This problem of the past singularity can be cured by adding the term to the Lagrangian in f (R) dark energy models [37]. Let us then consider the modified version of the model (4.83):

For this model one can easily show that the potential extends to the region and that the curvature singularity disappears accordingly. Also the scalaron mass approaches the finite value in the limit . The perturbation is bounded from above, which can evade the problem of the dominance of the oscillation mode in the past.Since the presence of the term can drive inflation in the early universe, one may anticipate that both inflation and the late-time acceleration can be realized for the model of the type (13.37). This is like a modified gravity version of quintessential inflation based on a single scalar field [486, 183, 187, 392]. However, we have to caution that the transition between two accelerated epochs needs to occur smoothly for successful cosmology. In other words, after inflation, we require a mechanism in which the universe is reheated and then the radiation/matter dominated epochs follow. However, for the model (13.37), the Ricci scalar evolves to the point and it enters the region . Crossing the point implies the divergence of the scalaron mass. Moreover, in the region , the Minkowski space is not a stable vacuum state. This is problematic for the particle creation from the vacuum during reheating. The similar problem arises for the models (4.84) and (4.89) in addition to the model proposed by Appleby and Battye [35]. Thus unified f (R) models of inflation and dark energy cannot be constructed easily in general (unlike a number of related works [456, 460, 462]). Brookfield et al. [104] studied the viability of the model () by using the constraints coming from Big Bang Nucleosynthesis and fifth-force experiments and showed that it is difficult to find a unique parameter range for consistency of this model.

In order to cure the above mentioned problem, Appleby et al. [37] proposed the f (R) model (11.40). Note that the case corresponds to the Starobinsky inflationary model [564] and the case corresponds to the model of Appleby and Battye [35] plus the term. Although the above mentioned problem can be evaded in this model, the reheating proceeds in a different way compared to that in the model [which we discussed in Section 3.3]. Since the Hubble parameter periodically evolves between and , the reheating mechanism does not occur very efficiently [37]. The reheating temperature can be significantly lower than that in the model . It will be of interest to study observational signatures in such unified models of inflation and dark energy.

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