  We point out some of the outstanding challenges for embedding inflationary
cosmology within string theory studying the process of reheating for models
where the inflaton is a closed string mode parameterising the size of an
internal cycle of the compactification manifold. A realistic model of inflation
must explain the tiny perturbations in the cosmic microwave background
radiation and also how to excite the ordinary matter degrees of freedom after
inflation, required for the success of Big Bang Nucleosynthesis. We study these
issues focusing on two promising inflationary models embedded in LARGE volume
type IIB flux compactifications. We show that phenomenological requirements and
consistency of the effective field theory treatment imply the presence at low
energies of a hidden sector together with a visible sector, where the Minimal
Supersymmetric Standard Model fields are residing. A detailed calculation of
the inflaton coupling to the fields of the hidden sector, visible sector, and
moduli sector, reveals that the inflaton fails to excite primarily the visible
sector fields, instead hidden sector fields are excited copiously after the end
of inflation. This sets severe constraints on hidden sector model building
where the most promising scenario emerges as a pure N=1 SYM theory, forbidding
the kinematical decay of the inflaton to the hidden sector. In this case it is
possible to reheat the Universe with the visible degrees of freedom even though
in some cases we discover a new tension between TeV scale SUSY and reheating on
top of the well-known tension between TeV scale SUSY and inflation.
