### 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

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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 $(M/L)_{\rm dyn}$ 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$^{\rm 3D}$ 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 $\log\alpha =(0.38\pm0.04)\times\log(\sigma_e/200\, km s^{-1})+(-0.06\pm0.01)$, where $\alpha=(M/L)_{\rm dyn}/(M/L)_{\rm pop}$ 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 $\sigma_e\approx 90-270\,km s^{-1}$.