iHOD quenching model: separating red and blue galaxies

Mapping stellar content to dark matter halos. II. Halo mass is the main driver of galaxy quenching

Ying Zu and Rachel Mandelbaum
McWilliams Center for Cosmology, Department of Physics, Carnegie Mellon University, 5000 Forbes Avenue, Pittsburgh, PA 15213, USA
E-mail: yzu@cmu.edu
Accepted XXX. Received YYY; in original form ZZZ

We develop a simple yet comprehensive method to distinguish the underlying drivers of galaxy quenching, using the clustering and galaxy-galaxy lensing of red and blue galaxies in SDSS. Building on the iHOD framework developed by zm15, we consider two quenching scenarios: 1) a “halo” quenching model in which halo mass is the sole driver for turning off star formation in both centrals and satellites; and 2) a “hybrid” quenching model in which the quenched fraction of galaxies depends on their stellar mass while the satellite quenching has an extra dependence on halo mass. The two best-fit models describe the red galaxy clustering and lensing equally well, but halo quenching provides significantly better fits to the blue galaxies above . The halo quenching model also correctly predicts the average halo mass of the red and blue centrals, showing excellent agreement with the direct weak lensing measurements of locally brightest galaxies. Models in which quenching is not tied to halo mass, including an age-matching model in which galaxy colour depends on halo age at fixed , fail to reproduce the observed halo mass for massive blue centrals. We find similar critical halo masses responsible for the quenching of centrals and satellites  (), hinting at a uniform quenching mechanism for both, e.g., the virial shock-heating of infalling gas. The success of the iHOD halo quenching model provides strong evidence that the physical mechanism that quenches star formation in galaxies is tied principally to the masses of their dark matter halos rather than the properties of their stellar components.

cosmology: observations — cosmology: large-scale structure of Universe — galaxies: luminosity function, mass function — gravitational lensing: weak — methods: statistical
pubyear: 2015pagerange: Mapping stellar content to dark matter halos. II. Halo mass is the main driver of galaxy quenchingMapping stellar content to dark matter halos. II. Halo mass is the main driver of galaxy quenching\savesymbol

iint \restoresymbolTXFiint

1 Introduction

The quenching of galaxies, namely, the relatively abrupt shutdown of star formation activities, gives rise to two distinctive populations of quiescent and active galaxies, most notably manifested in the strong bimodality of galaxy colours (strateva2001; baldry2006). The underlying driver of quenching, whether it be stellar mass, halo mass, or environment, should produce an equally distinct split in the spatial clustering and weak gravitational lensing between the red and blue galaxies. Recently, zm15 developed a powerful statistical framework, called the iHOD model, to interpret the spatial clustering (i.e., the projected galaxy autocorrelation function ) and the galaxy-galaxy (g-g) lensing (i.e., the projected surface density contrast ) of the overall galaxy population in the Sloan Digital Sky Survey (SDSS; york2000), while establishing a robust mapping between the observed distribution of stellar mass to that of the underlying dark matter halos. In this paper, by introducing two empirically-motivated and physically-meaningful quenching models within iHOD, we hope to robustly identify the dominant driver of galaxy quenching, while providing a self-consistent framework to explain the bimodality in the spatial distribution of galaxies.

Galaxies cease to form new stars and become quenched when there is no cold gas. Any physical process responsible for quenching has to operate in one of three following modes: 1) it heats up the gas to high temperatures and stops hot gas from cooling efficiently (e.g., gravitational collapse and various baryonic feedback; see benson2010, for a review); 2) it depletes the cold gas reservoir via secular stellar mass growth or sudden removal by external forces (e.g., tidal and ram pressure; gunn1972); and 3) it turns off gas supply by slowly shutting down accretion (e.g., strangulation; balogh2000). However, due to the enormous complexity in the formation history of individual galaxies, multiple quenching modes may play a role in the history of quiescent galaxies. Therefore, it is more promising to focus on the underlying physical driver of the average quenching process, which is eventually tied to either the dark matter mass of the host halos, the galaxy stellar mass, or the small/large-scale environment density that the galaxies reside in, hence the so-called “halo”, “stellar mass”, and “environment” quenching mechanisms, respectively.

Halo quenching has provided one of the most coherent quenching scenarios from the theoretical perspective. In halos above some critical mass (), virial shocks heat gas inflows from the intergalactic medium, preventing the accreted gas from directly fueling star formation (binney1977; birnboim2003; katz2003; binney2004; keres2005; keres2009). Additional heating from, e.g., the active galactic nuclei (AGNs) then maintains the gas coronae at high temperature (croton2006). For halos with , the incoming gas is never heated to the virial temperature due to rapid post-shock cooling, therefore penetrating the virial boundary into inner halos as cold flows. This picture, featuring a sharp switch from the efficient stellar mass buildup via filamentary cold flow into low mass halos, to the halt of star formation due to quasi-spherical hot-mode accretion in halos above , naturally explains the colour bimodality, particularly the paucity of galaxies transitioning from blue, star-forming galaxies to the red sequence of quiescent galaxies (cattaneo2006; dekel2006). To first order, halo quenching does not discriminate between centrals and satellites, as both are immersed in the same hot gas coronae that inhibits star formation. However, since the satellites generally lived in lower mass halos before their accretion and may have retained some cold gas after accretion, the dependence of satellite quenching on halo mass should have a softer transition across , unless the quenching by hot halos is instantaneous.

Observationally, by studying the dependence of the red galaxy fraction on stellar mass and galaxy environment  (i.e., using distance to the 5th nearest neighbour) in both the Sloan Digital Sky Survey (SDSS) and zCOSMOS, peng2010 found that can be empirically described by the product of two independent trends with and , suggesting that stellar mass and environment quenching are at play. By using a group catalogue constructed from the SDSS spectroscopic sample, peng2012 further argued that, while the stellar mass quenching is ubiquitous in both centrals and satellites, environment quenching mainly applies to the satellite galaxies.

However, despite the empirically robust trends revealed in P10, the interpretations for both the stellar mass and environment trends are obscured by the complex relation between the two observables and other physical quantities. In particular, since the observed of central galaxies is tightly correlated with halo mass  (with a scatter dex; see Paper I), a stellar mass trend of is almost indistinguishable with an underlying trend with halo mass. By examining the inter-relation among , , and , woo2013 found that the quenched fraction is more strongly correlated with at fixed than with at , and the satellite quenching by can be re-interpreted as halo quenching by taking into account the dependence of quenched fraction on the distances to the halo centres. The halo quenching interpretation of the stellar and environment quenching trends is further demonstrated by gabor2015, who implemented halo quenching in cosmological hydrodynamic simulations by triggering quenching in regions dominated by hot () gas. They reproduced a broad range of empirical trends detected in P10 and woo2013, suggesting that the halo mass remains the determining factor in the quenching of low-redshift galaxies.

Another alternative quenching model is the so-called “age-matching” prescription of hearin2013 and its recently updated version of hearin2014. Age-matching is an extension of the “subhalo abundance matching” (SHAM; conroy2006) technique, which assigns stellar masses to individual subhalos (including both main and subhalos) in the N-body simulations based on halo properties like the peak circular velocity (reddick2013). In practice, after assigning using SHAM, the age-matching method further matches the colours of galaxies at fixed to the ages of their matched halos, so that older halos host redder galaxies. In essence, the age-matching prescription effectively assumes a stellar mass quenching, as the colour assignment is done at fixed regardless of halo mass or environment, with a secondary quenching via halo formation time. Therefore, the age-matching quenching is very similar to the -dominated quenching of P10, except that the second variable is halo formation time rather than galaxy environment.

The key difference between the - and -dominated quenching scenarios lies in the way central galaxies become quiescent. One relies on the stellar mass while the other on the mass of the host halos, producing two very different sets of colour-segregated stellar-to-halo relations (SHMRs). At fixed halo mass, if stellar mass quenching dominates, the red centrals should have a higher average stellar mass than the blue centrals; in the halo quenching scenario the two coloured populations at fixed halo mass would have similar average stellar masses, but there is still a trend for massive galaxies to be red because higher mass halos host more massive galaxies. This difference in SHMRs directly translates to two distinctive ways the red and blue galaxies populate the underlying dark matter halos according to their and , hence two different spatial distributions of galaxy colours.

