I discuss a string inspired model of cosmology, characterised by
gravitational anomalies and torsion, in the early stages,
which may provide a geometric origin of the entire dark sector of the
Universe, from a running-vacuum-model inflation to
axionic dark matter, the axion degrees of freedom being associated with
torsion. During inflation, the model may, under some circumstances, lead to
enhanced gravitational-wave perturbations and produced densities of
primordial black holes. At post inflationary (radiation) eras, in models
with massive right-handed neutrinos, one may have CPT Violating
Leptogenesis, as a result of Lorentz-Violating backgrounds of the
torsion-related axion fields, generated during inflation from
condensates of primordial gravitational waves that induce, in turn,
gravitational-anomaly condensates.
In the current era, such a model may contribute to observable in
principle deviations from Lambda-CDM, and alleviation of the observed
tensions in the cosmological data, provided, of course, the latter are
not due to astrophysical/statistical uncertainties. Non-perturbative
effects in such models (e.g. string instantons) may also generate
non-derivative couplings between axions and right-handed neutrinos,
which in turn may affect properties of warm dark matter in galactic
structure.