  In this talk, I review briefly a scenario for the evolution of a
string-inspired cosmological model, in which condensates of primordial
gravitational waves (GW), formed at the very early eras after the Big Bang, are
considered responsible for inducing inflation and then a smooth exit to a
radiation dominated epoch. Primordial axion fields, that exist in the
fundamental massless gravitational (bosonic) string multiplet, couple to the
non-trivial GW-induced anomalies. As a result of this coupling, there exist
axion background configurations which violate (spontaneously) Lorentz symmetry,
and remain undiluted at the end of inflation. In models with heavy sterile
right-handed neutrinos (RHN), such backgrounds are linked to novel (Lorentz and
CPT Violating) mechanisms for the generation of matter-antimatter asymmetry in
the Cosmos, via the asymmetric decays of the RHN to standard model particles
and antiparticles. During the QCD epoch, the axions develop an
instanton-induced mass and can, thus, play the r\^ole of Dark Matter (DM). The
energy density of such a Universe, throughout its evolution, has the form of
that of a "running vacuum model", that is, it can be expanded in power series
of even powers of the Hubble parameter $H(t)$. The coefficients of those terms,
though, are different for the various cosmological epochs. For the
phenomenology of our model, which is consistent with the current cosmological
data, and could also help in alleviating (some of) the tensions, it suffices to
consider up to and including quartic powers of $H(t)$. In the early Universe
phase, it is the $H^4(t)$ term, induced by the GW condensate of the
gravitational anomaly, that drives inflation without the need for external
inflaton fields.
