 Solution

For the canonical scalar action and the stress-tensor convention in the paper, varying the inverse metric and volume element gives the stress-energy tensor of a canonical scalar fieldThe vacuum f(R) gravity equation, divided by $F$, isInsert this in the transformed Einstein tensor from part (a). Since $g_{\mu \nu }=-{\tilde{g}}_{\mu \nu }/F$, its algebraic term becomes ${\tilde{g}}_{\mu \nu }(RF-f)/(2F^2)$. Also $\varphi =\sqrt{3/2}\log (-F)$ gives ${\partial }_{\mu }\varphi =\sqrt{3/2}{\partial }_{\mu }F/F$. The remaining derivative tensor is exactly the kinetic part of $T_{\mu \nu }$: the combination $g_{\mu \nu }(\nabla F{)}^2$ equals ${\tilde{g}}_{\mu \nu }(\tilde{\nabla }F{)}^2$. ThusThis establishes the Einstein-frame representation of metric f(R) gravity in the specified normalization. To express the potential as a function of $\varphi$, choose a branch where $R\mapsto f^{\prime }(R)$ is locally invertible; the conformal metric already requires $f^{\prime }<0$.