THE EVOLUTION OF THE STELLAR MASS FUNCTIONS OF STAR-FORMING AND QUIESCENT GALAXIES TO z = 4 FROM THE COSMOS/UltraVISTA SURVEY^*

Each galaxy in the catalog has a z_phot determined by fitting the photometry in the 0.15–8.0 μm bands to template spectral energy distributions (SEDs) using the EAZY code (Brammer et al. 2008). In default mode, EAZY fits photometric redshifts using linear combinations of six templates from the PEGASE models (Fioc & Rocca-Volmerange 1999), as well as an additional red template from the Maraston (2005) models. In order to improve the accuracy of the z_phot values for high-redshift galaxies, we added two new templates to the default set; a ∼1 Gyr old post-starburst template, as well as a slightly dust-reddened Lyman break template (see Muzzin et al. 2013). Comparison of the z_phot values with 5100 spectroscopic redshifts from the zCOSMOS-bright 10k sample (Lilly et al. 2007), as well as 19 spectroscopic redshifts for red galaxies at z > 1 (van de Sande et al. 2011, 2013; Onodera et al. 2012; Bezanson et al. 2013) shows that the z_phot values have an rms dispersion of δz/(1 + z) = 0.013 and a >3σ catastrophic outlier fraction of 1.6%.

Stellar masses for all galaxies have been determined by fitting the SEDs of galaxies to stellar population synthesis (SPS) models using the FAST code (Kriek et al. 2009). It is well known that M_star derived from SED fitting depends on the assumptions made (metallicity, SPS model, dust law, initial mass function (IMF)) in this process (e.g., Marchesini et al. 2009; Muzzin et al. 2009a, 2009b; Conroy et al. 2009). These assumptions typically result in systematic changes to the SMFs, rather than larger random errors (e.g., Marchesini et al. 2009). Given the complexity of these systematic dependencies, in this paper we base the majority of the analysis on a default set of assumptions for the SED modeling and then in Appendix A we expand the range of SED modeling parameter space and explore the effects on the SMFs. In Appendix A, we also explore the effects of expanding the EAZY template set to include an old and dusty template, which provides a good fit for some of the bright high-redshift population (see also Marchesini et al. 2010).

For the default set of M_star we fit the SEDs to a set of models with exponentially declining star formation histories (SFHs) of the form SFR ∝ e^−t/τ, where t is the time since the onset of star formation, and τ sets the timescale of the decline in the star formation rate (SFR). We use the models of Bruzual & Charlot (2003), hereafter BC03, with solar metallicity, a Calzetti et al. (2000) dust law, and we assume a Kroupa (2001) IMF.⁹ We allow log(τ/Gyr) to range between 7.0 and 10.0, log(t/Gyr) to range between 7.0 and 10.1, and A_v to range between 0 and 4. The maximum allowed age of galaxies is set by the age of the universe at their z_phot. Further details on the default model set and the fitting process are discussed in Muzzin et al. (2013).

The Muzzin et al. (2013) catalog contains a total of 262,615 objects down to a 3σ limit of K_s < 24.35 in a 21 aperture. From that parent sample, we define a mass-complete sample for computing the SMFs by applying various cuts to the catalog.

Simulations of the catalog completeness (see Muzzin et al. 2013, Figure 4) show that the 90% point-source completeness limit in total magnitudes for the UltraVISTA data is K_{s, tot} = 23.4 after source blending is accounted for. This limit in K_{s, tot} also corresponds to the ∼5σ limit for the photometry in the 21 color aperture, and therefore is a sensible limiting magnitude for computing the SMFs.

As a demonstration of the quality of the SEDs near the 90% completeness limit, we plot in Figure 1 some randomly chosen examples of red and blue galaxy SEDs in three redshift bins: 2.5 < z < 3.0 (top row), 3.0 < z < 3.5 (middle row), and 3.5 < z < 4.0 (bottom row). We plot SEDs of galaxies that have fluxes near the 90% completeness limit (K_{s, tot} ∼ 23.4), as well as SEDs of galaxies that are ∼1 magnitude brighter (K_{s, tot} ∼ 22.4). Figure 1 shows that the SEDs of both red and blue galaxies at K_{s, tot} ∼ 22.4 are very well constrained. It also shows that at K_{s, tot} ∼ 23.4, the SEDs are also reasonably well constrained; however, the typical signal-to-noise ratio (S/N) in a 21 aperture is ∼5.

Zoom In Zoom Out Reset image size

It is possible to include galaxies fainter than the 90% K_{s, tot} completeness limit in the SMFs and correct for this incompleteness; however, given that the quality of the SEDs near K_{s, tot} ∼ 23.4 becomes marginal, we have chosen to restrict the sample to galaxies with good S/N photometry. This ensures that all galaxies included in the SMFs have reasonably well determined M_star and z_phot values.

When constructing the SMFs, we also exclude objects flagged as stars (star = 1) based on a color–color cut, as well as those with badly contaminated photometry (SExtractor flag K_flag > 4). Objects nearby very bright stars (contamination = 1) or bad regions (nan_contam > 3) are also excluded, and the reduction in area from these effects is taken account into the total survey volume.

Once these cuts are applied, the final sample of galaxies available for the analysis is 160,070. In Figure 2, we plot a grayscale representation of the M_star of this sample as a function of z_phot. In general, the sample is dominated by objects at z < 2; however, there are reliable sources out to z = 4.

Zoom In Zoom Out Reset image size

Figure 2 shows the M_star of galaxies down to 90% K_s-band completeness limit of the survey; however, in order to construct the SMFs, the limiting M_star above which the magnitude-limited sample is complete needs to be determined. In order to estimate the redshift-dependent completeness limit in M_star, we adopt the approach developed in Marchesini et al. (2009), which exploits the availability of other survey data that are deeper than UltraVISTA. Specifically, we employed the K-selected FIRES (Labbé et al. 2003; Förster Schreiber et al. 2006) and FIREWORKS (Wuyts et al. 2008) catalogs, already used in Marchesini et al. (2009, 2010) and the H₁₆₀-selected catalogs over the Hubble Ultra-Deep Field (HUDF) used in Marchesini et al. (2012). The FIRES-HDFS, FIRES-MS1054, FIREWORKS, and HUDF reach limiting magnitudes of $K_{\rm S,{\rm tot}}=25.6$, 24.1, 23.7, and 25.6, respectively. The M_star values in these catalogs have been calculated using the same SED modeling assumptions as in the UltraVISTA catalog.

Briefly, to estimate the redshift-dependent stellar mass-completeness limit of the UltraVISTA sample at K_{s, tot} = 23.4, we first selected galaxies belonging to the available deeper samples. We then scaled their fluxes and M_star values to match the K-band completeness limit of the UltraVISTA sample. The upper envelope of points in (M_{star, scaled}–z) space, encompassing 100% of the points, represents the most massive galaxies at K_s = 23.4 and so provides a redshift-dependent M_star completeness limit for the UltraVISTA sample. We refer the reader to Marchesini et al. (2009) for a more detailed description of this method. In Figure 2, we show this empirically derived 100% mass-completeness limit as a purple curve. Also, for reference we show the mass-completeness limit for a simple stellar population (SSP) formed at z = 10, which is extreme but indicative of a maximally old population.

Figure 2 shows that for galaxies at z < 1.5 and K_{s, tot} ∼ 23.4, the most extreme mass-to-light (M/L) ratios are less extreme than for a SSP. Around z = 1.5, the SSP curve and the empirical 100% completeness curve cross each other, which implies that there exist galaxies that have larger M/L ratios than a SSP. Such galaxies are typically galaxies with intermediate-to-old ages (for their redshift) with up to several magnitudes of dust extinction. More detailed SED modeling for galaxies with spectroscopic redshifts shows that these dusty and old galaxies are not uncommon among the massive galaxy population at z > 1.5 (see, e.g., Kriek et al. 2006, 2008; Muzzin et al. 2009a).

As Figure 2 shows, the empirically derived 100% mass-completeness limits are high due to the old and dusty population. Adopting the empirical 100% completeness limit for the SMFs therefore requires the exclusion of 57% of the magnitude-limited sample.

The 100% completeness limit is set by most extreme M/L ratio at any given redshift, regardless of the frequency of its occurrence. In principle, if only a small fraction of objects have these extreme M/L ratios, then adopting the 100% mass-completeness limit is an inefficient use of the data. In order to try to make better use of the dataset, we also derived 95% mass-completeness limits for the sample and this limit is plotted in Figure 2.

At all redshifts, the 95% mass-completeness limits are 0.2–0.3 dex lower, showing that it is only a small fraction of the overall population of galaxies that have extreme M/L ratios. If we adopt the 100% mass-completeness limits, the resulting sample of galaxies is 67,942. Adopting the 95% mass-completeness limits increases the sample by a factor of 1.4 to 95,675 galaxies. Given this substantial increase in statistics, and the advantage gained by probing further down the SMFs at higher redshift, we have adopted the 95% mass-completeness limits for the SMFs, but correct the lowest mass bin in each SMF by 5% in order to account for this.

It is well known that the overall galaxy population is bimodal in the distribution of colors and SFRs (e.g., Kauffmann et al. 2003; Hogg et al. 2004; Balogh et al. 2004; Blanton & Moustakas 2009) and that this bimodality persists out to high redshift (e.g., Bell et al. 2004; Taylor et al. 2009; Williams et al. 2009; Brammer et al. 2009, 2011). Given the bimodality, separating the evolution of the SMFs of star-forming and quiescent galaxies as a function of redshift is useful for understanding the relationship between the two populations.

