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The gravitational potential of initially Poisson distributed primordial blackholes (PBH) can induce a stochastic gravitational-wave background (SGWB) atsecond order in cosmological perturbation theory. This SGWB was previouslystudied in the context of general relativity (GR) and modified gravity setupsby assuming a monochromatic PBH mass function. Here we extend the previousanalysis in the context of GR by studying the aforementioned SGWB within morephysically realistic regimes where PBHs have different masses. In particular,starting from a power-law cosmologically motivated primordial curvature powerspectrum with a running spectral index we extract the extended PBH massfunction and the associated to it PBH gravitational potential which acts as thesource of the scalar induced SGWB. At the end, by taking into account thedynamical evolution of the PBH gravitational potential during the transitionfrom the matter era driven by PBHs to the radiation era we extract therespective GW signal today. Interestingly, in order to trigger an earlyPBH-dominated era and avoid the GW constraints at BBN we find that the runningof the spectral index $\alpha_\mathrm{s}$ of our primordial curvature powerspectrum should be within the narrow range$\alpha_\mathrm{s}\in[3.316,3.355]\times 10^{-3}$ while at the same time the GWsignal is found to be potentially detectable by LISA.

Gravitational waves induced from primordial black hole fluctuations: The effect of an extended mass function