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I review a cosmological model based on string-inspired gravity with anomaliesand torsion, which leads to a cosmology of running-vacuum-model (RVM) type. Itis argued that such a model can lead to observable, in principle, deviationsfrom the standard $\Lambda$CDM paradigm of cosmology at late (modern) eras ofthe Universe evolution, contributing to an alleviation of the currentlyobserved cosmological $H_0$ and structure-growth tensions, but also to apotential enhancement of the density of primordial black holes produced duringinflation. The latter can thus play a role as dark matter components, and leaveobservable imprints in gravitational-wave patterns during theradiation-dominance era of the Universe.

A string-inspired running-vacuum-model of cosmology, primordial-black-hole dark matter, and the current tensions in cosmological data