THE RISE OF MASSIVE RED GALAXIES: THE COLOR–MAGNITUDE AND COLOR–STELLAR MASS DIAGRAMS FOR z_phot ≲ 2 FROM THE MULTIWAVELENGTH SURVEY BY YALE–CHILE

This work is based on a K-selected catalog of the ECDFS from the MUSYC wide NIR imaging programme; these data are described and presented in Paper I. We will refer hereafter to this data set as "the" MUSYC ECDFS catalog, although it should be distinguished from the optical (B + V + R)-selected catalog, and the narrowband (5000 Å)-selected photometric catalogs described by Gronwall et al. (2007), and the spectroscopic catalog described by Treister et al. (2009).

The vital statistics of the imaging data that have gone into the MUSYC ECDFS catalog are summarized in Table 1. Unlike the three other MUSYC fields, the ECDFS data set was founded on existing, publicly available optical imaging: specifically, archival UU₃₈BVRI WFI data, including those taken as part of the COMBO-17 survey (Wolf et al. 2003) and ESO's Deep Public Survey (DPS; Arnouts et al. 2001), which have been rereduced as part of the GaBoDS project (Erben et al. 2005; Hildebrandt et al. 2006). We also include H-band imaging (P. Barmby 2005, private communication) from SofI on the 3.6 m NTT, covering ∼80% of the field, taken to complement the ESO DPS data, reduced and described by Moy et al. (2003).

Table 1. Summary of the Data Comprising the MUSYC ECDFS Catalog

Band (1)	λ0 (Å) (2)	Instrument (3)	Exposure Time (min) (4)	Area (□') (5)	FWHM (6)	5σ depth (AB) (7)
U	3560	WFI	1315	973	1 1	26.5
U38	3660	WFI	825	942	1 0	26.0
B	4600	WFI	1157	1011	1 1	26.9
V	5380	WFI	1743	1020	1 0	26.6
R	6510	WFI	1461	1016	0 9	26.3
I	8670	WFI	576	976	1 0	24.8
z'	9070	MOSAIC-II	78	997	1 1	24.0
J	12500	ISPI	≈ 80	882	< 1 5	23.3
H	16500	SofI	≈ 60	650	< 0 8	23.0
K	21300	ISPI	≈ 60	887	< 1 0	22.5

Notes.For each band, we give the filter identifier (Column 1) and effective wavelength (Column 2), as well as the detector name (Column 3). For the NIR data, exposure times (Column 4) are given per pointing; the effective seeing (Column 6) is given for the pointing with the broadest PSF. The 5σ limiting depths (Column 7) are as measured in 2 5 diameter apertures; the smallest apertures we use. The references for each set of imaging data are given in the main text (Section 2.1).

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2.1.1. Original Data Reduction and Calibration

2.1.2. Photometry

2.1.3. Completeness and Reliability

2.1.4. Sample Selection

We will investigate the z_phot ≲ 2 evolution in the massive galaxy population by comparing the situation at z ≫ 0 to that for z ≈ 0 galaxies in the Sloan Digital Sky Survey (SDSS; York et al. 2000); specifically, we use the "low-z" sample from the New York University (NYU) Value Added Galaxy Catalog (VAGC) of the SDSS presented by Blanton et al. (2005c). The Data Release 4 (DR4) low-z catalog contains ugriz photometry for 49,968 galaxies with 10 < D < 150 Mpc (z_spec ≲ 0.05), covering an effective area of 6670 □°; 2513 of these galaxies have M_*>10¹¹M_☉. Our analysis of these data closely follows that of the z ≫ 0 sample, and is described separately in Appendix A.

The technical crux on which any photometric look-back study rests is the determination of redshifts from broadband SEDs. We have computed our photometric redshifts using EAZY (Brammer et al. 2008), a new, fully-featured, user-friendly, and publicly-available photometric redshift code. By default, the z_phot calculation is based on all ten bands, although we do require that the effective weight in any given band is greater than 0.6; in practice, this requirement only affects the H band, where we do not have full coverage of the field. The characteristic filter response curves we use account for both atmospheric extinction and CCD response efficiency as a function of wavelength.

For our fiducial or default analysis, we simply adopt the recommended default settings for EAZY: namely, we adopt the EAZY default redshift and wavelength grids, template library, template combination method, template error function, K luminosity prior, etc. (See Brammer et al. 2008, for a complete description.) Note that, by default, EAZY assigns each object a redshift by marginalizing over the full probability distribution rather than, say, through χ² minimization. For this work, a key feature of EAZY is the control it offers over how the SED fitting is done: the user is able to specify whether and how the basic template spectra are combined, whether or not to include luminosity prior and/or a template error function, and how the output redshift is chosen. We will make use of EAZY's versatility in Section 9.4 to explore how these particular choices affect our results.

One of the unique aspects of the ECDFS is the high number of publicly available spectroscopic redshift determinations, which can be used to validate and/or calibrate our photometric redshifts. In Paper I, we describe a compilation of spectroscopic redshifts for 2914 unique objects in our catalog, including "robust" redshifts for 1656 galaxies in our main K < 22 sample. These redshifts come from some of the many literature sources available in the ECDFS, including those large surveys referred to in Section 1, as well as the X-ray-selected spectroscopic redshift catalogs of Szokoly et al. (2004) and Treister et al. (2009), a new survey by S. Koposov (2008, private communication), and a number of smaller projects. K20 is particularly useful in this regard, given its exceptionally high spectroscopic completeness, albeit over a very small area: 92% of K^(Vega) < 20 sources over 52 □'.

The main panel of Figure 1 shows our z_spec–z_phot plot. We prefer to quantify the photometric redshift quality in terms of the normalized median absolute deviation (NMAD¹¹) in Δz/(1 + z_spec), which we will abbreviate as σ_z; for this comparison sample, σ_z = 0.035. Further, the outlier fraction is acceptably small: 7.8%. Comparing only to the 241 redshifts from K20, we find σ_z = 0.033; for the Van der Wel et al. (2005) sample of 28 z ∼ 1 early-type galaxies the figure is 0.022. For 1 < z_spec < 2, we find σ_z = 0.059. For the 20% (269/1297) of 0.2 < z_phot < 1.8 galaxies from our mass-limited sample defined in Section 4 that have spectroscopic redshifts, we find σ_z = 0.043.

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Based on their catalog of the GOODS ACS, ISAAC and IRAC data, Grazian et al. (2006) have achieved a photometric redshift accuracy of 〈Δz/(1 + z_spec)〉 of 0.045. For comparison to Grazian et al. (2006), the inset panel shows the distribution of Δz = (z_phot − z_spec) for 0 < z_spec < 2; although offset by −0.046, the best-fit Gaussian to the distribution has a width of 0.046, as opposed to 0.06 for Grazian et al. (2006). For an identical sample of 938 galaxies with z_specs, we find σ_z = 0.043 for the Grazian et al. (2006) z_phots and σ_z = 0.035 for ours. In other words, our photometric redshift determinations are at least as good as the best published for K-selected samples at high redshifts. We also note in passing that our z_phot ≲ 1 photometric redshifts agree very well with those from COMBO-17 (Wolf et al. 2003); a detailed comparison to both these catalogs is presented in Paper I.

The lower panel of Figure 1 shows the redshift distributions for both our main galaxy sample (based on z_phot), and the spectroscopic comparison sample (based on z_spec). Note the presence of three prominent redshift spikes at 0.5 < z_spec < 0.8 (see also Vanzella et al. 2008); it appears that our redshift determinations may slightly underestimate the redshifts of these structures. The structures at z ∼ 1.0, 1.1, 1.2, 1.3, and 1.4 (Vanzella et al. 2008) are also visible in the z_spec distribution, but are "washed out" in the z_phot distribution.

