GALAXY STELLAR MASS FUNCTIONS FROM ZFOURGE/CANDELS: AN EXCESS OF LOW-MASS GALAXIES SINCE z = 2 AND THE RAPID BUILDUP OF QUIESCENT GALAXIES^*

We make use of the deep NIR imaging from ZFOURGE; (C. M. S. Straatman et al., in preparation), conducted using the FourStar imager (Persson et al. 2013) on the 6.5 m Magellan Baade telescope at Las Campanas Observatory. The use of medium-band filters in the NIR (van Dokkum et al. 2009) allows us to accurately sample wavelengths bracketing the Balmer break of galaxies leading to more well-constrained photometric redshifts at 1 < z < 4 than with broadband filters alone. In conjunction with existing optical through mid-IR photometry, this data set provides a comprehensive sampling of the 0.3–8 μm spectral energy distribution (SED) of galaxies.

The ZFOURGE data and photometry are described in detail by C. M. S. Straatman et al. (in preparation). Here, we provide a brief summary. ZFOURGE is composed of three 11' × 11' pointings with coverage in the CDFS (Giacconi et al. 2002), COSMOS (Capak et al. 2007), and UDS (Lawrence et al. 2007). The 5σ depth in a circular aperture of 06 diameter in K_s is 24.8, 25.2, and 24.6 in the CDFS, COSMOS, and UDS fields, respectively. Typical seeing was 08 or better in the ground-based bands. All optical–NIR images were convolved to a moffat point-spread function (PSF) with FWHM = 075; for some images, this meant deconvolution from an originally larger PSF. Fluxes were then measured within a circular aperture of 08. Since image quality is much lower in the Spitzer IRAC bands, this photometry was first deblended using the H₁₆₀ image with the techniques of Labbé et al. (2006). Apart from the Spitzer IRAC imaging, blending and source confusion is a minor issue.

The ZFOURGE fields also benefit from HST imaging taken as part of the Cosmic Assembly Near-IR Deep Extragalactic Legacy Survey (CANDELS; Grogin et al. 2011; Koekemoer et al. 2011). We utilize the J₁₂₅ and H₁₆₀ imaging which reach ∼26.5 mag, significantly deeper than our ground-based, medium-band data. The high signal-to-noise ratio (S/N) photometry aids in photometric redshift estimates even though the filters are broader than our ground-based, medium-band data. We also use the H₁₆₀ data as our detection image. But because some of the faintest sources in the H₁₆₀ images are not detected in our ground-based data, and thus will have poorly constrained SEDs, we limit our study to objects detected at S/N₁₆₀ > 10 (corresponding to H₁₆₀ ∼ 25.9) as a threshold to remove galaxies that are poorly detected at other wavelengths. In the Appendix, we show examples of galaxies near our adopted flux limit; as can be seen, these galaxies are strongly detected and have well-constrained photometric redshifts. The total area of our final sample with full coverage in ZFOURGE and CANDELS is ∼316 arcmin².

We also make use of data from the NEWFIRM Medium-Band Survey (NMBS; Whitaker et al. 2011) which includes imaging in the same set of medium-band, NIR filters as ZFOURGE. The similarity of the photometry helps reduce any intersurvey systematics. NMBS is composed of two 30' × 30' pointings in the AEGIS (Davis et al. 2007) and COSMOS (Capak et al. 2007) fields. The COSMOS pointing encompasses one of our ZFOURGE fields; in the region of overlap, we make use of the higher-quality ZFOURGE data as opposed to the NMBS data. Photometric redshifts from NMBS are shown to have a scatter of σ_z/(1 + z) = 0.017, 0.008 in the AEGIS and COSMOS fields, respectively, when compared to spectroscopic redshifts (Whitaker et al. 2011). Although NMBS is shallower than the rest of our sample, reaching depths of ∼24.5 mag in J₁, J₂, J₃ and ∼23.5 mag in H_s, H_l, K_s, including it increases our survey area by a factor of 5.3, allowing us to much better constrain the high-mass end of the SMF (see Brammer et al. 2011).

We use the public SED-fitting code EAZY (Brammer et al. 2008) to measure photometric redshifts and rest-frame colors. EAZY utilizes a default set of six spectral templates that include prescriptions for emission lines derived from the PEGASE models (Fioc & Rocca-Volmerange 1997) plus an additional dust-reddened template derived from the Maraston (2005) models. Linear combinations of these templates are fit to the 0.3–8 μm photometry for each galaxy to estimate redshifts.

Figure 1 demonstrates the accuracy of our photometric redshifts in comparison to available spectroscopic redshifts. Only sources reported with secure spectroscopic detections are considered. Overall, we find a normalized median absolute deviation (NMAD) scatter of 1.8% in Δz/(1 + z_spec). At z < 1.5, this scatter becomes 1.7% with about 2.7% of catastrophic failures (|Δz/(1 + z_spec)| > 0.15). As we push to z > 1.5, where the Balmer break of galaxies redshifts into the medium-band NIR filters, this scatter becomes 2.2% with about 9% of catastrophic failures. We note here that this scatter is likely biased upward since objects with secure spectroscopic redshifts tend to be strongly star-forming systems with weak Balmer breaks and thus do not benefit the most from the deep medium-band NIR photometry from ZFOURGE. Also shown in Figure 1 are redshift distributions in the three fields of ZFOURGE corresponding to our estimated magnitude limit as well as the magnitude limits of UltraVISTA (McCracken et al. 2012) and NMBS (Brammer et al. 2011) in black, purple, and orange, respectively. Spectroscopic redshifts from CDFS come from Vanzella et al. (2008), Le Fèvre et al. (2005), Szokoly et al. (2004), Doherty et al. (2005), Popesso et al. (2009), and Balestra et al. (2010). Spectroscopic redshifts from UDS come from Simpson et al. (2012) and Smail et al. (2008). Spectroscopic redshifts from COSMOS come from the NASA/IPAC Infrared Science Archive.¹⁰

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However, a comparison to spectroscopic samples can be of limited use, since the results of such comparisons depend strongly on how the spectroscopic objects were selected. Moreover, fainter objects and more distant objects, which are more difficult to detect spectroscopically, are also expected to have larger photometric redshift errors. We use the close-pairs analysis of Quadri & Williams (2010) to estimate the typical uncertainties for the full sample of objects in our catalog, finding σ ≈ 0.02 at z ∼ 0.5, and this increases to σ ≈ 0.05 at z ∼ 2.5.

