Classification of Cosmological Trajectories

In the context of effective Friedmann equation we classify the cosmologies in multi-scalar models with an arbitrary scalar potential $V$ according to their geometric properties. It is shown that all flat cosmologies are geodesics with respect to a conformally rescaled metric on the augmented' target space. Non-flat cosmologies with V=0 are also investigated. It is shown that geodesics in a doubly-augmented’ target space yield cosmological trajectories for any curvature $k$ when projected onto a given hypersurface.