AN INVENTORY OF THE STELLAR INITIAL MASS FUNCTION IN EARLY-TYPE GALAXIES

The stellar initial mass function (IMF) is fundamentally important to understanding both stellar populations and galaxies. The Milky Way (MW) IMF was originally characterized as a power-law mass distribution, dN/dM∝M^−α, with α ∼ 2.35 (Salpeter 1955), and subsequently refined to flatten at lower masses (M ≲ 0.5 M_☉; Kroupa 2001; Chabrier 2003).

Whether or not the MW IMF describes stellar populations elsewhere in the universe cannot yet be said through direct star counts. There have been some indirect observational hints of IMF variations and ample theoretical motivation for these, but no broadly convincing evidence has emerged (cf. Bastian et al. 2010).

This situation has recently changed, with a flurry of studies of early-type galaxies (ETGs) turning up indirect evidence for systematic IMF variations. These studies use models of stellar population synthesis (SPS) to fit integrated-light data (broadband colors and spectroscopic features) and fall into two broad categories: "pure" SPS and "hybrid" SPS+gravitating mass analyses.

The pure analyses rely on spectral lines that are differentially sensitive to giant or dwarf stars. These include the TiO feature at 6130 Å, the Na i doublet near 8190 Å, the Ca ii triplet near 8600 Å, the Wing–Ford [Fe/H] band at 9915 Å, and the Ca i line at 10345 Å (e.g., Cenarro et al. 2003; van Dokkum & Conroy 2010; Spiniello et al. 2012; Conroy & van Dokkum 2012, hereafter CvD12; Ferreras et al. 2013; Smith et al. 2012).

The hybrid analyses assume an IMF and infer a stellar mass-to-light ratio ϒ_⋆ using a more conventional SPS approach based on colors and age- and metallicity-sensitive spectral lines. Estimates of ϒ_⋆ are also derived using dynamics or gravitational lensing. Comparison of the independent results then yields a revised IMF (e.g., Cappellari et al. 2006, 2012a, 2012b; Ferreras et al. 2008; Tortora et al. 2009, 2010, 2012; Grillo & Gobat 2010; Grillo 2010; Treu et al. 2010; Napolitano et al. 2010, 2011; Auger et al. 2010; Thomas et al. 2011; Spiniello et al. 2011; Dutton et al. 2012a, 2012b; Sonnenfeld et al. 2012; Wegner et al. 2012).

One may characterize a revised IMF through its ϒ_⋆ relative to an MW disk IMF, δ_IMF ≡ ϒ_⋆/ϒ_⋆,MW (the "mismatch parameter"), where for reference we adopt the Chabrier IMF. Remarkably, almost all the above studies found "heavy" IMFs (δ_IMF ≫ 1) for the most massive ETGs. Less massive ETGs and spiral galaxies appear to have "normal/light" IMFs (δ_IMF ≲ 1; e.g., Bell & de Jong 2001; Bershady et al. 2011; Suyu et al. 2012; Brewer et al. 2012), and the bulge components of spirals may also have a similar mass dependence to ETGs (de Blok et al. 2008; Ferreras et al. 2010, hereafter F+10; Dutton et al. 2013, hereafter D+13).

These IMF findings are both potentially revolutionary and highly controversial and demand further investigation. In particular, with hybrid analyses there are lingering questions about degeneracies associated with the distribution of non-baryonic dark matter (DM). The time is also ripe to inventory the results to date and see if the apparent emerging consensus holds up under quantitative, systematic comparison—which could provide pressing motivation for understanding the physical origins of the trends. Comparisons were made for some hybrid analyses (Thomas et al. 2011; Dutton et al. 2012a; Wegner et al. 2012), but not for the pure SPS work.

In this paper we carry out such an inventory, while presenting our own novel results for a large sample of ETGs for reference, following the dynamical+SPS analyses of Tortora et al. (2012, hereafter T+12). We focus on the trends in δ_IMF with central stellar velocity dispersion, σ_⋆, and discuss some implications for the central DM fraction. σ_⋆ is widely considered as crucially connected to galaxy evolution and, unlike ϒ_⋆, is relatively independent of the bandpass and of δ_IMF.

We describe our data and analysis methods in Section 2. We present our results in Section 3 and make literature comparisons in Section 4. We summarize the conclusions and outlook in Section 5.

