MASS AND ENVIRONMENT AS DRIVERS OF GALAXY EVOLUTION IN SDSS AND zCOSMOS AND THE ORIGIN OF THE SCHECHTER FUNCTION^*

2.1.1. Sample

2.1.2. Star Formation Rates and Masses

2.1.3. Density Field

2.1.4. Treatment of Spectroscopic- and Mass-incompleteness

2.2.1. Construction of the Sample

2.2.2. Star Formation Rates and Masses

2.2.3. Construction of the Density Field

Several recent studies have emphasized the close relationship between the star formation rates of galaxies and their existing stellar mass, m, conveniently parameterized as the specific star formation rate, sSFR, defined as sSFR = SFR/m. In local SDSS samples, Salim et al. (2007) and Elbaz et al. (2007) have shown the existence of a tight "main sequence" of star-forming galaxies in which the sSFR is approximately constant over more than two decades of stellar mass, with a dispersion of only 0.3 dex about the mean relation. The relationship that is derived from the stellar masses and Hα-derived star formation rates of B04 is shown in Figure 1 for blue star-forming galaxies. The ridge line of this SDSS relation has the following relation log sSFR = −10.0–0.1 (log m–10.0) indicating only a weak dependence of sSFR on mass, i.e., sSFR ∝m^β with β = −0.1. Naturally, the inverse of the sSFR defines a timescale for the formation of the stellar population of a galaxy, τ = sSFR⁻¹. In the local universe, this is of order 10 Gyr, i.e., comparable to the Hubble time.

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This uniformity in the sSFR in "normal" star-forming galaxies is a striking feature of the galaxy population. It clearly, however, does not extend to the Ultra Luminous Infrared Galaxies (ULIRGs) which exhibit highly elevated star formation rates of 100 M_☉ yr⁻¹ or greater (Sanders & Mirabel 1996) in galaxies within the same range of stellar mass of normal galaxies. However, the ULIRGs are believed to be associated with rare major mergers (Sanders et al. 1988; Sanders & Mirabel 1996) and consequently distinct star formation processes. Although ULIRGs lie off the main sequence, their effect is in fact automatically incorporated into our analysis (as argued in Section 7.3 below) and their effect does not need to be considered separately.

The approximate constancy of sSFR with stellar mass in star-forming galaxies has also been seen at higher redshifts, e.g., Elbaz et al. (2007) at z ∼ 1 and Daddi et al. (2007a) at z ∼ 2, and a similar relation was derived by Pannella et al. (2009) using a completely independent indicator of SFR (radio stacking). We believe that contrary results in the literature (see, e.g., Maier et al. 2009) can often be ascribed to the inclusion of quenched very low SFR galaxies, to the use of star formation indicators that are more sensitive to the presence of dust, or to the selection of the sample, since an SFR-selected sample will generally produce a flattening of the sSFR–m relation.

In our analytic analysis below, we will follow the β dependence exactly. A constant sSFR (at a given epoch), i.e., β close to zero, is a good working hypothesis that we will adopt in our numerical simulations. Our conclusions do not actually depend on the accuracy of this assumption, and in fact our analysis provides some independent support for this hypothesis—e.g., the fact that the faint end slope of the mass function of star-forming galaxies does not change with redshift is a natural consequence of a very weak dependence of sSFR on galactic stellar mass.

The dependence of the star formation rate on environment has not been so well explored. The two panels of Figure 1 show the SDSS data from B04 split into the lowest (D1) and highest (D4) density quartiles of our SDSS density field constructed as described in Section 2.2.3. There is no detectable difference between the sSFR–mass relation for star-forming galaxies between the two environments. This is further shown in Figure 2, which shows the mean 〈log(SFR)〉 as a function of galactic mass and environment in the B04 sample.

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This invariance of the mean sSFR on environment should not be confused with the clear evidence (see Section 4 below) that the fraction of galaxies that are star forming does depend quite strongly on this same environmental measure, leading to a strong environment dependence of the average SFR for the overall population. This distinction emphasizes that the quenching of galaxies leading to the red sequence is a relatively sharp transition. Those galaxies that are not quenched evidently continue forming stars at the same rate, regardless of their environment, despite the fact that the chance of having been quenched evidently does depend strongly on the environment.

The same invariance of sSFR with environment is seen in the zCOSMOS 10k data to z ∼ 1. This is shown in Figure 3. Although this set of measurements is less complete and will be somewhat biased because of (mass-dependent) reddening, etc., we find no statistically significant dependence of the mean sSFR (of star-forming galaxies) on environment in zCOSMOS to z ∼ 1.

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It has also recently become clear that the uniformity of the sSFR described in the two previous subsections is also seen at much higher redshifts (e.g., Daddi et al. 2007a; Elbaz et al. 2007; Dunne et al. 2009; Pannella et al. 2009; Santini et al. 2009), but with a characteristic sSFR that is substantially elevated, by a factor of about 7 at z = 1 and by about 20 at z = 2. At z ∼ 2, the characteristic timescale τ = sSFR⁻¹ has thus fallen to about 0.5 Gyr, about seven times shorter than the Hubble time at that redshift. Despite the fact that individual star formation rates have reached those associated with ULIRGs in the local universe, it is clear that these galaxies are undergoing roughly steady star formation and are not associated with a short-lived burst of star formation associated with a merger event. Correspondingly, the extremely luminous sub-millimeter galaxies at these redshifts have elevated star formation rates above 1000 M_☉ yr⁻¹ and once again lie off the relation defined by "normal" star-forming galaxies (Daddi et al. 2007a).

Linking together the sSFR at z ∼ 2 and z ∼ 1 of Daddi et al. (2007a) and Elbaz et al. (2007) with the zCOSMOS data (see Figure 3), and the SDSS relation above, we adopt for the 〈sSFR〉 out to z ∼ 2 (see also Pannella et al. 2009), and incorporating the mass dependence discussed above,

$$\begin{eqnarray} {\rm sSFR}(m,t) &=& \frac{1}{{\tau (m,t)}}\nonumber\\ & = & 2.5\left({\frac{m}{{{\rm }10^{10} \;M_\odot }}} \right)^\beta {\rm }\left({\frac{t}{{3.5\;{\rm Gyr}}}} \right)^{ - 2.2} \;{\rm Gyr}^{ - 1}.\nonumber\\ \end{eqnarray} \tag{ 1 }$$

Beyond z ∼ 2, it appears that the characteristic sSFR flattens out and is roughly constant back to z ∼ 6 (e.g., González et al. 2010). The cause of this apparent change in behavior around z ∼ 2 is not well understood but is largely incidental to our discussion.

It should be noted that this change in sSFR is responsible for the evolution in the overall star formation rate density (SFRD) in the universe back to these redshifts, which Lilly et al. (1996) parameterized as t^−2.5. The dramatic change in SFRD back to z ∼ 2 is evidently not caused by an increase in the typical masses of star-forming galaxies, nor by an increase in the number of star-forming galaxies, since these change little with redshift (see Section 3.4 below), but rather by the large and uniform change in the SFR at a given galactic mass across the broad population of star-forming galaxies.

The similarity of the exponents suggests that the overall evolution of the sSFR at z < 2 reflects the evolution of the specific accretion rate of haloes (in both baryonic and dark matter) as they hierarchically grow. The flattening at high redshifts is poorly understood but may reflect a limit to the sSFR.

The mass (and luminosity) function(s) of blue star-forming galaxies can be well fit by a Schechter function (e.g., Lilly et al. 1995; Bell et al. 2003, 2007; Ilbert et al. 2005; Zucca et al. 2006). Surprisingly, it has become increasingly clear that the shape of the mass function stays remarkably constant over a large range of redshifts, despite the large increase in the masses of individual galaxies implied by the star formation law given in Equation (1). The characteristic mass M* and faint end slope α_s stay essentially constant, whereas the overall normalization ϕ* drifts upward with time especially at high redshifts z > 1. This constancy is clearly seen in the spectroscopic zCOSMOS sample to z ∼ 1 (see Figure 12 of Pozzetti et al. 2009) and in the photo-z COSMOS sample to z ∼ 2 (see Figure 18 of Ilbert et al. 2010), and has previously been remarked upon by others, including Bell et al. (2007, see their Figure 1).

An example of this remarkable fact is shown in Table 1 which shows the Schechter parameters obtained by re-fitting a single Schechter function to the sum of the components that are identified in the Ilbert et al. (2010) COSMOS analysis with "intermediate-" and "high-" activity galaxies, i.e., omitting the "quiescent population." Over the whole range 0.1 < z < 2, both M* and α_s remain essentially constant within 0.05 dex and 0.05, respectively, within this highly homogeneous data set (thereby avoiding any issues of mass determination from sample to sample). The normalization ϕ* is more variable, partly due to large-scale structure in COSMOS, but is more or less constant to z ∼ 1, but then declines by a factor of about 3 to z ∼ 2. Pérez-González et al. (2008) also constructed a mass function to z ∼ 4 which shows the same behavior, i.e., constant M* and α_s and with ϕ* slowly increasing with time, especially at z > 1.

Individual star-forming galaxies will be increasing their masses through the star formation described by Equation (1). Integration of the sSFR relation over time indicates that a galaxy which is not quenched and which remains on the blue "main sequence" will have increased its mass by about 2.0 dex since z = 2 (i.e., Δlog m = +2.0) and by 0.75 dex since z = 1, despite the rapid fall in sSFR. These star-forming galaxies are clearly a mobile population of galaxies moving steadily through the mass function, emerging to be quenched at the high mass end.

This paper is primarily concerned with the "quenching" of galaxies as a function of cosmic epoch, galactic stellar mass, and environment. Quenching is therefore distinct from the smooth and uniform cosmological evolution in the sSFR that is given by Equation (1). To a large degree the precise time evolution of the sSFR is incidental to our analysis and serves to set the "cosmic clock" for the process (see Section 7.2 below), rather than controlling the eventual outcome.

