  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.
