Scalaron dynamics from UV to IR

We consider a scenario where the scalaron of $f({\cal R})$ models is related to the volume modulus of string compactifications leaving only one scalar degree of freedom at low energy. The coefficient of the leading curvature squared contribution to the low energy effective action of gravity determines the mass of the scalaron. We impose that this mass is small enough to allow for the scalaron to drive Starobinski's inflation. After inflation, the renormalisation group evolution of the couplings of the $f({\cal R})$ theory, viewed as a scalar-tensor theory, provides the link with the Infra-Red regime. We consider a scenario where the corrections to the mass of the scalaron are large and reduce it below the electron mass in the Infra-Red, so that the scalaron plays a central role in the low energy dynamics of the Universe. In particular this leads to a connection between the scalaron mass and the measured vacuum energy provided its renormalisation group running at energies higher than the electron mass never drops below the present day value of the dark energy.