At any epoch and in any environment, the fraction of galaxies in the red and blue "states" are given by f_red and f_blue, which sum to unity. Empirically, the fraction f_red in the local universe is found to increase with galaxy stellar mass and with environmental density (see Figure 11 of Baldry et al. 2006).

To compare environments at different epochs we consider the over-density δ discussed in Section 2.1.3. This is equivalent to the comoving density ρ and we will use these interchangeably. It might be thought that the physical density would be a more useful quantity. However, our environmental scale of order 1 Mpc is outside of the virial radius of most haloes. The comoving density on this comoving scale best reflects broad environmental differences between voids, filaments, and clusters, which appear to control some properties of dark matter haloes (see, e.g., Section 4.3 below).

At a given epoch, we define the relative environmental quenching efficiency, ε_ρ, as follows: ε_ρ is the fraction of those galaxies at a given galactic mass, m, which would be forming stars (i.e., be blue) in some reference environment, ρ₀, but which are however progressively quenched (i.e., are red) in denser environments:

$$\begin{equation} \varepsilon _\rho (\rho,\rho _0,m) = \frac{{f_{{\rm red}} (\rho,m) - {\rm }f_{{\rm red}} (\rho _0,m)}}{{f_{{\rm blue}} (\rho _0,m)}}. \end{equation} \tag{ 3 }$$

It is convenient to choose ρ₀ to be the lowest density environment, i.e., the most void-like regions, where one might expect environmental effects to be minimum and where the dependence of galactic properties with environment is in any case seen to saturate (Baldry et al. 2006). This choice will henceforth be implied, so that ε_ρ will be always positive and never larger than unity.

It is important to stress that ε_ρ measures the differential quenching effect of the environment starting from the population of galaxies (at the same stellar mass) that is seen in the lowest density regions. The point of normalizing the change in the color distribution of the galaxy population in this way is that it makes ε_ρ insensitive to the addition of extra red galaxies provided that the size of this additional component is independent of environment.

There is also an equivalent relative mass-quenching efficiency, ε_m, which measures the differential effect of stellar mass in determining the red fraction at some fixed environmental density

$$\begin{equation} \varepsilon _m (m,m_0,\rho) = \frac{{f_{{\rm red}} (m,\rho) - {\rm }f_{{\rm red}} (m_0,\rho)}}{{f_{{\rm blue}} (m_0,\rho)}}. \end{equation} \tag{ 4 }$$

It is again convenient to set the reference mass m₀ to a very low mass where almost all galaxies (at least in the voids) are blue.

In this most general formalism, both ε_m and ε_ρ may be functions of both m and ρ. However, we show in the next section that in fact ε_m is independent of ρ and ε_ρ is independent of m.

Figure 5 shows the empirical values of ε_ρ and ε_m as functions of mass and environment in the SDSS sample. These are determined within moving boxes of size 0.3 dex in mass and 0.3 dex in (1 + δ). The gray hatched regions around selected lines show typical observational (sampling) uncertainties which have been simply derived from the binomial error of the fraction in the box (68% confidence level).

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To a remarkable degree, the relative mass-quenching efficiency function, ε_m, is found to be independent of environment, and the relative environmental quenching efficiency function ε_ρ is found to be independent of galactic stellar mass. The horizontal lines in the two panels of Figure 5 show the fitted relations

$$\begin{eqnarray} \varepsilon _\rho (\rho,\rho _0) &=& (1 - \exp (- (\rho /p_1)^{p_2 })) \nonumber\\ \varepsilon _m (m,m_0) &=& (1 - \exp (- (m/p_3){\rm }^{p_4 })) \end{eqnarray} \tag{ 5 }$$

with the values p₁ to p₄ given in Table 2, plotted at intervals of 0.2 dex in m and ρ.

The separation of the effects of mass and environment is naturally not perfect but holds over 2 orders of magnitude in both mass and environmental density, with local deviations from the horizontal lines that are comparable to the observational uncertainties. The limited excursions of the data show that deviations from this simple separable behavior in m and ρ are rather small, equivalent to no more than ±0.2 dex in either variable, a tenth or less of the overall range of each parameter.

In other words, the differential effect of the environment on the red–blue mix of galaxies in the SDSS is independent of galactic stellar mass and vice versa. This good empirical separability of mass and environment means that we can write the red fraction in terms of ε_m and ε_ρ, by either of the first two equations, which reduce to the third:

$$\begin{eqnarray} f_{{\rm red}} (\rho,m) &=& \varepsilon _m (m,m_0) + \varepsilon _\rho (\rho,\rho _0)[ {1 - \varepsilon _m (m,m_0)}] \nonumber\\ &=& \varepsilon _\rho (\rho,\rho _0) + \varepsilon _m (m,m_0){\rm }[ {1 - \varepsilon _\rho (\rho,\rho _0)} ] \nonumber\\ &=& \varepsilon _\rho + \varepsilon _m - \varepsilon _\rho \varepsilon _m \end{eqnarray} \tag{ 6 }$$

with ε_m independent of ρ and ε_ρ independent of m. This implies a simple symmetry to the f_red(ρ, m) surface, which is illustrated in Figure 6.

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Since ε_ρ is zero in the lowest density regions (i.e., the voids), this separability means that ε_m(m) is easily interpreted as the red fraction in these lowest density regions. Likewise, ε_ρ(ρ) is the red fraction for very low mass galaxies, for which ε_m is by construction zero.

By inserting the two fitted relations (5) into Equation (6), we recover

$$\begin{equation} f_{{\rm red}} (\rho,m) = 1 - \exp \left({ - \left({\rho /p_1 } \right)^{p_2 } - \left({m/p_3 } \right)^{p_4 } } \right), \end{equation} \tag{ 7 }$$

which was previously proposed by Baldry et al. (2006) as one of the two empirical fitting functions for the f_red(ρ, m) surface in SDSS.

The clear separability of the effects of environment and mass, when parameterized in this way, suggests that there are two distinct processes at work. We will henceforth refer to these as "environment quenching" and "mass quenching" to reflect their (independent) effects on f_red across the (ρ, m)-plane. These two quenching processes will be governed by rates (i.e., the probability of being quenched per galaxy per unit time) of λ_ρ and λ_m, respectively.

The distinction between the two effects will be even more clearly seen when we consider how, observationally, ε_m and ε_ρ depend on cosmic epoch. For this we turn to our zCOSMOS sample in the next section.

4.3.1. The Empirical Signature of Environment Quenching

Figure 7 shows the equivalent plots of ε_m and ε_ρ from the zCOSMOS-10k data for 0.3 < z < 0.6. Although the zCOSMOS data are inevitably noisier, a good degree of separation between the effects of mass and environment is again discernable in zCOSMOS. The horizontal lines again show the fits to functions of the form of Equation (5), with the values given in Table 2.

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Figure 8 then compares the form of ε_m(m) and ε_ρ(ρ), derived from zCOSMOS in three redshift bins, using a fitting function of the form of Equation (5), with the values given in Table 2. We omit the lowest redshift bin (0.1 < z < 0.35) because it suffers from both a small overall volume and more severe edge corrections to the densities, and is dominated by a few large structures (see Figure 9 below). It is clear that the environmental efficiency curve ε_ρ does not vary significantly with redshift, neither within the zCOSMOS data set, nor between zCOSMOS and the local SDSS sample. Evidently, the differential effects of the environment, at fixed over-density, are independent of both stellar mass and cosmic time.

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It is important to appreciate that the redshift independence of our environmental quenching efficiency parameter ε_ρ(ρ) does not imply that the effect of the environment on the galaxy population is unchanging, since the environments of almost all galaxies will be migrating to higher over-densities as large-scale structure develops through gravitational instability, since almost all galaxies occupy regions with δ > 0. This growth of structure is seen in Figure 9, where we split the zCOSMOS galaxy population into quartiles of density and plot the median density of the highest and lowest quartiles as a function of redshift. The median over-density of a given quartile increases steadily with cosmic epoch and does so at a faster rate for the higher density environments. The galaxy population therefore occupies a shifting locus in ρ on the unchanging ε_ρ(ρ) curve, progressively broadening in ρ and extending further up onto the steeper part of the ε_ρ(ρ) curve as time passes.

Environmental effects within the galaxy population therefore develop and accelerate over time as the galaxy population migrates to a broader range of densities. This can be seen in our earlier zCOSMOS analyses of f_red in Cucciati et al. (2009), and the analogous analysis of morphology in Tasca et al. (2009), in which we split the galaxy population by environmental density quartiles as here. Both analyses showed a progressive development of differences between the highest and lowest density quartiles as the redshift decreased from z ∼ 1 to locally. Environmental effects are much weaker at z ∼ 1 than today simply because many fewer galaxies inhabit the high δ regions where the (unchanging) environmental effects are strongest.

4.3.2. Physical Origin of Environment Quenching

In the previous subsection, we showed that the environment apparently imprints itself on the galaxy population in a way that is uniquely given by the environment (over-density), independent of epoch and of the mass of the galaxy.

A natural contender for this characteristic of the environmental effect is the quenching of galaxies as they fall into larger dark matter haloes (Larson et al. 1980; Balogh et al. 2000; Balogh & Morris 2000; van den Bosch et al. 2008). Examination of the 24 COSMOS mock catalogs (Kitzbichler & White 2007) shows that the fraction of galaxies, at a given mass, that are satellite galaxies, f_sat, is strongly environment dependent, but, at fixed ρ or δ, is almost entirely independent of epoch (at least since z = 0.7), and of galactic mass, m (especially for m < 10^10.9 M_☉), as shown in Figure 10. These are precisely the same two characteristics that we have identified empirically for our "environment-quenching" process.

