The temporal requirements of directly observing self-gravitating spiral waves in protoplanetary discs with ALMA

We investigate how the detectability of signatures of self-gravity in a protoplanetary disc depends on its temporal evolution. We run a one-dimensional model for secular timescales to follow the disc mass as a function of time. We then combine this with three-dimensional global hydrodynamics simulations that employ a hybrid radiative transfer method to approximate realistic heating and cooling. We simulate ALMA continuum observations of these systems, and find that structures induced by the gravitational instability (GI) are readily detectable when $q=M_\mathrm{disc}/M_*\gtrsim 0.25$ and $R_\mathrm{outer}\lesssim 100$ au. The high accretion rate generated by gravito-turbulence in such a massive disc drains its mass to below the detection threshold in $\sim10^4$ years, or approximately 1 % of the typical disc lifetime. Therefore, discs with spiral arms detected in ALMA dust observations, if generated by self-gravity, must either be still receiving infall to maintain a high $q$ value, or have just emerged from their natal envelope. Detection of substructure in systems with lower $q$ is possible, but would require a specialist integration with the most extended configuration over several days. This disfavours the possibility of GI-caused spiral structure in systems with $q<0.25$ being detected in relatively short integration times, such as those found in the DSHARP ALMA survey (Andrews et al. 2018; Huang et al. 2018). We find no temporal dependence of detectability on dynamical timescales