Rescaled Einstein-Hilbert Gravity from $f(R)$ Gravity: Inflation, Dark
  Energy and the Swampland Criteria
  We consider a class of exponential models of $f(R)$ gravity, in the presence
of a canonical scalar field, for which at early times the effective Lagrangian
of the theory becomes that of a rescaled canonical scalar field with the
Einstein-Hilbert term becoming $\sim \alpha R$, with $\alpha$ a dimensionless
constant. This rescaled Einstein-Hilbert scalar field theory at early times
alters the inflationary phenomenology of well-known scalar field models of
inflation, but more importantly, in the context of this rescaled theory, the
Swampland criteria are easily satisfied, assuming that the scalar field is
slowly rolling. We consider two models of inflation to exemplify our study, a
fibre inflation model and a model that belongs to the general class of
supergravity $\alpha$-attractor models. The inflationary phenomenology of the
models is demonstrated to be viable, and for the same set of values of the free
parameters of each model which ensure their inflationary viability, all the
known Swampland criteria are satisfied too, and we need to note that we assumed
that the first Swampland criterion is marginally satisfied by the scalar field,
so $\phi\sim M_p$ during the inflationary era. Finally, we examine the
late-time phenomenology of the fibre inflation potential in the presence of the
full $f(R)$ gravity, and we demonstrate that the resulting model produces a
viable dark energy era, which resembles the $\Lambda$-Cold-Dark-Mater model.
Thus in the modified gravity model we present, the Universe is described by a
rescaled Einstein-Hilbert gravity at early times, hence in some sense the
modified gravity effect is minimal primordially, and the scalar field controls
mainly the dynamics with a rescaled Ricci scalar gravity. However $f(R)$
gravity dominates at late-times, where it controls the dynamics,
synergistically with the scalar field.
