NEW CONSTRAINTS ON THE EVOLUTION OF THE STELLAR-TO-DARK MATTER CONNECTION: A COMBINED ANALYSIS OF GALAXY–GALAXY LENSING, CLUSTERING, AND STELLAR MASS FUNCTIONS FROM z = 0.2 to z = 1^*

The general methodology for the construction of the COSMOS ACS weak lensing catalog and our shape measurement procedure are presented in Leauthaud et al. (2007) and Rhodes et al. (2007). In this section, we present several updates to the pipeline that we have implemented since those publications. In Leauthaud et al. (2007), we used a parametric correction for the effects of charge transfer inefficiency (CTI) on galaxy shape measurements. Instead, in this paper, we use a physically motivated CTI correction scheme that operates on the raw data and returns electrons to pixels from which they were unintentionally dragged out during readout. This correction scheme has been shown by Massey et al. (2010) to reduce the CTI trails by a factor of ∼10 everywhere in the CCD and at all flux levels.

Following CTI correction in the raw images, image registration, geometric distortion, sky-subtraction, cosmic ray rejection, and the final combination of the dithered images are performed by the MultiDrizzle algorithm (Koekemoer et al. 2002). As described in Rhodes et al. (2007), a finer pixel scale of 003 pixel⁻¹ was used for the final co-added images. The lensing source catalog is constructed from 575 ACS/WFC tiles. Defects and diffraction spikes are carefully removed, leaving a total of 1.2 × 10⁶ objects to a limiting magnitude of I_F814W = 26.5.

The next step is to measure the shapes of galaxies and to correct them for the distortion induced by the time varying ACS point-spread function (PSF; see Rhodes et al. 2007). We continue to use a PSF model based on physical parameters rather than arbitrary principal components. In Leauthaud et al. (2007), we modeled the PSF as a multivariate polynomial in x, y, and focus. We now fit the PSF as a function of x, y, focus, and velocity aberration of the pointing (recorded in CALACS headers as "VAFACTOR"). Schrabback et al. (2010) found that VAFACTOR partially correlates with the higher-order PSF variations, motivating our use of this quantity. In the g–g lensing analysis presented here, the weak lensing shear is azimuthally averaged. Thus, any effects of PSF anisotropy cancel to leading order. Therefore, our science analysis is insensitive to subtle differences in the PSF modeling. We confirmed this by repeating our analysis with the independently obtained weak lensing catalog by Schrabback et al. (2010), which yields fully consistent results.

Finally, simulated images are used to derive the shear susceptibility factors that are necessary in order to transform shape measurements into unbiased shear estimators (Leauthaud et al. 2007). Representing a number density of 66 galaxies per arcminute², the final COSMOS weak lensing catalog contains 3.9 × 10⁵ galaxies with accurate shape measurements.

We use two updated versions (v1.8 dated from 2010 July 13 and v1.7 dated from 2009 August 1) of the photometric redshifts (hereafter photo-z's) presented in Ilbert et al. (2009) which have been computed with over 30 bands of multi-wavelength data. In particular, deep K_s, J, and u* band data allow for a good photo-z estimate at z > 1 via the 4000 Å break which is increasingly shifted into the near-infrared (IR). Further details regarding the data and the photometry can be found in Capak et al. (2007).

The photo-z catalog v1.8 has improved redshifts at z > 1 compared to v1.7. At z < 1, the difference between the two catalogs is minor. We use catalog v1.8 for g–g lensing measurements and v1.7 for the SMF and galaxy clustering. Interchanging the two catalogs does not affect our results.

Photo-z's were estimated using a χ² template fitting method (Le Phare; Ilbert et al. 2009) and calibrated with large spectroscopic samples from VLT-VIMOS (Lilly et al. 2007) and Keck-DEIMOS. The dispersion in the photo-z's as measured by comparing to the spectroscopic redshifts is $\sigma _{\Delta z/(1+z_{\rm spec})}=0.007$ at i⁺_AB < 22.5, where Δz = z_spec − z_phot. The deep IR and IRAC (Infrared Array Camera on the Spitzer Space Telescope) data enable the photo-z's to be calculated even at fainter magnitudes with a reasonable accuracy of $\sigma _{\Delta z/(1+z_{\rm spec})}=0.06$ at i⁺_AB ∼ 24 mag.

Figure 1 compares the spectroscopic and photometric redshifts of 8812 galaxies that belong to both the lensing source catalog and the zCOSMOS "bright" or "faint" programs. This figure also illustrates the sensitivity of g–g lensing signals to photometric redshift errors. As can be seen from Figure 1, g–g lensing signals are increasingly insensitive to photometric redshift errors for source galaxies at higher redshifts.

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Stellar masses are estimated using the Bayesian code described in Bundy et al. (2006) assuming a Chabrier IMF. Briefly, an observed galaxy's spectral energy distribution (SED) and redshift is referenced to a grid of models constructed using the Bruzual & Charlot (2003) synthesis code. The grid includes models that vary in age, star formation history, dust content, and metallicity. The assumed dust model is Charlot & Fall (2000). At each grid point, the probability that the observed SED fits the model is calculated, and the corresponding stellar mass to K-band luminosity ratio and stellar mass is stored. By marginalizing over all parameters in the grid, the stellar mass probability distribution is obtained. The median of this distribution is taken as the stellar mass estimate, and the width encodes the uncertainty due to degeneracies and uncertainties in the model parameter space (also described as "model error" in Section 4.2). The final uncertainty on the stellar mass also includes the K-band photometry uncertainty as well as the expected error on the luminosity distance that results from the photo-z uncertainty. The typical final uncertainty is 0.1–0.2 dex (also see Section 4.2 and Figure 4). Systematic uncertainties also come from the choice of stellar population templates used to fit the data as well as the assumption of a universal IMF. To first order these will cause global offsets in the mass estimates that will not affect comparisons within our sample but may impact comparisons to work by other authors. Since the primary goal of this paper is to study the redshift evolution of the SHMR derived from COSMOS data alone, we do not include these systematic uncertainties in our analysis and refer to Conroy et al. (2009) and Behroozi et al. (2010) for a broad discussion of systematic errors in stellar mass estimates.

Following the approach in Bundy et al. (2010), we obtained PSF-matched 30 diameter aperture photometry from the ground-based COSMOS catalogs (filters u*, B_J, V_J, g⁺, r⁺, i⁺, z⁺, K_s) described in Capak et al. (2007), Ilbert et al. (2009), and McCracken et al. (2010), after applying the photometric zero-point offsets tabulated in Capak et al. (2007). The depth in all bands reaches at least 25th magnitude (AB) with the K_s-band limited to K_s < 24 mag. Unlike Drory et al. (2009), we require K_s-band detections for all galaxies in the sample. We have found that the mass estimates in Bundy et al. (2010) agree with those of Drory et al. (2009) within the expected uncertainties (i.e., <0.2 dex). The mass estimates used in this work are slightly different from those in Bundy et al. (2010) in that they are based on updated redshift information (v1.7 of the photo-z catalog and the latest available spectroscopic redshifts as compiled by the COSMOS team) and use a slightly different cosmology (H₀ = 72 km s⁻¹ Mpc⁻¹ instead of H₀ = 70 km s⁻¹ Mpc⁻¹).

The bins in our analysis are defined by using two estimates of the mass completeness of the sample, as determined by the magnitude limits K_s < 24 mag and I_814W < 25 mag. The first estimate is more conservative and comes from estimating the observed magnitude of a maximal M_*/L stellar population model with solar metallicity, no dust, and a τ = 0.5 Gyr burst of star formation that occurred at z_form = 5. As a function of redshift, the stellar mass of such a population at the point where its observed K_s- and I_814W-band flux falls below the magnitude limits defines the mass completeness and roughly matches the 80% completeness limits determined when deeper samples are available (Bundy et al. 2006). This redshift-dependent limit is plotted as the dashed line in Figure 2. In practice, low-mass galaxies exhibit more star formation and therefore have lower M_*/L ratios than the passive template described above so we also define a second, more liberal mass limit (i.e., lower) that corresponds to a star-forming population. This is plotted as the dotted line. The majority of the sample bins lie above the more conservative limit, but the lowest mass bins are allowed to reach the star-forming mass limit under the assumption that passive stellar populations (potentially missed) at such low masses are extremely rare.

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We use two COSMOS galaxy catalogs: the Subaru (photo-z) catalog and the ACS (lensing) catalog. The ACS catalog corresponds to the COSMOS area that has been imaged with HST and covers a slightly smaller area than the Subaru catalog. The ACS coverage of COSMOS is 1.64 deg² and the Subaru coverage of COSMOS is 2.3 deg². The ACS catalog is used for our g–g lensing measurements and for the SMFs. The Subaru catalog is used to calculate the galaxy clustering.

Galaxies are selected with K_s < 24 mag to ensure that stellar masses can be computed for all galaxies in the sample. Note that for the g–g lensing analysis, these cuts only apply to the foreground lens sample. In addition to these cuts, we also reject galaxies that are in masked areas where the photometry is deemed unreliable. For this, we use the union of four masks in the B_J, V_J, i⁺, and z⁺ ("COSMOS.B.mask," "COSMOS.V.mask," "COSMOS.ip.mask," "COSMOS.zp.mask").

For the ACS sample, star–galaxy separation is performed using the morphological classifier described in Leauthaud et al. (2007). For the Subaru sample, we reject stars by imposing the condition that χ²_gal − χ_star² > 0.5 where χ²_gal represents the chi-square value of the best-fitting galaxy SED template and χ²_star represents the chi-square value of the best-fitting stellar SED template. In conjunction with the K_s cut, this stellar selection agrees with our ACS-based morphological classification at the 97% level. According to our ACS morphological classifier: 2% of the objects (or 5095 objects in the ACS area) in the final Subaru catalog are classified as stars and 0.1% of galaxies are misclassified as stars (or 246 objects in the ACS area) so the stellar sample reliability is 0.048.

