Non-commutative double geometry

We construct non-commutative theories with the Moyal-Weyl product in the Double Field Theory (DFT) framework. We deform the infinitesimal generalized diffeomorphisms and the Leibniz rule in a consistent way. The prescription requires a generalized star metric, which can be thought of as the fundamental double metric, in order to construct the action. Finally we use the generalized scalar field dynamics and the generalized scalar field-perfect fluid correspondence to construct the generalized energy momentum-tensor of a perfect fluid in the non-commutative double geometry. The present formalism paves the way to the study of string cosmologies scenarios including the Moyal-Weyl product in a T-duality invariant way.