High-energy effects on the spectrum of the inflationary gravitational wave background in braneworld cosmology

We discuss the cosmological evolution of the inflationary gravitational wave background in the Randall-Sundrum single-brane model. In braneworld cosmology, in which the three-dimensional spacelike hypersurface that we live in is embedded in five-dimensional anti-de-Sitter (AdS(5)) spacetime, the evolution of gravitational wave (GW) modes is affected by the nonstandard expansion of the universe and the excitation of the Kaluza-Klein modes. These are significant in the high-energy regime of the universe. We numerically evaluate these two effects by solving the evolution equation for GWs propagating through the AdS(5) spacetime. Using a plausible initial condition from inflation, we find that the excitation of Kaluza-Klein modes can be characterized by a simple scaling relation above the critical frequency f(crit) determined from the length scale of the fifth dimension l. The remarkable point is that this relation generally holds as long as the matter content of the universe is described by the perfect fluid with the equation of state p=w rho for 0 <= w <= 1. The resultant scaling relation is translated into the energy spectrum of the inflationary gravitational wave background as Omega(GW)proportional to f((3w-1)/(3w+2)) for f > f(crit). This indicates that in the radiation dominant case (w=1/3), the two high-energy effects accidentally compensate each other and the spectrum becomes almost the same as the one predicted in the four-dimensional theory, i.e., Omega(GW)proportional to f(0).