DARK MATTER CONTRACTION AND THE STELLAR CONTENT OF MASSIVE EARLY-TYPE GALAXIES: DISFAVORING "LIGHT" INITIAL MASS FUNCTIONS

Numerical simulations of cold dark matter (CDM) halos predict density distributions with an inner logarithmic slope d ln ρ/d ln r = −1 and a slope of d ln ρ/d ln r = −3 at large radii (Navarro et al. 1996, 2010, hereafter NFW). However, the inclusion of baryons—and the dissipational processes associated with them—in galaxy formation simulations may significantly alter the central dark matter (DM) distribution. The DM may show an increased central density due to radiative dissipation from the baryons that depends on the mass of stars formed in the galaxy (e.g., the adiabatic contraction, or AC, models of Blumenthal et al. 1986; Gnedin et al. 2004; hereafter B86 and G04, respectively) or dynamical friction may counterbalance this effect (Abadi et al. 2010) and could result in cored profiles (e.g., El-Zant et al. 2004; Nipoti et al. 2004). The form of the CDM halo is critical for understanding the relative importance of baryonic and total mass in governing the structure of early-type galaxies (ETGs).

Strong lensing can measure masses to a few percent precision and is therefore a particularly powerful probe of this regime, by itself and in combination with other techniques (Treu 2010, and references therein). Studies have previously used strong and weak gravitational lensing (SL and WL) to constrain the distribution of DM for massive ETGs and found that the data are broadly and on average consistent with an NFW halo (e.g., Gavazzi et al. 2007). However, the SL and WL analysis alone cannot simultaneously probe the stellar mass M_* and central DM slope due to degeneracies (Lagattuta et al. 2010); additional information must be included to disentangle the contributions of stars and CDM to the central density distribution (similar to the disk–halo degeneracy for spiral galaxies). Schulz et al. (2010) have recently used WL and stellar velocity dispersions to investigate evidence for AC in Sloan Digital Sky Survey (SDSS) ETGs. They find that the stellar velocity dispersions require more central mass than would be inferred from the stellar mass (assuming a Kroupa initial mass function, or IMF) and the central DM mass extrapolated from an NFW halo fit to the WL data. However, their findings are also consistent with no AC if a Salpeter IMF is assumed for the stellar component.

Jiang & Kochanek (2007) point out that the degeneracies between the central CDM slope and M_* can only be broken by using at least three mass probes. They use SL and dynamics to probe the central baryonic and CDM distribution of ETGs and include an ensemble measurement of the halo mass from WL data to find that AC models are favored over an NFW halo. However, Jiang & Kochanek (2007) assume a redshift-dependent but otherwise constant stellar mass-to-light ratio M_*/L for each of the galaxies, although the IMF (and therefore M_*/L) may be non-universal (Treu et al. 2010). A full understanding of the central mass distribution of galaxies requires constraints on both the stellar and CDM components as a function of mass. Although this joint analysis is more complex and there are residual degeneracies, it provides a rare opportunity to constrain directly the IMF of distant massive ETGs where resolved stellar population diagnostics are not available (see, e.g., Cappellari et al. 2009; Bastian et al. 2010, and references therein).

In this Letter, we present simultaneous constraints on M_* and the form of the central DM distribution for a sample of massive ETGs. We use SL, stellar dynamics, stellar population synthesis (SPS) stellar mass estimates M^SPS_* given an IMF, and WL shear measurements to constrain a model for the relationship between stellar and virial mass. We allow for several different DM halo profiles and we fit for an IMF mismatch parameter relating M_* to M^SPS_* (Treu et al. 2010). The two main improvements over the analysis by Treu et al. (2010) are the inclusion of WL data and the adoption of more flexible models. We use a standard flat ΛCDM cosmology with Ω_M = 0.3 and h = 0.7.