To obtain stellar masses, we use the FAST code (Kriek et al. 2009) which fits stellar population synthesis models to the measured SEDs of galaxies to infer various galactic properties. Specifically, we use models from Bruzual & Charlot (2003) following an exponentially declining star-formation history assuming a Chabrier (2003) initial mass function (IMF). We assume solar metallicity and allow A_v to vary between [0, 4].

We note here that stellar masses derived from SED-fitting are dependent on assumed parameters in the models (metallicity, dust law, stellar population models, etc.). Variations in these assumptions have been shown to lead to systematic offsets in stellar masses as opposed to random errors (e.g., Maraston 2005; Marchesini et al. 2009; Conroy et al. 2009); however, a full investigation of these effects is beyond the scope of this paper.

Understanding the mass-completeness limits of our data set is crucial to our analysis. Marchesini et al. (2009) describe a technique whereby a sample of galaxies below the nominal flux-completeness limit is taken from a deeper survey. These galaxies were then scaled up in flux and mass to the completeness limit of their survey. The resulting distribution in mass forms a representative sample of the most massive galaxies that could just escape detection in their sample. The upper envelope of this distribution will therefore represent an empirical determination of the redshift-dependent mass-completeness limit.

In the absence of deeper data, Quadri et al. (2012) modified this technique slightly by using a sample that lies above the flux completeness limit and scaling the fluxes and the masses down; this is the method adopted here. We start with all galaxies that are a factor of 2–3× above our S/N threshold (S/N₁₆₀ > 10) and scale down their masses by the appropriate factor. From this scaled down sample, we take the upper envelope that encompasses 80% of the galaxies as the redshift-dependent mass-completeness limit, shown in blue in Figure 2.

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To obtain another measurement of the mass-completeness limit, we employ a similar technique to that described by Chang et al. (2013). First, we estimate the magnitude limit corresponding to our S/N threshold to be H₁₆₀ ≈ 25.9. Then, in narrow bins of mass, we calculate the fraction of galaxies that are brighter than this magnitude at all S/Ns. At the highest stellar masses, this is 100% but gradually decreases as we probe toward lower masses. We search for the mass bin where this fraction is 80% at various redshifts, which we take as the mass-completeness limit. The results from this technique are shown in green in Figure 2.

Both of the empirical techniques above are performed on all galaxies (i.e., without distinguishing star-forming/quiescent) and yield nearly identical values which gives us confidence in our measurements. We use these mass-completeness limits for both the total and star-forming SMFs.

Finally, since quiescent galaxies have higher mass-to-light ratios than the general population, the corresponding mass-completeness limit will be higher. We calculate this mass-completeness limit from a stellar population synthesis model obtained from EzGal (Mancone & Gonzalez 2012). Specifically, we consider a single stellar population (SSP) following a Chabrier (2003) IMF of solar metallicity formed at a redshift of z_f = 5. The mass-completeness limit derived from this approach is representative of the oldest galaxies at a given redshift since z_f. We adopt this as the mass-completeness limit for quiescent galaxies, shown in red in Figure 2.

Our data provide a view of the SMF to depths that have previously been inaccessible over significant areas. In Figure 3, we plot an example SMF as measured by ZFOURGE, UltraVISTA (Muzzin et al. 2013), and NMBS (Brammer et al. 2011) which reach K_s-band 5σ depths of about 24.9, 23.4, and 22.8 mag, respectively. Furthermore, since ZFOURGE is split into three independent pointings, errors due to cosmic variance are suppressed compared to a survey of equal area composed of one pointing. We show an example of field-to-field variance in Figure 4 where we plot the SMF measured from each ZFOURGE pointing individually.

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In this work, we divide the full galaxy sample into star-forming and quiescent populations. We separate these populations in a rest-frame U − V versus V − J color–color diagram (hereafter UVJ diagram), which has been shown to effectively trace the galaxy color-bimodality as far as z = 3 (Labbé et al. 2005; Williams et al. 2009; Patel et al. 2012; Whitaker et al. 2011; Muzzin et al. 2013). The strength of this technique lies in its weak dependence on dust extinction, since the dust-reddening vector tends not to scatter galaxies across the selection boundary. This helps avoid contamination by dusty star-forming galaxies in typical red-sequence selection techniques.

We derive rest-frame U − V and V − J colors from the best-fit EAZY templates to the observed photometry. In Figure 5, we show UVJ diagrams for our galaxy sample at various redshifts. Only galaxies above their respective mass-completeness limit are shown. The bimodality can be seen to z ∼ 3.

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The accuracy with which we are able to measure the SMF is dependent on multiple steps, each having its own uncertainty. Poisson uncertainties (σ_poisson) are calculated using prescriptions from Gehrels (1986). We also include cosmic variance (σ_cv) and uncertainties in the SED modeling used to estimate photometric redshifts, rest-frame colors, and stellar masses (σ_sed).

We calculate cosmic variance as a function of redshift and mass using the getcv routine described in Moster et al. (2011). This yields cosmic variance uncertainties that range from ≈25% at 10¹¹ M_☉ to ≈8% at 10^8.5 M_☉. Cosmic variance can also be estimated from the scatter in the SMFs of the independent pointings from ZFOURGE (Figure 4). Overall, we find this scatter to be consistent with the predictions.

To estimate the uncertainty contribution from SED modeling, we conduct 100 Monte Carlo simulations on our catalogs. For each realization, we independently perturb photometric redshifts and stellar masses using the 68% confidence limits output from EAZY and FAST. SMFs are then recalculated over the same redshift ranges used throughout. The 1σ scatter in the resulting SMFs is then taken as the redshift- and mass-dependent uncertainty. These uncertainties range from 5%–15% over the span of redshifts in this study.

We consider another source of uncertainty involved in the classification of galaxies as star-forming versus quiescent. Here, we evaluate the statistical uncertainty associated with the UVJ classification based on photometric uncertainties (σ_uvj). To do this, we perform 100 Monte Carlo simulations on a sample of galaxies at 10 < S/N₁₆₀ < 200, perturbing fluxes according to a Gaussian probability density function based on 1σ photometric uncertainties. Photometric redshifts and rest-frame colors were remeasured for each iteration, from which galaxies were reclassified as being star-forming or quiescent. We find that at fixed S/N₁₆₀, more galaxies scatter into versus out of the quiescent region, boosting the quiescent fraction. However, this effect is small; we find that the quiescent fractions typically vary by <2%, and that this value has a scatter of <0.4% between the simulations. A particular concern may be that the number density of quiescent sources at low masses may be significantly affected by a small fraction of the (much more abundant) star-forming galaxies scattering into the quiescent region but we find that this is not a major concern.