The many degeneracies between SED shape and the intrinsic properties of the underlying stellar population, which are actually a help when deriving photometric redshifts, make the estimation of such properties from SED fitting highly problematic. Systematic uncertainties associated with parameterizations of the assumed star-formation history are at the level of 0.1 dex (Pozzetti et al. 2007), while uncertainties in the stellar population models themselves are generally accepted to be ≲0.3 dex; this is comparable to the uncertainty associated with the choice of stellar IMF. For these reasons, we have opted for a considerably simpler means of deriving rest-frame parameters.

Once the redshift is determined, we have interpolated rest-frame fluxes from the observed SED using a new utility dubbed InterRest, which is a slightly more sophisticated version of the algorithm described in Appendix C of Rudnick et al. (2003), and is described in detail in Paper I. InterRest is designed to dovetail with EAZY, and is also freely available.¹² We estimate the systematic errors in our interpolated fluxes (cf. colors) to be less than 2% (Paper I).

We then use this interpolated rest-frame photometry to estimate galaxies' stellar masses using a prescription from Bell & de Jong (2001), which is a simple linear relation between rest-frame (B − V) color and stellar mass-to-light ratio (M_*/L_V):

$$\begin{equation} \log _{10} M_*/L_V = -0.734 + 1.404 \times (B-V + 0.084), \end{equation} \tag{ 1 }$$

assuming M_V,☉ = 4.82. (Here, the factor of 0.084 is to transform from the Vega magnitude system used by Bell & de Jong 2001 to the AB system used in this work.) This prescription assumes a "diet Salpeter" IMF, which is defined to be 0.15 dex less massive than the standard Salpeter (1955) IMF, and is approximately 0.04 dex more massive than that of Kroupa (2001). Further, to prevent the most egregious overestimates of stellar masses, we limit M_*/L_V ⩽ 10 (see also Figure 4 of Borch et al. 2006). Although this limit affects just 1.2% of our main sample, we found it to be important for getting the high-mass end of the z ≈ 0 mass function right (Appendix A).

It is not immediately obvious whether using these color-derived M/Ls is significantly better or worse than, say, using stellar population synthesis to fit the whole observed SED. The prescription we use has been derived from SED-fit M/Ls; the scatter around this relation is on the order of 0.1 dex. By comparison, the precision of SED-fit M/L determinations is limited to 0.2 dex by degeneracies between, e.g., age, metallicity, and dust obscuration (see, e.g., Pozzetti et al. 2007; Conselice et al. 2007). Thus, the increase in the random error in M_* due to the use of color-derived rather than SED-fit stellar masses is ∼10%.

In addition to being both simpler and more transparent, however, the use of color-derived M/Ls has the major advantage of using the same rest-frame information for all galaxies, irrespective of redshift. This is especially important when it comes to the comparison between the high-z and low-z samples, where the available photometry samples quite different regions of the rest-frame spectrum. Since Equation (1) has ultimately been derived from SED-fit mass estimates, however, our color-derived mass estimates are still subject to the same systematic uncertainties. To the extent that such systematic effects are independent of color, redshift, etc., they can be accommodated within our results by simply scaling our limiting mass. On the other hand, if there is significant evolution in the color–M/L relation with redshift, then there is the risk that the use of color-derived M/Ls may introduce serious systematic errors with redshift. We investigate this issue further in Section 9.5, in which we also demonstrate that our results are essentially unchanged if we use conventional SED-fitting techniques to derive M/Ls.

A primary concern in this paper is the importance of systematic errors. To address this concern in the context of our photometric redshift determinations, we show in the top panels of Figure 2 the z_spec–z_phot agreement as a function of redshift, S/N in the K "color" aperture, and rest-frame color. In each panel, the points with error bars show the median and 15/85 percentiles in discrete bins.

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Looking first at our photometric redshifts: the first panel of Figure 2 shows the photometric redshift error, Δz/(1 + z_spec), as a function of z_spec. We see that there is a systematic offset between z_phot and z_spec for z_spec ≲ 1, such that our z_phots tend to be slightly too low (see also Figure 1); for z ≳ 1, this effect appears to be less. At least for K S/N ≳10, random errors in the photometric redshifts do not appear to be a strong function of S/N.

There is a clear systematic effect as a function of rest-frame color. For galaxies redder than (u − r) ≈2 (approximately the lower limit for z ≈ 0 red sequence galaxies), the agreement between z_phot and z_spec is very good. For galaxies with (u − r) ≲ 2, however, it seems that we systematically underestimate the true redshift by approximately Δz ≲ 0.02(1 + z_spec). It is plausible that this is in fact the driver of the weak apparent systematic effect with redshift, coupled with there being a greater proportion of blue galaxies in the spectroscopic redshift sample at lower redshifts.

How do these errors in redshift estimation play out in the derivation of rest-frame properties? We address this question with reference to the lower panels of Figure 2, which illustrate how redshift uncertainties affect our derivation of three basic rest-frame quantities (top to bottom): absolute luminosity, rest-frame color, and stellar mass. In each panel, we plot the difference between the values derived adopting the spectroscopic or photometric redshift, as a function of (left to right) redshift, K-band S/N, and rest-frame color, as well as photometric redshift error.

Quantitatively, for our z_spec comparison sample, the random photometric redshift error of Δz/(1 + z_spec) = 0.035 translates into a 0.360 mag error in absolute magnitude, a 0.134 mag error in rest-frame color, and a 0.107 dex error in stellar mass. (By contrast, the typical uncertainty in total flux for a K S/N = 10 galaxy is ΔK = 0.12–0.16 mag ≈ 0.05–0.07 dex.) Just as for the redshifts themselves, the clearest systematic effects in the derived quantities is with rest-frame color: there appear to be mild systematics with redshift for z ≲ 1, and no clear trend with S/N, at least for S/N > 10.

It is straightforward to understand why redshift errors play a larger role in the derivation of magnitudes rather than colors. When calculating magnitudes, the primary importance of the redshift is as a distance indicator. For the z_spec sample shown in Figure 2, the random scatter in ΔM_r due to distance errors alone (calculated by taking the difference in the distance modulus implied by z_phot versus that by z_spec) is 0.28 mag; i.e. ∼75% of the scatter seen in Figure 2. On the other hand, colors are distance independent, and this element of uncertainty is cancelled out.

What is surprising is the relative insensitivity of our stellar mass estimates to redshift errors. Focusing on the right panels of Figure 2, it can be seen that where the photometric redshift underestimates the true redshift/distance, we will infer both too-faint an absolute luminosity and too-red a rest-frame color. When it comes to computing a stellar mass, however, these two effects operate in opposite directions: although the luminosity is underestimated, the too-red color leads to an overestimate of the stellar mass-to-light ratio. The two effects thus partially cancel one another, leaving stellar mass estimates relatively robust to redshift errors.

In a photometric redshift survey, the measurement uncertainty due to random photometric redshift errors is considerably less for stellar masses than it is for absolute magnitudes. This conclusion remains unchanged using SED-fit stellar masses, rather than our favored color-derived ones. Conversely, we can say that (random) photometric redshift errors are not a dominant source of uncertainty in our stellar mass estimates. Indeed, as we have already noted the size of these errors is comparable with the uncertainties in our total flux measurements.

Figure 4 shows the CMD, plotted in terms of (u − r) color and absolute r magnitude, M_r, for z_phot ≲ 2.

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The first panel of Figure 4 is for z ≈ 0 galaxies from the "low-z" comparison sample; these data and our analysis of them are described in Appendix A. The basic features of the CMD—the red sequence, blue cloud, and green valley—are all easily discernible. The dotted line shows our characterization of the z ≈ 0 CMR for red galaxies (Equation (A1)), also discussed in Appendix A.

