We present J-band spectroscopy of passive galaxies focusing on the Na i doublet at 1.14 \mu m. Like the Na i 0.82 \mu m doublet , this feature is strong in low-mass stars and hence may provide a useful probe of the initial mass function ( IMF ) .
From high signal-to-noise composite spectra , we find that Na i 1.14 \mu m increases steeply with increasing velocity dispersion , \sigma , and for the most massive galaxies ( \sigma \ga 300 km s ^ { -1 } ) is much stronger than predicted from synthetic spectra with Milky-Way-like IMFs and solar abundances .
Reproducing Na i 1.14 \mu m at high- \sigma likely requires either a very high [ Na/H ] , or a bottom-heavy IMF , or a combination of both .
Using the Na D line to break the degeneracy between IMF and abundance , we infer [ Na/H ] \approx +0.5 and a steep IMF ( single-slope-equivalent x \approx 3.2 , where x = 2.35 for Salpeter ) , for the high- \sigma galaxies .
At lower mass ( \sigma = 50–100 km s ^ { -1 } ) , the line strengths are compatible with MW-like IMFs and near-solar [ Na/H ] .
We highlight two galaxies in our sample where strong gravitational lensing masses favour MW-like IMFs .
Like the high- \sigma sample on average , these galaxies have strong Na i 1.14 \mu m ; taken in isolation their sodium indices imply bottom-heavy IMFs which are hard to reconcile with the lensing masses .
An alternative full-spectrum-fitting approach , applied to the high- \sigma sample , recovers an IMF less heavy than Salpeter , but under-predicts the Na i 1.14 \mu m line at the 5 \sigma level .
We conclude that current models struggle to reproduce this feature in the most massive galaxies without breaking other constraints , and caution against over-reliance on the sodium lines in spectroscopic IMF studies .