Study of General Relativity and $f(R)$ Modified Gravity with
  Cosmological Constant for Einstein and Jordan Frames
  This thesis investigates a toy model for inflation in a class of modified
theories of gravity in the metric formalism. Instead of the standard procedure
-- assuming a non-linear Lagrangian $f(R)$ in the Jordan frame -- we start from
a simple {\phi}^2 potential in the Einstein frame and investigate the
corresponding $f(R)$ in the former picture. Such approach yields plenty of new
pieces of information, namely a self-terminating inflationary solution with a
linear Lagrangian, a robust criterion for stability of such theories, a
dynamical effective potential for the Ricci scalar $R$, the addition of an
ad-hoc Cosmological Constant in the Einstein frame leads to a Thermodynamical
interpretation of this physical system, which allows further insight on its
(meta)stability and evolution.