Therefore, by comparing the and predicted from each quenching model to the measurements from SDSS, we expect to robustly distinguish the two quenching scenarios. The iHOD framework we developed in Paper I is ideally suited for this task. The iHOD is a global “halo occupation distribution” (HOD) model defined on a 2D grid of and , which is crucial to modelling the segregation of red and blue galaxies in their distributions at fixed . The iHOD quenching constraint is fundamentally different and ultimately more meaningful compared to approaches in which colour-segregated populations are treated independently (e.g., tinker2013; puebla2015). Our iHOD quenching model automatically fulfills the consistency relation which requires that the sum of red and blue SHMRs is mathematically identical to the overall SHMR. More importantly, the iHOD quenching model employs only four additional parameters that are directly related to the average galaxy quenching, while most of the traditional approaches require additional parameters, rendering the interpretation of constraints difficult. Furthermore, the iHOD framework allows us to include more galaxies than the traditional HODs and take into account the incompleteness of stellar mass samples in a self-consistent manner.

This paper is organized as follows. We describe the selection of red and blue samples in Section 2. In Section 3 we introduce the parameterisations of the two quenching models and derive the iHODs for each colour. We also briefly describe the signal measurement and model prediction in Sections 2 and  3, respectively, but refer readers to Paper I for more details. The constraints from both quenching mode analyses are presented in Section 4. We perform a thorough model comparison using two independent criteria in Section 5 and discover that halo quenching model is strongly favored by the data. In Section 6 we discuss the physical implications of the halo quenching model and compare it to other works in 7. We conclude by summarising our key findings in Section 8.

Throughout this paper and Paper I, we assume a cosmology with . All the length and mass units in this paper are scaled as if the Hubble constant were . In particular, all the separations are co-moving distances in units of either or , and the stellar mass and halo mass are in units of and , respectively. Unless otherwise noted, the halo mass is defined by , where is the corresponding halo radius within which the average density of the enclosed mass is times the mean matter density of the Universe, . For the sake of simplicity, is used for the natural logarithm, and is used for the base- logarithm.

2 Sample Selection and Signal Measurement

In this section we describe the SDSS data used in this paper, especially the selection of the red and blue galaxies within the stellar mass samples, and the measurements of the galaxy clustering and the g-g lensing signals. We briefly describe the overall large-scale structure sample and the signal measurement, same as that used in Paper I, below in Section 2.1 and 2.3, respectively, and refer readers to Paper I for details. Here we focus more on the colour cut we employ to divide the galaxies into red and blue populations in Section 2.2.

2.1 NYU–VAGC and Stellar Mass Samples

Figure 1: The colour–stellar mass diagram of the SDSS Main Galaxy Sample at four different redshifts (from left to right: , , , and ). The colour maps indicate the co-moving galaxy number density at fixed colour and , normalized by the stellar mass function at that . The red dashed lines are the same across all panels, indicating little redshift evolution in the locus of red sequence on the colour– diagram. The black solid lines are the redshift-independent colour cut that we use to divides the galaxies into red and blue populations.

We make use of the final data release of the SDSS (DR7; abazajian2009), which contains the completed data set of the SDSS-I and the SDSS-II. In particular, we obtain the Main Galaxy Sample (MGS) data from the dr72 large–scale structure sample bright0 of the “New York University Value Added Catalogue” (NYU–VAGC), constructed as described in blanton2005. The bright0 sample includes galaxies with , where is the -band Petrosian apparent magnitude, corrected for Galactic extinction. We apply the “nearest-neighbour” scheme to correct for the galaxies that are without redshift due to fibre collision, and use data exclusively within the contiguous area in the North Galactic Cap and regions with angular completeness greater than . The final sample used for the galaxy clustering analysis includes galaxies over a sky area of deg. A further 5 per cent of the area is eliminated for the lensing analysis, due to the absence of source galaxies in that area.

As discussed in Paper I, we further restrict our analysis to galaxies above a “mixture limit”, defined as the stellar mass threshold above which the galaxy sample is relatively complete with a fair mix of red and blue galaxies. The functional form we adopt to describe the mixture limit is


shown as the thick yellow curves in Fig. 3 (discussed further below). By taking into account the sample incompleteness in a self-consistent way, the iHOD model is able to model the lensing and clustering statistics of all galaxies above the mixture limit, more than the traditional HOD models typically include from the same catalogue.

We employ the stellar mass estimates from the latest MPA/JHU value-added galaxy catalogue111http://home.strw.leidenuniv.nl/~jarle/SDSS/. The stellar masses were estimated based on fits to the SDSS photometry following the philosophy of kauffmann2003 and salim2007, and assuming the Chabrier (chabrier2003) initial mass function (IMF) and the bruzual2003 SPS model. The MPA/JHU stellar mass catalogue is then matched to the NYU-VAGC bright0 sample. We identify valid, unambiguous MPA/JHU stellar mass estimates for all but  () of the MGS galaxies. For those unmatched galaxies, we predict their stellar masses using the overall scaling between the two stellar mass estimates, depending on the colours (k-corrected to ).

As sources for the g-g lensing measurement, we use a catalogue of background galaxies (2012MNRAS.425.2610R) with a number density of 1.2 arcmin with weak lensing shears estimated using the re-Gaussianization method (2003MNRAS.343..459H) and photometric redshifts from Zurich Extragalactic Bayesian Redshift Analyzer (ZEBRA, 2006MNRAS.372..565F). The catalogue was characterised in several papers that describe the data, and use both the data and simulations to estimate systematic errors (see 2012MNRAS.425.2610R; 2012MNRAS.420.1518M; 2012MNRAS.420.3240N; mandelbaum2013).

2.2 Separating Sample into Red and Blue

Figure 2: Illustration of the agreement between using colour and star formation rate (SFR) as quenching indicators. Each panel shows the joint probability density distributions (PDFs) of the colour and the total star formation rate of red and blue galaxies at four different stellar masses (from left to right: , , , and ). Gray horizontal thick and vertical thin lines indicate the colour and the SFR cuts that divide the galaxies into red/blue and passive/active, respectively. The PDFs of red and blue galaxies, each normalized to unity, are represented by two separate sets of contour maps with their levels indicated by the corresponding colour bars on the right. The red galaxy samples include a very small fraction of dusty star-forming galaxies, with a contamination rate of depending on stellar mass (marked on the top right of each panel).

We define quenching by the colour (after K-correction to ) for three reasons: 1) colour bimodality is very stable across different environments and redshifts (baldry2006); 2) observationally colour is very easy to measure robustly, without the need to fit galaxy morphology or brightness profile; and 3) physically colour is the result of integrated star formation history, largely immune to incidental or minor star formation episodes. In addition, we aim to model and compare to the two separable quenching trends with stellar mass and environment that revealed in P10, who originally chose optical colour as the quenching indicator.

Figure 3: Selection of the red (left) and the blue (right) galaxy samples on the –redshift diagram. In each panel, the color map indicates the logarithmic number counts of galaxies at fixed and redshift. The thick yellow curve is the “mixture limit” above which the completeness of the galaxy samples is relatively high. By selecting almost all galaxies above this mixture limit, our iHOD analysis is able to include 84% more than galaxies than the traditional HOD methods.

Fig. 1 illustrates the colour-stellar mass diagrams (CSMDs) at four different redshifts and the stellar mass dependence of the colour cuts we applied to divide the red and blue galaxies. In each panel, the colour map indicates the distribution of the logarithmic comoving number density of galaxies in cells of and , normalized by the stellar mass function (SMF) at that . This normalization highlights the ranges with relatively high concentration of galaxies along the colour axis, enhancing the appearance of the “red sequence” on each panel. The CSMDs are cut off at different stellar masses due to the redshift dependence of the mixture limit, which is ultimately related to the flux limit of the spectroscopic survey. The red dashed lines going through the red sequences are uniform across all redshifts, indicating that the loci of the red sequence on the CSMD is independent of redshift within our sample. To divide the galaxies into red and blue, we therefore define the colour cut to be parallel to the red sequence on the CSMD, described by


and indicated by the black solid lines in Fig. 1. The weak stellar mass dependence in equation 2 causes a variation of between and within our sample, leading to differences in classification between blue vs. red of only a few per cent compared to a constant cut at .

