A universal initial mass function (IMF) is not intuitive, but so far no convincing evidence for a variable IMF exists. The detection of systematic variations of the IMF with star-forming conditions would be the Rosetta Stone for star formation.
In this contribution an average or Galactic-field IMF is defined, stressing that there is evidence for a change in the power-law index at only two masses: near 0.5 M ( and near 0.08 M ( . Using this supposed universal IMF, the uncertainty inherent in any observational estimate of the IMF is investigated by studying the scatter introduced by Poisson noise and the dynamical evolution of star clusters. It is found that this apparent scatter reproduces quite well the observed scatter in power-law index determinations, thus defining the fundamental limit within which any true variation becomes undetectable. The absence of evidence for a variable IMF means that any true variation of the IMF in well-studied populations must be smaller than this scatter.
Determinations of the power-law indices a are subject to systematic errors arising mostly from unresolved binaries. The systematic bias is quantified here, with the result that the single-star IMFs for young star clusters are systematically steeper by Da < 0X5 between 0.1 and 1 M ( than the Galactic-field IMF, which is populated by, on average, about 5-Gyr-old stars. The MFs in globular clusters appear to be, on average, systematically flatter than the Galactic-field IMF (Piotto & Zoccali; Paresce & De Marchi), and the recent detection of ancient white-dwarf candidates in the Galactic halo and the absence of associated low-mass stars (Ibata et al.; Me Γndez & Minniti) suggest a radically different IMF for this ancient population. Star formation in higher metallicity environments thus appears to produce relatively more low-mass stars. While still tentative, this is an interesting trend, being consistent with a systematic variation of the IMF as expected from theoretical arguments.