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If we suppose that "satellite quenching" quenches some fraction x of satellites as they fall into larger haloes, then it is easy to see that x will be given by the ratio of ε_ρ/f_sat. Inspection of Figure 10 shows that x takes a value that increases from about 30% at the lowest densities up to about 75% for our densest environments with log (1 + δ) ∼2. Interestingly, this is in the same range as the estimate (40%) of the fraction of satellites that are quenched from van den Bosch et al. (2008).

Ram pressure stripping and strangulation are usually considered as the physical mechanisms through which satellite quenching operates (see, e.g., Feldmann et al. 2010). Such processes may efficiently quench star formation, but would probably not lead to morphological transformations. Incorporation of morphological information into our picture could help us better understand this process, but this is beyond the scope of this paper.

4.4.1. Numbers of Galaxies

In Figure 8, the relative mass quenching efficiency ε_m(m) changes somewhat with cosmic epoch. More importantly, the individual masses of star-forming galaxies will be continuously growing due to their star formation. From both points of view, mass quenching must be a continuous process, and we therefore now consider the transformation rates of galaxies in this mass quenching process.

It is convenient to approach this via consideration of the numbers of galaxies. For a given set of galaxies, we define the transformation rate, λ, as the fraction of isolated (non-merging) blue galaxies that are quenched to form red galaxies per unit time. This may well be a function of mass and/or environment, and will have components due to both mass quenching and environment quenching, λ_m and λ_ρ. We have seen above that the λ_ρ term comes from the change in ρ with redshift, rather than the change in ε_ρ.

We must also consider the merging of galaxies, which we assume involves both blue and red galaxies (proportionally) as progenitors but which produces only red galaxies. For several reasons, it is convenient to characterize the merging rates κ in terms of the number of galaxies at the galactic mass of interest (rather than at the mass of the progenitors). We will therefore write the influx of new red galaxies into the mass bin per unit time as the product κ₊N_tot, while the efflux of red and blue galaxies out of the mass bin per unit time is assumed to be κ₋N_red and κ₋N_blue, respectively. The merging rates κ may well be functions of environment (Lin et al. 2010; L. de Ravel et al. 2010, in preparation; P. Kampczyk et al. 2010, in preparation) and possibly also of galactic stellar mass.

The number of blue galaxies in a given (fixed) mass bin will also be changing simply because ongoing star formation will bring up lower mass galaxies into the mass bin in question. Since the sSFR is roughly constant, the shift in logarithmic mass will be at most only weakly dependent on mass. We can therefore characterize this increase in the numbers via the logarithmic slopes of the mass function of star-forming galaxies and the sSFR–mass relations, the latter acting to compress more galaxies into a given interval of log mass. We henceforth define

$$\begin{equation} \begin{array}{l} \alpha = \dsty\frac{{d\log \phi _{{\rm blue}} }}{{d\log m}} \\\ms \beta = \dsty\frac{{d\left\langle {\log {\rm sSFR}} \right\rangle }}{{d\log m}}, \\ \end{array} \end{equation} \tag{ 8 }$$

with ϕ_blue being the number density of blue galaxies per unit log interval in mass. With this definition of β, the SFR is proportional to m^1+β. It should be noted that observationally, β is close to zero, but is probably slightly negative, β ∼ −0.1. We will follow β through in the following analysis, but will often assume for simplicity that it is zero. In the end, our conclusions are not very dependent on the precise value of β.

The parameter α will also be negative, strongly so at high masses, since for the well-known Schechter (1976) function given by ϕ(m) = ϕ*(m/M*)^1+αs exp(−m/M*), α will be given by

$$\begin{equation} \alpha = \left({1 + \alpha _s } \right) - \frac{m}{{M^*}}. \end{equation} \tag{ 9 }$$

Considering the changes in the numbers of red and blue galaxies, we get the following equations for the rate of change in the number of galaxies N per unit logarithmic mass bin, at fixed mass and environment,

$$\begin{eqnarray} \left. {\frac{{dN_{{\rm red}} }}{{dt}}} \right|_{m,\rho } &=& N_{{\rm blue}} \lambda _m + \kappa _ + N_{{\rm tot}} - \kappa _{-}N_{{\rm red}} \nonumber\\ \left. {\frac{{dN_{{\rm blue}} }}{{dt}}} \right|_{m,\rho } &=& N_{{\rm blue}} \left\{ { - \left({\alpha + \beta } \right){\rm sSFR} - \lambda _m - \kappa _- } \right\}, \nonumber\\ \left. {\frac{{dN_{{\rm tot}} }}{{dt}}} \right|_{m,\rho } &=& - N_{{\rm blue}} \left({\alpha + \beta } \right){\rm sSFR} + \left({\kappa _ + - \kappa _{-} } \right)N_{{\rm tot}}.\qquad \end{eqnarray} \tag{ 10 }$$

These exact continuity equations simply reflect the definitions of the quenching and merging rates λ and κ, once separability of the effects of mass and environment is adopted. The α × sSFR term accounts for the change in numbers of galaxies due to their increase in mass (bringing up galaxies from further down the star-forming mass function), while the β × sSFR term accounts for the compression or expansion of a given interval of logarithmic mass if β ≠ 0. They are valid at fixed m and ρ, which is why λ_ρ does not appear (since we have argued that environment quenching occurs as galaxies change environment). Later we will use the equation for dN_blue/dt for the population averaged over all ρ, and in this case, there will be an additional 〈λ_ρN_blue〉 term on the right-hand side of the first two equations (entering with plus and minus signs, respectively).

4.4.2. Information on Mass Quenching from the Red Fraction

For completeness, we look first at the change in the red fraction that is implied from these relations. However, we find that this is not as useful as a second approach (Section 4.4.3) and this section may be skipped if desired.

The change in f_red at fixed m and ρ is given by

$$\begin{eqnarray} \left. {\frac{{df_{{\rm red}} }}{{dt}}} \right|_{m,\rho } &=& \frac{1}{{N_{{\rm tot}} }}\left. {\frac{{dN_{{\rm red}} }}{{dt}}} \right|_{m,\rho } - \frac{{N_{{\rm red}} }}{{N^2 _{{\rm tot}} }}\left. {\frac{{dN_{{\rm tot}} }}{{dt}}} \right|_{m,\rho } \nonumber\\ &=& \left({1 - f_{{\rm red}} } \right)\left\{ {\lambda _m + \kappa _ + + f_{{\rm red}} \left({\alpha + \beta } \right){\rm sSFR}} \right\}.\qquad \end{eqnarray} \tag{ 11 }$$

The effect of merging out of the bin κ₋ has vanished in Equation (11), since merging was assumed to involve both blue and red galaxies proportionally. In terms of the simple red fraction, it can be seen that the effect of merging into the bin, κ₊, is indistinguishable from the blue to red quenching rate for isolated galaxies λ_m, and so we can lump these two together:

$$\begin{equation} \left({\lambda _m + \kappa _ + } \right) = \frac{1}{{\left({1 - f_{{\rm red}} } \right)}}\left. {\frac{{df_{{\rm red}} }}{{dt}}} \right|_{m,\rho } - f_{{\rm red}} (\alpha + \beta){\rm sSFR}{\rm.} \end{equation} \tag{ 12 }$$

The last term accounts for the fact that the number of blue galaxies (and therefore the red fraction) will change due to star formation increasing the mass of the blue galaxies. This will generally bring up more low mass galaxies than are moved out of the bin, unless the (α + β) combination is zero (for example, if the mass-function of blue galaxies is flat and the sSFR is independent of mass), or unless there is no star formation.

The df_red/dt at fixed m and ρ can be expressed in terms of the partial time derivatives of the observed quenching efficiencies introduced earlier:

$$\begin{equation} \left. {\frac{{df_{{\rm red}} }}{{dt}}} \right|_{m,\rho } = ({1 - \varepsilon _\rho })\frac{{\partial \varepsilon _m }}{{\partial t}} + ({1 - \varepsilon _m })\frac{{\partial \varepsilon _\rho }}{{\partial t}} \approx ({1 - \varepsilon _\rho })\frac{{\partial \varepsilon _m }}{{\partial t}}, \end{equation} \tag{ 13 }$$

the final approximation holding since ε_ρ is observed to be essentially time independent (see Section 4.3).

The environment-quenching rate λ_ρ that occurs as objects migrate toward higher densities (and thus higher ε_ρ) is straightforward since the numbers of objects will be conserved:

$$\begin{eqnarray} \left({1 - f_{{\rm red}} } \right)\lambda _\rho &=& \frac{{\partial f_{\rm red} }}{{\partial \log \rho }}\frac{{\partial \log \rho }}{{\partial t}} \nonumber\\ {\rm } &=& {\rm }\left({1 - \varepsilon _m } \right)\frac{{\partial \varepsilon _\rho }}{{\partial \log \rho }}\frac{{\partial \log \rho }}{{\partial t}}. \end{eqnarray} \tag{ 14 }$$

Putting together Equations (13) and (6), and then Equations (14) and (6), we therefore obtain the two transformation rates in terms of the observable quantities ε_m, ε_ρ, f_red, ρ, α, and β as follows:

$$\begin{eqnarray} \left({\lambda _m + \kappa _ + } \right) &=& \frac{1}{{\left({1 - \varepsilon _m } \right)}}\frac{{\partial \varepsilon _m }}{{\partial t}} - {\rm }f_{{\rm red}} \left({\alpha + \beta } \right){\rm sSFR} \nonumber\\ \lambda _\rho &=& \frac{1}{{({1 - \varepsilon _\rho })}}\frac{{\partial \varepsilon _\rho }}{{\partial \log \rho }}\frac{{\partial \log \rho }}{{\partial t}}. \end{eqnarray} \tag{ 15 }$$

The (λ_m + κ₊) combination could be functions of m and t, and also conceivably of ρ, since the merging rate may depend on ρ and because the f_red in the mass-growth term will also be higher in high density environments. The environmental transformation rate λ_ρ should not have any dependence on mass because of the demonstrated separability of ε_ρ and ε_m noted above. The simple form of these equations is a consequence of the separability of our two evolutionary processes in the (m, ρ)-plane.

