Using large galaxy surveys to distinguish quiescent galaxy models
One of the most striking properties of galaxies is the bimodality in their star-formation rates. A major puzzle is why any given galaxy is star-forming or quiescent, and a wide range of physical mechanisms have been proposed as solutions. We consider how observations, such as might be available in upcoming large galaxy surveys, might distinguish different galaxy quenching scenarios. To do this, we combine an -body simulation and multiple prescriptions from the literature to create several quiescent galaxy mock catalogues. Each prescription uses a different set of galaxy properties (such as history, environment, centrality) to assign individual simulation galaxies as quiescent. We find how and how much the resulting quiescent galaxy distributions differ from each other, both intrinsically and observationally. In addition to tracing observational consequences of different quenching mechanisms, our results indicate which sorts of quenching models might be most readily disentangled by upcoming observations and which combinations of observational quantities might provide the most discriminatory power.
Our observational measures are auto, cross, and marked correlation functions, projected density distributions, and group multiplicity functions, which rely upon galaxy positions, stellar masses and of course quiescence. Although degeneracies between models are present for individual observations, using multiple observations in concert allows us to distinguish between all ten models we consider. In addition to identifying intrinsic and observational consequences of quiescence prescriptions and testing these quiescence models against each other and observations, these methods can also be used to validate colors (or other history and environment dependent properties) in simulated mock catalogues.
July 14, 2019
As time marches forward, collapsed objects in the universe grow hierarchically via merging and accretion. Gas accreting onto halos serves as the fuel for star formation in the galaxies residing in dark matter halos. Not all galaxies are forming stars at a significant rate today however, so a central puzzle in galaxy formation is: why are some galaxies star-forming while others are quiescent? Several possible mechanisms to quench star-forming in galaxies have been identified. For instance, the gas may already be used up in stars, or might be stripped from the galaxies, or prevented from accreting onto them, or heated, and so on (see, e.g., the galaxy formation textbook by Mo, van den Bosch & White 2010). Determining which of these processes are most significant for quenching is a major challenge. Due to the vast range of physical processes and scales which can contribute to the quenching of star-formation (e.g. formation of stars, stellar feedback, and accretion onto and outflows from black-holes) empirical constraints can play a vital role.
Here we consider ways in which large statistical galaxy surveys can be used to distinguish between different models for galaxy quenching, motivated by the tremendous impact of the Sloan Digital Sky Survey (York et al., 2000) on studies of local galaxies, and other large surveys either beginning or being planned. We take such surveys to provide a set of galaxy positions and stellar masses and whether or not the star-formation in the galaxy is quenched. We shall not consider here detailed measurements of individual galaxies on smaller scales, such as galaxy structural properties, properties such as winds, or AGN signatures. While these are indeed valuable, our focus is on properties associated with large-scale structure as is available in large surveys.
We expect that using even relatively little information per galaxy can be highly informative when combined with large statistical samples, because many proposed galaxy quenching mechanisms differ in their dependence upon galaxy formation histories and environments. These differences lead to different overall statistical properties. For example, heating of a galaxy by an AGN may be more likely to occur when the galaxy is in the core of a massive halo, while stripping of gas could be more likely when a galaxy is moving through a very dense environment (e.g. as a satellite in a massive halo). The two resulting populations have different demographics.
In this work we create mock, quenched galaxy samples in -body simulations, based upon 10 different models and prescriptions drawn from or motivated by models in the literature. Each model assigns quiescence to individual mock galaxies based upon history and/or environmental criteria, which we expect to be correlated with the underlying physical (and baryonic) processes affecting star-formation. We identify similarities and differences between these models, both intrinsic and observable. Our mock observations of these data help elucidate which future observations might best disentangle such models, and how precise those observations need to be. How the observations interlock and constrain the models is also valuable information for refining mock catalogs – created via any method – which are used in the design and analysis of large surveys.
We shall only consider a single redshift here (); while connecting galaxies across time is a powerful technique, it also involves further assumptions. At low redshifts () the wealth of data and in depth analysis of rich data sets such as SDSS (www.sdss.org) has provided many powerful constraints on environments and other properties of quenched galaxies. Some of our models incorporate these insights. By going back to , we explore the implications of these mechanisms for a significantly earlier cosmic time, and one for which we expect large statistical samples in the near future111For example from DES (www.darkenergysurvey.org), HSC (www.naoj.org/Projects/HSC/surveyplan.html), JPAS (j-pas.org), and LSST (www.lsst.org).. Note that is approximately Gyr ago, which is longer than the main-sequence lifetime of stars. Any galaxy less massive than on the star forming main sequence would at least double its stellar mass in this period222Using Eq. (2) of Lilly et al. (2013).. It is also around that we see a rapid rise in the number density of intermediate-mass quiescent galaxies, making this a particularly interesting time to study.
We also focus exclusively on a single galaxy property, quiescence. Comparing measurements for one observed galaxy property, while fixing the others, makes this work similar to the approach of e.g. Hearin & Watson (2013); Masaki, Lin & Yoshida (2013) at . This forms a complement to more complex models, such as semi-analytic models, which provide more information on each galaxy across cosmic time but also more dependencies between inputs, assumptions and observations (see e.g. Lu et al., 2010; Neistein & Weinmann, 2010; Lu et al., 2013, for recent examples of fitting a suite of parameters in such models to a suite of observations).
The outline of the paper is as follows. In Section 2 we describe the -body simulation and our observation-based assignments of galaxy stellar mass and total quiescent fraction. In Section 3 we describe our 10 different prescriptions for selecting quiescent galaxies based on histories and environments, and in Section 4 and Section 5 we compare intrinsic and observational properties of the resulting catalogues respectively. Section 7 summarizes and concludes. Throughout we use to denote , express stellar masses in (with no factors of ), and halo masses in units of . All distances and volumes are comoving and expressed in Mpc or .
2 Simulations and stellar mass assignments
In order to compare different models and mechanisms for quiescence, we would like to have a sample of mock galaxies with environments, formation histories and stellar masses which are close to what is seen in observations. To this end we associate mock galaxies with dark matter subhalos (defined in more detail below) in an -body simulation. Such associations are currently the standard tool for analyzing galaxy surveys, as dark matter simulations of the cosmic web are well converged (e.g. Heitmann et al., 2008) and differences between dark matter subhalo definitions and merger trees are becoming similarly well characterized (Onions et al., 2012). The -body simulation provides subhalo properties such as positions, velocity, (dark matter) mass, environment and history. As dark matter is the dominant contribution to the mass density in the Universe, these quantities are expected to be close approximations333There are regimes where the approximation is known to break down, for instance on very small scales where baryons dominate; see Kuhlen, Vogelsberger & Angulo (e.g. 2012) for a recent review. See also Weinberg et al (2008); Simha et al. (2012); van Daalen et al. (2013) for detailed comparisons of subhalo properties with and without baryonic effects. to their values when baryons and all interactions are included (a comprehensive simulation of the latter is currently infeasible). We use the terms galaxy and subhalo interchangeably henceforth.
We employ an -body simulation performed in a periodic box of side Mpc. This box has a similar volume to the main galaxy survey of the SDSS, though without boundaries or gaps. At it would subtend around on a side in the plane of the sky. The cosmology is of the CDM family with . The simulation evolved equal mass particles from initial conditions generated at with second order Lagrangian perturbation theory, using the code described in White (2002). Phase space data for all of the particles were dumped starting at and for 45 times equally spaced in down to . We use the () output as our observation time. Further details about this simulation can be found in White, Cohn & Smit (2010).
For each output, halos are identified using the Friends of Friends (FoF) algorithm (Davis et al., 1985), with a linking length of times the mean inter-particle spacing. Halo masses quoted below are FoF halo masses unless otherwise specified. When halos merge, part of the smaller halo can survive as a self-bound substructure within the larger host halo. In such a situation we call the core of the larger halo the “central subhalo” (or central) of the final system and the core of the smaller halo which “fell in” a “satellite subhalo” (or satellite). We reserve the term “halo” to refer to the parent FoF structure which will host a central subhalo and possibly several satellite subhalos. The subhalos are tracked as overdensities in phase space using the FoF6D algorithm (Diemand, Kuhlen & Madau, 2006) as implemented in White, Cohn & Smit (2010). Merger trees and histories are calculated using the methods described in Wetzel, Cohn & White (2009); Wetzel & White (2010).
As in Wetzel & White (2010) we only consider halos (and their descendant subhalos) with masses above to avoid resolution issues (note that subhalos tracked using Subfind (Springel, Yoshida & White, 2001) require a higher minimum mass cut, as discussed in Guo & White 2012). This cut results in 310,687 galaxies in our box or (comoving). There are in addition 1, 18 and 110 (roughly cluster sized) halos with mass above , and .
2.1 Stellar mass assignments
Observationally, star formation is correlated with stellar mass, and many of the models we investigate use as an input the stellar mass of the galaxy. We thus need a way to assign stellar masses to our mock galaxies which results in the right demographics and environmental properties. We assign stellar mass to each galaxy using subhalo abundance matching (e.g. Vale & Ostriker, 2006; Conroy, Wechsler & Kravtsov, 2006) to the stellar mass function of Drory et al. (2009). We use this stellar mass function444Several other stellar mass functions exist in the literature including Perez-Gonzales et al. (2008); Ilbert et al. (2010); Pozzetti et al. (2010); Behroozi, Wechsler & Conroy (2013a); Davidzon et al (2013); Knobel et al. (2013); Marulli et al. (2013); Moustakas et al. (2013). These are significantly different and such differences would propagate into the details of our modeling. Some sources of the differences are discussed in Marchesini et al. (2009); Muzzin et al. (2009); Behroozi, Conroy & Wechsler (2010); Ilbert et al. (2010); Moustakas et al. (2013); Tinker et al. (2013). For example, Moustakas et al. (2013) find small effects from changing stellar population synthesis models between FSPS (Conroy, Gunn & White, 2009; Conroy, White & Gunn, 2010; Conroy & Gunn, 2010), Bruzual & Charlot (2003), Maraston (2005), and Pegase (Fioc & Rocca-Volmerange, 1997, 1999; Le Borgne et al., 2004), while Behroozi, Conroy & Wechsler (2010) find a shift of dex due to the choice of dust model. in part because several of our prescriptions start with the quiescent fractions of central and satellite galaxies proposed by Wetzel et al. (2013a, hereafter WTCB) based upon these data.