In recent years, several methods have been developed to classify galaxies into these categories. In this analysis, we classify between the types using the rest-frame U − V versus V − J color–color diagram (hereafter, the UVJ diagram). UVJ classification has been used in many previous studies (e.g., Labbé et al. 2005; Wuyts et al. 2007; Williams et al. 2009; Brammer et al. 2011; Patel et al. 2012). These previous studies have shown that separation of star-forming and quiescent galaxies in this color–color space is well-correlated with separation using UV+IR determined specific star formation rates (SSFRs; e.g., Williams et al. 2009) and SED fitting-determined SSFRs (e.g., Williams et al. 2010) up to z = 2.5. Separation in this color space is also correlated with the detection and non-detection of galaxies at 24 μm, down to implied SFRs of ∼40 M_☉ yr⁻¹ at z ∼ 2 (Wuyts et al. 2007; Brammer et al. 2011). We choose to separate galaxies based on a rest-frame color cut as opposed to a cut in a derived quantity such as SSFR because rest-frame colors can be calculated in a straightforward way for each galaxy in the sample. UV+IR SFRs can only be calculated for the most strongly star-forming galaxies at high redshift due to the limited depth of the 24 μm data.

In Figure 3, we plot the U − V versus V − J diagram for galaxies more massive than the 95% mass-completeness limits in several redshift bins. The galaxy bimodality is clearly visible in the UVJ diagram up to z = 2, but thereafter becomes less pronounced at the M_star completeness limits probed by the K_s-selected UltraVISTA catalog.

Zoom In Zoom Out Reset image size

To distinguish between star-forming and quiescent galaxies, we use box regions in the UVJ diagram that are similar, although not identical, to those defined in Williams et al. (2009), Whitaker et al. (2011), and Brammer et al. (2011). These regions are plotted as the solid lines in Figure 3. Quiescent galaxies are defined as

$$\begin{equation} U - V > 1.3, V - J < 1.5,\, \mbox{(all redshifts)} \end{equation} \tag{ 1 }$$

$$\begin{equation} U - V > (V - J)\times 0.88 + 0.69, \mbox{(0.0} < z < \mbox{1.0)} \end{equation} \tag{ 2 }$$

$$\begin{equation} U - V > (V - J)\times 0.88 + 0.59, \mbox{(1.0} < z < \mbox{4.0).} \end{equation} \tag{ 3 }$$

We note that these boxes are chosen arbitrarily, with the main criteria being that they lie roughly between the two modes of the population seen in Figure 3. They were originally defined by Williams et al. (2009), who defined them in such a way as to maximize the difference in SSFRs between the regions; however, our rest-frame color distribution is slightly different from that of Williams et al. (2009), which is the reason that we have adjusted the box locations. In Appendix B, we explore the effect on the SMFs of moving the location of the boxes in UVJ space. In general, we find that changing the UVJ box has little effect on the high-mass end of the quiescent SMF and the low-mass end of the star-forming SMF because those galaxies are dominated by very red and very blue galaxies, respectively. It does have a large effect on the SMF for intermediate mass galaxies, which is not unexpected given that these are typically the transition population at most redshifts.

With well-defined mass-completeness limits as a function of redshift and criteria for separating star-forming and quiescent galaxies, SMFs can now be computed. We employ two methods to determine the SMFs: the 1/V_max method and a maximum-likelihood method. These methods have different strengths and weaknesses. The 1/V_max method has the advantage that it does not assume a parametric form of the SMF, allowing a direct visualization of the data; however, it is a fully normalized solution and is susceptible to the effects of clustering. Conversely, the maximum-likelihood method has the advantage that it is not affected by density inhomogeneities (e.g., Efstathiou et al. 1988); however, it does assume a functional form for the fit.

3.4.1. The 1/V_max Method

3.4.2. The Maximum-likelihood Method

The SMFs are also determined using the maximum-likelihood method outlined by Sandage et al. (1979). For this method, it is assumed that the number density of galaxies (Φ(M_star)) is described by a Schechter (1976) function of the form

$$\begin{equation} \Phi (M) = (\ln 10)\Phi ^{*}[10^{(M - M^{*})(1+\alpha)}] \times \exp [-10^{(M - M^{*})}], \end{equation} \tag{ 4 }$$

where M = log(M_star/M_☉), α is the low-mass-end slope, M* = log($M^{*}_{{\rm star}}$/M_☉), where $M^{*}_{{\rm star}}$ is the characteristic mass, and Φ* is the normalization. For each possible combination of α and $M^{*}_{{\rm star}}$, the likelihood that each galaxy would be found in the survey is calculated. The best-fit solution for α and $M^{*}_{{\rm star}}$ in each redshift bin is obtained by maximizing the combined likelihoods of all galaxies (Λ) with respect to these parameters. Φ* is determined by requiring that the total number of observed galaxies is reproduced. The errors in Φ* are then determined from the minimum and maximum values of Φ* allowed by the confidence contours in the α versus $M^{*}_{{\rm star}}$ plane. Further details of the fitting process can be found in Marchesini et al. (2007, 2009).

3.4.3. The Low-mass-end Slope α

3.4.4. Determination of Uncertainties in the SMFs

In addition to the Poisson uncertainties, there are several other sources of uncertainty in the construction of SMFs that need to be taken into account. The largest are caused by the fact that M_star itself is not an observable quantity, but is derived from observables (i.e., multiwavelength photometry) using a set of models. The effect of photometric uncertainties on the derived values of z_phot and M_star is a non-trivial function of color, magnitude, and redshift caused by a range of data depths in various bands within the survey.

In order to calculate uncertainties in the SMFs due to photometric uncertainties, we perform 100 Monte Carlo (MC) realizations of the catalog. Within each realization, the photometry in the catalog is perturbed using the measured photometric uncertainties. New z_phot and M_star values are calculated for each galaxy using the perturbed catalog. The 100 MC catalogs are then used to recalculate the SMFs and the range of values gives an empirical estimate of the uncertainties in the SMFs due to uncertainties in M_star and z_phot that propagate from the photometric uncertainties.

In addition to these z_phot and M_star uncertainties, the uncertainty from cosmic variance is also included using the prescriptions of Moster et al. (2011). In Figure 4, we plot the uncertainty in the abundance of galaxies with log(M_star/M_☉) = 11.0 due to cosmic variance as a function of redshift. Cosmic variance is most pronounced at the high-mass end where galaxies are more clustered and at low redshift, where the survey volume is smallest. Also plotted in Figure 4 are the cosmic variance uncertainties from other NIR surveys such as FIREWORKS (Wuyts et al. 2008), MUYSC (Quadri et al. 2007; Marchesini et al. 2009), NMBS (Whitaker et al. 2011), and the UDS (Williams et al. 2009). These surveys cover areas that are factors of ∼50, 16, 4, and 2 smaller than UltraVISTA, respectively. Figure 4 shows that the larger area of UltraVISTA offers a factor of 1.5 improvement in the uncertainties in cosmic variance compared with even the best previous surveys; over the full redshift range, the uncertainty from cosmic variance is ∼8%–15% at log(M_star/M_☉) = 11.0.

Zoom In Zoom Out Reset image size

The total uncertainties in the determination of the SMFs are derived as follows. For the 1/V_max method, the total 1σ random error in each mass bin is the quadrature sum of the Poisson error, the error from photometric uncertainties as derived using the MC realizations, and the error due to cosmic variance. For the maximum-likelihood method, the total 1σ random errors of the Schechter function parameters α, $M^{*}_{{\rm star}}$, and Φ* are the quadrature sum of the errors from the maximum-likelihood analysis, the errors from photometric uncertainties as derived using the MC realizations, and the error due to cosmic variance (affecting only the normalization Φ*).

In Figure 5, we plot the best-fit maximum-likelihood SMFs for the star-forming, quiescent, and combined populations of galaxies. Figure 5 illustrates the redshift evolution of the SMFs of the individual populations, which we discuss in detail in Section 5. To better illustrate the relative contribution of both star-forming and quiescent galaxies to the combined SMF, we plot in Figure 6 the SMFs derived using the 1/V_max method (points), as well as the fits from the maximum-likelihood method (filled regions) in the same redshift bins. The SMFs of the combined population are plotted in the top panels, and the SMFs of the star-forming and quiescent populations are plotted in the middle panels. Within each of the higher redshift bins, the SMFs from the lowest redshift bin (0.2 < z < 0.5) are shown as the dotted line as a fiducial to demonstrate the relative evolution of the SMFs. The fraction of quiescent galaxies as a function of M_star is shown in the bottom panels and the best-fit Schechter function parameters for these redshift ranges are listed in Table 1.

