Cosmological Constraints on $f(G)$ Dark Energy Models

Modified gravity theories with the Gauss-Bonnet term $G=R^2-4R^{μν}R_{μν}+R^{μνρσ}R_{μνρσ}$ have recently gained a lot of attention as a possible explanation of dark energy. We perform a thorough phase space analysis on the so-called $f(G)$ models, where $f(G)$ is some general function of the Gauss-Bonnet term, and derive conditions for the cosmological viability of $f(G)$ dark energy models. Following the $f(R)$ case, we show that these conditions can be nicely presented as geometrical constraints on the derivatives of $f(G)$. We find that for general $f(G)$ models there are two kinds of stable accelerated solutions, a de Sitter solution and a phantom-like solution. They co-exist with each other and which solution the universe evolves to depends on the initial conditions. Finally, several toy models of $f(G)$ dark energy are explored. Cosmologically viable trajectories that mimic the $Λ$CDM model in the radiation and matter dominated periods, but have distinctive signatures at late times, are obtained.

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