  In a modest attempt to present potentially new paradigms in Cosmology,
including its inflationary epoch, and initiate discussions, I review in this
article some novel, string-inspired cosmological models, which entail a purely
geometrical origin of the Dark sector of the Universe but also of its observed
matter-antimatter asymmetry. The models contain gravitational (string-model
independent, Kalb-Ramond (KR)) axion fields coupled to primordial gravitational
anomalies via CP-violating interactions. The anomaly terms are
four-space-time-dimensional remnants of the Green-Schwarz counterterms
appearing in the definition of the field strength of the spin-one antisymmetric
tensor field of the (bosonic) massless gravitational string multiplet, which
also plays the r\^ole of a totally antisymmetric component of torsion. I show
how in such cosmologies the presence of primordial gravitational waves can lead
to anomaly condensates and dynamical inflation of a "running-vacuum-model"
type, without external inflatons, but also to leptogenesis in the radiation era
due to anomaly-induced Lorentz and CPT Violating KR axion backgrounds. I also
discuss how the torsion-related KR-axion could acquire a mass during the QCD
epoch, thus playing the role of (a component of) Dark Matter. Phenomenological
considerations of the inflationary and post-inflationary (in particular,
modern) eras of the model are briefly discussed, including its potential for
alleviating the observed tensions in the cosmological data of the current
epoch.