Our sample is derived from the Sloan Lens ACS Survey (SLACS; Bolton et al. 2006, 2008; Gavazzi et al. 2007; Auger et al. 2009), which consists of nearly 100 strong gravitational lensing galaxies. We exclude six galaxies which appear to be structurally different than the rest of the SLACS early-type lenses (Auger et al. 2010). These systems are significant outliers from the otherwise very tight relation between effective radius, velocity dispersion, total central mass, and stellar mass; this is likely due to systematic errors in the stellar velocity dispersions of these systems (also see Jiang & Kochanek 2007). We therefore use a subsample of 53 ETGs that have well-determined central stellar velocity dispersions, SPS stellar masses assuming Chabrier (2003) and Salpeter (1955) IMFs, and central projected mass estimates from SL. We also use WL shear measurements for a subset of 44 lenses with deep Advanced Camera for Surveys imaging. The details of how these data are derived can be found elsewhere (Gavazzi et al. 2007; Bolton et al. 2008; Auger et al. 2009, 2010). We emphasize that the SLACS lens galaxies are indistinguishable from twin ETGs selected from SDSS to have the same velocity dispersion and redshift and can therefore be considered as representative of the overall population of massive ETGs (Treu et al. 2006, 2009; Auger et al. 2009, 2010).

We model the lenses as two-component mass distributions consisting of stars and a CDM halo. Both components are modeled as spherical since our data do not constrain the halo flattening and our dynamical analysis is restricted to spherical Jeans modeling. Comparisons with non-spherical models show that the most relevant quantities such as mass–density slope and total mass are relatively insensitive to this approximation (e.g., Barnabè et al. 2009).

The stars are modeled with the Hernquist (1990) profile and the scale radius is set by the rest-frame V-band effective radius, r_a = 0.551r_e; our conclusions are unchanged if we adopt a Jaffe (1983) profile. We assume isotropic orbits, consistent with typical findings for massive ETGs (Barnabè et al. 2009). Mild radial anisotropy (Koopmans et al. 2009) only alters the inferred velocity dispersions by a few percent, i.e., an amount smaller than the errors, and therefore does not change our conclusions.

We investigate three different density profiles for the CDM halos. First, we consider the NFW profile with the concentration fixed by c(M_vir) ∝ M^−0.094_vir following Macciò et al. (2008), neglecting intrinsic scatter for computational simplicity. We also modify the NFW halo to include AC using the prescriptions of B86 and G04, with the baryon fraction set by Equation (1) and the scale radius given by r_a; these three profiles are the most widely adopted descriptions for CDM halos (e.g., Jiang & Kochanek 2007; Schulz et al. 2010; Napolitano et al. 2010), and the NFW (B86) halo has the lowest (highest) central DM density.

We fix the relationship between stellar and virial mass to have the form suggested by Moster et al. (2010, who use a Kroupa IMF) based upon abundance matching techniques. We assume

$$\begin{equation} {\it M}_* \equiv {\it M}_*^{\rm SPS} = {\it M}_{*,0} \frac{({M}_{\rm vir}/{M}_{\rm vir,0})^{\gamma _1}}{\left[1 + ({M}_{\rm vir}/{M}_{\rm vir,0})^\beta \right]^{(\gamma _1-\gamma _2)/\beta }}, \end{equation} \tag{ 1 }$$

where M^SPS_* is the stellar mass inferred from the SPS models assuming a Salpeter IMF, γ₁ = 7.17, γ₂ = 0.201, and β = 0.557, and we fit for M_*,0 and M_vir,0. We note that our results are qualitatively the same if we use different values for γ₁, γ₂, and β (e.g., from Dutton et al. 2010), although the goodness of fit changes; we will explicitly fit for these parameters in a future paper to ascertain the extent to which one model prefers a given set of parameters over another.

