  $f(R)$ gravity models belong to an important class of modified gravity models
where the late time cosmic accelerated expansion is considered as the
manifestation of the large scale modification of the force of gravity. $f(R)$
gravity models can be expressed in terms of a scalar degree of freedom by
redefinition of models variable. The conformal transformation of the action
from Jordan frame to Einstein frame makes the scalar degree of freedom more
explicit and can be studied conveniently. We have investigated the features of
the scalar degree of freedoms and the consequent cosmological implications of
the power-law ($\xi R^n$) and the Starobinsky (disappearing cosmological
constant) $f(R)$ gravity models numerically in the Einstein frame. Both the
models show interesting behaviour of their scalar degree of freedom and could
produce the accelerated expansion of the Universe in the Einstein frame with
the negative equation of state of the scalar field. However the scalar field
potential for the power-law model is the well behaved function of the field,
whereas the potential becomes flat for higher value of field in the case of the
Starobinsky model. Moreover, the equation of state of the scalar field for the
power-law model is always negative and less than -1/3, which corresponds to the
behaviour of the dark energy that produces the accelerated expansion of the
Universe. This is not always the case for the Starobinsky model. At late times
Starobinsky model behaves as cosmological constant $\Lambda$ as behaves by
power-law model for the values of $n\rightarrow 2$ at all times.
