  One of the goals of gravitational-wave astronomy is to quantify the evolution
of the compact binary merger rate with redshift. The redshift distribution of
black hole mergers would offer considerable information about their
evolutionary history, including their progenitor formation rate, the dependence
of black hole formation on stellar metallicity, and the time delay distribution
between formation and merger. Efforts to measure the binary redshift
distribution are currently limited, however, by the detection range of existing
instruments, which can individually resolve compact binary merger events only
out to $z\lesssim1$. We present a novel strategy with which to measure the
redshift distribution of binary black hole mergers well beyond the detection
range of current instruments. By synthesizing direct detections of individually
resolved mergers with \textit{indirect} searches for the stochastic
gravitational-wave background due to unresolved distant sources, we can glean
information about the peak redshift, $z_p$, at which the binary black hole
merger rate attains its maximum, even when this redshift is beyond the
detection horizon. Using data from Advanced LIGO and Virgo's first and second
observing runs, we employ this strategy to place joint constraints on $z_p$ and
the slope $\alpha$ with which the binary merger rate increases at low
redshifts, ruling out merger rates that grow faster than $\alpha \gtrsim 7$ and
peak beyond $z_p \gtrsim 1.5$. Looking ahead, we project that approximately one
year of observation with design-sensitivity Advanced LIGO will further break
remaining degeneracies, enabling a direct measurement of the peak redshift of
the binary black hole merger history.
