On-shell Homogeneous Curvature Modes in Quadratic $F(R,\mathcal G)$ Cosmology
5We study homogeneous cosmological perturbations in the quadratic model $F(R,\mathcal G)=R+αR^2+β\mathcal G^2$. Linearization of the modified Friedmann equation together with dust conservation yields a second-order equation for the homogeneous Hubble perturbation, or equivalently a third-order equation for the matter contrast. We then restrict the asymptotic analysis to backgrounds that satisfy the zeroth-order equations of the same quadratic theory. Besides the low-curvature dust branch, which approaches $H=2/(3t)$, the model admits a nontrivial vacuum de Sitter solution with $βH_{\rm dS}^6=1/96$. Its homogeneous curvature modes are stable only when $α<-1/(6H_{\rm dS}^2)$. At high curvature the background equation also admits a vacuum big-rip asymptotic branch beginning with $H=7/(t_s-t)$; the first corrections are fixed by the quadratic couplings. The corresponding $\mathcal G^2$-dominated perturbation has one divergent homogeneous mode. The simultaneous Einstein limit is taken only as a consistency check and recovers the standard dust behavior.