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I review a string-inspired cosmological model, with gravitational anomaliespresent 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 primordialgravitational waves (GW), that are created at a pre-inflationary era, the modelleads to inflation of the so-called running-vacuum-model (RVM) type, which isrealised without the need for external inflaton fields, being due to thenon-linearities that characterise the gravitational theory. The model providesan alternative to the $\Lambda$CDM paradigm, and is also capable of inducingmatter-antimatter asymmetry in the Universe, during the post-inflationaryradiation era, in models with right-handed neutrinos in their matter spectra.The so-induced asymmetry is triggered by a Lorentz-symmetry-violating KR axionbackground that is generated during the RVM-inflationary period, as aconsequence of the GW condensate. The modern era of this cosmology is argued tobe characterised by observable (RVM-type) deviations from $\Lambda$CDM, withthe potential of alleviating the observed tensions in the cosmological data.

Going beyond the Standard paradigm of Cosmology: Torsion, gravitational anomalies and inflation without inflaton fields