We choose to abundance match stellar mass to the maximum mass in a subhalo’s history. For a satellite subhalo, this maximum mass is often the mass right before it becomes a satellite, sometimes known as the infall mass. (Using the maximum mass is along the lines recommended by Reddick et al. (2013), who compared several possible abundance matching proxies for stellar masses and luminosities at , although the peak velocity was an even better proxy in their case.) We include a dex scatter in stellar mass at fixed maximum mass (again motivated by the choices of WTCB, ) but we clip the scatter at to avoid outliers in the more massive halos which are rare in our simulation box. After assigning , we keep only galaxies with . Of the 310,687 mock galaxies in the box making the halo resolution mass cut, 254,950 then remain (of which 185,532 are central). This galaxy sample is starting point for each of our catalogues below. The stellar mass functions for Drory et al. (2009) and for the simulation box are shown in Fig. 1 where we see good agreement, as expected.
2.2 Quiescent fraction as a function of stellar mass
Each prescription we consider below assigns a subset of the galaxies in our simulation box to be quiescent. Matching observations of this fraction is clearly a key property, and is built into the specification of several of our models. For example, many of our models are taken from prescriptions of WTCB, which are in turn based upon the Drory et al. (2009) quiescent fraction as a function of stellar mass, (see Fig. 2). Others require a quiescent fraction as function of stellar mass as input, again we choose that of Drory et al. (2009). Specifically, we take as the number of quiescent galaxies divided by the sum of quiescent and active galaxies in the double Schechter function fits of Eq. (1) and Table 3 of Drory et al. (2009).
There is a large diversity in observed quiescent fractions as a function of stellar mass. The origins of some of these differences are understood, in particular, the Drory et al. (2009) quiescent fractions are defined through SED fitting, and thus differ from those employing other definitions (e.g. specific star formation rate, color cuts, morphology or some combination). Pozzetti et al. (2010) compare samples of quiescent galaxies selected by several different definitions for a data set at . The galaxy samples differ physically and in number density, with color cut based samples tending to include more galaxies because of dust. Two of our models are based upon non-SED based criteria for quiescence. To take this additional (in part definitional) complexity out of our comparisons, we modify our prescriptions, if needed, to improve agreement with the SED based Drory et al. (2009) . Unfortunately there is not always a unique modification to match to the SED quiescent fraction, or matching it exactly is difficult. We shall discuss these cases in detail below. Our quiescent fraction choice produces approximately 55,000-65,000 quiescent galaxies for each catalogue, depending upon model.
3 Prescriptions for assigning quiescent galaxies
We now describe the different quiescent galaxy prescriptions we apply and the 10 resulting catalogues.
Our starting point is the box of mock galaxies with stellar masses assigned as described above. (By fixing this underlying galaxy distribution for all 10 catalogues, any differences in quiescent galaxy catalogues are due solely to the differences in assigning quiescence.)
The first and simplest model is motivated by a method by Skibba & Sheth (2009) used at to assign colors to mock galaxies. The method assumes we have a luminosity and central/satellite assignment for each object, and that the color depends only on these properties. Such a method gives good agreement with a number of observations at . In our case, we use each galaxy’s stellar mass, , and whether it is a satellite or central galaxy to determine the probability that it is quiescent. Individual galaxies are then marked, at random, using these probabilities. With our assumptions, there is only one free function that we must specify to determine the model, and we can take it to be the probability that a satellite galaxy of stellar mass is quiescent: . The central quiescent fraction then is given by the requirement that we match the overall quiescent fraction of Drory et al. (2009). In more detail, and because conventions can differ, we define as the fraction of satellites which are quiescent, i.e. the total number of satellite galaxies of a given which are quiescent is , where is the fraction of galaxies which are satellites and is the total number of galaxies with stellar mass . The definition of is analogous.
These functions have all been well constrained at , but we expect them to change with redshift. Unfortunately, a precise measurement of these functions at higher redshift is difficult as it requires a good group catalog and quiescence classification over a cosmologically representative volume555Group catalogues, for small volumes (e.g., Gerke et al. 2012; George et al. 2013), and estimates for associated quiescent fractions (e.g. Knobel et al. 2013; Tinker et al. 2013) are just beginning to be made, based upon a variety of quiescence definitions.. To overcome this, WTCB suggest extrapolating from to higher in two ways and using the difference as a measure of uncertainty in the model. As more observations become available we can use the measured rather than these extrapolations. In the meantime, including both models helps us to understand the observational signatures we see later.
The first extrapolation to redshift , which we refer to as “fix sat”, assumes (Wetzel et al., 2013a)
It is shown for our redshift in Fig. 1 as the dotted (red) line. For the stellar masses of interest to us, the overall quiescent fraction of galaxies is observed to drop rapidly to increasing redshift, while the satellite fraction evolves more modestly (see e.g. table 1 of Wetzel et al., 2013a, for a recent compilation). If we hold fixed, the decrease in must be obtained by decreasing . Thus, in order to match the observed evolution of the total quiescent fraction, the fraction of central galaxies which are quiescent in this prescription drops rapidly with increasing redshift.
To implement Eq. (1) and the related models described below, we assign the specified quiescent fractions in 100 bins covering . If either or go outside the region , we clip its value and choose the other fraction to reach the desired . For our first prescription, we thus randomly assign galaxies to be quiescent only requiring that Eq. (1) above and the corresponding are satisfied in each stellar mass bin.
We extended this prescription in 2 ways, to get 2 more catalogues. The first extension is a version of “galactic conformity”666Several definitions of conformity are in use, some of which not only affect galaxies sharing the same halo, but also galaxies within a larger region. Ours is most similar to that used by Weinmann et al. (2006) and Ross & Brunner (2009) to describe correlation functions for SDSS at low redshift. See also recent work by Kauffmann et al. (2013); Phillips et al. (2013).. The quiescent centrals are taken to be the same as those in the “fix sat” case above. Satellites of quiescent centrals are initially all taken to be quiescent as well. If the resulting number of quiescent satellites is too small to satisfy Eq. (1) in a given bin, more satellites are randomly assigned to be quiescent. If the number of quiescent satellites is instead too large, quiescent satellites are randomly changed to active to get agreement. Both cases occur, for higher and lower respectively. (As we shall see later, in a similar model with more quiescent centrals, many more halos have star-forming satellites but quiescent centrals, while there are no cases of quiescent satellites with star-forming centrals. We shall explore the observational consequences of this below.) On average, of the satellites in high mass halos are quiescent in this variant, compared to about 60% for the (random) “fix sat” model. We call this prescription “fix sat-conf”.
We explored a stronger form of galactic conformity as well, which we did not include in our family of catalogues. With our fixed choice of quiescent centrals, inherited from the “fix sat” prescription, we assigned a rank to each halo. Halos with quiescent centrals were ranked first, in order of descending (total) richness. Halos with star-forming centrals followed the centrally quiescent halos but were ranked in ascending order of richness. Each satellite galaxy then inherited the ranking of its parent halo. Within a bin of stellar mass the first ranked of the satellites were marked as quiescent, with the rest being star-forming. This prescription resulted in no star forming galaxies in the richest halos which have quiescent centrals (down to halo masses below ). As this is in clear conflict with observations, we work with the less extreme “fix sat-conf” prescription described above.
The second extension based upon Eq. (1), “fix sat-h”, introduces a quenching ordering (and thus an implied time scale). Central galaxies from the “fix sat” model are again left unchanged. Each satellite is then ranked by its infall time. To deal with the discreteness in the output times from the simulation we take the infall time to be random, uniformly distributed between the time step when the galaxy was last a central and when it became a satellite.777This does add scatter between infall times of galaxies which were part of the same halo when falling into the larger halo, however we expect some scatter as well in the quenching process. We take some satellites as quiescent upon infall with a probability based on the extrapolations in WTCB for in this model as a function of redshift. (They give estimates for 4 stellar mass bins and we extend the redshift scaling of their highest mass bin to apply to the most massive satellites.) We then take the remaining satellites in each stellar mass bin to be quiescent in order of infall time (earliest first) to reach in each stellar mass bin. The quenched satellite which fell in most recently of the latter set determines a quenching time, shown as solid (red) triangles in Fig. 3. (At the highest stellar masses there are very few satellites, leading to the larger scatter in quenching time.)
These first three models differ in how they include environment or history, but the quiescent central galaxies are identical, as is the quiescent satellite fraction as a function of . In particular these models all have a relatively low central quiescent fraction, as a direct consequence of Eq. (1). We now consider the second extrapolation of from low redshift of WTCB, to illustrate the impact of this assumption.
An alternate way to evolve the quiescent satellite fraction, also suggested by WTCB, is to fix not but the ratio of the central and satellite quiescent fractions. To match the observed drop in towards higher , both and decrease. However, since the drop is shared between the two sources this prescription leads to a larger quiescent central fraction than the assumption of Eq. (1). Specifically we take
Note, for the ratio is negative because the quiescent central fraction at goes negative. For these values of we take all central galaxies to be active () and set the quiescent satellite fraction to obtain of Drory et al. (2009).
