$F(Q)$ gravity with Gauss-Bonnet corrections: from early-time inflation to late-time acceleration

We show that in the $f(Q)$ gravity with a non-metricity scalar $Q$, the curvatures in Einstein's gravity, that is, the Riemann curvature constructed from the standard Levi-Civita connection, could not be excluded or naturally appear. The first observation is that even in $f(Q)$ gravity, the conservation of the matter energy-momentum tensor is not described by the covariant derivatives in the non-metricity gravity but that is given by the Levi-Civita connection. The commutator of the covariant derivatives in Einstein's gravity inevitably induces the Riemann curvature. There is no symmetry nor principle which prohibits the Riemann curvature in non-metricity gravity. Based on this observation, we propose and investigate $f\left(Q, \mathcal{G} \right)$ gravity with the Gauss-Bonnet invariant $\mathcal{G}$ and its generalisations. We show how $f\left(Q, \mathcal{G} \right)$ models realising any given the Friedmann-Lema\^{i}tre-Robertson-Walker (FLRW) spacetime can be reconstructed. We apply the reconstruction formalism to cosmology. Explicitly, the gravity models which realise slow roll or constant roll inflation, dark energy epoch as well as the unification of the inflation and dark energy are found. The dynamical autonomous system and the gravitational wave in the theory under investigation are discussed. It is found the condition that the de Sitter spacetime becomes the (stable) fixed point of the system.