Quadratic Lagrangians and horizons

It has been known for some time that the standard isotropic cosmological model with an R² Lagrangian has horizon-breaking solutions in the t→0 singularity limit. We search for horizonless power-law solutions in the vacuum Bianchi type-I and orthogonal type-V cosmological models. We find that type V with an R² Lagrangian does contain such solutions. At first glance this would suggest that quadratic theories might provide an alternative resolution to the horizon problem. However, for both the horizon-less isotropic and type-V solutions we show that the eigenvalues of the linear vector field near the stationary points in phase space have opposite signs, demonstrating that such solutions are not Lyapunov stable. The addition of free quantum fields may stabilize the solutions.