Variations in the stellar CMF and IMF: from bottom to top

We use a recently developed analytic model for the interstellar medium (ISM) structure from scales of giant molecular clouds (GMCs) through star-forming cores to explore how the pre-stellar core mass function (CMF) and, by extrapolation, stellar initial mass function (IMF) should depend on both local and galactic properties. If the ISM is supersonically turbulent, the statistical properties of the density field follow from the turbulent velocity spectrum, and the excursion set formalism can be applied to analytically calculate the mass function of collapsing cores on the smallest scales on which they are self-gravitating (non-fragmenting). Two parameters determine the model: the disc-scale Mach number M-h (which sets the shape of the CMF) and the absolute velocity/surface density (to assign an absolute scale). We show that, for normal variation in disc properties and gas temperatures in cores in the Milky Way and local galaxies, there is almost no variation in the high-mass behaviour of the CMF/IMF. The slope is always close to Salpeter down to less than or similar to 1 M-circle dot. We predict modest variation in the sub-solar regime, mostly from variation in M-h, but this is consistent with the similar to 1 sigma observed scatter in sub-solar IMFs in local regions. For fixed global (galaxy) properties, there is little variation in shape or 'upper mass limit' with parent GMC mass or density. However, in extreme starbursts - ultra-luminous infrared galaxies (ULIRGs) and merging galaxy nuclei - we predict a bottom-heavy CMF. This agrees well with the IMF recently inferred for the centres of Virgo ellipticals, believed to have formed in such a process. The CMF is bottom heavy despite the gas temperature being an order of magnitude larger, because M-h is also much larger. Larger M-h values make the 'parent' cloud mass (the turbulent Jeans mass) larger, but promote fragmentation to smaller scales (set by the sonic mass, not the Jeans mass); this shifts the turnover mass and also steepens the slope of the low-mass CMF. The model may also predict a top-heavy CMF for the in situ star formation in the sub-pc disc around Sgr A*, but the relevant input parameters are uncertain.