 Einstein-frame representation of metric f(R) gravity

In the mostly-minus convention with vacuum gravitational action proportional to $\int \sqrt{-g}f(R)$ and $f^{\prime }<0$, the displayed conformal transformation turns its vacuum equation into ${\tilde{G}}_{\mu \nu }=T_{\mu \nu }(\varphi )$, with potential $V=(R{f}^{\prime }-f)/(2f^{\prime 2})$ and the stress-energy tensor of a canonical scalar field. The conformal Ricci formula cancels the second derivatives of $f^{\prime }$ against the original field equation. Its remaining derivative tensor is ${\partial }_{\mu }\varphi {\partial }_{\nu }\varphi -{\tilde{g}}_{\mu \nu }(\tilde{\nabla }\varphi {)}^2/2$. Expressing $V$ as a function of $\varphi$ requires a locally invertible $R\mapsto f^{\prime }(R)$. When $f^{\prime \prime }=0$, this scalar-tensor parametrization degenerates and introduces no new propagating scalar degree of freedom.