# V. CONCLUSIONS

In this paper we studied a modified gravity theoretical framework which extends the vacuum f(R) gravity theory, and it consists of higher order scalar field kinetic terms that are added to the standard f(R) gravity Lagrangian density. Due to the form of the extra terms in the action, we called this theory k-essence f(R) gravity theory, and our main aim was to investigate the inflationary aspects of this theory, in the slow-roll approximation. Actually, the class of k-essence f(R) gravity theory which we studied in this paper gets very much simplified if the slow-roll condition is imposed on the scalar field, and we investigated the dynamics of inflation in the resulting theory. By using standard formulas for the slow-roll indices coming from generalized \( f ( R , \phi , X ) \) theories studied some time ago, we derived the slow-roll indices for a general f(R) gravity, and then we applied the formalism for an f(R) gravity of the form \( R \! + \! \alpha R ^ { n } \). This theory without the k-essence part is not compatible with the latest Planck observational data, so we questioned the viability of the theory in view of the presence of the k-essence terms. As we demonstrated, there is a range of values of the free parameters for which the phenomenological viability of the theory can be achieved, for both the phantom and ghost-free models which we used. Since the result might be model dependent, we used another approach in order to see whether the k-essence f(R) gravity can produce viable phenomenology. To this end, we fixed the Hubble rate as a function of the e-foldings number, and we modified standard f(R) gravity reconstruction techniques to accommodate the presence of the k-essence terms, always in the slow-roll approximation. Using the resulting reconstruction techniques we derived the k-essence f(R) gravity which can realize the given Hubble rate, and then we provided general formulas for the slow-roll indices as functions of the e-foldings number, always in the slow-roll approximation. Accordingly, we calculated the slow-roll indices and the corresponding observational indices and we demonstrated that the resulting theory can be compatible with the Planck data, however the result is strongly model dependent. Thus we validated that the k-essence f(R) gravity theory can produce phenomenologically viable cosmologies in the slow-roll approximation. The latter is a vital ingredient of the formalism we employed, so the basic question is, does this theory have inflationary attractors in the absence of the slow-roll condition? The vacuum f(R) gravity theory has stable and unstable de Sitter attractors without the slow-roll condition implied, as was explicitly demonstrated in [31], by using the dynamical system approach, so the question is does a general non-slow-roll k-essence f(R) gravity possesses inflationary attractors? This question is non-trivial and no one can guarantee this, before a consistent autonomous dynamical system is derived for the theory in question. For example, in the case of Gauss-Bonnet gravity there exist inflationary attractors even if the slow-roll condition does not hold true, although these are unstable, as was proved in Ref. [32], and the same applies for vacuum f(R) gravity theories in the presence of a non-flat metric [33]. Moreover, the existence of unstable de Sitter attractors is a feature of f(R)-\( \phi \) theories [34].However the latter type of theory contains potential terms, which are absent in the k-essence f(R) gravity, so the next major task is to question the existence of inflationary attractors in the non-slow-roll k-essence f(R) gravity theory. To this end one should appropriately construct a consistent autonomous dynamical system, study its fixed points, and test their stability, analytically if these are hyperbolic fixed points, or at least numerically if the fixed points are non-hyperbolic. The interpretation of the existence of unstable inflationary attractors is a major issue in these theories, which in some sense can be viewed as an inherent mechanism for the graceful exit from inflation, but this is a highly non-trivial issue to discuss here, and of course out of the context of this work. Work is in progress along the above research lines.

Finally, it is noteworthy mentioning that even in this k-essence framework, it is unavoidable having the initial Big Bang singularity, when inflationary scenarios are considered. However, it is interesting to note that, if the underlying theory can go beyond the k-essence f(R) gravity type inflation, namely a torsional based f(T ) modified gravity[59, 60], or a Horndeski scalar [61] one may not only realize inflationary cosmology, but also a non-singular bouncing phase that can be applied to avoid the big bang singularity. In fact, it would be interesting to extend the formalism we developed in this paper to find an appropriate k-essence f(R) gravity type theory that may realize a bouncing cosmology. In the context of other extensions of f(R) gravity this is also possible [62], so the question remains if there are k-essence modified gravities that may realize cosmological bounces. We hope to address this issue in a future work.

# Acknowledgments

This work is supported by MINECO (Spain), FIS2016-76363-P, and by project 2017 SGR247 (AGAUR, Catalonia) (S.D.O). This work is also supported by MEXT KAKENHI Grant-in-Aid for Scientific Research on Innovative Areas “Cosmic Acceleration” No. 15H05890 (S.N.) and the JSPS Grant-in-Aid for Scientific Research (C) No. 18K03615