The other eight panels show the 0.2 < z_phot < 1.8 MUSYC ECDFS data. Except for the two highest redshift bins, which are twice as large, the z_phot ≫ 0 bins have been chosen to have an equal comoving volume.¹³ For the z ≈ 0 bin, we plot only a random subsample of the low-z catalog, chosen to effectively match the volume of the higher-redshift bins. Thus, the density of points in the color–magnitude plane is directly related to changes in the bivariate comoving number density.

In the bottom-right corner of each panel, we show representative error bars for a M_* ≈ 10¹¹M_☉ galaxy with (u − r) ≈ 2.0, near the mean redshift of each bin. In order to derive these errors, we have created 100 Monte Carlo realizations of our catalog, in which we have perturbed the catalog photometry according to the photometric errors, and repeated our analysis for each: the error bars show the scatter in the values so derived. The shaded gray regions show approximately how our K < 22 completeness limit projects onto color–magnitude space through each redshift bin, derived using synthetic single stellar population (SSP) spectra.

Examining this diagram it is clear that, in the most general terms possible, bright/massive galaxies were considerably bluer in the past. At a fixed magnitude, the entire z ∼ 1 galaxy population is a few tenths of a magnitude bluer than at z ≈ 0. At the same time, particularly for z ≳ 1, there is a growing population of galaxies with M_r < −22 and (u − r) < 2 that has no local analog. While there are some indications of a red sequence within the z ≫ 0 data, particularly for z_phot ≲ 1, it is certainly not so easily distinguishable as locally.

There are a number of advantages to using stellar mass as a basic parameter, rather than absolute magnitude. Principal among these is the fact that stellar mass is more directly linked to a galaxy's growth and/or assembly: while a galaxy's brightness will wax and wane with successive star-formation episodes, a galaxy's evolution in stellar mass is more nearly monotonic. On the other hand, it must be remembered that the necessary assumptions in the derivation of stellar mass estimates produce greater systematic uncertainties than for absolute luminosities. As bursts of star formation and other aspects of differing star-formation histories among galaxies are likely greater at higher redshifts, however, we will focus on the color–stellar mass diagram in this and following sections.

Figure 5 shows the CM_*D for z ≲ 2. As in Figure 4, the first panel shows a random subsample of the low-z sample; the other panels show the MUSYC ECDFS data. The dotted line in each panel shows the z ≈ 0 color–stellar mass relation (CM_*R), which we have derived in Appendix A, given in Equation (A1).

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Each of the basic features of the CMD are also seen in the CM_*D. We see an increasing number of galaxies with z_phot ≳ 1 and with M_*>10¹¹M_☉ and (u − r) ≲ 2 which have no analogs in the local universe. For an SSP, the colors of these galaxies would suggest ages of ≲1 Gyr: these massive galaxies appear to be in the throes of their final star-formation episodes. For z_phot ≳ 1.2, these galaxies may even dominate the massive galaxy population. Some evidence for a distinct red sequence is visible in the CM_*D for z_phot ≲ 1, but not much beyond.

The next three sections are devoted to more quantitative discussion of each of the following three specific observations:

• 1.  
  We see evidence for a red galaxy sequence for z_phot ≲ 1.2; beyond this redshift, whether due to physical evolution or to observational errors, it becomes impossible to unambiguously identify a distinct red sequence on the basis of the present data (Section 6).
• 2.  
  At a fixed mass, a red sequence galaxy at z ∼ 1 is a few tenths of a magnitude bluer than its z ∼ 0 counterpart (Section 7).
• 3.  
  At higher redshifts, there appear to be fewer massive galaxies on the red sequence. Further, it appears that the proportion of blue cloud galaxies among the most-massive galaxies increases; conversely, the red fraction is lower at higher redshifts (Section 8).

Locally, red sequence galaxies totally dominate the massive galaxy population: what bimodality exists between the red and blue populations is weak. (As the name suggests, "bimodality" implies two distinct local maxima in the distribution.) This is a reflection of the apparent "transition mass" between red and blue galaxies observed by Kauffmann et al. (2003); the bimodality is stronger in a luminosity-limited sample, including a greater proportion of bright but less-massive blue cloud galaxies (see also Figure 15; Appendix A). At slightly higher redshifts, where some progenitors of z ≈ 0 red sequence galaxies are still forming stars in the blue cloud, we may then expect the bimodality to actually become stronger, before weakening again as the fraction of those galaxies already on the red sequence becomes small at moderate-to-high redshifts. In general, however, the color distributions shown in Figure 6 are not clearly bimodal.

With this in mind, as a simple means of separating red from blue galaxies, we have fitted the observed distributions in each redshift bin with double Gaussian functions. These fits are shown by the smooth curves in each panel of Figure 6. For the most part, these fits provide a reasonable description of the 0.2 < z_phot ≲ 1.2 data (see also Borch et al. 2006).

In contrast to lower redshifts, for z_phot ≳ 1.3, we are no longer able to reliably fit the color distributions in this manner: on the basis of the sensitivity tests presented in Section 9, neither the red/blue separation nor the fits to the distributions of these populations are robust. Looking at the typical errors shown in each panel of Figure 5, the measurement errors on the (u − r) colors of red galaxies rises sharply from Δ(u − r) ≲ 0.1 mag for z_phot ≲ 1.2 to 0.2 mag for z_phot ∼ 1.3, 0.3 mag for z_phot ∼ 1.5, and 0.4 mag for z_phot ∼ 1.7. This would suggest that our inability to reliably distinguish both a red and a blue population at these higher redshifts may very well be due to the large errors in rest-frame optical colors for higher-redshift galaxies.

In order to help interpret our z_phot ≳ 1 results, we have tested our ability to recover a single-color galaxy population, given our observational and analytical errors. To this end, we have generated a mock photometric catalog containing only red sequence galaxies: beginning with our main galaxy sample, we have replaced each galaxy's photometry using synthetic spectra for an SSP formed over a short period beginning at z_form = 5, assuming the catalog values of z_phot and M_*, and then adding typical MUSYC ECDFS photometric errors. We have then reanalyzed this catalog using the same methods and procedures as for the main analysis, including recomputing photometric redshifts, rest-frame colors, and stellar masses.

The results of these tests are shown in Figure 7, which plots the observed width of the color distribution for this intrinsically single color population, as a function of photometric redshifts, assuming photometric errors typical for the MUSYC ECDFS (squares). The measurement errors on the (u − r) colors of red galaxies rise sharply from 0.05–0.07 mag for z_phot ≲ 1 to 0.10 mag for z_phot ≈ 1, and then continue to increase for higher redshifts. In order to demonstrate that this is a product of photometric errors per se and not redshift errors, we have also repeated this analysis holding the redshifts of each object fixed; the results of this test are shown as the dotted line. Even with spectroscopic redshifts, the depth of our NIR data would seem to preclude the detection of a distinct red sequence at z ≳ 1.3.

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What then would be required in order to confirm the non/existence of a red sequence at z_phot ≳ 1.5? We have also constructed mock galaxy catalogs with S/N that is 2.5 and 10 times greater than typical values for the MUSYC ECDFS catalog; the results of these tests are shown as the triangles and circles, respectively. Even pushing a full magnitude deeper, it would be difficult to identify a red sequence at z ∼ 1.5, assuming that observational errors of Δ(u − r) ≲ 0.1 mag would be required to robustly identify a red sequence. In order to probe z ≳ 1.5, an order of magnitude improvement is required. This would suggest that the detection of a red sequence at z ≳ 1.5 would require a J-band (5σ point source) limit of ∼25.8, roughly the final target depth for the Ultra Deep component of the UKIRT Infrared Deep Sky Survey (Lawrence et al. 2007).

