CONNECTION BETWEEN DYNAMICALLY DERIVED INITIAL MASS FUNCTION NORMALIZATION AND STELLAR POPULATION PARAMETERS

The stellar initial mass function (IMF) of massive galaxies has been the focus of much attention recently, triggered by findings that suggest a non-universal and systematically varying form of the IMF among galaxies in the current day universe (van Dokkum & Conroy 2010; Cappellari et al. 2012). These findings differ from studies of resolved stellar systems in and near the Milky Way, which indicate that the IMF has a universal form (Kroupa 2002; Bastian et al. 2010). Strong evidence for a non-universal IMF has come from the application of various independent techniques, including gravitational lensing (Auger et al. 2010), stellar dynamical modeling (Cappellari et al. 2012, 2013a; Tortora et al. 2013; Conroy et al. 2013), and spectral synthesis (van Dokkum & Conroy 2010, 2011; Spiniello et al. 2012, 2014; La Barbera et al. 2013; Ferreras et al. 2013). With few exceptions (e.g., Smith et al. 2012; Smith & Lucey 2013; Peacock et al. 2014), such studies so far agree that, in general, galaxies with higher velocity dispersion require a "heavy" form of the IMF in order to account fully for the measured total mass-to-light ratio. While this general picture is one of qualitative agreement, there is ongoing debate as to what key parameters drive the IMF variations. In particular, Smith (2014) find a notable discrepancy between methods on a galaxy-by-galaxy basis, and conclude that the dynamically derived IMF normalization for 34 objects in Atlas3D does not correlate with [Mg/Fe] after controlling for velocity dispersion.

In this Letter, we report on how the stellar population properties of the complete ATLAS^3D sample are related to the dynamically derived IMF normalization. This expands the sample of 34 objects of Smith (2014) to a total of 212 galaxies, studying age, metallicity, and [α/Fe], and spanning a significantly larger range of stellar population parameter values.

All observations come from the ATLAS^3D Survey (Cappellari et al. 2011), and comprise optical integral-field spectroscopy covering half the stellar light, on average (i.e., one effective radius, R_e). The spectral data cubes were spatially integrated to form an effective aperture corresponding to a radius of one-eighth of an effective radius, R_e/8. Such an aperture is similar to those used in other (generally long-slit or single-fiber spectrograph) studies in the literature, and allows our results to be more directly compared to them. The single stellar population (SSP) models of Schiavon (2007) were used to measure the SSP-equivalent age, metallicity [Z/H], and alpha-element abundance [α/Fe] (with IMF fixed to that of a unimodal power law of the form: ζ(m)∝m^−2.35, Salpeter 1955) using the chi-squared approach detailed in McDermid et al. (2006), finding the model from an interpolated grid which simultaneously best approximates our measured Hβ, Fe5015 and Mg b line indices on the Lick/IDS system (Worthey & Ottaviani 1997). A full presentation of the stellar populations for the ATLAS^3D Survey is given in McDermid et al. (2014).

Stellar kinematics were measured from our integral-field spectroscopy using the pixel fitting code pPXF (Cappellari & Emsellem 2004), and the resulting maps of stellar line-of-sight mean velocity and velocity dispersion were fitted using general anisotropic Jeans models (Cappellari 2008), in conjunction with multi-Gaussian mass models (Emsellem et al. 1994; Cappellari 2002) from Sloan Digital Sky Survey imaging, which are reported in Scott et al. (2013). The stellar mass-to-light ratios (M/L)_stars we use here were derived accounting for a standard dark matter halo (Navarro et al. 1996), and correspond to models "B" as described in detail in Cappellari et al. (2013b).

The same integral-field spectroscopy was used to measure the spectroscopic stellar mass-to-light ratio (assuming a Salpeter IMF), (M/L)_Salp. These were measured independently from the SSP parameters (which also assume a Salpeter IMF), using a regularized pPXF spectral fit of the MIUSCAT stellar population models from Vazdekis et al. (2012) as templates. To be consistent with the stellar kinematics used in the dynamical modeling, the spectra within an aperture extending to one effective radius were summed and used in the spectral fit. In all, 264 templates were simultaneously fitted to this aperture spectrum, giving the mass-weighted mean mass-to-light ratio for each galaxy taking into account a smooth, non-parametric star-formation history. The resulting r-band (M/L)_stars and (M/L)_Salp values are given in Table 1 of Cappellari et al. (2013a).

