General aspects of Gauss-Bonnet models without potential in dimension four

In the present work, the isotropic and homogenous solutions with spatial curvature $k=0$ of four dimensional Gauss-Bonnet models are characterized. The main assumption is that the scalar field $\phi$ which is coupled to the Gauss-Bonnet term has no potential [50]-[51]. Some singular and some eternal solutions are described. The evolution of the universe is given in terms of a curve $\gamma=(H(\phi), \phi)$ which is the solution of a polynomial equation $P(H^2, \phi)=0$ with $\phi$ dependent coefficients. In addition, it is shown that the initial conditions in these models put several restrictions on the evolution. For instance, an universe initially contracting will be contracting always for future times and an universe that is expanding was always expanding at past times. Thus, there are no cyclic cosmological solutions for this model. These results are universal, that is, independent on the form of the coupling $f(\phi)$ between the scalar field and the Gauss-Bonnet term. In addition, a proof that at a turning point $\dot{\phi}\to0$ a singularity necessarily emerges is presented. This is valid unless the Hubble constant $H\to 0$ at this point. This proof is based on the Raychaudhuri equation for the model. The description presented here is in part inspired in the works [32]-[34]. However, the mathematical methods that are implemented are complementary of those in these references, and they may be helpful for study more complicated situations in a future.