THE SFR–M_* RELATION AND EMPIRICAL STAR FORMATION HISTORIES FROM ZFOURGE AT 0.5 < z < 4^*

The FourStar Galaxy Evolution Survey (ZFOURGE¹⁶ : Straatman et al. 2015) is a deep near-IR survey conducted with the FourStar imager (Persson et al. 2013) covering one 11' × 11' pointing in each of the three legacy fields CDF-S (Giacconi et al. 2002), COSMOS (Capak et al. 2007) and UDS (Lawrence et al. 2007) reaching depths of ∼26 mag in J₁, J₂, J₃, and ∼25 mag in H_s, H_l, and K_s (5σ in d = 06 apertures). The medium-bandwidth filters utilized by this survey offer spectral resolutions λ/Δλ ≈ 10, roughly twice that of their broadband counterparts. This increase provides for finer sampling of the Balmer/4000 Å spectral break at 1 < z < 4, leading to well-constrained photometric redshifts. In combination with ancillary imaging, the full photometric data set covers the observed 0.3−8 μm wavelength range.

Photometric redshifts and rest-frame colors were measured using the public SED-fitting code EAZY (Brammer et al. 2008) on PSF-matched optical-NIR photometry. 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.

A comparison of our derived photometric redshifts to a sample of 1437 galaxies with secure spectroscopic redshifts is shown in Figure 1. We calculate a scatter of Δz/(1 + z_spec) = 1.8% at z < 1.5 and fraction of catastrophic outliers ($| {\rm{\Delta }}z/(1+{z}_{\mathrm{spec}})| \gt 0.15$) of 2.7%. At z > 1.5 these rise to 2.2% and 9% respectively. An additional analysis of z_phot accuracy can be found in Section 2 of Kawinwanichakij et al. (2014) and Straatman et al. (2015). Spectroscopic redshifts from CDF-S are taken 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). For COSMOS spectroscopic redshifts come from Lilly et al. (2009) and Trump et al. (2009). Spectroscopic redshifts for UDS come from Simpson et al. (2012) and Smail et al. (2008).

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Stellar masses were derived by fitting stellar population synthesis templates to the 0.3–8 μm photometry using the SED-fitting code FAST (Kriek et al. 2009). FAST was run using a grid of Bruzual & Charlot (2003) models assuming a Chabrier (2003) IMF and solar metallicity. Exponentially declining star formation histories (SFHs) (Ψ ∝ e^−t/τ) are used with log(τ/year) ranging between 7 and 11 in steps of 0.2 and allowing log(age/year) to vary between 7.5 and 10.1 in steps of 0.1. A Calzetti et al. (2000) extinction law is also incorporated with values of A_V varying between 0 and 4 in steps of 0.1.

Mass-completeness limits are estimated using a method similar to Quadri et al. (2012). Briefly, we estimate the distribution of mass-to-light ratios of galaxies that are somewhat above our K_s = 25 mag limit, and use this distribution to estimate the 90% mass-completeness limit of galaxies at K_s = 25. These mass-completeness limits are shown in Figure 1 along with the distribution of stellar masses and redshifts of galaxies in the ZFOURGE catalogs. A more complete discussion of the mass-completeness limits will be presented by Straatman et al. (2015).

We make use of Spizer/MIPS (GOODS-S: PI Dickinson, COSMOS: PI Scoville, UDS: PI Dunlop) and Herschel/PACS data (GOODS-S: Elbaz et al. 2011, COSMOS & UDS: PI Dickinson) for measuring total infrared luminosities (L_IR) to derive SFRs. Imaging from these observatories used in this study include 24, 100 and 160 μm. Median 1σ flux uncertainties for CDF-S/COSMOS/UDS are approximately 3.9/10.3/10.1 μJy in the 24 μm imaging, 0.20/0.43/0.45 mJy in the 100 μm imaging and 0.35/0.70/0.93 mJy in the 160 μm imaging respectively.

