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 ($\omega =0$). In this treatment, four $f(R)$ toy models are used to analyze the early-universe cosmology, when the scalar field $\varphi$ dominates over standard matter. We have obtained solutions to the Klein-Gordon equation for those models. It is found that for the first model $\left(f(R)=\beta R^n\right)$, as time increases the scalar field decreases and decays asymptotically. For the second model $\left(f(R)=\alpha R+\beta R^n\right)$ it was found that the function $\varphi (t)$ crosses the $t$-axis at different values for different values of $\beta$. For the third model $\left(f(R)=R-\frac{{\nu }^4}{R}\right)$, when the value of $\nu$ is small the potential $V(\varphi )$ behaves like the standard inflationary potential. For the fourth model $\left(f(R)=R-(1-m){\nu }^2(\frac{R}{{\nu }^2}{)}^m-2\Lambda \right)$, 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 $n_s$ and the tensor-to-scalar ratio $r$.

arXiv comment27 pages, 7 figures, Published in IJGMMP

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• Perturbations of quasi-Newtonian universes in scalar-tensor gravity
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