In total, our uncertainty budgets become

$$\begin{eqnarray} \sigma _{\rm tot} &= & \sqrt{ \sigma ^2_{{\rm poisson}} + \sigma ^2_{{\rm cv}} + \sigma ^2_{{\rm sed}} }\nonumber \\ \sigma _{\rm sf/qui} &= & \sqrt{ \sigma ^2_{{\rm poisson}} + \sigma ^2_{{\rm cv}} + \sigma ^2_{{\rm sed}} + \sigma ^2_{uvj} }. \end{eqnarray} \tag{ 1 }$$

In Figure 6, we show our measurements of the total SMF over 0.2 < z < 3. For comparison, we have included corresponding measurements at similar redshift intervals from recent works (Santini et al. 2012; Moustakas et al. 2013; Ilbert et al. 2013; Muzzin et al. 2013). We find excellent agreement in the regions of overlap, except with Santini et al. (2012) who measure higher densities of galaxies at z > 2.

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In Figure 7, we subdivide the total SMF into star-forming and quiescent populations over the same range of redshifts as in Figure 6. The data are also presented in Table 1. We reiterate that these mass functions have been supplemented by NMBS to provide better constraints at the high-mass end. Orange arrows show the mass limits for the contribution of NMBS to each SMF in Figure 7. We also show the growth of each SMF (total, star-forming, and quiescent) in Figure 8 over our entire redshift range.

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Table 1. Stellar Mass Functions

