Cosmological inflation in $F(R,\mathcal{G})$ gravity

Cosmological inflation is discussed in the framework of $F(R,\mathcal{G})$ gravity where $F$ is a generic function of the curvature scalar $R$ and the Gauss-Bonnet topological invariant $\mathcal{G}$. The main feature that emerges in this analysis is the fact that this kind of theory can exhaust all the curvature budget related to curvature invariants without considering derivatives of $R,$ $R_{\mu \nu }$, $R_{\sigma \mu \nu }^{\lambda }$ etc. in the action. Cosmological dynamics results driven by two effective masses (lenghts) related to the $R$ scalaron and the $\mathcal{G}$ scalaron working respectively at early and very early epochs of cosmic evolution. In this sense, a double inflationary scenario naturally emerges.