We show that cosmological observables can constrain the topology of the
compact additional dimensions predicted by string theory. To do this, we
develop a general strategy for relating cosmological observables to the
microscopic parameters of the potentials and field-dependent kinetic terms of
the multiple scalar fields that arise in the low-energy limit of string theory.
We apply this formalism to the Large Volume Scenarios in Type IIB flux
compactifications where analytical calculations are possible. Our methods
generalize to other settings.