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 are better explained with 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 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 γ₁=2.40±0.09, γ₂=2.00±0.14, and γ₃=1.85±0.11 in the regimes 0.1-0.5 Msun, 0.5-1 Msun, and >1 Msun 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 ~300 km/s, 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: γ₁≈1.3+4.3logσ₁₆₀; γ₂≈2.3-1.2logσ₁₆₀; and γ₃≈2.3-1.7logσ₁₆₀, with σ₁₆₀=σ/160 km/s. If galaxies grow primarily inside-out, so that velocity dispersions are relatively stable, these relations should also hold at high redshift.