Singularity-free cosmological solutions in quadratic gravity

We study a general field theory of a scalar field coupled to gravity through a quadratic Gauss-Bonnet term $\xi (\phi {)R}_{\mathrm{GB}}^2.$ The coupling function has the form $\xi \left(\phi \right)={\phi }^n,$ where n is a positive integer. In the absence of the Gauss-Bonnet term, the cosmological solutions for an empty universe and a universe dominated by the energy-momentum tensor of a scalar field are always characterized by the occurrence of a true cosmological singularity. By employing analytical and numerical methods, we show that, in the presence of the quadratic Gauss-Bonnet term, for the dual case of even n, the set of solutions of the classical equations of motion in a curved FRW background includes singularity-free cosmological solutions. The singular solutions are shown to be confined in a part of the phase space of the theory allowing the non-singular solutions to fill the rest of the space. We conjecture that the same theory with a general coupling function that satisfies certain criteria may lead to non-singular cosmological solutions.