  Recent work suggests that Type Ia supernovae (SNe) are composed of two
distinct populations: prompt and delayed. By explicitly incorporating
properties of host galaxies, it may be possible to target and eliminate
systematic differences between these two putative populations. However, any
resulting {\em post}-calibration shift in luminosity between the components
will cause a redshift-dependent systematic shift in the Hubble diagram.
Utilizing an existing sample of 192 SNe Ia, we find that the average luminosity
difference between prompt and delayed SNe is constrained to be $(4.5 \pm
8.9)%$. If the absolute difference between the two populations is 0.025 mag,
and this is ignored when fitting for cosmological parameters, then the dark
energy equation of state (EOS) determined from a sample of 2300 SNe Ia is
biased at $\sim1\sigma$. By incorporating the possibility of a two-population
systematic, this bias can be eliminated. However, assuming no prior on the
strength of the two-population effect, the uncertainty in the best-fit EOS is
increased by a factor of 2.5, when compared to the equivalent sample with no
underlying two-population systematic. To avoid introducing a bias in the EOS
parameters, or significantly degrading the measurement accuracy, it is
necessary to control the post-calibration luminosity difference between prompt
and delayed SN populations to better than 0.025 mag.