Total
	0.2 < z < 0.5	0.5 < z < 0.75	0.75 < z < 1.0	1.0 < z < 1.25	1.25 < z < 1.5	1.5 < z < 2.0	2.0 < z < 2.5	2.5 < z < 3.0
Log(M/ M☉)	Log(Φ)	Log(Φ)	Log(Φ)	Log(Φ)	Log(Φ)	Log(Φ)	Log(Φ)	Log(Φ)
8.00	$-1.37^{+0.06}_{-0.07}$	...	...	...	...	...	...	...
8.25	$-1.53^{+0.06}_{-0.07}$	$-1.53^{+0.06}_{-0.07}$	...	...	...	...	...	...
8.50	$-1.71^{+0.07}_{-0.08}$	$-1.60^{+0.05}_{-0.06}$	$-1.70^{+0.05}_{-0.06}$	...	...	...	...	...
8.75	$-1.86^{+0.07}_{-0.08}$	$-1.76^{+0.06}_{-0.06}$	$-1.86^{+0.05}_{-0.06}$	$-1.99^{+0.06}_{-0.06}$	$-2.02^{+0.06}_{-0.07}$	...	...	...
9.00	$-2.03^{+0.08}_{-0.09}$	$-1.86^{+0.06}_{-0.07}$	$-2.01^{+0.06}_{-0.06}$	$-2.14^{+0.06}_{-0.07}$	$-2.14^{+0.06}_{-0.07}$	$-2.20^{+0.05}_{-0.06}$	...	...
9.25	$-2.01^{+0.07}_{-0.08}$	$-2.00^{+0.06}_{-0.07}$	$-2.10^{+0.06}_{-0.07}$	$-2.24^{+0.06}_{-0.07}$	$-2.28^{+0.06}_{-0.07}$	$-2.31^{+0.05}_{-0.06}$	$-2.53^{+0.06}_{-0.07}$	...
9.50	$-2.10^{+0.07}_{-0.09}$	$-2.12^{+0.07}_{-0.08}$	$-2.23^{+0.06}_{-0.07}$	$-2.29^{+0.06}_{-0.07}$	$-2.46^{+0.07}_{-0.08}$	$-2.41^{+0.05}_{-0.06}$	$-2.50^{+0.06}_{-0.07}$	$-2.65^{+0.06}_{-0.07}$
9.75	$-2.17^{+0.08}_{-0.10}$	$-2.21^{+0.06}_{-0.07}$	$-2.39^{+0.07}_{-0.08}$	$-2.48^{+0.07}_{-0.08}$	$-2.53^{+0.07}_{-0.08}$	$-2.54^{+0.06}_{-0.06}$	$-2.63^{+0.06}_{-0.07}$	$-2.78^{+0.07}_{-0.08}$
10.00	$-2.24^{+0.08}_{-0.10}$	$-2.25^{+0.06}_{-0.08}$	$-2.45^{+0.07}_{-0.09}$	$-2.59^{+0.08}_{-0.09}$	$-2.61^{+0.08}_{-0.09}$	$-2.67^{+0.06}_{-0.07}$	$-2.74^{+0.07}_{-0.08}$	$-3.02^{+0.08}_{-0.09}$
10.25	$-2.31^{+0.08}_{-0.09}$	$-2.35^{+0.07}_{-0.08}$	$-2.45^{+0.07}_{-0.09}$	$-2.73^{+0.08}_{-0.10}$	$-2.68^{+0.08}_{-0.09}$	$-2.76^{+0.06}_{-0.07}$	$-2.91^{+0.08}_{-0.09}$	$-3.21^{+0.09}_{-0.10}$
10.50	$-2.41^{+0.08}_{-0.10}$	$-2.45^{+0.07}_{-0.09}$	$-2.52^{+0.08}_{-0.09}$	$-2.64^{+0.07}_{-0.09}$	$-2.71^{+0.08}_{-0.09}$	$-2.87^{+0.07}_{-0.08}$	$-3.07^{+0.09}_{-0.10}$	$-3.35^{+0.10}_{-0.13}$
10.75	$-2.53^{+0.09}_{-0.11}$	$-2.55^{+0.08}_{-0.09}$	$-2.59^{+0.08}_{-0.10}$	$-2.72^{+0.08}_{-0.10}$	$-2.84^{+0.08}_{-0.10}$	$-3.03^{+0.08}_{-0.09}$	$-3.35^{+0.10}_{-0.13}$	$-3.74^{+0.13}_{-0.17}$
11.00	$-2.91^{+0.11}_{-0.15}$	$-2.82^{+0.09}_{-0.11}$	$-2.93^{+0.10}_{-0.13}$	$-3.01^{+0.10}_{-0.12}$	$-3.12^{+0.10}_{-0.13}$	$-3.13^{+0.08}_{-0.10}$	$-3.54^{+0.12}_{-0.16}$	$-4.00^{+0.18}_{-0.25}$
11.25	$-3.46^{+0.14}_{-0.18}$	$-3.32^{+0.10}_{-0.13}$	$-3.47^{+0.11}_{-0.15}$	$-3.62^{+0.11}_{-0.15}$	$-3.65^{+0.12}_{-0.16}$	$-3.56^{+0.10}_{-0.13}$	$-3.89^{+0.12}_{-0.17}$	$-4.14^{+0.17}_{-0.28}$
11.50	...	...	...	...	$-4.99^{+0.30}_{-0.41}$	$-4.27^{+0.12}_{-0.15}$	$-4.41^{+0.14}_{-0.19}$	$-4.73^{+0.31}_{-2.00}$
Star-forming
	0.2 < z < 0.5	0.5 < z < 0.75	0.75 < z < 1.0	1.0 < z < 1.25	1.25 < z < 1.5	1.5 < z < 2.0	2.0 < z < 2.5	2.5 < z < 3.0
Log(M/ M☉)	Log(Φ)	Log(Φ)	Log(Φ)	Log(Φ)	Log(Φ)	Log(Φ)	Log(Φ)	Log(Φ)
8.00	$-1.42^{+0.06}_{-0.07}$	...	...	...	...	...	...	...
8.25	$-1.59^{+0.06}_{-0.07}$	$-1.60^{+0.06}_{-0.07}$	...	...	...	...	...	...
8.50	$-1.76^{+0.07}_{-0.08}$	$-1.67^{+0.05}_{-0.06}$	$-1.72^{+0.05}_{-0.06}$	...	...	...	...	...
8.75	$-1.91^{+0.07}_{-0.08}$	$-1.83^{+0.06}_{-0.06}$	$-1.88^{+0.05}_{-0.06}$	$-2.00^{+0.06}_{-0.06}$	$-2.03^{+0.06}_{-0.07}$	...	...	...
9.00	$-2.08^{+0.08}_{-0.09}$	$-1.92^{+0.06}_{-0.07}$	$-2.04^{+0.06}_{-0.06}$	$-2.16^{+0.06}_{-0.07}$	$-2.15^{+0.06}_{-0.07}$	$-2.20^{+0.05}_{-0.06}$	...	...
9.25	$-2.06^{+0.07}_{-0.08}$	$-2.09^{+0.06}_{-0.07}$	$-2.14^{+0.06}_{-0.07}$	$-2.26^{+0.06}_{-0.07}$	$-2.29^{+0.06}_{-0.07}$	$-2.32^{+0.05}_{-0.06}$	$-2.53^{+0.06}_{-0.07}$	...
9.50	$-2.17^{+0.07}_{-0.09}$	$-2.19^{+0.07}_{-0.08}$	$-2.27^{+0.06}_{-0.07}$	$-2.32^{+0.06}_{-0.07}$	$-2.48^{+0.07}_{-0.08}$	$-2.42^{+0.05}_{-0.06}$	$-2.51^{+0.06}_{-0.07}$	$-2.66^{+0.06}_{-0.07}$
