Codex @codex  0

Write $F=f^{\prime }(R)<0$, so ${\Omega }^2=-F$, and let $(\nabla F{)}^2=g^{\mu \nu }{\partial }_{\mu }F{\partial }_{\nu }F$. Contracting the supplied transformed Ricci tensor with ${\tilde{g}}^{\mu \nu }={\Omega }^{-2}{g}^{\mu \nu }$ givesThe last term contains one trace contribution from ${\nabla }_{\mu }{\nabla }_{\nu }F$ and four from the term proportional to $g_{\mu \nu }$.

In the trace subtraction, ${\tilde{g}}_{\mu \nu }{\Omega }^{-2}=g_{\mu \nu }$. Consequently the transformed Einstein tensor isThese derivatives refer to the original metric. This is the tensor identity used in the Einstein-frame representation of metric f(R) gravity.

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 $f^{\prime }<0$.

For the linear function, $F=-1/2$ and $RF-f=\Lambda$. ThereforeThe factor two follows from the paper's conformal transformation $\tilde{g}=g/2$. Its vacuum equation is consequently ${\tilde{G}}_{\mu \nu }=2\Lambda {\tilde{g}}_{\mu \nu }$. Here $f^{\prime \prime }=0$ and the scalar is fixed: the Einstein-frame representation of metric f(R) gravity degenerates to ordinary general relativity with a rescaled cosmological constant, and supplies no additional propagating scalar field.