On $f(R)$ gravity in scalar-tensor theories

Abstract

We study $f(R)$ gravity models in the language of scalar-tensor theories. The correspondence between $f(R)$ gravity and scalar-tensor 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 $\left(f(R)=βR^{n}\right)$, as time increases the scalar field decreases and decays asymptotically. For the second model $\left(f(R)=αR+βR^{n}\right)$ it was found that the function $ϕ(t)$ crosses the $t$-axis at different values for different values of $β$. For the third model $\left(f(R)=R-\frac{ν^{4}}{R}\right)$, when the value of $ν$ is small the potential $V(ϕ)$ behaves like the standard inflationary potential. For the fourth model $\left(f(R)=R-(1-m)ν^{2}\Big(\frac{R}{ν^{2}}\Big)^{m}-2Λ\right)$, we show that there is a transition between $1.5 1.55$. The slow-roll approximation is applied to each of the four $f(R)$ models and we obtain the respective expressions for the spectral index $n_{s}$ and the tensor-to-scalar ratio $r$.