Constraining supermassive primordial black holes with magnetically induced gravitational waves

Primordial black holes (PBHs) can answer a plethora of cosmic conundra, among which the origin of the cosmic magnetic fields. In particular, supermassive PBHs with masses $M_\mathrm{PBH}>10^{10} M_\odot$ and furnished with a plasma-disk moving around them can generate through the Biermann battery mechanism a seed primordial magnetic field which can later be amplified so as to provide the magnetic field threading the intergalactic medium. In this work, we derive the gravitational wave (GW) signal induced by the magnetic anisotropic stress of such a population of magnetised PBHs. Interestingly enough, by using GW constraints from Big Bang Nucleosynthesis (BBN) and an effective model for the galactic/turbulent dynamo amplification of the magnetic field, we set a conservative upper bound constraint on the abundances of supermassive PBHs at formation time, $\Omega_\mathrm{PBH,f}$ as a function of the their masses, namely that $\Omega_\mathrm{PBH,f}\leq 2.5\times 10^{-10}\left(\frac{M}{10^{10}M_\odot}\right)^{45/22}$. Remarkably, these constraints are comparable, and, in some mass ranges, even tighter compared to the constraints on $\Omega_\mathrm{PBH,f}$ from large-scale structure (LSS) probes; hence promoting the portal of magnetically induced GWs as a new probe to explore the enigmatic nature of supermassive PBHs.