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In a modest attempt to present potentially new paradigms in Cosmology,including its inflationary epoch, and initiate discussions, I review in thisarticle some novel, string-inspired cosmological models, which entail a purelygeometrical origin of the Dark sector of the Universe but also of its observedmatter-antimatter asymmetry. The models contain gravitational (string-modelindependent, Kalb-Ramond (KR)) axion fields coupled to primordial gravitationalanomalies via CP-violating interactions. The anomaly terms arefour-space-time-dimensional remnants of the Green-Schwarz countertermsappearing in the definition of the field strength of the spin-one antisymmetrictensor field of the (bosonic) massless gravitational string multiplet, whichalso plays the r\^ole of a totally antisymmetric component of torsion. I showhow in such cosmologies the presence of primordial gravitational waves can leadto anomaly condensates and dynamical inflation of a "running-vacuum-model"type, without external inflatons, but also to leptogenesis in the radiation eradue to anomaly-induced Lorentz and CPT Violating KR axion backgrounds. I alsodiscuss how the torsion-related KR-axion could acquire a mass during the QCDepoch, thus playing the role of (a component of) Dark Matter. Phenomenologicalconsiderations of the inflationary and post-inflationary (in particular,modern) eras of the model are briefly discussed, including its potential foralleviating the observed tensions in the cosmological data of the currentepoch.

philosophical transactions - royal society. mathematical, physical and engineering sciences/philosophical transactions - royal society. mathematical, physical and engineering sciences

Geometrical origins of the Universe dark sector: string-inspired torsion and anomalies as seeds for inflation and dark matter