Whether a galaxy is quiescent or star-forming, however, is never a clear-cut choice. Galaxy bimodality shows in nearly every aspect of galaxy properties, including broad-band colour, star formation rate (SFR), morphology (e.g., late/early-type, De Vaucouleurs/exponential profile), and concentration (e.g., Sérsic index). bernardi2010 found that many late-type (Sb and later) galaxies lie above the red galaxy colour cut (similar to equation 2) and they tend to be edge-on discs reddened by dust. Conversely, some early-type galaxies lie below the cut, either showing star-forming AGN or post-starburst spectrum, with their star formation history well described by a recent minor and short starburst superimposed on old stellar component (huang2009). As a result, woo2013 advocated the use of SFR as the quenching indicator, and they claimed that one third of the red galaxies are star forming. However, the large fraction of star-forming contaminants in the “red” population in woo2013 is mainly caused by the rest-frame colour woo2013 adopted in defining red galaxies, derived from magnitudes that are K-corrected from the SDSS photometry. Using the native colour largely eliminates the star-formers from the red galaxies.

Fig. 2 clearly demonstrates the good consistency between using colour and SFR as quenching indicators. The four panels show the joint 2D probability density distributions (PDFs) of galaxy colour and the logarithmic SFR at four different  (from left to right: , , , and ). In each panel, the thick horizontal line represents the colour cut defined in equation 2, while the thin vertical line indicates the SFR value that saddles the separate SFR distributions of passive and active galaxies, which can be well described by


in parallel to the star-forming sequence defined in salim2007. The fraction of dusty, star-forming galaxies in the red population decreases from to as stellar mass increases from to , significantly lower than the one third reported in woo2013 using colours. Therefore, we conclude that it is robust to use red fraction as a proxy for quenching efficiency, and the results of our analysis should stay the same if SFR were used for the selection of quenched galaxies.

As described in Paper I, the iHOD model constructs individual HODs within very narrow redshift slices (we use ), so that the sample selection does not require a single uniform stellar mass range among all the redshift slices within that sample, i.e., having a rectangular shape on the - diagrams. Fig. 3 illustrates the galaxy samples selected on the - diagram within each coloured population for the iHOD quenching analysis. The colour intensity represents the logarithmic galaxy number counts in cells of and . As mentioned in Section 2.1, all selected samples have the “wedge”-like stellar mass thresholds described by the mixture limit, and thus contain extra galaxies at the far end of the redshift range that are usually unused in traditional HOD analysis. Additionally, since those high redshift wedges have a larger comoving volume per unit redshift than the low redshifts, they include the most abundant regions on the - diagram, corresponding to the reddest regions on both panels of Fig. 3. The resultant increase in the selected galaxy sample sizes is more than compared to traditional selections.

Above the mixture limit, the red galaxies (left panel) are two times more abundant than the blue galaxies (right panel), despite the fact that the ratio of the two colours is close to unity in the spectroscopic survey. Therefore, we can afford finer binning in in the red galaxy sample than in the blue galaxy sample, especially at the high mass end. Table 1 summarises the basic information of the two sets of sample selections used by the iHOD quenching analysis. In total, we divide red galaxies and blue galaxies into eight and six subsamples, respectively.

8.59.4 0.010.04
9.49.8 0.020.06
9.810.2 0.020.09
10.210.6 0.020.13
10.611.0 0.040.18
11.011.2 0.080.22
11.211.4 0.080.26
11.412.0 0.080.30
11.012.0 0.080.30
Table 1: Red and blue stellar mass bins used for the iHOD quenching analysis, corresponding to the selections in the left and right panels of Figure 3, respectively. The red selection includes three stellar mass samples above as opposed to the single sample in the blue selection, while the five lower stellar mass red and blue samples share the same binning in and .

2.3 Measuring Galaxy Clustering and Galaxy-Galaxy Lensing

We measure the projected correlation function for each galaxy sample by integrating the 2D redshift–space correlation function ,


where and are the projected and the line-of-sight (LOS) comoving distances between two galaxies. We measure the signal out to a maximum projected distance of , where the galaxy bias is approximately linear. For the integration limit, we adopt a maximum LOS distance of . We only use the values down to the physical distance that corresponds to the fibre radius at the maximum redshift of each sample, with fewer data points for higher stellar mass (hence larger maximum fibre radius) samples.

The Landy–Szalay estimator (landy1993) is employed for computing the 2D correlation The error covariance matrix for each measurement is estimated via the jackknife re-sampling technique. We divide the entire footprint into spatially contiguous, roughly equal–size patches on the sky and compute the for each of the jackknife subsamples by leaving out one patch at a time. For each stellar mass sample, we adopt the sample mean of the subsample measurements as our final estimate of , and the sample covariance matrix as an approximate to the underlying error covariance.

For the surface density contrast , we measure the projected mass density in each radial bin by summing over lens-source pairs “” and random lens-source pairs “”,


where is the tangential ellipticity component of the source galaxy with respect to the lens position, the factor of converts our definition of ellipticity to the tangential shear , and is the inverse variance weight assigned to each lens-source pair (including shot noise and measurement error terms in the variance). is the so-called critical surface mass density, defined as


where and are the angular diameter distances to lens and source, and is the distance between them. We use the estimated photometric redshift each source to compute and . The factor of comes from our use of comoving coordinates. We subtract off a similar signal measured around random lenses, to subtract off any coherent systematic shear contributions (2005MNRAS.361.1287M); this signal is statistically consistent with zero for all scales used in this work. Finally, we correct a bias in the signal caused by the uncertainties in the photometric redshift using the method from 2012MNRAS.420.3240N.

To calculate the error bars, we also used the jackknife re-sampling method. As shown in 2005MNRAS.361.1287M, internal estimators of error bars (in that case, bootstrap rather than jackknife) perform consistently with external estimators of errorbars for on small scales due to its being dominated by shape noise.

3 Quenching Models and Signal Predictions

Figure 4: Illustration of the difference between hybrid (top row) and halo (bottom row) quenching models. For each quenching model, we show the red galaxy fraction as a function of both stellar mass and halo mass for the central (left) and satellite (right) populations, respectively, predicted from the best-fit parameters. The long arrow in each panel indicates the direction of increasing red fraction, driven by either stellar mass (top left), halo mass (bottom left and right), or the combination of both (top right) within the two quenching models.

In this section, we introduce the mathematical descriptions of the hybrid (Section 3.1) and the halo (Section 3.2) quenching models. We also briefly describe how to infer the iHODs for the red and blue galaxies in Section 3.3, but refer reader to Paper I for more details on the iHOD framework. The prediction of and from each coloured iHOD is rather complex but exactly the same as that for the overall galaxy populations, therefore we directly refer readers to the relevant sections (4 and 5) in Paper I for details. We ignore quenching via mergers in both quenching models considered below, as merging-induced quenching is negligible at  (peng2010).

3.1 Hybrid Quenching Model

The hybrid quenching model parameterizes the red fraction as a function of both and , aiming to mimic the empirical stellar mass and environment quenching trends observed in P10. In the physical picture implied by this model, every quiescent galaxy had spent some portion of its life on the star-forming “main sequencing” as a central before the eventual quenching (daddi2007; noeske2007; speagle2014), due to either the depletion of gas supply (i.e., stellar mass quenching) or entering another halo as a satellite, when environment quenching kicked in.

While the stellar mass trend is straightforward to parameterize, it is unclear whether the environment quenching trend among satellites can be mimicked by a trend in halo mass, as the environment–halo mass relation is very complex and depends strongly on the definition of that environment. The P10 environment of satellites, as defined by , shows strong correlation with group richness when the richness is below five. In richer systems, however, the correlation is mostly smeared out and instead anti-correlates with the halo-centric distance  (peng2012). This apparent transition between the two richness regimes is caused by the increase of , the ratio between the typical distance to the fifth nearest neighbour to the halo virial radius, from below to above unity. When , is roughly proportional to , thus more tied to . When , is essentially an intra-halo overdensity measured at , which depends more strongly on than due to the steep declining slope of the NFW-like halo density profile.