It is clear from this analysis that we cannot confirm or exclude the existence of a red galaxy sequence at z ≳ 1.3 on the basis of the present data. This is in good accord with the recent detections of a z ≲ 1.5 red galaxy sequence by Cirasuolo et al. (2007) and Franzetti et al. (2007; see also Kriek et al. 2008). Moreover, we note in passing that if we were to subtract away the broadening effect of photometric errors, as derived from the test described above, then the implied intrinsic width of the red sequence is ≈0.1 mag for all z_phot ≲ 1.2, consistent with the z_spec < 1.0 findings of Ruhland et al. (2009).

Our inability to robustly distinguish separate red and blue galaxy populations on the basis of the observed color distributions for z_phot ≳ 1.2 forces us to devise some alternative means of separating "red" galaxies from the general field population.

We have already seen that the color evolution of the red galaxy population is roughly consistent with ancient stars at all redshifts (Figure 9). Our simple solution is therefore to use the predicted color evolution for a passively evolving stellar population formed at high redshift to define a redshift-dependent "red" selection criterion, namely,

$$\begin{eqnarray} &&(u-r) > 2.57 + 0.24 \times \log _{10} (M_* / 10^{11} \mbox{$M$$_{\odot }$}) \nonumber \\ &&\quad\quad\quad\hspace{7pt} + \delta (z_{\mathrm{phot}}) - 0.25, \end{eqnarray} \tag{ 2 }$$

where δ(z_phot) is the (u − r) color evolution of an SSP with z_form = 5, as shown in Figure 9. This selection limit is shown as the dashed lines in each of Figures 5, 6, and 8.

How does this definition of "red" relate to things like membership of the red sequence, star-formation rate and/or history, etc.? As we remarked in the first paragraph, a galaxy's optical color is a reflection of its mean stellar age, modulo the complicating factors of metallicity and dust extinction. In addition to "red and dead" galaxies, therefore, simply selecting "red" galaxies can potentially catch a significant number of star-forming galaxies with high dust obscuration. In other words, while all passive galaxies are red, not all red galaxies are passive.

In this sense, it is not unreasonable to interpret the redshift evolution of the number and fraction of red, massive galaxies as placing an upper limit on the numbers of "fully formed" (in the sense that they have essentially completed their star formation and/or assembly) massive galaxies. These results can thus be used to constrain the epoch at which the star-formation quenching mechanism operates.

Figure 10 shows the evolving number density of M_*>10¹¹M_☉ galaxies for z_phot < 1.8. As before, the z ≈ 0 point comes from our analysis of the low-z sample discussed in Appendix A; the histograms are for the main MUSYC ECDFS sample. For the z ≫ 0 galaxies, since we have used bins of equal comoving volume, the observed numbers (right axes) can be directly related to a comoving number density (left axes), modulo uncertainties in the cosmological model. The black point/histograms refer to the total M_*>10¹¹ M_☉ population; the red point/histograms refer to red galaxies only; the dotted lines show where our results are significantly affected by incompleteness.

The error bars on the z ≫ 0 histograms include the estimated measurement uncertainty due to field-to-field variation, derived as in Somerville et al. (2004), but modified for cuboid rather than spherical volumes (R. Somerville 2008, private communication). For any single measurement, this is the dominant source of uncertainty: typically ∼30%, as compared to random photometric errors, which are at the ∼10% level. Note, however, that each of the z_phot ≳ 1 bins contains its own "spike" in the z_spec distribution (Vanzella et al. 2008); the 0.6 < z_phot < 0.8 bin contains two, and the 0.8 < z_phot < 1.0 bin none. Further, note that for z_phot ≳ 1, our redshift errors are comparable to the size of the bins themselves; in this sense, the densities in neighboring bins are correlated, and the variations due to large-scale structure are to a certain extent masked.

The observed evolution in the number density of massive galaxies is moderate: the number density of M_*>10¹¹M_☉ galaxies at 1.5 < z_phot < 1.8 is 52 ± 8% of the z ≈ 0 value. We see a very similar trend in the mass density, although a handful of galaxies with inferred M_* ≫ 10¹² M_☉ make this measurement considerably noisier.

In order to quantify the observed evolution, we have made a parametric fit to our measured number densities of the form:

$$\begin{equation} n_{\mathrm{tot}}(z) = n_{0} (1 + z)^{\gamma _{\mathrm{tot}}}. \end{equation} \tag{ 3 }$$

In practice, since the dominant source of uncertainty in any single point is from field-to-field variance, we perform a linear fit in log n–log(1 + z) space, weighting each point according to its uncertainty as shown in Figure 10. Further, we constrain the fits to pass through the z ≈ 0 point, effectively eliminating n₀ as a free parameter. In this way, we find γ_tot = −0.52 ± 0.12. The fit itself is shown in Figure 10 as the smooth black curve; the shaded region shows the 1σ uncertainty in the fit.

We now turn our attention to the red galaxy population. From Figure 10, it is clear that the observed evolution is much stronger for red galaxies than it is for the total population: the number density of red galaxies at 1.5 < z_phot < 1.8 is 18 ± 3% of the z ≈ 0 value. Making a fit to the red galaxy number densities to quantify this evolution, in analogy to the previous section, we find γ_red = −1.60 ± 0.14 (smooth red curve).

A complementary way of characterizing the rise of massive red galaxies is to look at the evolution of the red galaxy fraction. There are several advantages to focusing on the red galaxy fraction, rather than the number density of red galaxies per se. First and foremost, the uncertainty in the red fraction due to large-scale structure and field-to-field variation should be considerably smaller than those in the number density, especially at z ≳ 1 (Cooper et al. 2007).

We show the red fraction as a function of photometric redshift in the upper panel of Figure 10. Fitting these results (smooth red curve) with the same functional form as in Equation (3), we find γ_frac = −1.06 ±  0.16. Taken together, the results encapsulated in Figure 10 suggest that ≲20% of M_*>10¹¹M_☉ galaxies in the local universe were already on the red sequence by z_phot ≈ 1.6 (9.5 Gyr ago). By the same token, approximately 50% of these galaxies only (re-)joined the red sequence after z = 0.5 (5.0 Gyr ago).

It is clear from Figure 10 that almost all the signal for z ≫ 0 evolution in the red fraction comes from the z ≈ 0 comparison point (see also Borch et al. 2006). Fitting to the z ≫ 0 data alone, we find f_red = (0.53 ± 0.05)(1 + z_phot)^{−0.29±0.17}; that is, less than a 2σ detection of evolution. The same is also true for the number density measurements (see also Borch et al. 2006). The reasons for this are not just the relatively mild evolution, but also to the relative size of the error bars on the low- and high-redshift measurements. Fitting only to the z ≫ 0 points, we find γ_tot = −1.55 ± 0.84 and γ_red = −1.77 ± 1.84; these fits "overpredict" the z ≈ 0 number densities by factors of 2.2 and 1.3, respectively.

In this sense, then, rather than quantifying the absolute evolution in the z ≫ 0 population, we are performing a difference measurement between the situations at z ≈ 0 and z ≫ 0. For this reason, it is imperative that care is taken when deriving the z ≈ 0 comparison values to ensure that the low- and high-redshift samples have been analyzed in a uniform way.

In comparison to more sophisticated analyses by Bell et al. (2003) and Cole et al. (2001), our z ≈ 0 number densities are approximately 10% higher and 15% lower, respectively. Adopting these values in place of our own determinations, we find γ_tot = −0.72 ± 0.12 using the Bell et al. (2003) mass function, and γ_tot = −0.42 ± 0.12 using the Cole et al. (2001) mass function, changes of −0.2 and +0.1 with respect to our default analysis. These rather large discrepancies underline the importance of uniformity in the analysis of high- and low-redshift galaxies.