Using the same nomenclature as Cappellari et al. (2013a), we quantify the mass normalization of the IMF using the IMF parameter α_dyn ≡ (M/L)_stars/(M/L)_Salp, such that α_dyn > 1 implies an IMF "heavier" than Salpeter. For the results that follow, we apply the same sample selection as that paper,²³ excluding objects with Hβ > 2.3 Å within an effective radius, on the basis that they exhibit very strong age gradients, thus breaking the constant mass-to-light ratio assumption of the dynamical modeling approach, and making (M/L)_Salp ill-defined. In addition, we exclude the object PGC071531, as the quality of the kinematic data was too poor to allow a secure measure of α_dyn. The following analysis uses the remaining 212 galaxies satisfying these criteria.

3.1. IMF and Line-strengths

We begin by showing the link between the IMF mass normalization (via α_dyn) to simple empirical quantities to make our results independent of stellar population models. Figure 1 presents the distribution of the IMF parameter plotted in the two-dimensional plane of line strength indices Hβ and [MgFe50]'.²⁴ The former index is sensitive to stellar age; the latter is sensitive to metallicity in a way that is not strongly dependent on [α/Fe] (Kuntschner et al. 2010). The values of the IMF parameter in this plane have been adaptively smoothed using the two-dimensional locally weighted regression robust technique (dubbed LOESS) of Cleveland & Devlin (1988), as implemented in Cappellari et al. (2013a).²⁵ The LOESS-smoothed distribution tries to remove observational errors and intrinsic scatter to estimate the mean values of the underlying galaxy population, thus approximating the mean values one would obtain from simple binning of much larger samples.

Zoom In Zoom Out Reset image size

Overplotted on these points is a grid of SSP parameter predictions from Schiavon (2007) for a super-solar [α/Fe] =0.2. There is a trend of larger values of the IMF parameter (corresponding to "heavier" IMFs) toward older, metal-rich objects. The trend, however, is not monotonic, with lower metallicity objects also showing relatively high IMF parameter values on average.

The LOESS technique averages co-spatial data points and, as with any binning/averaging approach, the absolute range of values is reduced compared to the original data via reduction of intrinsic and measurement scatter (e.g., compare the color scale of Figure 1 with the intrinsic IMF values implied in Figure 2). The LOESS-averaged color-map plots are useful to not only uncover the underlying trends, but also quantitatively treat correlations of the actual values. In the following section we analyze the two-dimensional projections of IMF with stellar population parameters individually.

Zoom In Zoom Out Reset image size

3.2. IMF and SSP-parameters

Each panel in Figure 2 presents our two mass-to-light ratios (M/L)_stars and (M/L)_Salp plotted against each other, overplotted with diagonal identity lines to indicate the corresponding IMF normalization, such that the IMF varies perpendicular to these diagonal lines. The thick black line shows the result of a one-dimensional locally weighted regression (LOESS) analysis in order to trace the central ridge-line of the mass-to-light ratio points. The measured SSP parameters are indicated by colored points with the parameter name given in the plot title. Again, the SSP parameter values have been smoothed using the two-dimensional LOESS algorithm used in Section 3.1 in order to show the average trend expected from a larger sample of galaxies. These plots are directly comparable to Figure 11 of Cappellari et al. (2013a) where the color scale was used to show how velocity dispersion varies in this plane.

It can be seen that the general trends agree with what was inferred above, namely that the IMF becomes systematically "heavier," on average, for galaxies that are older, more metal-rich, and additionally, more enhanced in alpha elements. It is clear from these plots, however, that stellar population is not a good predictor of α_dyn. The average trend appears due to the fact that, while older, or alpha-enhanced, galaxies span a larger range of α_dyn values going from Kroupa to heavier than Salpeter, the range of α_dyn appears largely limited to Kroupa-like values for galaxies with young ages and low [α/Fe]. A similar conclusion can be drawn from Figure 11 of Cappellari et al. (2013a), where galaxies with low velocity dispersion are similarly limited to Kroupa-like IMF normalization. Metallicity shows a broadly similar trend, however, the detailed distribution is different from that with age and [α/Fe], with the iso-metallicity bands running more parallel to the IMF normalization. Notably, there are objects with low metallicity that require an IMF heavier than Salpeter, as in Figure 1.