Due to the large PSFs of the MIPS/PACS imaging (FWHM ≳ 4'') source blending is a considerable effect. Therefore we use the Multi-resolution Object PHotometry oN Galaxy Observations (MOPHONGO) code written by I. Labbé to extract deblended photometry in these far-IR data (for a detailed discussion see Labbé et al. 2006; Wuyts et al. 2007). The algorithm uses higher resolution imaging to generate a segmentation map containing information on the locations, sizes and extents of objects. In this work we use deep K_s band as the prior (FWHM = 046). Point-sources coincident in both images are used to construct a convolution kernel that maps between the high and low resolution PSFs. Objects used to construct this kernel need to be hand selected as many point-sources in the K_s imaging are frequently undetected at far-IR wavelengths. A model of each far-IR image is generated by convolving the high-resolution segmentation map with the corresponding kernel allowing the intensities of individual objects to vary freely. Background and rms maps are generated locally for each object on scales that are three times the 30'' tile-size used. By subtracting the modeled light of neighboring sources, "cleaned" image tiles of individual objects are produced which will be used in the stacking analysis discussed in the following section.

Modern near-infrared galaxy surveys have made it possible to detect approximately mass-complete samples of galaxies to high redshifts (z ≈ 4). Unfortunately however, imaging used to probe obscured star formation (typically far-IR and radio) rarely ever reach complementary depths. Thus, many studies over the past several years have turned to measuring SFRs from stacked data in order to compensate for this disparity (e.g., Dunne et al. 2009; Rodighiero et al. 2010; Karim et al. 2011; Whitaker et al. 2014; Schreiber et al. 2015). However it is important to keep in mind that the interpretation of stacked results may be complicated by the fact that the intrinsic distribution of SFRs may not be unimodal or symmetric.

We classify galaxies as either actively star-forming or quiescent using the UVJ color–color diagram (Labbé et al. 2005; Wuyts et al. 2007; Williams et al. 2009). The rest-frame (U − V) and (V − J) colors are estimated using EAZY (Section 2.2). The advantage of this diagram is that it effectively separates the two reddening vectors caused by aging and dust extinction, decreasing the likelihood of dust-enshrouded star-forming galaxies being identified as quiescent. The UVJ diagram is thus a more effective tool for categorizing galaxies into star-forming and quiescent subsamples than a simple color–magnitude criterion.

The deep near-IR photometry (K_s ≈ 25) of ZFOURGE allows us to reliably select galaxies based on stellar mass. Across the entire redshift range considered in this work (0. 5 < z < 4) we detect 12,433 galaxies in the K_s band imaging that lie above our estimated mass-completeness limits. From this mass-complete sample, we find that 5875 (47%), 8542 (69%) and 8630 (69%) are not detected in the 24, 100 and 160 μm images respectively (where detection is defined as S/N > 1). As such, we resort to stacking of the far-IR photometry for our K_s-selected sample in order to more precisely measure fluxes for ensembles of galaxies. In bins of redshift and stellar mass, we average-combine "cleaned" image tiles (see Section 2.3) of individual galaxies for each of the far-IR bandpasses. Stacking of "cleaned" imaging has been shown to significantly decrease contamination from blended sources (see also Fumagalli et al. 2014; Whitaker et al. 2014). Finally, photometry is measured in apertures of 35, 40 and 60 for the 24, 100 and 160 μm respectively with a background subtraction as measured from an annulus of radii 15''−19'' on each stack. PSFs generated from bright objects are used to derive aperture corrections of 2.21, 1.76, and 1.61 respectively for the 24, 100 and 160 μm imaging. Because these PSFs were constructed on the same 30'' tile-size these are not corrections to total flux, thus, we adopt additional correction factors of 1.2, 1.38, and 1.54 to account for flux that falls outside the tile.

In order to estimate the error on the mean flux measured for each stack we perform 100 bootstrap resamplings on each mass-redshift subsample. Stacks of the 24, 100, and 160 μm tiles are generated from these resamplings from which fluxes are measured as described above. We take the inter-68th percentile of these flux distributions as the corresponding uncertainties on the estimated IR fluxes. In Section 3 we describe how these uncertainties are propagated to estimates of L_IR. It is worth mentioning that fluxes measured from stacking are subject to biases due to the clustering of galaxies. However, detailed simulations have shown this effect to be negligible at the image resolution of our data set (Viero et al. 2013; Schreiber et al. 2015) and using "cleaned" image tiles also helps to minimize contamination from neighboring sources.

