We measure equivalent widths ( EW ) – focussing on two unique features ( NaI and TiO2 ) of low-mass stars ( \lesssim 0.3 M _ { \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 _ { * } ) 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 ( \mathrm { [ 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 \mathrm { km s ^ { -1 } } .