The stellar initial mass function of early type galaxies from low to
  high stellar velocity dispersion: homogeneous analysis of ATLAS$^{\rm 3D}$
  and Sloan Lens ACS galaxies
  We present an investigation about the shape of the initial mass function
(IMF) of early-type galaxies (ETGs), based on a joint lensing and dynamical
analysis, and on stellar population synthesis models, for a sample of 55 lens
ETGs identified by the Sloan Lens ACS (SLACS) Survey. We construct axisymmetric
dynamical models based on the Jeans equations which allow for orbital
anisotropy and include a dark matter halo. The models reproduce in detail the
observed \textit{HST} photometry and are constrained by the total projected
mass within the Einstein radius and the stellar velocity dispersion ($\sigma$)
within the SDSS fibers. Comparing the dynamically-derived stellar mass-to-light
ratios $(M_*/L)_{\rm dyn}$, obtained for an assumed halo slope $\rho_{\rm
h}\propto r^{-1}$, to the stellar population ones $(M_*/L)_{\rm pop}$, derived
from full-spectrum fitting and assuming a Salpeter IMF, we infer the mass
normalization of the IMF. Our results confirm the previous analysis by the
SLACS team that the mass normalization of the IMF of high $\sigma$ galaxies is
consistent on average with a Salpeter slope. Our study allows for a fully
consistent study of the trend between IMF and $\sigma$ for both the SLACS and
\ATLAS samples, which explore quite different $\sigma$ ranges. The two samples
are highly complementary, the first being essentially $\sigma$ selected, and
the latter volume-limited and nearly mass selected. We find that the two
samples merge smoothly into a single trend of the form $\log\alpha
=(0.38\pm0.04)\times\log(\sigma_{\rm
e}/200\,\mathrm{km~s}^{-1})+(-0.06\pm0.01)$, where $\alpha=(M_*/L)_{\rm
dyn}/(M_*/L)_{\rm pop}$ and $\sigma_{\rm e}$ is the luminosity averaged
$\sigma$ within one effective radius $R_{\rm e}$. This is consistent with a
systematic variation of the IMF normalization from Kroupa to Salpeter in the
interval $\sigma_{\rm e}\approx90-270\,\mathrm{km~s}^{-1}$.