We compute the g–g lensing, galaxy clustering, and stellar mass in three redshift intervals: z₁ = [0.22, 0.48], z₂ = [0.48, 0.74], and z₃ = [0.74, 1]. For each redshift bin, we define a lower limit on the stellar mass to ensure that all samples are complete in terms of stellar mass. These cuts correspond to M_* > 10^8.7 M_☉ for z₁, M_* > 10^9.3 M_☉ for z₂, and M_* > 10^9.8 M_☉ for z₃ (see Table 1) and are defined by the stellar mass completeness limit at the far edge of the redshift bin (see Figure 2). All galaxy samples used in this paper are complete.

The SMFs are calculated using the ACS catalog. As in Bundy et al. (2010), we compute mass functions using the V_max technique (Schmidt 1968). We weight galaxies by the maximum volume in which they would be detected within the K_s-band and I_814W-band limits in a given redshift interval. For each host galaxy i in the redshift interval j, the value of Vⁱ_max is given by the minimum redshift at which the galaxy would drop out of the sample,

$$\begin{equation} V^i_{{\rm max}} = \int _{z_{{\rm low}}}^{z_{{\rm high}}} d \Omega \frac{dV}{dz} dz, \end{equation} \tag{ 1 }$$

where dΩ is the solid angle subtended by the survey area and dV/dz is the comoving volume element. The redshift limits are given as

$$\begin{equation} z_{{\rm high}} = {\rm min}\big(z^j_{{\rm max}}, z^j_{K_{{\rm lim}}}, z^j_{I_{{\rm lim}}}\big), \end{equation} \tag{ 2 }$$

$$\begin{equation} z_{{\rm low}} = z^j_{{\rm min}}, \end{equation} \tag{ 3 }$$

where the redshift interval, j, is defined by [z^j_min, z^j_max] and $z^j_{K_{{\rm lim}}}$ and $z^j_{I_{{\rm lim}}}$ refer to the redshift (see Table 1 for the redshift limits) at which the galaxy would still be detected below the K_s- and I_814W-band limits. We use the best-fit SED template as determined by the stellar mass estimator to calculate these values, thereby accounting for the k-corrections necessary to compute V_max values (no evolutionary correction is applied).

The galaxy clustering samples are defined by a series of stellar mass thresholds rather than bins. For galaxy clustering, threshold samples are the most straightforward to model in the HOD context. At small scales, the signal-to-noise ratio (S/N) in galaxy clustering is due mostly to Poisson noise and at large scales it is subject to sample ("cosmic") variance. The optimal binning scheme for clustering can be different than that for g–g lensing. For example, massive galaxies produce the strongest shear signals, thus a relatively small sample is required to produce a robust measurement relative to a simple pair-counting statistic like the two-point correlation function whereas the inverse would be true for low-mass galaxies. The threshold samples we employ for clustering measurements are listed in Table 2. Except at the high-mass end where galaxies become scarce, the binning scheme employed for the clustering is a constant 0.5 dex in log₁₀(M_*).

Since we do not need galaxy shape information to measure the angular correlation function w(θ), we are not restricted to the ACS coverage of the COSMOS field. We therefore use the COSMOS Subaru catalog to calculate w(θ). For each threshold sample we measure w(θ) using the well-known Landy & Szalay (1993) estimator of

$$\begin{equation} w(\theta) = \frac{{\it DD} - 2{\it DR} + {\it RR}}{{\it RR}}, \end{equation} \tag{ 4 }$$

where DD are the number of data–data pairs in a given bin of angle θ, RR are the number of random–random pairs, and DR are the number of data–random pairs. Data and random pairs are normalized by the total number of galaxies and random points, respectively. In all measurements we use 10⁵ random points. The distribution of the random points is taken from a combination of four COSMOS masks (B_J, V_J, i⁺, and z⁺), thus mimicking the angular completeness and geometry of the survey.

Because the volumes probed in each sample are small, our clustering measurements are subject to the effect of the integral constraint (IC; Groth & Peebles 1977). Due to spatial fluctuations in the number density of galaxies, the mean correlation function measured from an ensemble of samples will be smaller than the correlation function measured from a single contiguous sample of the same volume as the sum of the ensemble sample. This attenuation of w(θ) becomes relevant on angular scales significant with respect to the sample size. We estimate the IC correction to our w(θ) measurements through the use of mock galaxy distributions described in Paper I and Section 4. In practice, we do not modify the measurements for the IC but rather adjust the theoretical models to account for the finite sample size.

Finally, in order to compute w(θ) from our model, we need to know the normalized redshift distribution of the galaxy sample, N(z) (see Equation (32) in Paper I). For this we use the probability distribution functions of the photometric redshifts to estimate the true N(z) for our photo-z slices. However, the COSMOS photo-z errors are accurate enough such that in tests we find a minimal effect when using a flat top-hat z-bin for N(z).

In the weak gravitational lensing limit, the observed shape ε_obs of a source galaxy is directly related to the lensing induced shear γ according to

$$\begin{equation} \varepsilon _{\rm obs} = \varepsilon _{\rm int}+\gamma, \end{equation} \tag{ 5 }$$

where ε_int is the source galaxy's intrinsic shape that would be observed in the absence of gravitational lensing. In our notation, ε_int, ε_obs, and γ are spin-2 tensors. The above relationship indicates that galaxies would be ideal tracers of the distortions caused by gravitational lensing if the intrinsic shape ε_int of each source galaxy was known a priori. However, lensing measurements exhibit an intrinsic limitation, encoded in the width of the ellipticity distribution of the galaxy population, denoted here as σ_int, and often referred to as the "intrinsic shape noise." Because the intrinsic shape noise (of order σ_int ∼ 0.27; Leauthaud et al. 2007) is significantly larger than γ, shears must be estimated by averaging over a large number of source galaxies.

Throughout this paper, the gravitational shear is noted as γ whereas $\tilde{\gamma }$ represents our estimator of γ. The uncertainty in the shear estimator is a combination of unavoidable intrinsic shape noise, $\sigma _{\rm int}=\sqrt{\langle \varepsilon _{\rm int}^2\rangle }$, and of shape measurement error, σ_meas:

$$\begin{equation} \sigma _{\rm \tilde{\gamma }}^2 = \sigma _{\rm int}^2+\sigma _{\rm meas}^2. \end{equation} \tag{ 6 }$$

We will refer to $\sigma _{\rm \tilde{\gamma }}$ as "shape noise" whereas σ_int will be called the "intrinsic shape noise." The former includes shape measurement error and will vary according to each specific data set and shape measurement method. Averaged over the whole COSMOS field, the weak lensing distortions represent a negligible perturbation to Equation (6).

The derivation of our shear estimator is presented in Leauthaud et al. (2007). We employ the RRG method (see Rhodes et al. 2000 for further details) for galaxy shape measurements. Briefly, we form $\tilde{\gamma }$ from the PSF-corrected ellipticity according to

$$\begin{equation} \tilde{\gamma }=C \times \frac{\varepsilon _{\rm obs}}{G}, \end{equation} \tag{ 7 }$$

where the shear susceptibility factor,¹⁹ G, is measured from moments of the global distribution of ε_obs and other, higher-order shape parameters (see Equation (28) in Rhodes et al. 2000). Using a set of simulated images similar to those of Shear TEsting Program (STEP; Heymans et al. 2006b; Massey et al. 2007) but tailored exclusively to this data set, we find that, in order to correctly measure the input shear on COSMOS-like images, the RRG method requires an overall calibration factor of C = (0.86^+0.07_−0.05)⁻¹. Shear calibration factors are a generic feature of all shape measurement algorithms. This is because no shear measurement technique is independent of the observing conditions and the underlying galaxy population (Massey et al. 2007; Bernstein 2010; Zhang & Komatsu 2011). Thus, it is a common practice in weak lensing measurements to derive survey-specific calibration factors using simulations designed to mimic both the observing conditions and the surveyed galaxy population.

The shear signal induced by a given foreground mass distribution on a background source galaxy will depend on the transverse proper distance between the lens and the source and on the redshift configuration of the lens–source system. A lens with a projected surface mass density, Σ(r), will create a shear that is proportional to the surface mass density contrast, ΔΣ(r):

$$\begin{equation} \Delta \Sigma (r)\equiv \overline{\Sigma }({< }r)-\overline{\Sigma }(r)=\Sigma _{\rm crit}\times \gamma _t(r). \end{equation} \tag{ 8 }$$

Here, $\overline{\Sigma }({< }r)$ is the mean surface density within proper radius r, $\overline{\Sigma }(r)$ is the azimuthally averaged surface density at radius r (e.g., Miralda-Escude 1991; Wilson et al. 2001), and γ_t is the tangentially projected shear. The geometry of the lens–source system intervenes through the critical surface mass density,²⁰ Σ_crit, which depends on the angular diameter distances to the lens (D_OL), to the source (D_OS), and between the lens and source (D_LS):

$$\begin{equation} \Sigma _\mathrm{crit} = \frac{c^2}{4\pi G_{{{\rm N}}}}\, \frac{D_\mathrm{OS}}{D_\mathrm{OL}\,D_\mathrm{LS}}\;, \end{equation} \tag{ 9 }$$

where G_N represents Newton's constant. Hence, if redshift information is available for every lens–source pair, each estimate of γ_t can be directly converted to an estimate of ΔΣ(r) which is a more desirable quantity than γ_t because it depends only on the mass distribution of the lens.

