  The search for signs of life on other worlds has largely focused on
terrestrial planets. Recent work, however, argues that life could exist in the
atmospheres of temperate sub-Neptunes. Here, we evaluate the usefulness of
carbon dioxide isotopologues as evidence of aerial life. Carbon isotopes are of
particular interest as metabolic processes preferentially use the lighter
$^{12}$C over $^{13}$C. In principle, the upcoming James Webb Space Telescope
(JWST) will be able to spectrally resolve the $^{12}$C and $^{13}$C
isotopologues of CO$_{2}$, but not CO and CH$_{4}$. We simulated observations
of CO$_{2}$ isotopologues in the H$_{2}$-dominated atmospheres of our nearest
($< 40$ pc), temperate (equilibrium temperature of 250-350 K) sub-Neptunes with
M dwarf host stars. We find $^{13}$CO$_{2}$ and $^{12}$CO$_{2}$ distinguishable
if the atmosphere is H$_{2}$-dominated with a few percentage points of CO$_{2}$
for the most idealized target with an Earth-like composition of the two most
abundant isotopologues, $^{12}$CO$_{2}$ and $^{13}$CO$_{2}$. With a
Neptune-like metallicity of 100$\times$ solar and a C/O of 0.55, we are unable
to distinguish between $^{13}$CO$_{2}$ and $^{12}$CO$_{2}$ in the atmospheres
of temperate sub-Neptunes. If atmospheric composition largely follows
metallicity scaling, the concentration of CO$_{2}$ in a H$_{2}$-dominated
atmosphere will be too low to distinguish CO$_{2}$ isotopologues with JWST. In
contrast, at higher metallicities, there will be more CO$_{2}$, but the smaller
atmospheric scale height makes the measurement impossible. Carbon dioxide
isotopologues are unlikely to be useful biosignature gases for the JWST era.
Instead, isotopologue measurements should be used to evaluate formation
mechanisms of planets and exoplanetary systems.