In Figure 11, we compare our results to a selection of the steadily growing number of similar measurements; we show results from: the COMBO-17 survey (Borch et al. 2006), the GOODS-MUSIC catalog (Fontana et al. 2006), the K20 survey (Fontana et al. 2004), the VVDS (Pozzetti et al. 2007), the DEEP2 survey (Bundy et al. 2006), and MUNICS (Drory et al. 2004). (Note that all of these results have come within the last five years.) In all cases, the points shown in Figure 11 have been obtained by integrating up a fit mass function, taking into account different choices of cosmology and IMF. We note that the strong redshift spike at z_phot ∼ 0.7 is also present in the GOODS-MUSIC results, which are based on the CDFS.

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The comparison with the COMBO-17 results deserves some further comment. While our results agree reasonably well with the combined results from three of the COMBO-17 fields (large diamonds), there is significant disagreement between our results and the COMBO-17 results in the ECDFS (small diamonds). In Appendix B, we present a detailed comparison between the MUSYC and COMBO-17 data sets, and demonstrate that the discrepancy between the two results is due to systematic calibration errors in the COMBO-17 photometry, rather than differences in our analyses. We note in passing that these calibration errors have only a small effect on the COMBO-17 photometric redshifts, and affect the ECDFS only; not the other COMBO-17 fields (Wolf et al. 2008).

Particularly given the significant uncertainties in these measurements, and with the possible exception of the MUNICS results, the degree of agreement between these surveys is impressive. However, in all of these surveys, field–to–field variance is a—if not the—major source of uncertainty. More and larger fields are necessary to lock down the growth rate of massive galaxies.

How sensitive or robust are our results to errors in the basic photometric calibration? And how accurate are the calibrations of each of the different bands, relative to one another, and in an absolute sense? We address these two questions in this section.

9.1.1. The Effect of Perturbing Individual Bands

9.1.2. Testing the MUSYC ECDFS Photometric Calibration

While we rely on SExtractor for our basic photometry, we have introduced three sophistications in our analysis (see Section 2.1.2). In this section, we investigate the effects that these three changes have on our results.

9.2.1. Background Subtraction

9.2.2. Total Magnitudes

9.2.3. SED Apertures and Color Gradients

In Section 4, we have argued that we are approximately complete for M_*>10¹¹M_☉ galaxies for z_phot < 1.8. In this section, we examine the effects of incompleteness due to both the K < 22 detection threshold and the K S/N >5 "analysis" selection, by deriving simple completeness corrections.

9.3.1. Incompleteness Due to Low S/N

9.3.2. Correcting for Undetected Sources

The next major aspect of our analysis that we will explore in detail is systematic effects associated with the derivation of photometric redshifts; we split this discussion into two parts. In the first part (Section 9.4.1) we explore how our results depend on the choice of templates used in the z_phot calculation. Then, in the second (Section 9.4.2), we investigate how our results depend on the exact method used for deriving photometric redshifts, by varying individual aspects of the EAZY algorithm. Some illustrative results from a selection of these sensitivity tests are given in Figure 12.

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9.4.1. Trialing Different Template Sets

9.4.2. Variations in our Default Photometric Redshift Calculation

We have relied on a rather simple method for estimating stellar mass-to-light ratios, based on rest-frame (B − V) colors. Certainly this method does not make use of the full amount of information available in the full, ten-band SED; on the other hand, it does ensure that the same information is used for all galaxies, irrespective of redshift.

Nevertheless, we have trialed reanalyzing our data using standard population synthesis SED-fitting techniques to estimate stellar masses for our main sample, in order to assess the potential scale of biases arising from this aspect of our experimental design. Specifically, we have fitted Bruzual & Charlot (2003) synthetic spectra to the observed photometry, with redshifts fixed to the fiducial z_phot, using the hyperzspec utility, which is a part of the hyperz v1.2 release package (M. Bolzonella 2005, private communication).

While the random scatter in comparison to the color-derived stellar mass estimates is 0.18 dex, after correcting for IMF, cosmology, etc., the SED-fit masses are systematically lower by 0.3 dex. This is also true for both the FIREWORKS (Wuyts et al. 2008) and GOODS-MUSIC (Grazian et al. 2006) catalogs of the CDFS-GOODS data. Besides this offset, we do not see evidence for strong evolution in the normalization of the color relation in comparison with the SED-fit masses (but see Lin et al. 2007). Since we have not been able to identify the source of this offset, we have simply corrected for it. Note, however, that we have not refitted the z ≈ 0 SEDs for this test.

Using these SED-fit stellar masses, we find γ_tot = −0.49,  γ_red = −1.62, and γ_frac = −1.11. These represent differences of just +0.03, −0.02, and −0.05 with respect to our fiducial values. Similarly, using the prescription for M/L as a function of (B − V) from Bell et al. (2003) rather than that from Bell & de Jong (2001), we find γ_tot = −0.41, γ_red = −1.55, and γ_frac = −1.12; changes of +0.11, +0.05, and −0.06, respectively.

This suggests that our results are not grossly affected by our choice of method for estimating stellar masses, and especially not by the use of color- as opposed to SED-derived stellar masses.

Large-Scale Structure and Field-to-Field Variance. One interesting test of the effect of field-to-field variance is to restrict our attention only to the MUSYC ECDFS coverage of the GOODS-CDFS region. For this 143 □' subfield, we find γ_tot = −0.62, γ_red = −1.37, and γ_frac = −1.00, differences of −0.10, +0.23, and +0.06 with respect to the full 816 □' ECDFS. In comparison with the full ECDFS, the dearth of galaxies at z_phot ∼ 0.8 is even more pronounced, and seems to extend over the range 0.8 < z_phot < 1.2; this appears to be the main driver of the strong change in γ_red. Conversely, if we exclude the significantly underdense GOODS area, we find γ_tot = −0.49, γ_red = −1.60, and γ_frac = −1.10; differences of +0.03, −0.00, and −0.04, respectively. Again, this analysis underscores the fact that measurements of the red fraction are more robust against the effects of large-scale structure than those of the number densities themselves.

The z ≈ 0 Value. In Section 8.4, we have shown that the z ≈ 0 comparison point is critical in providing most of the signal for z ≫ 0 evolution in the massive galaxy population. We have also considered how our results vary if we change the way we treat the z ≈ 0 point. In our default analysis, we constrain the fits so that they pass through the z ≈ 0 points, effectively using the z ≈ 0 point to normalize the high-redshift measurements. Since the (Poisson statistical) errors on the z ≈ 0 points are just a few percent, our results do not change significantly if we include the z ≈ 0 points in the fits.

For our default analysis, we have approximately accounted for the effect of photometric redshift errors, as they apply to the z ≫ 0 sample, on the z ≈ 0 measurements; we have done this by randomly perturbing the masses and colors of z ≈ 0 galaxies using the typical uncertainties for z ≫ 0 galaxies, due to the use of photometric redshifts, and given in Figure 2. Systematic and random errors in the number densities both of all and of red galaxies are on the order of 5%. If we do not account for the Eddington bias in this way, we find that γ_tot = −0.46 and γ_frac = −1.16, changes of 0.06 and −0.10, respectively.

Correcting for Galactic Dust Extinction. Note that we have not included specific corrections for Galactic foreground dust extinction. The CDFS was specifically chosen for its very low Galactic gas and dust column density; the suggested corrections for the optical bands are ≲0.05 mag. These corrections are typically as large as or larger than the uncertainties on the photometric zero-points themselves; it is for this reason that we have chosen not to apply these corrections. If we were to include these corrections, however, our derived values of γ_tot and γ_frac change by −0.03 and +0.04, respectively.