We remove sources suspected of hosting active galactic nuclei (AGN) from all samples based on radio, X-ray and IR indicators. Radio AGN are identified as sources with 1.4 GHz excess having Ψ_1.4/Ψ_IR ≥ 3 where Ψ_1.4 is the radio-inferred SFR based on Equation (6) of Bell (2003) and Ψ_IR is the IR-inferred SFR discussed in Section 2.5. More discussion of the selection and properties of radio AGN will be provided in Rees et al. (2015). Unobscured X-ray AGN are identified as having 10⁴² ≤ L_X ≤ 10⁴⁴ and HR < −0.2 where L_X and HR are the rest-frame X-ray luminosity in erg s⁻¹ and hardness ratio respectively. All objects with L_X > 10⁴⁴ are classified as QSOs and also rejected. Infrared AGN are identified based on an adaptation from the criteria of Messias et al. (2012) and will be presented in more detail by Cowley et al. (2015).

We calculate total star formation rates by adding contributions from UV and IR light. This approach assumes that the IR emission of galaxies (L_IR) originates from dust heated by the obscured UV light of young, massive stars. Thus by adding its contribution to that of the unobscured UV luminosity (L_UV) the total SFR for galaxies can be calculated. We use the conversion from Bell et al. (2005) scaled to a Chabrier (2003) IMF to derive SFRs from our data:

$$\begin{eqnarray}&&{{\rm{\Psi }}}_{\mathrm{UV}+\mathrm{IR}}\ [{M}_{\odot }\;{\mathrm{yr}}^{-1}]=1.09\times {10}^{-10}\ ({L}_{\mathrm{IR}}+2.2{L}_{\mathrm{UV}})\end{eqnarray} \tag{ 1 }$$

where L_IR is the integrated 8–1000 μm luminosity and L_UV = 1.5 ν L_ν,2800 represents the rest-frame 1216–3000 Å luminosity, both in units of L_⊙.

For our stacking analysis we aim to measure the average star formation rate of galaxies in bins of redshift and stellar mass. In each mass-redshift bin we use the median rest-frame 2800 Å luminosity output by EAZY for L_UV and estimate 1σ uncertainties from 100 bootstrap resamplings. We estimate bolometric infrared luminosities (L_IR ≡ L_{8–1000 μm}) by fitting an IR spectral template to the stacked 24–160 μm photometry. The template introduced by Wuyts et al. (2008), hereafter referred to as the W08 template, was constructed by averaging the logarithm of the spectral template library from Dale & Helou (2002) motivated by results from Papovich et al. (2007). The validity of this luminosity-independent conversion has been demonstrated by Muzzin et al. (2010) through comparison SFRs derived from Hα versus 24 μm fluxes for a sample of galaxies at z ∼ 2.

Furthermore, Wuyts et al. (2011) find that at 0 < z < 3 this luminosity-independent conversion yields consistent L_IR estimates from 24 μm when compared to L_IR derived from PACS photometry from the Herschel PEP survey (Lutz et al. 2011). For our stacking analysis, we smooth the W08 template by the redshift distribution of the galaxies in each mass-redshift bin prior to fitting.

Errors on L_IR were estimated from 100 Monte Carlo simulations of the stacked IR fluxes. For each mass-redshift bin we perturb the stacked 24–160 μm fluxes by a normal probability density function (PDF) of width given by the estimated uncertainties described in Section 2.4. Infrared luminosities are calculated for each iteration in the same way as described above. Errors for L_IR are derived from the 68th percentile range of each L_IR distribution. We combine these with the uncertainties estimated for L_UV to derive uncertainties on Ψ_UV+IR. All measurements of UV and IR luminosities, star formation rates, and corresponding errors can be found in Table 1.