To measure ΔΣ(r) with high S/N, the lensing signal must be stacked over many foreground lenses and background sources. For every ith lens and jth source separated by a proper distance r_ij, an estimator of the mean excess projected surface mass density ($\Delta \tilde{\Sigma }_{ij}$) at r_ij is computed according to

$$\begin{equation} \Delta \tilde{\Sigma }_{ij}(r_{ij})=\tilde{\gamma }_{t,ij}\times \Sigma _{{\rm crit},ij}, \end{equation} \tag{ 10 }$$

where $\tilde{\gamma }_{t,ij}$ is the tangential shear of the source relative to the lens. The COSMOS photometric redshifts described in Section 2.2 are used to estimate Σ_{crit, ij} for every lens–source pair. In order to optimize the S/N, an inverse variance weighting scheme is employed when ΔΣ_ij is summed over many lens–source pairs. Each lens–source pair is attributed a weight that is equal to the estimated variance of the measurement:

$$\begin{equation} w_{ij} = \frac{1}{ (\Sigma _{{\rm crit},ij} \times \sigma _{\tilde{\gamma },ij} )^2}. \end{equation} \tag{ 11 }$$

In this manner, faint small galaxies which have large measurement errors are downweighted with respect to sources that have well-measured shapes.

For the types of lenses studied in this paper, the S/N per lens is not high enough to measure ΔΣ on an object-by-object basis so instead we stack the signal over many lenses. For a given sample of lenses, the total excess projected surface mass density is the weighted sum over all lens–source pairs:

$$\begin{equation} \Delta \Sigma = {\sum _{j=1}^{N_{{\rm Lens}}} \sum _{i=1}^{N_{{\rm Source}}} w_{ij} \times \tilde{\gamma }_{t,ij}\times \Sigma _{{\rm crit},ij} \over \sum _{j=1}^{N_{{\rm Lens}}} \sum _{i=1}^{N_{{\rm Source}}}w_{ij}}. \end{equation} \tag{ 12 }$$

We only give a brief outline of the overall methodology used to compute the g–g lensing signals since this has already been presented in detail in Leauthaud et al. (2010). Foreground lens galaxies are divided into three redshift samples and then are further binned by stellar mass (see Figure 2 and Table 3). For each lens sample, ΔΣ is computed according to Equation (12) from 25 kpc (physical distance) to 1.5 Mpc in logarithmically spaced radial bins of 1.8 dex. In Leauthaud et al. (2010), we used a theoretical estimate of the shape measurement error in order to derive the inverse variance for each source galaxy. Instead, in this paper, the dispersion of each shear component is measured directly from the data in bins of S/N and magnitude. The measured shear dispersion is equal to the quadratic sum of the intrinsic shape noise and of the shape measurement error (Equation (6)). Our empirical derivation of the shear dispersion varies from $\sigma _{\rm \tilde{\gamma }} \sim 0.25$ for bright galaxies with high S/N to $\sigma _{\rm \tilde{\gamma }} \sim 0.4$ for faint galaxies with low S/N. Overall, we find that the theoretical and the empirical schemes yield very similar results with the latter method resulting in slightly larger error bars because the theoretical scheme tends to somewhat underestimate the shape measurement error for faint galaxies.

Photometric redshifts are used to derive Σ_crit for each lens–source pair. The lower 68% confidence bound on each source redshift is used to select background galaxies. For each lens–source pair, we demand that $z_{\rm source}-z_{\rm lens}> \sigma _{68\%}(z_{\rm source})$ so as to minimize foreground contamination. The g–g lensing signal is most sensitive to redshift errors when z_source is only slightly larger then z_lens (see Figure 1). For this reason, in addition to the previous cut, we also implement a fixed cut so that z_source − z_lens > 0.1. Furthermore, in order to minimize the effects of signal dilution caused by catastrophic errors, we also reject all source galaxies with a secondary peak in the redshift probability distribution function (i.e., the parameter zp₂ is non zero in the Ilbert et al. 2009 catalog). This cut is aimed to reduce the number of catastrophic errors in the source catalog. After all cuts have been applied, the g–g lensing source catalog that we use represents 35 galaxies per arcmin².

Finally, we re-compute all our g–g lensing signals using the Schrabback et al. (2010) COSMOS shear catalog which has been independently derived from ours. We find identical g–g lensing signals from both shear catalogs, indicating that any relative shear calibration differences between the two shear catalogs has no impact on these results. This test provides an independent validation of our g–g lensing results.

Following Behroozi et al. (2010, hereafter B10), $f_{\textsc {shmr}}(M_h)$ is mathematically defined via its inverse function:

$$\begin{eqnarray} \log _{10}\big(f_{\textsc {shmr}}^{-1}(M_\ast)\big) &=& \log _{10}(M_h)\nonumber\\ &=& \log _{10}(M_1) + \beta \,\log _{10}\left(\frac{M_\ast }{M_{\ast,0}}\right)\nonumber\\ && + \frac{\left(\frac{M_\ast }{M_{\ast,0}}\right)^\delta }{1 + \left(\frac{M_{\ast }}{M_{\ast,0}}\right)^{-\gamma }} - \frac{1}{2}, \end{eqnarray} \tag{ 13 }$$

where M₁ is a characteristic halo mass, M_{*, 0} is a characteristic stellar mass, β is the low-mass slope, and δ and γ control the high-mass slope. We refer to B10 for a more detailed justification of this functional form. Briefly, we expect that at least four parameters are required to model the SHMR: a normalization, break, a low-mass slope, and a bright end slope. In addition, B10 have found that the SHMR displays sub-exponential behavior at large M_*. This is modeled by the δ parameter which leads to a total of five parameters. Figure 3 illustrates the influence of each parameter on the shape on the SHMR and further details on the role of each parameter can be found in Section 2.1 of Paper I.

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In contrast to B10, we do not parameterize the redshift evolution of this functional form. Rather, we bin the data into three redshift bins and check for redshift evolution in the parameters a posteriori. We also assume that Equation (13) is only relevant for central galaxies. Following the HOD ansatz, satellite galaxies within groups and clusters occupy subhalos—bound, virialized halos that are contained within the radius of a larger halo. Abundance matching models like B10 assume that the halos and subhalos of the same mass (or circular velocity) contain galaxies of the same stellar mass, where the subhalo mass is taken as the mass at the last time the galaxy was a central galaxy. Here we put no such prior on the galaxy–subhalo connection. Rather, the halo occupation of satellite galaxies is constrained by the data.

The measured scatter in stellar mass at fixed halo mass has an intrinsic component (denoted $\sigma _{{\log M_{*}}}^{\rm i}$), but also includes a stellar mass measurement error due to redshift, photometry, and modeling uncertainties (denoted $\sigma _{{\log M_{*}}}^{\rm m}$). Ideally, we would measure both components but unfortunately we can only constrain the quadratic sum of these two sources of scatter. Nonetheless, given a model for $\sigma _{{\log M_{*}}}^{\rm m}$, we could in principle extract $\sigma _{{\log M_{*}}}^{\rm i}$ from $\sigma _{\log M_*}$.

Previous work suggests that $\sigma _{\log M_*}$is independent of halo mass. For example, Yang et al. (2009) find that $\sigma _{{\log M_{*}}}=0.17$ dex and More et al. (2009) find a scatter in luminosity at fixed halo mass of 0.16 ± 0.04 dex. Both Moster et al. (2010) and B10 are able to fit the Sloan Digital Sky Survey (SDSS) galaxy SMF assuming $\sigma _{{\log M_{*}}}=0.15$ dex and $\sigma _{{\log M_{*}}}=0.175$ dex, respectively. However, these results are derived with spectroscopic samples of galaxies. In contrast to these surveys, we expect a larger measurement error for the COSMOS stellar masses due to the use of photometric redshifts. In addition, since photo-z errors increase for fainter galaxies, we might also expect that $\sigma _{{\log M_{*}}}^{\rm m}$ (and thus $\sigma _{\log M_*}$) will depend on M_*.

To test if the assumption that $\sigma _{\log M_*}$is constant with M_* has any impact on our results, we implement two models for $\sigma _{\log M_*}$. In the first case (called "sig_mod1"), $\sigma _{{\log M_{*}}}$ is assumed to be constant (this is our base-line model). In the second case (called "sig_mod2"), we explicitly model $\sigma _{{\log M_{*}}}^{\rm m}$ to reflect stellar mass measurement errors. Note that the goal of this exercise is not to perform a careful and thorough error analysis, but simply to build a realistic enough model to asses whether or not an M_*-dependent error has any strong impact on our conclusions.

For the sig_mod2 model, we consider three contributions to the stellar mass error budget. The first is called "model error": this is measured by the 68% confidence interval of the mass probability distribution determined for each galaxy by the mass estimator. It represents the range of model templates (each with its own M/L ratio) that provide reasonable fits to the observed SED. This range is determined by the photometric uncertainty in the observed SED, degeneracies in the grid of models used to fit the data, and how well the grid of models represents the true parameter space of observed galaxy populations as well as their colors.²² The second term is the photo-z error, which derives from the uncertainty in the luminosity distance owing to the error on a given photometric redshift. The final component is the photometric uncertainty from the observed K-band magnitude, which translates into an uncertainty in luminosity and therefore stellar mass. The total measurement error, $\sigma _{{\log M_{*}}}^{\rm m}$, is the sum in quadrature of these three sources of error. The results are shown in Figure 4 for the three redshift bins.

