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.