Evidence For A Mild Steepening And Bottom-Heavy IMF In Massive Galaxies From Sodium And Titanium-Oxide Indicators

We measure equivalent widths (EW) - focussing on two unique features (NaI and TiO2) of low-mass stars (<0.3M\odot) - for luminous red galaxy spectra from the 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 (\sim {\sigma}\ast) early-type galaxies (ETGs) but deviate increasingly for more massive ETGs, due do 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 (SDSSJ0912+0029) and an SDSS ETG (SDSSJ0041-0914) 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 \leq 3.0 at the 90% C.L. We conclude that NaI and TiO features, in comparison with state-of-the-art SSP models, suggest a mildly steepening IMF from Salpeter (dn/dm \propto m-x with x = 2.35) to x \approx 3.0 for ETGs in the range {\sigma} = 200 - 335 km s-1.