We also allow for an offset between the model stellar mass and the SPS stellar mass, equivalent to a mass-dependent IMF (referred to as the "Free" IMF model). The offset has the form

$$\begin{equation} {\rm log}\frac{\it {M}_*^{\rm SPS}}{{\it M_*}} = -\eta {\rm log}\frac{{\it M_*}}{10^{11}\,{{\it M}}_\odot } - \log \alpha, \end{equation} \tag{ 2 }$$

where α is equivalent to that introduced by Treu et al. (2010) and η is equivalent to the slope of Equation (5) in Treu et al. (2010). We adopt the M^SPS_* derived by Auger et al. (2009), although our results are robust with respect to changes in the SPS models and choice of priors (Treu et al. 2010) due to the relative simplicity of the older stellar populations of the SLACS ETGs. Uncertainties in SPS models of evolved galaxies (Conroy & Gunn 2010; Maraston et al. 2009) may lead to systematic offsets in M^SPS_* of order ∼0.05 dex (e.g., Treu et al. 2010).

We perform a Markov chain Monte Carlo simulation to fit the models to the data; we propose M_vir for each lens and use Equation (1) (and Equation (2) for the Free IMF model) to predict a stellar mass. These masses normalize the halo and bulge, and we use these profiles to predict the stellar kinematics, SL, and WL signals for each galaxy. We compare these quantities and the model stellar masses with the observed quantities on a galaxy-by-galaxy basis (i.e., we do not bin or average measurements, including the WL) to optimize the free model parameters and determine goodness-of-fit values.

The best-fit parameters and χ² for each model are listed in Table 1, and we show the best-fit mass distributions for a characteristic lens system (L_V = 10^10.85 L_☉) in Figure 1. This illustrates that the low normalization of the Chabrier IMF requires a more massive CDM halo to increase the central projected mass and be consistent with the mass required by the SL data (this is true for the NFW and AC models). However, increasing the halo mass also increases the scale radius because the virial radius increases and the concentration becomes smaller (e.g., Macciò et al. 2008); therefore, the halo mass grows at a faster rate than the projected DM mass inside the Einstein radius. The net effect is that very massive halos are needed to match the SL data if a Chabrier IMF is used (see Figure 2).

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Table 1. Best-fitting Parameters and Goodness of Fits

Halo	IMF	log[M*,0/M☉]	log[Mvir,0/M☉]	η	log α	χ2/dof	$\langle {\rm ln\, \mbox{\it P}}\rangle$
NFW	Chab	11.38 ± 0.09	11.32 ± 0.24	...	...	2.68	−19.16
	Salp	11.29 ± 0.04	10.38 ± 0.06	...	...	1.18	266.54
	Free	11.44 ± 0.06	10.52 ± 0.07	0.08 ± 0.04	0.03 ± 0.03	1.04	284.16
G04	Chab	10.97 ± 0.04	10.11 ± 0.11	...	...	1.66	189.76
	Salp	11.36 ± 0.05	10.43 ± 0.05	...	...	1.12	268.98
	Free	11.26 ± 0.06	10.34 ± 0.06	0.03 ± 0.05	−0.08 ± 0.02	1.08	281.58
B86	Chab	10.94 ± 0.04	10.07 ± 0.10	...	...	1.42	228.43
	Salp	11.47 ± 0.06	10.54 ± 0.06	...	...	1.26	234.00
	Free	11.25 ± 0.06	10.33 ± 0.07	0.01 ± 0.05	−0.11 ± 0.02	1.13	269.05

Notes. The "Free" IMF is from Equation (2); for η = 0, log α = 0 for a Salpeter IMF, and log α ≈ −0.25 (log α ≈ −0.2) for a Chabrier (Kroupa) IMF. The χ² is given for a model with the mean parameters listed in the table, although the interpretation of Δχ² is difficult due to the large number of degrees of freedom; $\langle {\rm ln \mbox{\it P}}\rangle$ is the mean of the natural log of the posterior probability. It is also difficult to directly interpret Δ〈lnP〉, but values of several tens clearly indicate a strong preference for one model over another.

