Natural coordinates and horizontal approximations in two-field cosmological models

We construct natural local coordinate systems on the phase space of two-field cosmological models with orientable target space, which allow for a description of cosmological flows through quantities of direct physical interest. Such coordinates are induced by the fundamental observables of the model, which we formulate geometrically using the tautological bundle of the tangent bundle of the scalar manifold. We also describe a large class of geometric dynamical approximations induced by the choice of an Ehresmann connection in the tangent bundle of the scalar manifold. Such approximations take a conceptually simple form in natural coordinates and we illustrate one of them as an application.