  I review a string-inspired cosmological model, with gravitational anomalies
present at very early epochs, which includes a totally antisymmetric torsion,
that in (3+1)-dimensions is equivalent to a pseudoscalar (Kalb-Ramond,
string-model independent) axion field. Upon condensation of primordial
gravitational waves (GW), that are created at a pre-inflationary era, the model
leads to inflation of the so-called running-vacuum-model (RVM) type, which is
realised without the need for external inflaton fields, being due to the
non-linearities that characterise the gravitational theory. The model provides
an alternative to the $\Lambda$CDM paradigm, and is also capable of inducing
matter-antimatter asymmetry in the Universe, during the post-inflationary
radiation era, in models with right-handed neutrinos in their matter spectra.
The so-induced asymmetry is triggered by a Lorentz-symmetry-violating KR axion
background that is generated during the RVM-inflationary period, as a
consequence of the GW condensate. The modern era of this cosmology is argued to
be characterised by observable (RVM-type) deviations from $\Lambda$CDM, with
the potential of alleviating the observed tensions in the cosmological data.
