13.3 Curing the curvature singularity in f (R) dark energy models, unified models of inflation and dark energy

In Sections 5.2 and 8.1 we showed that there is a curvature singularity for viable f (R) models such as (4.83View Equation) and (4.84View Equation). More precisely this singularity appears for the models having the asymptotic behavior (5.19View Equation) in the region of high density (R ≫ Rc). As we see in Figure 3View Image, the field potential V (ϕ) = (F R − f )∕ (2 κ2F ) has a finite value μRc∕ (2 κ2) in the limit ∘ -------- ϕ = 3∕(16π )mplln F → 0. In this limit one has f → 0 ,RR, so that the scalaron mass 1∕(3f ) ,RR goes to infinity.

This problem of the past singularity can be cured by adding the term 2 2 R ∕(6M ) to the Lagrangian in f (R) dark energy models [37Jump To The Next Citation Point]. Let us then consider the modified version of the model (4.83View Equation):

(R ∕Rc )2n R2 f(R ) = R − μRc (R-∕R--)2n-+-1-+ 6M-2. (13.37 ) c
For this model one can easily show that the potential V (ϕ) = (F R − f )∕(2κ2F ) extends to the region ϕ > 0 and that the curvature singularity disappears accordingly. Also the scalaron mass approaches the finite value M in the limit ϕ → ∞. The perturbation δR 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 R2∕ (6M 2) 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 (13.37View Equation). This is like a modified gravity version of quintessential inflation based on a single scalar field [486183187392]. 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 (13.37View Equation), the Ricci scalar R evolves to the point f,RR = 0 and it enters the region f,RR < 0. Crossing the point f,RR = 0 implies the divergence of the scalaron mass. Moreover, in the region f,RR < 0, 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 (4.84View Equation) and (4.89View Equation) in addition to the model proposed by Appleby and Battye [35Jump To The Next Citation Point]. Thus unified f (R) models of inflation and dark energy cannot be constructed easily in general (unlike a number of related works [456460462]). Brookfield et al. [104] studied the viability of the model f (R ) = R − α∕Rn + βRm (n, m > 0) 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. [37Jump To The Next Citation Point] proposed the f (R) model (11.40View Equation). Note that the case c = 0 corresponds to the Starobinsky inflationary model f(R ) = R + R2 ∕(6M 2) [564] and the case c = 1∕2 corresponds to the model of Appleby and Battye [35Jump To The Next Citation Point] plus the R2 ∕(6M 2) 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 2 2 f(R) = R + R ∕ (6M ) [which we discussed in Section 3.3]. Since the Hubble parameter periodically evolves between H = 1 ∕(2t) and H = 𝜖∕M, the reheating mechanism does not occur very efficiently [37]. The reheating temperature can be significantly lower than that in the model f(R ) = R + R2 ∕(6M 2). It will be of interest to study observational signatures in such unified models of inflation and dark energy.

  Go to previous page Go up Go to next page