Dynamical system approach to running Λ cosmological models

We study the dynamics of cosmological models with a time dependent cosmological term. We consider five classes of models; two with the non-covariant parametrization of the cosmological term Λ: Λ(H)CDM cosmologies, Λ(a)CDM cosmologies, and three with the covariant parametrization of Λ: Λ(R)CDM cosmologies, where R(t) is the Ricci scalar, Λ(φ)-cosmologies with diffusion, Λ(X)-cosmologies, where X = (1)/(2)g^αβ∇_α∇_βφ is a kinetic part of the density of the scalar field. We also consider the case of an emergent Λ(a) relation obtained from the behaviour of trajectories in a neighbourhood of an invariant submanifold. In the study of the dynamics we used dynamical system methods for investigating how an evolutionary scenario can depend on the choice of special initial conditions. We show that the methods of dynamical systems allow one to investigate all admissible solutions of a running Λ cosmology for all initial conditions. We interpret Alcaniz and Lima's approach as a scaling cosmology. We formulate the idea of an emergent cosmological term derived directly from an approximation of the exact dynamics. We show that some non-covariant parametrization of the cosmological term like Λ(a), Λ(H) gives rise to the non-physical behaviour of trajectories in the phase space. This behaviour disappears if the term Λ(a) is emergent from the covariant parametrization. (orig.)