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The paper extends basic Einstein--Hilbert action by adding a newly proposedinvariant constructed from a specific contraction between the Einstein tensorand the energy momentum tensor, encoding a non--minimal coupling between thespace--time geometry and the matter fields. The fundamental Einstein--Hilbertaction is extended by considering a generic function ${f}(R,G_{\mu \nu}T^{\mu\nu})$ which is further decomposed into its main constituents, a geometriccomponent which depends on the scalar curvature, and a second element embeddingthe interplay between geometry and matter fields. Specific cosmological modelsare established at the level of background dynamics, based on particularcouplings between the matter energy--momentum tensor and the Einstein tensor.After deducing the resulting field equations, the physical aspects for thecosmological model are investigated by employing a dynamical system analysisfor various coupling functions. The investigation showed that the present modelis compatible with different epochs in the evolution of our Universe, possibleexplaining various late time historical stages.

Physical aspects of ${f}(R,G_{\mu \nu}T^{\mu \nu})$ modified gravity theories