Reconciling M/L Ratios Across Cosmic Time: a Concordance IMF for Massive Galaxies

The stellar initial mass function (IMF) is thought to be bottom heavy in the cores of the most massive galaxies, with an excess of low-mass stars compared to the Milky Way. However, studies of the kinematics of quiescent galaxies at 2 < z < 5 find M/L ratios that indicate lighter IMFs. Light IMFs have also been proposed for the unexpected populations of luminous galaxies that JWST has uncovered at z > 7 to reduce tensions with galaxy formation models. Here we explore “ski slope” IMFs that are simultaneously bottom heavy, with a steep slope at low stellar masses, and top heavy, with a shallow slope at high masses. We derive a form of the IMF for massive galaxies that is consistent with measurements in the local universe and yet produces relatively low M/L ratios at high redshift. This concordance IMF has slopes γ 1 = 2.40 ± 0.09, γ 2 = 2.00 ± 0.14, and γ 3 = 1.85 ± 0.11 in the regimes 0.08 M ⊙ – 0.5 M ⊙, 0.5 M⊙ – 1 M ⊙, and >1 M ⊙, respectively. The IMF parameter α, the mass excess compared to a Milky Way IMF, ranges from log(α)≈+0.3 for present-day galaxies to log(α)≈−0.1 for their star-forming progenitors. The concordance IMF applies only to the central regions of the most massive galaxies, with velocity dispersions σ ∼ 300 km s−1, and their progenitors. However, it can be generalized using a previously measured relation between α and σ. We arrive at the following modification to the Kroupa (2001) IMF for galaxies with σ ≳ 160 km s−1: γ1≈1.3+4.3logσ160 , γ2≈2.3−1.2logσ160 , and γ3≈2.3−1.7logσ160 , with σ 160 = σ/160 km s−1. If galaxies grow primarily inside out, so that velocity dispersions are relatively stable, these relations should also hold at high redshift.