  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, $H^2$.