We can construct the same three variant models, random, conformity, and satellite infall, using Eq. (2) above as a base instead of Eq. (1) as was done earlier. Some differences are immediately notable. As just described, in comparison to the “fix sat” model, the “fix c/s” model has more (or the same number of) quiescent centrals in all bins at . Due to the increase in number of quiescent centrals relative to the “fix sat” models, the “fix c/s-conf” variation has many more (roughly four times as many) satellites which are active in halos with quiescent centrals, relative to the “fix sat-conf” variation. Furthermore, unlike the “fix sat-conf” case, there are no quiescent satellites in halos with active centrals. We shall see the implications of this below. For the “-h” variation, we again find a the quenching time by assigning galaxies with the earliest infall time as quiescent, but now up to the fraction required by Eq. (2). This quenching time is much longer than that found for the “fix sat” models, hovering above 4 Gyr for almost all stellar masses, as can be seen in Fig. 3. Satellites take longer to quench because many more are quiescent upon infall, and because many fewer are needed as quiescent overall, given Eq. (2).
3.3 Other models
Our four other prescriptions for when a galaxy becomes quiescent did not use satellite and central quiescent fractions as a constraint. All are based on models presented in the literature, although we have modified or applied them to obtain the Drory et al. (2009) quiescent fraction as a function of stellar mass.
3.3.1 Maximum mass
In the “maxmass” prescription, satellites are quenched in order of the earliest time since their maximum mass, up to the number needed to reach the required in each bin. Similarly to our infall times for satellites, we smooth out the discreteness of the output times in the -body simulation by assigning the time of maximum mass for each galaxy randomly and uniformly between the relevant bracketing time steps (again this will erase some correlations between galaxies sharing the same halo upon infall). For satellites this maximum mass is also taken to be at or before infall time – we thus exclude mass gains through satellite merging, which can be significant (e.g. Angulo et al., 2009; Simha et al, 2009; Wetzel, Cohn & White, 2009). This implicitly assumes that mass gains due to merging after infall do not induce further star formation, which seems reasonable. Matching in each bin then determines a quenching time, which is comparable to that of the “fix sat-h” model (open cyan circles and red filled triangles respectively in Fig. 3).
As the time of maximum mass is correlated with the time of first infall into any larger halo (Wetzel et al., 2013b), this model is similar to the WTCB model for quenching. For comparison with their model, we also show their quenching times in Fig. 3, both for (which they use, solid line) and one estimate (Tinker & Wetzel, 2010) for the extension (dashed line).
3.3.2 Halo growth
The second of these prescriptions, which we label , is based on a property of average star formation histories highlighted by Behroozi, Wechsler & Conroy (2013a, b). They calculated the average star formation rate as a function of stellar mass from , and compared with simulations to find an approximate relation
i.e. star formation rate proportional to baryon accretion rate, where the baryon accretion rate is and in their cosmology. For independent of redshift, Eq. (3) would imply a redshift-independent relation. While the relation does not evolve strongly, it is not completely redshift independent (e.g. Yang et al., 2012; Wetzel et al., 2013a; Moster, Naab & White, 2013; Behroozi, Wechsler & Conroy, 2013b; Watson & Conroy, 2013), and thus this model can only be an approximation888Eq. (3) was also calibrated to simulations with a certain step size which turns out to be different from ours, and thus is approximate for this reason as well. We thank F. van den Bosch for emphasizing this point to us.. We make further approximations by using Eq. (3) to estimate the sSFR for individual galaxies and then apply a cut on sSFR to classify a galaxy as quiescent or star-forming, which is similar but not identical to an SED-based classification (as mentioned above, and expanded upon below).
The halo mass change in Eq. (3), , is the change in halo mass. We approximate the mass gain by the FoF mass gain as the two mass definitions are close at this redshift. In addition, the change in observed stellar mass (the observed star formation rate, of interest to us) has a factor of 1.11 relative to the change in the true stellar mass given above (from Eq. (7) in Behroozi, Wechsler & Conroy (2013a)).
Two additional assumptions are needed to identify which galaxies are quiescent. First of all, this approach does not indicate what to do with satellite galaxies. In addition, many central halos have their last significant mass gain earlier than . Setting all galaxies in these categories to have zero star formation well exceeds the Drory et al. (2009) quiescent fraction constraint. We instead make an ansatz, again based upon the models in WTCB: we take galaxies to have a star formation rate from their most recent halo mass gain (using for that step and for that step), but if that time is before the present time step, we add a damping factor assuming that the star formation is decaying from an earlier time. (Again, mass gains for satellites after infall, due to merging, are not included.) Following WTCB, we assume star formation actually started at and that its value at the time of most recent mass gain (given by Eq. 3) has decayed by the time of observation. Taking the star formation history for central galaxies , with and , WTCB find within Gyr. We take Gyr. The star-formation rate at is then evolved from the value given by Eq. (3) at the time of most recent mass gain using . With only this prescription, however, the model does not have enough quenched galaxies. So in addition to increasing the star formation by using earlier time steps, some quenching is needed to decrease the star formation by the current time. We thus augment the model by setting for any central or satellite galaxy which had its more recent mass gain more than a quenching time ago, assuming this quenching time depends upon stellar mass, similar to our earlier models. We have three examples of quenching times from our above constructions: that of the “fix sat-h”, “fix c/s-h” and “maxmass” models, in Fig. 3. We chose to use the “fix sat-h” quenching times as they gave the closest agreement to the desired quiescent fraction when combined with the sSFR cut below. (This allows some interesting model comparisons below, as for“fix sat-h” model the times were only defined by and used for the satellites, with centrals were assigned randomly; and here the times are used for all galaxies.) The quenching time is responsible for the quiescence of about 1/5 of the quiescent centrals and about 4/5 of the quiescent satellites (these latter overlap with the “fix sat-h” satellites as a result of using that prescription’s time scale).
The last piece needed to get some estimate of quiescence is a comparison of this sSFR estimate to a quiescence classification based on SEDs. The sSFR assignments in WTCB suggest a cut999Note that the bimodality of star formation rates seen in observations (and in WTCB) is not strong in our prescription, as more than half of the quiescent galaxies are such because their mass gains are before . The remainder of the quiescent galaxies are just the low end tail of the sSFR distribution. There is a similar issue in Mutch, Croton & Poole (2013). between active and quiescent galaxies at sSFR. A comparison of SED and sSFR cuts for galaxies at similar redshifts is found in fig. 1 of Pozzetti et al. (2010). Taking those numbers at face value, a sSFR cut of does not include any active galaxies by the SED definition, but also neglects some quiescent ones. We change the maximum specific star formation rate for quiescent galaxies to , which in Pozzetti et al. (2010) would allow some admixture of active SED galaxies but also includes more quiescent SED galaxies. The resulting galaxy sample gives better agreement numerically with the SED determined quiescent fraction as a function of in Drory et al. (2009).
Finally, we note that we do not include any scatter in the sSFR, taking it directly from Eq. (3), and we do not self-consistently integrate over time, instead applying it instantaneously at . A forward integration of Eq. (3) may couple and more closely, while inclusion of scatter would obviously weaken such a correlation.
3.3.3 Stellar mass and density
Our third additional prescription, which we label “”, is based on Peng et al. (2010, see also ). These authors found that the red and blue fractions of the COSMOS sample could be described by the product of two factors, depending upon projected density and stellar mass,
This model is particularly interesting for our purposes because it does not specifically refer to halo mass or to central/satellite designation. However we need to modify it slightly. Although we have for each galaxy, the projected density assignment as used by Peng et al. (2010) relies upon a sample selected with , and depends upon color, which is related to quiescence, which we are trying to find. In addition, the parameters in Eq. (4) are tuned to separate galaxies into red and blue (i.e. using observations based upon a color cut), rather than quiescent and active. We make two modifications as a result.
Instead of using an -limited sample to define projected density, we use an -limited sample. We choose the minimum to give the same galaxy number density as the number of galaxies passing the cut of Peng et al. (2010). At , the limit is . This corresponds to a density of using the COMBO-17, DEEP2 and VVDS -band luminosity functions quoted in Faber et al. (2007). Matching this number density fixes our threshold . The projected density is then the inverse square of the distance to the nearest neighbor within the cylinder divided by its average value for a sample of random positions within the box.
Using and in Eq. 4 gives each galaxy in our box a probability of being red, defined by the color cut chosen in Peng et al. (2010). The resulting well exceeds that of Drory et al. (2009). This is not surprising. As mentioned earlier, color cuts tend to classify more galaxies as quiescent (red) than an SED (our comparison Drory et al. (2009) sample) or sSFR based cut.101010See also Tinker et al. (2013) for discussion of different quiescent fractions in these models. Our assignment based upon rather than might be also a factor, however, this red fraction in our box is a good match to that in Knobel et al. (2013), including central and satellite red fractions as a function of . (For this test we also use the Knobel et al. (2013) stellar mass function for consistency.)
Our prescription is thus to keep the form of the Peng et al. (2010) model, but to modify the parameters to match the of Drory et al. (2009). As Peng et al. (2010) tuned their parameters using the measured distributions of quiescence as a function of both and , while we only have , our modifications cannot be unique. (Some degeneracy is expected as there is some correlation of high with high .) We found fit our fiducial quite well, which corresponds to multiplying the Peng et al. (2010) default values (from their table 2, averaged for two redshift bins) by .
Our last prescription is based upon a proposal by Hearin & Watson (2013), see also Zentner, Hearin & van den Bosch (2011), of “age matching”. These authors assign -band luminosities to a simulation similar to ours using subhalo abundance matching, and then order galaxies in luminosity bins according to a redshift (roughly a quenching time). Galaxies with the earliest are taken to be quiescent, up to the total number required (for us set by Drory et al. 2009). Hearin & Watson (2013) define as the maximum (earliest) of three possible redshifts:
-when the host halo equals or exceeds .