Given the small changes in ε_m with cosmic epoch (see Figure 8), the dominant term in the quenching of galaxies (λ_m + κ₊) in Equation (15) is the second term, due to the increase in mass of the star-forming galaxies, and not the first, which arises from the change in red fraction at fixed mass. The precise time dependence of the ε_m curve is also quite sensitive to the choice for the redshift dependence of the color cut, and it is noticeable that the curves for zCOSMOS show even less change within their redshift range than the difference between zCOSMOS and SDSS. The apparent changes in ε_m in Figure 8 should therefore be treated with caution.

For both these reasons, it is therefore more revealing to look directly at the evolution of the mass function of star-forming galaxies, which looks at the galaxies that have not been quenched. This topic is therefore examined in the next section.

4.4.3. Information on Mass Quenching from the Mass Function of Star-forming Galaxies

The mass function of star-forming galaxies reflects the survival of those star-forming galaxies that have not been quenched, giving a clearer view of the action of quenching. Consideration of the star-forming mass function, rather than the color mix, will require us to average the effects of environment quenching since we will lose direct information on the environment. This is almost unavoidable because the definition of environment (e.g., quartiles of the density distribution of galaxies) will be intrinsically linked to the amplitude of the observed mass (or luminosity) functions.

We commented above (Section 3.4) on the remarkable stability of the mass function of star-forming galaxies over a broad range of epochs, despite the large increase in the masses of star-forming galaxies obeying Equation (1) (see, e.g., Renzini 2009). We show in this section that the observed constancy of the exponential cutoff M* for the star-forming mass function demands a strikingly simple form of the mass-quenching law in which the mass-quenching rate is directly proportional to the SFR alone, independent of epoch. This must operate for masses around and above M*. In Section 5.2, we show that this mass quenching law also naturally explains the Schechter form of the passive mass function over a much wider range of masses, establishing its viability over about two decades of galactic mass.

It is easy to see how this requirement comes about at high masses in the regime of the exponential cutoff: the rate of increase in log m of a galaxy is proportional to its individual sSFR. If the sSFR across the population is at least approximately independent of mass, then the mass function of star-forming galaxies will shift to higher masses at a speed (in log m space) that is proportional to the sSFR. This shift must be counteracted by quenching and so the quenching rate must be proportional to the sSFR. Second, in order to maintain the exponential cutoff at the same mass, the quenching rate must also be proportional to mass, because more rapid quenching is required where the mass function of star-forming galaxies is steepest, i.e., where the effect of increasing mass will most strongly affect the numbers of objects. The logarithmic slope α of the Schechter function is proportional to mass, at high masses (see Equation (9)). The product m × sSFR is clearly just the SFR.

We can also see this analytically by looking at what would be required if the mass function of blue star-forming galaxies had a Schechter form that was observed to be absolutely constant with epoch, i.e., with constant M*, α_s, and ϕ*. In this heuristic case with a simple analytic solution (which we stress is not observed in the sky), the rate of cessation of star formation is obtained by simply setting dN_blue/dt = 0 in Equation (10) and inserting expression (9) for α for a Schechter function:

$$\begin{eqnarray} \lambda + \kappa _ - &=& - {\rm sSFR}(\alpha + \beta) \nonumber\\ &=& - {\rm sSFR}\left({{\rm 1\, +\ }\alpha _s \,+\, \beta - \frac{m}{{M^*}}} \right). \end{eqnarray} \tag{ 16 }$$

There is now a degeneracy between the quenching of isolated galaxies and the merging rate out of the mass bin, κ₋, since clearly, in terms of depleting the star-forming population, there is no difference between cessation of star formation in an isolated galaxy and cessation due to a merger. We will refer to the combination λ + κ₋ as the combined "death function," η, of star-forming galaxies, which is simply the probability that star formation ceases in unit time, for whatever reason. The inverse η⁻¹ is a statistical timescale for quenching to occur (which should not be confused with the time required for the physical transformation to take place, once started).

It is instructive to consider the origin of the two terms on the right-hand side of the second part of Equation (16) in this heuristic example. The constant 1 + α_s + β term acts to keep the normalization of the Schechter function constant even though star formation is bringing up a (generally larger) number of lower mass galaxies. The steepening of the star-forming mass function that would occur if β is not zero is counteracted directly by the small mass dependence of the sSFR term. The most crucial point, however, is that the sSFR × m/M* term acts to keep the characteristic M* in the Schechter function the same, as introduced above. As noted above, the m dependence is required to produce the exponential cutoff at M*. The sSFR dependence comes because this controls the rate of logarithmic increase in stellar mass. Since the sSFR is simply the SFR/m, this term reduces to SFR/M*.

In the high mass regime around and above M*, Equation (16) clearly reduces to the following very simple form for the (total) quenching rate (which will be dominated by mass quenching) because (1 + α_s + β) is small and κ is small compared to λ:

$$\begin{equation} \eta \sim \lambda _m \sim \frac{{{\rm SFR}}}{{M^*}} = \mu {\rm SFR}{\rm.} \end{equation} \tag{ 17 }$$

Having established this at high masses, we can then look at what happens to the mass function of star-forming galaxies if we were to postulate that the mass quenching rate has this simple form (Equation (17)) for galaxies of all masses, in all environments, and at all epochs.

Putting the expression for the slope α(m) of the Schechter function from Equation (9) into the continuity Equation (10), it can be seen that the mass function of star-forming galaxies would drift up in normalization, unless (1 + α_s + β) was zero (whereas it probably takes a value around −0.5). The rate of increase in ϕ* will be proportional to the overall sSFR, and thus most of the evolution would occur at high redshifts where the sSFR is highest. Encouragingly, this is exactly what is seen in the data (Pérez-González et al. 2008; Pozzetti et al. 2009; Ilbert et al. 2010) as discussed in Section 3.4 above. In addition, adoption of this simple death function for all galaxies would imply that the faint end slope would gradually steepen with time, depending on the value of β (with no change for the simple case of β = 0). As noted in Section 3.4, there is even evidence in the data for a small shift in this direction, especially at z > 1.

Since these changes to the star-forming mass function are exactly what are observed, we are encouraged to postulate Equation (17) to be the universal form of mass quenching λ_m for all galaxies. Although this very simple form of the mass quenching is only required (by the constancy of M* of star-forming galaxies) at high masses, we show below (in Section 5.2) that it in fact reproduces the observed Schechter mass function of passive galaxies over more than 2 orders of magnitude in mass, i.e., extending well below M* into the regime where the Schechter mass function becomes a power law. We will therefore conclude below that our postulated mass quenching law (17) must be valid over this much broader (2 dex) range in mass and will assume this for the remainder of Section 4.

4.4.4. The Combined Rate of the Three Quenching Mechanisms

We can now combine the "environment quenching" and "merger quenching" with the "mass quenching" explored in Section 4.4.3. We demonstrated above that environment quenching must be independent of mass because ε_ρ is independent of mass (and time). Observationally, there is no clear indication of how the merging rate of galaxies depends on the galactic mass and we will assume for simplicity that this is also independent of mass, as with environment quenching. Since both are the results of the growth of structure, this seems a reasonable assumption. On the other hand, mass quenching will be roughly proportional to mass at any given epoch because of the dependence of SFR on mass.

There is therefore a good motivation to consider a more general form for the combined quenching rate. We write the overall death function, η, acting on the population as a whole, as the sum of the star formation driven term given in Equation (17), plus the mass-independent term η_ρ that is naturally associated with the environment quenching rate λ_ρ, defined in Equation (15), plus a (mass-independent) merging term κ₋. This may be written as

$$\begin{eqnarray} \eta &=& \lambda _m + ({\lambda _\rho + \kappa _ - }) \nonumber\\ &=& \mu {\rm SFR} + \left({\frac{1}{{1 - \varepsilon _\rho }}\frac{{\partial \varepsilon _\rho }}{{\partial \log \rho }}\frac{{\partial \log \rho }}{{\partial t}} + \kappa _ - } \right) \nonumber\\ &=& \eta _m + \eta _\rho. \end{eqnarray} \tag{ 18 }$$

Note that η_ρ is independent of m and η_m is independent of ρ. At a given epoch, η_m is proportional to m^(1+β) ∼ m for β ∼ 0. Figure 11 shows the derived dependences of η_m, λ_ρ, and κ₋ on time and, for η_m, also on galactic stellar mass. The λ_ρ and κ₋ are plotted for both the first and fourth environmental density quartiles, as derived above.

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Comparing all of the rates, we conclude that mass quenching dominates at all epochs for the highest mass galaxies, m > 10^10.5 M_☉, i.e., at and above M*, while for low mass galaxies below 10¹⁰ M_☉, environmental effects dominate at late epochs, with merging likely dominating at higher redshifts z > 0.5. We will return to this point below, but point out here the close connection between "merging" and "satellite quenching." Both result from the merger of two dark matter halos. If the baryonic galaxies merge, we will see a "merger." If they do not, we will see a new satellite galaxy that may be quenched through satellite quenching.

