We derive the first constraints on the time delay distribution of binary black hole (BBH) mergers using the LIGO-Virgo Gravitational-Wave Transient Catalog GWTC-2. Assuming that the progenitor formation rate follows the star formation rate (SFR), the data favor that $43$--$100%$ of mergers have delay times $<4.5$ Gyr (90% credibility). Adopting a model for the metallicity evolution, we derive joint constraints for the metallicity-dependence of the BBH formation efficiency and the distribution of time delays between formation and merger. Short time delays are favored regardless of the assumed metallicity dependence, although the preference for short delays weakens as we consider stricter low-metallicity thresholds for BBH formation. For a $p(tau) propto tau^{-1}$ time delay distribution and a progenitor formation rate that follows the SFR without metallicity dependence, we find that $tau_mathrm{min}<2.2$ Gyr, whereas considering only the low-metallicity $Z < 0.3,Z_odot$ SFR, $tau_mathrm{min} < 3.0$ Gyr (90% credibility). Alternatively, if we assume long time delays, the progenitor formation rate must peak at higher redshifts than the SFR. For example, for a $p(tau) propto tau^{-1}$ time delay distribution with $tau_mathrm{min} = 4$ Gyr, the inferred progenitor rate peaks at $z > 3.9$ (90% credibility). Finally, we explore whether the inferred formation rate and time delay distribution vary with BBH mass.
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