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As detailed in Section 5, we find that our results are largely unchanged, regardless of which form we adopt for $\sigma _{\log M_*}$. This can be explained as follows. Since the data are binned by M_*, the observables are in fact sensitive to the scatter in halo mass at fixed stellar mass, denoted $\sigma _{\log M_h}$. Given a model for the SHMR, $\sigma _{\log M_h}$can be mathematically derived from $\sigma _{\log M_*}$. Further details on the mathematical connection between $\sigma _{\log M_*}$and $\sigma _{\log M_h}$can be found in Paper I. We find that the slope of the SHMR increases steeply at M_* > 10¹¹ M_☉ so that $\sigma _{\log M_h}$becomes quite large at the high-mass end. For example, $\sigma _{\log M_h}\sim 0.46$ dex at M_* = 10¹¹ M_☉ and $\sigma _{\log M_h}\sim 0.7$ dex at M_* = 10^11.5 M_☉. As a result, the data are particularly sensitive to $\sigma _{\log M_*}$at large M_* but not very sensitive to $\sigma _{\log M_*}$at low M_*. Therefore, accounting for the mass dependence of $\sigma _{\log M_*}$has little impact on the results because the constraints on $\sigma _{\log M_*}$mainly originate from high-mass galaxies anyway.

Finally, we note that our derived values for the parameters of the SHMR should be independent of $\sigma _{\log M_*}$(we will show that $\sigma _{\log M_*}$is not largely degenerate with any of the other nine parameters in Section 5). The observables (g–g lensing, clustering, and SMF) do themselves depend on $\sigma _{\log M_*}$but since we account for $\sigma _{\log M_*}$in the model, the extracted SHMR should reflect the true underlying physical relationship between halo and stellar mass (this however would not be the case if one were to neglect $\sigma _{\log M_*}$). In other terms, our model fully accounts for Eddington bias in all three observables (also see discussion in Paper I).

To model the central occupation function, we use five free parameters (M₁, M_{*, 0}, β, δ, γ) to model $f_{\textsc {shmr}}$ and we leave $\sigma _{{\log M_{*}}}$ as an additional free parameter. As described in Paper I, the central occupation function for a sample of galaxies more massive than M^t1_* (where M^t1_* represents some threshold in M_*) is expressed as

$$\begin{eqnarray} && \langle N_{\rm cen}(M_h|M_{*}^{t1}) \rangle \nonumber\\ && = \frac{1}{2} \left[ 1-\mbox{erf}\left(\frac{\log _{10}(M_{*}^{t1}) - \log _{10}(f_{\textsc {shmr}}(M_h)) }{\sqrt{2}\sigma _{{\log M_{*}}}} \right)\right].\quad\quad \end{eqnarray} \tag{ 14 }$$

Our model also has an additional five parameters that are necessary to model the satellite occupation function, 〈N_sat〉. For a set of galaxies more massive than threshold M^t_*, 〈N_sat〉 is

$$\begin{equation} \langle N_{\rm sat}(M_h|M_{*}^{t}) \rangle = \langle N_{\rm cen}(M_h|M_{*}^{t}) \rangle \left(\frac{M_h}{M_{\rm sat}}\right)^{\alpha _{\rm sat}} \exp \left(\frac{-M_{\rm cut}}{M_h}\right). \end{equation} \tag{ 15 }$$

The free parameters that determine satellite occupation as a function of stellar mass are β_sat, B_sat, β_cut, B_cut, and α_sat. The first two parameters, β_sat and B_sat, determine the amplitude of 〈N_sat〉. The second two parameters, β_cut and B_cut, set the scale of the exponential cutoff. These parameters enter into 〈N_sat〉 as

$$\begin{equation} \frac{M_{\rm sat}}{10^{12}\, M_{\odot }}= B_{\rm sat} \left(\frac{f^{-1}_{\textsc {shmr}}(M_\ast ^{t_1})}{10^{12}\, M_{\odot }}\right)^{\beta _{\rm sat}}, \end{equation} \tag{ 16 }$$

and

$$\begin{equation} \frac{M_{\rm cut}}{10^{12}\, M_{\odot }}= B_{\rm cut} \left(\frac{f^{-1}_{\textsc {shmr}}(M_\ast ^{t_1})}{10^{12}\, M_{\odot }}\right)^{\beta _{\rm cut}}. \end{equation} \tag{ 17 }$$

Finally, α_sat represents the power-law slope of the satellite mean occupation function. We set α_sat = 1 for all samples which should be a good choice because the theoretical expectation is that the number of sub-halos above a given mass scales linearly with halo mass (Kravtsov et al. 2004; Conroy et al. 2006; Moster et al. 2010; Tinker et al. 2010). Results from group catalogs also indicate that α_sat ∼ 1 (Collister & Lahav 2005; Yang et al. 2009). Previous HOD analyses of clustering results at varying redshifts also vary little from a value of unity (Zehavi et al. 2005, 2011b; Zheng et al. 2007; van den Bosch et al. 2007; Tinker et al. 2007).

In total, our model contains ten free parameters and one fixed parameter (α_sat). A summary and description of these parameters can be found in Table 4 and also in Paper I.

Table 4. Parameters in Model

Parameter	Unit	Description	〈Ncen〉 or 〈Nsat〉	Free/Fixed
M1	M☉	Characteristic halo mass in the SHMR	〈Ncen〉	Free
M*, 0	M☉	Characteristic stellar mass in the SHMR	〈Ncen〉	Free
β	None	Low-mass slope in the SHMR	〈Ncen〉	Free
δ	None	Controls high-mass slope in the SHMR	〈Ncen〉	Free
γ	None	Controls the transition regime in the SHMR	〈Ncen〉	Free
$\sigma _{{\log M_{*}}}$	dex	Log-normal scatter in stellar mass at fixed halo mass	〈Ncen〉	Free
βsat	None	Slope of the scaling of Msat	〈Nsat〉	Free
Bsat	None	Normalization of the scaling of Msat	〈Nsat〉	Free
βcut	None	Slope of the scaling of Mcut	〈Nsat〉	Free
Bcut	None	Normalization of the scaling of Mcut	〈Nsat〉	Free
αsat	None	Power-law slope of the satellite occupation function	〈Nsat〉	Fixed at 1

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The COSMOS survey covers a relatively small volume and therefore sample variance effects must be taken into account. The volumes probed in the three redshift bins are given in Table 1 and vary from 0.88 × 10⁶ Mpc³ for the ACS region in z₁ (0.22 < z < 0.48) to 4.39 × 10⁶ Mpc³ for the Subaru region in z₃ (0.74 < z < 1.0). The volumes sampled by COSMOS are too small to obtain an accurate estimate of the sample variance from the data itself.

A series of mock catalogs are used to calculate the covariance matrices for all three observables. Details regarding the construction of these mocks can be found in Paper I. Briefly, COSMOS-like mocks are created from a single simulation (named "Consuelo") 420 h⁻¹ Mpc on a side, resolved with 1400³ particles, and a particle mass of 1.87 × 10⁹ h⁻¹M_☉.²³ This simulation can robustly resolve halos with masses above ∼10¹¹ h⁻¹M_☉ and is part of the Las Damas suite²⁴ (J. McBride et al., in preparation). We create mocks for the three redshift intervals: z₁ = [0.22, 0.48], z₂ = [0.48, 0.74], and z₃ = [0.74, 1]. For each redshift interval, we construct a series of mocks created from random lines of sight through the simulation volume that have the same area as COSMOS and the same comoving length for the given redshift slice. This yields 405 independent mocks for the z₁ bin, 172 mocks for the z₂ bin, and 109 mocks for the z₃ bin. For each redshift bin, mocks are created from the simulation output at the median redshift of the bin (see Table 1).

We converge on the mocks used for estimating errors using an iterative method. To begin with, we find an initial best-fit model to the data, without using any covariance matrices. This initial fit is used to populate the mocks and to create a first set of covariance matrices. We then re-fit the data using these covariance matrices and use the best-fit HOD models to create our final covariance matrices. All results presented here use this final set of covariance matrices.

In order to fit the model to the data, we minimize

$$\begin{equation} \chi ^2_{\rm tot} = \chi ^2_{\Phi _{\rm SMF}}+\sum _i\chi ^2_{\Delta \Sigma,i}+\sum _j\chi ^2_{w(\theta),j}, \end{equation} \tag{ 18 }$$

where the sum over i and j indicates summation over the different stellar mass bins and thresholds, respectively. For the SMF and each stellar mass sample in w(θ) and ΔΣ, χ² is calculated by

$$\begin{equation} \chi ^2 = \sum _{n,l}\left(x_n-y_n\right)C^{-1}_{nl}\left(x_l-y_l\right), \end{equation} \tag{ 19 }$$

where x_n is the model calculation of the quantity for data point n, y_n is the measurement, and C⁻¹ is the inversion of the covariance matrix.

To obtain the posterior probability distributions for the parameter set (M₁, M_{*, 0}, β, δ, γ, $\sigma _{\log M_{*}}$, β_sat, B_sat, β_cut, B_cut), we implement a Markov Chain Monte Carlo (MCMC) algorithm as follows.

• 1.  
  We sample the region of interest around the best fit by initializing all chains close to the minimum χ² solution with varying initial random spreads.
• 2.  
  We apply conservative limits on all 10 parameters. As shown in Figure 8, the recovered posterior distributions are independent of these limits.
• 3.  
  The covariance matrix is updated for the first 3000 elements in the chain, and then held fixed for the remainder of the chain.
• 4.  
  We run six chains per case, with ∼ 40,000 steps. We discard all elements before the covariance matrix is constant (the first 3000), leaving ∼37,000 elements per chain.
• 5.  
  We use the GetDist package provided with CosmoMC (Lewis & Bridle 2002) for computing convergence diagnostics. The chains are tested for convergence and mixing with the Gelman–Rubin criterion (Gelman & Rubin 1992; Gilks et al. 1996). We impose a limit on the worst R − 1 statistic of R − 1 < 0.01 and observe that for all cases under study the mean and marginalized posterior distributions are well constrained.
• 6.  
  We use GetDist output statistics to estimate confidence limits on all parameters.

