  We prove that inflation is forbidden in the most well understood class of
semi-realistic type IIA string compactifications: Calabi-Yau compactifications
with only standard NS-NS 3-form flux, R-R fluxes, D6-branes and O6-planes at
large volume and small string coupling. With these ingredients, the first
slow-roll parameter satisfies epsilon >= 27/13 whenever V > 0, ruling out both
inflation (including brane/anti-brane inflation) and de Sitter vacua in this
limit. Our proof is based on the dependence of the 4-dimensional potential on
the volume and dilaton moduli in the presence of fluxes and branes. We also
describe broader classes of IIA models which may include cosmologies with
inflation and/or de Sitter vacua. The inclusion of extra ingredients, such as
NS 5-branes and geometric or non-geometric NS-NS fluxes, evades the assumptions
used in deriving the no-go theorem. We focus on NS 5-branes and outline how
such ingredients may prove fruitful for cosmology, but we do not provide an
explicit model. We contrast the results of our IIA analysis with the rather
different situation in IIB.
