I review a string-inspired cosmological model with gravitational anomalies in its early epochs, which is based on fields from the (bosonic) massless gravitational multiplet of strings, in particular gravitons and Kalb Ramond (KR), string-model independent, axions (the dilaton is assumed constant). I show how condensation of primordial gravitational waves, which are generared at the very early eras immediately after the big bang, can lead to inflation of the so called running vacuum model (RVM) type, without external inflatons. The role of the slow-roll field is played by the KR axion, but it does not drive inflation. The non-linearities in the anomaly terms do. Chiral fermionic matter excitations appear at the end of this RVM inflation, as a result of the decay of the RVM vacuum, and are held responsible for the cancellation of the primordial gravitational anomalies. Chiral anomalies, however, survive in the post-inflationary epochs, and can lead to the generation of a non perturbative mass for the KR axion, which could thus play the role of dark matter in this Universe. As a result of the condensed gravitational anomaly, there is a Lorentz-invariance violating KR axion background, which remains undiluted during the RVM inflation, and can lead to baryogenesis through leptogenesis in the radiation era, in models with sterile right-handed neutrinos. I also discuss the phenomenology of the model in the modern era, paying particular attention to linking it with a version of RVM, called type II RVM, which arguably can alleviate observed tensions in the current-epoch cosmological data.
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