Future dynamics in $$f(R)$$ f ( R ) theories

The $$f(R)$$ gravity theories provide an alternative way to explain the current cosmic acceleration without invoking a dark energy matter component used in the cosmological modeling in the framework of general relativity. However, the freedom in the choice of the functional forms of $$f(R)$$ gives rise to the problem of the degeneracy among these gravity theories on theoretical and (or) observational grounds. In this paper 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 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 Aviles et al., whose motivation comes from the cosmographic approach to $$f(R)$$ gravity. We perform a detailed numerical study of the dynamics of these theories taking into account the recent constraints on the cosmological parameters made by the Planck Collaboration. We demonstrate that besides being useful for discriminating between these two $$f(R)$$ gravity theories, the future dynamics technique can also be used to determine the finite-time behavior as well as the fate of the Universe in the framework of these $$f(R)$$ gravity theories. There also emerges from our analysis the result that one still can have a dust flat FLRW solution with a big rip, if gravity is governed by $$f(R) = R + \alpha R^n $$ . We also show that FLRW dust solutions with $$f''<0$$ do not necessarily lead to singularities.