Protoplanetary disc population synthesis I. Constraining disc parameters to reproduce disc observations

Context. Protoplanetary discs are the birthplaces of planets. Recent studies highlight the role of stellar mass sampling in determining disc lifetimes from the observed fraction of stars with discs. Low-mass stars tend to host longer-lived discs, allowing planet formation via solid accretion. Observations also reveal a strong correlation between stellar (and substellar) mass and accretion rate, typically following $\dot{M}\propto M_\star^2$. Aims. We aim to identify the optimal parameters of a disc evolution model that reproduces both the observed disc fractions and accretion rates in young stellar populations. Methods. We performed a population synthesis study exploring different dependencies of the viscosity parameter $\alpha$ on stellar mass. Disc evolution includes viscous accretion and photoevaporation (internal and external). Initial disc masses and radii were drawn from observationally motivated distributions, while stellar masses followed a given distribution and a time-dependent star formation rate (SFR) was introduced. Results. Matching observed disc fractions and accretion trends requires $\alpha$ to increase with stellar mass. External photoevaporation is necessary to produce low-mass discs with high accretion rates, and a time-dependent SFR enhances accretion in young clusters while extending disc lifetimes in older ones. A stellar mass cut-off reproduces the distance-dependent biases in observed disc fractions. Conclusions. Both stellar and environmental effects are essential to explain the observed properties of protoplanetary discs. A stellar-mass-dependent viscosity reproduces the $\dot{M}$-$M_\star$ relation, while external photoevaporation and extended star formation histories shape the accretion rate distribution across environments.