We note that the population parameters vary more tightly with (M/L)_Salp than (M/L)_stars, reflecting the strong dependence of stellar population M/L estimates on stellar population parameters. The dynamical M/L estimate is independent of stellar evolutionary effects, which may explain the broader range of population parameters at fixed (M/L)_stars.

In Figure 3 we present the two-dimensional relations of the IMF parameter and population parameters, where points represent our individual measurements. A robust linear fit of the form y = a + bx is made in each panel using the LTS_LINEFIT routine described in Cappellari et al. (2013b), which combines the Least Trimmed Squares robust technique of Rousseeuw & Driessen (2006) into a least-squares fitting algorithm which allows for intrinsic scatter. Fit parameters are given in the upper-left of each panel. Our relations have around 25% scatter due to observational errors, with 12% intrinsic scatter. The Spearman's rank correlation coefficient (r) and the two-sided significance of its deviation from zero (0 < p < 1, with smaller values implying higher significance) are given in the upper right. The correlation coefficient values, which are independent from the assumption of a linear relationship, confirm the presence of weak but statistically significant relationships of α_dyn with age and [α/Fe] and no evidence of a correlation with total metallicity.

Zoom In Zoom Out Reset image size

The main result of this Letter is that the trends between the dynamically derived IMF normalization parameter α_dyn and stellar population parameters are weak. We find that old or alpha-enhanced galaxies have on average heavier IMF normalization, but a heavy IMF is not found in only those galaxies. Old (and high [α/Fe]) galaxies span a large range of IMF normalization going from Kroupa to Salpeter, while young (and low [α/Fe]) ones seem to almost exclusively have a Kroupa normalization. Metallicity shows a less clear distinction, having a broad range of mass normalizations at all metallicity values (Figure 3, middle panel). Conroy & van Dokkum (2012) also find a weaker relationship between IMF and total metallicity, [Z/H], concluding, as here, that there is no compelling evidence of a correlation.

We have explored several avenues to verify the weak trends we find. For example, we find fully consistent results using the mass-weighted ages and metallicities from the spectral fits used to derive our (M/L)_Salp instead of the SSP values, showing that the different models (Schiavon SSP models versus Vazdekis model spectra) and methods (chi-squared fitting for the best SSP versus pPXF spectral fitting of star formation histories for the mass-weighted values) yield consistent results. Using the spectrum integrated within a full effective radius R_e to derive the SSP parameters (instead of R_e/8) gives trends equivalent to Figure 3 within the 1σ uncertainties, with comparable scatter, correlation coefficient values, and significance (using this larger aperture, the Spearman (r, p) values become (0.23, 1.1 × 10⁻³), (0.05, 0.5), and (0.22, 2.1 × 10⁻³) for the equivalent relations of IMF with age, metallicity, and [α/Fe], respectively). This reassures us that our choice of aperture is not important and that possible radial variations in the IMF (Pastorello et al. 2014) do not directly affect our conclusions.

It is well established that stellar population parameters correlate positively with velocity dispersion (Thomas et al. 2005; Graves et al. 2009; Thomas et al. 2010). Given the positive correlation between velocity dispersion and IMF presented in Cappellari et al. (2013a), it is tempting to conclude that the trends shown in Figure 3 are simply tracing this underlying trend of IMF with velocity dispersion, as argued in Smith (2014). However, the tightest correlation of our stellar population parameters with velocity dispersion is with total metallicity (McDermid et al. 2014). Conversely, metallicity shows no evidence for any direct, linear correlation with the IMF. This contrary trend of IMF with metallicity compared with age and [α/Fe] suggests that our findings are not purely a result of the underlying correlation of IMF with velocity dispersion, as this would give rise to consistent behavior between all three stellar population parameters, which all positively correlate with velocity dispersion.