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These massive halos cannot fit the SL masses and the WL shears (the inferred M_* values are also much higher than the measurements), and the data therefore clearly disfavor a Chabrier IMF (Table 1), consistent with the results from SL and dynamical constraints only (Treu et al. 2010). The G04 model is slightly preferred over the NFW halo for the Salpeter IMF while the B86 model is disfavored. The normalization and break point of the M_vir–M_* relation are consistent between the NFW and G04 model with a Salpeter IMF, and both are also consistent with the results from Moster et al. (2010) when the ∼0.20 dex offset between Kroupa and Salpeter IMFs is taken into account.

Figure 2 illustrates that the favored models require significantly higher star formation efficiencies compared to the values of ≈0.02 − 0.1 (for a Chabrier IMF; double this for Salpeter) found for massive ETGs from galaxy–galaxy WL studies (e.g., Mandelbaum et al. 2006), which may point to a generic inadequacy of the family of models considered here. Interestingly, simulations of galaxy formation may require similarly high star formation efficiencies to reproduce the observed isothermal central mass distributions of massive ETGs (Duffy et al. 2010) and suggest that the high efficiency may not be due to our simplifying assumptions.

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The results for the Free IMF model are somewhat more difficult to interpret due to the addition of two extra parameters; the fit is significantly improved compared with the fixed IMF models, as is expected when allowing for a more flexible model. The NFW model and G04 model fit the data equally well, but again the B86 model is disfavored. Note, however, that the form of the M_vir–M_* relation that we are using was derived assuming a constant IMF; allowing for a varying IMF would necessarily change the structural parameters γ₁, γ₂, and β. Nevertheless, the present model indicates that even the AC halos require the IMF to have a slight trend with mass in the sense that more massive galaxies require more Salpeter-like IMFs. The trend is stronger with the NFW halo (Figure 3) and is consistent with the relation found by Treu et al. (2010).

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There is a growing consensus that the fraction of DM at the center of massive ETGs increases with galaxy mass (Tortora et al. 2009; Napolitano et al. 2010; Graves & Faber 2010; Auger et al. 2010). This increase could be due to a genuine increase in CDM, perhaps as the result of AC mediated by the detailed accretion history of halos (e.g., Abadi et al. 2010; Lackner & Ostriker 2010). Alternatively, the trends may be the result of a mass-dependent IMF leading to more baryonic DM (in the form of low-mass stars or stellar remnants, depending on whether the IMF is bottom heavy or top heavy, respectively) in more massive galaxies (e.g., Treu et al. 2010), perhaps due to cosmic evolution of the IMF (van Dokkum et al. 2008) and a mass-dependent formation redshift for the stellar mass (e.g., Thomas et al. 2005; van der Wel et al. 2009).

4.1. Ruling Out "Light" IMFs for Massive ETGs

4.2. Adiabatic Contraction and a Varying IMF

4.3. Beyond Structural Constraints

We have demonstrated that, given our standard assumptions, the IMF of massive ETGs cannot be "light," even in the presence of significant adiabatic contraction. Furthermore, the IMF is only consistent with being universal if the central DM profile is mass dependent, as in the AC models presented here. It is clear that if one wants to preserve a light and universal IMF, one has to abandon standard assumptions about DM halos, like the NFW profile. However, there are two important caveats. First, better constraints on the star formation efficiency must be obtained from the data in order to draw definitive conclusions about the role of a mass-dependent IMF relative to CDM halo contraction. Second, although our conclusions are robust with respect to a wide variety of changes in our assumptions, we have not exhausted all possible families of theoretical models.

We will address these caveats, quantify the relationship between the central DM profile and the IMF in more detail, and quantify the consequences of our various assumptions in a forthcoming paper (M. W. Auger et al. 2010, in preparation).

We thank the referee for his/her helpful comments, and Aaron Dutton and Matteo Barnabè for their suggestions. We acknowledge support from the NSF through CAREER award NSF-0642621, from the Packard Foundation through a Packard Fellowship to T.T., and from NASA, through HST grants 11202, 10798, and 10494. R.G. is supported by the Centre National des Etudes Spatiales.