We discuss the cosmological evolution of the inflationary gravitational wave background ( IGWB ) in the Randall-Sundrum single-brane model .
In braneworld cosmology , in which three-dimensional space-like 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 non-standard expansion of the universe and the excitation of the Kaluza-Klein modes ( KK-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 KK-modes can be characterized by a simple scaling relation above the critical frequency f _ { crit } determined from the length scale of the fifth dimension \ell .
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 ( EOS ) p = w \rho for 0 \leq w \leq 1 .
The resultant scaling relation is translated into the energy spectrum of the IGWB as \Omega _ { GW } \propto f ^ { ( 3 w - 1 ) / ( 3 w + 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 } \propto f ^ { 0 } .