Figures 5–7 show the best-fit models for each of the three redshift bins for the galaxy clustering, the g–g lensing, and the SMF. The upper panels show the angular correlation function w(θ) for stellar mass threshold samples. The middle right panel shows the COSMOS SMF which is measured for all galaxies with M_* > 10^8.7 M_☉ for z₁, M_* > 10^9.3 M_☉ for z₂, and M_* > 10^9.8 M_☉ for z₃. The dotted blue line shows the SMF of satellite galaxies from our model. The lower panels show the g–g lensing signals for the stellar mass bins defined in Table 3. When looking at Figures 5–7, one must keep in mind that the data points are correlated (see Paper I for the covariance matrices) and so it is difficult to evaluate "by eye" whether or not the fits are adequate. The reduced χ² for the fits are 1.7, 1.6, and 1.9 for z₁, z₂, and z₃, respectively.

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For our z₁ sample, we have compared the COSMOS mass function with previously published mass functions from SDSS (Li & White 2009; Baldry et al. 2008; Panter et al. 2007). Because of the use of photometric redshifts in COSMOS, we expect a larger stellar mass measurement error which should cause Eddington bias and lead to an inflated observed SMF at the high-mass end in COSMOS compared to SDSS. A more in-depth comparison of the mass functions is discussed further in Section 5.6.

Table 5 gives the best-fit values from GetDist for all 10 parameters and for the three redshift bins. Figure 8 shows the one-dimensional and two-dimensional joint marginalized constraints on the parameters for the z₂ bin. All of the parameters have well-behaved, uni-modal distributions. In the interest of brevity, we have not included equivalent figures for z₁ and z₃ but they are similar to Figure 8. All parameters are well constrained in the three redshift bins.

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Table 5. Best-fit Parameters for the Three Redshift Bins

Parameter	z1 = [0.22, 0.48]	z2 = [0.48, 0.74]	z3 = [0.74, 1]
sig_mod1
log10(M1)	12.520 ± 0.037	12.725 ± 0.032	12.722 ± 0.027
log10(M*, 0)	10.916 ± 0.020	11.038 ± 0.019	11.100 ± 0.018
β	0.457 ± 0.009	0.466 ± 0.009	0.470 ± 0.008
δ	0.566 ± 0.086	0.61 ± 0.13	0.393 ± 0.088
γ	1.53 ± 0.18	1.95 ± 0.25	2.51 ± 0.25
$\sigma _{{\log M_{*}}}$	0.206 ± 0.031	0.249 ± 0.019	0.227 ± 0.020
Bcut	1.47 ± 0.73	1.65 ± 0.65	2.46 ± 0.53
Bsat	10.62 ± 0.87	9.04 ± 0.81	8.72 ± 0.53
βcut	−0.13 ± 0.28	0.59 ± 0.28	0.57 ± 0.20
βsat	0.859 ± 0.038	0.740 ± 0.059	0.863 ± 0.053
sig_mod2
log10(M1)	12.518 ± 0.038	12.724 ± 0.033	12.726 ± 0.028
log10(M*, 0)	10.917 ± 0.020	11.038 ± 0.019	11.100 ± 0.017
β	0.456 ± 0.009	0.466 ± 0.009	0.470 ± 0.008
δ	0.582 ± 0.083	0.62 ± 0.12	0.47 ± 0.10
γ	1.48 ± 0.17	1.93 ± 0.25	2.38 ± 0.24
$\sigma _{{\log M_{*}}}^i$a	0.192 ± 0.031	0.245 ± 0.019	0.220 ± 0.019
Bcut	1.52 ± 0.79	1.69 ± 0.65	2.57 ± 0.56
Bsat	10.69 ± 0.89	9.01 ± 0.81	8.66 ± 0.53
βcut	−0.11 ± 0.29	0.60 ± 0.27	0.58 ± 0.20
βsat	0.860 ± 0.039	0.740 ± 0.059	0.863 ± 0.053

Note.^a In the sig_mod2 case we fit for $\sigma _{\log M_{*}}^i$ whereas in the sig_mod1 case we fit for $\sigma _{{\log M_{*}}}$.

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Table 5 lists the best-fit values for sig_mod1 where we have assumed that $\sigma _{\log M_*}$is constant and for sig_mod2 where we have explicitly accounted for stellar mass-dependent errors. We find no strong difference in our results, regardless of which model we adopt. We conclude that our results are robust to the effects of mass-dependent scatter. In the sig_mod2 case, we model $\sigma _{{\log M_{*}}}^{\rm m}$ and assume that $\sigma _{{\log M_{*}}}$ is the sum in quadrature of $\sigma _{{\log M_{*}}}^{\rm i}$ (which is assumed to be constant) and $\sigma _{{\log M_{*}}}^{\rm m}$. Therefore, the quantity that we actually fit for in the sig_mod2 case is $\sigma _{\log M_{*}}^{\rm i}$ (whereas in sig_mod1 we fit for $\sigma _{{\log M_{*}}}$). This is why, as expected, the best-fit scatter in Table 5 is slightly lower for sig_mod2 compared to sig_mod1. Note that we are not claiming to actually extract meaningful values for the intrinsic scatter in stellar mass at fixed halo mass. To do so would require a more thorough error analysis which is beyond the scope of this paper. Overall, our conclusions regarding $\sigma _{{\log M_{*}}}$ are twofold. First, we can safely assume that $\sigma _{{\log M_{*}}}$ is constant and ignore any mass-dependent effects induced by measurement error. This is due to the fact that all three observables are primarily sensitive to the effects of $\sigma _{{\log M_{*}}}$ at large M_* where the slope of the SHMR increases sharply. Similar conclusions were reached by B10 who find that the effects of scatter are insignificant below M_* = 10^10.5 M_☉ where the slope of the SHMR is not steep enough to have a significant impact. Second, we find that $\sigma _{{\log M_{*}}} \sim 0.23 \pm 0.03$ dex, in broad agreement with previous results. For example, B10 estimate that $\sigma _{{\log M_{*}}}^{m}=0.07$ dex and $\sigma _{\log M_{*}}^{i}=0.16$ dex, resulting in a total scatter of $\sigma _{{\log M_{*}}}=0.175$ dex (their total scatter is lower than ours as expected because we have a larger $\sigma _{{\log M_{*}}}^{{\rm m}}$ component).

Figure 9 shows the measured redshift evolution for all 10 parameters. Previous work on this topic has been strongly limited by systematic differences in stellar mass estimates between low- and high-z surveys. The main factors that contribute to this systematic uncertainty arise from the choice of an IMF, a stellar population synthesis (SPS) model, a dust model, and a population history model (estimates for the magnitude of each effect can be found in B10). According to B10, the total systematic uncertainty of stellar mass estimates (excluding IMF uncertainties) is of order 0.25 dex. In addition, the choice of an IMF results in another 0.25 dex uncertainty. We stress that the results in this paper have been derived in a homogeneous fashion from high to low redshift. Our conclusions regarding redshift evolution should thus be more robust than those from work that combine results from distinct low- and high-z surveys. While we believe that a homogeneous analysis globally reduces the systematic errors associated with redshift trends, we should note that our results might still be affected at some level by redshift-dependence systematic errors in stellar mass estimators. These could be caused, for example, by a redshift evolution of the IMF. Currently, however, the magnitude of such redshift-dependence systematic errors is very poorly known. For this reason, we do not discuss such errors further in this paper, however, this is clearly an issue that requires further investigation.

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The most striking result from Figure 9 is the redshift evolution in the two parameters M₁ and M_{*, 0}. This is one of the major results of this paper which we will discuss in more detail in the following section. Apart from these two parameters, there is marginal evidence for some evolution in γ and B_sat. No strong evolution is detected in the remaining six parameters. It is interesting to note that β (which controls the low-mass slope of the SHMR) remains constant at β ∼ 0.46. We will provide an interpretation of this result in the discussion section.

Figure 10 shows the best-fit SHMR for z₁ compared to a variety of low-redshift constraints from weak lensing (Mandelbaum et al. 2006b; Leauthaud et al. 2010; Hoekstra 2007), abundance matching (Moster et al. 2010; Behroozi et al. 2010), satellite kinematics (Conroy et al. 2007; More et al. 2010), and the Tully–Fisher relation (Geha et al. 2006; Pizagno 2006; Springob et al. 2005; Blanton et al. 2008) (see Section 5.5 for a more in-depth comparison). Most of the data are in broad agreement and show clear evidence for a variation in the dark-to-stellar mass ratio with a minimum of M_h/M_* ∼ 27 at M_* ∼ 4.5 × 10¹⁰ M_☉ and M_h ∼ 1.2 × 10¹² M_☉. As demonstrated by B10, however, meaningful and detailed comparisons between various data sets are hampered by systematic uncertainties in stellar mass estimates. For this reason, we will mainly focus in what follows on the conclusions that can be drawn by inter-comparing the three COSMOS redshift bins.

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At low stellar mass, M_h scales as M_h∝(M_*)^0.46 and this scaling does not evolve significantly with redshift from z = 0.2 to z = 1. At high stellar mass, the SHMR rises sharply at M_* > 10^10.5 M_☉ as a result of which $\sigma _{\log M_h}$(the scatter in halo mass at fixed stellar mass) also increases and M_* is clearly no longer a good tracer of halo mass. For example, a galaxy with M_* ∼ 10^11.3 M_☉ may be the central galaxy of group with M_h ∼ 10¹³–10¹⁴ M_☉ or may also be the central galaxy of a cluster with M_h > 10¹⁵ M_☉.

A quantity that is of particular interest is the mass at which the ratio M_h/M_* reaches a minimum. This minimum is noteworthy for models of galaxy formation because it marks the mass at which the accumulated stellar growth of the central galaxy has been the most efficient. We describe the SHMR at this minimum in terms of the "pivot stellar mass," M^piv_*, the "pivot halo mass," M^piv_h, and the "pivot ratio," (M_h/M_*)^piv. Note that M^piv_* and M^piv_h are not simply equal to M₁ and M_{*, 0}. Indeed, the mathematical formulation of the SHMR is such that the pivot masses depend on all five parameters. The three parameters that have the strongest effect on the pivot masses are M₁, M_{*, 0}, and γ (see Paper I).

