Evidence for a Mild Steepening and Bottom-Heavy Initial Mass Function 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) - 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, [{α}/Fe], and elemental abundances. We find that models are able to simultaneously reproduce the observed NaD {λ}5895 and Na I {λ}8190 features for lower-mass (~ {σ}) early-type galaxies (ETGs) but deviate increasingly for more massive ETGs, due do strongly mismatching NaD EWs. The TiO2 {λ}6230 and the Na I {λ}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 ≤ 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 ∝ m-x with x = 2.35) to x ≈ 3.0 for ETGs in the range {σ} = 200 - 335 km s-1.