  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 $z\sim 1$.
