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

I review a cosmological model based on string-inspired gravity with anomalies and torsion, which leads to a cosmology of running-vacuum-model (RVM) type. It is argued that such a model can lead to observable, in principle, deviations from the standard $\mathrm{\Lambda }$CDM paradigm of cosmology at late (modern) eras of the Universe evolution, contributing to an alleviation of the currently observed cosmological $H_0$ and structure-growth tensions, but also to a potential enhancement of the density of primordial black holes produced during inflation. The latter can thus play a role as dark matter components, and leave observable imprints in gravitational-wave patterns during the radiation-dominance era of the Universe.