Flux compactifications: stability and implications for cosmology

We study the dynamics of the size of an extra-dimensional manifold stabilized by fluxes. Inspecting the potential for the 4D field associated with this size ( the radion), we obtain the conditions under which it can be stabilized and show that stable compactifications on hyperbolic manifolds necessarily have a negative four-dimensional cosmological constant, in contradiction with experimental observations. Assuming compactification on a positively curved ( spherical) manifold we find that the radion has a mass of the order of the compactification scale, M-c, and Planck suppressed couplings. We also show that the model becomes unstable and the extra dimensions decompactify when the four-dimensional curvature is higher than a maximum value. This in particular sets an upper bound on the scale of inflation in these models: V-max similar to (McMP2)-M-2, independently of whether the radion or other field is responsible for inflation. We comment on other possible contributions to the radion potential as well as finite temperature effects and their impact on the bounds obtained.