4.4.5. Establishing the Schechter Function Exponential Cutoff

To see clearly the role of the quenching law (Equation (18)) in establishing, as well as maintaining, the value of M*, we consider the change in the number of galaxies that is given by Equation (10), together with the quenching law (Equation (18)) applied to a Schechter function that initially has some different M*(t). We can then write

$$\begin{eqnarray} \frac{1}{{N_{{\rm blue}} }}\left. {\frac{{dN_{{\rm blue}} }}{{dt}}} \right|_m &=& - {\rm sSFR}(\alpha + \beta) - ({\lambda _m + \lambda _\rho + \kappa _ - }) \nonumber\\ &=& - {\rm sSFR}({1 + \alpha _s + \beta}\nonumber\\ && {- m({M^{*}(t)^{ - 1} - \mu })}) - \eta _\rho. \end{eqnarray} \tag{ 19 }$$

Clearly, if M*(t) is less than μ⁻¹, the number of galaxies at high m will increase strongly due to the term in the inner brackets. If M*(t) is greater than μ⁻¹, the number of galaxies at high m will likewise drop strongly. Only when M*(t) = μ⁻¹ will the term in the inner brackets become zero and the change in the mass function of star-forming galaxies be independent of mass (if β = 0). Thereafter, the evolution in the mass function will be confined to changes in ϕ* (with mild changes in α_s if β ≠ 0).

In other words, if we start with some IMF with generally low galactic masses, the mass function of star-forming galaxies will initially traverse across to higher masses (with uniform shape if β = 0) due to star formation. As significant numbers of galaxies approach m ∼ μ⁻¹, the quenching law acts to establish the exponential cutoff in the star-forming mass function at M* = μ⁻¹. At that point, and thereafter, a "garbage-pile" of passive galaxies will build up around this value of M* as star formation continues to feed galaxies into the "buzz-saw" of the quenching. The overall mass function will subsequently evolve only vertically in density, with fixed shape (if β = 0). The action of the mass-independent quenching terms, i.e., those due to environment and merger quenching, will be to cause additional passive galaxies to rain down from all masses.

To look at the evolution in ϕ* once M* stabilizes, we simply integrate Equation (19) with M*(t) = μ⁻¹

$$\begin{eqnarray} \ln \left. {N_{{\rm blue}} } \right|_m &=& - \int_{t_{{\rm init}} }^t {({{\rm sSFR}({1 + \alpha _s + \beta }) + \eta _\rho })} dt, \nonumber\\ \left. {N_{{\rm blue}} } \right|_m &=& {\rm }\left. {N_{{\rm blue,init}} } \right|_m \left({\frac{m}{{m_{{\rm init}} }}} \right)^{ - (1 + \alpha _s + \beta)} ({1 - f_{\eta _\rho } }). \qquad \end{eqnarray} \tag{ 20 }$$

Here f_ηρ is just the fraction of galaxies that have suffered quenching (or merging) from the mass-independent η_ρ term, and m_init is the mass that galaxies seen at time t would have had at time t_init, i.e., m/m_init is the mass increase of these galaxies in the time interval of interest. If we substitute into Equation (20) a Schechter mass function at some initial time t_init, with a value of M* = μ⁻¹ as in the death function (Equation (18)), then we find that the mass function remains, as expected, a Schechter function with exactly the same characteristic M*, a normalization ϕ* that gradually increases with time by an amount that is related to the fractional increase in mass of star-forming galaxies, modulated by a decrease due to the mass-independent quenching due to the average η_ρ in the population. Unless β ∼ 0, the Schechter faint-end slope α_s will steepen, i.e., become slightly more negative, due to the mass dependence of the m/m_init term, which is of course independent of m for β = 0. Both effects will be more prominent at high redshift where the sSFR is much higher and the mass growth term correspondingly larger. These results are of course the same as we argued above. If the initial M* is smaller than the μ⁻¹ in our quenching law, then the star-forming mass function shifts to higher masses until M* reaches this value.

4.4.6. Characteristics of Mass Quenching

We therefore argue that the observed evolution of the mass function of star-forming galaxies, i.e., constant M* and a gradually increasing ϕ* (Bell et al. 2007; Pérez-González et al. 2008; Pozzetti et al. 2009; Ilbert et al. 2010) demands the remarkably simple form (Equation (17)) for the mass quenching of star-forming galaxies: one that is proportional only to the SFR and independent of stellar mass and epoch (except insofar as these two determine the SFR). To this must be added a mass-independent term, which will of course dominate at low masses and which can be identified by the environment-quenching plus mass-independent merging.

We should stress that although we have referred to a "mass-quenching" effect because it produces the strong dependence of f_red on galactic mass, the mass quenching process is, paradoxically, independent of galactic mass per se. At a given epoch, the SFR of course depends strongly on mass since the sSFR has β ∼ 0, but this is in our analysis a secondary correlation, although one which has an important effect on the shape of the mass function, as discussed below. The primary dependence of quenching on SFR evidently holds rather well since z ∼ 2, during which period the SFR, at a given stellar mass, has declined by a factor of 20.

We stress that the "M*" in our mass-quenching law Equation (17) is a constant across the whole population of galaxies, and across a wide range of cosmic time. It therefore has nothing to do with the masses of individual galaxies, either in stars or in gas. Our mass-quenching law (Equation (17)) does not therefore have much to do with the consumption timescales of gas in a particular galaxy or halo, and evidently has little to do with any properties of individual galaxies, except for their individual instantaneous star formation rates.

The mass-quenching death rate is evidently simply the star formation rate, divided by the (constant) characteristic mass M* of the Schechter mass function of star-forming galaxies. The value of M* is empirically around 10^10.6 M_☉, so from Equation (17) we obtain quantitatively

$$\begin{equation} \lambda _m \sim \left({\frac{{{\rm SFR}}}{{40\;M_{{\odot}} \;{\rm yr}^{ - 1} }}} \right){\rm Gyr}^{ - 1}. \end{equation} \tag{ 21 }$$

Implicit in this scenario is a strong link between the star formation rate and the star-forming lifetime of a galaxy. Applying Equation (21) with star formation rates of order 1000 M_☉ yr⁻¹ implies a lifetime η_m⁻¹ ∼ 40 million yr, comparable to the implied lifetimes of the ULIRG phase (see, e.g., Sanders & Mirabel 1996; Caputi et al. 2009). The massive (m ∼ 10^10.7 M_☉) galaxies at z = 2 that are seen with high star formation rates (SFR ∼ 100 M_☉ yr⁻¹) must be close to the point of death, with quenching timescale τ_q ∼ η_m⁻¹ ∼ 0.4 Gyr, a few dynamical times at best. It will be interesting to see whether the characteristic dynamical properties of these objects (in-spiraling disk instabilities) that have recently been revealed by high-resolution integral field spectroscopy (Genzel et al. 2008) are confined to these galaxies that are evidently on the point of being quenched, or are a more general phenomenon that is exhibited by less terminal galaxies at the same sSFR, but lower masses, at the same redshifts.

4.4.7. Possible Physical Origins of Mass Quenching

The conclusion from the extended discussion in the previous sections is very simple: a quenching rate that is simply proportional to the SFR establishes and maintains a Schechter mass function for star-forming galaxies with a value of M* that is constant, despite the increase in individual stellar masses.

As we have stressed, our analysis is purely empirical and there may or may not be a direct causal link with star formation. As a trivial example, if the mass-quenching rate was for some reason proportional to galactic mass and to some other quantity that was proportional to the cosmic epoch to the inverse 2.2 power, it would look exactly as if it was proportional to the SFR, even if there was no direct physical link with star formation. We explore this aspect further in Section 7.3 below, where we show the links between our scenario, involving linking the quenching rates and star formation rates, and a picture involving a static "mass limit" for galaxies.

This statistical analysis cannot establish such a causal link. Rather the goal has been to identify the signatures of the evolution that any more physically based model must obey. Having said that, what physical process could cause a quenching rate that is indeed causally proportional to the star formation rate, independent of stellar mass and epoch? One obvious possibility is some feedback process involving energy injection from supernovae and the creation of superwinds, or some other process such as the local ionization model of Cantalupo (2010). Nevertheless, it is surprising that the depth of the potential well, or some other properties of the local environment, do not appear to enter.

Another possibility could be related to AGN feedback. In Silverman et al. (2009a; see also Daddi et al. 2007b) we showed that the ratio of the black-hole accretion rate (BHAR) dm_BH/dt and the star formation rate in zCOSMOS X-ray selected AGNs was roughly constant, at least back to z ∼ 1, with a mass ratio of order 0.01. A similar ratio was found in the SDSS by Netzer (2009). Multiplying this ratio by the observed AGN fraction of about 0.03 for these same objects (Bundy et al. 2008; Silverman et al. 2009b) gives an average BHAR of 0.0003 SFR. This means that the chance of having a quasar-like outburst could be (statistically) proportional to the SFR. Equation (21) could therefore be re-written (at least statistically) in terms of a BHAR

$$\begin{equation} \eta _m = 80\;\left(\frac{\rm BHAR} {M_{\odot}\;{\rm yr}^{ - 1}} \right)\;{\rm Gyr}^{ - 1}. \end{equation} \tag{ 22 }$$

We note that if this connection with AGNs is valid, this would imply that powerful quasars with BHAR ∼ 1 M_☉ yr⁻¹ should lead to rapid quenching with τ_q ∼ 10⁷ yr.

Regardless of the actual physical mechanism, our model can be used to make a clear prediction for the statistical properties (i.e., their mass function) of those objects that are being seen in the process of being quenched. This prediction, which may help in identifying plausible transient objects, and thus lead to a physical understanding of mass quenching, is developed in Section 5.6 below.

We showed in Section 4.4.3 how the observed constancy of M* of star-forming galaxies requires a particular form of the mass-quenching rate, i.e., one proportional to the star formation rate alone. We can turn the argument around and suggest that a mass-quenching rate for star-forming galaxies that is proportional simply to the instantaneous star formation rate could be the physical mechanism for producing the distinctive Schechter function for star-forming galaxies, and the observed constant characteristic mass M*, independent of whether and how the actual sSFR changes with time. This idea is supported by the analysis in Section 4.4.5, which shows how this quenching law can also establish the exponential cutoff at this particular mass, as well as simply maintaining it.