Figure 11 shows the redshift evolution of the SHMR. Three points are of particular interest in this figure. First, we detect no strong redshift evolution in the low-mass slope of the SHMR (M_* < 10^10.2 M_☉). Indeed, as highlighted in the previous section already, β is remarkably constant out to z = 1. In Paper I, we have shown that β regulates the low-mass slope of the SMF so in other terms, we could also state that the faint end slope of the SMF shows remarkably little redshift evolution. We do however find that the amplitude of the low-mass (M_* < 10^10.2M_☉) SHMR was higher at earlier times so that galaxies at fixed stellar mass live in more massive halos at earlier epochs. We will return to this result in the discussion section.

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Second, we detect a strong redshift evolution in M^piv_* and M^piv_h which is shown more explicitly in Figure 12. The detected evolution is such that both the halo mass and the stellar mass for which the accumulated stellar growth of the central galaxy has been the most efficient are smaller at late times than at earlier times. This trend is a manifestation of at least one meaning of the term "downsizing" (Cowie et al. 1996; Brinchmann & Ellis 2000; Juneau et al. 2005). Originally, the term downsizing referred to the notion that the maximum K-band luminosity of galaxies above a specific star formation rate (SSFR) threshold decreases with time (Cowie et al. 1996). Since then, downsizing has been widely employed to more generally describe the behavior in which a certain parameter that regulates galaxy formation decreases with time (for recent discussions on downsizing see Fontanot et al. 2009 and Conroy & Wechsler 2009). Our results show strong evidence for downsizing in both the pivot halo mass and the pivot stellar mass. We have already remarked in the previous section that a strong evolution in M₁ and M_{*, 0} is seen in Figure 9. Although these two parameters are not strictly equal to M^piv_* and M^piv_h, they do have a strong impact on the location of the pivot masses. Thus, the evolution seen in M₁ and M_{*, 0} is directly related to the observed downsizing behavior in the pivot masses that is apparent in Figures 11 and 12. The pivot stellar mass evolves from M^piv_* = 5.75 ± 0.13 × 10¹⁰ M_☉ at z = 0.88 to M^piv_* = 3.55  ±  0.17 × 10¹⁰ M_☉ at z = 0.37 with an evolution detected at 10σ. We note that all errors have been derived by marginalizing over all other parameters.

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The evolution in M^piv_* varies smoothly in the three redshift bins, however the evolution in M^piv_h is less smooth, in particular in the z₂ bin. We suggest that M^piv_h is more sensitive to sample variance than M^piv_*. Indeed, the first-order effect of sample variance is to change the normalization of the SMF (see Paper I); this will directly affect M₁ and thus M^piv_h. In summary: M^piv_h is sensitive to sample variance between redshift bins whereas M^piv_* is sensitive to systematic errors in stellar mass measurements between redshift bins.

Finally, at high masses (M_* > 10¹¹ M_☉) there is an interesting hint that the amplitude of the SHMR is decreasing at higher redshifts, but we lack the statistics for a clear detection, mainly due to the small volume probed by COSMOS.

Figure 10 compares our z₁ SHMR to previous work on this topic at low redshift. The general picture that emerges from Figure 10 is one of remarkable broad agreement between various methods on the overall shape of the SHMR. In detail, however, meaningful comparison between various surveys is severely limited by systematic differences in stellar mass estimates (∼0.25 dex between different surveys). For this reason, we mainly focus on qualitative comparisons in this section. All results have been adjusted to our assumed value of H₀ = 72 km s⁻¹ Mpc⁻¹ and unless stated otherwise, halo masses are converted to M_200b assuming a Navarro–Frenk–White (NFW) profile and a Muñoz-Cuartas et al. (2011) mass–concentration relation for a WMAP5 cosmology when necessary. All results quoted here assume either a Kroupa or a Chabrier IMF. Since the systematic shift in M_* between these two IMFs is small (∼0.05 dex), we do not adjust for this difference. We also do not attempt to correct for differences in the assumed cosmological model.

5.5.1. Comparison with Previous Work: Low Redshift

5.5.2. Comparison with Previous Work: High Redshift

5.5.3. Comparison with Moster et al.

5.5.4. Comparison with Behroozi et al.

5.5.5. Testing the Redshift Evolution of the Pivot Masses

Given the striking downsizing signal that we detect for the pivot masses, we would like to check that this result is not an artifact of the method we are using. As such we have also re-analyzed the three COSMOS SMFs using the abundance matching method of B10. In this test, the method followed here is identical to that of B10 but with different constraints on the SMF. Specifically, we use the three cosmos SMFs to constrain the redshift evolution of the SHMR and do not include any SDSS data. The results are shown in Figure 13. We find that the downsizing signal for M^piv_h and M^piv_* is clearly detected in the COSMOS data using the methods of B10. This provides an independent test on our detected evolution of the pivot quantities and suggests that the detected downsizing signal is robust to the methodology that is employed.

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This test also raises the interesting question of how the method used in this paper (an HOD-based model that includes fits to clustering and g–g lensing) compares to the method of B10 (abundance matching using only the SMF). As can be seen in Figure 13, we find that the two methods yield very similar results for the pivot quantities. We do however find subtle differences in the actual SHMRs between the two methods. Tracking down the cause of the exact differences, although a very interesting question in itself, is beyond the scope of this paper and we defer this study to follow-up work. For the purposes of this paper, we will simply emphasize that the analysis of B10 applied to the COSMOS results fully agrees with our claims concerning the evolution of the pivot quantities.

In our analysis, the errors on the SMF are small compared to the clustering and the lensing. It is always the case that a measurement of a one-point statistic from a given set of data is more precise than a measurement of a two-point (or higher) statistic. Thus, the SMF plays an important role in constraining our parameter set. Therefore, we investigate the SMF in further detail in this section, and in particular, we show a more in-depth comparison with SDSS mass functions.

Figure 14 shows the COSMOS mass functions compared to various SDSS mass functions that have been commonly employed in the literature (Panter et al. 2007; Baldry et al. 2008; Li & White 2009). The main difference that we may expect between the COSMOS mass functions and the SDSS ones (besides sample variance and systematic error) is that the high end of the mass function may be inflated due to a larger value of $\sigma _{{\log M_{*}}}$ in COSMOS. To gauge how much the COSMOS mass functions are affected by Eddington bias compared to SDSS, we use our model to predict the COSMOS mass functions, de-convolved to the expected scatter for SDSS ($\sigma _{{\log M_{*}}}\sim 0.17$ dex). The results are shown in the right-hand panel of Figure 14. We find that the difference in scatter is not significant enough to explain the differences between the COSMOS and SDSS mass functions. It is more likely that the differences are due, for example, to varying assumptions regarding stellar population and dust models.

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The difference between COSMOS and Li & White (2009) corresponds roughly "by eye" to a "left/right" shift along the X-axis (log₁₀(M^li_*) ∼ log₁₀(M^cosmos_*) − 0.2). This difference is within the estimated systematic uncertainties (0.25 dex according to B10). However, this type of systematic shift will be reflected directly in the SHMR (Figures 10 and 11) by a "left/right" shift along the X-axis and will also affect the normalization of M^piv_* and (M_h/M_*)^piv. This ∼0.2 dex shift would bring the normalization of B10 into closer agreement with our results. Tracking down the exact source of this systematic shift is beyond the scope of this paper but it could be associated with differences in the assumed dust model²⁶ for example.

We conclude that in order to use the low-z SDSS data as a z ∼ 0 anchor to study the redshift evolution of the SHMR, a homogeneous analysis of both the SDSS and the COSMOS data is critical. This will be the focus of a future paper.

Figure 15 shows the conditional SMFs for various halo masses and redshifts from our best-fit model. We can use these functions to calculate the total amount of stellar material locked up in galaxies as a function of halo mass, noted hereafter M^tot_* (see Equation (16) in Paper I). Investigating M^tot_* is of interest because it reveals the efficiency with which dark matter halos accumulate stellar mass from the combined effects of in situ star formation and accretion via merging.

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One might worry that calculating the total amount of stellar material locked up in satellite galaxies requires extrapolating our model beyond the lower and upper stellar mass bounds for which our model has been calibrated. As discussed in Section 5.5.1, the extrapolation of our model is in good agreement with results from Blanton et al. (2008) at low M_* and with Hoekstra (2007) at high M_*. Thus, to first order, this extrapolation does not appear unreasonable. Let us therefore make the assumption that the extrapolation of our SHMR is not wildly incorrect. We will now investigate which mass range of satellite galaxies contributes most to M^tot_*.

From Figure 15, it is clear that satellite galaxies are a sub- dominant component of the total stellar mass at M_h = 10¹²M_☉. Our present concern is therefore only relevant for M_h > 10¹² M_☉. Let us consider the total stellar mass associated with satellite galaxies as a function of M_h in a fixed stellar mass bin: M^{tot, sat}_*(M_h|M^t1_*, M_*^t2). As shown in Paper I, the expression for M^{tot, sat}_*(M_h|M^t1_*, M_*^t2) is given by

$$\begin{equation} M_{*}^{\rm tot,sat}(M_h|M_*^{t1},M_*^{t2})=\int _{M_*^{t1}}^{M_*^{t2}}\Phi _s(M_*|M_h)M_*{\rm d}M_*. \end{equation} \tag{ 20 }$$

We have tested how M^{tot, sat}_*(M_h|M^t1_*, M_*^t2) varies with the integral limits, M^t1_* and M^t2_*. We find that at fixed halo mass, most of the stellar mass associated with satellite galaxies arises from a relatively narrow range in stellar mass. In particular, for halos with M_h > 10¹¹ M_☉, the bulk of M^{tot, sat}_* is built from satellite galaxies in the range 10¹⁰ M_☉ < M_* < 10¹¹ M_☉. Therefore, provided that the extrapolation of our model is not wildly incorrect, the bulk of M^{tot, sat}_* arises from satellites that are within the tested limits of our model.

