SPIDER VIII - constraints on the stellar initial mass function of early-type galaxies from a variety of spectral features

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 Sloan Digital Sky Survey-based Spheroids Panchromatic Investigation in Different Environmental Regions 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) similar to 100 km s(-1) towards a more bottom-heavy IMF with increasing Sigma(0), becoming steeper than the Salpeter function at Sigma(0) greater than or similar to 220 km s(-1). We analyse 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 analysing 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 cannot discriminate between a single power-law (unimodal) IMF and a low-mass (less than or similar to 0.5 M-circle dot) 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 similar to 20 per cent at Sigma(0) similar to 100 km s(-1), up to similar to 80 per cent at Sigma(0) similar to 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).