-when the galaxy becomes a satellite.
-when the host halo growth rate slows down.
Each of these redshifts is taken to be zero if it never occurs. Hearin & Watson (2013) use in the first condition, while we shall instead use . The two definitions are close at the redshifts of interest. The halo growth rate is taken from Wechsler et al. (2002). While Hearin & Watson (2013) use a relation between concentration and halo growth rate, we instead use the definition of Wechsler et al. (2002): . We search through the time steps in the simulation to find the earliest time that the halo growth rate dropped below, and subsequently stayed below, the threshold growth rate.
As in our previous models, we bin on (rather than -band luminosity). We assign quiescent galaxies based upon the galaxies with the highest in each of 100 stellar mass bins. The last condition (slow down in growth) accounts for slightly more than 60 per cent of the , with the remainder of the assignments split approximately equally between the other two conditions.
A fifth additional model which we implemented, but which we do not include below, is that of Lu et al. (2013). These authors assign star formation rate based upon halo mass, infall time for satellites, and stellar mass. Using their preferred model we found quiescent fractions too large to be consistent with Drory et al. (2009). In particular their model had a very short satellite quenching time, making essentially all satellites quenched. While there are combinations of parameters in their prescription which give longer quenching times, without repeating their full likelihood analysis we could not identify combinations which matched the our required quiescent fraction and the other constraints they imposed.111111We thank Z. Lu for discussions of their work.
3.4 Summary of the models
|Model||cen or sat||sat infall time||nearby galaxies|
|“fix sat” “fix c/s”||x|
|“fix sat-conf” “fix c/s-conf”||x||x|
|“fix sat-h” “fix c/s-h”||x||x|
The quiescence criteria above result in 10 galaxy catalogues, differing only by which galaxies are marked as quiescent. The next step is to compare measurements on all these catalogues, however it is first useful to summarize how the prescriptions differ. A comparison of the criteria used in determining quiescence (central or satellite in halo, infall time, halo mass change, etc.) is given in Table 1. In words, the criteria for quiescence can be briefly described as:
- “fix sat” and “fix c/s”
- “fix sat-conf” and “fix c/s-conf”
as above but put quiescent satellites preferentially in halos with quiescent centrals (the rest are random)
- “fix sat-h” and “fix c/s-h”
randomly assign some satellites as quiescent at infall using proposed higher redshift behavior, assign rest of quiescent satellites in order of infall time to reach the desired quiescent fraction
assign galaxies which reach their maximum mass the earliest as quiescent, up to the total quiescent fraction desired
take sSFR from most recent halo mass gain; also assign galaxies as quiescent if the most recent halo mass gain is before time scale found in “fix sat-h”
quiescent probability from the stellar and (projected) local density using Eq. (4).
ranks galaxies by earliest of three possible starvation times; assigns quiescence to galaxies in order, up to the number needed.
4 Comparisons: intrinsic
With these catalogs in hand we wish to see in what ways the different prescriptions lead to different populations of quiescent galaxies. We begin with a discussion of the intrinsic properties, before turning to the observational consequences in Section 5.
Two intrinsic properties are imposed on the catalogues by construction. The first is the stellar mass function, shown in Fig. 1 and implemented identically in all the catalogues. The second, the quiescent fraction of all galaxies as a function of stellar mass, , was used implicitly or explicitly in the catalogue construction as well (and used to exclude the model of Lu et al. 2013 which did not provide a good match to this function). In Fig. 2, the observations of Drory et al. (2009) are shown along with the measurements from the models. While the models generally reproduce quite well, the match is not perfect for all of them. In particular, the and prescriptions have slight excesses at low , and the model is also high at high , although in-between it matches the others fairly well.
We now consider measurement other than these two intentionally degenerate properties to highlight differences between the catalogues.
4.1 Central and Satellite Quiescent Fractions
The quiescent fraction of all galaxies as a function of stellar mass, above, can be decomposed into central and satellite contributions. These are challenging to obtain observationally, which is in part the reason why WTCB propose two different quiescent satellite and central decompositions (the starting points for our “fix sat” and “fix c/s” models). However, understanding the way in which these functions differ between the models will help us to understand the observational trends in the next section, and which observations bear most directly upon this separation.
The breakdowns into quiescent central () and quiescent satellite () galaxies for our catalogues are shown in Fig. 4. For context we also reproduce the total quiescent galaxy fraction of Drory et al. (2009) from Fig. 2. The solid and dot-dashed lines for the “fix sat” (red) and “fix c/s” (blue) models show the “fix c/s” model almost always has the same number or more quiescent central galaxies than the “fix sat” model, as we have remarked before. This is a direct consequence of the assumed redshift dependence of . If the decrease in quenched fraction with increasing redshift is borne entirely by the central galaxies, rather than being shared equally by the central and satellite galaxies, then is very small.
Note that the models separate into two groups, with the prescription lying in-between. The trends in the central, quiescent galaxies are mirrored for the quiescent satellites as the sum is fixed. The models with conformity and the “-h” models have the same fractions as their parent models, because those models simply “shuffle” the quiescent or satellite galaxies into different halos.
Three of the four models which do not specify the satellite and central quiescent fractions directly have strong similarities to the “fix sat” or “fix c/s” models. In part this is because the quenching prescriptions lead to overlap of the quenched populations. The model uses the quenching time after satellite infall of the “fix sat-h” model, 90 per cent of the model quenched satellites coincide with 80 per cent of the “fix sat-h” quenched satellites. The “maxmass” model also has 90 per cent of its satellites in common with the “fix sat-h” model (and a similar percentage in common with the satellites). These models have much less overlap in their quiescent centrals: quiescent centrals are randomly chosen for “fix sat-h”, while those for are quiescent because recent mass gain was too small or too long ago, and for “maxmass” because largest mass was too long ago. About 2/3 of the “maxmass” quiescent centrals are also centrals, but only about 1/3 of and “maxmass” centrals overlap with “fix sat-h” quiescent centrals. As about three times as many central galaxies as satellite galaxies differ between the “maxmass” and models relative to the “fix sat” models, central galaxy population differences might be the source of observational differences between the “maxmass” and models and the “fix sat” model.
At lower the satellite fraction is relatively large, and satellites can saturate the required red fraction. Since the halos of central galaxies tend to gain mass more rapidly than those of satellites (whose last mass gain was prior to their infall) the and “maxmass” models predict a very small fraction of quiescent, low centrals.
By contrast, the model lies close to the initial “fix c/s” fractions. This model has more low-, quiescent centrals than most of the others. In practice, the model makes little distinction between centrals and satellites, as the parameters which we found to fit the constraint imply . These fractions are also fairly close for the “fix c/s” models in Fig. 1. If the satellite and central quiescent fractions are close, more central galaxies will be quiescent because they outnumber satellites at any given . (This satellite/central blindness is not necessarily part of the form for assigning quiescence. At fixed stellar mass, there is a difference on average between satellites and centrals, and parameters such as those in Peng et al. (2010) can lead to a large difference between and .) But by sampling a wide range of parameters, it seems that getting low enough at low requires raising at low , thus taking towards as we found.)
We now turn from quiescent satellite and central stellar mass functions to the question of which halos these satellites and central galaxies occupy.
4.2 Quiescent Galaxy Halo Occupation
The manner in which the low- and high-mass quiescent central and satellite galaxies are distributed among halos differs significantly between the models. In this section we ask: “in which types of halos are quiescent galaxies found?”
Fig. 5 shows the average number of quiescent galaxies per halo (the “halo occupation number” or HON; Peacock & Smith 2000; Seljak 2000; Cooray & Sheth 2002), as a function of halo mass. At left we plot all quiescent galaxies, at right only central quiescent galaxies. The average number of all galaxies per halo is also shown for reference as the black line (and is the same for all models). The quiescent fraction as a function of halo mass is just the ratio of each model line to this black line, it is largest at high mass for the “fix sat-conf”, “maxmass” and models. (The active galaxy HON – the difference of the black solid line and any of the other curves – follows a rising power law for all of the models as well121212We thank P. Behroozi for discussing this measurement with us.).
As was the case for the quiescent fractions, the quiescent galaxy HON’s split roughly into two groupings, which cross around .
The “fix c/s” model has more quiescent, central galaxies at low than the “fix sat” model. This in turn implies more quiescent galaxies in low mass halos and a shallower HON. Within the “fix sat” and “fix c/s” classes, the “-conf” models have more quiescent satellites in more massive halos, as these are the halos most likely to host quiescent centrals. Similarly, the “-h” models have more quiescent galaxies in massive halos than their random counterparts because such halos typically host more satellites (of a given stellar mass) that fell in long ago – arising from a combination of reduced dynamical friction and earlier halo formation time. A similar effect operates for the “maxmass” prescription, because satellites which have reached their highest mass long ago tend to have fallen in long ago, and thus are again more prevalent in high mass halos.
The model HON again seems very similar to the random “fix c/s” model. In both, the satellite and central quiescent fractions are much closer to each other than the other models. This large central quiescent fraction at low can be seen in Fig. 4 for both models. Since low centrals live in low mass halos, this in turn explains the low behavior of the HON. Again, this behavior seems to be built into the model by requiring the form of Eq. (4) to have low enough quiescent fraction at low (to match our Drory et al. (2009) constraint).131313 Even though quiescence tends to go with high density and satellites tend to have higher densities, there are many more low density centrals at low , which somewhat cancel out the preference for quiescent satellites.