Including Spectroscopic Redshift Determinations. From among the thousands of spectroscopic redshift determinations that are publicly available in the ECDFS, we have robust z_specs for just over 20% (1669/7840) of galaxies in our main sample, and for 20% (269/1297) of those with 0.2 < z_phot < 2.0 and M_*>10¹¹M_☉. Repeating our analysis with this additional information included, our results do not change significantly: we find γ_tot = −0.50 and γ_red = −1.70, differences of just +0.02 and +0.00, respectively. Similarly, we show in Appendix B that our z ≲ 0.8 results do not change by more than a few percent if we adopt photometric redshifts from COMBO-17 (Wolf et al. 2004; σ_z = 0.035).

Excluding X-Ray-Selected Galaxies. Note that we have made no attempt to exclude Type I active galactic nuclei (AGNs) or QSOs from our analysis. In order to discover how significant an omission this is, we have repeated our analysis excluding all those objects appearing in the X-ray-selected catalogs of Szokoly et al. (2004) and Treister et al. (2009). This excludes 3.2% (250/7840) of galaxies from our main sample, and 6.5% (96/1482) from our M_*>10¹¹M_☉ sample. Given these numbers, it is perhaps unsurprising that the exclusion of these objects does not greatly affect our results: γ_tot and γ_frac change by −0.07 and +0.03, respectively.

How, and how much, do our results depend on our analytical methods and experimental design? We have now described a rather large number of sensitivity tests, designed both to identify which aspects of our experimental design are crucial in determining our results, as well as to estimate the size of lingering systematic errors in our results. The results of many of these tests are summarized in Table 3. Clearly, these sorts of tests can provide a staggering array of "metadata," offering insights to guide not only the interpretation of the present data and results, but also the design of future experiments.

As we have shown in Section 8.4, the systematic uncertainties in the γs due to discrepancies between the treatment of z ≈ 0 and z ≫ 0 galaxies is on the order of ±0.2; this is the single greatest potential source of systematic uncertainty. Since we have treated these two samples in a uniform manner, we do not consider this as a lingering source of systematic uncertainty.

The dominant sources of systematic uncertainty are, in order of importance: the method of deriving photometric redshifts (Δγ_tot ≈ 0.15, Δγ_frac ≈ 0.15); photometric calibration errors (Δγ_tot ≈ 0.05; Δγ_frac ≈ 0.10); incompleteness (Δγ_tot ≈ 0.1, Δγ_frac ≈ 0); the choice of photometric redshift template set (Δγ_tot ≈ 0.06; Δγ_frac ≈ 0.06); photometric methods (Δγ_tot ≈ 0.07; Δγ_frac ≈ 0); and stellar mass estimates (Δγ_tot ≈ 0.05; Δγ_frac ≈ 0.05). We can also, for example, dismiss incompleteness due to our K S/N criterion, contamination of the sample by QSOs, and minor details of our photometric redshift calculation as significant sources of systematic uncertainty.

In summary, we estimate the systematic errors on the measured values to be Δγ_tot = 0.21 and Δγ_frac = 0.20; these values have been obtained by adding the dominant sources of systematic error in quadrature. We note that previous studies have not generally taken (all) sources of systematic error into account in their analysis.

Figure 13 shows evolutionary tracks for a passively evolving stellar population, in terms of observed colors in the MUSYC ECDFS bands. Specifically, we have used Pégase V2.0 (Le Borgne & Rocca-Volmerange 2002) models with a short star-formation episode (an e-folding timescale of 100 Myr), beginning with Z = 0.004 at z_form = 5, and ending with Z≈ Z_☉. This (approximately) maximally old model describes a "red envelope" for the color–redshift relation for observed galaxies: at a given redshift only extreme dust extinction can produce observed colors redder than these tracks.

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As is evident in this figure, as the Balmer and/or 4000 Å breaks are first redshifted between the observed bands, there is a sharp rise in the observed color—by selecting objects that are very red in a certain color, it is thus possible to select red galaxies that lie beyond a certain redshift. This is completely analogous to the ERO (McCarthy 2004, and references therein) or DRG (Franx et al. 2003) selection criteria for red galaxies at moderate-to-high redshifts. The particular selections we have adopted in Figure 13 (and given in column 1 of Table 4) translate to minimum redshifts of approximately 0.16, 0.40, 0.66, 1.12, and 2.05.

By applying several of these color criteria in concert, with the reddest criteria given primacy, it is then possible to select red galaxies in rough redshift intervals: the highest redshift galaxies are selected by the criterion (J − K)>1.4; then, the next highest redshift galaxies are then selected from the remaining set (i.e., (J − K) < 1.4) by the criterion (I − J)>1.4, and so on. In the current context, this can be thought of as a two-color, binned photometric redshift for red galaxies.

The prevalence of objects selected in this way are given in column 4 of Table 4. To our limit of K < 22, we find 386, 655, 690, 304, and 286 objects selected by this tiered set of color criteria.

It is clear that the exact redshift range over which a galaxy might satisfy a given color criterion depends on that galaxy's SED: as well as passively evolving galaxies, these selections will also include galaxies whose red observed colors are due to, e.g., dust obscuration or considerably higher redshifts. Thus, the details of the evolving, bivariate color–magnitude distribution is folded into the numbers of color-selected galaxies. For this reason, we are forced to interpret the numbers of color-selected galaxies with reference to a particular model for the evolution of red galaxies.

This is done as follows: we construct a mock catalog in which galaxies' luminosities are distributed according to the z ≈ 0 luminosity function for red galaxies, which we have determined from the low-z sample, as analyzed in Appendix A. This luminosity function is nearly identical to that of Blanton et al. (2005b); our results do not change significantly adopting the red galaxy luminosity functions from either Bell et al. (2003) or Cole et al. (2001). Galaxies are placed randomly (i.e., uniform comoving density) in the volume z < 5. Next, we generate synthetic photometry for each object in the catalog using the set of Pégase V2.0 (Le Borgne & Rocca-Volmerange 2002) models shown in Figure 13, which are essentially passively evolving from z_form = 5. We also include typical errors for the MUSYC ECDFS catalog, to approximately account for photometric scatter. Finally, we apply our color selection criteria to this catalog, exactly as for the observed ECDFS catalog. The predicted prevalence of color-selected galaxies, to be compared with those observed, are given in column 5 of Table 4.

Whereas all red galaxies are assumed to evolve passively in the synthetic reference catalog, the real color-selected samples will include dusty or high-redshift star-forming galaxies: while all passive galaxies are red, not all red galaxies are passive. As we argue in Section 8.1, the number of color-selected galaxies can therefore be used to place an upper limit on the number of passively evolving galaxies, since these color criteria should select a complete but contaminated sample of genuinely passive galaxies.

The results presented in Table 4 thus suggest that the number density of bright, passively evolving galaxies has increased by a factor of at least ∼2 since z ∼ 1. Moreover, we find that the number density of passive galaxies in the range 1 ≲ z ≲ 2 is at most ∼ 25% of the present-day value. Taken at face value, this would suggest that at most 1/4 present-day red sequence galaxies can have evolved passively from z ∼ 2; the remainder were still forming, whether through star formation or by the hierarchical assembly of undetected, and thus fainter, subunits, or some combination of the two.

Finally, we emphasize the essential simplicity of this analysis, and thus its suitability for comparisons between different data sets, which may use different strategies for the computation of photometric redshifts, rest-frame colors, stellar masses, and so on.

