We discuss a connection between gravitational-wave physics, quantum theory anomalies, right-handed (sterile) neutrinos, (spontaneous) CPT Violation and Leptogenesis, within the framework of string-inspired cosmological models. In particular, we present a scenario, according to which (primordial) gravitational waves induce gravitational anomalies during inflation. This, in turn, results in the existence of an undiluted (at the exit of inflation/beginning of radiation era) bakcground of the Kalb-Ramond (KR) axion of the massless bosonic string gravitational multiplet. The latter may violate spontaneously CP and CPT symmetries, and induce leptogenesis during the radiation-dominated era in models involving right-handed neutrinos. The so-generated lepton asymmetry may then be communicated to the baryon sector by appropriate baryon-minus-lepton-number (B - L)-conserving, but (B + L)-violating, (sphaleron) processes in the Standard Model sector, thus leading to matter dominance over antimatter in the Universe.In the current (approximately de Sitter) era, the KR axion background may provide a source for an axionic dark matter in the Universe, through its mixing with other axions that are abundant in string models. As an interesting byproduct of our analysis, we demonstrate that the anomalies contribute to the vacuum energy density of the Universe terms of running-vacuum type, proportional to the square of the Hubble parameter, $H^2$.
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