On the occasion of a century from the proposal of General relativity by Einstein, I attempt to tackle some open issues in modern cosmology, via a toy but non-trivial model. Specifically, I would like to link together: (i) the smallness of the cosmological constant today, (ii) the evolution of the universe from an inflationary era after the bigbang till now, and (iii) local supersymmetry in the gravitational sector (supergravity) with a broken spectrum at early eras, by making use of the concept of the “running vacuum” in the context of a simple toy model of four-dimensional N = 1 supergravity. The model is characterised by dynamically broken local supersymmetry, induced by the formation of gravitino condensates in the early universe. As I will argue, there is a Starobinsky-type inflationary era characterising the broken supersymmetry phase in this model, which is compatible with the current cosmological data, provided a given constraint is satisfied among some tree-level parameters of the model and the renormalised cosmological constant of the de Sitter background used in the analysis. Applying the “running vacuum” concept, then, to the effective field theory at the exit of inflation, makes a smooth connection (in cosmic time) with the radiation dominance epoch and subsequently with the current era of the Universe, characterised by a small (but dominant) cosmological-constant contribution to the cosmic energy density. In this approach, the smallness of the cosmological constant today is attributed to the failure (due to quantum gravity non-perturbative effects) of the aforementioned constraint.
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