Zoom In Zoom Out Reset image size

Zoom In Zoom Out Reset image size

Table 1. Best-fit Schechter Function Parameters for the SMFs

Redshift	Sample	Number	$\log {M^{\rm lim}_{\rm star}}$	$\log {M^{\star }_{\rm star}}$	Φ⋆	α	$\Phi ^{\star }_{\rm 2}$	α2
			(M☉)	(M☉)	(10−4 Mpc−3)		(10−4 Mpc−3)
0.2 ⩽ z < 0.5	All	18546	8.37	11.22$^{+0.03}_{-0.03}$(0.03)	12.16$^{+0.52}_{-0.50}$($^{+1.75}_{-1.55}$)	−1.29 ± 0.01(0.01)	⋅⋅⋅	⋅⋅⋅
0.2 ⩽ z < 0.5	All	18546	8.37	11.06 ± 0.01(0.01)	19.02 ± 0.14(2.31)	−1.2	⋅⋅⋅	⋅⋅⋅
0.2 ⩽ z < 0.5	All	18546	8.37	10.97 ± 0.06(0.06)	16.27$^{+3.12}_{-1.39}$($^{+3.88}_{-2.41}$)	−0.53$^{+0.16}_{-0.27}$($^{+0.16}_{-0.28}$)	9.47$^{+2.02}_{-3.64}$($^{+2.32}_{-3.83}$)	−1.37$^{+0.01}_{-0.06}$($^{+0.01}_{-0.06}$)
0.2 ⩽ z < 0.5	Quiescent	4364	8.37	11.21 ± 0.03(0.04)	10.09 ± 0.54($^{+1.69}_{-1.33}$)	−0.92 ± 0.02($^{+0.04}_{-0.02}$)	⋅⋅⋅	⋅⋅⋅
0.2 ⩽ z < 0.5	Quiescent	4364	8.37	10.75 ± 0.01(0.01)	30.65 ± 0.01(3.71)	−0.4	⋅⋅⋅	⋅⋅⋅
0.2 ⩽ z < 0.5	Quiescent	4364	8.37	10.92$^{+0.06}_{-0.02}$($^{+0.06}_{-0.02}$)	19.68$^{+1.09}_{-1.73}$($^{+2.64}_{-2.96}$)	−0.38$^{+0.06}_{-0.11}$($^{+0.06}_{-0.12}$)	0.58$^{+0.26}_{-0.31}$($^{+0.27}_{-0.32}$)	−1.52$^{+0.06}_{-0.16}$($^{+0.06}_{-0.16}$)
0.2 ⩽ z < 0.5	Star-forming	14182	8.37	10.81 ± 0.03(0.03)	11.35$^{+0.76}_{-0.67}$($^{+1.60}_{-1.51}$)	−1.34 ± 0.01(0.01)	⋅⋅⋅	⋅⋅⋅
0.2 ⩽ z < 0.5	Star-forming	14182	8.37	10.75 ± 0.01(0.01)	13.58$^{+0.15}_{-0.13}$($^{+1.62}_{-1.59}$)	−1.3	⋅⋅⋅	⋅⋅⋅
0.5 ⩽ z < 1.0	All	42019	8.92	11.00$^{+0.02}_{-0.01}$($^{+0.02}_{-0.01}$)	16.25$^{+0.28}_{-0.62}$($^{+1.17}_{-1.28}$)	−1.17 ± 0.01(0.01)	⋅⋅⋅	⋅⋅⋅
0.5 ⩽ z < 1.0	All	42019	8.92	11.04 ± 0.01(0.01)	14.48 ± 0.07(1.00)	−1.2	⋅⋅⋅	⋅⋅⋅
0.5 ⩽ z < 1.0	Quiescent	9127	8.92	10.87$^{+0.02}_{-0.01}$($^{+0.02}_{-0.01}$)	13.68$^{+0.26}_{-0.36}$($^{+1.09}_{-1.01}$)	−0.44 ± 0.02($^{+0.04}_{-0.02}$)	⋅⋅⋅	⋅⋅⋅
0.5 ⩽ z < 1.0	Quiescent	9127	8.92	10.84 ± 0.01(0.02)	14.38 ± 0.02(0.99)	−0.4	⋅⋅⋅	⋅⋅⋅
0.5 ⩽ z < 1.0	Quiescent	9127	8.92	10.84 ± 0.02(0.03)	14.55$^{+0.56}_{-0.42}$($^{+1.21}_{-1.09}$)	−0.36$^{+0.06}_{-0.03}$($^{+0.06}_{-0.04}$)	0.005$^{+0.021}_{-0.003}$($^{+0.021}_{-0.004}$)	−2.32$^{+0.40}_{-0.32}$($^{+0.41}_{-0.38}$)
0.5 ⩽ z < 1.0	Star-forming	32892	8.92	10.78$^{+0.01}_{-0.02}$	12.71$^{+0.51}_{-0.29}$($^{+1.07}_{-0.91}$)	−1.26 ± 0.01($^{+0.02}_{-0.01}$)	⋅⋅⋅	⋅⋅⋅
0.5 ⩽ z < 1.0	Star-forming	32892	8.92	10.82 ± 0.01(0.02)	10.95$^{+0.06}_{-0.08}$(0.73)	−1.3	⋅⋅⋅	⋅⋅⋅
1.0 ⩽ z < 1.5	All	22959	9.48	10.87$^{+0.02}_{-0.01}$($^{+0.02}_{-0.01}$)	13.91$^{+0.43}_{-0.59}$($^{+1.05}_{-1.23}$)	−1.02 ± 0.02(0.02)	⋅⋅⋅	⋅⋅⋅
1.0 ⩽ z < 1.5	All	22959	9.48	10.99 ± 0.01(0.01)	9.30 ± 0.06(0.66)	−1.2	⋅⋅⋅	⋅⋅⋅
1.0 ⩽ z < 1.5	Quiescent	6455	9.48	10.73 ± 0.02(0.02)	8.81 ± 0.19(0.63)	−0.17 ± 0.04($^{+0.06}_{-0.04}$)	⋅⋅⋅	⋅⋅⋅
1.0 ⩽ z < 1.5	Quiescent	6455	9.48	10.83 ± 0.01(0.02)	7.48 ± 0.02($^{+0.51}_{-0.56}$)	−0.4	⋅⋅⋅	⋅⋅⋅
1.0 ⩽ z < 1.5	Star-forming	16504	9.48	10.76 ± 0.02(0.02)	8.87$^{+0.50}_{-0.54}$($^{+0.79}_{-0.86}$)	−1.21 ± 0.03(0.03)	⋅⋅⋅	⋅⋅⋅
1.0 ⩽ z < 1.5	Star-forming	16504	9.48	10.82 ± 0.01(0.01)	7.20$^{+0.06}_{-0.11}$(0.49)	−1.3	⋅⋅⋅	⋅⋅⋅
1.5 ⩽ z < 2.0	All	8927	10.03	10.81 ± 0.02($^{+0.02}_{-0.03}$)	10.13$^{+0.58}_{-0.56}$($^{+1.11}_{-1.02}$)	−0.86 ± 0.06($^{+0.11}_{-0.06}$)	⋅⋅⋅	⋅⋅⋅
1.5 ⩽ z < 2.0	All	8927	10.03	10.96 ± 0.01($^{+0.03}_{-0.01}$)	6.33$^{+0.06}_{-0.11}$($^{+0.53}_{-0.71}$)	−1.2	⋅⋅⋅	⋅⋅⋅
1.5 ⩽ z < 2.0	Quiescent	2656	10.03	10.67 ± 0.03(0.04)	4.15$^{+0.06}_{-0.08}$($^{+0.35}_{-0.36}$)	0.03 ± 0.11(0.12)	⋅⋅⋅	⋅⋅⋅
1.5 ⩽ z < 2.0	Quiescent	2656	10.03	10.80 ± 0.01(0.02)	3.61$^{+0.02}_{-0.04}$(0.30)	−0.4	⋅⋅⋅	⋅⋅⋅
1.5 ⩽ z < 2.0	Star-forming	6271	10.03	10.85$^{+0.02}_{-0.03}$($^{+0.02}_{-0.04}$)	5.68$^{+0.60}_{-0.40}$($^{+0.89}_{-0.65}$)	−1.16$^{+0.07}_{-0.05}$($^{+0.14}_{-0.06}$)	⋅⋅⋅	⋅⋅⋅
1.5 ⩽ z < 2.0	Star-forming	6271	10.03	10.91 ± 0.01($^{+0.04}_{-0.01}$)	4.49 ± 0.09($^{+0.39}_{-0.60}$)	−1.3	⋅⋅⋅	⋅⋅⋅
2.0 ⩽ z < 2.5	All	2236	10.54	10.81 ± 0.05($^{+0.06}_{-0.05}$)	4.79$^{+0.28}_{-0.41}$($^{+0.70}_{-0.76}$)	−0.55$^{+0.16}_{-0.19}$($^{+0.22}_{-0.24}$)	⋅⋅⋅	⋅⋅⋅
2.0 ⩽ z < 2.5	All	2236	10.54	11.00 ± 0.02(0.02)	2.94$^{+0.07}_{-0.11}$(0.40)	−1.2	⋅⋅⋅	⋅⋅⋅
2.0 ⩽ z < 2.5	Quiescent	528	10.54	10.87 ± 0.10($^{+0.11}_{-0.17}$)	1.02$^{+0.17}_{-0.23}$($^{+0.30}_{-0.27}$)	−0.71$^{+0.37}_{-0.33}$($^{+0.67}_{-0.39}$)	⋅⋅⋅	⋅⋅⋅
2.0 ⩽ z < 2.5	Quiescent	528	10.54	10.79 ± 0.02(0.03)	1.14 ± 0.05(0.18)	−0.4	⋅⋅⋅	⋅⋅⋅
2.0 ⩽ z < 2.5	Star-forming	1708	10.54	10.80 ± 0.05(0.06)	3.72$^{+0.25}_{-0.32}$($^{+0.58}_{-0.66}$)	−0.53$^{+0.22}_{-0.19}$($^{+0.28}_{-0.24}$)	⋅⋅⋅	⋅⋅⋅
2.0 ⩽ z < 2.5	Star-forming	1708	10.54	11.03 ± 0.02($^{+0.03}_{-0.02}$)	2.01$^{+0.08}_{-0.10}$($^{+0.28}_{-0.30}$)	−1.3	⋅⋅⋅	⋅⋅⋅
2.5 ⩽ z < 3.0	All	814	10.76	11.03$^{+0.10}_{-0.09}$($^{+0.12}_{-0.11}$)	1.93$^{+0.43}_{-0.51}$($^{+0.62}_{-0.68}$()	−1.01$^{+0.37}_{-0.34}$($^{+0.45}_{-0.41}$)	⋅⋅⋅	⋅⋅⋅
2.5 ⩽ z < 3.0	All	814	10.76	11.09 ± 0.02(0.03)	1.66 ± 0.10(0.34)	−1.2	⋅⋅⋅	⋅⋅⋅
2.5 ⩽ z < 3.0	Quiescent	178	10.76	10.80$^{+0.23}_{-0.17}$($^{+0.27}_{-0.21}$)	0.65$^{+0.12}_{-0.24}$($^{+0.18}_{-0.27}$)	−0.39$^{+1.03}_{-0.95}$($^{+1.18}_{-1.11}$)	⋅⋅⋅	⋅⋅⋅
2.5 ⩽ z < 3.0	Quiescent	178	10.76	10.81 ± 0.04(0.05)	0.66$^{+0.08}_{-0.07}$($^{+0.17}_{-0.14}$)	−0.4	⋅⋅⋅	⋅⋅⋅
2.5 ⩽ z < 3.0	Star-forming	636	10.76	11.06$^{+0.13}_{-0.10}$($^{+0.14}_{-0.12}$)	1.39$^{+0.35}_{-0.48}$($^{+0.47}_{-0.57}$)	−1.03 ± 0.39(0.47)	⋅⋅⋅	⋅⋅⋅
2.5 ⩽ z < 3.0	Star-forming	636	10.76	11.14 ± 0.03($^{+0.05}_{-0.03}$)	1.09 ± 0.09($^{+0.22}_{-0.26}$)	−1.3	⋅⋅⋅	⋅⋅⋅
3.0 ⩽ z < 4.0	All	174	10.94	11.49$^{+0.36}_{-0.22}$($^{+0.37}_{-0.28}$)	0.09$^{+0.09}_{-0.07}$($^{+0.15}_{-0.08}$)	−1.45$^{+0.59}_{-0.54}$($^{+0.70}_{-0.60}$)	⋅⋅⋅	⋅⋅⋅
3.0 ⩽ z < 4.0	All	174	10.94	11.40 ± 0.06(0.08)	0.13 ± 0.02($^{+0.08}_{-0.04}$)	−1.2	⋅⋅⋅	⋅⋅⋅
3.0 ⩽ z < 4.0	Quiescent	28	10.94	10.85$^{+0.64}_{-0.32}$($^{+0.76}_{-0.43}$)	0.04$^{+0.03}_{-0.04}$($^{+0.05}_{-0.04}$)	0.46$^{+3.16}_{-2.41}$($^{+3.30}_{-2.78}$)	⋅⋅⋅	⋅⋅⋅
3.0 ⩽ z < 4.0	Quiescent	28	10.94	11.00 ± 0.10($^{+0.14}_{-0.11}$)	0.05$^{+0.02}_{-0.01}$($^{+0.05}_{-0.02}$)	−0.4	⋅⋅⋅	⋅⋅⋅
3.0 ⩽ z < 4.0	Star-forming	146	10.94	11.56$^{+0.44}_{-0.25}$($^{+0.45}_{-0.33}$)	0.06$^{+0.08}_{-0.05}$($^{+0.12}_{-0.05}$)	−1.51$^{+0.61}_{-0.55}$($^{+0.77}_{-0.62}$)	⋅⋅⋅	⋅⋅⋅
3.0 ⩽ z < 4.0	Star-forming	146	10.94	11.47 ± 0.07($^{+0.08}_{-0.10}$)	0.08 ± 0.02($^{+0.05}_{-0.03}$)	−1.3	⋅⋅⋅	⋅⋅⋅

