EVIDENCE FOR A MILD STEEPENING AND BOTTOM-HEAVY INITIAL MASS FUNCTION IN MASSIVE GALAXIES FROM SODIUM AND TITANIUM-OXIDE INDICATORS

We measure equivalent widths (EWs)-focusing on two unique features (NaI and TiO2) of low-mass stars (less than or similar to 0.3 M-circle dot)-for luminous red galaxy spectra from the Sloan Digital Sky Survey (SDSS) and X-Shooter Lens Survey (XLENS) in order to study the low-mass end of the initial mass function (IMF). We compare these EWs to those derived from simple stellar population models computed with different IMFs, ages, [alpha/Fe], and elemental abundances. We find that models are able to simultaneously reproduce the observed NaD lambda 5895 and Na I lambda 8190 features for lower-mass (similar to sigma(*)) early-type galaxies (ETGs) but deviate increasingly for more massive ETGs, due to strongly mismatching NaD EWs. The TiO2 lambda 6230 and the Na I lambda 8190 features together appear to be a powerful IMF diagnostic, with age and metallicity effects orthogonal to the effect of IMF. We find that both features correlate strongly with galaxy velocity dispersion. The XLENS ETG (SDSS J0912+0029) and an SDSS ETG (SDSS J0041-0914) appear to require both an extreme dwarf-rich IMF and a high sodium enhancement ([Na/Fe] = +0.4). In addition, lensing constraints on the total mass of the XLENS system within its Einstein radius limit a bottom-heavy IMF with a power-law slope to x <= 3.0 at the 90% CL. We conclude that NaI and TiO features, in comparison with state-of-the-art simple stellar population models, suggest a mildly steepening IMF from Salpeter (dn/dm proportional to m(-x) with x = 2.35) to x approximate to 3.0 for ETGs in the range sigma = 200-335 km s(-1).