We explore the effects of the deflagration to detonation transition (DDT) density on the production of Ni-56 in thermonuclear supernova explosions (type Ia supernovae). Within the DDT paradigm, the transition density sets the amount of expansion during the deflagration phase of the explosion and therefore the amount of nuclear statistical equilibrium (NSE) material produced. We employ a theoretical framework for a well-controlled statistical study of two-dimensional simulations of thermonuclear supernovae with randomized initial conditions that can, with a particular choice of transition density, produce a similar average and range of Ni-56 masses to those inferred from observations. Within this framework, we utilize a more realistic simmered white dwarf progenitor model with a flame model and energetics scheme to calculate the amount of Ni-56 and NSE material synthesized for a suite of simulated explosions in which the transition density is varied in the range 1-3x10^7 g/cc. We find a quadratic dependence of the NSE yield on the log of the transition density, which is determined by the competition between plume rise and stellar expansion. By considering the effect of metallicity on the transition density, we find the NSE yield decreases by 0.055 +/- 0.004 solar masses for a 1 solar metallicity increase evaluated about solar metallicity. For the same change in metallicity, this result translates to a 0.067 +/- 0.004 solar mass decrease in the Ni-56 yield, slightly stronger than that due to the variation in electron fraction from the initial composition. Observations testing the dependence of the yield on metallicity remain somewhat ambiguous, but the dependence we find is comparable to that inferred from some studies.
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