  Primordial Black Holes (PBH) could be the cold dark matter of the universe.
They could have arisen from large (order one) curvature fluctuations produced
during inflation that reentered the horizon in the radiation era. At reentry,
these fluctuations source gravitational waves (GW) via second order anisotropic
stresses. These GW, together with those (possibly) sourced during inflation by
the same mechanism responsible for the large curvature fluctuations, constitute
a primordial stochastic GW background (SGWB) that unavoidably accompanies the
PBH formation. We study how the amplitude and the range of frequencies of this
signal depend on the statistics (Gaussian versus $\chi^2$) of the primordial
curvature fluctuations, and on the evolution of the PBH mass function due to
accretion and merging. We then compare this signal with the sensitivity of
present and future detectors, at PTA and LISA scales. We find that this SGWB
will help to probe, or strongly constrain, the early universe mechanism of PBH
production. The comparison between the peak mass of the PBH distribution and
the peak frequency of this SGWB will provide important information on the
merging and accretion evolution of the PBH mass distribution from their
formation to the present era. Different assumptions on the statistics and on
the PBH evolution also result in different amounts of CMB $\mu$-distortions.
Therefore the above results can be complemented by the detection (or the
absence) of $\mu$-distortions with an experiment such as PIXIE.