Note. The listed errors are the 1σ Poisson errors, whereas the values in parentheses list the total 1σ errors, including Poisson uncertainties, the uncertainties from photometric redshift random errors, and cosmic variance.

Download table as:  ASCIITypeset image

For reference, in the lowest redshift panel (0.2 < z < 0.5) of Figure 6 we plot the SMFs at z ∼ 0.1 for the total population from the of studies of Cole et al. (2001), Bell et al. (2003), and Baldry et al. (2012). Plotted in the middle panel of the lowest redshift bin are the SMFs of star-forming and quiescent galaxies from Bell et al. (2003) and Baldry et al. (2012). Qualitatively, these are similar to our measurements, although we note that the selection of star-forming and quiescent galaxies is done differently than the UVJ selection in UltraVISTA.

In the left panel of Figure 7, we plot the integrated SMD of all galaxies as a function of redshift. For consistency with other studies in the literature, the SMDs have been calculated by integrating the maximum-likelihood Schechter function fits down to a limit of log(M_star/M_☉) = 8.0 at each redshift. We perform this integration using the full maximum-likelihood fit, even though at z > 2 α is not well constrained. In order to account for possible systematic errors caused by an underestimate of α due to the limited data depth, we also include at all redshifts the uncertainties from the maximum-likelihood fits with α = −1.2. Therefore, the quoted uncertainties in Figure 7, and all subsequent figures that use integration of the SMFs, span the full range of uncertainties for both the α-free and α-fixed SMFs.

Zoom In Zoom Out Reset image size

Overplotted in Figure 7 are measurements of the SMD at various redshifts from previous studies by Cole et al. (2001, C01), Bell et al. (2003, B03), Drory et al. (2005, D05), Rudnick et al. (2006, R06), Fontana et al. (2006, F06), Elsner et al. (2008, E08), Pérez-González et al. (2008, P08), Marchesini et al. (2009, M09), Kajisawa et al. (2009, K09), Drory et al. (2009, D09), Marchesini et al. (2010, M10), Ilbert et al. (2010, I10), Mortlock et al. (2011, M11), Baldry et al. (2012, B12), Bielby et al. (2012, Bi12), Santini et al. (2012, S12), and Moustakas et al. (2013, M13). The substantially larger volume covered by UltraVISTA allows for an impressive improvement in the uncertainties in the evolution of the SMDs. Within the uncertainties, the majority of previous measurements agree reasonably well with the UltraVISTA measurements, particularly at z < 2. At z > 2, there is less agreement with previous datasets; the UltraVISTA SMDs are lower than some previous works such as Elsner et al. (2008), Pérez-González et al. (2008), and Santini et al. (2012). The disagreement with Elsner et al. (2008) and Pérez-González et al. (2008) is because those studies measure a larger Φ* than UltraVISTA. The discrepancy with Santini et al. (2012) is primarily because they measure a steep value of α at z > 2.

SMDs and their uncertainties for star-forming and quiescent galaxies have also been computed using the same integration method as for the total SMD. These are plotted in the left panel of Figure 8 as a function of redshift. In general, it is clear that at z < 3.5, the SMD of quiescent galaxies grows faster than that of star-forming galaxies. We explore the implications of this further in Section 5. All SMDs are listed in Table 2.

