We investigate the generation of primordial gravitational waves from inflation in braneworld cosmologies with extra dimensions. Advantage of using primordial gravitational waves to probe extra dimensions is that their theory depends only on the geometry, not on the microscopic models of inflation and stabilization. D(D-3)/2 degrees of freedom of the free bulk gravitons are projected onto the 3d brane as tensor, vector and scalar modes. We found the following no-go results for a generic geometry of a five (or D) dimensional warped metric with four dimensional de Sitter (inflationary) slices and two (or one) edge of the world branes: Massive KK graviton modes are not generated from inflation (with the Hubble parameter H) due to the gap in the KK spectrum; the universal lower bound on the gap is sqrt{3/2} H. Massless scalar and vector projections of the bulk gravitons are absent, unlike in geometries with KK compactification. A massless 4d tensor mode is generated from inflation with the amplitude H/M_P, where M_P is the effective Planck mass during inflation, derived from the D dimensional fundamental mass M_S and the volume of the inner dimensions. However, M_P for a curved dS braneworld may differ from that of the flat brane at low energies, either due to the H-dependence of the inner space volume or variations in the brane separation before stabilization. Thus the amplitude of gravitational waves from inflation in braneworld cosmology may be different from that predicted by inflation in 4d theory.
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