The Mass Density of Merging Binary Black Holes Over Cosmic Time

The connection between the binary black hole (BBH) mergers observed by LIGO-Virgo-KAGRA (LVK) and their stellar progenitors remains uncertain. Specifically, the fraction $\epsilon$ of stellar mass that ends up in BBH mergers and the delay time $\tau$ between star formation and BBH merger carry information about the astrophysical processes that give rise to merging BBHs. We model the BBH merger rate in terms of the cosmic star formation history, coupled with a metallicity-dependent efficiency $\epsilon$ and a distribution of delay times $\tau$, and infer these parameters with data from the Third Gravitational-Wave Transient Catalog (GWTC-3). We find that the progenitors to merging BBHs preferentially form in low metallicity environments with a low metallicity efficiency of $\log_{10}\epsilon_{<Z_t}=-3.99^{+0.68}_{-0.87}$ and a high metallicity efficiency of $\log_{10}\epsilon_{<Z_t}=-4.60^{+0.30}_{-0.34}$ at the 90% credible level. The data also prefer short delay times. For a power-law distribution $p(\tau)\propto \tau^\alpha$, we find $\tau_\text{min}<1.9 $ Gyr and $\alpha<-1.32$ at 90% credibility. Our model allows us to extrapolate the mass density in BBHs out to high redshifts. We cumulatively integrate our modelled density rate over cosmic time to get the total mass density of merging stellar mass BBHs as a function of redshift. Today, stellar-mass BBH mergers make up only $\sim 0.01\%$ of the total stellar mass density created by high-mass ($>10\,M_\odot$) progenitors. However, because massive stars are so short-lived, there may be more mass in merging BBHs than in living massive stars as early as $\sim 2.5$ Gyr ago. We also compare to the mass in supermassive BHs, finding that the mass densities were comparable $\sim 12.5$ Gyr ago, but the mass density in SMBHs quickly increased to $\sim 75$ times the mass density in merging stellar mass BBHs by $z\sim 1$.