The Stellar and Sub-Stellar Initial Mass Function of Simple and Composite Populations

The current knowledge on the stellar IMF is documented. It is usuallydescribed as being invariant, but evidence to the contrary has emerged: it appears to become top-heavy when the star-formation rate density surpasses about 0.1 M_{odot}/(year {pc}³) ona pc scale and it may become increasingly bottom-heavy withincreasing metallicity and in increasingly massive elliptical galaxies. Itdeclines quite steeply below about 0.07 M_{odot } with brown dwarfs (BDs) and very low mass stars having their own IMF. The most massive star of mass m _max formed in an embedded cluster with stellar mass M _ecl correlates strongly with M _ecl being a result of gravitation-driven but resource-limited growth and fragmentation-induced starvation. There is no convincing evidence whatsoeverthat massive stars do form in isolation. Massive stars form above a density threshold in embedded clusters which become saturated when m_{max} = m_{max{ast}}≈ 150 M_{odot } which appears to be the canonical physical upper mass limit of stars. Super-canonical massive stars arise naturally due to stellar mergers induced bystellar-dynamical encounters in binary-rich very young dense clusters.