  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