The weak trends we find here are consistent with the similarly weak trend between α_dyn and velocity dispersion (r = 0.36, p = 5.7 × 10⁻⁸), reported in Cappellari et al. (2013a). A comparison of the ATLAS^3D velocity dispersion–IMF relation with several published spectral studies (La Barbera et al. 2013; Conroy & van Dokkum 2012; Treu et al. 2010; Spiniello et al. 2014) is given in Figure 12 of Spiniello et al. (2014). While there is overlap between the various spectral studies and ATLAS^3D over the range of velocity dispersion in common (σ > 130 km s⁻¹), the overall relationship for the entire sample is significantly more shallow than that suggested by the spectral studies. Restricting our sample to galaxies with this range of velocity dispersion (where the agreement is best) results in shallower gradients in the α_dyn–SSP relations presented here, with larger uncertainties (a zero gradient with age and [α/Fe] is excluded with only 2σ confidence rather than 5σ), and smaller correlation coefficients (r <0.16) with less significance (p >0.02), consistent with no significant correlation of α_dyn with any population parameter.

Other explanations for the weak trends are possible biases in the dynamical and/or population IMF estimates. An obvious potential systematic problem with the dynamically derived IMF, which has often been invoked in the past, is the treatment of the dark matter content in the dynamical models. We showed in Cappellari et al. (2012) that for dark matter to explain the IMF trends, it would have to follow the stellar distribution much closer than any current model predicts, making this explanation very unlikely.

Other sources of biases include general problems in the (M/L)_Salp derived from stellar population models. Using the SSP models of Vazdekis et al. (2012) and Bruzual & Charlot (2003) to derive (M/L)_Salp from our line-strengths, we find that the weak age trend is removed due to the biased reduction in (M/L)_Salp by young populations when using the (luminosity-weighted) SSP approach instead of (mass-weighted) spectral fitting. Any relationship with [Z/H] remains absent using this SSP approach, and the weak trend with [α/Fe], though present, becomes less significant, with best-fit gradients excluding zero at only a 1σ–4σ level. Our weak trends are therefore a general outcome from different models and approaches.

Finally, we note that a direct comparison of the IMF parameters for galaxies in common between our study and Conroy & van Dokkum (2012) yields only a weak relationship between the two studies (Smith 2014). The relatively small overlapping sample (34 objects), and differences in spatial apertures used in determining the IMF normalization (Conroy & van Dokkum 2012 use a small central aperture, whereas our IMF analysis uses data from within, on average, one effective radius) preclude firm conclusions on how this lack of agreement relates to the systematic uncertainties of either study. Removing these remaining issues is the focus of future work.

We present the observed trends of stellar population parameters with the dynamically determined IMF for 212 early-type galaxies from the ATLAS^3D survey. Using adaptive smoothing to highlight the average trends comparable to studies of larger samples, we find that the IMF normalization tends on average to be "heavy" for both weak Hβ and/or weak metal absorption line strengths. Using SSP models, we show that the stellar population properties of age, metallicity, and [α/Fe] span a broad range of values for any given IMF normalization.

We present the observed relations of IMF with the SSP properties of the individual galaxies of our sample. We find notably weak relations between the IMF and all three stellar population parameters, with mildly positive correlations of IMF "heaviness" with age and [α/Fe], and no significant correlation with total metallicity. Taken together with the weak relation between the IMF normalization and velocity dispersion already presented in Cappellari et al. (2013a), these results are somewhat at variance with those from various recent spectroscopic studies. Future efforts for obtaining large samples of individual objects with integral-field spectroscopy that also provides IMF-sensitive spectral features are now critical to permit the responsible systematic uncertainties in local galaxy IMF studies to be resolved.

The authors thank the referee for helping to improve this article. The research leading to these results has received funding from the European Community's Seventh Framework Programme (/FP7/2007-2013/) under grant agreement No. 229517. M.C. acknowledges support from a Royal Society University Research Fellowship. This work was supported by the rolling grants PP/E001114/1 and ST/H002456/1 and visitor's grants PPA/V/S/2002/00553, PP/E001564/1 and ST/H504862/1 from the UK Research Councils. R.L.D. acknowledges travel and computer grants from Christ Church, Oxford and support from the Royal Society in the form of a Wolfson Merit Award 502011.K502/jd. S.K. acknowledges support from the Royal Society Joint Projects Grant JP0869822. T.N. and M. Bois acknowledge support from the DFG Cluster of Excellence "Origin and Structure of the Universe." M.S. acknowledges support from a STFC Advanced Fellowship ST/F009186/1. During this research, M. Bois has received funding from the European Research Council under the Advanced Grant Program Num. 267399-Momentum. L.Y. acknowledges support from NSF grant AST-1109803. The authors acknowledge financial support from ESO.