Upper: distribution of ${\log }_{10}(M_{\mathrm{c}\mathrm{o}\mathrm{r}\mathrm{e}}^{\mathrm{m}\mathrm{a}\mathrm{x}})$ as a function of t for the ALMAGAL sample. Black error bars indicate the mean and standard deviation of ${\log }_{10}(M_{\mathrm{c}\mathrm{o}\mathrm{r}\mathrm{e}}^{\mathrm{m}\mathrm{a}\mathrm{x}})$ within each time bin, while blue error bars represent the standard error of the mean (i.e., stderr = std/√N, with N being the number of sources in each bin). The red line shows the exponential fit between $M_{\mathrm{c}\mathrm{o}\mathrm{r}\mathrm{e}}^{\mathrm{m}\mathrm{a}\mathrm{x}}$ and t. Lower: distribution of L_bol as a function of t for the ATLASGAL sample. The red line represents the best-fit model of Eq. (13). The dashed and solid black lines indicate the contributions from intrinsic stellar luminosity and accretion luminosity, respectively (see Sect. 3.3).