Our final task is to compare the results of this section with those in Section 8; we do this in Figure 14. In this plot, the abscissa is the effective wavelength of the bluer of the two filters used in each color criterion; the ordinate is the number of color-selected galaxies observed in the MUSYC ECDFS catalog, relative to the expectation for passive evolution. The yellow squares refer to the z_form = 5 model shown in Figure 13 and presented in Table 4. The blue triangles and red pentagons show how these results would vary had we assumed z_form = 4 and z_form = 6, respectively.

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The specific redshift range that is selected by each color criterion depends on the particular tracks used. For a given color criterion, the mean redshift of selected sources (derived from the synthetic "passive evolution" catalog) thus varies for different choices of z_form: approximate transformations are given for each scenario at the top of the figure.

Finally, the solid line shows our fit results for the z_phot ≲ 2 number density evolution of red galaxies, shown in Figure 10. In order to put this curve on this plot, we have used the mean redshift of (synthetic, passive) color-selected sources assuming passive evolution from z_form = 5. Given the very different assumptions and methods lying behind these two results, the agreement is very impressive.

Both analyses clearly indicate that the number of red galaxies at z ≈ 0.7 is approximately 50% of the present-day value; at z ≈ 1.5, it is approximately 25%. In other words, at most, approximately 1/2 local red sequence galaxies were already "in place" by z ≈ 0.7, and have evolved passively since that time; at z ≈ 1.5, at most 1/4 were in place.

For each galaxy in the low-z sample, we have constructed a ugriz SED using "model" magnitudes (measured by fitting either an exponential or R^1/4 profile in each band, with structural parameters derived from the i-band image) as given in the low-z catalog. In analog to high-z galaxies, these SEDs are then scaled up to the "Petrosian" r magnitude. The Petrosian apertures are designed to measure a fixed fraction of a galaxy's light, irrespective of brightness or distance, but with some dependence on the light distribution: neglecting the effects of seeing the Petrosian aperture captures 99% of the total light for an exponential profile, compared to ≳80% for a de Vaucouleurs profile (Blanton et al. 2005c; Strauss et al. 2002). We make no attempt to correct for missed flux, but note that similar effects are present at a similar level in our own data (see Section 9.2.2). Following Blanton et al. (2005c), we use the following factors to correct the basic SDSS calibration to AB magnitudes: (−0.042, +0.036, +0.015, +0.013, −0.002) for (u, g, r, i, z).

We have derived rest-frame photometry and stellar masses for low-z galaxies using exactly the same machinery as for the high-z sample, adopting the heliocentric redshifts given in the low-z catalog. Our interpolated rest-frame ugriz photometry agrees quite well with that given by Blanton et al. (2005c) in the low-z catalog. (Whereas we use galaxy color–color relations, based on six empirical galaxy template spectra, to interpolate rest-frame photometry, Blanton et al. (2005c) determine rest-frame fluxes by fitting a non-negative linear combination of five carefully chosen synthetic spectra, and then integrating the best-fit spectrum.) On average, in comparison with Blanton et al. (2005c), our (u − r) colors are ∼0.02 mag bluer for blue galaxies and ∼0.03 mag redder for red galaxies. We calculate the distance modulus using the proper-motion-corrected redshifts given in the catalog. Finally, we have derived stellar masses based on the interpolated rest-frame B and V fluxes, using Equation (1).

Figure 15 shows our CMD (left) and CM_*D (right) for the general z ≈ 0 field galaxy population; in each panel, the logarithmic gray scales show the data density. We note that these two plots are very nearly equivalent: the CM_*D can be seen as simply a "sheared" version of the CMD, with the bluer galaxies dragged further toward the left. A separate red galaxy sequence is easily discernible in each panel.

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We have characterized this red galaxy sequence, as a function of magnitude or of mass, by taking a narrow slices in either absolute magnitude or stellar mass, and fitting double Gaussians to the color distribution in each bin. The results of these fits are shown in each panel of Figure 15 as the red and blue contours. This gives some indication of the relative numbers of red and blue galaxies as a function of magnitude/mass. In order to characterize the separate red and blue populations, we also show the center and width of each Gaussian fit as the yellow and cyan points and error bars.

In order to determine the slope of the high luminosity/mass end of the red sequence CMR and CM_*R, we have simply made linear fits to the centers of the M_r < −21 and M_*>10^10.5M_☉ "red" distributions. The results of these fits are

$$\begin{eqnarray} (u-r) &=& 2.612 - 0.090 \times (M_r + 22)\nonumber\\ & =& 2.573 + 0.237 \times \log _{10} (M_* / 10^{11} \mbox{ $M_{\odot }$}). \end{eqnarray} \tag{ A1 }$$

These relations are shown as the solid black lines in each panel. In order to make some distinction between "red" and "blue" galaxies which could be easily applied to the high-z sample, we simply use a cut 0.25 mag bluer than this relation. In each panel, this cut can be seen to approximately coincide with the so-called "green valley" between the red and blue galaxy populations.

Finally, in the inset of each diagram, we show the color distribution for the brightest (M_r < −21) or most massive (M_*>10¹¹) galaxies, after subtracting away the color–magnitude or color–stellar mass relations given by Equation (A1). (See also Figure 6.) We also show double Gaussian fits to these distributions, which are the basis for the z ≈ 0 points shown in Figures 8 and 9. We note in passing that, at least for −19 ≳ M_r ≳ −21.5, the width of the color distributions of red and blue galaxies is not a strong function of magnitude: Δ(u − r) ∼ 0.12 and Δ(u − r) ≈ 0.20 for red and blue galaxies, respectively.

For our purposes (see Section 8), the crucial final quantity is the number density of massive galaxies at z ≈ 0. We find that the number density of M_*>10¹¹M_☉ galaxies is (6.64 ± 0.38) × 10⁻⁴h³ Mpc⁻³; for (u, r) red galaxies, the number is (6.07 ± 0.36) × 10⁻⁴h³ Mpc⁻³. This measurement is not significantly affected by either volume or surface brightness incompleteness. Compared with the more sophisticated analyses of Bell et al. (2003) and Cole et al. (2001), our value for the number density of massive galaxies is 10% higher and 25% lower, respectively.

As we have repeatedly stressed, our prime concern is uniformity in the analysis of the low- and high-z samples. Photometric redshift errors are a major source of uncertainty for the high-z sample—how would comparable errors affect the low-z measurements? Given the redshift range of our low-z galaxies (0.003 < z < 0.05) and the estimated photometric redshift errors among high-z galaxies (Δz/(1 + z) ∼ 0.035), it would be inappropriate to simply apply typical redshift errors and repeat our calculations: this would effectively throw away all distance information. Instead, what we have done is to apply the typical uncertainties on M_r, (u − r), and M_* due to photometric redshift errors, as shown in Figure 2. For simplicity, we have ignored correlations between these errors.

In line with our earlier findings, including the effects of photometric redshift errors makes less of a difference to measurements based on stellar masses than to those based on absolute magnitudes. Our fits to the red sequence become

$$\begin{eqnarray} (u-r)& = &2.608 - 0.113 \times (M_r + 22) \nonumber\\ &=& 2.577 + 0.223 \times\, \log _{10} (M_* / 10^{11} \mbox{$M_{\odot }$}). \end{eqnarray} \tag{ A2 }$$

Our measurements of the z ≈ 0 number density of galaxies more massive than 10¹¹M_☉ become (6.93 ± 0.38) × 10⁻⁴h³ Mpc⁻³ in total, and (5.86 ± 0.35) × 10⁻⁴h³ Mpc⁻³ for red sequence galaxies. That is, the effects of the redshift errors expected among z ≫ 0 galaxies affect the measurement of the z ≈ 0 number density by less than 5%.