In this picture, there is thus a very simple relation between the constant of proportionality between the star formation rate and the mass-quenching death rate, μ, and the resulting characteristic mass of the Schechter function. If the mass-quenching death rate is given by

$$\begin{equation} \eta_m = \mu {\rm SFR} \end{equation} \tag{ 23 }$$

with μ being some constant reflecting the physics of the quenching process, then the result will be to produce a Schechter mass function for the star-forming galaxies, with a characteristic mass M* given by

$$\begin{equation} M^* = \mu ^{ - 1}. \end{equation} \tag{ 24 }$$

Although we cannot identify the value of μ in physical terms, we suggest that it is this parameter that actually determines M* (and not vice versa).

We established in Section 4.4.3 that the action of our mass quenching law is to produce a single Schechter function for the star-forming galaxies with this characteristic M*. Any additional quenching that is independent of mass will cull galaxies uniformly from the mass function and will not change its shape.

Quenching will halt further mass growth, except through the effects of merging, and so the mass function of passive galaxies should be closely related to that of the blue star-forming galaxies which in turn, we have argued, should be determined by the mass-quenching parameter μ in Equation (18).

At any epoch, the star formation rate (and thus quenching rate) in star-forming galaxies will be proportional to m^1+β. Thus, if the star-forming population at some epoch obeys a Schechter function with some M*_blue and α_s,blue, then the passive galaxies produced (per unit time) by the mass quenching of this population will have the same mass-function shape, multiplied by m^1+β. Given the form of the Schechter function,

$$\begin{equation} \phi (m)dm = \phi^*\left({\frac{m}{{M^*}}} \right)^\alpha e^{ - m/M^*} dm, \end{equation} \tag{ 25 }$$

the passive mass function will therefore be a new Schechter function with exactly the same value of M*, but with a faint-end slope α_s,red that is modified by (1 + β), i.e.,

$$\begin{equation} \alpha _{s,{\rm red}} = \alpha _{s,{\rm blue}} + (1 + \beta). \end{equation} \tag{ 26 }$$

This will be true at a given epoch and true at all epochs if the α_s and M* of the star-forming galaxies are unchanging, as is more or less observed, as in Section 3.4 and Table 1.

We will therefore get a new Schechter function with exactly the same M* but with a modified faint end slope which will differ by an amount that is related to the slope of the sSFR–m relation, Δα_s = (1 + β) ∼ 0.9, or Δα_s = 1 for the simple case of β = 0. This is a clear consequence of the simple mass-quenching law that we have adopted, provided that it extends down to masses well below M*, as we have postulated.

It has been known for many years that the Schechter function for (massive) passive galaxies has a faint end slope that is substantially less negative than that of star-forming galaxies (e.g., Binggeli et al. 1988; Lilly et al. 1995; Folkes et al. 1999; Blanton et al. 2001; Ball et al. 2006). Already a decade ago (Folkes et al. 1999), it was established that the faint end slope differs from that of star-forming galaxies by Δα ∼ 1. Our quenching law, derived (and extended) from the time dependence of the mass function of star-forming galaxies at high masses, naturally produces this striking feature of the passive galaxy population over a much broader range of masses.

The result of the mass-independent term(s) in the overall death function given by Equation (18), which we associated with environmental quenching and/or mass-independent merging of galaxies, will be to produce a second population of passive galaxies whose mass function simply shadows the mass function of star-forming galaxies, with exactly the same M* and α_s, but (generally) lower ϕ*. This second Schechter component will therefore dominate the mass function of passive galaxies at low masses.

There is good evidence for the existence of this second component of passive galaxies. An upturn in the mass function of passive galaxies at low masses is clearly seen in the SDSS r-band luminosity function of Blanton et al. (2001; see also our own SDSS analysis below) and has also been highlighted in the deep photo-z based COSMOS analysis of Drory et al. (2009, see their Figure 5). We would expect the size (i.e., ϕ*) of this second lower mass population of passive galaxies to be larger in higher density environments because of the larger λ_ρ term in Equation (18), the higher merging rate, and the requirement that f_red increase with density for all masses, including these very low ones. All other things being equal, we predict that the strength of the second (low mass) component would be a few (approximately four) times stronger in the highest density quartile than in the lowest quartile, reflecting the higher environment quenching and merging rates in the denser environments.

To summarize, if we neglect for the moment the effects of any subsequent merging of quenched galaxies (see Section 5.5 below), then we would expect to see a double (two-component) Schechter function for the passive galaxies. The two components will have exactly the same M* but have two α_s that differ by (1 + β), one of which will be the same as that of star-forming galaxies and the other shallower (i.e., less negative) by an amount related to the slope of the sSFR–m relation. Furthermore, the common M* value should precisely match that of the star-forming galaxies.

The two Schechter components in the mass function of passive galaxies therefore reflect the two distinct quenching routes. At any given epoch, one is roughly proportional to mass, because of the flat sSFR–mass relation, and the other is quite independent of mass, through the separability of mass and environment established in this paper.

If we now add together the mass functions of both the star-forming and passive galaxies, we would therefore simply predict another double (i.e., two-component) Schechter function. Both components will again share the same M* with the two α_s differing as above by 1 + β ∼ 0.9. This is again a simple prediction of the adopted death function given by Equation (18).

The above discussion considered the effect of merging in the quenching of galaxies, but did not include the changes in the mass functions due to either the increase in the masses and reduction in the number densities of the merger remnants. These two effects can be thought of as translating some fraction of the mass function to lower number densities and higher masses. For galaxies at low masses, which were generally quenched by merging or by our environment-quenching process, and where the mass function is a power law, these two effects will not substantially change the shape of the mass function.

However, for galaxies which were originally mass quenched, i.e., those around M* where the mass function is strongly peaked (with α_s ∼ −0.4), the effect of subsequent merging on the mass function may be significant, especially at the highest masses where the mass function steepens dramatically. These galaxies also generally quenched early, and therefore have more time for subsequent merging. Most of the mergers at these masses will be "dry," i.e., involve already-quenched galaxies, simply because these dominate the galaxy population. If the merger rate is independent of mass (and mass-ratio), this subsequent dry-merging will produce shadow mass functions that are simply displaced to higher masses and lower densities. These will likely dominate the resulting total mass function at very high masses where the mass function is steep.

The resulting mass function of passive galaxies will therefore be more complex than the single Schechter function(s) described above. However, if the composite mass function (summing the non-merged and the displaced merged populations) is force fit by a single Schechter function, then the result is generally to increase M* and decrease α_s (i.e., make it more negative). To illustrate this effect, we construct a heuristic composite mass function in which we assume that 15% of galaxies have undergone a 1:1 merger. This is done by adding a 7.5% component of galaxies with Δm = +0.3 dex to an 85% unmerged population that is described by a single Schechter function with α_s = −0.4. We then force fit this new composite population with a single Schechter function. In this particular case, α_s decreases by 0.15 and M* increases by 0.09 dex, i.e., Δα_s ∼ 1.6 ΔM* for small amounts of merging.

Interestingly, a lot of this shift in M* is associated with the degeneracy between M* and α_s in the Schechter fits, since the average mass of the galaxies in this simple illustration has evidently increased by only 0.03 dex, i.e., only a third of the change seen in M*. Merging at larger mass ratios would require a larger fraction of mergers to produce the same distortion of the mass function and would produce a shift in M* that was closer to the change in average mass.

Provided that the amount of post-quenching merging of passive galaxies is modest, it evidently produces a relatively small perturbation to the predictions described in the previous section. It should be noted that even though the merger rate for passive galaxies is small overall, the most massive galaxies on the steepest part of the mass function will almost all have undergone significant merging, as we discuss further in Section 6 (see Figure 16 below).

The above analysis made five simple predictions for the forms and inter-relationships of the mass functions of star-forming and passive galaxies in different environments in our simple empirically driven model.

• 1.  
  The mass function of star-forming galaxies should be a pure single Schechter function, but that of passive galaxies should be a double Schechter function.
• 2.  
  The mass functions of star-forming galaxies in different environments should quantitatively have the same values of M* and α_s.
• 3.  
  The mass functions of star-forming galaxies and of the passive galaxies in the lowest density D1 quartile (where post-quenching merging can be neglected) should have exactly the same M*, but values of those differ by (1 + β) ∼ 1.
• 4.  
  The secondary component of passive galaxies, which dominates at the lowest masses, should have the same M* and α as the star-forming galaxies, and (easier to verify) an amplitude ϕ* that is higher (by about a factor of 4) in the high density D4 environment compared with D1.
• 5.  
  The primary component of the mass function of passive galaxies should have been modified in D4 through post-quenching merging, producing a higher M* and less negative α_s, with Δα_s ∼ 1.6 ΔM*.

We have tested the first five simple predictions by constructing the mass functions of the SDSS red and blue samples, plus the overall mass function, in the highest (D4) and lowest (D1) quartiles of environmental density, using the same V_max approach that we described above. These are shown in Table 3 and Figure 12. We also show in Figure 12 the overall mass function constructed from the SDSS DR4 by Baldry et al. (2008), which shows excellent agreement with our own overall mass function.

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For the blue galaxies we use only a single Schechter function since this provides a fully acceptable fit, while for the red galaxies we use a double (two-component) Schechter function, with the M* of the two components constrained to always have the same value, it being almost impossible to constrain the M* of the weaker component independently. As is clear from Figure 12, the mass function of passive galaxies indeed requires a two-component Schechter function. We also find that we have to fix the α_s of the weaker of the two passive components to be the faint end slope of the blue population since this is also poorly constrained by the data.