But for our purposes, what matters is the mean satellite quenching efficiency as a function of halo mass , averaged over galaxies at all within that halo. As pointed out by woo2013, the density profile of more massive halos falls off less steeply with distance than that of less massive systems, so the probability of finding the -th nearest neighbour increases with halo mass. Therefore, the P10 environment quenching trend can be potentially encapsulated within the halo model as a satellite quenching dependence on halo mass.

Assuming stellar mass as the main driver of central galaxy quenching, we parameterize the red fraction of centrals as


where is a characteristic stellar mass () and dictates how fast the quenching efficiency increases with , with being exponential. The satellites are subject to an extra halo quenching term , so that




where is a characteristic halo mass and controls the pace of satellite quenching. The above equations, including and as powered exponential functions, are very similar to the fitting formula adopted in baldry2006 and peng2012.

The top left and right panels of Fig. 4 illustrate the central and satellite red fractions, computed from the best-fit hybrid quenching model via Equations 7 and 8, respectively. The arrow in each panel points in the direction of increasing quenching efficiency on the 2D plane of and . For the central galaxies, although the quenching is driven by along the horizontal axis, the red fraction still shows strong increasing trend with due to the tight correlation between and , i.e., the SHMR of central galaxies. The 2D distribution of satellite red fractions displays a “boxy” pattern, echoing the separate stellar mass and environment quenching trends detected in P10.

Combining the central and the satellite terms, the red fraction of galaxies with stellar mass inside halos of total mass is


where is the satellite fraction that can be predicted by the overall iHOD model from Paper I. For the hybrid model, equation 10 can be reduced to


3.2 The Halo Quenching Model

As described in the introduction, the halo quenching model relies on halo mass alone to quench both central and satellite galaxies, and gabor2015 demonstrated that it also naturally explains the stellar mass and environment quenching trends seen in P10, by embedding galaxies in massive halos filled with hot gas via virial heating. However, depending on the exact physical processes driven by , halo quenching may apply to the central and satellites differently. For instance, while the central galaxies in halos above could be quenched by shocked-heated gas and then maintain a high gas temperature via the “raido”-mode feedback from AGNs, the satellite galaxies in the those halos may still retain some cold gas as the “central” galaxies of its own coherent sub-group. Therefore, the satellite galaxies continue to accrete gas and convert it to stars over a period of Gyr after entering into a larger halo (simha2009). Similar processes like slow strangulation (assuming no accretion onto satellites) also produce prolonged quenching actions (peng2015). In this case, the halo quenching of centrals and satellites are somewhat decoupled, and the quenching of satellites is a more gradual process than that of centrals.

Therefore, unlike the hybrid model, we describe the red fractions of centrals and satellites as two independent functions of :




where and are the critical halo masses responsible for triggering quenching of central and satellites, respectively, and and are the respective powered-exponential indices controlling the transitional behavior of halo quenching across the critical halo masses.

Similarly, the bottom left and right panels of Fig. 4 illustrate the central and satellite red fractions, computed from the best-fit halo quenching model via Equations 12 and 13, respectively, with arrows indicating halo mass as the sole driver for quenching in both populations. The orthogonality of the hybrid and halo quenching directions for central galaxies is the key distinction that we look to exploit in this paper, by identifying its imprint on the clustering and g-g lensing signals of red and blue galaxies. The total red fraction can be obtained by substituting equations 12 and  13 into equation 10.

Finally, we emphasize that in reality the true quenching arrow could be pointing anywhere between the two orthogonal directions, i.e., a more generalized quenching model consisting of a linear mixture of the two, with the linear coefficients varying as functions of and as well — schematically,


However, as a first step of constraining quenching, the goal of this paper is to find out if is closer to zero (i.e., halo-quenching dominated) or unity (i.e, hybrid-quenching dominated).

Figure 5: The iHOD, i.e., the average number of galaxies per dex in stellar mass within halos at fixed mass of the red (left column) and blue (right column) populations, predicted from the best-fit hybrid (top row) and halo (bottom row) quenching models based on the same iHOD of the overall population. Black and white contour lines in each panel indicate the iHOD distributions for the central and satellite galaxies, respectively, while the colour contours show the iHOD for the total galaxy population. The two best-fit models produce similar overall patterns of galaxy occupation, with very subtle but important differences — compared to hybrid quenching, halo quenching produces stronger segregation between red and blue centrals in high vs. low mass halos.

3.3 From Quenching Models to Colour-segregated iHODs

In order to predict the and for the red and blue galaxies in each quenching model, we construct iHODs for both coloured populations by combining the overall iHOD with predicted by that quenching model.

Let us start with the red galaxies. The key is to derive , the 2D joint PDF of the red galaxies of stellar mass sitting in halos of mass , given the 2D PDF of the overall galaxy population inferred from Paper I,


where is the overall red faction of all galaxies, obtained via


As described in Paper I, iHOD predicts the and signals for a given galaxy sample by combining the predicted signals from individual narrow redshift slices, each of which is described by a single standard HOD.

For deriving standard HODs within redshift slices for the red galaxies, we need


while is the predicted parent (i.e., including observed and unobserved galaxies) SMF of the red galaxies normalized by their total number density ,


Finally, we arrive at the HOD of red galaxies at any redshift as


where is the observed SMF of red galaxies at redshift , directly accessible from the survey. For modelling the samples defined in Figure 3 for the iHOD analysis, we measure the observed galaxy SMF at each redshift, and then obtain the HOD for that redshift slice using Equation (19). In this way, we avoid the need to explicitly model the sample incompleteness as a function of and/or .

For the blue galaxies, we apply the same procedures above to obtain from , by substituting with in equations 1519.

Fig. 5 illustrates the two sets of coloured iHODs derived from the best-fit hybrid (top row) and halo (bottom row) quenching models. In each row, the left and right panels display , the average log-number of galaxies per dex in stellar mass within halos at fixed mass, for the red and blue populations, respectively. The white and black contour lines highlight the central and satellite galaxy occupations separately on the - plane. All panels reveal the same generic pattern, consisting of a tight sequence that corresponds to the SHMR of the central galaxies, and a cloud underneath occupied by the satellite galaxies. The level of similarity exhibited by the red galaxies is especially high between the two quenching models (left column).

However, comparing the left and right panels in the same row (i.e., red vs. blue galaxies in the same quenching model), the red centrals are more preferentially sitting in the high- and high- region than in the low- and low- one, while the opposite is true for the blue centrals. This segregation happens regardless of quenching models, confirming our notion that it is difficult to unambiguously disentangle the two quenching directions, despite their orthogonality, by merely examining the quenching trend with , or some surrogate of that has substantial scatter about the true  (e.g., group richness).

The satellites are quenched by in both models, but are also partially by in the hybrid model. Thus, there are more high- blue satellite galaxies within massive halos in the halo quenching model (bottom right panel) than in the hybrid model (top right panel). In addition, the low mass halos in the hybrid quenching model are more likely to host blue dwarf satellites than in the halo quenching model.

The two sets of iHOD models, presented in this Section and in Fig. 5, are the analytical foundation that allow us to predict the and signals as functions of the four parameters in each quenching model. Any difference shown in Fig. 5 between the two models will be propagated to the different behaviours in the final predictions of and , and is thus detectable by comparing the two sets of predictions to the measurements from SDSS galaxies.