It is notable that the quiescent galaxy richness of clusters can vary by over a factor of two, i.e. the quiescent galaxy HON is very model dependent. This implies that the HON, or the cluster luminosity function for clusters of a wide range of masses, can provide a strong discriminant between models (this has been considered in e.g. Tinker et al. 2012). The differences in the number of quiescent galaxies as a function of host halo mass has implications for the physical role halo environments and membership play in turning galaxies from active to quiescent.
4.3 Quiescent Galaxy Distribution in Clusters
In addition to measuring the number of galaxies in a halo of a given mass, we can ask how these galaxies are distributed within the halos (i.e. the profile). Aside from the prescription, and to some extent the “-conf” models, spatial properties are not used to define which galaxies are quiescent – so any dependence arises due to correlations with other properties which are used in the models.
For the 18 most massive clusters in the simulation () we stacked the counts of quiescent galaxies in bins of radial distance from the most bound particle within the halo (which is very close to the minimum of the halo potential and the density peak). We used logarithmically spaced bins in , where is the virial radius141414We take , where is the radius at which the average density within the cluster reaches 180 times the average background density. This radius is measured from the halo center using all of the particles in the simulation (not just group members). The factor is the conversion between and for a Navarro, Frenk & White (1996) halo at , calculated as described in White (2001).. The total number of quiescent galaxies in clusters changes from model to model, as already shown in Fig. 5. To highlight the additional information given by the profile, we normalize the stacked counts for each model at . The resulting profiles are shown in Fig. 6.
In all of the models the quiescent galaxy radial profiles tend to be more centrally concentrated than those of all galaxies (the solid black line), reminiscent of the well-known morphology- and color-density relations (e.g. Dressler et al, 1997; Treu et al, 2003; Balogh et al, 2004; Kauffmann et al, 2004; Postman et al, 2005; Christlein & Zabludoff, 2005; Loh et al, 2008; Skibba et al, 2009; Cibinel et al, 2013; Lackner & Gunn, 2013; Muzzin et al., 2012). Amongst the models, the “fix c/s-h” and models are the most centrally concentrated (this increase in concentration is due to a quicker drop off in quiescent galaxy density at high radius compared to many of the other models). The weak but non-zero correlation between infall time and radius (Oman, Hudson & Behroozi, 2013)151515Also, Wetzel et al to appear. imprints a radial dependence for the model. The large number of quiescent galaxies in the model near the cluster centers is a direct consequence of the increase of the quiescent fraction with projected density (Eq. 4).
5 Comparisons: observational
In summary, the models produce different numbers of quiescent centrals and satellites, place them in different halos (HON) and in different places in halos (profile). These results are very useful for understanding the differences between the models but are challenging to access observationally. They do however drive differences in quantities which can be probed observationally, for example the change in the HON leads to changes in the galaxy correlation function or in the richness distribution of groups. We now turn to such statistics and describe how the differences we have seen above translate into differences in the observables.
5.1 Quiescent galaxy correlation function
One of the most basic measurements that can be made on any population of objects is its two-point correlation function. In recent years it has become standard to use measurements of the correlation function to infer the halo occupation distribution (shown in Fig. 5). We thus expect the correlation function to be a useful diagnostic of these models.
We use the Landy & Szalay (1993) estimator to measure the (auto-)correlation functions for our catalogs. The results for the real-space correlation function are shown at the top of Fig. 7, with the ratios of each model to the full galaxy real-space correlation function below (to highlight the differences between models). Errors are shown for only 4 cases to avoid clutter in this and subsequent figures. The box is split into octants to compute the errors, which represent the error in the mean of the octants. They thus indicate how well a survey of comparable volume to our box could determine .161616While we primarily show real space auto-, cross- and marked correlations here and below, the trends we focus on are possible from combinations of their redshift space counterparts such as Fig. 8, as well as being accessible via construction of real space quantities themselves. Clearly, several of our prescriptions cleanly separate with a volume comparable to our simulation. The difference between the curves in comparison to the errors can be used to indicate the observational requirements to distinguish them.
The correlation function trends are as expected from the halo occupations in Fig. 5: models with more galaxies in higher mass halos cluster more strongly on both large and small scales. The large-scale clustering is set by the bias, which is determined by the mean, galaxy-weighted, halo mass. The small-scale clustering is set by the number of central-satellite and satellite-satellite pairs within a single halo. Satellites with the longest time since infall (taken as quiescent for “fix sat-h”) or since maximum mass (taken as quiescent for “maxmass”) reside in the most massive clusters, so the corresponding quiescent samples cluster more strongly than the randomly chosen quiescent galaxy sample does. In addition, the “-conf” models tend to put more quiescent satellites in high mass halos, as most of the high mass halo central galaxies are quiescent. (This model also has the largest variance as changing one central galaxy to or from quiescent in a rare halo can significantly change the number of quiescent satellites for that halo mass and thus the number of quiescent galaxies with many nearby neighbors. This suggests that surveys aimed at ruling out this model need to sample a large volume with a representative sample of high-mass halos.)
The impact of the radial profile within massive halos on is not very strong. While the models separate in Fig. 6 we see two degeneracies seen in the HON in as well: the and random“fix c/s” models, and and “fix sat-h” models are also close (the latter two have significant overlap in their populations as mentioned earlier).
In practice the three dimensional real-space correlation function is found by “deprojecting” either the projected correlation function (to minimize the influence of redshift-space distortions) or the angular correlation function (if only coarse distance estimates are available). If accurate redshifts are available for the galaxies, the redshift space correlation function can in principle carry additional information, at least in part because the small-scale clustering is affected by fingers-of-god and thus is sensitive to the satellite fraction in massive halos. To bring the comparison closer to the observational plane, the projected correlation function
is shown in Fig. 8, where we see it exhibits similar behavior to . In Eq. (5), is the separation in the plane of the sky, is the separation along the line-of-sight in redshift space and is a cutoff in the line-of-sight direction. Ideally is very large so that the the cutoff doesn’t matter, however, our Mpc on-a-side octants are relatively small which argues against using very large . One can still define and measure with a small , but the should be matched when comparing theory and observation. Throughout we use Mpc, thus considering cylinders Mpc deep in the redshift (velocity plus position) direction within each of the eight Mpc octants in the box.
We also computed the angle-averaged, or monopole, redshift correlation function, . The trends and differences between the models in closely followed those seen in , indicating that both statistics are comparable in their discriminating power and the additional sensitivity to e.g. the fingers-of-god is small. (The most noticeable difference in is that the “fix c/s-conf” model increases its relative clustering at low separation, making it easier to differentiate it from the “fix c/s-h” model.)
While it is not as commonly employed, it is also possible to measure the cross-correlation171717We thank F. van den Bosch for suggesting this measurement. between quiescent and star-forming galaxies (or between either population and the full sample). As an example we show the cross-correlation function, , between quiescent and star-forming galaxies at top in Fig. 9. The differences between models are not large, and appear mostly at small scales. This cross-correlation is most useful in testing the models which invoke conformity (especially the “fix sat-conf” model from the rest). Models which exhibit galactic conformity have fewer pairs of quiescent and star-forming galaxies within a single halo (preferring to have all of one type, the same as the central) so the cross-correlation is suppressed on small scales. The trend is less pronounced in the “fix c/s-conf” model, which has more halos where central and satellite galaxy quiescence are mismatched. The cross correlation divided by a reference power law model, shown in the lower half of Fig. 9, more clearly separates the models.
5.2 Group multiplicity function
One way to constrain the manner in which galaxies occupy halos is to model the galaxy correlation function, assuming a particular profile and parameterized halo occupation distribution. An alternative method is to measure the number of groups of objects of a given richness, i.e. the group multiplicity function. Differences between models or between models and observations are of course easier to interpret the more closely the richness measure (or other observable) tracks halo mass, but even relatively coarse measures can be interpreted with the aid of mock catalogs.
There are numerous methods for constructing group catalogs, which rely on a wide array of differing assumptions. It is not our intention here to perform an exhaustive comparison, but rather to illustrate the potential of this measurement as a diagnostic. For our example we assume we have redshifts for each of our quiescent galaxies and find groups using a Friends-of-Friends algorithm in redshift space (e.g. Huchra & Geller, 1982). We follow Berlind et al. (2006, see also , who used it to study abundance matching) and set the linking lengths in the line-of-sight and perpendicular directions as and , both measured in units of the mean inter-galaxy separation and including only quiescent galaxies in the input catalog. We define richness to be the number of (quiescent) galaxies associated with each group. Fig. 11 shows the cumulative number density of groups as a function of richness, for the different catalogs.
The basic trends in Fig. 11 can be understood by reference to the HON in Fig. 5. The “fix c/s” and models have the fewest rich groups, because they have the fewest quiescent galaxies per massive halo (Fig. 5). By contrast the “maxmass” and “fix sat-conf” models, which have many quiescent galaxies in massive halos, have a relatively large number density of rich groups. In general we see that the imperfections involved in constructing the group finder (i.e. the impurity and incompleteness) do not qualitatively change the trends present in the underlying models. This suggests that a sufficiently accurate group catalog would provide a strong discriminant between the models. Because it opens up a large parameter space, we leave an examination of other group finding methods, and methods which can work with photometric redshifts, for future work.
5.3 Two dimensional profile in clusters
One can also consider the observational counterpart of the three dimensional cluster profile (Fig. 6), the two dimensional cluster profile. Here galaxies are counted within a redshift cylinder of the cluster center. However, accurately estimating the observational scatter is more challenging with our mock catalogues, as it relies upon the accuracy of the cluster mass (determining ) and cluster centering, both of which include significant assumptions besides those used in the construction of the catalogue. To bound the information one could obtain from this measurement we consider the case where the group center and are known perfectly. In this limit the most notable distinction this measurement provides is between the and “fix c/s” models. However as we will see below, this can be obtained with other more direct observations.
