Abstract
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.
Highlights
A wide range of cosmological observations coming from different sources, including the supernovae type Ia (SNe Ia)[1,2,3], the cosmic microwave background radiation (CMBR) [4,5], and baryon acoustic oscillation (BAO) surveys [6,7,8,9,10,11], clearly indicate that the Universe is currently expanding with an accelerating rate
To proceed with the investigation on the potentialities, difficulties, and limitations of f (R) gravity, we examine this question by investigating the future dynamics of Friedmann–Lemaître–Robertson–Walker (FLRW) dust flat model in two f (R) gravity theories, namely the well-known f (R) = R + α Rn gravity, for which many results are available in the literature [88,89,90,91], and another by Aviles, Bravetti, Capozziello, and Luongo [92] (ABCL gravity for short), whose motivation comes from the cosmographic approach to f (R) gravity [93,94,95,96,97]
There have been a great deal of recent papers on f (R) gravity motivated by the attempts to explain the current cosmic acceleration with no need of invoking a dark energy component
Summary
A wide range of cosmological observations coming from different sources, including the supernovae type Ia (SNe Ia). The fact that f (R) theories can potentially be used to explain the observed accelerating expansion has given birth to a number of articles on these gravity theories, in which several features of f (R) gravity have been discussed [19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,42], including the stability conditions [43,44,45,46,47], compatibility with solar-system tests [48,49,50,51,52,53,54,55,56], energy conditions [57,58,59,60,61,62,63,64], nonlocal causal structure [65,66,67,68], and observational constraints from a diverse set of cosmological observations [69,70,71,72,73,74,75,76,77,78,79,80,81]. The freedom in the choice of the functional forms of f (R) has motivated many different suggestions of f (R) gravity theories, which account for the accelerating expansion and are compatible with the solar-system tests, it gives rise to the problem of how to constrain and break the degeneracy among these gravity theories on theo-
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