Having underlined this caveat, we have calculated M^tot_* using the best-fit parameters for each of the three redshift bins and the results are shown in Figure 16. This figure will be discussed in detail in the following section.

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Figure 16 separates the stellar content of the average dark matter halo into a contribution from the central galaxy and a contribution from the sum of satellite galaxies. Central galaxies show an M_*/M_h ratio that rises steeply to a maximum at M_h ∼ 10¹² M_☉ before decreasing somewhat more gradually in halos of higher mass. The fact that halos above this mass scale (at the redshifts considered) have cooling times longer than their dynamical times has been invoked by modelers for some time to help explain why cooling and star formation shut down at the highest masses, with some refinement due to the presence of so-called cold-mode accretion (Birnboim & Dekel 2003; Kereš et al. 2005; Birnboim et al. 2007; Cattaneo et al. 2006). We will return to the evolution of this mass scale at a later point in the discussion.

Central galaxies strongly dominate the total stellar mass content at M_h ≲ 2 × 10¹³ M_☉ ("the central dominated regime"), including at the peak mass, M_h ∼ 10¹² M_☉, while the stellar mass in satellites dominates at M_h ≳ 2 × 10¹³ M_☉ ("the satellite dominated regime").

The transition between the two regimes is driven by the steep decline in M^cen_*/M_h at M_h > 10¹² M_☉. This decline occurs as the contribution from satellites begins to rise. One might then naturally ask if central galaxies in group-scale halos experience stunted growth simply because stellar mass is accumulating within the halo in the form of satellite galaxies, instead of merging onto the central galaxy. Figure 16 reveals that this is not the case. Indeed, the solid line in this figure demonstrates that the total stellar mass fraction of halos declines at M_h > 10¹²M_☉. Thus, even if all satellite galaxies were allowed to rapidly coalesce at the center of the potential well, the central galaxies of group-scale halos would still have lower M_*/M_h ratios than those in halos of M_h ∼ 10¹² M_☉. Thus, we conclude that dark matter halos globally decline in the efficiency by which they accumulate stellar mass at M_h > 10¹² M_☉.

We note that in massive halos, the intra-cluster light (ICL; not accounted for in this analysis) is estimated to contribute an additional 20%–30% of the total stellar mass (Feldmeier et al. 2004; Zibetti et al. 2005; Gonzalez et al. 2005; Krick et al. 2006). Adding this to the satellite component in Figure 16 does little to bridge the factor of two to four gap in M^tot_*/M_h between the satellite component in high-mass halos and centrals at the peak mass.

The majority of the total mass in an average dark matter halo is built from halo–halo mergers with mass ratios above 1:10 (e.g., Hopkins et al. 2010a). At halo masses below 10¹²M_☉, the steep rise in M^tot_*/M_h with M_h implies that the typical stellar mass ratio of galaxy mergers will be less than the typical mass ratio of the dark matter halos hosting these galaxies. In other words, major halo mergers are minor galaxy mergers in this regime. Thus, the accumulation of stellar mass through the effects of merging will be limited compared to the growth in total mass of such halos. The steep rise of M^tot_*/M_h must therefore reflect the greater importance of star formation at masses below M_h ∼ 10¹² M_☉ over assembly from galaxy mergers (Bundy et al. 2009). Similar conclusions have also been reached by Conroy & Wechsler (2009) (see their Figures 2 and 3 in particular).

Simple arguments suggest, however, that once halos grow past the pivot mass and in the absence of significant star formation, M^tot_*/M_h should dip below the peak value since these halos can only grow by merging with halos with lower values of M^tot_*/M_h. At slightly higher mass, the decline in M^tot_*/M_h now means that stellar mass ratios are enhanced with respect to halo mass ratios, and the trend must reverse again. This repeating pattern should cause a flattening of M^tot_*/M_h above the pivot mass. While this behavior is certainly apparent in Figure 16, our different redshift bins also reveal that at fixed mass among high-mass halos (M_h > 4 × 10¹³), the total stellar mass content declines at later epochs. We speculate that this trend could arise from the smooth accretion of dark matter, which brings no new stellar mass, and amounts to as much as 40% of the growth of dark matter halos (e.g., Fakhouri & Ma 2010). One way to test this hypothesis would be to populate a z = 0.88 N-body simulation with our z₃ HOD and evolve the subhalo and halo populations to z = 0, assuming no star formation. This would reveal the amount of stars that are acquired through mergers in this redshift range (Zentner et al. 2005). We note that the destruction of satellites and a growing ICL component could also contribute to this trend given that the ICL at z = 0 could make up 20%–30% of the stellar content of massive halos, roughly the amount by which M^tot_*/M_h declines over our redshift range.

The location at which halos reach their maximum accumulated stellar mass efficiency is encoded by the pivot mass quantities, M^piv_*, M^piv_h, and (M_h/M_*)^piv. While we observe downsizing trends for both M^piv_* and M^piv_h, the co-evolution of these two parameters leaves (M_h/M_*)^piv constant with redshift. This can be seen in the right panel of Figure 16: the pivot ratio evolves very little over our redshift range, while the mass scale of the peak (M^piv_h) does evolve downward by nearly a factor of two. Given the low satellite content of halos at these masses, merging is not likely to play a dominant role in driving growth in M_* below the pivot peak, arguing instead that the regulation of star formation is key to understanding this behavior. The physical process that sets the pivot peak at M_h = 10¹² M_☉, and drives the subsequent decline in M^cen_*/M_h at higher halo masses, must be linked to the shutdown of star formation in central galaxies. In the reminder of this discussion, we will focus on interpreting the downsizing behavior of the pivot quantities in this context.

A complete and comprehensive interpretation of our results requires modeling and accounting for dark matter accretion histories, galaxy merger rates, and star formation rates (SFRs) as a function of redshift (for example, see Conroy & Wechsler 2009). Nonetheless, we will introduce some toy models based on simple arguments to provide a first interpretation of our results. Our goal here is to evaluate our results in the context of other observations and theoretical work on galaxy formation models and to set the stage for a more detailed treatment in subsequent work. We begin with a general treatment of evolution in the SHMR and then will focus on applying this treatment to interpret the evolution we observe in the pivot quantities.

The physical basis for evolution in M_*/M_h versus M_h must be considered carefully because the stellar mass and associated halo mass of a galaxy can evolve independently, depending on the mass scale involved and on processes including merging, smooth (diffuse) dark matter accretion, star formation, and even tidal stripping. The sum of these processes on the growth of M_h and M_* shapes the behavior of M_*/M_h versus M_h in different ways, as illustrated by the schematic diagram in Figure 17. Here we let η₁ represent the ratio M_*/M_h at redshift z_high and η₀ represent the equivalent ratio at redshift z_low with z_high > z_low. We further consider a dark matter halo of mass M_h that has grown by a relative factor of $\lambda _{M_h}$ from z_high to z_low. We can write that $M_h(z_{\rm low})=\lambda _{M_h}\times M_h(z_{\rm high})$. Characterizing growth in the stellar mass of the central galaxy (although similar arguments apply to M^tot_*) by a factor of $\lambda _{M_*}$, we can simply write that

$$\begin{equation} \lambda _{M_*} = \frac{\eta _0}{\eta _1}\times \lambda _{M_h}. \end{equation} \tag{ 21 }$$

If η₀ > η₁, we infer that $\lambda _{M_*}>\lambda _{M_h}$ and that the stellar mass has experienced a stronger relative amount of growth compared to that of the dark matter from z_high to z_low. If on the contrary, η₀ < η₁ then the relative growth of the stellar mass is less than the dark matter. Note that this schematic view demonstrates that identical curves for M_*/M_h versus M_h at different redshifts do not necessarily indicate a lack of evolution, since $\lambda _{M_h}$ is always greater than zero.

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We now apply this intuitive framework to the evolution observed in Figure 16. Considering values of η₁ and η₀ applied to the total stellar mass curves in Figure 16, we see that below M_h = 10¹², the fractional growth in stellar mass outweighs the growth in halo mass. This reflects the greater importance of star formation at low masses over assembly from galaxy mergers, the same conclusion reached above by simply considering the shape of the SHMR. This evolutionary trend reverses above M_h = 10¹² M_☉, consistent with the notion that star formation is largely shut down in centrals above this mass.

We now apply these simple arguments to the evolution in the pivot quantities. We focus only on central galaxies, neglecting the minor contribution from satellites near the pivot mass. The aim here is to explore several simple models for the quenching of star formation and to investigate which models might reproduce the observed evolution of the pivot quantities, namely, a pivot halo and stellar mass that decrease at later epochs (downsizing) but leave the pivot ratio constant.

Shown schematically in Figure 18, we consider how our high-redshift (z = 0.88) SHMR relation would evolve toward lower redshifts under several prescriptions for stellar and halo growth. We begin with no assumptions about the SFR but adopt a halo growth rate ($\lambda _{M_h}$) that is roughly constant over the mass range spanned by the peak. This assumption is well justified by dark matter mass accretion rates derived from N-body simulations (Wechsler et al. 2002; McBride et al. 2009; Fakhouri et al. 2010). For example, Fakhouri et al. (2010) find that $\dot{M_h}/M_h$ is only weakly dependent on halo mass with $\dot{M_h}/M_h\propto M_h^{0.1}$.

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We now consider several different quenching models and investigate their impact on the redshift evolution of the pivot quantities.

