Variations in the initial mass function in early-type galaxies: A
  critical comparison between dynamical and spectroscopic results
  I present a comparison between published dynamical (ATLAS3D) and
spectroscopic (Conroy & van Dokkum) constraints on the stellar initial mass
function (IMF) in early-type galaxies, using the 34 galaxies in common between
the two works. Both studies infer an average IMF mass factor $\alpha$ (the
stellar mass relative to a Kroupa-IMF population of similar age and
metallicity) greater than unity, i.e. both methods favour an IMF which is
heavier than that of the Milky Way, on average over the sample. However, on a
galaxy-by-galaxy basis, there is no correlation between $\alpha$ inferred from
the two approaches. I investigate how the two estimates of $\alpha$ are
correlated systematically with the galaxy velocity dispersion, $\sigma$, and
with the Mg/Fe abundance ratio. The spectroscopic method, based on the
strengths of metal absorption lines, yields a correlation only with metal
abundance ratios: at fixed Mg/Fe, there is no residual correlation with
$\sigma$. The dynamical method, applied to exactly the same galaxy sample,
yields the opposite result: the IMF variation correlates only with dynamics,
with no residual correlation with Mg/Fe after controlling for $\sigma$. Hence
although both methods indicate a heavy IMF on average in ellipticals, they lead
to incompatible results for the systematic trends, when applied to the same set
of galaxies. The sense of the disagreement could suggest that one (or both) of
the methods has not accounted fully for the main confounding factors, i.e.
element abundance ratios or dark matter contributions. Alternatively, the poor
agreement might indicate additional variation in the detailed shape of the IMF,
beyond what can currently be inferred from the spectroscopic features.