Zoom In Zoom Out Reset image size

Table 2. Number and Stellar Mass Densities

Redshift	Density	$\log {\left (\frac{M_{\rm star}}{M_{\odot }}\right)}>8$	$\log {\left (\frac{M_{\rm star}}{M_{\odot }}\right)}>9$	$\log {\left (\frac{M_{\rm star}}{M_{\odot }}\right)}>10$	$\log {\left (\frac{M_{\rm star}}{M_{\odot }}\right)}>11$	$\log {\left (\frac{M_{\rm star}}{M_{\odot }}\right)}>11.5$
0.2 ⩽ z < 0.5	log (ηA)	−1.49 ± 0.06	−1.89 ± 0.07	−2.35 ± 0.09	−3.25 ± 0.13	−4.52$^{+0.18}_{-0.23}$
	log (ηQ)	−2.19$^{+0.07}_{-0.06}$	−2.45 ± 0.07	−2.66 ± 0.09	−3.33 ± 0.13	−4.60$^{+0.20}_{-0.17}$
	log (ηSF)	−1.58 ± 0.06	−2.03 ± 0.07	−2.65 ± 0.09	−4.08$^{+0.13}_{-0.15}$	−6.07$^{+0.21}_{-0.36}$
	log (ρA)	8.41 ± 0.06	8.40 ± 0.07	8.35 ± 0.09	7.98$^{+0.13}_{-0.14}$	7.08$^{+0.18}_{-0.23}$
	log (ρQ)	8.19 ± 0.06	8.19 ± 0.07	8.18 ± 0.09	7.91 ± 0.13	7.00$^{+0.21}_{-0.17}$
	log (ρSF)	7.99 ± 0.06	7.97 ± 0.07	7.85 ± 0.09	7.08$^{+0.14}_{-0.16}$	5.50$^{+0.21}_{-0.35}$
0.5 ⩽ z < 1.0	log (ηA)	−1.69 ± 0.04	−2.00 ± 0.04	−2.45 ± 0.06	−3.49 ± 0.11	−4.88$^{+0.18}_{-0.14}$
	log (ηQ)	−2.36$^{+0.26}_{-0.22}$	−2.70 ± 0.04	−2.85 ± 0.06	−3.60 ± 0.10	−5.10$^{+0.16}_{-0.17}$
	log (ηSF)	−1.70$^{+0.04}_{-0.03}$	−2.09 ± 0.04	−2.66 ± 0.06	−4.10$^{+0.11}_{-0.10}$	−6.18$^{+0.24}_{-0.16}$
	log (ρA)	8.26 ± 0.03	8.25 ± 0.04	8.19 ± 0.06	7.73$^{+0.11}_{-0.10}$	6.72$^{+0.19}_{-0.14}$
	log (ρQ)	7.96 ± 0.03	7.96 ± 0.04	7.94 ± 0.06	7.61 ± 0.10	6.48$^{+0.16}_{-0.17}$
	log (ρSF)	7.97 ± 0.03	7.95 ± 0.04	7.84 ± 0.06	7.06 ± 0.11	5.39$^{+0.24}_{-0.16}$
1.0 ⩽ z < 1.5	log (ηA)	−2.04$^{+0.17}_{-0.03}$	−2.26$^{+0.07}_{-0.04}$	−2.66 ± 0.06	−3.77 ± 0.12	−5.45$^{+0.35}_{-0.18}$
	log (ηQ)	−3.01$^{+0.05}_{-0.03}$	−3.01$^{+0.05}_{-0.04}$	−3.12 ± 0.06	−3.93 ± 0.12	−5.74$^{+0.33}_{-0.19}$
	log (ηSF)	−1.94$^{+0.10}_{-0.04}$	−2.30$^{+0.05}_{-0.04}$	−2.85 ± 0.06	−4.27 ± 0.12	−6.39$^{+0.29}_{-0.19}$
	log (ρA)	8.02 ± 0.03	8.01 ± 0.04	7.95 ± 0.06	7.41$^{+0.13}_{-0.12}$	6.13$^{+0.36}_{-0.18}$
	log (ρQ)	7.65 ± 0.03	7.65 ± 0.04	7.64 ± 0.06	7.25 ± 0.12	5.83$^{+0.34}_{-0.19}$
	log (ρSF)	7.78 ± 0.03	7.76 ± 0.04	7.66 ± 0.06	6.88$^{+0.13}_{-0.12}$	5.17$^{+0.30}_{-0.19}$
1.5 ⩽ z < 2.0	log (ηA)	−2.41$^{+0.37}_{-0.11}$	−2.56$^{+0.19}_{-0.08}$	−2.88 ± 0.07	−3.97 ± 0.14	−5.76$^{+0.38}_{-0.21}$
	log (ηQ)	−3.39$^{+0.12}_{-0.04}$	−3.40$^{+0.10}_{-0.05}$	−3.48 ± 0.07	−4.29 ± 0.14	−6.27$^{+0.40}_{-0.24}$
	log (ηSF)	−2.21$^{+0.21}_{-0.19}$	−2.52 ± 0.11	−3.01 ± 0.07	−4.25 ± 0.14	−6.05$^{+0.26}_{-0.23}$
	log (ρA)	7.79$^{+0.05}_{-0.03}$	7.79$^{+0.05}_{-0.04}$	7.74 ± 0.07	7.20$^{+0.14}_{-0.13}$	5.81$^{+0.39}_{-0.21}$
	log (ρQ)	7.29 ± 0.03	7.30 ± 0.04	7.29 ± 0.07	6.88 ± 0.14	5.29$^{+0.42}_{-0.24}$
	log (ρSF)	7.65 ± 0.04	7.64 ± 0.05	7.56 ± 0.07	6.93 ± 0.14	5.53$^{+0.26}_{-0.23}$
2.0 ⩽ z < 2.5	log (ηA)	−3.06$^{+0.70}_{-0.13}$	−3.11$^{+0.43}_{-0.11}$	−3.30$^{+0.16}_{-0.09}$	−4.20 ± 0.16	−5.89$^{+0.38}_{-0.26}$
	log (ηQ)	−3.58$^{+0.35}_{-0.41}$	−3.67$^{+0.22}_{-0.30}$	−3.91$^{+0.11}_{-0.14}$	−4.82 ± 0.17	−6.37$^{+0.31}_{-0.40}$
	log (ηSF)	−3.18$^{+0.87}_{-0.16}$	−3.23$^{+0.52}_{-0.12}$	−3.42$^{+0.18}_{-0.09}$	−4.32 ± 0.16	−6.04$^{+0.42}_{-0.27}$
	log (ρA)	7.43$^{+0.11}_{-0.04}$	7.43$^{+0.11}_{-0.06}$	7.41$^{+0.10}_{-0.08}$	6.98 ± 0.16	5.69$^{+0.40}_{-0.26}$
	log (ρQ)	6.83 ± 0.07	6.82 ± 0.07	6.80 ± 0.09	6.38 ± 0.17	5.21$^{+0.32}_{-0.41}$
	log (ρSF)	7.31$^{+0.14}_{-0.04}$	7.31$^{+0.13}_{-0.06}$	7.29$^{+0.10}_{-0.08}$	6.86 ± 0.16	5.53$^{+0.45}_{-0.27}$
2.5 ⩽ z < 3.0	log (ηA)	−2.88$^{+0.69}_{-0.52}$	−3.09$^{+0.41}_{-0.35}$	−3.43$^{+0.19}_{-0.18}$	−4.32 ± 0.18	−5.57$^{+0.28}_{-0.27}$
	log (ηQ)	−4.02$^{+1.29}_{-0.68}$	−4.05$^{+0.84}_{-0.52}$	−4.19$^{+0.39}_{-0.29}$	−5.00 ± 0.20	−6.63$^{+0.45}_{-0.44}$
	log (ηSF)	−2.99$^{+0.84}_{-0.53}$	−3.20$^{+0.49}_{-0.35}$	−3.55$^{+0.21}_{-0.18}$	−4.42 ± 0.18	−5.61$^{+0.28}_{-0.27}$
	log (ρA)	7.32$^{+0.13}_{-0.09}$	7.32$^{+0.13}_{-0.10}$	7.28$^{+0.12}_{-0.11}$	6.91 ± 0.18	6.04 ± 0.28
	log (ρQ)	6.58$^{+0.29}_{-0.17}$	6.58$^{+0.27}_{-0.17}$	6.57$^{+0.22}_{-0.15}$	6.19 ± 0.20	4.95$^{+0.48}_{-0.46}$
	log (ρSF)	7.21$^{+0.16}_{-0.10}$	7.21$^{+0.15}_{-0.11}$	7.17$^{+0.13}_{-0.12}$	6.82 ± 0.18	6.00 ± 0.28
3.0 ⩽ z < 4.0	log (ηA)	−3.14$^{+1.41}_{-1.25}$	−3.65$^{+0.84}_{-0.74}$	−4.20$^{+0.36}_{-0.34}$	−5.04$^{+0.22}_{-0.16}$	−5.80$^{+0.25}_{-0.24}$
	log (ηQ)	−5.44$^{+3.26}_{-0.94}$	−5.44$^{+2.19}_{-0.67}$	−5.46$^{+1.15}_{-0.42}$	−5.83$^{+0.38}_{-0.18}$	−6.97$^{+0.69}_{-0.49}$
	log (ηSF)	−3.11$^{+1.47}_{-1.38}$	−3.67$^{+0.88}_{-0.81}$	−4.27$^{+0.38}_{-0.35}$	−5.11$^{+0.19}_{-0.17}$	−5.85 ± 0.24
	log (ρA)	6.64$^{+0.43}_{-0.19}$	6.63$^{+0.31}_{-0.18}$	6.57$^{+0.20}_{-0.15}$	6.31$^{+0.21}_{-0.17}$	5.89 ± 0.26
	log (ρQ)	5.58$^{+0.93}_{-0.22}$	5.58$^{+0.81}_{-0.22}$	5.58$^{+0.62}_{-0.20}$	5.44$^{+0.42}_{-0.19}$	4.62$^{+0.77}_{-0.53}$
	log (ρSF)	6.59$^{+0.49}_{-0.20}$	6.58$^{+0.34}_{-0.19}$	6.50$^{+0.19}_{-0.15}$	6.25$^{+0.19}_{-0.17}$	5.86 ± 0.27

Notes. Number densities η are in units of Mpc⁻³, while stellar mass densities ρ are in units of M_☉ Mpc⁻³. Both the number and stellar mass densities are calculated for log (M_star/M_☉) < 13. The listed errors are the total 1σ errors, including Poisson uncertainties, the uncertainties from photometric redshift random errors, and cosmic variance.

Download table as:  ASCIITypeset image

In the right panel of Figure 7, we plot the evolution of the integrated number densities of galaxies calculated using the same Schechter function fits. The number densities are determined down to limiting masses of log(M_star/M_☉) = 11.5, 11.0, 10.0, and 8.0, and these points are labeled in the figure. The number densities measured from the NMBS survey (Brammer et al. 2011) for mass limits of log(M_star/M_☉) = 11.0 and 10.0 are plotted as black points. These points were measured using a catalog constructed in a similar way as the UltraVISTA catalog and agree well with the number densities in UltraVISTA. Also shown in Figure 7 are the integrated number densities at z < 1 calculated from the PRIMUS survey (Moustakas et al. 2013), which covers an area ∼3× larger than UltraVISTA. These are also consistent with the UltraVISTA measurements. Similar integrated number densities for both the star-forming and quiescent populations are shown in the right panel of Figure 8 and the values are listed in Table 2.

In Figure 9, we plot the evolution of the Schechter parameters $M^{*}_{{\rm star}}$, Φ*, and α as a function of redshift for the combined population of both star-forming and quiescent galaxies. All points plotted are from the single Schechter function fits except for the 0.2 < z < 0.5 bin, where we have plotted the $M^{*}_{{\rm star}}$ and combined Φ*s of the double Schechter function fits. We do not plot α at z > 2 in Figure 9 because the limiting mass of the data at these redshifts is too high to provide constraints. The lack of constraints, combined with the correlation between $M^{*}_{{\rm star}}$ and α, means that the uncertainties in α at z > 2 are likely to be underestimated. Also plotted in Figure 9 are the best-fit Schechter parameters from the low-redshift SMFs from Cole et al. (2001) and Bell et al. (2003) and the high-redshift rest-frame-optical-selected SMFs from Pérez-González et al. (2008), Marchesini et al. (2009), Marchesini et al. (2010), and Caputi et al. (2011).

