Testing Primordial Black Holes with multi-band observations of the stochastic gravitational wave background

The mass distribution of Primordial Black Holes (PBHs) is affected by drops in the pressure of the early Universe plasma. For example, events in the standard model of particle physics, such as the $W^\pm/Z^0$ decoupling, the quark-hadron transition, the muon and pion becoming non-relativistic, and the annihilation of electrons and positrons, cause a suppression in the Equation of State parameter and leave peaks in the PBH mass function around $10^{-6},\,2,\,60$, and $10^6\, M_\odot$, respectively, in the case of a nearly scale-invariant primordial power spectrum. The superposition of unresolved mergers of such PBHs results in a stochastic gravitational-wave background (SGWB) that covers a wide range of frequencies and can be tested with future gravitational wave (GW) detectors. In this paper, we discuss how its spectral shape can be used to infer properties about inflation, the thermal history of the Universe, and the dynamics of binary formation in dense halos encoded in their merger rate formula. Although many of these physical effects are degenerate within the sensitivity of a single detector, they can be disentangled by the simultaneous observation of the SGWB at different frequencies, highlighting the importance of multi-frequency observations of GWs to characterize the physics of PBHs from the early to the late time Universe.