Linear cosmological perturbations in almost scale-invariant fourth-order gravity

We study a class of almost scale-invariant modified gravity theories, using a particular form of $f(R,G)=\alpha R^2+\beta G\log G$ where $R$ and $G$ are the Ricci and Gauss-Bonnet scalars, respectively and $\alpha$, $\beta$ are arbitrary constants. We derive the Einstein-like field equations to first order in cosmological perturbation theory in longitudinal gauge.