  The James Webb Space Telescope (JWST) may be capable of finding biogenic
gases in the atmospheres of habitable exoplanets around low mass stars.
Considerable attention has been given to the detectability of biogenic oxygen,
which could be found using an ozone proxy, but ozone detection with JWST will
be extremely challenging, even for the most favorable targets. Here, we
investigate the detectability of biosignatures in anoxic atmospheres analogous
to those that likely existed on the early Earth. Arguably, such anoxic
biosignatures could be more prevalent than oxygen biosignatures if life exists
elsewhere. Specifically, we simulate JWST retrievals of TRAPPIST-1e to
determine whether the methane plus carbon dioxide disequilibrium biosignature
pair is detectable in transit transmission. We find that ~10 transits using the
Near InfraRed Spectrograph (NIRSpec) prism instrument may be sufficient to
detect carbon dioxide and constrain methane abundances sufficiently well to
rule out known, non-biological CH$_{4}$ production scenarios to ~90%
confidence. Furthermore, it might be possible to put an upper limit on carbon
monoxide abundances that would help rule out non-biological methane-production
scenarios, assuming the surface biosphere would efficiently drawdown
atmospheric CO. Our results are relatively insensitive to high altitude clouds
and instrument noise floor assumptions, although stellar heterogeneity and
variability may present challenges.
