f(R) theories

TL;DR

The review surveys f(R) gravity as a minimal modification of GR, clarifying its metric and Palatini formalisms, the scalaron degree of freedom, and the Brans–Dicke correspondence. It synthesizes inflationary and dark-energy applications, detailing viable models (e.g., Starobinsky, Hu–Sawicki) and local gravity tests enabled by the chameleon mechanism. A comprehensive perturbation framework is developed for both background evolution and linear/nonlinear structure formation, highlighting predictions for CMB, LSS, and ISW signals. Extensions to Brans–Dicke and Gauss–Bonnet theories are discussed, along with the behavior of relativistic stars and the role of higher-curvature invariants, underscoring both observational opportunities and theoretical constraints in modified gravity.

Abstract

Over the past decade, f(R) theories have been extensively studied as one of the simplest modifications to General Relativity. In this article we review various applications of f(R) theories to cosmology and gravity - such as inflation, dark energy, local gravity constraints, cosmological perturbations, and spherically symmetric solutions in weak and strong gravitational backgrounds. We present a number of ways to distinguish those theories from General Relativity observationally and experimentally. We also discuss the extension to other modified gravity theories such as Brans-Dicke theory and Gauss-Bonnet gravity, and address models that can satisfy both cosmological and local gravity constraints.

Figures (14)

• Figure 1: The field potential (\ref{['poein']}) in the Einstein frame corresponding to the model (\ref{['stamodel']}). Inflation is realized in the regime $\kappa \phi \gg 1$.
• Figure 2: Four trajectories in the $(r,m)$ plane. Each trajectory corresponds to the models: (i) $\Lambda$CDM, (ii) $f(R)=(R^{b}-\Lambda)^{c}$, (iii) $f(R)=R-\alpha R^{n}$ with $\alpha>0,0<n<1$, and (iv) $m(r)=-C(r+1)(r^{2}+ar+b)$. From AmenTsuji07.
• Figure 3: (Top) The potential $V(\phi)=(FR-f)/(2\kappa^2F^2)$ versus the field $\phi=\sqrt{3/(16\pi)}m_{\mathrm{pl}}\,\ln F$ for the Starobinsky's dark energy model (\ref{['Bmodel']}) with $n=1$ and $\mu=2$. (Bottom) The inverted effective potential $-V_{\mathrm{eff}}$ for the same model parameters as the top with $\rho^*=10R_cm_{\mathrm{pl}}^2$. The field value, at which the inverted effective potential has a maximum, is different depending on the density $\rho^*$, see Eq. (\ref{['kappaphiM']}). In the upper panel "de Sitter" corresponds to the minimum of the potential, whereas "singular" means that the curvature diverges at $\phi=0$.
• Figure 4: Evolution of $\gamma$ versus the redshift $z$ in the model (\ref{['Amodel']}) with $n=1$ and $\mu=1.55$ for four different values of $k$. For these model parameters the dispersion of $\gamma$ with respect to $k$ is very small. All the perturbation modes shown in the figure have reached the scalar-tensor regime ($M^2 \ll k^2/a^2$) by today. From Moraes09.
• Figure 5: The regions (i), (ii) and (iii) for the model (\ref{['Bmodel']}). We also show the bound $n>0.9$ coming from the local gravity constraints as well as the condition (\ref{['Bmodelcon']}) coming from the stability of the de Sitter point. From Moraes09.