5.4 Distribution of projected density:
The model of Peng et al. (2010) used a projected density, , in determining which galaxies are classified as quiescent. This is a property that can be used to help discriminate amongst models, even when the models make no explicit reference to it. There are many types of density one could define (for a comparison, see, e.g., Haas, Schaye & Jeeson-Daniel (2012); Muldrew et al. (2012)), we shall consider because it has been studied in this context by Peng et al. (2010) and is used as input to one of our models.
The distribution of across the whole galaxy sample is approximately lognormal, as expected. However the distributions of for the quiescent galaxies, in raw counts and as a fraction of all galaxies (Fig. 12), discriminate quite strongly among the models. Models split into two groups, mostly determined by the quiescent satellite fraction as a function of stellar mass. Models with a large number of quiescent central galaxies tend to have overall more galaxies at lower (Fig. 12, top). (Central galaxies are a minority population for high densities, only reaching at least half of the galaxies for projected densities .) A slightly different division between the models is seen in the quiescent galaxy fraction (Fig. 12, bottom). Differences become the most pronounced at the high density tail (), composed almost entirely of satellites in rich groups or clusters. The general shape of the quiescent fraction (Fig. 12, bottom) is expected: the rise of the quiescent fraction to high implies that higher satellites are generally quiescent and such satellites lie predominantly in massive halos.
The “fix c/s” model has the lowest quiescent fraction at high – Eq. (2) requires many low-, central galaxies to be quiescent. This means the satellite galaxies in the same bin must be star-forming, removing the high- population in Fig. 12. The “fix c/s-conf” model favors galaxies in high mass halos with a star-forming central to be star-forming, removing quiescent galaxies from these high density halos. This is somewhat counteracted for the “fix c/s-h” models as the longest lived satellites tend to be in the higher mass halos. The model has the highest quiescent fraction for highest , partly by construction. Next highest are the “fix sat-h”, and “maxmass” models. These models turn galaxies quiescent based upon infall time and galaxies that fell in earlier tend to be in higher mass halos and thus denser environments.
In is encouraging that two models which were quite close in their HON, and “fix c/s”, separate cleanly by looking at the high projected density tail of the quenched fraction.
5.5 Marked Correlation Functions
In addition to looking at the distribution of , we can use the projected density as a weight when computing the two point function, creating a “marked” correlation function (Sheth, Connolly & Skibba, 2005; Harker et al., 2006; Skibba et al, 2013). Such a function is straightforward to compute if can be computed and can be used to break degeneracies in modeling the occupation distribution (White & Padmanabhan, 2009).
The marked correlation function, , is defined as
where the sum is over all galaxy pairs, and , which are separated by and and are the marks, is the number of pairs in the bin and is the average mark in the sample. on scales where the clustering of objects depends upon the mark. One of the advantages of a marked correlation function, from an observational perspective, is that it is simple to modify a code which computes to also compute . No further information is needed (beyond the marks and galaxy positions). In fact, one does not even need a random catalogue due to the cancellation between the numerator and denominator. It is also relatively easy to estimate the errors (Sheth, Connolly & Skibba, 2005). We will focus here on the three dimensional marked correlation function but there is no reason one cannot use the projected or angular version instead.
If we take our mark, , to be the projected density of the th galaxy, the resulting is shown in Fig. 13. For comparison, we also show the marked correlation function for all (not just quiescent) galaxies as the solid black line. As expected, the quiescent galaxies cluster differently, as the mark was used in their selection. There is a feature between and Mpc, roughly at the virial radius of massive halos. Some similarities between the model ordering at small radius and the stacked and rescaled cluster profiles (Fig. 6) is apparent.
If the dynamic range in the mark is very large, then there is some concern that can become sensitive to rare outliers. A simple way around this is to modify the mark from to
with for and . Fig. 12 suggests that is a reasonable choice, and we take that as our fiducial value. The resulting is shown in Fig. 14. Similar behavior is seen as we vary from 30 to 300 suggesting the features we see in Fig. 13 are stable and robustly present in the catalogs. As and combine projected density and galaxy number, an enhancement at small scales could be due to many galaxies with a small mark, or few galaxies with a large mark. To differentiate, we removed the 3-7 per cent of galaxies with from the catalog. This only decreases the radius below which these models dominate (to kpc/h), but does not erase the separation. Thus we infer the small-scale enhancement is driven by a large number of galaxies, not simply the dense tail of the distribution.
In this section we considered several observational measurements and how they separated out the different quiescent galaxy catalogues. Now we turn to general results and implications of the intrinsic and observational measurements the the ensemble of catalogues.
We generated a set of mock galaxies by assigning stellar masses to subhalos in an -body simulation via abundance matching. These mock galaxies match, by construction, the stellar mass function of Drory et al. (2009), as shown in Fig. 1. We then marked a fraction of these galaxies as quiescent, using 10 prescriptions adapted from the literature. All of the prescriptions are tuned give a good fit to the total quiescent fraction as a function of , as shown in Fig. 2, but differ in their assumptions about which properties are important in determining quiescence and in how they incorporate centrality, environment and history.
The number of implied quenching mechanisms differs as well. Two are used in the “fix sat”, “fix sat-h”,“fix c/s”, “fix c/s-h” models (for satellite and for central galaxies), and for the model (stellar mass and projected density), while “fix sat-conf”, “fix c/s-conf” might allow one mechanism across a shared halo. The “maxmass”, models have one mechanism, combined with no mass gains for satellites, while the model has three mechanisms.
We investigated the consequences of these different quiescence assumptions for intrinsic galaxy properties and observational measurements, at one fixed redshift. Models differed in intrinsic properties such as halo occupation and galaxy profiles within massive halos, although the central/satellite split as a function of stellar mass was degenerate for several of them. Observationally, we considered measurements derived from galaxy positions and stellar masses. Promisingly, we found that almost all of the models could be separated from each other by some combination of observations, if measurement errors could be made small enough. While many of the models could be distinguished based on very traditional measurements such as the auto- and cross-correlation functions, we also saw that less often considered statistics such as the group multiplicity function and the (projected) density-marked correlation function allowed us to further break degeneracies between models. We advocate that these statistics be reported in future.
We now turn to intrinsic and observational properties and how they relate to each other and to mechanisms/models for quenching star formation. We saw that the fraction of quenched galaxies which were central or satellite roughly divided the models into two groups. The strongest division was between different choices for the evolution of the quenched satellite fraction with redshift, as expected, but this division also strongly separated the model from the ones in which or where quiescence was based upon the time at which the subhalo achieved its maximum mass, for example. This division further implied a change in the way quiescent galaxies occupied halos, i.e. the HON (Fig. 5) which showed up clearly in the relative clustering (Fig. 7). This argues that measures of quiescent galaxy clustering can be used to constrain the redshift evolution of the quiescent fraction of satellite and central galaxies and thus the timescale over which star-formation quenching acts (see also Wetzel et al., 2013a; Tinker et al., 2013, for similar conclusions).
Models which are degenerate in their satellite or central quenched galaxy fractions as a function of stellar mass can still differ significantly in other measures, depending upon environment and history. Those which invoke conformity tend to have the largest number of quiescent satellites in massive halos within their class. Observationally this leads to enhanced auto-correlation, since massive halos have a large bias. These models also have a depressed cross-correlation between active and quiescent galaxies at small separations, as galaxies in a halo tend to all be either active or quiescent. Conformity might point to quenching properties acting on larger scales than those of a halo, as for fixed stellar mass it can give similar quenching likelihoods to satellites at the outskirts and satellites near the center.
The models based upon conformity have central galaxy assignments identical to those of their associated models based on satellite infall or random selection. It was therefore encouraging that there were significant, observable differences associated with the variations in satellite quiescence mechanisms. (This is in part why conformity was introduced at lower redshifts in the SDSS (Ross & Brunner, 2009), but here it also separated out the models where satellite infall was related to quiescence.)
The associated “-h” models are best further considered along with the other three models using infall time (where satellite infall stops mass gain or generally increases quiescence probability – our “maxmass”, and models). If halo growth is considered a proxy for baryonic accretion (and satellite infall a proxy for strangulation, or cessation of baryonic accretion), these can be thought of as “feeding limited” models. Observationally, these five models most prominently group together in the projected density marked correlation functions and . They are large at short distances, only being surpassed by the model (which uses in its construction). This observation thus seems to separate out models which are feeding limited or in which group pre-processing played an important role.
The differences between the three “feeding limited” models which have similar satellite populations (“fix sat-h”, “maxmass”, ) are interesting as well. All quench satellites after infall (with similar although not identical decay times). The first assigns the central galaxies randomly, while the other two depend upon halo mass gain or time of maximum mass. Differences between these models might thus indicate which intrinsic properties and observations are impacted by the history of central mass gains. (Although the central and satellite quiescent fractions are not identical, in raw numbers there are more differences in the quiescent central populations.) They are quite close in the full HON at high mass, but differ very slightly at low stellar mass (also seen in the central galaxy HON). The three models separate out most clearly (but not by much) from each other in the auto-correlation function. The “maxmass” is slightly larger at high richness in the group multiplicity function, and the model also has a relatively depressed quiescent-active galaxy cross-correlation at short distances. Otherwise, the history of central mass gains seems to not have much of an effect on the observations we considered.
All of the models make an explicit distinction between central and satellite galaxies except for the model. Due to the way it is constructed, this model shows a very different distribution of for quiescent galaxies than the other models, which results in a strong enhancement of the projected density-marked correlation function at small scales, and the quiescent fraction as a function of projected density. In the other observations it is close to degenerate with the model sharing its HON and profile, which assigns quiescent galaxies depending only upon centrality (“fix c/s”).
