  Graviton production due to collapsing extra dimensions is studied. The
momenta lying in the extra dimensions are taken into account. A $D$-dimensional
background is matched to an effectively four-dimensional standard radiation
dominated universe. Using observational constraints on the present
gravitational wave spectrum, a bound on the maximal temperature at the
beginning of the radiation era is derived. This expression depends on the
number of extra dimensions, as well as on the $D$-dimensional Planck mass.
Furthermore, it is found that the extra dimensions have to be large.
