A low-mass cut-off near the hydrogen burning limit for Salpeter-like
  initial mass functions in early-type galaxies
  We conduct a detailed investigation of the properties of the stellar initial
mass function (IMF) in two massive early-type lens galaxies with velocity
dispersions of sigma ~245 km/s and sigma ~325 km/s, for which both HST imaging
and X-Shooter spectra are available. We compare the inferences obtained from
two fully independent methods: (i) a combined gravitational lensing and stellar
dynamics (L&D) analysis of the data sets employing self-consistent axisymmetric
models, and (ii) a spectroscopic simple stellar population (SSP) analysis of
optical line-strength indices, assuming single power-law IMFs. The results from
the two approaches are found to be in agreement within the 1-sigma
uncertainties. Both galaxies are consistent with having a Salpeter IMF
(power-law slope of x = 2.35), which is strongly favoured over a Chabrier IMF
(x = 1.8), with probabilities inferred from the joint analysis of 89% and 99%,
respectively. Bottom-heavy IMFs significantly steeper than Salpeter (x >= 3.0)
are ruled out with decisive evidence (Bayes factor B > 1000) for both galaxies,
as they exceed the total mass derived from the L&D constraints. Our analysis
allows, for the first time, the inference of the low-mass cut-off of the IMF
(M_low). Combining the joint L&D and SSP analyses of both galaxies, we infer an
IMF slope of x = 2.22 +/- 0.14, consistent with Salpeter IMF, and a low-mass
limit M_low = 0.13 +/- 0.03 M_sun, just above the hydrogen burning limit.
