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 } , obtained for an assumed halo slope \rho _ { h } \propto r ^ { -1 } , to the stellar population ones ( M _ { * } / L ) _ { Salp } , 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 ^ { \mathrm { 3 D } } 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 \pm 0.04 ) \times \log ( \sigma _ { e } / 200 \mathrm { km \leavevmode% \nobreak s } ^ { -1 } ) + ( -0.06 \pm 0.01 ) , where \alpha = ( M _ { * } / L ) _ { dyn } / ( M _ { * } / L ) _ { Salp } 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 \mathrm { km \leavevmode \nobreak s } ^ { -1 } .