  We have studied cosmological constraints on the number and radii of possible
large extra dimensions. If such dimensions exist, Kaluza-Klein (KK) modes are
copiously produced at high temperatures in the early universe, and can
potentially lead to unacceptable cosmological effects. We show that during
reheating, large numbers of KK modes are produced. These modes are not diluted
completely by the entropy production during reheating because they are produced
non-relativistically. This means that the modes produced during reheating can
easily be the dominant component. For instance, for two extra dimensions the
bound on their radii from considering only the thermally produced KK modes is R
< 1.1 x 10^-4 mm. If the modes produced during reheating are also accounted
for, the bound is strengthened to R < 2.2 x 10^-5 mm. This bound is stronger
than all other known astrophysical or laboratory limits.
