  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 big-bang 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.
