On f(R) gravity in scalar-tensor theories

Abstract

We study f(R) gravity models in the language of scalar-tensor (ST) theories. The correspondence between f(R) gravity and ST theories is revisited since f(R) gravity is a subclass of Brans-Dicke models, with a vanishing coupling constant (ω=0). In this treatment, four f(R) toy models are used to analyze the early-universe cosmology, when the scalar field ϕ dominates over standard matter. We have obtained solutions to the Klein-Gordon equation for those models. It is found that for the first model (f(R)=βRn), as time increases the scalar field decreases and decays asymptotically. For the second model (f(R)=αR+βRn), it was found that the function ϕ(t) crosses the t-axis at different values for different values of β. For the third model (f(R)=R−ν4R), when the value of ν is small, the potential V(ϕ) behaves like the standard inflationary potential. For the fourth model (f(R)=R−(1−m)ν2(Rν2)m−2Λ), we show that there is a transition between 1.5<m<1.55. The behavior of the potentials with m<1.5 is totally different from those with m>1.55. The slow-roll approximation is applied to each of the four f(R) models and we obtain the respective expressions for the spectral index ns and the tensor-to-scalar ratio r.