  In models of the Universe with extra dimensions gravity propagates in the
whole space-time. Graviton production by matter on the brane is significant in
the early hot Universe. In a model of 3-brane with matter embedded in 5D
space-time conditions for gravitons emitted from the brane to the bulk to
return back to the brane are found. For a given 5-momentum of graviton falling
back to the brane the interval between the times of emission and return to the
brane is calculated. A method to calculate contribution to the energy-momentum
tensor from multiple graviton bouncings is developed. Explicit expressions for
contributions to the energy-momentum tensor of gravitons which have made one,
two and three bounces are obtained and their magnitudes are numerically
calculated. These expressions are used to solve the evolution equation for dark
radiation. A relation connecting reheating temperature and the scale of extra
dimension is obtained. For the reheating temperature $T_R\sim 10^6 GeV$ we
estimate the scale of extra dimension $\m$ to be of order $10^{-9} GeV\,\,\,
(\m^{-1}\sim 10^{-5} cm )$.