Zoom In Zoom Out Reset image size

A comparison of the parameters in our lowest redshift bin (0.2 < z < 0.5) with those parameters from Cole et al. (2001) and Bell et al. (2003) shows good agreement for both M_star and Φ*. There is some disagreement in α, with our data having a steeper low-mass-end slope than the local SMFs. It is unclear why this is, as the UltraVISTA data reach a comparable depth in M_star as local studies. Part of the discrepancy may be because Cole et al. (2001) and Bell et al. (2003) fit a single Schechter function when a double function is required. If we compare our double Schechter function fits (α₁ = −0.53$^{+0.16}_{-0.28}$, α₂ = −1.37$^{+0.01}_{-0.06}$) with those derived from Baldry et al. (2012) (α₁ = −0.35 ± 0.18, α₂ = −1.47 ± 0.05), we find good agreement.

Figure 9 also shows that there is good agreement between our SMFs and previous high-redshift SMFs in the literature. There is a significant improvement in the uncertainties in the SMFs derived from the UltraVISTA catalog, mostly due to the fact it covers an area that is a factor of 8.8 and 11.4 larger than the areas used in the Pérez-González et al. (2008) and Marchesini et al. (2009) studies, respectively. Figure 9 confirms the results of those previous works and shows that within the substantially smaller uncertainties, there is still no significant evolution in $M^{*}_{{\rm star}}$ out to z ∼ 3.0. This lack of evolution implies that whatever process causes the exponential tail of the Schechter function does so in a consistent way over much of cosmic time. At z > 3.5, we find some evidence for a change in $M^{*}_{{\rm star}}$; however, given the lack of constraints on α at this redshift and the correlation between $M^{*}_{{\rm star}}$ and α, the uncertainties are still large.

Although there is no significant evolution in $M^{*}_{{\rm star}}$, there is a substantial evolution in Φ* from z = 3.5 to z = 0.0. If we compare with the Φ* at z = 0 from Cole et al. (2001), we find that it evolves by 2.58$^{+1.01}_{-0.37}$ dex between z ∼ 3.5 and z ∼ 0.0. As Figure 9 shows, this evolution is stronger than the values of 1.22 ± 0.43, 1.76$^{+0.40}_{-0.82}$, 1.92$^{+0.39}_{-0.36}$, and 1.89$^{+0.14}_{-0.19}$ dex measured previously by Pérez-González et al. (2008), Marchesini et al. (2009), Marchesini et al. (2010), and Caputi et al. (2011), respectively.

Interestingly, it appears that there is a statistically significant evolution in α up to z = 2, the redshift where the data are deep enough that there are still reasonable constraints on α. A flattening of the slope with redshift was also seen in the SMFs of Marchesini et al. (2009), which probe to slightly lower M_star. Such a flattening in the combined population is a natural consequence of the fact that there appears to be little evolution in the α of the star-forming population, but a flattening in α for the quiescent population with increasing redshift (e.g., Figure 5; see Section 5.2). UV-selected samples suggest a steep α at high redshift (e.g., Reddy & Steidel 2009; Stark et al. 2009; González et al. 2011; Lee et al. 2012), which, taken at face value, disagrees with the flattening of the slope observed for the K_s-selected SMF. As we show in Section 5.3, UV selection misses a fraction of the massive galaxy population due to their quiescence and/or dustiness, so α may be overestimated for the combined population.

Recently, Santini et al. (2012) measured the faint-end slope using ultra-deep HAWK-I K_s data, which provides a better comparison with the UltraVISTA SMFs than UV-selected SMFs. They also find a steep faint-end slope (α = −1.84 ± 0.06) at z ∼ 2, which, taken at face value, does not agree well with our measurement. Of course, α and $M^{*}_{{\rm star}}$ are correlated and Santini et al. (2012) cover an area that is two orders of magnitude smaller than that of UltraVISTA and therefore they have poor constraints on $M^{*}_{{\rm star}}$. At z ∼ 2, they find a value of log($M^{*}_{{\rm star}}$/M_☉) = 11.82 ± 0.28, which is almost an order of magnitude larger than the UltraVISTA value of log($M^{*}_{{\rm star}}$/M_☉) = 10.81$^{+0.06}_{-0.05}$. This makes it clear that in order to simultaneously fit the Santini et al. (2012) measurements and the UltraVISTA measurements, a double Schechter function is required.

This implies that the decline in α with increasing redshift in the UltraVISTA SMFs is more a reflection of the changing limiting M_star with increasing redshift. At higher redshift, the UltraVISTA Schechter function fits become increasingly dominated by the high-mass end and do not constrain the double Schechter function upturn. Probing further down the low-mass end will most likely result in steeper α values than have been measured with the current depth. We note that given the limiting M_star of the UltraVISTA SMFs and the fact that α is not well constrained at high redshift, we have also performed all the fits with fixed α = −1.2, which is closer to the Santini et al. (2012) and UV-selected SMF measurements. These fixed-α fits are incorporated into the uncertainties in quantities such as the SMDs and so these should still be representative.

If the α values of star-forming and quiescent galaxy SMFs shown in Figure 5 are representative of the α values at lower masses than are probed by the current catalog, then it suggests that the single Schechter function fit we have used will be a poor representation of the faint end at increasing redshift. Because the α values of the star-forming and quiescent SMFs are so different, and the mix of the two populations is a strong function of mass (e.g., Figure 6), a double Schechter function parameterization will be necessary to describe the faint end of the combined population below the mass limit of the current data. If so, then the flattening of α with redshift that we measure for the combined population is most likely a consequence of the fact that the mass limit of the survey is near the M_star where the number densities transition between being dominated by the quiescent population to being dominated by the star-forming population. A more thorough understanding of the differences in the slope of rest-frame-optical-selected and UV-selected SMFs will have to wait until deeper rest-frame-optical data are available and can be used to select galaxies with comparable limits in stellar mass as the UV-selected samples.

The SMF of the quiescent population shows significant evolution since z = 3.5. Although there is little change in $M^{*}_{{\rm star}}$, Φ* has increased by 0.57$^{+0.03}_{-0.04}$, 1.39$^{+0.07}_{-0.07}$, and 2.75$^{+0.36}_{-0.23}$ dex since z ∼ 1.0, 2.0, and 3.5, respectively. With the improved uncertainties from UltraVISTA, there is also evidence for a steady increase in the number density of even the most massive quiescent galaxies (log(M_star/M_☉ > 11.5) since z = 3.5 (Figure 8). The uncertainties from earlier studies (e.g., Brammer et al. 2011) were large enough that they could accommodate no growth in the number densities of these galaxies since z = 2.2.

At z < 1, the data are complete to a sufficiently small M_star such that the upturn in the number density of quiescent galaxies at log(M_star/M_☉) < 9.5 can be clearly seen. This upturn is also seen at z = 0 in the quiescent population (e.g., Bell et al. 2003; Baldry et al. 2012) and the UltraVISTA data now confirm that it persists to at least z = 1. In Appendix B, we, we show that the existence of the upturn is robust to the definition of quiescent galaxies, although its prominence can be increased or decreased depending on the strictness of the selection. Interestingly, the location of the upturn seems to evolve with mass, from log(M_star/M_☉) ∼ 9.2 at z = 0.75 to log(M_star/M_☉) ∼ 9.5 at z = 0.35.

It has been suggested that the upturn is the result of a population of star-forming satellite galaxies that have been quenched in high-density environments (e.g., Peng et al. 2010, 2012). The Peng et al. (2010) model separates the quenching process into two forms: "environmental quenching," and "mass quenching"; the former is caused by high-density environments, and the latter is caused by processes internal to the galaxies themselves. In their model, the M_star where the upturn occurs is determined by the relative αs of the mass-quenched quiescent population and the environmentally quenched quiescent population (formerly a recently star-forming population). In the case that the α values of the self-quenched quiescent populations and the star-forming populations do not evolve with redshift—which is consistent with our measurements—an evolution in the M_star of the upturn would imply a change in the fraction of galaxies that self-quench as compared to those that environmentally quench. An evolution to higher masses with decreasing redshift would imply an increase in the fraction of galaxies that are environmental-quenched with decreasing redshift, i.e., environmental-quenching becomes increasingly more important at lower redshift.

It is also interesting to examine the evolution of the fraction of quiescent galaxies as a function of both M_star and z. These fractions are plotted in the bottom panels of Figure 6. At z < 1, we recover the well-known result that quiescent galaxies dominate the high-mass end of the SMF (e.g., Bundy et al. 2006; Ilbert et al. 2010; Pozzetti et al. 2010; Brammer et al. 2011), but thereafter some interesting trends emerge.

The fraction of quiescent galaxies with log(M_star/M_☉) > 11.0 continues to decline slowly with increasing redshift up to z = 1.5. After z > 1.5, that decline accelerates significantly and we find that by z = 2.5 the fraction of quiescent galaxies has decreased to the point where star-forming galaxies dominate the SMF at all stellar masses. This result is shown in Figure 10 where we plot the M_star at which quiescent galaxies dominate the SMF (denoted M_cross) as a function of redshift. Also plotted in Figure 10 are measurements of M_cross from other studies (Bell et al. 2003; Bundy et al. 2006; Pozzetti et al. 2010; Baldry et al. 2012). These studies use different definitions of star-forming and quiescent galaxies, so a direct comparison with the UltraVISTA measurements is difficult; however, we note that they are largely consistent with the UltraVISTA measurements in the redshift range where the data overlap.