We apply a combination of SPS and stellar dynamical models to a sample of ∼4500 giant ETGs, in the redshift range of z = 0.05–0.1, drawn from the SPIDER project (La Barbera et al. 2010b). Our data include optical+near-infrared photometry [from the Sloan Digital Sky Survey (SDSS) and the UKIRT Infrared Deep Sky Survey-Large Area Survey],⁵ high-quality measurements of galactic structural parameters (effective radius R_e and Sérsic index n), and SDSS central-aperture velocity dispersions σ_Ap. The sample galaxies are defined as bulge-dominated systems with passive spectra, while late-type systems are efficiently removed through the SDSS classification parameters based on the spectral type and the fraction of light that is better described by a de Vaucouleurs (1948) profile (see T+12 for further details). The structural parameters are measured using 2DPHOT (La Barbera et al. 2008) and are found to be significantly different from the SDSS estimates (La Barbera et al. 2010b). The sample is 95% complete at a stellar mass of M_⋆ = 3 × 10¹⁰  M_☉, which corresponds to σ_Ap ∼ 160 km s^-1.

The SPS-based ϒ_⋆ values were derived by fitting Bruzual & Charlot (2003) models to the multi-band photometry, assuming a Chabrier IMF. These results have been shown to be consistent with independent literature (e.g., MPA masses in Dutton et al. 2012a), while possible systematics in stellar mass (M_⋆) estimates are discussed in Swindle et al. (2011) and T+12. Despite the well-known age–metallicity degeneracies in photometric data, these conspire to keep the stellar mass-to-light ratio well constrained, with scatter of 0.05–0.15 dex. The agreement is also excellent when our color-derived masses are compared with spectroscopic estimates. The largest systematical uncertainty in our analysis comes from our ignorance about the IMF shape, which can produce variations of the stellar M/L of a factor as large as ∼2. For this reason the IMF is a key issue in stellar population analysis and the central topic of this paper.

Our dynamical-mass estimates use spherical isotropic Jeans equations fitted to the σ_Ap data. We also extend the T+12 analysis using two-component mass models: a Sérsic-based stellar distribution following the K-band light, and a standard DM profile. For the latter we adopt a series of plausible assumptions (cf. Cappellari et al. 2012a), as the data do not allow us to constrain both components simultaneously.

Our DM models are based on the Navarro et al. (1996, hereafter NFW) profile, with an adjustable degree of baryon-induced adiabatic contraction (AC). For the virial mass and concentration (M_vir, c_vir), we adopt mean trends for a WMAP5 cosmology (Macciò et al. 2008), and for the M_vir–M_⋆ relation we used Moster et al. (2010, hereafter M+10). Each galaxy's mass model then has one free parameter, ϒ_⋆, plus optional AC (Gnedin et al. 2004, hereafter G+04), providing our no-AC-NFW and AC-NFW base models.

We explore the sensitivity of our results to these assumptions by doing the analyses with the following alternatives: (1) AC recipes of varying strengths (Blumenthal et al. 1986, hereafter B+86; Abadi et al. 2010, hereafter A+10); (2) M_vir with a fixed value for the entire sample: M_vir = 10¹² M_⊙, 10¹³ M_☉, or 10¹⁴ M_☉; (3) WMAP3-based c_vir–M_vir relation (Macciò et al. 2008); (4) no DM is present; (5) mild kinematic anisotropy, with β = −0.2 or +0.2; (6) c_vir–M_vir relation altered to mimic a warm dark matter (WDM) cosmology, assuming different particle masses (Schneider et al. 2012, hereafter S+12a).

To study the mean trends of ϒ_⋆ with velocity dispersion, we construct "average" galaxies by dividing our sample into 10 σ_Ap bins, for which we compute median values of M_⋆, R_e  n, R_Ap, and σ_Ap. We show these values and their 25–75 percentile scatter in Figure 1. For each σ_Ap-bin and a given DM model, we solve the radial Jeans equation for the ϒ_⋆ value that matches the observed σ_Ap.

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Our final analysis products, for each galaxy bin and mass model, will be the SPS-determined ϒ_{⋆, MW}, the dynamically determined ϒ_⋆, the inferred δ_IMF, and the inferred central DM fraction, f_DM ≡ 1 − ϒ_⋆/ϒ_dyn. For homogeneity, we convert σ_Ap to σ_e (the value at R_e), using the best-fitting relation in Cappellari et al. (2006; also done for the literature results later).

Our main IMF results are shown in Figure 2. The two thick black curves correspond to our standard no-AC-NFW (solid) and AC-NFW (long-dashed) models, and the suite of alternative models are also plotted, as labeled.

