Strong constraint on large extra dimensions from cosmology

We study 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 nonrelativistically. 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 $R<~1.1\times {10}^{-4}\hspace{0.5em}\mathrm{mm}.$ If the modes produced during reheating are also accounted for, the bound is strengthened to $R<~2.2\times {10}^{-5}\hspace{0.5em}\mathrm{mm}.$ This bound is stronger than all other known astrophysical or laboratory limits.