The $H_0$ tension alleviated through ultra-light primordial black holes: an information insight through gravitational waves

The Hawking evaporation of ultra-light primordial black holes (PBH) dominating the early Universe before Big Bang Nucleosynthesis can potentially increase the effective number of extra neutrino species $\mathrm{\Delta }{N}_{\mathrm{e}\mathrm{f}\mathrm{f}}$ through the emission of dark radiation degrees of freedom alleviating in this way the $H_0$ tension problem. Interestingly, these light PBHs can form a gas of Poisson distributed compact objects which can induce a gravitational-wave (GW) background due to second order gravitational interactions. Therefore, by considering the contribution to $\mathrm{\Delta }{N}_{\mathrm{e}\mathrm{f}\mathrm{f}}$ due to the production of the aforementioned GW background we revisit in this work the constraints on the relevant parameters at hand, namely the PBH mass, $m_{\mathrm{P}\mathrm{B}\mathrm{H}}$, the initial PBH abundance at PBH formation time, ${\mathrm{\Omega }}_{\mathrm{P}\mathrm{B}\mathrm{H},\mathrm{f}}$ and the number of DR radiation degrees of freedom, $g_{\mathrm{D}\mathrm{R}}$ by accounting at the same time for the relevant upper bounds constraints on $\mathrm{\Delta }{N}_{\mathrm{e}\mathrm{f}\mathrm{f}}$ from the Planck collaboration.