### abstract

- We consider the history of the universe from the big bang to the late period in the context of a unified interacting dark energy - dark matter model. The model is based on the Two Measures Theories, which introduces metric independent volume elements and this allows to construct unified Dark Matter Dark Energy, where the cosmological constant appears as an integration constant associated to the equation of motion of the measure fields. The Dynamical space time Theories generalize the Two Measure Theories by introducing a vector field, whose equation of motion guarantees the conservation of a source of a certain Energy Momentum tensor, which may be related, but in general is not the same as the gravitational Energy Momentum tensor. By considering the dynamical space field appearing in another part of the action or by demanding that the dynamical space time vector be a gradient of a scalar, we obtain a diffusive non conservation of an energy momentum tensor. These generalizations leads at the end to a formulation of interacting DE-DM dust models in the form of a diffusive type interacting Unified Dark Energy and Dark Matter scenario. A numerical solution of the theories shows that in some cases the evolution of the very early universe is governed by Stiff equation of state, and undergoes to $\Lambda$CDM. In another cases the universe bounces to hyper inflation. But all of these solutions have the another final transition to $\Lambda$CDM as a fixed point for the late universe.