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On the occasion of a century from the proposal of General relativity byEinstein, I attempt to tackle some open issues in modern cosmology, via a toybut non-trivial model. Specifically, I would like to link together: (i) thesmallness of the cosmological constant today, (ii) the evolution of theuniverse from an inflationary era after the big-bang till now, and (iii) localsupersymmetry in the gravitational sector (supergravity) with a broken spectrumat early eras, by making use of the concept of the "running vacuum" in thecontext of a simple toy model of four-dimensional N=1 supergravity. The modelis characterised by dynamically broken local supersymmetry, induced by theformation of gravitino condensates in the early universe. As I will argue,there is a Starobinsky-type inflationary era characterising the brokensupersymmetry phase in this model, which is compatible with the currentcosmological data, provided a given constraint is satisfied among sometree-level parameters of the model and the renormalised cosmological constantof 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 asmooth connection (in cosmic time) with the radiation dominance epoch andsubsequently 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 isattributed to the failure (due to quantum gravity non-perturbative effects) ofthe aforementioned constraint.

Supersymmetry, Cosmological Constant and Inflation: Towards a fundamental cosmic picture via "running vacuum"