CO isotopolog line fluxes of viscously evolving disks: cold CO conversion insufficient to explain observed low fluxes

Protoplanetary disks are thought to evolve viscously, where the disk mass - the reservoir available for planet formation - decreases over time as material is accreted onto the central star. Observations show a correlation between dust mass and the stellar accretion rate, as expected from viscous theory. However, the gas mass inferred from 13CO and C18O line fluxes, which should be a more direct measure, shows no such correlation. Using thermochemical DALI models, we investigate how 13CO and C18O J=3-2 line fluxes change over time in a viscously evolving disk. We also investigate if the chemical conversion of CO through grain-surface chemistry combined with viscous evolution can explain the observations of disks in Lupus. The 13CO and C18O 3-2 line fluxes increase over time due to their optically thick emitting regions growing in size as the disk expands viscously. The C18O 3-2 emission is optically thin throughout the disk for only a subset of our models (Mdisk (t = 1 Myr) < 1e-3 Msun). For these disks the integrated C18O flux decreases with time, similar to the disk mass. The C18O 3-2 fluxes for the bulk of the disks in Lupus (with Mdust < 5e-5 Msun) can be reproduced to within a factor of ~2 with viscously evolving disks in which CO is converted into other species through grain-surface chemistry driven by a cosmic-ray ionization rate zeta_cr ~ 5e-17 - 1e-16 s^-1. However, explaining the stacked C18O upper limits requires a lower average abundance than our models can produce and they cannot explain the observed 13CO fluxes, which, for most disks, are more than an order of magnitude fainter than what our models predict. Reconciling the 13CO fluxes of viscously evolving disks with the observations requires either a combination of efficient vertical mixing and a high zeta_cr or low mass disks (Mdust < 3e-5 Msun) being much thinner and/or smaller than their more massive counterparts.