9.75	$-2.25^{+0.08}_{-0.10}$	$-2.28^{+0.06}_{-0.07}$	$-2.47^{+0.07}_{-0.08}$	$-2.52^{+0.07}_{-0.08}$	$-2.55^{+0.07}_{-0.08}$	$-2.56^{+0.06}_{-0.06}$	$-2.67^{+0.06}_{-0.07}$	$-2.79^{+0.07}_{-0.08}$
10.00	$-2.36^{+0.08}_{-0.10}$	$-2.39^{+0.07}_{-0.08}$	$-2.55^{+0.08}_{-0.09}$	$-2.68^{+0.08}_{-0.09}$	$-2.68^{+0.08}_{-0.09}$	$-2.73^{+0.06}_{-0.07}$	$-2.78^{+0.07}_{-0.08}$	$-3.06^{+0.08}_{-0.09}$
10.25	$-2.50^{+0.08}_{-0.09}$	$-2.55^{+0.07}_{-0.08}$	$-2.60^{+0.07}_{-0.09}$	$-2.88^{+0.09}_{-0.10}$	$-2.75^{+0.08}_{-0.10}$	$-2.89^{+0.07}_{-0.07}$	$-3.00^{+0.08}_{-0.09}$	$-3.32^{+0.09}_{-0.11}$
10.50	$-2.63^{+0.09}_{-0.11}$	$-2.76^{+0.08}_{-0.09}$	$-2.77^{+0.08}_{-0.09}$	$-2.81^{+0.07}_{-0.09}$	$-2.87^{+0.08}_{-0.09}$	$-3.07^{+0.07}_{-0.09}$	$-3.26^{+0.09}_{-0.11}$	$-3.59^{+0.11}_{-0.14}$
10.75	$-2.91^{+0.10}_{-0.12}$	$-3.00^{+0.08}_{-0.10}$	$-2.91^{+0.09}_{-0.11}$	$-2.99^{+0.08}_{-0.10}$	$-3.07^{+0.08}_{-0.10}$	$-3.26^{+0.09}_{-0.10}$	$-3.54^{+0.11}_{-0.14}$	$-3.97^{+0.16}_{-0.20}$
11.00	$-3.43^{+0.13}_{-0.18}$	$-3.46^{+0.10}_{-0.13}$	$-3.37^{+0.10}_{-0.13}$	$-3.29^{+0.10}_{-0.13}$	$-3.39^{+0.10}_{-0.13}$	$-3.35^{+0.09}_{-0.11}$	$-3.69^{+0.13}_{-0.17}$	$-4.16^{+0.20}_{-0.28}$
11.25	$-4.39^{+0.30}_{-0.41}$	$-4.30^{+0.20}_{-0.25}$	$-4.17^{+0.16}_{-0.20}$	$-4.21^{+0.15}_{-0.20}$	$-3.95^{+0.13}_{-0.17}$	$-3.85^{+0.10}_{-0.13}$	$-4.00^{+0.13}_{-0.17}$	$-4.32^{+0.18}_{-0.29}$
11.50	...	...	...	...	$-5.17^{+0.37}_{-0.52}$	$-4.78^{+0.17}_{-0.21}$	$-4.59^{+0.15}_{-0.21}$	$-4.94^{+0.32}_{-2.00}$
Quiescent
	0.2 < z < 0.5	0.5 < z < 0.75	0.75 < z < 1.0	1.0 < z < 1.25	1.25 < z < 1.5	1.5 < z < 2.0	2.0 < z < 2.5	2.5 < z < 3.0
Log(M/ M☉)	Log(Φ)	Log(Φ)	Log(Φ)	Log(Φ)	Log(Φ)	Log(Φ)	Log(Φ)	Log(Φ)
8.25	$-2.41^{+0.08}_{-0.10}$	...	...	...	...	...	...	...
8.50	$-2.62^{+0.10}_{-0.11}$	$-2.42^{+0.07}_{-0.08}$	...	...	...	...	...	...
8.75	$-2.82^{+0.12}_{-0.14}$	$-2.58^{+0.07}_{-0.08}$	...	...	...	...	...	...
9.00	$-2.96^{+0.14}_{-0.16}$	$-2.77^{+0.09}_{-0.10}$	$-3.19^{+0.11}_{-0.12}$	$-3.46^{+0.12}_{-0.14}$	...	...	...	...
9.25	$-2.96^{+0.08}_{-0.10}$	$-2.75^{+0.09}_{-0.10}$	$-3.17^{+0.10}_{-0.12}$	$-3.65^{+0.15}_{-0.17}$	$-3.97^{+0.21}_{-0.24}$	...	...	...
9.50	$-2.98^{+0.09}_{-0.10}$	$-2.94^{+0.10}_{-0.11}$	$-3.33^{+0.12}_{-0.14}$	$-3.46^{+0.13}_{-0.14}$	$-3.79^{+0.17}_{-0.19}$	$-4.14^{+0.17}_{-0.19}$	...	...
9.75	$-2.91^{+0.09}_{-0.11}$	$-2.99^{+0.07}_{-0.08}$	$-3.16^{+0.11}_{-0.12}$	$-3.57^{+0.14}_{-0.16}$	$-3.75^{+0.16}_{-0.18}$	$-3.95^{+0.14}_{-0.15}$	$-3.72^{+0.11}_{-0.12}$	$-4.16^{+0.17}_{-0.20}$
10.00	$-2.86^{+0.09}_{-0.11}$	$-2.83^{+0.07}_{-0.08}$	$-3.16^{+0.11}_{-0.12}$	$-3.37^{+0.12}_{-0.14}$	$-3.45^{+0.12}_{-0.14}$	$-3.55^{+0.09}_{-0.11}$	$-3.76^{+0.11}_{-0.13}$	$-4.08^{+0.16}_{-0.18}$
10.25	$-2.78^{+0.08}_{-0.10}$	$-2.78^{+0.07}_{-0.09}$	$-2.97^{+0.08}_{-0.09}$	$-3.26^{+0.11}_{-0.13}$	$-3.52^{+0.13}_{-0.15}$	$-3.35^{+0.08}_{-0.09}$	$-3.64^{+0.11}_{-0.12}$	$-3.89^{+0.13}_{-0.15}$
10.50	$-2.80^{+0.09}_{-0.11}$	$-2.75^{+0.08}_{-0.09}$	$-2.89^{+0.08}_{-0.10}$	$-3.11^{+0.08}_{-0.09}$	$-3.24^{+0.08}_{-0.10}$	$-3.30^{+0.08}_{-0.09}$	$-3.53^{+0.10}_{-0.12}$	$-3.74^{+0.12}_{-0.15}$
10.75	$-2.76^{+0.09}_{-0.12}$	$-2.75^{+0.08}_{-0.10}$	$-2.87^{+0.09}_{-0.11}$	$-3.05^{+0.08}_{-0.10}$	$-3.23^{+0.09}_{-0.11}$	$-3.40^{+0.09}_{-0.11}$	$-3.82^{+0.13}_{-0.16}$	$-4.12^{+0.18}_{-0.22}$
11.00	$-3.07^{+0.12}_{-0.16}$	$-2.93^{+0.09}_{-0.11}$	$-3.12^{+0.10}_{-0.13}$	$-3.33^{+0.10}_{-0.13}$	$-3.46^{+0.10}_{-0.13}$	$-3.54^{+0.09}_{-0.11}$	$-4.08^{+0.17}_{-0.22}$	$-4.51^{+0.27}_{-0.38}$
11.25	$-3.52^{+0.14}_{-0.19}$	$-3.37^{+0.11}_{-0.14}$	$-3.57^{+0.12}_{-0.15}$	$-3.75^{+0.12}_{-0.16}$	$-3.95^{+0.13}_{-0.17}$	$-3.87^{+0.10}_{-0.13}$	$-4.54^{+0.15}_{-0.21}$	$-4.61^{+0.19}_{-0.32}$
11.50	...	...	...	...	$-5.47^{+0.52}_{-0.90}$	$-4.44^{+0.13}_{-0.16}$	$-4.89^{+0.19}_{-0.26}$	$-5.14^{+0.34}_{-2.00}$

