Construction of cosmologically viable f (G) gravity models

We derive conditions under which f (G) gravity models, whose Lagrangian densities f are written in terms of a Gauss-Bonnet term G, are cosmologically viable. The most crucial condition to be satisfied is d² f / d G² > 0, which is required to ensure the stability of a late-time de Sitter solution as well as the existence of standard radiation/matter dominated epochs. We present a number of explicit f (G) models in which a cosmic acceleration is followed by the matter era. We find that the equation of state of dark energy can cross the phantom divide before reaching the present Universe. The viable models have asymptotic behavior d² f / d G² → + 0 for | G | → ∞, in which case a rapid oscillation of perturbations occurs unless such an oscillating degree of freedom is suppressed relative to a homogeneous mode in the early universe. We also introduce an iterative method to avoid numerical instabilities associated with a large mass of the oscillating mode.