  If quantum fields exist in extra compact dimensions, they will give rise to a
quantum vacuum or Casimir energy. That vacuum energy will manifest itself as a
cosmological constant. The fact that supernova and cosmic microwave background
data indicate that the cosmological constant is of the same order as the
critical mass density to close the universe supplies a lower bound on the size
of the extra dimensions. Recent laboratory constraints on deviations from
Newton's law place an upper limit. The allowed region is so small as to suggest
that either extra compact dimensions do not exist, or their properties are
about to be tightly constrained by experimental data.