4 Constraints on the Two Quenching Models

Parameter Description Uniform Prior Range prior case fixed case
Halo Quenching Model
Characteristic halo mass for central galaxy quenching [11.0, 15.5]
Pace of central galaxy quenching with halo mass [0.0, 3.0]
Characteristic halo mass for satellite galaxy quenching [11.0, 15.5]
Pace of satellite galaxy quenching with halo mass [0.0, 3.0]
Hybrid Quenching Model
Characteristic stellar mass for central and satellite quenching [9.0, 12.0]
Pace of galaxy quenching with stellar mass [0.0, 3.0]
Characteristic halo mass for satellite galaxy quenching [11.0, 15.5]
Pace of satellite galaxy quenching with halo mass [0.0, 3.0]
Table 2: Description, prior specifications, and posterior constraints of the parameters in the halo (top) and hyrbid (bottom) quenching models. All the priors are uniform distributions running across the entire range of possible values for the parameters, and the uncertainties are the confidence regions derived from the 1D posterior probability distributions.
Figure 6: Confidence regions from our halo-quenching model analysis of the clustering and lensing of the red and blue galaxies in the 2D planes that comprised of all the pair sets of the four quenching parameters. Histograms in the diagonal panels show 1D posterior distributions of individual parameters. Contour levels run through confidence limits of 95 per cent (light brown) and 68 per cent (dark brown) inwards. The filled contours show the constraints from our fiducial model where the iHOD parameters that describe the overall galaxy population are drawn from priors informed by the analysis in Paper I, while the open contours in the panels of the upper triangle are the constraints from a simpler model where the overall iHOD parameters are kept fixed at the best-fit values derived from Paper I.
Figure 7: Similar to Fig. 6, but for the hybrid quenching model.
Figure 8: Comparison between the measurements of clustering (top row) and lensing (bottom row) signals to the predictions by the two best-fit quenching models, for the eight red stellar mass samples (left column) and the six blue samples (right column). In each panel, the signals from bottom to top are in ascending order of the average stellar mass of the galaxy samples, each shifted from its adjacent sample by dex to avoid clutter. Points with errorbars are the measurements, while the solid and the dashed curves are the predictions from the best-fit halo and hybrid quenching models, respectively. The two best-fit models provide equally adequate fits to the measurements for the red galaxies, but predict different g-g lensing signals of high- blue galaxies (bottom right; the two high mass samples).
Figure 9: Comparison between the clustering (left) and lensing (right) signals predicted by the halo (solid) and hybrid (dashed) quenching models for the red (red), and blue (blue) galaxies. The thick gray curves indicate the best-fit iHOD model prediction for the overall galaxy population, with the stellar mass ranges marked on the right panel. The two quenching models generally predict similar scale and stellar mass dependences of the red and blue deviations from predictions of the overall population (comparing solid and dashed lines of the same colour), except for the high mass blue lenses (comparing the blue solid lines to the blue dashed lines in the right panel).
Figure 10: Decomposition of the clustering (top) and lensing (bottom) signals of the high mass blue galaxy sample (), by the best-fit hybrid (left) and halo (right) quenching models. In each panel, the green curve shows the predicted signal for the overall galaxy population at this stellar mass. The measurement and the predicted signal for blue galaxies are shown by the points with errorbars and the thick blue curve, respectively, with the reduced marked on the top. The predicted signal is then decomposed into different components involving separate contributions from centrals and satellites (see text for details). In particular, the “1-halo non-satellite” terms in the panels indicate the underlying dark matter density profiles of the host (sub)halos. The halo quenching model provides a much better fit to the and measurements by allowing substantially fewer massive halos to host these high- blue galaxies as centrals, and a higher satellite fraction among these massive blue galaxies.

4.1 Constraints of the Quenching Parameters

Ideally one would constrain both the iHOD parameters and the quenching parameters together, by simultaneously fitting to the and measurements of the overall, red, and blue galaxies. However, since the measurements of the overall population have the highest signal-to-noise ratio and the overall iHOD does not include quenching, it is conceptually more reasonable to adopt a two-step scheme. In the first step we constrain the iHOD parameters using only the measurements of the galaxy samples without dividing by colour (i.e., Paper I). In the second step, when constraining the quenching parameters, we either fix the best-fit iHOD parameters (i.e., the “fixed case”) or input the iHOD constraints from Paper I as priors (i.e., the “prior case”). In particular, for the prior case we draw the iHOD parameters from the joint prior distribution represented by the Markov Chain Monte Carlo (MCMC) samples derived in Paper I. For each quenching model, we adopt the results from the prior case as our fiducial constraint in the following analysis.

In addition to the powerful statistical features of the iHOD framework inherited from Paper I, our quenching analysis also adds two important advantages compared to the traditional HOD modelling of red and blue galaxies. Firstly, traditional HOD studies of red and blue galaxies treat the two populations independently, so that the total number of HOD parameters inevitably doubles compared to the modelling of the overall galaxy population (e.g., tinker2013; puebla2015). In our analysis, the red and blue populations are derived not from scratch, but by filtering the overall iHOD with the red fraction predicted by each quenching model, which is described by only four simple yet physically meaningful parameters. Our method also guarantees that the sum of the red and blue SHMRs is mathematically identical to the overall SHMR. Secondly, the traditional method usually parameterizes the red galaxy fraction as a 1D function of halo mass, while our method affords a 2D function of defined on the plane, which is crucial to the task of examining stellar mass as a potential driver for quenching.

For each quenching model, we infer the posterior probability distributions of the four model parameters from the and measurements of the eight red and six blue galaxy samples within a Bayesian framework. We model the combinatorial vector of the and the components of the red and blue galaxies as a multivariate Gaussian, which is fully specified by its mean vector () and covariance matrix (). The Gaussian likelihood is thus




in hybrid quenching, and


in halo quenching.

We adopt flat priors on the model parameters, with a uniform distribution over a broad interval that covers the entire possible range of each parameter (see the 2nd column of Table 2). The final covariance matrix is assembled by aligning the error matrices of and measured for individual coloured samples along the diagonal blocks of the full matrix. We ignore the weak covariance between and (with the covariance being weak due to the fact that is dominated by shape noise), and between any two measurements of the same type but for different stellar mass or coloured samples.

Fig. 6 presents a summary of the inferences from the halo quenching model analysis, showing the 1D posterior distribution for each of the four model parameters (diagonal panels), and the and confidence regions for all the parameter pairs (off–diagonal panels). In the panels of the lower triangle, we highlight the results from our fiducial model, i.e., the prior case, employing the iHOD parameter constraints from Paper I as priors. In each panel of the upper triangle, we compare the constraints from the fiducial analysis (filled contours) to that of the fixed case analysis, which keeps the iHOD parameters at their best-fit values derived from Paper I. The two analyses are consistent with each other, implying that the explanation of the red and blue signals does not require any modification in the description of the overall galaxy population. The two inferred characteristic halo mass scales are very similar to the critical shock heating mass scale, , while the two powered-exponential indices, and , indicate that the central and the satellite quenching transition differently across that shared characteristic halo mass. We defer the detailed discussion of the physical implications of the halo quenching constraints in Section 6. The confidence regions of the 1D posterior constraints are listed in Table 2.

Similarly, Fig. 7 presents the constraints on the hybrid quenching model. The critical stellar mass for quenching all galaxies is , echoing the characteristic stellar mass for downsizing at the low redshift. The stellar mass quenching index is slightly below unity, the value required for maintaining the observed redshift-independence of Schechter and faint-end slope of the star-forming galaxies in the stellar mass quenching formalism proposed in P10. The characteristic halo mass for the quenching of satellites is much higher than , albeit with a similar quenching index of .

4.2 Best-fit Model Predictions

Fig. 8 compares the clustering (top row) and g-g lensing (bottom row) signals measured from SDSS (points with errorbars) to those predicted by the best-fit halo (solid lines) and hybrid (dashed line) quenching models, for the eight red (left column) and the six blue (right column) stellar mass samples. In terms of the overall goodness-of-fit, the best-fit halo quenching model yields a of , while the hybrid quenching model has a worse of . The reduced values are thus and for the halo and hybrid quenching, respectively, both providing reasonable fits to the data, considering that the uncertainties in the measurements of the low- samples are under-estimated. We defer a discussion of the statistical significance of both best fits to the upcoming section.

For the red galaxy samples, the two quenching models predict very similar signals except for the two lowest stellar mass bins. Unfortunately the measurements in these two bins are severely affected by the underestimated cosmic variance due to the small volumes, with highly correlated uncertainties on all scales. Therefore, neither quenching model gives an adequate fit to their signals. The signals of the two lowest mass bins are less affected by cosmic variance because the measurement error is dominated by shape noise, and are thus better described by the two quenching model predictions.

The two quenching models also predict very similar signals for the blue galaxies, except for the high mass ones with . While both quenching models give adequate fits to the signals of these massive blue galaxies, the halo quenching model produces much better fit to their signals than the hybrid quenching model, driving most of the difference in the log-likelihoods (i.e., the values) of the two best-fit models. This difference revealed by the massive blue galaxies, as will be discussed further later, is the key to distinguishing the two quenching models.

