specific rate, can be important in deriving the DTD. At low $z$, the gas-phase metallicity is typically higher than that of the stellar population [@Gallazzi:2005df]. The relation between galaxy mass and gas-phase metallicity is well determined at low $z$ [@Tremonti:2004et] and has been measured to evolve at higher redshifts, so that the typical metallicity decreases by $\sim0.15$ dex per $z$ up to at least $z\approx2$ (e.g., @Kewley [@Savaglio:2005hi; @Erb:2006qy; @Maiolino:2008gh]). We use $$Z(z) = 0.03 \times 10^{-0.15\,z}
\label{Zztop}$$ to account for stars forming from gas that is increasingly metal poor at higher $z$, with a resulting change in rate arising through either the relation approximated by Eq. (\[umfit\]) for the $\dot{n}_{0.85}$ model or Eq. (\[umfit2\]) for the $\dot{n}_{0.70}$ model, again normalizing each to unity at $Z=0.025$ to be consistent with our specific rate models. We also use the same values of $\phi_*$ and DTD slopes as in the corresponding specific rate models.

Fig. \[Iahist\] shows the resulting cosmic rates for the $\dot{n}_{0.85}$ (with $\Delta t_{\rm Gyr}^{-0.8}$; thick dashed line) and $\dot{n}_{0.70}$ ($\Delta t_{\rm Gyr}^{-0.9}$; thick dotted line) models. Both models yield similar histories as the metallicity-independent case, with parametrizations for all three included in the Appendix. This is due to 