  We consider how accelerated expansion, whether due to inflation or dark
energy, imposes strong constraints on fundamental theories obtained by
compactification from higher dimensions. For theories that obey the null energy
condition (NEC), we find that inflationary cosmology is impossible for a wide
range of compactifications; and a dark energy phase consistent with
observations is only possible if both Newton's gravitational constant and the
dark energy equation-of-state vary with time. If the theory violates the NEC,
inflation and dark energy are only possible if the NEC-violating elements are
inhomogeneously distributed in thecompact dimensions and vary with time in
precise synchrony with the matter and energy density in the non-compact
dimensions. Although our proofs are derived assuming general relativity applies
in both four and higher dimensions and certain forms of metrics, we argue that
similar constraints must apply for more general compactifications.