Fig. 9 highlights the split between the red and blue galaxies from the overall population in the  (left) and  (right) signals, predicted by the two best-fit quenching models for the eight stellar mass bins marked in the right panel. In each panel, the thick gray curves are the iHOD predictions for the overall galaxy samples, which bifurcate into the thin red and blue curves, i.e., predictions for the red and blue galaxies. Solid and dashed line styles indicate the halo and hybrid quenching models, respectively. As seen in Fig. 8, the two quenching models predict very similar bifurcation signatures, except for the high mass bins where the hybrid quenching predicts a stronger large-scale bias, a weaker small-scale clustering strength, but a stronger small-scale g-g lensing amplitude, than the halo quenching for the blue galaxies. Unfortunately the measured signals for the high mass galaxies are cut off at small scales due to fibre collision, and the measurement uncertainties in the large-scale are not small enough to distinguish the two quenching predictions (top right panel of Fig. 8).

Therefore, the g-g lensing of the massive blue galaxies clearly provides the most discriminative information, as shown in the right panel of Fig. 9. For blue galaxies above , the halo quenching model predicts substantially lower weak lensing amplitudes than the hybrid model on all distance scales, and thus provides a much better fit to the measurements (see bottom right panel of Fig. 8).

To understand the discrepancy between the two quenching predictions for the massive blue galaxies, we show the decomposition of  (top row) and  (bottom row) signals predicted by the best-fit hybrid (left column) and halo (right column) quenching models for the blue sample in Fig. 10. In each panel, the data points with errorbars and the thick blue curve are the measured and predicted signals for the blue sample, while the iHOD prediction for the overall sample is shown by the thin green curve. The best-fit quenching model prediction is then decomposed into contributions from the 1-halo and 2-halo (thin dotted) terms. For the 1-halo term includes the contributions from centrel-satellite pairs (thin solid; “1-h c-s”) and satellite-satellite (thin dashed; “1-h s-s”) pairs; For the 1-halo term consists of a satellite term (thin dashed) and a non-satellite term (thin solid). We also include a point source stellar mass term in , which is model-independent and negligible on most of the relevant scales (not shown here). Most importantly, the 1-halo non-satellite term is directly related to the average dark matter density profile of the host halos (including both the main halos for centrals and the subhalos for satellites), and its amplitude is proportional to the average mass of those halos. The halo quenching model clearly provides a much better fit to the data than the hybrid model, with factors of two and four improvement in for and , respectively. In addition, Fig. 10 shows the crucial advantage of including g-g lensing in the joint analysis — since the best-fit hybrid quenching model adequately describes the galaxy clustering (), its deficiency would not be exposed unless we compare the predictions to data ().

Fig. 10 reveals two major differences between the two quenching model predictions: 1) the halo quenching model predicts a much higher satellite fraction among the massive blue galaxies than the hybrid model, hence the more prominent “1-halo satellite” terms; and 2) the average (sub)halo mass of those massive blue galaxies predicted by the halo quenching model is much lower compared to the hybrid model prediction, hence the lower g-g lensing amplitudes and better fit to the data. Roughly speaking, since the hybrid quenching model relies on the stellar mass to quench central galaxies, it tends to place central galaxies at fixed into similar halos regardless of their colours. However, in the halo quenching model any galaxies that are unquenched have to live in lower mass halos than their quenched counterparts with similar . In the section below we will argue that, the discrepancy between the average halo masses of the massive blue galaxies predicted by the two quenching models is insensitive to the details in the model parameters, therefore can be used as a robust feature for identifying the dominant quenching driver.

4.3 Origin of Host Halo Mass Segregation between Red and Blue Centrals

Figure 11: Comparison between the stellar-to-halo mass relations (SHMRs) derived by the hybrid (top) and halo quenching (bottom) models. On the left panel of each row, red solid, blue solid, and green dashed curves indicate the average logarithmic stellar mass of red, blue, and any central galaxies at fixed halo mass, respectively, predicted by the corresponding quenching model. The shaded band of each color shows the log-normal scatter about the average relation. The three colored ticks on the top indicates the three halo masses for which we show the central galaxy stellar mass distributions in the right panel. The integral of the distribution for each halo mass (shown in the legend) is the expected total number of central galaxies at that halo mass, which can be decomposed into the contributions from red (red shaded histogram) and blue (blue dashed histogram or blue shaded area) centrals. The halo quenching model produces exactly the same SHMRs for the two coloured populations as the overall relation, due to the lack of correlation between quenching and stellar mass at fixed halo mass, whereas the hybrid quenching model makes fractionally more red centrals at higher .

Comparison between the two best-fit predictions in Section 4.2 reveals that , the average host halo mass at fixed stellar mass (i.e., the mean halo-to-stellar mass relation; HSMR), is potentially the key discriminator of the two types of quenching models. In particular, by predicting a lower for the blue centrals, the halo quenching model provides a much better fit to the and signals of the massive blue galaxies than the hybrid quenching model. But before going any further, we need to understand the cause of this discrepancy between the two quenching models, especially to answer the following questions. Firstly, what is the origin of the host halo mass segregation between the two colours? Secondly, is the halo quenching necessary for predicting the strong segregation in between the red and blue centrals, and can the stellar mass quenching process produce an equally low halo mass for the massive blue centrals with a different ?

For red or blue central galaxies, the conversion from the mean SHMR (i.e., ) to its inverse relation, the HSMR, is highly non-trivial. Using the blue centrals as an example, the HSMR can be computed from




In the above equation, is the PDF of blue central galaxy stellar mass at fixed , determined by the mean SHMR of the blue centrals and its scatter, is the blue fraction of centrals, and is the halo mass function. Therefore, for given cosmology the HSMR of the blue central galaxies has two components, the blue central SHMR (both mean and scatter) and the blue fraction of centrals. To understand for both colours more quantitatively, we start by examining the red and blue SHMRs predicted by the two models.

The top and bottom panels in the left column of Fig. 11 show the mean SHMRs of the total, red, and blue central galaxies, predicted by the best-fit hybrid and halo quenching models, respectively. Coloured bands indicate the logarithmic scatters about the mean relations. The hybrid quenching model predicts a segregation in between the red and blue central galaxies at fixed halo mass, as the high galaxies are more likely to be quenched. The halo quenching model, however, predicts exactly the same SHMRs for all three populations, as galaxies at fixed halo mass are equally likely to be quenched regardless of stellar mass. The red and blue segregation in , or the lack thereof, is best illustrated in the two right panels of Fig. 11, using three halo masses as examples (, , ).

In each panel, the total filled area for each halo shows the stellar mass distribution of central galaxies in that halo. The width of the distribution decreases with halo mass due to the flattening of SHMR on the high mass end. Under each distribution, the red and blue shaded areas represent the contributions from the red and blue centrals, so that the sum of the red and blue SHMRs exactly recovers the total SHMR. In the hybrid quenching model for any given halo mass, the red galaxy distributions are shifted to higher compared to the blue distributions, which are indicated by the dashed histograms and are equivalent to the blue shaded regions. The halo quenching model produces zero such shift. The non-zero shift in the hybrid model drives the SHMR of the blue central galaxies to become shallower than that of the red centrals as seen in the top left panel of Fig. 11. Naively, one might think that this shift will also cause the blue centrals to reside in more massive halos than the red centrals if we simply compare the inverse functions of the two SHMRs — a shallower (blue) SHMR maps the same on the y-axis to a higher halo mass on the x-axis.

Figure 12: Left panel: blue galaxy fractions as functions of halo mass predicted by the two best-fit quenching models. Black and Blue thick solid curves are the blue fractions within centrals predicted by the best-fit halo and hybrid quenching models, respectively. Black thin curve indicates the blue satellite fraction predicted by the best-fit halo quenching model. Right panel: the magnitude of the logarithmic slopes of , characterised by in the two quenching models. The hybrid quenching model predicts a rapid decline of at low masses and then slows drastically at high masses, while the halo quenching model maintains a steady decline over all masses. The two dotted lines are the asymptotic behaviors of the hybrid model at very low and high masses, derived in equation 31. The slow decline of in the hybrid model over-predicts the fraction of blue centrals among massive halos.

