Future dynamics in f(R) theories
  The $f(R)$ gravity theories provide an alternative way to explain the current
cosmic acceleration without invoking dark energy matter component. However, the
freedom in the choice of the functional forms of $f(R)$ gives rise to the
problem of how to constrain and break the degeneracy among these gravity
theories on theoretical and/or observational grounds. In this paper to proceed
further with the investigation on the potentialities, difficulties and
limitations of $f(R)$ gravity, we examine the question as to whether the future
dynamics can be used to break the degeneracy between $f(R)$ gravity theories by
investigating the future dynamics of spatially homogeneous and isotropic dust
flat models in two $f(R)$ gravity theories, namely the well known $f(R) = R +
\alpha R^{n}$ gravity and another by A. Aviles et al., whose motivation comes
from the cosmographic approach to $f(R)$ gravity. To this end we perform a
detailed numerical study of the future dynamic of these flat model in these
theories taking into account the recent constraints on the cosmological
parameters made by the Planck team. We show that besides being powerful for
discriminating between $f(R)$ gravity theories, the future dynamics technique
can also be used to determine the fate of the Universe in the framework of
these $f(R)$ gravity theories. Moreover, there emerges from our numerical
analysis that if we do not invoke a dark energy component with
equation-of-state parameter $\omega < -1$ one still has dust flat FLRW solution
with a big rip, if gravity deviates from general relativity via $f(R) = R +
\alpha R^n $. We also show that FLRW dust solutions with $f''<0$ do not
necessarily lead to singularity.
