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Running Dark Energy and Dark Matter models are candidates to resolve theHubble constant tension. However the model does not consider a Lagrangianformulation directly. In this paper we formulate an action principle where theRunning Vacuum Model (RVM) is obtained from an action principle, with a scalarfield model for the whole dark components. The Dynamical Spacetime vector field$\chi_\mu$ is a Lagrange multiplier that forces the kinetic term of the scalarfield to behave as the modified dark matter. When we replace the vector fieldby a derivative of a scalar the model predicts diffusion interactions betweenthe dark components with a different correspondence to the RVM. We test themodels with the Cosmic Chronometers, Type Ia Supernova, Quasars, Gamma rayBursts and the Baryon Acoustic Oscillations data sets. We find that$\Lambda$CDM is still the best model. However this formulation suggests anaction principle for the $\Lambda$CDM, the RVM model and other extensions.

Running Dark Energy and Dark Matter from Dynamical Spacetime