The SWELLS survey - VI. Hierarchical inference of the initial mass functions of bulges and discs

The long-standing assumption that the stellar initial mass function (IMF) is universal has recently been challenged by a number of observations. Several studies have shown that a 'heavy' IMF (e. g. with a Salpeter-like abundance of low-mass stars and thus normalization) is preferred for massive early-type galaxies, while this IMF is inconsistent with the properties of less massive, later-type galaxies. These discoveries motivate the hypothesis that the IMF may vary (possibly very slightly) across galaxies and across components of individual galaxies (e. g. bulges versus discs). In this paper, we use a sample of 19 late-type strong gravitational lenses from the Sloan WFC Edge-on Late-type Lens Survey (SWELLS) to investigate the IMFs of the bulges and discs in late-type galaxies. We perform a joint analysis of the galaxies' total masses (constrained by strong gravitational lensing) and stellar masses (constrained by optical and near-infrared colours in the context of a stellar population synthesis model, up to an IMF normalization parameter). Using minimal assumptions apart from the physical constraint that the total stellar mass m* within any aperture must be less than the total mass m(tot) within the aperture, we find that the bulges of the galaxies cannot have IMFs heavier (i.e. implying high mass per unit luminosity) than Salpeter, while the disc IMFs are not well constrained by this data set. We also discuss the necessity for hierarchical modelling when combining incomplete information about multiple astronomical objects. This modelling approach allows us to place upper limits on the size of any departures from universality. More data, including spatially resolved kinematics (as in Paper V) and stellar population diagnostics over a range of bulge and disc masses, are needed to robustly quantify how the IMF varies within galaxies.