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In calculating the SMF, we include only galaxies that lie above the mass-completeness limit corresponding to the upper redshift limit of each subsample. We follow the procedures outlined in Avni & Bahcall (1980) to combine the multiple fields of our survey in calculating SMFs. The SMF (Φ) is then simply calculated as

$$\begin{equation} \Phi (M) = \; \frac{1}{\Delta M} \; \sum _{i=1}^N \frac{1}{V_c}, \end{equation} \tag{ 2 }$$

where M = Log(M/M_☉), ΔM is the size of the mass bin, N is the number of galaxies in the mass bin between the redshift limits (z_min, z_max), and V_c is the comoving volume based on the survey area and redshift limits. We refrain from using the 1/V_max formalism (Avni & Bahcall 1980) to avoid introducing any potential bias associated with evolution in the SMF over our relatively wide redshift bins. Since we do not apply a 1/V_max correction, V_c is the same for all galaxies in a given redshift bin.

The depth of our survey allows us to test for the shape of the SMF; namely, we fit both single- and double-Schechter functions to determine the best fit. The single-Schechter (1976) function is defined as

$$\begin{equation} \Phi (M) dM = \: \mathrm{ln} (10) \; \Phi ^* [ 10^{(M-M^*)(1+\alpha)} ] \mathrm{exp}(-10^{(M-M^*)}) dM, \end{equation} \tag{ 3 }$$

where again M = Log(M/M_☉), α is the slope of the power law at low masses, Φ* is the normalization, and M* is the characteristic mass. The double-Schechter function is defined as

$$\begin{eqnarray} \Phi (M) dM &=& \Phi _1 (M) dM + \Phi _2 (M) dM\nonumber \\ &=& \mathrm{ln} (10) \; \mathrm{exp} (-10^{(M-M^*)}) 10^{(M-M^*)}\nonumber \\ &&\times [ \Phi ^*_1 10^{(M-M^*) \alpha _1} + \Phi ^*_2 10^{(M-M^*) \alpha _2} ] dM,\quad\quad \end{eqnarray} \tag{ 4 }$$

where again M = Log(M/M_☉), (α₁, α₂) are the slopes, ($\Phi ^*_1$, $\Phi ^*_2$) are the normalizations of the constituent Schechter functions, and M* again is the characteristic mass. Note that one value for M* is used for both constituents in the double-Schechter function. This functional form of the double-Schechter function is the same as in Baldry et al. (2008).

Recent measurements of the total SMF at z < 1.5 have shown that the SMF steepens at M  <  10¹⁰ M_☉ (e.g., Baldry et al. 2008; Li & White 2009; Drory et al. 2009; Pozzetti et al. 2010; Moustakas et al. 2013; Ilbert et al. 2013; Muzzin et al. 2013). We fit each of our mass functions with both single- and double-Schechter functions. We show best-fit parameters as well as reduced chi-squared values for each in Tables 2 and 3. From the reduced chi-squared values, we find that the total SMF is a much better fit by a double-Schechter function at z ⩽ 2. At z > 2, we find that a single-Schechter function is sufficient; however, this may be because we do not go deep enough to detect significant structure at low masses.

Table 2. Best-fit Double-Schechter Parameters

Total
Redshift	Log(M*)a	α1	Log($\Phi ^*_1$)b	α2	Log($\Phi ^*_2$)b	$\chi ^2_{\mathrm{red}}$
0.20 < z < 0.50	10.78 ± 0.11	−0.98 ± 0.24	−2.54 ± 0.12	−1.90 ± 0.36	−4.29 ± 0.55	0.3
0.50 < z < 0.75	10.70 ± 0.10	−0.39 ± 0.50	−2.55 ± 0.09	−1.53 ± 0.12	−3.15 ± 0.23	0.5
0.75 < z < 1.00	10.66 ± 0.13	−0.37 ± 0.49	−2.56 ± 0.09	−1.61 ± 0.16	−3.39 ± 0.28	0.6
1.00 < z < 1.25	10.54 ± 0.12	0.30 ± 0.65	−2.72 ± 0.10	−1.45 ± 0.12	−3.17 ± 0.19	0.8
1.25 < z < 1.50	10.61 ± 0.08	−0.12 ± 0.49	−2.78 ± 0.08	−1.56 ± 0.16	−3.43 ± 0.23	0.3
1.50 < z < 2.00	10.74 ± 0.09	0.04 ± 0.62	−3.05 ± 0.11	−1.49 ± 0.14	−3.38 ± 0.20	0.8
2.00 < z < 2.50	10.69 ± 0.29	1.03 ± 1.64	−3.80 ± 0.30	−1.33 ± 0.18	−3.26 ± 0.23	0.4
2.50 < z < 3.00	10.74 ± 0.31	1.62 ± 1.88	−4.54 ± 0.41	−1.57 ± 0.20	−3.69 ± 0.28	1.3
Star-forming
Redshift	Log(M*)	α1	Log($\Phi ^*_1$)	α2	Log($\Phi ^*_2$)	$\chi ^2_{\mathrm{red}}$
0.20 < z < 0.50	10.59 ± 0.09	−1.08 ± 0.23	−2.67 ± 0.11	−2.00 ± 0.49	−4.46 ± 0.63	0.3
0.50 < z < 0.75	10.65 ± 0.23	−0.97 ± 1.32	−2.97 ± 0.28	−1.58 ± 0.54	−3.34 ± 0.67	0.3
0.75 < z < 1.00	10.56 ± 0.13	−0.46 ± 0.63	−2.81 ± 0.10	−1.61 ± 0.17	−3.36 ± 0.28	0.9
1.00 < z < 1.25	10.44 ± 0.11	0.53 ± 0.73	−2.98 ± 0.14	−1.44 ± 0.11	−3.11 ± 0.16	0.8
1.25 < z < 1.50	10.69 ± 0.12	−0.55 ± 0.69	−3.04 ± 0.12	−1.62 ± 0.24	−3.59 ± 0.35	0.2
1.50 < z < 2.00	10.59 ± 0.10	0.75 ± 0.70	−3.37 ± 0.16	−1.47 ± 0.10	−3.28 ± 0.13	0.9
2.00 < z < 2.50	10.58 ± 0.18	2.06 ± 1.43	−4.30 ± 0.39	−1.38 ± 0.15	−3.28 ± 0.18	0.7
2.50 < z < 3.00	10.61 ± 0.22	2.36 ± 1.84	−4.95 ± 0.49	−1.67 ± 0.19	−3.71 ± 0.25	0.8
Quiescent
Redshift	Log(M*)	α1	Log($\Phi ^*_1$)	α2	Log($\Phi ^*_2$)	$\chi ^2_{\mathrm{red}}$
0.20 < z < 0.50	10.75 ± 0.10	−0.47 ± 0.20	−2.76 ± 0.09	−1.97 ± 0.34	−5.21 ± 0.48	0.2
0.50 < z < 0.75	10.68 ± 0.07	−0.10 ± 0.27	−2.67 ± 0.05	−1.69 ± 0.24	−4.29 ± 0.33	0.9
0.75 < z < 1.00	10.63 ± 0.12	0.04 ± 0.44	−2.81 ± 0.05	−1.51 ± 0.67	−4.40 ± 0.56	0.4
1.00 < z < 1.25	10.63 ± 0.12	0.11 ± 0.44	−3.03 ± 0.05	−1.57 ± 0.81	−4.80 ± 0.61	0.8
1.25 < z < 1.50	10.49 ± 0.11	0.85 ± 1.07	−3.36 ± 0.30	−0.54 ± 0.66	−3.72 ± 0.44	0.6
1.50 < z < 2.00	10.77 ± 0.18	−0.19 ± 0.96	−3.41 ± 0.23	−0.18 ± 1.21	−3.91 ± 0.51	1.9
2.00 < z < 2.50	10.69 ± 0.14	−0.37 ± 0.52	−3.59 ± 0.10	−3.07 ± 16.13	−6.95 ± 1.66	0.7
2.50 < z < 3.00	9.95 ± 0.23	−0.62 ± 2.63	−4.22 ± 0.41	2.51 ± 2.43	−4.51 ± 0.62	1.7

Notes. ^aIn units of M_☉. ^bIn units of Mpc⁻³ dex⁻¹.

