On the time evolution of the $M_{\rm d} - M_\star$ and $\dot M - M_\star$ correlations for protoplanetary discs: the viscous timescale increases with stellar mass

Large surveys of star-forming regions have unveiled power-law correlations between the stellar mass and the disc parameters, such as the disc mass $M_{\mathrm{d}} \propto {M_{\star}}^{\lambda_{\mathrm{m}}}$ and the accretion rate $\dot M \propto {M_{\star}}^{\lambda_{\mathrm{acc}}}$. The observed slopes appear to be increasing with time, but the reason behind the establishment of these correlations and their subsequent evolution is still uncertain. We conduct a theoretical analysis of the impact of viscous evolution on power-law initial conditions for a population of protoplanetary discs. We find that, for evolved populations, viscous evolution enforces the two correlations to have the same slope, $\lambda_{\mathrm{m}}$ = $\lambda_{\mathrm{acc}}$, and that this limit is uniquely determined by the initial slopes $\lambda_{\mathrm{m}, 0}$ and $\lambda_{\mathrm{acc}, 0}$. We recover the increasing trend claimed from the observations when the difference in the initial values, $\delta_0 = \lambda_{\mathrm{m}, 0} - \lambda_{\mathrm{acc}, 0}$, is larger than $1/2$; moreover, we find that this increasing trend is a consequence of a positive correlation between the viscous timescale and the stellar mass. We also present the results of disc population synthesis numerical simulations, that allow us to introduce a spread and analyse the effect of sampling, which show a good agreement with our analytical predictions. Finally, we perform a preliminary comparison of our numerical results with observational data, which allows us to constrain the parameter space of the initial conditions to $\lambda_{\mathrm{m}, 0} \in [1.2, 2.1]$, $\lambda_{\mathrm{acc}, 0} \in [0.7, 1.5]$.