  I review a (3+1)-dimensional, string-inspired cosmological model with
gravitational anomalies (of Chern-Simons (CS) type) at early epochs, and a
totally-antisymmetric torsion, dual to a massless axion-like field
("gravitational axion"), which couples to the CS term. Under appropriate
conditions, primordial gravitational waves can condense, leading to a
condensate of the CS anomaly term. As a consequence, one obtains inflation in
this theory, of running-vacuum-model (RVM) type, without the need for external
inflatons. At the end of the inflationary era, chiral fermionic matter is
generated, whose gravitational anomalies cancel the primordial ones. On the
other hand, chiral anomalies of gauge type, which are also generated by the
chiral matter, remain present during the post-inflationary epochs and become
responsible for the generation of a non-perturbative mass for the
torsion-related gravitational axion, which, in this way, might play the role of
a Dark Matter component of geometrical origin. Moreover, in this model, stringy
non-perturbative effects during the RVM inflationary phase generate periodic
structures for the potential of axion-like particles that arise due to
compactification, and co-exist with the gravitational axions. Such periodic
potential modulations may lead to an enhanced production of primordial black
holes during inflation, which in turn affects the profile of the generated
gravitational waves during the radiation era, with potentially observable
consequences. This model also entails an unconventional mechanism for
Leptogenesis, due to Lorentz-violating backgrounds of the gravitational axions
that are generated during inflation, as a consequence of the anomaly
condensates, and remain undiluted in the radiation era.
