Abstract
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 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
$(M/L)_dyn$ to the stellar population ones $(M/L)_pop$, 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$^3D$ 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 $łog\alpha
=(0.38\pm0.04)\timesłog(\sigma_e/200\, km s^-1)+(-0.06\pm0.01)$, where
$\alpha=(M/L)_dyn/(M/L)_pop$ and $\sigma_e$ is the luminosity
averaged $\sigma$ within one effective radius $R_e$. This is consistent with a
systematic variation of the IMF normalization from Kroupa to Salpeter in the
interval $\sigma_e90-270\,km s^-1$.
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