• 1.  
  No quenching model. To begin with, we consider a model with no quenching of star formation and in which the stellar growth rate ($\lambda _{M_*}$) is constant over the halo mass range spanned by the peak (Row A in Figure 18). We consider the redshift evolution of the SHMR for three values of $\lambda _{M_*}$ defined with respect to $\lambda _{M_h}$. In all three cases, this model leads to an increase in M^piv_h with time (contrary to what we observe). We can therefore conclude that $\lambda _{M_*}$ must vary with M_h, not a surprise given the expectation that the SFR shuts down above M_h = 10¹²M_☉.
• 2.  
  Fixed halo mass for quenching. We next consider a model in Row B in which star formation is quenched as galaxies cross above a fixed halo mass, M_q. However, this instantaneous quenching model fails to reproduce (not surprisingly) the downward evolution of M^piv_h and also yields evolution in the pivot ratio, which is not detected.
• 3.  
  Redshift-dependent halo mass for quenching. Row C shows a model in which M_q shifts downward with time. This model leads to downsizing in the pivot halo mass if $\lambda _{M_*}>\lambda _{M_h}$ below M_q (Row C, middle panel). However, in order to keep the pivot ratio fixed in this scenario, the growth rate, $\lambda _{M_*}$, must be tuned with respect to the rate at which M_q declines. This would require a fortuitous coincidence, but obviously cannot be dismissed as an explanation.
• 4.  
  Critical M_*/M_h ratio for quenching. We finally consider an alternative scenario in which star formation is limited by a critical mass ratio, η_crit ≡ M_*/M_h ≈ 0.04 (Row E). In this model, we can qualitatively reproduce our main results, including the downsizing trends in the pivot stellar and halo mass, and, by construction, a constant pivot ratio set by η_crit (compare the middle panel of Row D to the middle panel of Row E). However, in order to produce downsizing behavior in this model, $\lambda _{M_*}$ must be larger than $\lambda _{M_h}$ below η_crit. It is important to note that if $\lambda _{M_*}\le \lambda _{M_h}$ below η_crit, this model would fail to produce downsizing.

The model explored in Row E seems a promising and simple mechanism that can explain the observed evolution of the pivot quantities. We now test if observations are consistent with the requirement that $\lambda _{M_*}>\lambda _{M_h}$ below M_h ∼ 10¹² M_☉. A halo of mass M_h ∼ 2 × 10¹¹ M_☉ grows by a factor of $\lambda _{M_h}\sim 1.4$ from z = 0.88 to z = 0.37 (Fakhouri et al. 2010). SFRs as a function of M_* and redshift have recently been measured by Noeske et al. (2007), Cowie & Barger (2008), and Gilbank et al. (2010). These measurements indicate that galaxies of mass M_* ∼ 10¹⁰ M_☉ grow by roughly a factor of $\lambda _{M_*}=2\hbox{--}3$ from z = 0.88 to z = 0.37. Therefore, at z < 1 and for M_h < M^piv_h, current estimates for halo growth coupled with estimates for stellar growth are indeed consistent with our observation that stellar mass has experienced a stronger relative amount of growth compared to that of the dark matter ($\lambda _{M_h}<\lambda _{M_*}$ below η_crit).

We further note that the relatively weak dependence of $\lambda _{M_*}$ on M_h required to maintain a constant slope in M_*/M_h versus M_h with redshift (again for M_h < M^piv_h) is implied by the weak SSFR–M_* relation observed for galaxies with M_* < M^piv_* (a typical power-law fit gives SSFR ∼M^0.4_*). This could explain why β (the low-mass slope of the SHMR) is observed to remain remarkably constant at β = 0.46 at z < 1.

Given the simple arguments outlined above, we argue that the fact that the pivot ratio remains constant may suggest that star formation is fundamentally limited by a critical mass ratio, η_crit ≡ M_*/M_h ≈ 0.04. A very elegant and compelling consequence of this model is that the observed downsizing trends in M^piv_* and M^piv_h can be automatically explained given the observation that $\lambda _{M_*}>\lambda _{M_h}$ below M_h ∼ 10¹² M_☉. Previous work by Bundy et al. (2006) on the evolution of the galaxy SMF has found a similar downsizing trend for the "transition mass," M_tr, which is defined in terms of M_* by the declining fraction of star-forming galaxies at the highest masses. The transition mass from Bundy et al. (2006) evolves from M_tr ∼ 9 × 10¹⁰M_☉ at z ∼ 0.9 to M_tr ∼ 5 × 10¹⁰M_☉ at z ∼ 0.6. This transition mass is similar to M^piv_*, reinforcing the notion that the pivot mass marks the end of rapid star formation among halos. The obvious difference between Bundy et al. (2006) and this paper is that our work adds a key missing ingredient, which is the evolution of the pivot halo mass and the pivot ratio. If the quenching of star formation depends on a critical M_*/M_h ratio then the fact that low-mass galaxies grow more rapidly than dark matter below the pivot scale provides a simple explanation for the observed downsizing in the sites of star formation observed by studies such as Bundy et al. (2006).

In the previous section, we demonstrated that a constant pivot ratio provides important clues concerning the physical mechanisms that quench star formation. We now discuss possible mechanisms that might tie quenching to M_*/M_h.

The notion of a fixed maximum stellar-to-dark matter ratio, η_crit, has been relatively unexplored in the literature. Theoretical arguments tend to favor a relatively fixed (if broad) critical halo mass (see Birnboim & Dekel 2003) at z ≲ 1, with significant modifications from cold-mode accretion occurring mostly at higher redshifts. But, quenching at fixed halo mass alone is not sufficient to reproduce the local SMF and red-sequence fraction, and also fails from simple arguments to produce downsizing in the pivot masses, as shown by Row B of Figure 18. As a result, most semi-analytic models include a quenching channel initiated by mergers or disk instabilities among galaxies in halos below the critical halo mass threshold (e.g., Bower et al. 2006; Croton et al. 2006; Cattaneo et al. 2006). Above the critical halo mass, once a quasi-static halo of hot gas has formed, low-luminosity feedback (i.e., radio-mode AGN feedback) is often invoked to prevent cooling and star formation in massive halos at late times. Using simple arguments, we have shown that mergers do not provide a likely explanation for determining the scale of the pivot mass. We therefore focus on AGN feedback and disk instabilities as possible mechanisms.

It is common practice to consider low-luminosity AGN feedback only in halos above a fixed halo mass. However, in practice, efficient AGN feedback is more complex and requires at least two ingredients. First, a quasi-static halo of hot gas must exist to which the AGN jets can couple. However, AGN feedback also requires a sufficiently large black hole to produce jets powerful enough to initiate this coupling. Given that gas cooling rates scale roughly with halo mass and that black hole mass scales roughly with galaxy mass, via the M_*–σ relation (Gebhardt et al. 2000; Ferrarese & Merritt 2000), it is reasonable to speculate that AGN feedback efficiency would depend on M_*/M_h.

Considering the galaxy population broadly, this scenario requires a sufficiently large bulge component for AGN quenching to be effective. While bulges may be built stochastically in galaxy mergers, a further link may tie secular bulge formation via disk instabilities to the value of M_*/M_h, thereby cementing the relationship between quenching and η_crit. It has been shown that disks become unstable to bar modes if the disk mass dominates the gravitational potential (Efstathiou et al. 1982; Mo et al. 1998). Semi-analytic models typically consider that disk instabilities occur if

$$\begin{equation} V_{\rm max}/(G_N M_{\rm disk}/r_{\rm disk})^{0.5} \leqslant 1, \end{equation} \tag{ 22 }$$

where M_disk represents disk mass, r_disk is the disk radius, and V_max is the maximum of the rotation curve. Depending on the implementation of this criterion, V_max may be equated either to the halo virial velocity or the disk velocity at its half-mass radius (see discussion in Parry et al. 2009). An instability will cause either a partial or a total collapse of the disk, leading to a burst of star formation at the center, the formation of a spheroid, and also possibly fueling the central black hole. The disk instability criterion in Equation (22) shows a dependence on (M_h/M_disk) (relating V_max to M_h) which could be reflected in M_h/M_* as the instability converts cold gas into stars. Disk instabilities might therefore play a role in setting the pivot masses and enforcing η_crit. The coincident fueling of the central black hole during the instability may help initiate AGN quenching and regulate the global decline in M^cen_*/M_h beyond the pivot mass.

Assuming the disk instability framework, we can test whether the predicted sizes of stable galactic disks given our critical pivot ratio, η_crit ≡ M_*/M_h ≈ 0.04, are consistent with observations. Approximating the halo virial velocity for V_max, we rewrite Equation (22) to derive the maximum size of stable disks:

$$\begin{equation} r_{\rm disk} < R_h \times (M_*/M_h). \end{equation} \tag{ 23 }$$

Applying this criterion to the pivot masses yields the condition that r_disk < 8 kpc. Interestingly, this condition is very well satisfied by the observed size distributions of disk galaxies which tend to fall off rapidly just below r_disk = 8 kpc at both z ∼ 0 (see Figure 11 in Shen et al. 2003) and at z  >  0 (see Figure 10 in Sargent et al. 2007). We conclude that, via the growth of bulges and initialization of AGN feedback, disk instabilities provide a promising link between our observed critical pivot ratio and quenching, although further investigation is clearly needed.

Finally, we note that η_crit might also be related to the competition between the cooling and accretion of cold gas in central starbursts and the resulting feedback from either star formation itself or a co-evolving quasar-mode AGN. This Eddington-like limit has been explored in the context of stellar systems by Hopkins et al. (2010b) who derive a maximum stellar surface density from simple arguments. In the context of dark matter halos explored here, similar arguments might naturally yield a fixed value for η_crit similar to that obtained by our analysis.