We perform a spectroscopic study to constrain the stellar Initial Mass Function ( IMF ) by using a large sample of 24 , 781 early-type galaxies from the SDSS-based SPIDER survey .
Clear evidence is found of a trend between IMF and central velocity dispersion ( \sigma _ { 0 } ) , evolving from a standard Kroupa/Chabrier IMF at \sigma _ { 0 } \sim 100 km s ^ { -1 } towards a more bottom-heavy IMF with increasing \sigma _ { 0 } , becoming steeper than the Salpeter function at \sigma _ { 0 } \lower 2.15 pt \hbox { $ \buildrel > \over { \sim } $ } 220 km s ^ { -1 } .
We analyze a variety of spectral indices , combining gravity-sensitive features , with age- and metallicity-sensitive indices , and we also consider the effect of non solar abundance variations .
The indices , corrected to solar scale by means of semi-empirical correlations , are fitted simultaneously with the ( nearly solar-scaled ) extended MILES ( MIUSCAT ) stellar population models .
Similar conclusions are reached when analyzing the spectra with a hybrid approach , combining constraints from direct spectral fitting in the optical with those from IMF-sensitive indices .
Our analysis suggests that \sigma _ { 0 } , rather than [ \alpha / { Fe } ] , drives the variation of the IMF .
Although our analysis can not discriminate between a single power law ( unimodal ) IMF and a low-mass ( \lower 2.15 pt \hbox { $ \buildrel < \over { \sim } $ } 0.5 M _ { \odot } ) tapered ( bimodal ) IMF , robust constraints can be inferred for the fraction in low-mass stars at birth .
This fraction ( by mass ) is found to increase from \sim 20 \% at \sigma _ { 0 } \sim 100 km s ^ { -1 } , up to \sim 80 \% at \sigma _ { 0 } \sim 300 km s ^ { -1 } .
However , additional constraints can be provided with stellar mass-to-light ( M / L ) ratios : unimodal models predict M / L significantly larger than dynamical M / L , across the whole \sigma _ { 0 } range , whereas a bimodal IMF is compatible .
Our results are robust against individual abundance variations .
No significant variation is found in Na and Ca in addition to the expected change from the correlation between [ \alpha / { Fe } ] and \sigma _ { 0 } .