Unified dark energy and dark matter from dynamical space time

A unification of dark matter and dark energy based on a dynamical space time theory is discussed. By introducing a dynamical space time vector field χ_μ as a Lagrange multiplier, a conservation of an energy momentum tensor Tμν(χ) is implemented in addition to the conservation of the metric energy momentum tensor. This Lagrangian generalizes the Unified dark energy and dark matter from a scalar field different from quintessence which did not consider a Lagrangian formulation. This generalization allows the solutions which were found previously, in addition to a-non singular bouncing solutions that rapidly approach to the ΛCDM model. The dynamical time vector field exactly coincides with the cosmic time for the a ΛCDM solution and suffers a slight shift (advances slower) with respect to the cosmic time in the region close to the bounce for the bouncing non singular solutions. For some exponential potential which gives a possible interaction between DE and DM and could explain the coincidence problem.