De Sitter solutions in Einstein-Gauss-Bonnet gravity

We consider de Sitter solutions in models with the Gauss–Bonnet term, including $L(R,G)$ gravity models. We investigate evolution equations of a scalar field nonminimally coupled both with the curvature and with the Gauss–Bonnet term and look for the fixed points of scalar field dynamics which correspond to de Sitter solutions. We show that, in the case of a positive coupling function, it is possible to introduce an effective potential $V_{eff}$ which can be expressed through the function $U$ of nonminimal coupling with the curvature, the scalar field potential $V$, and the coupling function with the Gauss–Bonnet term denoted by $F$. We show that it is convenient to investigate the structure of fixed points using the effective potential because the stable de Sitter solutions correspond to minima of the effective potential. We have found de Sitter solutions in a few $L(R,G)$ models to demonstrate the effective potential method.