  Galactic and extragalactic studies have shown that metal-rich globular
clusters (GCs) are approximately three times more likely to host bright
low-mass X-ray binaries (LMXBs) than metal-poor GCs. There is no satisfactory
explanation for this metallicity effect. We tested the hypothesis that the
number density of red giant branch (RGB) stars is larger in metal-rich GCs, and
thus potentially the cause of the metallicity effect. Using Hubble Space
Telescope photometry for 109 unique Milky Way GCs, we investigated whether RGB
star density was correlated with GC metallicity. Isochrone fitting was used to
calculate the number of RGB stars, which were normalized by the GC mass and
fraction of observed GC luminosity, and determined density using the volume at
the half-light radius $r_{h}$. The RGB star number density was weakly
correlated with metallicity [Fe/H], giving Spearman and Kendall Rank test
$p$-values of 0.00016 and 0.00021 and coefficients $r_{s} = 0.35$ and $\tau =
0.24$ respectively. This correlation may be biased by a possible dependence of
$r_{h}$ on [Fe/H], although studies have shown that $r_{h}$ is correlated with
Galactocentric distance and independent of [Fe/H]. The dynamical origin of the
$r_{h}$-metallicity correlation (tidal stripping) suggests that metal-rich GCs
may have had more active dynamical histories, which would promote LMXB
formation. No correlation between the RGB star number density and metallicity
was found when using only the GCs that hosted quiescent LMXBs. A complete
census of quiescent LMXBs in our Galaxy is needed to further probe the
metallicity effect, which will be possible with the upcoming launch of eROSITA.