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It is clear that the overall normalization of δ_IMF is degenerate with the adopted halo model, as DM can be traded against stellar mass. This degeneracy is most severe when allowing for uncertainties in the halo response to baryons (more so than with the virial mass assumptions). However, for a given flavor of the halo model, there is always a strong correlation between δ_IMF and σ_e (the δ_IMF–M_⋆ trend is weaker). We quantify this correlation for each model with a log–log fit, reporting the best-fit parameters in Table 1 (also plotted in Figure 2 for the two reference models⁶); the typical relation is $\delta _{\rm IMF} {\;\raise 0.1ex\hbox{$\propto $}\kern -0.78em \lower 1.2ex \hbox{$\sim $}\;} \sigma _{\rm e}$.

Table 1. Best-fit Parameters for the Relation $\log \delta _{\rm IMF} = a + b \log \frac{{\sigma _{\rm e}}}{200 \, {\rm km}\,{\rm s^{-1}}}$, for Model Suite (See the Main Text)

Model	a	b
Mvir–M⋆ (M+10)	0.18	0.86
Mvir–M⋆ (M+10) + AC (G+04)	0.11	1.07
Mvir–M⋆ (M+10) + AC (A+10)	0.16	0.93
Mvir–M⋆ (M+10) + AC (B+86)	0.04	1.29
${M_{\rm vir}}= 10^{12} {\,M_\odot }$	0.19	0.88
${M_{\rm vir}}= 10^{13} {\,M_\odot }$	0.18	0.97
${M_{\rm vir}}= 10^{14} {\,M_\odot }$	0.16	1.09
Mvir–M⋆ (M+10), WMAP3	0.18	0.85
No DM	0.21	0.79
β = +0.2	0.17	0.87
β = −0.2	0.19	0.88
CDM (S+12a)	0.17	0.90
mWDM = 1 keV (S+12a)	0.18	0.88
mWDM = 0.5 keV (S+12a)	0.18	0.84
mWDM = 0.25 keV (S+12a)	0.19	0.80

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Other models of potential interest are low-density DM cores (Burkert 1995) and alternatives to DM (e.g., Milgrom 1983). These would imply similar δ_IMF–σ_e slopes to our limiting no-DM model. The no-DM assumption produces the uppermost curve in Figure 2 and is qualitatively consistent with a slightly expanded NFW model. The same results, using the same data set but somewhat different mass modeling assumptions, have been anticipated in Tortora et al. (2012).

One could alter these slopes with additional model tuning; e.g., with strong AC at high σ_e  and halo expansion at low σ_e, one could completely flatten out the trend. On the other hand, decreasing the AC at high σ_e and increasing the strength of the AC toward lower σ_e  would make the trend even steeper. We currently have no a priori motivation for either direction for the AC variations, and thus we cannot argue in favor of an IMF universality for the former case or for a very strong IMF non-universality in the latter one.

We next examine the overall IMF normalization, with the no-AC-NFW and AC-NFW cases bracketing the most plausible range of models. For reference, we show δ_IMF predictions for several standard IMFs (Salpeter, Kroupa, Chabrier). We note that at a fixed IMF, age, and metallicity, the ϒ_⋆ and δ_IMF values are uniquely predicted, but the reverse is not true. A given δ_IMF result can imply multiple IMF solutions, particularly if one allows for mass functions more complicated than a pure power law.

If we adopt a no-AC-NFW model, an MW-like IMF is implied for low-σ_e galaxies and a Salpeter IMF for high-σ_e ones. For AC-NFW, the low-σ_e galaxies have sub-MW IMFs, and the high-σ_e ones have IMFs intermediate to Kroupa and Salpeter.

In all cases, extremely bottom-heavy IMFs (assumed single power law, α ≳ 2.6) are ruled out, on average. Even if one assumed no DM, such IMFs would violate the dynamical constraints on the overall mass-to-light ratio.

Although resolving the remaining IMF–DM degeneracy will require more extensive analysis, we carry out a simple exercise to provide initial clues, inspired by Dutton et al. (2012b). As in that paper, we select the galaxies with mean central stellar surface densities Σ_⋆ > 2500 M_☉ pc⁻² and analyze them the same as the full sample. The rationale is that such galaxies are the most star dominated and the least sensitive to DM uncertainties. The results are shown by the gray curves in Figure 2, where, as expected, the model curves are closer together. The implied δ_IMF normalization is fairly low—similar to the full-sample AC results. We may then apply this IMF result to the full sample if we assume no additional systematic δ_IMF–Σ_⋆ correlation (cf. Schulz et al. 2010).

