Constraining changes in the merger history of (P)BH 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 $z\sim 4$, 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 ($z\simeq 1$ 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, 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.
