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

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 2 &lt; z &lt; 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 &gt; 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 &gt;1 M ⊙ , respectively. The IMF parameter α , the mass excess compared to a Milky Way IMF, ranges from <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" overflow="scroll"> <mml:mi>log</mml:mi> <mml:mo stretchy="false">(</mml:mo> <mml:mi>α</mml:mi> <mml:mo stretchy="false">)</mml:mo> <mml:mo>≈</mml:mo> <mml:mo>+</mml:mo> <mml:mn>0.3</mml:mn> </mml:math> for present-day galaxies to <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" overflow="scroll"> <mml:mi>log</mml:mi> <mml:mo stretchy="false">(</mml:mo> <mml:mi>α</mml:mi> <mml:mo stretchy="false">)</mml:mo> <mml:mo>≈</mml:mo> <mml:mo>−</mml:mo> <mml:mn>0.1</mml:mn> </mml:math> 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 : <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" overflow="scroll"> <mml:msub> <mml:mi>γ</mml:mi> <mml:mn>1</mml:mn> </mml:msub> <mml:mo>≈</mml:mo> <mml:mn>1.3</mml:mn> <mml:mo>+</mml:mo> <mml:mn>4.3</mml:mn> <mml:mi>log</mml:mi> <mml:msub> <mml:mi>σ</mml:mi> <mml:mn>160</mml:mn> </mml:msub> </mml:math> , <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" overflow="scroll"> <mml:msub> <mml:mi>γ</mml:mi> <mml:mn>2</mml:mn> </mml:msub> <mml:mo>≈</mml:mo> <mml:mn>2.3</mml:mn> <mml:mo>−</mml:mo> <mml:mn>1.2</mml:mn> <mml:mi>log</mml:mi> <mml:msub> <mml:mi>σ</mml:mi> <mml:mn>160</mml:mn> </mml:msub> </mml:math> , and <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" overflow="scroll"> <mml:msub> <mml:mrow> <mml:mi>γ</mml:mi> </mml:mrow> <mml:mrow> <mml:mn>3</mml:mn> </mml:mrow> </mml:msub> <mml:mo>≈</mml:mo> <mml:mn>2.3</mml:mn> <mml:mo>−</mml:mo> <mml:mn>1.7</mml:mn> <mml:mi>log</mml:mi> <mml:msub> <mml:mrow> <mml:mi>σ</mml:mi> </mml:mrow> <mml:mrow> <mml:mn>160</mml:mn> </mml:mrow> </mml:msub> </mml:math> , 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.