These numbers, approximately accounting for the Eddington bias due to photometric redshift errors (as they apply to the z ≫ 0 sample), are then what have used as a local reference value to compare with the z ≫ 0 data.

The essential differences between the COMBO-17 and MUSYC data sets are COMBO-17's 12 medium bands, allowing much more precise photometric redshift determinations for z ≲ 1, and the MUSYC z'JHK data, which open the door to the z ≳ 1 universe. The first and most obvious concern is thus that the difference between the two surveys' z_phot ≲ 1 results are a product of their different photometric redshift accuracies. To test this, we have tried simply adopting the COMBO-17 photometric redshifts in place of our own determinations, and repeated our analysis.

The results of this test are shown in Figure 16: the main panel shows our primary result (namely, the number density of M_*>10¹¹M_☉ galaxies as a function of redshift; see Figure 10). The solid histograms show this trial reanalysis of the MUSYC data adopting COMBO-17 redshifts; this should be compared with the fiducial results from MUSYC (dashed histogram) and COMBO-17 (dotted histograms). We have derived these "COMBO-17" results from the public catalog presented by Wolf et al. (2003), supplemented with the stellar mass determinations used by Borch et al. (2006).

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At least for z_phot < 0.8, the use of COMBO-17 redshifts in place of our own does not have a great effect on our results: the solid histograms lie quite close to the dashed ones. Quantitatively, adopting COMBO-17 redshifts in place of our own leaves the number of M_*>10¹¹M_☉ galaxies over 0.2 < z_phot < 0.8 unchanged to within 1%; the effect is slightly greater for the red population, leading to a red fraction that is 17% higher over the same interval. This implies that, at least for z_phot ≲ 1, our results are not seriously affected by our lower photometric redshift accuracy (σ_z≈ 0.03 for MUSYC vs. 0.02 for COMBO-17, for R < 24 and z_spec < 1).

To the left of the main panel in Figure 16, we show the difference in (from top to bottom) the photometric redshifts, absolute B, and V magnitudes, rest-frame (B − V) colors, and stellar masses used to produce the solid histograms, and those from the COMBO-17 catalog. In all cases, the "Δ" is in the sense of "MUSYC reanalysis" minus "COMBO-17 catalog," and is plotted as a function of the COMBO-17 catalog redshift. Within each panel, we also give the biweight mean offset for each quantity, evaluated for sources with R < 23 (to reduce random scatter due to photometric uncertainties) and 0.2 < z_phot < 0.5 (where the COMBO-17 photometry still samples rest-frame V) in the COMBO-17 catalog.

Looking now briefly at these panels, we see that for z_phot ≳ 0.5, where the COMBO-17 value for the rest-frame V magnitude comes from an extrapolation of the best-fit template spectrum, there is a progressive offset between the rest-frame V luminosities inferred by the MUSYC and COMBO-17 data and analyses—even while using the same redshifts. Even for z_phot < 0.5, however, we see a systematic offset of 0.05 mag in (B − V) color. Finally, we note that the greater number of 0.6 < z_phot < 0.8 galaxies in comparison with 0.2 < z_phot < 0.6 that we see in the MUSYC data is also present in the COMBO-17 photometric redshift distribution; the difference is that these galaxies do not have M_*>10¹¹M_☉ in the COMBO-17 catalog.

We therefore conclude that the difference between the COMBO-17 and MUSYC results is not a product of the two surveys' different photometric redshift accuracies: the use of COMBO-17 rather than MUSYC redshifts affects our results only by a few percent. This leaves the possibilities that the difference between the two results is due to the different methods used to infer galaxies' rest-frame properties, or to differences in the data themselves.

Are the discrepancies between the COMBO-17 and MUSYC results due to different systematic effects inherent in our different methods for deriving rest-frame photometry and stellar masses? In order to address this concern, we have reanalyzed the COMBO-17 data using the procedures described in the main text. For this test, in order to isolate the effect of the different analytical methods, we also adopt the COMBO-17 redshift determinations; the difference between this test and the previous one is thus the use of the COMBO-17, rather than the MUSYC, photometry.

The results of this reanalysis are shown as the solid histograms in Figure 17; all other symbols and their meanings are as in Figure 16. The results of the MUSYC reanalysis of the COMBO-17 data agree very well with the COMBO-17 analysis proper: the solid black histogram lies very near the dotted black histogram. Quantitatively, the MUSYC analysis of the COMBO-17 data leads to a red fraction which is 33% higher than for COMBO-17's own analysis; the total number density of M_*>10¹¹M_☉ galaxies agree to better than 1%.

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Comparing the MUSYC- and COMBO-17-derived rest-frame properties—again, based on the same photometry and redshifts—we do see systematic differences. We make three specific observations: (1) there appear to be discrete redshift regimes where the MUSYC- and COMBO-17-derived rest-frame fluxes compare differently; (2) for 0.2 ≲ z_phot ≲ 0.35, we see a "bimodal" offset in the MUSYC- and COMBO-17-derived (B − V) colors, corresponding to red and blue galaxies; (3) the MUSYC-derived rest-frame fluxes appear to be systematically brighter than those derived by COMBO-17, even for the same photometry. Remarkably, even despite these differences, our stellar mass estimates agree to within 0.04 dex, with a scatter of just 0.11 dex. Finally, we note that the progressive offset in extrapolated V luminosities for z_phot ≳ 0.5 seen in the previous test is not seen here; that is, applied to the same data, the two techniques for extrapolating the rest-frame photometry yield similar results.

We therefore conclude that the difference between the COMBO-17 and MUSYC results in the ECDFS cannot be explained by differences in the analytical methods employed by each team: the MUSYC reanalysis of the COMBO-17 data agrees with the COMBO-17 fiducial results. Instead, it seems that the different results are really in the data themselves.

A direct, object-by-object comparison of COMBO-17 and MUSYC photometry reveals significant differences in the photometric calibration of the two surveys (Paper I). Specifically, we see differences of 0.00, 0.06, 0.08, 0.08, and 0.14 mag between the MUSYC and COMBO-17 UBVRI-band photometry for galaxies (cf. stars), such that the MUSYC photometry is systematically brighter and redder. This has subsequently been confirmed to be due to an error in the photometric calibration of the COMBO-17 data (Wolf et al. 2008). Can this difference in photometric calibration explain the different results found by COMBO-17 and MUSYC?

To address this issue, we have simply scaled the COMBO-17 photometry to match the MUSYC photometry, and repeated our analysis. The COMBO-17 team has calibrated each of the medium bands relative to the nearest broad band; accordingly we have scaled each of the medium bands by the MUSYC-minus-COMBO-17 offset for the nearest broad band. The results of this test are shown in Figure 18. The MUSYC reanalysis of the recalibrated COMBO-17 photometry agrees extremely well with the fiducial MUSYC results: the measured number density of M_*>10¹¹ M_☉ galaxies at 0.2 < z_phot < 0.8 agrees to better than 1%.

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There are two separate aspects to this recalibration: galaxies are both brighter and redder in the MUSYC catalog than in COMBO-17. Since COMBO-17 measures the total fluxes from their R-band image, the 0.08 mag offset between the COMBO-17 and MUSYC zero-points makes galaxies appear 7.6% brighter (and thus more massive) in the MUSYC catalog; this effect is responsible for approximately 70% of the change in the measured number density. At the same time, the reddening of the SED shape implies a higher mass-to-light ratio after recalibration; this effect is responsible for the other 30% of the change in the measurements.

We therefore conclude that the primary cause for the disagreement between the results of the MUSYC and COMBO-17 surveys in the ECDFS is the different photometric calibrations of the two surveys: reanalyzing the COMBO-17 data set, recalibrated to match the MUSYC photometry, the results agree with the MUSYC fiducial analysis.