  Primordial black holes (PBHs) with a wide mass distribution imprinted by the
thermal history of the Universe, which naturally produces a high peak at the
solar mass scale, could explain the gravitational-wave events seen by
LIGO/Virgo and up to the totality of the dark matter. We show that compared to
monochromatic or log-normal mass functions, the gravitational wave backgrounds
(GWBs) from early PBH binaries and from late binaries in clusters are strongly
enhanced at low frequency and could even explain the NANOGrav observations.
This enhancement comes from binaries with very low mass ratios, involving
solar-mass and intermediate-mass PBHs at low frequency, solar-mass and
subsolar-mass at high frequency. LISA could distinguish the various models,
while in the frequency band of ground-based detectors, we find that the GWB
from early binaries is just below the current LIGO/Virgo limits and above the
astrophysical background, if they also explain black hole mergers. The GWB from
binaries in clusters is less boosted but has a different spectral index than
for neutron stars, astrophysical black holes or early PBH binaries. It is
detectable with Einstein Telescope or even with the LIGO/Virgo design
sensitivity.