However, a more careful inspection of the segregation patterns reveals a second, and much more important difference in the predicted fraction of blue galaxies among centrals — blue centrals persist in all halo masses in the hybrid quenching model, but barely show up in the halos in the halo quenching model. The left panel of Fig.12 illustrates the blue fractions as functions of predicted by the best-fit halo (thick black solid) and hybrid (thick blue dashed) quenching models. The amplitude of in hybrid quenching also depends on and the blue dashed curve is the average blue fraction over all galaxies above . While in the halo quenching case strictly follows the powered exponential form (i.e., equation 12), in the hybrid case it is affected by both the stellar mass quenching and the slope of the SHMR. We also show the blue fraction of satellites (thin solid) derived by the halo quenching analysis, which exhibits a slower decline with compared to that of centrals.

The blue galaxy fractions of centrals decline rapidly with halo mass in both quenching models, but the speed of decline varies differently as a function of halo mass between the two models. To investigate this quantitatively, we define the “central galaxy quenching rate” as a function of halo mass, , as the logarithmic rate at which the blue fraction declines with halo mass, , which is shown in the right panel of Fig.12 for each model. As expected, the halo quenching produces a steady increase of with


For the hybrid quenching case, experiences a rapid decline at low and then a gradual one at high . This shift in gear can be understood as follows. The central galaxy quenching rate depends on both the and the derivative of the SHMR, so that


Since also has a powered exponential form (see equation 7),


The slope of the SHMR is tightly constrained by Paper I, which described the SHMR as the inverse of


where , and are the characteristic stellar and halo mass that separate the behaviours in the low and high mass ends, and the remaining parameters control the running slopes of the SHMR. Assuming reasonable values of the slope parameters (i.e., , , ; see Paper I), equation 28 can be approximated by


Clearly, the SHMR is a steep power-law relation at the low mass end, with , whereas at the high mass end the slope of SHMR is very shallow, with .

Therefore, the slope of the SHMR is


Combining equations 26, 27, and 30, we arrive at


Assuming from the best-fit hybrid quenching model, we have


The above equation is shown as the dotted lines on the right panel of Fig. 12, roughly reproducing the two distinctive asymptotic behaviours of at the low and high mass ends. The actual slope of is steeper than predicted by equation 32 at high masses, where equation 30 becomes less accurate.

The comparison between and in Fig. 12 (i.e., equations 32 and  25) clearly reveals that, the halo quenching model does not quench central galaxies in the low mass halos as efficiently as the hybrid model, but by maintaining a steady quenching rate at the halo quenching model is able to quench almost all centrals in the very massive halos. The hybrid quenching model, however, is relatively inefficient to quench massive central galaxies in the very high mass halos. When calculating the HSMR using equation 24, this difference in completely dominates the effect due to the slight difference between the two coloured SHMRs. Therefore, the stellar mass quenching, due to its slow central galaxy quenching rate on the high mass end, is incapable of producing a strong segregation in the HSMR between the two colours. In order for the hybrid quenching model to mimic the steeper slope of , the stellar mass quenching trend would have to drop so precipitously that the abundance of blue galaxies is cut off beyond some maximum stellar mass, which is ruled out by the observed SMFs of blue galaxies (see Fig. 14). Therefore, we further emphasize that this slow quenching rate with halo mass in the hybrid model is caused by the changing slope of SHMR across , and is thus insensitive to the stellar mass quenching prescriptions, e.g., the value of .

To summarize the findings above using the quenching diagram of Fig. 4, the steep slope () of the SHMR below makes the SHMR more aligned with the quenching arrow along the -axis (i.e., stellar mass quenching, see top left panel of Fig. 4), causing progressively more galaxies to be quenched at higher halo mass. Above , however, the SHMR becomes shallower () and is almost perpendicular to the quenching arrow, leaving a substantial number of blue centrals in massive halos. As a result, the massive blue centrals are extremely scarce in the halos in the halo quenching model (bottom right panel of Fig. 11), but have a much stronger presence within those halos in the hybrid quenching model (top right panel of Fig. 11). By the same token, a strong segregation in the host halo mass between the red and blue centrals would points to the necessity of a dominant halo mass quenching for the central galaxies.

5 Comparing the Hybrid and Halo Quenching Models

Figure 13: The average weak lensing mass of the host halos of red and blue central galaxies as functions of stellar mass. Solid and dashed thick curves are the predictions from the halo and hybrid quenching models, respectively. The colored bands show the the 1- uncertainties about the two mean halo-to-stellar mass relations from the fiducial halo quenching model. The circles and squares with errorbars are the directly measured halo masses for the red and blue locally brightest galaxies (LBGs), respectively. The halo quenching model provides excellent agreement with both the red and blue LBG measurements, while the hybrid quenching model predicts a much higher halo mass for the blue LBGs.

In this section we perform a robust comparison between the two quenching models in two ways, an internal one based on Bayesian Information Criterion (BIC) described in Section 5.1, and an external one based on cross-validation (Section 5.2), which is motivated by the quenching impact on the average halo mass of the massive blue galaxies quantitatively explained in Section 4.3.

5.1 Internal Model Comparison: Bayesian Information Criterion

In Bayesian applications, pairwise comparisons between models and are often based on the Bayes factor , which is defined as the ratio of the posterior odds, , to the prior odds, . In our case, the bayes factor is


so that a above unity indicates the data favor halo quenching and a below points to hybrid quenching. However, in most practical settings (as is the case here) the prior odds are hard to set precisely, and model selection based on BIC is widely employed as a rough equivalent to selection based on Bayes factors. The BIC (a.k.a., “Schwarz information criterion”), is defined as


where is the maximum likelihood value, is the number of parameters, and is the number of data points. kass95 argued that in the limit of large  ( in our analyses),


i.e., can be viewed as a rough approximation to , so that indicates that is favored (strongly) and points (strongly) to .

The between the two quenching models is , which corresponds to an asymptotic value of according to equation 35. Therefore, based on the BIC test, the clustering and the g-g lensing measurements of the red and blue galaxies strongly favor the halo quenching model against the hybrid quenching model, and the halo mass is the more statistically dominant driver of galaxy quenching than stellar mass.

The two quenching models are non-nested models with the same and , so the second term of equation 34 that penalizes model complexities is the same in both quenching models. The BIC test is then equivalent to the alternative Akaike information criterion that is based on relative likelihoods, or a simple test (i.e., ). These tests all point to the halo mass as the main driver of quenching.

5.2 External Model Comparison: Halo Masses of Massive Blue Centrals

The discussion in Section 4.3 points to a potentially smoking-gun test of the two quenching models, by comparing the host halo mass of the massive red and blue central galaxies predicted from the two best-fit quenching models, to other mass measurements for observed groups/clusters with red and blue centrals within the same redshift range. Unfortunately, clusters with blue centrals are systematically under-selected by most of the photometric cluster finders based on matching to the red sequence, while spectroscopic group catalogues constructed from friends-of-friends algorithms do not have large enough volume for finding many massive clusters.

Recently, mandelbaum2015 constructed a sample of locally brightest galaxies (LBGs) from the SDSS main galaxy sample, by adopting a set of isolation criteria carefully calibrated against semi-analytic mock galaxy catalogues to minimize the satellite contamination rate (wang2015). The resulting LBG sample is thus a subset of all massive central galaxies, but with excellent purity of central galaxy membership and zero bias against blue colour. Therefore, the LBGs are ideal for our purpose of measuring the segregation in halo mass between the red and blue centrals.

mandelbaum2015 measured the average host halo mass of the LBGs directly by fitting an NFW density profile (after projection to 2D) to the weak lensing signals measured below . Fig. 13 compares the host halo mass measured as a function of LBG stellar mass (data points with errorbars) to that predicted by the best-fit halo (solid) and hybrid (dashed) quenching models. The errorbars on the LBG measurements are the 1- uncertainties on the mean halo mass, derived from bootstrap-resampled datasets. The coloured bands about the solid curves are the uncertainties on the mean halo mass predicted from the confidence regions. To avoid clutter, we do not show the uncertainties on the hybrid quenching predictions, which are comparable to the halo quenching uncertainties. The average halo mass predicted by the halo quenching model is in excellent agreement with the measurements from the LBG sample, while the hybrid quenching model, as expected, grossly over-predicts the halo mass for the massive blue galaxies.

