[Paper Review] Reconciling M/L Ratios Across Cosmic Time: a Concordance IMF for Massive Galaxies

Pieter van Dokkum, Charlie Conroy|arXiv (Cornell University)|Jul 8, 2024

Astronomy and Astrophysical Research4 citations

TL;DR

This paper proposes a 'concordance' initial mass function (IMF) for massive galaxies that is simultaneously bottom-heavy at low masses (γ₁ ≈ 2.4) and top-heavy at high masses (γ₃ ≈ 1.85), reconciling low M/L ratios at high redshift (z ≈ 2–5) with high M/L ratios in local massive ellipticals. The IMF evolves from α ≈ 10^0.3 (1.2× more low-mass stars) locally to α ≈ 10^{-0.1} (near-Milky-Way) in high-redshift progenitors, with slopes tied to velocity dispersion via σ-based scaling laws.

ABSTRACT

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 27, 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 $γ_1=2.40\pm0.09$, $γ_2=2.00\pm0.14$, and $γ_3=1.85\pm0.11$ in the regimes 0.08-0.5 Msun, 0.5-1 Msun, and >1 Msun respectively. The IMF parameter $α$, the mass excess compared to a Milky Way IMF, ranges from $\log(α)\approx+0.3$ for present-day galaxies to $\log(α)\approx-0.1$ 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 $σ\gtrsim 160$ km/s: $γ_1\approx1.3+4.3\logσ_{160}$; $γ_2\approx2.3-1.2\logσ_{160}$; and $γ_3\approx2.3-1.7\logσ_{160}$, with $σ_{160}=σ/160$ km/s. If galaxies grow primarily inside-out, so that velocity dispersions are relatively stable, these relations should also hold at high redshift.

Motivation & Objective

• To reconcile the apparent contradiction between low M/L ratios in high-redshift quiescent galaxies and high M/L ratios in local massive ellipticals.
• To develop an IMF that is simultaneously bottom-heavy at low masses and top-heavy at high masses, consistent with both local and high-redshift observations.
• To derive a physically motivated, velocity dispersion-dependent IMF that generalizes the concordance IMF beyond local galaxies.
• To provide a framework for testing IMF variations using upcoming JWST observations of low-mass stars and molecular features.

Proposed method

• Uses gravity-sensitive absorption lines in integrated spectra to infer low-mass IMF slopes in local massive galaxies.
• Applies dynamical mass measurements (M_dyn/M_*) from kinematics to constrain IMF at high redshift (z ≈ 2–5).
• Derives a three-segment IMF with slopes γ₁ = 2.40±0.09 (0.1–0.5 M⊙), γ₂ = 2.00±0.14 (0.5–1 M⊙), γ₃ = 1.85±0.11 (>1 M⊙).
• Establishes a scaling relation between IMF parameters and velocity dispersion σ, using σ₁₆₀ = σ/160 km s⁻¹.
• Proposes a generalized IMF: γ₁ ≈ 1.3 + 4.3 log σ₁₆₀, γ₂ ≈ 2.3 – 1.2 log σ₁₆₀, γ₃ ≈ 2.3 – 1.7 log σ₁₆₀ for σ ≳ 160 km s⁻¹.
• Proposes JWST programs (GO-5629, GO-4757) to test the low-mass IMF via near-IR H₂O features and resolved stellar populations.

Experimental results

Research questions

• RQ1 Can an IMF that is both bottom-heavy and top-heavy simultaneously reconcile M/L ratios across cosmic time in massive galaxies?
• RQ2 What is the velocity dispersion dependence of the IMF in massive galaxies, and how does it evolve with redshift?
• RQ3 How do the observed M/L ratios of high-redshift quiescent galaxies (z ≈ 2–5) compare to predictions from a standard IMF versus a concordance IMF?
• RQ4 Can independent JWST observations of low-mass stars and molecular features confirm the proposed IMF structure?
• RQ5 What physical mechanisms could produce a hybrid IMF with steep low-mass and shallow high-mass slopes?

Key findings

• The concordance IMF has slopes γ₁ = 2.40 ± 0.09, γ₂ = 2.00 ± 0.14, and γ₃ = 1.85 ± 0.11 in the mass ranges 0.1–0.5 M⊙, 0.5–1 M⊙, and >1 M⊙, respectively.
• The IMF parameter α, representing mass excess relative to the Milky Way IMF, is log(α) ≈ +0.3 for present-day massive galaxies and log(α) ≈ -0.1 for their high-redshift progenitors.
• The IMF slopes scale with velocity dispersion via γ₁ ≈ 1.3 + 4.3 log σ₁₆₀, γ₂ ≈ 2.3 – 1.2 log σ₁₆₀, and γ₃ ≈ 2.3 – 1.7 log σ₁₆₀ for σ ≳ 160 km s⁻¹.
• The concordance IMF is consistent with dynamical M/L ratios ≈1 at z ≈ 3–5, implying a top-heavy high-mass slope, while matching high M/L in local galaxies via a bottom-heavy low-mass slope.
• The model predicts that the mass excess in low-mass stars (and remnants) is not the dominant contributor to high M/L in local galaxies, challenging assumptions in previous IMF studies.
• JWST programs (GO-5629, GO-4757) are proposed to test the low-mass IMF directly via near-IR H₂O features and resolved stellar populations at z ≈ 0.7.