Constraining changes in the merger history of BH and PBH binaries with the stochastic gravitational wave background

Black holes binaries coming from a distribution of primordial black might exhibit large merger rate up to redshifts. Using phenomenological model for the rate, we show that changes in its slope, redshifts $z\sim 4$, are constrained by current limits on amplitude stochastic gravitational wave background LIGO/Virgo O3 run. This shows constrains larger than single event horizon detection ($z\simeq 1$ same detector). Moreover, steep rates shape signal at intermediate frequencies differs usual $2/3$ IR scaling. We discuss implications our future experiments wide range frequencies, as design array, Einstein Telescope, LISA and PTA. Additionally, i) present provide equally constraining bounds abundance PBHs arising Gaussian inhomogeneities ii) Degeneracies level (i.e. different histories leading similar background) can be broken considering complementary direct observation number counts detectors such Telescope.