Table 1.  SFR–M_* Relations Data

(1)	(2)	(3)	(4)	(5)	(6)	(7)	(8)	(9)	(10)
Redshift	log(M*)	log(LUV)all	log(LIR)all	log(Ψ)all	log(LUV)sf	log(LIR)sf	log(Ψ)sf
Range	(M⊙)	(L⊙)	(L⊙)	(M⊙/year)	(L⊙)	(L⊙)	(M⊙/year)	${N}_{\mathrm{all}}$	Nsf
0.50 < z < 0.75	8.625	${9.05}_{-0.02}^{+0.01}$	${9.18}_{-0.11}^{+0.10}$	$-{0.36}_{-0.04}^{+0.04}$	${9.07}_{-0.02}^{+0.02}$	${9.25}_{-0.09}^{+0.08}$	$-{0.32}_{-0.04}^{+0.03}$	493	456
	8.875	${9.22}_{-0.02}^{+0.04}$	${9.43}_{-0.04}^{+0.05}$	$-{0.16}_{-0.02}^{+0.03}$	${9.28}_{-0.01}^{+0.02}$	${9.46}_{-0.05}^{+0.04}$	$-{0.11}_{-0.02}^{+0.02}$	391	343
	9.125	${9.38}_{-0.02}^{+0.06}$	${9.75}_{-0.04}^{+0.04}$	${0.08}_{-0.03}^{+0.03}$	${9.47}_{-0.02}^{+0.02}$	${9.82}_{-0.03}^{+0.04}$	${0.15}_{-0.02}^{+0.03}$	300	260
	9.375	${9.50}_{-0.02}^{+0.06}$	${10.04}_{-0.04}^{+0.03}$	${0.29}_{-0.03}^{+0.02}$	${9.57}_{-0.03}^{+0.03}$	${10.09}_{-0.03}^{+0.03}$	${0.35}_{-0.02}^{+0.02}$	261	234
	9.625	${9.49}_{-0.04}^{+0.10}$	${10.42}_{-0.02}^{+0.02}$	${0.55}_{-0.02}^{+0.02}$	${9.64}_{-0.06}^{+0.04}$	${10.47}_{-0.02}^{+0.02}$	${0.63}_{-0.02}^{+0.02}$	203	175
	9.875	${9.36}_{-0.08}^{+0.03}$	${10.56}_{-0.02}^{+0.02}$	${0.66}_{-0.02}^{+0.02}$	${9.51}_{-0.05}^{+0.03}$	${10.70}_{-0.02}^{+0.02}$	${0.79}_{-0.02}^{+0.02}$	146	111
	10.125	${9.31}_{-0.07}^{+0.07}$	${10.64}_{-0.03}^{+0.04}$	${0.72}_{-0.03}^{+0.03}$	${9.60}_{-0.05}^{+0.06}$	${10.83}_{-0.03}^{+0.02}$	${0.92}_{-0.02}^{+0.02}$	147	93
	10.375	${9.39}_{-0.04}^{+0.03}$	${10.74}_{-0.06}^{+0.04}$	${0.81}_{-0.05}^{+0.04}$	${9.65}_{-0.05}^{+0.07}$	${11.01}_{-0.05}^{+0.05}$	${1.08}_{-0.05}^{+0.04}$	101	56
	10.625	${9.49}_{-0.02}^{+0.04}$	${10.66}_{-0.08}^{+0.06}$	${0.75}_{-0.07}^{+0.06}$	${9.67}_{-0.08}^{+0.05}$	${11.01}_{-0.07}^{+0.07}$	${1.09}_{-0.08}^{+0.06}$	67	30
	10.875	${9.66}_{-0.03}^{+0.02}$	${10.57}_{-0.07}^{+0.06}$	${0.71}_{-0.05}^{+0.05}$	${9.74}_{-0.07}^{+0.04}$	${10.79}_{-0.12}^{+0.08}$	${0.91}_{-0.10}^{+0.07}$	46	27
	11.125	${9.73}_{-0.04}^{+0.05}$	${10.46}_{-0.16}^{+0.10}$	${0.65}_{-0.11}^{+0.09}$	${9.84}_{-0.06}^{+0.05}$	${10.81}_{-0.09}^{+0.06}$	${0.94}_{-0.06}^{+0.07}$	18	8
$0.75\lt z\lt 1.00$	8.625	${9.25}_{-0.02}^{+0.01}$	${8.57}_{-0.38}^{+0.23}$	$-{0.33}_{-0.04}^{+0.04}$	${9.27}_{-0.02}^{+0.00}$	${8.61}_{-0.26}^{+0.24}$	$-{0.31}_{-0.03}^{+0.03}$	599	583