All of the basic quantitative relationships predicted by our simple model are seen to hold rather well. For example, the M* for passive and star-forming galaxies in the overall mass function (not differentiating at all by environment) differs by only 0.05 ± 0.02 dex.

Turning to the lowest density D1 quartile, where we could expect the effects of merging to be minimal, we find that the two M* (blue and red) are essentially identical, i.e., differing by 0.01 ± 0.02 dex, i.e., well within the observational uncertainties. The α_s of the dominant population of passive galaxies has Δα_s = 1.03 ± 0.05 relative to the blue galaxies, compared with the prediction of Δα_s = (1 + β) = 0.9 for β = −0.1, and of course Δα_s = 1.0 for the simple case of β = 0. We take these results as a compelling vindication of our simple model.

Looking at the overall mass function for all galaxies, we find Δα_s = 1.04 ± 0.13 for the two Schechter components (with M* constrained to the same). In their analysis of the SDSS DR4, Baldry et al. (2008) also found a double Schechter function with α_s of −1.58 and −0.46, i.e., with Δα_s ∼ 1.12, i.e., consistent with our own analysis, and also with the expected value of Δα_s from our model.

Finally, turning to the highest density quartile, D4, it can be seen that the α_s of the dominant passive population is a little more negative, by Δα_s ∼ 0.2 ± 0.08, in the densest D4 quartile compared with D1, and that M* is also bigger by about 0.16 ± 0.02 dex. As discussed in Section 5.4, these small differences can easily be explained by a modest amount (≤30%, and possibly <15%) of dry merging of the passive galaxies in the D4 environment (and none in D1). In contrast, the blue population in the different environments is very similar, i.e., the two M* are within 0.04 ± 0.03 dex and the two α_s are within 0.02 ± 0.04. This compelling similarity is again expected in our simple model, since the shape of the mass function for the blue galaxies will not be at all affected by merging, provided that the merger rate is independent of mass and that mergers remove galaxies from the blue main sequence.

We draw from the above three important conclusions:

• 1.  
  our simple model works extremely well, especially in the low-density D1 quartile where the effect of merging around M* is negligible;
• 2.  
  the success in correctly predicting the mass function of passive galaxies over a broad range of masses extending well below M* shows that our mass-quenching law must be valid over a much broader range of masses (of star-forming galaxies) than it was originally derived from (see Section 4.4.3); and
• 3.  
  the amount of merging even in dense environments is quite constrained, if our model is correct. We would be very surprised, given the observed increase in M* of 0.16 dex, if the average mass of typical passive galaxies (around M*) in the densest quartile had increased by more than this same amount (i.e., 40%), and, in fact, given the factor of 3 gain in ΔM* relative to the mean increase in galactic stellar mass discussed in Section 5.4, we suspect that the change in the average mass of these galaxies due to dry merging is probably considerably smaller, i.e., possibly as low as 0.06 dex (15%).

Given that passive galaxies clearly dominate the high mass end of the mass function (e.g., Figures 4 and 12), readers may be surprised by our conclusion that the blue sequence of star-forming galaxies can populate the red sequence with little or no post-quenching dry merging, completely so in D1, and with only 15%−40% merging in D4 (cf. the different scenarios of Faber et al. 2007). We stress that this conclusion follows (if our model is correct) from the observation that the M* of active and passive galaxies is essentially identical. Although passive galaxies clearly vastly outnumber star-forming ones at the highest masses, the most massive star-forming galaxies are so rapidly quenched that they are visible for only a short period of time, making it fully possible for the red sequence to be populated directly from the blue sequence, at least in D1.

It is often thought that the color distribution of galaxies as a function of mass (e.g., Figure 4) indicates a sharp mass "threshold" between star-forming (lower mass) and passive (higher mass) galaxies and there have been speculations as to the evolution of this transition mass. We stress however that red and blue galaxies in fact form a completely overlapping population with essentially the same M*. In this context, it should be appreciated that two Schechter functions that have the same M* but very different α_s (as in the blue and red populations) "look" quite different, even around M*. As an example, the "typical" mass of galaxies, i.e., the mass at which mϕ(m) peaks, differs by 0.6 dex for two Schechter functions with the same M* but with Δα_s = 1. Likewise, the relative numbers of galaxies in the two populations shift from one population to the other as m^Δα.

To gain an appreciation of the relative sizes of the different components of the galaxy population in different environments, we fix all of the Schechter parameters except ϕ* (to avoid the well-known degeneracies). These are shown in the second part of Table 3. The values of ϕ* in each quartile of density are arbitrary, since they depend on the definition of the quartile. The most useful approach is to normalize each component to the ϕ* value for the dominant (α_s = −0.4) passive population in the two density quartiles. This yields relative normalizations for the two α_s = −1.4 components of 0.45 (blue D1), 0.17 (blue D4), 0.028 (red D1), and 0.13 (red D4).

Normalizing in this way, we find that the secondary passive component is some 4.5 times larger in D4 compared with D1. This reflects the higher environment-quenching and merger rates in the denser environment building up this component.

It can also be seen that the sum of the two α_s = −1.4 components (blue + red) is 60% larger in D1 (0.48) than it is in D4 (0.30), when both are normalized to the respective α_s = −0.4 passive populations. This is the effect we drew attention to in Bolzonella et al. (2009), in which the "hump" of the double Schechter function is more pronounced in the higher density environments. We suspect that it reflects a difference in the mass function of star-forming galaxies at very early times in the denser environments which leads to more galaxies reaching the mass-quenching regime around M* before the sSFR starts to drop. This is illustrated in our numerical model in Section 6.

Finally, although they are most beautifully established in SDSS, we note that there is quite good evidence in favor of these basic relationships at higher redshifts also. If we average the fitted mass-function parameters in zCOSMOS for 0.1 < z < 0.7 from Bolzonella et al. (2009), which mostly samples relatively massive galaxies, we find the same values of M*, with log M*/M_☉ = 10.77 and 10.82 for red and blue galaxies, respectively, and with α_s = −0.32 and −1.4, i.e., a Δα_s slope difference of 1.1, close to our prediction of about 1.0.

Regardless of the physical origin of mass-quenching, there is a clear prediction from our model for the mass function of those galaxies that are in the process of being mass-quenched. This prediction may help in identifying plausible candidates for such objects, be they associated with AGN or some other transient phenomenon such as unusual color–morphology combinations, etc.

We assume that the process of mass quenching is associated with some recognizable transient signature that is visible for some period of time τ_trans. The simplest assumption would be that τ_trans is independent of our variables of m, ρ, and t. In this case, it is easy to see from Equation (10), and setting κ = 0, that the mass function of the transient "being-quenched" objects should follow:

$$\begin{eqnarray} \phi_{{\rm trans}} (m,t) &=& \frac{{dN_{{\rm red}} }}{{dt}}\tau _{{\rm trans}} \nonumber\\ &=& \phi _{{\rm blue}} (m,t) \times {\rm SFR}(m,t) \times \mu \tau _{{\rm trans}} \nonumber\\ &=& \phi _{{\rm blue}} (m,t) \times {\rm sSFR}(m,t) \times \left({\frac{m}{{M^*}}} \right)\tau _{{\rm trans}}. \qquad \end{eqnarray} \tag{ 27 }$$

Therefore, the mass function of the transient objects will be given, for the simple case of τ_trans = constant, by a single Schechter function with parameters

$$\begin{eqnarray} M^* _{{\rm trans}} &=& M^* _{{\rm blue}} = M^* _{{\rm red}} \nonumber\\ \alpha _{s,{\rm trans}} &=& \alpha _{s,{\rm blue}} + (1 + \beta) = \alpha _{s,{\rm red}} \nonumber\\ \phi ^* _{{\rm trans}} &=& \phi ^* _{{\rm blue}} \left. {{\rm sSFR}(t)} \right|_{M^* } \tau _{{\rm trans}} = \phi ^* _{{\rm blue}} \frac{{\tau _{{\rm trans}} }}{{\left. {\tau (t)} \right|_{M^* } }}, \end{eqnarray} \tag{ 28 }$$

where sSFR(t)$|_{M^{*}}$ and τ(t)$|_{M^{*}}$ are the evolving sSFR and star formation timescale, respectively (τ = sSFR⁻¹), at the galactic mass corresponding to M*.

The shape of the transient mass function should therefore be exactly the same as for the population of already mass-quenched passive galaxies, but the number density normalization ϕ*_trans will be the product of the ϕ*_blue of the currently star-forming galaxies multiplied by the dimensionless ratio of the visibility timescale τ_trans and the star formation timescale τ at M. This ratio will strongly evolve with redshift due to the change in sSFR given by Equation (1). More general cases with a non-constant τ_trans(m, t) may be easily derived.

As would be expected, this formalism naturally produces a close linkage between the evolution of the characteristic sSFR in the universe as a whole and the number density of the transient objects involved in the mass quenching process.

An important consequence of our model is that all of these statements regarding the mass function of transitory objects should be strictly independent of the Mpc-scale environment. This is simply because of the independence of mass quenching from the environment that was established in Section 4.2.

Having established these inter-connections, we could ask whether the transient objects are actually seen to be still on the star-forming main sequence, or are already passive galaxies, or intermediate objects, or some combination of all three. This information would tell us when in the mass-quenching process the relevant "signature" was produced, but the fundamental relationships in the mass functions that we have outlined above would be unchanged.

We define the "age" in this context to signify the V-band light-weighted age of the stellar populations (using the Bruzual & Charlot 2003 models to define the evolving M/L of a given stellar population). The more massive passive galaxies are produced by the mass-quenching star formation term in the overall death function (Equation (18)). Their rate of production at a given mass therefore follows the average sSFR in the universe and thus falls strongly at redshifts below z ∼ 2. In contrast, the less massive passive galaxies with m ≪ M* are primarily produced through the action of the mass-independent environmental quenching and merging processes, which together have a much weaker dependence on cosmic epoch. This therefore automatically produces a much broader distribution of ages and a lower mean age. This is shown in Figure 17 for the simple model defined in Section 5.3.

