  Chemical disequilibrium in planetary atmospheres has been proposed as a
generalized method for detecting life on exoplanets through remote
spectroscopy. Among solar system planets with substantial atmospheres, the
modern Earth has the largest thermodynamic chemical disequilibrium due to the
presence of life. However, how this disequilibrium changed over time and, in
particular, the biogenic disequilibria maintained in the anoxic Archean or less
oxic Proterozoic eons are unknown. We calculate the atmosphere-ocean
disequilibrium in the Precambrian using conservative proxy- and model-based
estimates of early atmospheric and oceanic compositions. We omit crustal solids
because subsurface composition is not detectable on exoplanets, unlike
above-surface volatiles. We find that (i) disequilibrium increased through time
in step with the rise of oxygen; (ii) both the Proterozoic and Phanerozoic may
have had remotely detectable biogenic disequilibria due to the coexistence of
O$_{2}$, N$_{2}$, and liquid water; and (iii) the Archean had a biogenic
disequilibrium caused by the coexistence of N$_2$, CH$_4$, CO$_2$, and liquid
water, which, for an exoplanet twin, may be remotely detectable. On the basis
of this disequilibrium, we argue that the simultaneous detection of abundant
CH$_{4}$ and CO$_{2}$ in a habitable exoplanet's atmosphere is a potential
biosignature. Specifically, we show that methane mixing ratios greater than
0.001 are potentially biogenic, whereas those exceeding 0.01 are likely
biogenic due to the difficulty in maintaining large abiotic methane fluxes to
support high methane levels in anoxic atmospheres. Biogenicity would be
strengthened by the absence of abundant CO, which should not coexist in a
biological scenario.
