4 Dark Energy in f (R) Theories

In this section we apply f (R) theories to dark energy. Our interest is to construct viable f (R) models that can realize the sequence of radiation, matter, and accelerated epochs. In this section we do not attempt to find unified models of inflation and dark energy based on f (R) theories.

Originally the model f (R) = R − α∕Rn (α > 0, n > 0) was proposed to explain the late-time cosmic acceleration [113Jump To The Next Citation Point120114143] (see also [456Jump To The Next Citation Point559172232121613762Jump To The Next Citation Point] for related works). However, this model suffers from a number of problems such as matter instability [215244], the instability of cosmological perturbations [146Jump To The Next Citation Point74Jump To The Next Citation Point544Jump To The Next Citation Point526Jump To The Next Citation Point251Jump To The Next Citation Point], the absence of the matter era [2829239], and the inability to satisfy local gravity constraints [469Jump To The Next Citation Point470Jump To The Next Citation Point245Jump To The Next Citation Point233Jump To The Next Citation Point154Jump To The Next Citation Point448Jump To The Next Citation Point134Jump To The Next Citation Point]. The main reason why this model does not work is that the quantity f,RR ≡ ∂2f∕∂R2 is negative. As we will see later, the violation of the condition f,RR > 0 gives rise to the negative mass squared M 2 for the scalaron field. Hence we require that f,RR > 0 to avoid a tachyonic instability. The condition f,R ≡ ∂f ∕∂R > 0 is also required to avoid the appearance of ghosts (see Section 7.4). Thus viable f (R) dark energy models need to satisfy [568Jump To The Next Citation Point]

f,R > 0, f,RR > 0, for R ≥ R0 (> 0), (4.56 )
where R0 is the Ricci scalar today.

In the following we shall derive other conditions for the cosmological viability of f (R) models. This is based on the analysis of [26Jump To The Next Citation Point]. For the matter Lagrangian ℒM in Eq. (2.1View Equation) we take into account non-relativistic matter and radiation, whose energy densities ρm and ρr satisfy

ρ˙m + 3H ρm = 0, (4.57 ) ρ˙r + 4H ρr = 0, (4.58 )
respectively. From Eqs. (2.15View Equation) and (2.16View Equation) it follows that
3F H2 = (F R − f )∕2 − 3H F˙+ κ2 (ρm + ρr), (4.59 ) ˙ ¨ ˙ 2 − 2F H = F − H F + κ [ρm + (4∕3)ρr]. (4.60 )

 4.1 Dynamical equations
 4.2 Viable f (R) dark energy models
 4.3 Equation of state of dark energy

  Go to previous page Go up Go to next page