Zoom In Zoom Out Reset image size

As Figure 10 shows, the crossing mass at z = 0.35 is log(M_star/M_☉) = 10.55 and the crossing mass evolves as log(M_cross/M_☉) ∝ (1+z)^0.9 up to z = 1.5. Thereafter, it evolves much more rapidly log(M_cross/M_☉) ∝ (1+z)^5.6 up to z = 2.5, after which star-forming galaxies dominate the full population.

Despite the sharp change in the evolution of M_cross at z = 1.5, the overall increase in the number density of the quiescent population (i.e., Φ*) is fairly smooth with redshift. Therefore, the rapid evolution of M_cross at z > 1.5 is really a reflection of the fact that M_cross moves onto the exponential part of the Schechter function at high redshift, whereas it occurs on the power-law part of the Schechter function at low redshift.

Although their fraction diminishes substantially at high redshift, it is noteworthy that we find a non-zero fraction of quiescent galaxies (∼10%–20%) up to z = 3.5. A similarly small but non-zero fraction was also found by Marchesini et al. (2010) in the NMBS. We will discuss the SEDs of the quiescent population in more detail in a future paper (D. Marchesini, in preparation), but we note that for some, the best-fit ages are ⩽1.0 Gyr. If they formed most of their stars in a rapid burst, as suggested by the best-fit τ model, it suggests that there may be a non-zero population of quiescent galaxies that extends to as high as z = 6–8. Substantially deeper wide-field data (or longer wavelength selection) will be needed to find such galaxies at z > 4, as the K_s-band moves blueward of the Balmer break and quickly becomes inefficient at detecting red galaxies.

Figure 5 shows that in contrast to the SMF of quiescent galaxies, the evolution of the SMF of star-forming galaxies from z = 3.5 to z = 0 is fairly modest. $M^{*}_{{\rm star}}$ and α show no significant evolution up to z = 3.5, albeit with large uncertainties in the latter. Unlike the quiescent population, the number density of log(M_star/M_☉) > 11.5 galaxies shows almost no evolution up to z = 3.5 (Figure 8). The only significant evolution is in Φ*, which evolves by 0.45$^{+0.03}_{-0.03}$, 1.01$^{+0.06}_{-0.06}$, and 2.40$^{+0.21}_{-0.21}$ dex since z ∼ 1.0, 2.0, and 3.5, respectively. If we compare this to the evolution of the quiescent population at the same redshifts, we find that the quiescent population has grown faster in Φ* by factors of ∼1.3, 2.4, and 2.2 since z ∼ 1.0, 2.0, and 3.5, respectively. This shows that at all redshifts the majority of the growth in the combined SMF is due to the increase in the quiescent population.

The non-evolution in $M^{*}_{{\rm star}}$ and α and the rather slow evolution in Φ* for the star-forming population is remarkable if considered in the context of the evolution of the SFR per unit M_star (SSFR) over the same redshift range. The SSFR of star-forming galaxies with log(M_star/M_☉) = 10.0 (11.0) declines by a factor of ∼20 (25) since z ∼ 2 (e.g., Muzzin et al. 2013), which means that the growth rate of these galaxies is evolving substantially with redshift. That the process of galaxy quenching evolves in such a way to keep the shape and normalization of the star-forming SMF roughly constant over this redshift range, while there is a significant decrease in the SSFRs, implies a carefully orchestrated balance between galaxy growth and quenching with redshift.

Given that with the UltraVISTA data we now have a reasonably well determined SMF at z = 3.5 for star-forming galaxies, it is interesting to compare this SMF with measurements of the SMFs of UV-selected star-forming galaxies. In principle, our SMF of rest-frame-optical-selected star-forming galaxies should be more complete, as both UV-bright and UV-faint star-forming galaxies will be selected, where the latter are likely to be highly dust-obscured galaxies. With few rest-frame-optical-selected SMFs for star-forming galaxies at z > 3 in the literature, it is still unclear how large the population of UV-faint, star-forming galaxies is.

In the left panel of Figure 11, we plot the SMF of K_s-selected star-forming galaxies at 2.0 < z < 2.5, as well as the SMF of BM/BX-selected galaxies by Reddy & Steidel (2009). We have converted the Reddy & Steidel (2009) SMF from a Salpeter IMF to a Kroupa IMF to match UltraVISTA (N. Reddy 2013, private communication). The BM/BX galaxies typically span the redshift range 1.9 < z < 2.7 (Reddy & Steidel 2009), which is a reasonable match to the redshift range of the K_s-selected galaxies.

Zoom In Zoom Out Reset image size

Considering the sizable systematic differences that can arise from different ways of measuring the SMF (see, e.g., the inter-comparison of SMFs in the literature in the Appendix of Marchesini et al. 2009), the SMFs do agree reasonably well, particularly at the highest masses. This suggests either that the BM/BX selection effectively selects the majority of massive star-forming galaxies, regardless of their UV luminosity, or that massive dusty star-forming galaxies are less abundant than UV-bright ones at z ∼ 2.3. Qualitatively speaking, the UVJ diagram (Figure 3) suggests that the colors of star-forming galaxies above the M_star completeness limit are primarily red as compared to blue, so the high completeness of the BM/BX selection for this population is somewhat unexpected.

The low-mass end of the SMFs do not agree well; the BM/BX population is more abundant than the rest-frame-optical-selected population. This is difficult to reconcile given that we have adopted strict M_star completeness limits and that the K_s selection should be more robust than UV selection. Although we cannot be conclusive, the discrepancy could arise from systematic effects due to the fact that these SMFs are determined in quite different ways. Reddy & Steidel (2009) determine the BM/BX SMF from the UV LF using a subsample of galaxies for which they have spectroscopic redshifts and SED-determined M_star values. They convert the UV LF to a SMF assuming that the distribution of M_star in the subsample is representative of the entire population. This is different than our approach of fitting z_phot and M_star for each galaxy individually. Interestingly, Reddy & Steidel (2009) also determine a much steeper faint-end slope (α = −1.73 ± 0.07) for the SMF than other UV-selected samples that measure the SMF with z_phot and SED-fitting techniques similar to those used for the UltraVISTA catalog (e.g., Stark et al. 2009; González et al. 2011; Lee et al. 2012).

In the right panel of Figure 11, we compare the K_s-selected star-forming SMF at 2.5 < z < 3.0 with the SMF of Lyman break galaxies (LBGs) from Reddy & Steidel (2009). Down to the mass limit of UltraVISTA, the shapes of those SMFs agree reasonably well, although the number density of K_s-selected star-forming galaxies is a factor of ∼2 higher than the number density of LBGs at log(M_star/M_☉) = 11.0. Taken at face value, this result suggests the LBG selection may miss approximately half of the massive galaxies at z > 2.5. A similar result was also obtained by Marchesini et al. (2010), who found that only 8/14 galaxies in their mass-complete sample (log(M_star/M_☉) > 11.4) of K_s-selected galaxies at 3.0 < z < 4.0 would be selected with U- and B-dropout selection. Using the VVDS spectroscopic sample, Le Fèvre et al. (2005) and Cucciati et al. (2012) also found that ∼50% of the star-forming population at z > 3 may meet the LBG selection criteria, although we note that this result is based on a comparison of LFs, not SMFs.

In Figure 12, we expand the comparison of the K_s-selected SMFs and UV-selected SMFs using more recent determinations in the literature from Stark et al. (2009), González et al. (2011), and Lee et al. (2012). We also compare with the K_s-selected total SMF at 3.0 < z < 4.0 determined from the MUSYC/FIREWORKS/FIRES surveys (Marchesini et al. 2009) and the NMBS (Marchesini et al. 2010), as well as the IRAC-selected SMFs in the UDS from Caputi et al. (2011). In general, most K_s-selected galaxies at this redshift are star-forming galaxies (e.g., Figure 6), so comparing the total SMFs from Marchesini et al. (2009, 2010) and Caputi et al. (2011) with the star-forming SMFs is a reasonable comparison. We note that all of the SMFs in Figure 12 have been determined with a method similar to the UltraVISTA K_s-selected catalog, e.g., z_phot from broadband photometry, and M_star from SED fitting with similar assumptions about SFHs. These SMFs have a slightly higher median redshift than the BM/BX and LBG SMFs, so we compare to the highest-redshift SMF in UltraVISTA, 3.0 < z < 4.0.

Zoom In Zoom Out Reset image size

Beginning with the K_s-selected samples, we find that, within the errors, the K_s- and IRAC-selected SMFs agree reasonably well, although the region of comparison is limited to fairly high M_star. Interestingly, all three show little evolution in the number density of the most massive galaxies (at log(M_star/M_☉) > 11.5) from z = 3.5 to z = 0. The largest discrepancy between the K_s-selected SMFs is for UltraVISTA and MUSYC at log(M_star/M_☉) = 11.0, where the number density from UltraVISTA is a factor of ∼3 less abundant in such galaxies than MUSYC. Comparison of the 1/V_max points shows that this difference is not significant.

If confirmed, the result that the abundance of log(M_star/M_☉) > 11.5 star-forming galaxies is unchanged since z = 3.5 is quite interesting. We note that at the present time, we cannot be 100% confident of how real this population is. Examination of their SEDs shows that the fits are reasonable; however, they are almost all extremely red and dusty. It is also difficult to tell if emission from an AGN causes their M_star values to be inflated.

