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

A boosted gravitational-wave background for primordial black holes with broad mass distributions and thermal features