Revisiting string-inspired running-vacuum models under the lens of light primordial black holes

Light primordial black holes (PBHs) with masses $M_\mathrm{PBH}<10^9\mathrm{g}$ can interestingly dominate the Universe's energy budget and give rise to early matter-dominated (eMD) eras before Big Bang Nucleosyntesis (BBN). During this eMD era, one is met with an abundant production of induced gravitational waves (GWs) serving as a portal to constrain the underlying theory of gravity. In this work, we study this type of induced GWs within the context of string-inspired running-vaccuum models (StRVMs), which, when expanded around de Sitter backgrounds, include logarithmic corrections of the space-time curvature. In particular, we discuss in detail the effects of StRVMs on the source as well as on the propagation of these PBH-induced GWs. Remarkably, under the assumption that the logarithmic terms represent quantum gravity corrections in the PBH era, we show that GW overproduction can be avoided if one assumes a coefficient of these logarithmic corrections that is much larger than the square of the reduced Planck mass. The latter cannot characterise quantum gravity corrections, though, prompting the need for revision of the quantisation of StRVMs in different than de Sitter backgrounds, such as those characterising PBH-driven eMD eras. This non trivial result suggests the importance of light PBHs as probes of new physics.