  A four-dimensional universe, arising from a flux compactification of Type IIB
string theory, contains scalar fields with a potential determined by
topological and geometric parameters of the internal -hidden- dimensions. We
show that inflation can be realized via rolling towards the large internal
volume minima that are generic in these scenarios, and we give explicit
formulae relating the microscopic parameters (e.g., the Euler number of the
internal space) to the cosmological observables (e.g., the spectral index). We
find that the tensor-to-scalar ratio, the running of the spectral index, and
the potential energy density at the minimum are related by consistency
relations and are exponentially small in the number of e-foldings. Further,
requiring that these models arise as low-energy limits of string theory
eliminates most of them, even if they are phenomenologically valid. In this
context, this approach provides a strategy for systematically falsifying
stringy inflation models.