	8.875	${9.42}_{-0.02}^{+0.01}$	${9.48}_{-0.10}^{+0.08}$	$-{0.02}_{-0.03}^{+0.03}$	${9.43}_{-0.01}^{+0.02}$	${9.52}_{-0.10}^{+0.07}$	${0.00}_{-0.03}^{+0.03}$	477	452
	9.125	${9.59}_{-0.03}^{+0.02}$	${9.93}_{-0.06}^{+0.03}$	${0.27}_{-0.03}^{+0.02}$	${9.62}_{-0.02}^{+0.03}$	${9.95}_{-0.05}^{+0.05}$	${0.30}_{-0.03}^{+0.02}$	376	351
	9.375	${9.63}_{-0.04}^{+0.02}$	${10.27}_{-0.02}^{+0.03}$	${0.48}_{-0.02}^{+0.02}$	${9.66}_{-0.02}^{+0.04}$	${10.31}_{-0.03}^{+0.02}$	${0.52}_{-0.02}^{+0.02}$	298	268
	9.625	${9.79}_{-0.05}^{+0.02}$	${10.48}_{-0.03}^{+0.02}$	${0.68}_{-0.02}^{+0.02}$	${9.83}_{-0.02}^{+0.04}$	${10.54}_{-0.03}^{+0.02}$	${0.73}_{-0.02}^{+0.02}$	202	178
	9.875	${9.66}_{-0.08}^{+0.03}$	${10.75}_{-0.02}^{+0.02}$	${0.86}_{-0.02}^{+0.02}$	${9.78}_{-0.06}^{+0.07}$	${10.82}_{-0.03}^{+0.02}$	${0.94}_{-0.02}^{+0.02}$	162	137
	10.125	${9.53}_{-0.07}^{+0.04}$	${10.91}_{-0.03}^{+0.03}$	${0.99}_{-0.03}^{+0.03}$	${9.74}_{-0.05}^{+0.15}$	${11.07}_{-0.02}^{+0.02}$	${1.15}_{-0.02}^{+0.02}$	128	85
	10.375	${9.52}_{-0.07}^{+0.04}$	${10.89}_{-0.05}^{+0.03}$	${0.96}_{-0.04}^{+0.04}$	${9.74}_{-0.04}^{+0.04}$	${11.09}_{-0.04}^{+0.03}$	${1.17}_{-0.04}^{+0.03}$	107	64
	10.625	${9.72}_{-0.06}^{+0.03}$	${11.02}_{-0.05}^{+0.04}$	${1.11}_{-0.04}^{+0.03}$	${9.83}_{-0.07}^{+0.10}$	${11.28}_{-0.02}^{+0.02}$	${1.35}_{-0.02}^{+0.03}$	77	44
	10.875	${9.66}_{-0.02}^{+0.05}$	${10.94}_{-0.06}^{+0.06}$	${1.03}_{-0.06}^{+0.05}$	${9.93}_{-0.15}^{+0.07}$	${11.20}_{-0.09}^{+0.09}$	${1.29}_{-0.09}^{+0.08}$	42	22
	11.125	${9.89}_{-0.09}^{+0.02}$	${10.92}_{-0.08}^{+0.07}$	${1.04}_{-0.06}^{+0.06}$	${9.97}_{-0.06}^{+0.06}$	${11.19}_{-0.14}^{+0.08}$	${1.28}_{-0.11}^{+0.08}$	23	12
$1.00\lt z\lt 1.25$	8.625	${9.40}_{-0.02}^{+0.01}$	${8.98}_{-0.62}^{+0.29}$	$-{0.15}_{-0.05}^{+0.04}$	${9.40}_{-0.01}^{+0.01}$	${9.00}_{-0.46}^{+0.17}$	$-{0.15}_{-0.05}^{+0.04}$	371	368
	8.875	${9.55}_{-0.02}^{+0.01}$	${9.33}_{-0.20}^{+0.11}$	${0.04}_{-0.04}^{+0.03}$	${9.55}_{-0.02}^{+0.01}$	${9.32}_{-0.12}^{+0.16}$	${0.04}_{-0.04}^{+0.04}$	379	376
	9.125	${9.76}_{-0.02}^{+0.01}$	${10.04}_{-0.05}^{+0.08}$	${0.41}_{-0.04}^{+0.04}$	${9.77}_{-0.01}^{+0.01}$	${10.05}_{-0.08}^{+0.07}$	${0.42}_{-0.05}^{+0.03}$	284	282
	9.375	${9.84}_{-0.03}^{+0.01}$	${10.36}_{-0.03}^{+0.05}$	${0.62}_{-0.03}^{+0.02}$	${9.85}_{-0.02}^{+0.02}$	${10.37}_{-0.03}^{+0.04}$	${0.63}_{-0.03}^{+0.03}$	248	239
