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 . 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, \citet[][hereafter Paper I]{zm15} developed a powerful statistical framework, called the model, to interpret the spatial clustering (i.e., the projected galaxy autocorrelation function wp) and the galaxy-galaxy (g-g) lensing (i.e., the projected surface density contrast $\ds$) of the overall galaxy population in the Sloan Digital Sky Survey \citep[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 , 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 ; 2) it depletes the cold gas reservoir via secular stellar mass growth or sudden removal by external forces \citep[e.g., tidal and ram pressure;][]{gunn1972}; and 3) it turns off gas supply by slowly shutting down accretion \citep[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 ($M_{}{\sim}10^{12}\hmsol$), virial shocks heat gas inflows from the intergalactic medium, preventing the accreted gas from directly fueling star formation . Additional heating from, e.g., the active galactic nuclei (AGNs) then maintains the gas coronae at high temperature . For halos with Mh < Mshock, 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 Mshock, naturally explains the colour bimodality, particularly the paucity of galaxies transitioning from blue, star-forming galaxies to the red sequence of quiescent galaxies . 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 Mshock, unless the quenching by hot halos is instantaneous. Observationally, by studying the dependence of the red galaxy fraction $f\red$ on stellar mass $\ms$ and galaxy environment δ5NN (i.e., using distance to the 5th nearest neighbour) in both the Sloan Digital Sky Survey (SDSS) and zCOSMOS, \citet[][hereafter P10]{peng2010} found that $f\red$ can be empirically described by the product of two independent trends with $\ms$ and δ5NN, suggesting that stellar mass and environment quenching are at play. By using a group catalogue constructed from the SDSS spectroscopic sample, 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 $\ms$ of central galaxies is tightly correlated with halo mass $\mh$ (with a scatter ∼0.22 dex; see Paper I), a stellar mass trend of $f\red$ is almost indistinguishable with an underlying trend with halo mass. By examining the inter-relation among $\ms$, $\mh$, and δ5NN, found that the quenched fraction is more strongly correlated with $\mh$ at fixed $\ms$ than with $\ms$ at $\mh$, and the satellite quenching by δ5NN 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 , who implemented halo quenching in cosmological hydrodynamic simulations by triggering quenching in regions dominated by hot (105.4K) gas. They reproduced a broad range of empirical trends detected in P10 and , 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 and its recently updated version of . Age-matching is an extension of the “subhalo abundance matching” \citep[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 . In practice, after assigning $\ms$ using SHAM, the age-matching method further matches the colours of galaxies at fixed $\ms$ 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 $\ms$ regardless of halo mass or environment, with a secondary quenching via halo formation time. Therefore, the age-matching quenching is very similar to the $\ms$-dominated quenching of P10, except that the second variable is halo formation time rather than galaxy environment. The key difference between the $\mh$- and $\ms$-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 $\ms$ and $\mh$, hence two different spatial distributions of galaxy colours. Therefore, by comparing the wp and $\ds$ predicted from each quenching model to the measurements from SDSS, we expect to robustly distinguish the two quenching scenarios. The framework we developed in Paper I is ideally suited for this task. The is a global “halo occupation distribution” (HOD) model defined on a 2D grid of $\ms$ and $\mh$, which is crucial to modelling the segregation of red and blue galaxies in their $\ms$ distributions at fixed $\mh$. The quenching constraint is fundamentally different and ultimately more meaningful compared to approaches in which colour-segregated populations are treated independently \citep[e.g.,][]{tinker2013, puebla2015}. Our 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 quenching model employs only four additional parameters that are directly related to the average galaxy quenching, while most of the traditional approaches require ∼20 additional parameters, rendering the interpretation of constraints difficult. Furthermore, the framework allows us to include ∼80% 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 [sec:data]. In Section [sec:model] we introduce the parameterisations of the two quenching models and derive the s for each colour. We also briefly describe the signal measurement and model prediction in Sections [sec:data] and [sec:model], respectively, but refer readers to Paper I for more details. The constraints from both quenching mode analyses are presented in Section [sec:constraint]. We perform a thorough model comparison using two independent criteria in Section [sec:result] and discover that halo quenching model is strongly favored by the data. In Section [sec:physics] we discuss the physical implications of the halo quenching model and compare it to other works in [sec:compare]. We conclude by summarising our key findings in Section [sec:conclusion]. Throughout this paper and Paper I, we assume a $\lcdm$ cosmology with (Ωm, ΩΛ, σ₈, h) = (0.26, 0.74, 0.77, 0.72). All the length and mass units in this paper are scaled as if the Hubble constant were $100\,\kms\mpc^{-1}$. In particular, all the separations are co-moving distances in units of either $\hkpc$ or $\hmpc$, and the stellar mass and halo mass are in units of $\hhmsol$ and $\hmsol$, respectively. Unless otherwise noted, the halo mass is defined by $\mh\,{\equiv}\,M_{200m}\,{=}\,200_m(4\pi/3)r_{200m}^3$, where r200m is the corresponding halo radius within which the average density of the enclosed mass is 200 times the mean matter density of the Universe, $_m$. For the sake of simplicity, lnx = logex is used for the natural logarithm, and lgx = log₁₀x is used for the base-10 logarithm.
