An inventory of the stellar initial mass function in early-type galaxies
  Given a flurry of recent claims for systematic variations in the stellar
initial mass function (IMF), we carry out the first inventory of the
observational evidence using different approaches. This includes literature
results, as well as our own new findings from combined stellar-populations
synthesis (SPS) and Jeans dynamical analyses of data on $\sim$~4500 early-type
galaxies (ETGs) from the SPIDER project. We focus on the mass-to-light ratio
mismatch relative to the Milky Way IMF, \dimf, correlated against the central
stellar velocity dispersion, \sigs. We find a strong correlation between \dimf\
and \sigs, for a wide set of dark matter (DM) model profiles. These results are
robust if a uniform halo response to baryons is adopted across the sample. The
overall normalization of \dimf, and the detailed DM profile, are less certain,
but the data are consistent with standard cold-DM halos, and a central DM
fraction that is roughly constant with \sigs. For a variety of related studies
in the literature, using SPS, dynamics, and gravitational lensing, similar
results are found. Studies based solely on spectroscopic line diagnostics agree
on a Salpeter-like IMF at high \sigs, but differ at low \sigs. Overall, we find
that multiple independent lines of evidence appear to be converging on a
systematic variation in the IMF, such that high-\sigs\ ETGs have an excess of
low-mass stars relative to spirals and low-\sigs\ ETGs. Robust verification of
super-Salpeter IMFs in the highest-\sigs\ galaxies will require additional
scrutiny of scatter and systematic uncertainties. The implications for the
distribution of DM are still inconclusive.