At present, there are no spectroscopic redshifts for such galaxies so we cannot rule out the possibility that they are extremely dusty galaxies at a lower redshift (see also the discussion in Marchesini et al. 2010). We explore this possibility further in Appendix A and find that the use of an additional very dusty template in z_phot fitting can reduce the number density of this population by a factor of ∼2. Given this, at present it may be best to consider their abundance an upper limit on the true abundance.

Returning to the UV-selected samples, comparison of the K_s-selected SMFs with the UV-selected SMFs also shows reasonable agreement in the limited mass range where the surveys overlap. At log(M_star/M_☉) = 11.0, the K_s-selected SMF lies between the Stark et al. (2009) and Lee et al. (2012) SMFs. Extrapolation of the best-fit Schechter functions of both of those samples does not agree well with the abundance of massive star-forming galaxies in UltraVISTA. A simultaneous Schechter function fit to the Lee et al. (2012) data and the UltraVISTA data does not produce a good fit. There seem to be two possible reasons for this. Either UV selection misses most of the most massive star-forming galaxies at z > 3 due to dust obscuration or the abundance of these galaxies is overestimated in the K_s-selected sample due to a very dusty population at lower redshift.

Overall, the SMFs of rest-frame-optical-selected samples and UV-selected samples compare reasonably well in the mass range where they overlap. There is some evidence that UV selection is not complete for galaxies with log(M_star/M_☉) > 11.0; however, spectroscopic verification of the massive population of K_s-selected galaxies will be important to verify this result.

The left panel of Figure 7 shows that the measured evolution of the SMD for the combined population is well-determined and in excellent agreement with previous determinations. Our data show that the SMD of the universe was only 50%, 10%, and 1% of its current value at z ∼ 1.0, 2.0, and 3.5, respectively.

The left panel of Figure 8 shows the evolution of the SMD of the quiescent and star-forming populations separately. The SMDs of these populations have evolved quite differently since z = 3.5, a conclusion that was already apparent from the comparison of the SMFs themselves. The SMD of star-forming galaxies increases at a rate ρ_star∝(1 + z)^{−2.3 ± 0.2}, whereas the SMD of quiescent galaxies evolves much faster, as ρ_star∝(1 + z)^{−4.7 ± 0.4}. This strong differential evolution in the SMD can also be seen in the inset panel of Figure 8, where we plot the fraction of the total SMD in star-forming and quiescent galaxies as a function of redshift.

As Figure 7 shows, the universe contained only ∼1% of the total M_star formed by z = 0 at z = 3.5. The inset of Figure 8 shows that at that time the fraction of quiescent galaxies was small and approximately 90% of the total SMD was contained within star-forming galaxies. Since z = 3.5, the fraction of the SMD in quiescent galaxies has grown continuously and at z ∼ 0.75 the SMD in quiescent galaxies became approximately equal to the SMD in star-forming galaxies. Perhaps coincidently, this equality in SMD between the two types also occurs precisely at the redshift when ∼50% of the total SMD of the universe has formed. Thereafter, the SMD in quiescent galaxies exceeds the SMD in star-forming galaxies, although we note that the details of this statement depend on the low-redshift comparison samples, as there are notable differences between the local studies of Bell et al. (2003) and Baldry et al. (2012).

Due to the superior area and depth of the UltraVISTA data, we can also compare SMDs determined from K_s-selected samples at z = 3.5 with those determined from UV-selected samples at the same redshift. In Figure 13, we plot the SMDs from the UltraVISTA data as well as from the UV-selected samples of Stark et al. (2009), Labbé et al. (2010), González et al. (2011), and Lee et al. (2012). We have also included the SMDs at z = 0 from Cole et al. (2001), Bell et al. (2003), and Baldry et al. (2012); the measurements of the SMD in Figure 13 span an impressive redshift baseline of z = 0.0–8.5.

Zoom In Zoom Out Reset image size

If we compare the K_s-selected SMD at z = 3.5 with the UV-selected SMDs at z ∼ 3.7 determined from Stark et al. (2009), González et al. (2011), and Lee et al. (2012) there is reasonable agreement. The SMDs from the UV-selected samples agree well with each other and are systematically ∼0.5 dex higher than the K_s-selected measurement. The uncertainties in the K_s-selected SMD at z = 3.5 are quite large because the data just reach $M^{*}_{{\rm star}}$ and require a substantial extrapolation of the Schechter function. Although lower, the K_s-selected SMD does agree with the UV-selected SMD within the 1σ uncertainties. As shown in Figure 12, the UV-selected SMFs underpredict the number density of the most massive galaxies, so the agreement of the total SMDs demonstrates that the total M_star contained in the most massive galaxies is negligible compared with lower mass galaxies.

The reasonable agreement between the total SMD of the K_s-selected sample and the UV-selected sample illustrates a key point, namely, that at z > 3.5, UV-selected samples do select samples that account for most of the SMD of the universe. Even though direct a comparison of the SMFs in Section 5.3 showed that UV selection misses approximately half of the massive star-forming galaxies and all of the massive quiescent galaxies, Figure 13 shows that the SMD in such galaxies at z = 3.5 is fairly small. Therefore, UV selection appears to be quite complete for the majority of star-forming galaxies at these redshifts.

Given that our results suggest that UV-selected samples should be representative of the SMD at z > 3.5, it seems reasonable to fit the SMD over a large redshift baseline. The SFR density shows a clear decline at z < 1.5 (e.g., Hopkins & Beacom 2006; Bouwens et al. 2011; Sobral et al. 2013), so we fit the data down to that redshift. Including the UV-selected samples with UltraVISTA, we find that the total SMD evolves as ρ_star∝(1 + z)^{−2.7 ± 0.2} from z = 8.5–1.5. This fit is also in good agreement with a fit to just the UV-selected samples, which evolve as ρ_star∝(1 + z)^{−2.4 ± 0.6}.

The exponent in the overall fit is tantalizingly similar to the volume growth of the universe. Why the density of stars in galaxies increases at such a rate may be purely coincidental, but obviously needs to be investigated in more detail and in the context of models of galaxy evolution.

The evolution of the SMFs and SMDs illustrate the cosmological evolution of the distribution of galaxies as a function of M_star and the integrated M_star within the overall galaxy population. While important quantities from the standpoint of modeling the process of galaxy evolution, measurements of how individual galaxies assemble their mass may be more useful. Such measurements are difficult, as they requires the non-trivial task of linking galaxies and their descendents through cosmic time.

Several studies have argued that one approach for linking galaxies to their descendents is to select galaxies at a fixed constant number density (e.g., van Dokkum et al. 2010; Papovich et al. 2011). More recently, Brammer et al. (2011) argued that selecting galaxies at a fixed cumulative number density is a better approach, as it is single-valued at all M_star. In a recent paper, Leja et al. (2013) tested this method on semi-analytic models (SAMs) of galaxy formation. They found that selection at a fixed cumulative number density recovered the mass evolution of the population with an accuracy of ∼0.15 dex. They also found that the fixed cumulative number density approach may underpredict the mass growth of high-mass galaxies and overpredict the mass growth of lower-mass galaxies, although they note that this result depends on the evolution of galaxies in the SAM, which does not properly reproduce the SMF as a function of redshift.

Here, we perform fixed cumulative number density selection using the UltraVISTA SMFs in order to measure the average mass evolution of the population. We have chosen four fixed cumulative number densities to follow. These cumulative number densities are chosen to correspond to the cumulative number density for galaxies with log(M_star/M_☉) = 11.5, 11.0, 10.5, and 10.0, in the lowest redshift SMF bin (0.2 < z < 0.5).

In Figure 14, we plot M_star at these four fixed cumulative number densities out to z = 3. The solid shaded region in Figure 14 represents the M_star completeness limits of the survey. Cumulative number densities that require extrapolation of the Schechter function to M_star below the completeness limits are shown as open circles.

Zoom In Zoom Out Reset image size

The UltraVISTA data are sufficiently complete, and the M_star evolution sufficiently slow such that we measure the mass growth of the log(M_star/M_☉) = 11.5 population out to z ∼ 3.0. This population demonstrates a remarkably slow growth in M_star, increasing by only by 0.3 dex since z ∼ 3 and 0.2 dex since z ∼ 2. Using the same method, Brammer et al. (2011) measured a growth of 0.17 dex for this population since z ∼ 2, in excellent agreement with our finding.

With the caveat that increasingly larger extrapolations are required, Figure 14 suggests that the growth of galaxies with lower M_star is always faster than galaxies with higher M_star at z < 2. This can be seen more clearly in the right panel of Figure 14, where we plot the derivative of the M_star growth curves. These curves present a remarkable sequence at all z < 3, where the derivatives are always higher for lower mass galaxies.

Figure 14 encapsulates what has already been shown by many previous studies, namely that there is a "downsizing" of the galaxy population such that the highest mass galaxies assemble most of their M_star at high redshift, whereas lower mass galaxies grow more slowly over cosmic time. One of the most interesting implications of the curves in Figure 14 is that although low-mass galaxies grow at higher rates than higher mass galaxies at z < 2, they do not "overtake" their more massive counterparts. This is simply because the most massive galaxies have assembled such significant amounts of M_star at very early times. It also implies that at redshifts higher than those probed by the current data, the mass assembly rate of the most massive galaxies must be extremely rapid.

For example, galaxies with log(M_star/M_☉) = 11.5 at z ∼ 0.35 have to assemble as much mass at z < 3 as they do at z > 3. The amount of cosmic time between 0 < z < 3 is ∼5 × more than at 3 < z < ∞, which implies that even if those galaxies begin forming stars shortly after the big bang, their mass assembly rates must be substantially higher in the past and that they must also be substantially higher than those of low-mass galaxies at any epoch.