  In previous works we have derived a Running Vacuum Model (RVM) for a string
Universe, which provides an effective description of the evolution of
4-dimensional string-inspired cosmologies from inflation till the present
epoch. In the context of this "stringy RVM" version, it is assumed that the
early Universe is characterised by purely gravitational degrees of freedom,
from the massless gravitational string multiplet, including the antisymmetric
tensor field. The latter plays an important role, since its dual gives rise to
a `stiff' gravitational-axion "matter", which in turn couples to the
gravitational anomaly terms, assumed to be non-trivial at early epochs. In the
presence of primordial gravitational wave (GW) perturbations, such anomalous
couplings lead to an RVM-like dynamical inflation, without external inflatons.
We review here this framework and discuss potential scenarios for the
generation of such primordial GW, among which the formation of unstable domain
walls, which eventually collapse in a non-spherical-symmetric manner, giving
rise to GW. We also remark that the same type of "stiff" axionic matter could
provide, upon the generation of appropriate potentials during the
post-inflationary eras, (part of) the Dark Matter (DM) in the Universe, which
could well be ultralight, depending on the parameters of the string-inspired
model. All in all, the new (stringy) mechanism for RVM-inflation preserves the
basic structure of the original (and more phenomenological) RVM, as well as its
main advantages: namely, a mechanism for graceful exit and for generating a
huge amount of entropy capable of explaining the horizon problem. It also
predicts axionic DM and the existence of mild dynamical Dark Energy (DE) of
quintessence type in the present universe, both being "living fossils" of the
inflationary stages of the cosmic evolution.