Several observations were quite good proxies for the intrinsic differences we saw between the models. For example, the group multiplicity function we used orders models in almost the same way as the HON at high richness or halo mass (only two models reverse). The two dimensional stacked cluster profile has information similar to the three dimensional stacked cluster profile and the projected density distribution (for ) showed a similar ranking of models to the quiescent satellite fraction at intermediate . If one can reliably infer which galaxies belong to which dark matter halo from a group catalog, and has a reliable method of determining the central galaxy, then the central and satellite quiescent fractions become directly observable. As we saw, these functions were extremely helpful in partitioning the models. Conversely the differences in quiescent fraction at high projected density and differences in the radial profile may impact the calculations of purity and completeness for color-selected cluster finders which are trained on mock catalogs (which make either implicit or explicit assumptions about these properties).
We used ten catalogues with quiescence based upon dark matter subhalo histories, environment and centrality. However, these tests could be expected to shed light on many other proposed models, including those based upon fewer simulation properties (e.g. without subhalos, such as Peacock & Smith 2000; Seljak 2000; Cooray & Sheth 2002), more simulation properties (e.g. including histories, for instance, Wetzel et al. 2013a; Moster, Naab & White 2013; Mutch, Croton & Poole 2013; Cattaneo et al. 2013; Baugh 2006; Benson 2012, or baryons, e.g., Schaye et al. 2010; Almgren et al 2013; Vogelsberger et al. 2013), or other simulation properties (e.g., density as in Scoville et al. 2013). Other tests may also add more information (e.g. lensing was used by Masaki, Lin & Yoshida (2013) for similar tests at lower redshifts, we did not consider it here as galaxy shapes are required as well). Some tests to constrain quiescence were recently reported by Tinker et al. (2013) at similar redshifts (clustering, weak lensing, group catalogue) as this work was being prepared for publication.
One of the most striking features of the galaxy population is that it exhibits bimodality in color, morphology and star-formation rate. In particular, the existence of two broad types of galaxies (those which are actively star-forming and those which are quiescent) cries out for a theoretical explanation. Unfortunately the range of physics and of physical scales involved in setting the star-formation rate in a galaxy is enormous, making empirical models of the phenomenon of great importance. Many such models have been proposed to explain the color or star-formation rate bimodality at . Motivated by the wealth of data on the higher Universe which we expect to have soon from large surveys, and using high-resolution cosmological -body simulations, we have investigated the predictions for a range of models for star-formation quenching to see in what way they differ and what observations (of what accuracy) can be used to discriminate amongst them. The advantage of using large surveys is that statistical errors can be controlled and are more accessible and one can disentangle the many correlations between galaxy properties which limit inferences from small samples. The disadvantage is that information is more circumstantial and so requires a different sort of detective work.
We have focused our attention at , which is at high enough that we expect significant galaxy evolution but low enough that we expect to have large statistical samples of galaxies in the near future. Note that is approximately Gyr ago, which is longer than the main-sequence lifetime of stars. Any galaxy less massive than on the star forming main sequence would at least double its stellar mass in this period. It is also around that we see a rapid rise in the number density of intermediate-mass, quiescent galaxies, making this a particularly interesting time to study.
We considered several different galaxy properties which are expected to correlate with star-formation quenching and investigated how statistical measurements might be used to distinguish them. We created different mock catalogues, all sharing the same stellar mass for each galaxy and close to the same overall quenched fraction as a function of stellar mass, but with different criteria for classifying a galaxy as quiescent. Many of these models had degeneracies in basic intrinsic properties (satellite/central fraction, for instance), or halo occupation (HON), but the suite of observations we used could nonetheless separate them out, given small enough measurement errors. Fixing the quenched central galaxies and varying the satellite quenching gave models which were distinguishable, similarly, models where the majority of differences were in the central galaxies also could be differentiated.
More specifically, in addition to the quiescent galaxy auto and cross-correlation functions and quiescent fraction as a function of stellar mass, we found that projected density counts, the projected density marked correlation function, and the group multiplicity function could further serve to separate models, and also seemed to correlate with centrality in host halo, feeding limited quenching and halo occupation respectively. These observational measurements increase the galaxy formation information available from surveys possessing galaxy stellar masses and positions.
We have mostly considered how these observational measurements can be used to distinguish models from each other, using observations. Alternatively this provides a particularly discriminating set of tests for validating mock catalogues, constructed by any means. They can also be used to decode associations of other galaxy properties with galaxy histories, environments and centrality, for instance AGN activity or morphology.
While we were completing this work we became aware of Tinker et al. (2013) which compared several different measurements of a survey at similar redshifts to constrain quiescent fraction evolution and of Hearin et al. (2013b) which considered measurements at low redshift to constrain color and developed a formalism for comparing dark matter simulation properties which are correlated with color.
We thank E. Bell, F. van den Bosch, K. Bundy, N. Dalal, C. Knobel, M. George, A. Hearin, C. Heymans, C. Lackner, Z. Lu, P. Behroozi, D. Watson, A. Wetzel for conversations, and D. Watson, A. Hearin, R. Skibba, A. Wetzel and the referee for helpful suggestions on the draft. JDC also thanks the Royal Observatory, Edinburgh, and Kavli IPMU for their hospitality and invitations to speak on this work as it was being completed. JDC was supported in part by DOE. MW was supported in part by NASA.
- Almgren et al (2013) Almgren, A., Bell, J., Lijewski, M., Lukic, Z., Van Andel, E., 2013, ApJ, 765, 39
- Angulo et al. (2009) Angulo, R.E., Lacey, C.G., Baugh,C.M., Frenk, C.S., 2009, MNRAS, 3999, 983
- Balogh et al (2004) Balogh, M.L., Baldry, I.K., Nichol, R., Miller, C., Bower, R., Glazebrook, K., 2004, ApJL, 615, 101
- Baugh (2006) Baugh, C.M., 2006, RPPh, 69, 3101
- Behroozi, Conroy & Wechsler (2010) Behroozi, P.S., Conroy, C., Wechsler, R.H., 2010, ApJ, 717, 379
- Behroozi, Wechsler & Conroy (2013a) Behroozi, Wechsler, Conroy, 2013, ApJ, 770, 57
- Behroozi, Wechsler & Conroy (2013b) Behroozi, Wechsler, Conroy, 2013, ApJL, 762,31
- Benson (2012) Benson, A., 2012, New Astronomy, 17, 175
- Berlind et al. (2006) Berlind, A. A., et al., 2006, ApJS, 167, 1
- Bruzual & Charlot (2003) Bruzual, G., & Charlot, S. 2003, MNRAS, 344, 1000
- Cattaneo et al. (2013) Cattaneo, A., Woo, J., Dekel, A., Faber, S.M., 2013, MNRAS, 430, 686
- Christlein & Zabludoff (2005) Christlein, D., Zabludoff, A.I., 2005, ApJ, 621, 201
- Cibinel et al (2013) Cibinel, A., et al., 2013, ApJ, 777, 116
- Conroy, Wechsler & Kravtsov (2006) Conroy, C., Wechsler, R., Kravtsov, A.R., 2006, ApJ, 647, 201
- Conroy, Gunn & White (2009) Conroy, C., Gunn, J. E., & White, M. 2009, ApJ, 699, 486
