the bulk geometry can be neglected. We also show that, in contrast with the effect to the bulk geometry, the back reaction of the perturbation significantly alters the intrinsic geometry of the brane for the $\left(\downarrow \right)$ solution.