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Table 3. Best-fit Single-Schechter Parameters

Total
Redshift	Log(M*)a	α	Log(Φ*)b	$\chi ^2_{\mathrm{red}}$
0.20 < z < 0.50	11.05 ± 0.10	−1.35 ± 0.04	−2.96 ± 0.10	3.3
0.50 < z < 0.75	11.00 ± 0.06	−1.35 ± 0.04	−2.93 ± 0.07	4.6
0.75 < z < 1.00	11.16 ± 0.12	−1.38 ± 0.04	−3.17 ± 0.11	4.5
1.00 < z < 1.25	11.09 ± 0.10	−1.33 ± 0.05	−3.19 ± 0.11	4.2
1.25 < z < 1.50	10.88 ± 0.05	−1.29 ± 0.05	−3.11 ± 0.08	5.6
1.50 < z < 2.00	11.03 ± 0.05	−1.33 ± 0.05	−3.28 ± 0.08	4.5
2.00 < z < 2.50	11.13 ± 0.13	−1.43 ± 0.08	−3.59 ± 0.14	0.3
2.50 < z < 3.00	11.35 ± 0.33	−1.74 ± 0.12	−4.36 ± 0.29	1.0
Star-forming
Redshift	Log(M*)	α	Log(Φ*)	$\chi ^2_{\mathrm{red}}$
0.20 < z < 0.50	10.73 ± 0.06	−1.37 ± 0.04	−2.94 ± 0.08	2.4
0.50 < z < 0.75	10.79 ± 0.07	−1.42 ± 0.04	−3.04 ± 0.08	0.7
0.75 < z < 1.00	10.86 ± 0.07	−1.43 ± 0.04	−3.16 ± 0.09	3.0
1.00 < z < 1.25	10.85 ± 0.07	−1.37 ± 0.05	−3.20 ± 0.09	3.2
1.25 < z < 1.50	10.89 ± 0.05	−1.38 ± 0.05	−3.27 ± 0.08	2.1
1.50 < z < 2.00	10.97 ± 0.05	−1.45 ± 0.05	−3.44 ± 0.08	4.2
2.00 < z < 2.50	11.28 ± 0.19	−1.60 ± 0.08	−3.96 ± 0.19	1.3
2.50 < z < 3.00	11.49 ± 0.46	−1.93 ± 0.12	−4.82 ± 0.38	1.4
Quiescent
Redshift	Log(M*)	α	Log(Φ*)	$\chi ^2_{\mathrm{red}}$
0.20 < z < 0.50	11.11 ± 0.14	−0.98 ± 0.07	−3.18 ± 0.10	4.1
0.50 < z < 0.75	11.03 ± 0.08	−0.98 ± 0.07	−3.15 ± 0.09	9.9
0.75 < z < 1.00	10.88 ± 0.09	−0.59 ± 0.10	−3.00 ± 0.08	2.1
1.00 < z < 1.25	10.84 ± 0.09	−0.47 ± 0.11	−3.16 ± 0.07	2.1
1.25 < z < 1.50	10.60 ± 0.04	−0.03 ± 0.14	−3.17 ± 0.05	0.9
1.50 < z < 2.00	10.76 ± 0.05	−0.14 ± 0.12	−3.29 ± 0.05	1.8
2.00 < z < 2.50	10.73 ± 0.08	−0.49 ± 0.18	−3.63 ± 0.09	0.4
2.50 < z < 3.00	10.65 ± 0.19	−0.43 ± 0.34	−3.92 ± 0.14	1.6

Notes. ^aIn units of M_☉. ^bIn units of Mpc⁻³ dex⁻¹.

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This is clearly shown in Figure 9 where we plot the residuals of both single- and double-Schechter fits to the total SMF. A prominent upturn is revealed in the top three panels where we fit single-Schechter functions at Log(M/M_☉) >10 only. In the bottom three panels, we show the residuals from fitting double-Schechter functions at all masses, which are consistent within our measurement uncertainties. However, although the double-Schechter provides a good fit at all redshifts, we find that a single-Schechter works just as well at z > 1.5 for the quiescent SMF and at z > 2 for the total and star-forming SMFs. We observe the same behavior even if the NMBS data is excluded from the calculation, proving that the steepening of the low-mass slope is not caused by a systematic offset between the surveys we use. In fact, there is evidence for a steepening in each of the three ZFOURGE fields independently.

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The left panel of Figure 10 shows the best-fit values for M* as a function of redshift. There is little statistically significant evolution in M* at z  <  2, in agreement with other studies (Marchesini et al. 2009; Santini et al. 2012; Muzzin et al. 2013). We note that our values of M* are ∼0.2 dex lower than these previous studies. We find that this offset is the result of comparing single- vs. double-Schechter fits to the SMF. The weak evolution in M* suggests that the physical mechanism(s) responsible for the exponential cutoff in the SMF has a mass scale that is independent of redshift (see also Peng et al. 2010, 2012).

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We show the best-fit values for the faint-end slope α as a function of redshift in the middle panel of Figure 10. We plot only the steeper slope of (α₁, α₂), which dominates at the lowest masses. We find no statistically significant evolution in the low-mass slope within our redshift range. Some evolution in alpha may be suggested when comparing to the z ∼ 0 SMF from Moustakas et al. (2013); however, we note that those authors do not probe below 10⁹ M_☉and thus do not strongly constrain the slope at the lowest masses. We do find better agreement with the z ∼ 0 SMF from Baldry et al. (2012), who reach lower masses.

