# $f(R)$ gravity $f(R)$ gravity refers to a special class of [[0006 Higher-derivative gravity|higher-curvature gravity]] where the Lagrangian takes the form$S=\frac{1}{2 \kappa} \int d^4 x \sqrt{-g} f(R)+S_M,$where $f(R)$ is a general function of the Ricci scalar, and its equation of motion is given by $f^{\prime}(R) R_{\mu \nu}-\frac{1}{2} f(R) g_{\mu \nu}-\left[\nabla_{\mu} \nabla_{\nu}-g_{\mu \nu} \square\right] f^{\prime}(R)=\kappa T_{\mu \nu}.$ ## [[0161 Palatini formalism|Palatini formulation]] - treat the connection as an independent field - the action looks the same but the degrees of freedom are different ## Equivalence to a [[0140 Scalar-tensor theory|scalar-tensor]] theory - $S=\int d^{D} x \sqrt{-g}\left[f(\phi)+f^{\prime}(\phi)(R-\phi)\right]$ - using EOM for the scalar $f^{\prime\prime}(\phi)(R-\phi)=0$ - get back $f(R)$ as long as $f^{\prime\prime}(\phi)\ne0$ ## Refs - reviews - [[SotiriouFaraoni2008]][](https://arxiv.org/pdf/0805.1726.pdf) - shorter version [[Sotiriou2009Review]][](https://iopscience.iop.org/article/10.1088/1742-6596/189/1/012039/pdf) - [[DeFeliceTsujikawa2010Review]][](https://arxiv.org/abs/1002.4928) - [[EzawaKajiharaKiminamiSodaYano1998]] - [[DeruelleSendoudaYoussef2009]] - equivalent to a [[0140 Scalar-tensor theory|scalar-tensor]] - [[1983#Teyssandier, Tourrenc]] - in the Einstein limit [[2007#Olmo]] - boundary term of f(R) by direct variation - [[GuarnizoCastanedaTejeiro2010]][](https://arxiv.org/abs/1002.0617) - [[AlhamzawiAlhamzawi2014]] - Jordan frame - [[DeruelleSendoudaYoussef2009]][](https://arxiv.org/abs/0906.4983) - [[KlusonMatous2022]][](https://arxiv.org/pdf/2209.14560.pdf)