	9.625	${9.93}_{-0.05}^{+0.05}$	${10.57}_{-0.04}^{+0.03}$	${0.78}_{-0.03}^{+0.03}$	${9.99}_{-0.06}^{+0.03}$	${10.61}_{-0.04}^{+0.04}$	${0.83}_{-0.03}^{+0.03}$	157	146
	9.875	${9.78}_{-0.03}^{+0.05}$	${10.83}_{-0.03}^{+0.02}$	${0.95}_{-0.02}^{+0.02}$	${9.85}_{-0.06}^{+0.06}$	${10.88}_{-0.02}^{+0.02}$	${1.00}_{-0.02}^{+0.02}$	113	99
	10.125	${9.72}_{-0.05}^{+0.07}$	${11.03}_{-0.02}^{+0.03}$	${1.11}_{-0.02}^{+0.02}$	${9.85}_{-0.05}^{+0.05}$	${11.14}_{-0.02}^{+0.02}$	${1.22}_{-0.02}^{+0.02}$	123	95
	10.375	${9.60}_{-0.05}^{+0.05}$	${11.10}_{-0.03}^{+0.03}$	${1.17}_{-0.03}^{+0.03}$	${9.85}_{-0.07}^{+0.04}$	${11.31}_{-0.04}^{+0.03}$	${1.38}_{-0.04}^{+0.03}$	92	57
	10.625	${9.74}_{-0.02}^{+0.02}$	${11.26}_{-0.03}^{+0.03}$	${1.32}_{-0.03}^{+0.03}$	${9.82}_{-0.08}^{+0.21}$	${11.39}_{-0.04}^{+0.03}$	${1.46}_{-0.03}^{+0.03}$	83	59
	10.875	${9.86}_{-0.04}^{+0.08}$	${11.28}_{-0.06}^{+0.06}$	${1.35}_{-0.06}^{+0.06}$	${9.96}_{-0.01}^{+0.07}$	${11.42}_{-0.08}^{+0.08}$	${1.49}_{-0.08}^{+0.07}$	36	22
	11.125	${9.89}_{-0.01}^{+0.00}$	${11.61}_{-0.09}^{+0.05}$	${1.55}_{-0.07}^{+0.06}$	${9.93}_{-0.06}^{+0.09}$	${11.76}_{-0.05}^{+0.06}$	${1.71}_{-0.05}^{+0.06}$	12	8
$1.25\lt z\lt 1.50$	8.625	${9.52}_{-0.01}^{+0.02}$	${8.18}_{-0.93}^{+0.95}$	$-{0.09}_{-0.06}^{+0.05}$	${9.52}_{-0.01}^{+0.02}$	${8.25}_{-0.63}^{+0.71}$	$-{0.09}_{-0.07}^{+0.06}$	287	286
	8.875	${9.65}_{-0.01}^{+0.01}$	${9.41}_{-0.17}^{+0.13}$	${0.13}_{-0.04}^{+0.03}$	${9.65}_{-0.01}^{+0.01}$	${9.41}_{-0.19}^{+0.13}$	${0.13}_{-0.04}^{+0.03}$	429	426
	9.125	${9.80}_{-0.02}^{+0.01}$	${10.07}_{-0.13}^{+0.07}$	${0.45}_{-0.05}^{+0.04}$	${9.81}_{-0.01}^{+0.01}$	${10.09}_{-0.11}^{+0.09}$	${0.46}_{-0.05}^{+0.05}$	354	348
	9.375	${9.94}_{-0.03}^{+0.03}$	${10.37}_{-0.06}^{+0.05}$	${0.67}_{-0.03}^{+0.03}$	${9.94}_{-0.03}^{+0.03}$	${10.37}_{-0.06}^{+0.05}$	${0.67}_{-0.03}^{+0.03}$	269	265
	9.625	${9.98}_{-0.02}^{+0.02}$	${10.74}_{-0.03}^{+0.03}$	${0.92}_{-0.02}^{+0.02}$	${9.99}_{-0.02}^{+0.04}$	${10.76}_{-0.04}^{+0.02}$	${0.93}_{-0.03}^{+0.02}$	205	197
	9.875	${10.01}_{-0.04}^{+0.04}$	${10.88}_{-0.04}^{+0.03}$	${1.03}_{-0.03}^{+0.02}$	${10.04}_{-0.02}^{+0.04}$	${10.93}_{-0.03}^{+0.02}$	${1.07}_{-0.03}^{+0.02}$	151	141
