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Toward physical cosmology: focus on inhomogeneous geometry and its non-perturbative effects

We outline the key-steps toward the construction of a physical, fully relativistic cosmology. The influence of inhomogeneities on the effective evolution history of the Universe is encoded in backreaction terms and expressed through spatially averaged geometrical invariants. These are absent and potential candidates for the missing dark sources in the standard model. Since they can be interpreted as energies of an emerging scalar field (the morphon), we are in the position to propose a strategy of how phenomenological scalar field models for Dark Energy, Dark Matter and Inflation, that are usually added as fundamental sources to a homogeneous-geometry (FLRW) cosmology, can be potentially traced back to inhomogeneous geometrical properties of space and its embedding into spacetime. We lay down a line of arguments that is - thus far only qualitatively - conclusive, and we address open problems of quantitative nature, related to the interpretation of observations. We discuss within a covariant framework ... [abridged]