The difference in the host halo mass between red and blue centrals, as predicted by the hybrid quenching model, can be understood simply as the outcome of a larger differential growth between dark matter and stellar mass in quiescent systems than in star-forming ones. More specifically, in quenched systems the dark matter halos usually continued to grow after the shutdown of stellar mass growth, while in star-forming systems the two often grew in sync, creating a bimodality in host halo mass between two colours without invoking halo quenching. However, this differential growth effect causes at most a factor of two difference in the average host halo masses, too small to explain the factor of several difference observed at the high mass end (quadri2015).

The LBG experiment in Fig. 13 further demonstrates that the halo quenching model, employing halo mass as the driver for galaxy quenching, is superior to the hybrid quenching model, which relies on stellar mass to quench central galaxies. As we explained in Section 4.3, the deficiency of the hybrid model in describing the signals of the massive blue galaxies is intrinsic to the stellar mass quenching mechanism, which fails to explain the rare occurrence of blue centrals in massive clusters. The combined evidence from the BIC model comparison and the LBG experiment strongly suggests that the halo mass is the main driver for quenching the galaxies observed in SDSS.

6 Physical Implications of the Constraints on Halo Quenching Model

Figure 14: Comparison between the color-segregated stellar mass functions predicted by the best-fit halo quenching model (thick solid curves with shaded 1- uncertainty bands) and that observed in the SDSS (shaded histograms) at three different redshifts (from left to right: , , and ). Within each color, the total SMF is further decomposed into the central (dashed) and satellite (dotted) contributions. The excellent agreement between the predicted and observed SMFs for both the red and blue populations is not a result of fitting, but the consistency between galaxy abundance and their clustering and lensing signals predicted by our iHOD analysis using the halo quenching model within . Note that above , the observed SMF in the lowest redshift bin (left; ) is subject to slight incompleteness while the other two are relatively complete, hence a slight deviation from our best-fit prediction.

With the halo quenching model being established as the more viable scenario, we now focus back on the physical implications of our constraints on halo quenching.

6.1 Uniform Characteristic Halo Masses for Quenching Centrals and Satellites

Although the halo quenching formula for centrals and satellites are decoupled in the analysis, our fiducial constraint nonetheless recovers two very similar characteristic halo masses ( and ) for both species at around . It is very tempting to associate this uniform quenching mass scale for both central and satellites to , the critical halo mass responsible for the turning-on of shock heating. Analytical calculations and hydrodynamic simulations both favor a of few times  (birnboim2003; keres2005; dekel2006), providing one of the most plausible explanation for the similar values of and derived statistically in our analyses.

Conservatively speaking, even if the similarity between our inferred characteristic halo masses and were coincidental, the consistency between and still indicates that the quenching of centrals and satellites are somewhat coupled, most likely driven by processes that are both tied to the potential well of the halos. For instance, the supermassive black holes (SMBHs) could provide the thermal or mechanical feedback required to stop the halo gas from cooling and feeding the satellites (dimatteo2005; croton2006; somerville2008), while regulating the growth of the central galaxies (ferrarese2000; gebhardt2000; tremaine2002). hopkins2007 suggested that the SMBH mass is largely determined by the depth of the potential well in the central regions of the system, which precedes the assembly of halo mass, i.e., the maximum circular velocity is already half the present-day value by the time the halo has accreted only two per cent of its final mass (bosch2014).

6.2 Implications for Satellite Quenching and Galactic Conformity

The halo quenching of satellites has a slower transition across than that of the central galaxies (left panel of Fig. 12). This rules out the possibility that halo quenching does not distinguish between centrals and satellites. As mentioned in the introduction, even in the hot halo quenching scenario where gas cooling of centrals and satellites were equally inhibited, the satellites might experience significant delays in their quenching, due to a shorter exposure to the hot halo and/or a spell of star formation from pockets of cold gas they carried across the virial radius of the larger, hotter halo (simha2009; wetzel2012; wetzel2013; bosch2008). Additionally, recent observations suggest that other processes like pre-processing during infall (haines2015), strangulation (peng2015), and ram pressure stripping (muzzin2014) are all at play, contributing to the satellite quenching trend with halo mass. The inefficiency of satellite quenching is also seen in dwarf galaxies below the stellar mass scale we probed here (wheeler2014).

Another interesting aspect of the halo quenching scenario is that it may help explain galactic conformity, i.e., the observed correlation between colours of the central galaxies and their surrounding satellites (weinmann2006; knobel2015), because the quiescent pairs of centrals and satellites are quenched by the common halos they share. However, this halo quenching-induced conformity only occurs among central-satellite pairs at fixed of the centrals. To explain the galactic conformity observed at fixed , there either has to be a substantial scatter between observed and true , or a secondary process that couples the quenching of centrals and satellites within the same halo. For instance, halos formed earlier with higher concentration may be more likely to host quenched pairs of centrals and satellites than their younger and less concentrated counterparts at the same  (paranjape2015). Galactic conformity does not appear when the centrals and satellites were quenched independently, e.g., in the hybrid quenching scenario.

The combination of this intra-halo conformity and the correlation between clustering bias and halo mass, could potentially explain the inter-halo conformity observed in the galaxy marked correlation statistics (skibba2006; cohn2014) and hinted by galaxy pairs in the local volume (; see kauffmann2013), although a secondary quenching induced by either formation time or halo concentration at fixed may be required (paranjape2015). We will explore the conformity prediction of the halo quenching model in the upcoming third paper of this series.

6.3 Colour-segregated Stellar Mass Functions of Central and Satellite Galaxies

Figure 15: Red (left) and blue (right) satellite stellar mass functions conditioned at six different host halo masses, predicted by the best-fit halo quenching model. For each conditional SMF curve, the shaded band indicates the 1- uncertainty and the accompanying dashed curve indicate the conditional SMF of the combined red and blue satellites. The two inset panels show similar sets of curves as the main panel, but normalized by the mass of the corresponding host halos.

As another consistency-check of our analysis, Fig.14 shows the underlying SMFs of the red and blue galaxies predicted by our best-fit halo quenching model at three redshifts (, , ). In each panel, the shaded bands are the 1- uncertainties on the predicted SMFs, and the shaded histograms show the observed SMFs of each colour, i.e., direct galaxy number counts at each redshift without the weighting. The SMF of each colour is further decomposed into contributions from the central (dashed) and satellite (dotted) galaxies. The observed SMF at  (left panel) is more incomplete at the high end due to photometric confusions about bright sources in SDSS (see Paper I for details), therefore lying further below our predictions than that at the two higher redshifts. Since the observed SMFs are not used as input data to the constraints, the excellent agreement between our predictions and the direct number counts on the high stellar mass end is very encouraging — it demonstrates the great consistency between the three key observables (i.e., galaxy clustering, g-g lensing, and the SMFs) measured for the two coloured populations and that predicted by the best-fit halo quenching model within the iHOD framework. Furthermore, the successful recovery of the red and blue SMFs means that the halo quenching model naturally recovers the stellar mass quenching trend observed in P10.

The best-fit halo quenching also provides concrete prediction for the conditional SMF of the red and blue satellite galaxies, shown on the left and right panels in Fig.15, respectively. The conditional SMF is defined as the average number of satellites per dex in stellar mass at fixed halo mass, , the integration of which over gives the commonly-used satellite HOD, i.e., the average number of galaxies per halo above some stellar mass limit . In each panel, the solid curves are the red/blue satellite SMFs within halos of six different masses ( ), with their 1- uncertainties indicated by the shaded bands. The dashed curves are the same in both panels, showing the two-colour combined satellite SMFs. Each inset panel shows the same set of curves as in the main panel, but each normalized by the corresponding halo mass. Overall, the satellite population above is dominated by the red galaxies, and the number of blue satellite galaxies per halo mass decreases with increasing halo mass due to progressively stronger halo quenching effect, while the total number of satellites per halo mass remains roughly constant.

7 Comparison to Previous Works

Figure 16: Comparison of various predicted and measured