  Short-period double white dwarf (DWD) binaries will be the most prolific
source of gravitational waves (GWs) for the Laser Interferometer Space Antenna
(LISA). DWDs with GW frequencies below $\sim1$ mHz will be the dominant
contributor to a stochastic foreground caused by overlapping GW signals.
Population modeling of Galactic DWDs typically assumes a binary fraction of 50%
and a log-uniform Zero Age Main Sequence (ZAMS) orbital period distribution.
However, recent observations have shown that the binary fraction of close,
solar-type stars exhibits a strong anti-correlation with metallicity which
modulates the ZAMS orbital period distribution below $10^4$ days. In this study
we perform the first simulation of the Galactic DWD population observable by
LISA which incorporates an empirically-derived metallicity-dependent binary
fraction, using the binary population synthesis suite COSMIC and a
metallicity-dependent star formation history. We compare two models: one which
assumes a metallicity-dependent binary fraction, and one with a binary fraction
of 50%. We repeat our analysis for three different assumptions for Roche-lobe
overflow interactions. We find that while metallicity impacts the evolution and
intrinsic properties of our simulated DWD progenitor binaries, the
LISA-resolvable populations of the two models remain roughly indistinguishable.
However, the size of the total Galactic DWD population orbiting in the LISA
frequency band is reduced by more than half when accounting for a
metallicity-dependent binary fraction for two of our four variations, which
also lowers the effective foreground. The LISA population remains unchanged in
number for two variations, highlighting the sensitivity of the population to
binary evolution prescriptions.