In the last panel of Figure 10, we show the redshift evolution of $\Phi ^*_1 + \Phi ^*_2$. In contrast with the apparent constancy of M* and α, we find clear evolution in Φ*. Thus, to rough approximation, the shape of the total SMF does not evolve over 0 < z < 2 but the normalization does. Moustakas et al. (2013) do not report parameters for functional fits to their measured SMF; therefore, we fit our own doube-Schechter function to their z ≈ 0.1 SMF. The best-fit parameters we find for Log(M*), α₁, Log($\Phi ^*_1$), α₂, and Log($\Phi ^*_2$) are 10.79, −0.74, −2.44, −1.75, and −3.69, respectively.

In Figure 11, we show the growth in the number density of galaxies as a function of mass in several redshift bins for the star-forming and quiescent subpopulations. We show this growth by normalizing our star-forming/quiescent SMFs to the most recent measurements of the star-forming/quiescent SMFs at z ≈ 0 from SDSS (Moustakas et al. 2013). The redshift ranges at z > 0.4 in Figure 11 are chosen to track the evolution in similar time intervals of approximately 1.2 Gyr.

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At Log(M/M_☉) <11, where we have sufficient statistics to trace the evolution of the mass function, we find that the SMF of star forming galaxies grows moderately with cosmic time, by 1.5–2.5× since z ∼ 2. There is a hint that it actually decreases with time at z < 0.6. Only between 2 < z < 3 do we observe a large jump in the number of star-forming galaxies at Log(M/M_☉) > 10. These results are consistent with previous works which have generally found that the star-forming SMF evolves relatively weakly with redshift (Arnouts et al. 2007; Bell et al. 2007; Pozzetti et al. 2010; Brammer et al. 2011; Muzzin et al. 2013).

The growth of quiescent galaxies since z ≈ 2 is much more rapid than that of star-forming galaxies (e.g., Arnouts et al. 2007; Bell et al. 2007). At masses greater than 10¹⁰ M_☉  we find roughly a factor of six increase between z = 2 and z = 0, in agreement with previous studies; however, at lower masses, there is a 15–30× increase. This is the first clear detection of a decline in the low-mass quiescent population toward high-redshift that is not affected by incompleteness. This rapid evolution causes the quiescent fraction to increase by about a factor of five for low-mass galaxies (<10¹⁰ M_☉) from ≈7% at z = 2 to ≈34% at z = 0.

Obtaining a precise estimate of the integrated stellar mass density in the universe requires probing the SMF well below M*. Most recent attempts at intermediate redshifts have been made using NIR selected surveys, which make it possible to define highly complete samples down to some stellar mass limit. However, if this limit does not reach significantly below M*  then the integrated stellar mass density depends on an extrapolation of the observed SMF using the best-fit Schechter parameters (e.g., Marchesini et al. 2009; Santini et al. 2012; Ilbert et al. 2013; Muzzin et al. 2013), which may be poorly constrained and may depend sensitively on the exact—and uncertain—level of completeness near the nominal mass-completeness limit.

In Figure 12, we show our measurements for the evolution of the cosmic stellar mass densities (ρ) of all, star-forming, and quiescent galaxies. Previous studies have typically integrated best-fit Schechter functions between 8 < Log(M/M_☉) <13, extrapolating below mass-completeness limits wherever necessary. We choose to integrate our best fits between 9 < Log(M/M_☉) <13 since this is only marginally below our completeness limit in our highest redshift bin. We note here that using 10⁹ M_☉ as opposed to 10⁸ M_☉ as a lower-limit decreases ρ by <5%.

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Uncertainties are evaluated from 500 Monte Carlo simulations of the measured SMFs. For each iteration, we perturb all data points using the combined uncertainties as described in Section 2.5. We then refit Schechter functions to recalculate ρ, taking the resulting scatter as the uncertainty. We parameterize our measurements of the redshift evolution of the total stellar mass density as follows:

$$\begin{equation} { \rm Log}(\rho) = a (1+z) + b, \end{equation} \tag{ 5 }$$

where ρ is the total stellar mass density in units of M_☉ Mpc⁻³. From a least-squares fit, we find best-fit values of a = −0.33 ± 0.03 and b = 8.75 ± 0.07.

Figure 12 also shows results from recent deep and large-area surveys, which are in overall agreement with our measurements. Santini et al. (2012) present results using data from CANDELS Early Release Science program in conjunction with deep (K_s ∼ 25.5) imaging from Hawk-I. Although their work covers significantly less area than we present here (33 arcmin² versus 316 arcmin²), our measurements agree within 1σ uncertainties. Measurements from the recent UltraVISTA survey (McCracken et al. 2012), which covers ∼1.6 deg² to a depth of K_s = 23.4, are presented in Ilbert et al. (2013) and Muzzin et al. (2013). Our results are in excellent agreement at all redshifts except 1.5 < z < 2.5 with Muzzin et al. (2013). The difference between our result and Muzzin et al. (2013) is mostly due to the large difference in the faint end slope: Muzzin et al. (2013) measure a slope of ∼ − 0.9, whereas we find −1.33 for the best-fit single-Schechter function at 1.5 < z < 2.0. Muzzin et al. (2013) note that α is not well constrained by their data and do not rule out a low-mass slope as steep as ours.

Another estimate of the stellar mass density was provided by Reddy & Steidel (2009), who used an optically selected sample of star-forming galaxies at 1.9 < z < 3.4 to argue that the low-mass end of the SMF is quite steep and may have been underestimated by previous studies; they concluded that a large fraction of the stellar mass budget of the universe was locked up in dwarf galaxies. However, these authors were not able to probe the SMF directly given the nature of their sample and their limited NIR and IR data, so they inferred the SMF by performing large corrections for incompleteness. Given the depth of our NIR-selected sample, we are able to probe down to similarly low masses (∼10⁹ M_☉) for complete samples.

We compare our z > 2 measurements to estimates based on the Reddy & Steidel (2009) measurements in Figure 12. We obtain their value by integrating the SMF shown in their Figure 12 and after converting from a Salpeter IMF (N. Reddy 2013, private communication). The agreement is excellent, however, as noted by those authors, their sample is incomplete for galaxies with red colors. Thus, the good agreement that we find is partially due to the steeper slope of their inferred SMF which is balanced by incompleteness at high masses.¹¹ Nonetheless, it is encouraging that similar results are obtained using very different types of data sets and different methods.