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The model would also predict that at a given stellar mass, passive galaxies that are satellites of other galaxies would be on average younger than passive galaxies that are "centrals." There is some evidence that this is indeed the case (e.g., Rogers et al. 2010). If our environmental-quenching effect is indeed associated with satellite quenching as we suspect, then it follows that central galaxies can only have been quenched through mass quenching. Satellite galaxies on the other hand may also have been quenched through environment quenching, i.e., satellite or merger quenching, which as shown in Figure 16, is a more recent phenomenon with a roughly flat rate over time.

Our quenching law (17) also naturally explains qualitatively the broad run of α-element abundances with the masses of passive galaxies (Thomas et al. 2005). The α-element abundance of a given set of stars in a galaxy will be related to the inverse sSFR of the galaxy at the time that those stars were formed, since this quantity sets the timescale for the chemical enrichment of those particular stars. In the simple picture sketched above, all passive galaxies that were quenched prior to z ∼ 2 will have formed all of their stars at high sSFR, specifically with τ_sSFR ∼ 0.5 Gyr. This is sufficiently short for them to be α-enhanced (Renzini 2009). Lower mass galaxies which form over a longer range of cosmic time (Figure 16) through our environment-quenching effects will have generally formed a significant fraction of their stars at later times (see Figure 15) when the sSFR in the galaxy population has fallen to lower values. The τ_sSFR will be longer, above 1 Gyr and the galaxies should exhibit normal α/Fe ratios. In Figure 18, we show the distributions of the light weighted 〈sSFR⁻¹〉 for galaxies, i.e., averaging over the stars in a given galaxy, in the numerical model of Section 5.3 as a function of their final passive mass, again averaging over all environments.

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These natural consequences of the death law given by Equation (18) are in broad qualitative agreement with measurements of the ages and chemical compositions of passive galaxies (Carollo et al. 1993, 1994; Gallazzi et al. 2005, 2006; Thomas et al. 2005, 2010).

It will be clear from the foregoing discussion that the precise cosmic sSFR history is more or less incidental to much of the discussion. Because the mass-quenching death rate is proportional to the star formation rate, changes in the latter will be precisely balanced by the corresponding changes in the former. The resulting changes to the mass function(s) will be driven by Δln m (see Equation (20)), independent of the timescale over which this occurs. If the death rate were indeed dominated by η_m alone (i.e., without environmental effects) then a lower sSFR would simply cause a slower evolution, and not a different outcome as far as the build-up of a Schechter mass function is concerned. The high and constant sSFR at early times ensures that the exponential growth of galaxies is sufficient to produce large numbers of star-forming galaxies around M* by the epoch corresponding to z ∼ 2.

In practice, the juxtaposition of the star formation driven η_m (which depends strongly on mass at a given epoch) and the mass-independent η_ρ terms in the composite η does introduce a dependence on the sSFR history. A change in the relative time dependence of these two death rates would for instance change the relative size of the two components in the double Schechter function discussed in Section 5.2, even though the double form of the mass function and the relationships between the M* and the α_s values would stay exactly the same.

One important consequence of the independence of our model on the precise history of the sSFR is that the model is not dependent on the precise estimates of the star formation rates in individual galaxies, which continue to have some measurement uncertainty. The basic form of the mass quenching law is set by the observed constancy of M* of star-forming galaxies, while the numerical value of the constant μ is set by M*⁻¹. Likewise, we could argue that the value of β must be close to zero simply because the mass function of star-forming galaxies has almost constant α_s since z ∼ 2. This conclusion is therefore actually independent of the actual measurements of SFR in individual galaxies and the precise form of sSFR(m, t, ρ) is not so important.

We have tried to stress throughout the paper the empirical nature of our analysis, without attempting to ascribe strong causal connections to particular physical processes. One of the great successes of our model is clearly the natural emergence of the various Schechter functions that describe star-forming and passive galaxies. In this section, we explore the links to other possible mechanisms, and in particular the idea of a "limiting maximum mass" for galaxies. Such a limit could be related, for example, to the well-established idea of cooling of gas in dark matter halos as a function of mass (Silk 1976; Rees & Ostriker 1977).

The role of Equation (23) in producing the Schechter function exponential cutoff can be viewed in terms of survival probabilities. We introduced η as a probabilistic mass-quenching rate for a given galaxy at a given time. If we consider some galaxy that is growing in mass via star formation, then the probability P(m) that it will survive to a particular stellar mass m, will simply be given by

$$\begin{eqnarray} \frac{{dP}}{{dt}} &=& - \eta P = - \mu \frac{{dm}}{{dt}}P \nonumber\\ \frac{{dP}}{P} &=& - \mu dm \nonumber\\ P \propto \exp (- \mu m) &=& \exp ({ - m/M^*}). \end{eqnarray} \tag{ 29 }$$

It can be seen that Equation (27) naturally gives the Schechter cutoff to the mass function of galaxies at M* = μ⁻¹ in terms of the survival probability distribution function of individual galaxies. It should be noted how P(m) is completely independent of the form or history of dm/dt. This is also why our model does not depend on the detailed cosmic evolution of the sSFR. It also ensures that the model is independent of the scatter around the sSFR–mass relation and of the observational estimates of star formation rates in distant galaxies.

This independence of the outcome on the details of individual star formation histories also means that we do not need to consider explicitly any extreme "star-bursting" galaxies, such as the ULIRGs or sub-millimeter galaxies, which will have individual sSFR above the "main-sequence" relation. Provided that they are quenched according to the same quenching law that is given by Equation (17) they will automatically be included in the framework. This is the justification for not considering their role(s) earlier in the paper.

Exploring this further, it is interesting to look at the mass and SFR of a particular galaxy that follows the changing sSFR described by Equation (1) (see also Renzini 2009). This is shown in Figure 19. During the period when the sSFR is constant, at z > 2, the galaxy will increase its mass exponentially. However, as the sSFR starts to decline, the rate of mass growth slows, but the SFR, and thus the quenching probability, for this individual galaxy stays broadly similar over an extended period of time. This is because the build-up of stellar mass largely compensates for the decline in sSFR. Because of the balancing of mass and sSFR, one could therefore view quenching as a quasi-random process after z ∼ 2. A more massive galaxy, following a higher track in Figure 19, will also experience a more or less constant risk of being quenched per unit time, after z ∼ 2, but at a higher level. A lower mass galaxy will continue to grow without serious risk of quenching until its mass approaches M*. In the end, the total mass of stars that is produced (statistically) by the time all three galaxies are finally quenched, will be the same, and be roughly equal to M* as in Equation (25). This is again a consequence of the survival probability analysis given in Equation (25).

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It might therefore be thought that there is something of a tautology in our quenching law. A clear consequence of it is that the statistical lifetimes of individual galaxies are simply set by the time that is needed to build up an M*-worth of stellar mass, regardless of how fast or slowly this is done. Since most galaxies will be the dominant galaxies in their halos, and since we are anyway in a sense taking care of the satellites through our "environment-quenching" term, we could also broadly translate this to simply requiring quenching when the mass of the dark matter halo reaches a certain limiting value. This limit could be implied by well-established cooling time arguments (Silk 1976; Rees & Ostriker 1977; Dekel & Birnboim 2006; Cattaneo et al. 2006).

It is clear from the above that our empirical "star-formation-rate" quenching law and a "mass-threshold" quenching law are rather similar in their outcomes, even if they are presumably based on very different physics. Both involve a rather impressive constancy since z ∼ 2: SFR quenching requires that the physical relationship of the quenching rate to the star formation rate be independent of epoch (as well as the mass and environment of a galaxy), while the "threshold quenching" requires that the mass threshold be likewise rather precisely constant over a very wide range of time and environments, although this is apparently achieved in some models (e.g., Birnboim & Dekel, 2003).

The only differences might be in the detailed outcomes. We refer back to the point made earlier in the paper. Our SFR-quenching law, because of its implicit mass dependence (roughly proportional at any epoch to mass, with β ∼ 0) and its implicit sSFR linkage, naturally produces precisely the Schechter forms of the mass function of passive and active galaxies. This is true both for the exponential cutoff and for the change in the faint end power-law slope with Δα_s ∼ 1 of the passive galaxies relative to the star-forming galaxies. The Schechter forms for the mass-function(s) are seen over about 2 orders of magnitude in galactic mass.

If the build-up of passive galaxies was simply related to some threshold in mass beyond which galaxies cannot grow, then we might well have expected some other form for the mass functions of the resulting galaxies. An absolute "brick-wall" for galaxy growth would have resulted in a step function for the blue galaxies and a delta function for the passive ones. More realistically, we might have expected a Gaussian distribution for the mass function of passive galaxies, reflecting the statistical variations in the limiting mass due to different random factors. The fact that we get an exact Schechter function for both the star-forming and for the mass-quenched passive galaxies, which clearly holds in the SDSS mass-function (Figure 12) over about 2 orders of magnitude in stellar mass, is a natural consequence of a probabilistic death rate that operates over a very broad range of masses in a way that is, at any epoch, closely proportional to the product of mass and specific star formation rate, i.e., to the star formation rate alone.

It is not clear whether a "mass-limit" quenching law can so successfully account for the Schechter shapes of the different mass functions and the precise inter-relationships shown above. Not least, we emphasize that the basic separability between mass and environment, which was established in this paper, requires that any "second-parameter" producing a scatter in the dark halo mass limit must be strictly independent of the Mpc-scale environment.