Shouts and Murmurs: Combining Individual Gravitational-wave Sources with the Stochastic Background to Measure the History of Binary Black Hole Mergers

<jats:title>Abstract</jats:title> <jats:p>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 <jats:inline-formula> <jats:tex-math> <?CDATA $z\lesssim 1$?> </jats:tex-math> <jats:inline-graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="apjlab9743ieqn1.gif" xlink:type="simple" /> </jats:inline-formula>. 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 <jats:italic>indirect</jats:italic> searches for the stochastic gravitational-wave background due to unresolved distant sources, we can glean information about the peak redshift, <jats:italic>z</jats:italic> <jats:sub> <jats:italic>p</jats:italic> </jats:sub>, 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 <jats:italic>z</jats:italic> <jats:sub> <jats:italic>p</jats:italic> </jats:sub> and the slope <jats:italic>α</jats:italic> with which the binary merger rate increases at low redshifts, ruling out merger rates that grow faster than <jats:inline-formula> <jats:tex-math> <?CDATA $\alpha \gtrsim 7$?> </jats:tex-math> <jats:inline-graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="apjlab9743ieqn2.gif" xlink:type="simple" /> </jats:inline-formula> and peak beyond <jats:inline-formula> <jats:tex-math> <?CDATA ${z}_{p}\gtrsim 1.5$?> </jats:tex-math> <jats:inline-graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="apjlab9743ieqn3.gif" xlink:type="simple" /> </jats:inline-formula>. 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.</jats:p>