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