- Conroy & Gunn (2010) Conroy, C., & Gunn, J. E. 2010, ApJ, 712, 833
- Conroy, White & Gunn (2010) Conroy, C., White, M., & Gunn, J. E. 2010, ApJ, 708, 58
- Cooray & Sheth (2002) Cooray, A., Sheth, R., 2002, PhR, 372, 1
- van Daalen et al. (2013) van Daalen, M.P., Schaye, J., McCarthy, I.G., Booth, C.M., Dalla Vecchia, C., 2013, preprint [arXiv:1310.7571]
- Davidzon et al (2013) Davidzon et al., 2013, A& A, 558, A23
- Davis et al. (1985) Davis M., Efstathiou G., Frenk C.S., White S.D.M., 1985, ApJ, 292, 371
- Diemand, Kuhlen & Madau (2006) Diemand J., Kuhlen M., Madau P., 2006, ApJ, 649, 1
- Dressler et al (1997) Dressler, A., et al., 1997, ApJ, 490, 577
- Drory et al. (2009) Drory, N., et al., 2009, ApJ, 707, 1595
- Faber et al. (2007) Faber S.M., Willmer C.N.A., Wolf C., et al., 2007, ApJ, 665, 265
- Fioc & Rocca-Volmerange (1997) Fioc, M., Rocca-Volmerange, B., 1997, A&A, 326, 950
- Fioc & Rocca-Volmerange (1999) Fioc, M., Rocca-Volmerange, B., 1999, astro-ph/9912179
- Gerke et al. (2012) Gerke, B.F., et al., 2012, Ap J, 751, 50
- George et al. (2013) George, M., Ma, C.-P., Bundy, K., Leauthaud, A., Tinker, J., Wechsler, R.H., Finoguenov, A., 2013, ApJ, 770, 113
- Guo & White (2012) Guo, Q, White, S.D.M., 2013, preprint [arXiv:1303.3586]
- Haas, Schaye & Jeeson-Daniel (2012) Haas, M.R., Schaye, Joop, Jeeson-Daniel, A., 2012, MNRAS, 419,2133
- Harker et al. (2006) Harker, G., Cole, S., Helly, J., Frenk, C., Jenkins, A., 2006, MNRAS, 367, 1039,
- Hearin et al. (2013a) Hearin, A.P., Zentner, A.R., Berlind, A.A., Newman, J.A., 2013, MNRAS, 433, 659
- Hearin & Watson (2013) Hearin, A.P., Watson, D.F., 2013, MNRAS, 435, 1313
- Hearin et al. (2013b) Hearin, A.P., Watson, D.F., Becker, M.R., Reyes, R., Berlind, A.A., Zentner, A.R., 2013, arxiv:1310.6747
- Heitmann et al. (2008) Heitmann K., et al., 2008, CS&D, 1, 15003
- Huchra & Geller (1982) Huchra J.P., Geller M.J., 1982, ApJ, 257, 423
- Ilbert et al. (2010) Ilbert O., et al., 2010, ApJ, 709, 644
- Kauffmann et al (2004) Kauffmann, G., White, S.D.M., Heckman, T.M., Menard, B., Brinchmann, J., Charlot, S., Tremonti, C., Brinkmann, J., 2004, MNRAS, 353, 713
- Kauffmann et al. (2013) Kauffmann, G., Li, C., Zhang, W., Weinmann, S., 2013, MNRAS, 430, 1447
- Knobel et al. (2013) Knobel, C., et al., 2013, ApJ, 769, 24
- Kovac et al. (2010) Kovac, K., et al, 2010, ApJ, 708, 505,
- Kovac et al. (2013) Kovac, K., et al, 2013, preprint [arXiv:1307.4402]
- Kuhlen, Vogelsberger & Angulo (2012) Kuhlen, M., Vogelsberge, M., Angulo, R., 2012, Physics of the Dark Universe, 1, 50
- Lackner & Gunn (2013) Lackner, C.N., Gunn, J.E., 2013, MNRAS, 428, 2141
- Landy & Szalay (1993) Landy,S.D., Szalay, A.S., ApJL, 412, 64
- Le Borgne et al. (2004) Le Borgne, D., et al. 2004, A&A, 425, 881
- Lilly et al. (2013) Lilly S.J., Carollo C.M., Pipino A., Renzini A., Peng Y., 2013, ApJ, 772,119
- Loh et al (2008) Loh, Y.-S., Ellingson, E., Yee, H.K.C., Gilbank, D.G., Gladders, M.D., Barrientos, L.F., 2008, ApJ, 680, 214
- Lu et al. (2010) Lu, Y., Mo, H.J., Weinberg, M.D., Katz, N., 2011 MNRAS, 416, 1949
- Lu et al. (2013) Lu, Z., Mo, H.J., Lu, Y, Katz, N., Weinberg, M., van den Bosch, F.C., Yang, X., 2013, preprint [arXiv:1306.0650]
- Maraston (2005) Maraston, C., 2005, MNRAS, 362, 799
- Marchesini et al. (2009) Marchesini, D., van Dokkum, P.G., Forster Schreiber, N.M., Franx, M., Labbe, I., Wuyts, S., 2009, ApJ, 701, 1765,
- Marulli et al. (2013) Marulli, F., et al., 2013, A & A, 557, 17
- Masaki, Lin & Yoshida (2013) Masaki, S., Lin, Y-T, Yoshida, N., 2013, preprint [arXiv:1301.1217],
- Mo, van den Bosch & White (2010) Mo, H., van den Bosch, F., White, S.D.M., 2010, Galaxy Formation and Evolution, Cambridge University Press
- Moster et al. (2010) Moster, B.P., Somerville, R.S., Maulbetsch, C., van den Bosch, F.C., Maccio, A.V., Naoob, T., Oser, L., 2010, ApJ, 710, 903
- Moster, Naab & White (2013) Moster, B.P., Naab, T., White, S.D.M., 2013, MNRAS, 428, 3121
- Moustakas et al. (2013) Moustakas, J., et al., 2013, ApJ, 767, 50
- Muldrew et al. (2012) Muldrew, S.I., et al, 2012, MNRAS, 419, 2670
- Mutch, Croton & Poole (2013) Mutch, S.J., Croton, D.J., Poole, G.B., 2013, MNRAS, 435, 2445
- Muzzin et al. (2009) Muzzin, A., Marchesini, D., van Dokkum, P.G., Labbe, I., Kriek, M., Franx, M., 2009, ApJ, 701, 1839
- Muzzin et al. (2012) Muzzin, A., et al, 2012, ApJ, 746, 188
- Navarro, Frenk & White (1996) Navarro J., Frenk C.S., White S.D.M., 1996, ApJ, 462, 563
- Neistein & Weinmann (2010) Neistein, E., Weinmann, S.M., 2010, MNRAS, 405, 2717
- Oman, Hudson & Behroozi (2013) Oman, K.A., Hudson, M.J., Behroozi, P.S., 2013, MNRAS, 431, 2307
- Onions et al. (2012) Onions, J., et al, 2012, MNRAS, 423, 1200
- Peacock & Smith (2000) Peacock, J.A., Smith, R.E., 2000, MNRAS, 318, 1144
- Peng et al. (2010) Peng, Y.-J., et al., 2010, ApJ, 721, 193
- Perez-Gonzales et al. (2008) Perez-Gonzalez, P.G., et al, 2008, ApJ, 675, 234
- Phillips et al. (2013) Phillips, J.I., Wheeler, C., Boylan-Kolchin, M., Bullock, J.S., Cooper, M.C., Tollerud, E.J., 2103, preprint [arXiv:1307.3552]
- Postman et al (2005) Postman, M., et al., 2005, ApJ, 623, 721
- Pozzetti et al. (2010) Pozzetti, L, et al, 2010, A&A 523, A 13
- Reddick et al. (2013) Reddick, R.M., Wechsler, R.H., Tinker, J.L., Behroozi, P.S., 2013, ApJ, 771, 30
- Ross & Brunner (2009) Ross, A.J., Brunner, R.J., 2009, MNRAS, 399, 878
- Schaye et al. (2010) Schaye, J., et al., 2010, MNRAS, 402, 1536
- Scoville et al. (2013) Scoville, N., et al., 2013, APJS, 206, 3
- Seljak (2000) Seljak, U., 2000, MNRAS, 318, 203
- Sheth, Connolly & Skibba (2005) Sheth, R.K., Connolly, A.J., Skibba, R., 2005, astro.ph/0511773
- Simha et al (2009) Simha, V., Weinberg, D.H., Dave, R., Gnedin, O.Y., Katz, N., Keres, D., 2009, MNRAS, 399, 650
- Simha et al. (2012) Simha, V., Weinberg, D. H., Dave, R., Fardal, M., Katz, N., Oppenheimer, B. D., 2012, MNRAS, 423, 3458
- Skibba & Sheth (2009) Skibba R.A., Sheth R.K., 2009, MNRAS, 392, 1080
- Skibba et al (2009) Skibba, R.A., et al., 2009, MNRAS, 399, 966
- Skibba et al (2013) Skibba, R. A., Sheth, R. K., Croton, D. J., Muldrew, S. I., Abbas, U., Pearce, F. R., Shattow, G. M., 2013, MNRAS, 429, 458
- Springel, Yoshida & White (2001) Springel V., Yoshida N., White S. D. M., 2001, New Astronomy, 6, 51
- Tinker & Wetzel (2010) Tinker, J.L., Wetzel, A.R., 2010, ApJ, 719, 88
- Tinker et al. (2012) Tinker, J.L., et al., 2012, ApJ, 745,16
- Tinker et al. (2013) Tinker, J.L., Leauthaud, A., Bundy, K., George, M.R., Behroozi, P., Massey, R., Rhodes, J., Wechsler, R., 2013, preprint [arXiv:1308.2974]
- Treu et al (2003) Treu, T., Ellis, R.S., Kneib, J.-P., Dressler, A., Smail, I., Czoske, O., Oemler, A., Natarajan, P., 2003, ApJ, 591, 53
- Vale & Ostriker (2006) Vale, A., Ostriker, J.P., 2006, MNRAS, 371, 1173
- Vogelsberger et al. (2013) Vogelsberger, M., Genel, S.,Sijacki, D., Torrey, P., Springel, V., Hernquist, L., 2013, preprint [arXiv:1305.2913]
- Watson & Conroy (2013) Watson, D.F., Conroy, C., 2013, AjP, 772, 139
- Wechsler et al. (2002) Wechsler, R.H., Bullock, J.S., Primack, J.R., Kravtsov, A.V., Dekel, A., 2002, ApJ, 568, 52
- Weinberg et al (2008) Weinberg, D.H., Colombi, S., Dave, R., Katz, N., 2008, ApJ, 678, 6
- Weinmann et al. (2006) Weinmann, S.M., van den Bosch, F.C., Yang, X., Mo, H.J., 2006, MNRAS, 366, 2
- Wetzel, Cohn & White (2009) Wetzel, A.R., Cohn, J.D., White, 2009, MNRAS, 395, 1376
- Wetzel & White (2010) Wetzel, A.R., White, M., 2010, MNRAS, 403, 1072
- Wetzel et al. (2013a) Wetzel, A. R., Tinker, J.L, Conroy, C., van den Bosch, F.C., 2013a, MNRAS, 432, 336
- Wetzel et al. (2013b) Wetzel, A. R., Tinker, J.L, Conroy, C., van den Bosch, F.C., 2013b, preprint [arXiv:1303.7231]
- White (2001) White M., 2001, A&A, 367, 27
- White (2002) White M., 2002, ApJS, 143, 241
- White & Padmanabhan (2009) White, M., Padmanabhan, N., 2009, MNRAS, 395, 2381
- White, Cohn & Smit (2010) White, M., Cohn, J.D., Smit, R., 2010, MNRAS, 408, 1818
- Yang et al. (2012) Yang, X., Mo, H.J., van den Bosch, F.C., Zhang, Y., Han, J., 2012, ApJ, 752, 41
- York et al. (2000) York D.G., et al., 2000, AJ, 120, 1579
- Zentner, Hearin & van den Bosch (2011) Zentner, A.R., Hearin, A.P., van den Bosch, F.C., 2013, arXiv:1311.1818