 Solution

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.