  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.
