Gravitational waves and cosmological braneworlds: A characteristic evolution scheme

Abstract

Motivated by the problem of the evolution of bulk gravitational waves in Randall-Sundrum cosmology, we develop a characteristic numerical scheme to solve $1+1$ dimensional wave equations in the presence of a moving timelike boundary. The scheme exhibits quadratic convergence, is capable of handling arbitrary brane trajectories, and is easily extendible to non-AdS bulk geometries. We use our method to contrast two different prescriptions for the bulk fluctuation initial conditions found in the literature; namely, those of Hiramatsu et al. and Ichiki and Nakamura. We find that, if the initial data surface is set far enough in the past, the late-time waveform on the brane is insensitive to the choice between the two possibilities; and we present numeric and analytic evidence that this phenomenon generalizes to more generic initial data. Observationally, the main consequence of this work is to reaffirm previous claims that the stochastic gravitational wave spectrum is predominantly flat ${\ensuremath{\Omega}}_{\mathrm{GW}}\ensuremath{\propto}{f}^{0}$, in contradiction with naive predictions from the effective 4-dimensional theory. Furthermore, this flat spectrum result is predicted to be robust against uncertainties in (or modifications of) the bulk initial data, provided that the energy scale of brane inflation is high enough.