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 \simeq 245 km s ^ { -1 } and \sigma \simeq 325 km s ^ { -1 } , 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 \geq 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 _ { \mathrm { low } } ) .
Combining the joint L & D and SSP analyses of both galaxies , we infer an IMF slope of x = 2.22 \pm 0.14 , consistent with Salpeter IMF , and a low-mass limit M _ { \mathrm { low } } = 0.13 \pm 0.03 { M _ { \odot } } , just above the hydrogen burning limit .