  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.