Although this paper is primarily concerned with the IMF, we briefly examine some implications for the central DM content. Our main mass models are, by construction, fully consistent with current expectations for ΛCDM halo profiles, while also agreeing with the observations for a plausible IMF range (δ_IMF ∼ 0.5–2.0).

We show the implied f_DM within 1 R_e in Figure 3. We find fairly universal values of f_DM ∼ 0.2 and ∼0.5 for the no-AC-NFW and AC-NFW models, respectively. These results are not altered appreciably in the alternative models explored above. Note also that the high-Σ_⋆ test above prefers the AC-NFW model.

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The figure also shows that if we adopted a constant IMF, then we would infer a strong increase of f_DM with σ_e. Such behavior has been invoked as the driver for the "tilt" of the ETG fundamental plane (see Tortora et al. 2009, and references therein), but now with renewed comparison to realistic DM halo models, we find that the tilt is driven at least in part (and perhaps wholly) by the IMF. Put differently, the observed ϒ_dyn–σ_e relation is too steep to explain through standard DM models and requires an additional factor.

We now compare our SPIDER-based results with an inventory of other literature results for ETGs in Figure 4 and late-type galaxies in Figure 5.

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4.1. Early-type Galaxies

4.2. Late-type Galaxies

We have analyzed the dynamics and stellar populations of a large sample of ETGs from the SPIDER project and found compelling evidence for heavier IMFs in the central regions of higher-σ_⋆ galaxies. The IMF mismatch relative to Chabrier is δ_IMF ∼ 0.5–1.1 at σ_⋆ ∼ 125 km s⁻¹ and ∼1.2–1.8 at ∼250 km s⁻¹. The δ_IMF–σ_⋆ trend is robust to a wide set of modeling assumptions and accounts for much of the tilt in the fundamental plane. The distribution of DM is degenerate with the overall IMF normalization and difficult to constrain, and therefore we have assumed that any halo contraction is invariant with σ_⋆. Some ways to break the degeneracy between IMF and halo contraction are to (1) analyze cases with extended velocity dispersion or X-ray emission profiles (e.g., Napolitano et al. 2009, 2011) or (2) incorporate complementary data from strong/weak gravitational lensing (e.g., Auger et al. 2010). However, we have argued that the only way to preserve the IMF universality is to allow for halo contraction at high σ_⋆  and halo expansion at low σ_⋆.

We have performed the first general inventory of IMF results from a variety of studies in the literature, using both pure and hybrid techniques, and found that these generally agree well with our SPIDER results. There is remarkably widespread agreement on a Salpeter-like IMF for massive ETGs (σ_⋆ ≳ 200 km s⁻¹). At lower σ_⋆, the data are still somewhat limited and the different results have not yet converged, but most of the studies point to MW-like IMFs. Agreement on a super-Salpeter IMF in the most massive galaxies (σ_⋆ ≳ 275 km s⁻¹) is also not yet universal.

These results appear to be fully compatible with underlying ΛCDM halos. However, more detailed conclusions about halo contraction or expansion are still elusive, and the data on galaxy centers do not clearly rule out alternative DM models once the variable IMF is accounted for.

More work is clearly needed to understand the systematics in the different analyses; to build up better statistics on a wide range of galaxy types, environments, and redshifts; and to determine which parameters correlate best with IMF variations (e.g., metallicity or starburst intensity; CvD12; Smith et al. 2012). It may also be particularly helpful to venture beyond the centers of galaxies, using data from a wide baseline in radius to help break the IMF–DM degeneracies.

It appears we are nearing convergence on determining what the basic components of galaxies are (distributions of stars and DM). The next challenge will be to understand why these arrive at their distributions. What drives the power spectrum in cloud fragmentation and star formation? How do baryonic processes interact with and re-shape their surrounding DM halos?

We thank the referees for their constructive comments. We thank Matt Auger, Michele Cappellari, Reinaldo de Carvalho, Aaron Dutton, Francesco La Barbera, and Tommaso Treu for helpful discussions. We also thank Charlie Conroy for helpful discussions and for providing us with his updated results in tabular form. C.T. was supported by the Swiss National Science Foundation and the Forschungskredit at the University of Zurich. A.J.R. was supported by the National Science Foundation Grant AST-0909237.