	10.125	${9.90}_{-0.05}^{+0.05}$	${11.17}_{-0.03}^{+0.03}$	${1.26}_{-0.03}^{+0.03}$	${10.0}_{-0.06}^{+0.04}$	${11.23}_{-0.03}^{+0.03}$	${1.32}_{-0.03}^{+0.03}$	148	128
	10.375	${9.70}_{-0.02}^{+0.07}$	${11.27}_{-0.03}^{+0.04}$	${1.33}_{-0.04}^{+0.04}$	${9.83}_{-0.05}^{+0.06}$	${11.40}_{-0.04}^{+0.03}$	${1.46}_{-0.04}^{+0.03}$	121	90
	10.625	${9.82}_{-0.06}^{+0.06}$	${11.30}_{-0.06}^{+0.06}$	${1.37}_{-0.06}^{+0.05}$	${9.96}_{-0.06}^{+0.02}$	${11.56}_{-0.05}^{+0.04}$	${1.63}_{-0.05}^{+0.04}$	77	45
	10.875	${9.98}_{-0.05}^{+0.03}$	${11.54}_{-0.11}^{+0.09}$	${1.61}_{-0.11}^{+0.09}$	${10.03}_{-0.03}^{+0.05}$	${11.78}_{-0.06}^{+0.07}$	${1.83}_{-0.06}^{+0.07}$	56	37
	11.125	${10.0}_{-0.01}^{+0.04}$	${11.59}_{-0.09}^{+0.07}$	${1.65}_{-0.09}^{+0.07}$	${9.97}_{-0.04}^{+0.24}$	${11.88}_{-0.05}^{+0.05}$	${1.93}_{-0.05}^{+0.05}$	14	8
$1.50\lt z\lt 2.00$	8.875	${9.76}_{-0.01}^{+0.01}$	${8.51}_{-0.55}^{+0.71}$	${0.11}_{-0.06}^{+0.06}$	${9.76}_{-0.01}^{+0.01}$	${8.53}_{-0.54}^{+0.68}$	${0.11}_{-0.08}^{+0.06}$	645	644
	9.125	${9.86}_{-0.01}^{+0.01}$	${10.22}_{-0.09}^{+0.10}$	${0.55}_{-0.05}^{+0.05}$	${9.86}_{-0.01}^{+0.02}$	${10.23}_{-0.09}^{+0.09}$	${0.56}_{-0.06}^{+0.05}$	752	747
	9.375	${10.03}_{-0.01}^{+0.01}$	${10.47}_{-0.05}^{+0.04}$	${0.76}_{-0.03}^{+0.03}$	${10.03}_{-0.01}^{+0.02}$	${10.47}_{-0.05}^{+0.04}$	${0.76}_{-0.03}^{+0.03}$	616	607
	9.625	${10.10}_{-0.03}^{+0.01}$	${10.87}_{-0.02}^{+0.03}$	${1.04}_{-0.02}^{+0.02}$	${10.11}_{-0.01}^{+0.02}$	${10.90}_{-0.02}^{+0.03}$	${1.07}_{-0.02}^{+0.02}$	438	416
	9.875	${10.06}_{-0.02}^{+0.02}$	${11.04}_{-0.03}^{+0.03}$	${1.17}_{-0.03}^{+0.03}$	${10.12}_{-0.03}^{+0.04}$	${11.10}_{-0.02}^{+0.02}$	${1.23}_{-0.02}^{+0.02}$	316	287
	10.125	${9.97}_{-0.02}^{+0.05}$	${11.30}_{-0.02}^{+0.02}$	${1.38}_{-0.02}^{+0.02}$	${10.10}_{-0.05}^{+0.04}$	${11.38}_{-0.02}^{+0.02}$	${1.47}_{-0.02}^{+0.02}$	239	199
	10.375	${9.74}_{-0.05}^{+0.04}$	${11.43}_{-0.04}^{+0.02}$	${1.48}_{-0.03}^{+0.03}$	${9.88}_{-0.03}^{+0.07}$	${11.56}_{-0.02}^{+0.03}$	${1.62}_{-0.02}^{+0.02}$	179	133
	10.625	${9.84}_{-0.04}^{+0.02}$	${11.51}_{-0.03}^{+0.03}$	${1.57}_{-0.03}^{+0.03}$	${10.00}_{-0.09}^{+0.02}$	${11.75}_{-0.03}^{+0.02}$	${1.80}_{-0.03}^{+0.03}$	142	94
	10.875	${10.06}_{-0.06}^{+0.03}$	${11.61}_{-0.05}^{+0.04}$	${1.67}_{-0.05}^{+0.04}$	${10.07}_{-0.06}^{+0.02}$	${11.80}_{-0.05}^{+0.04}$	${1.86}_{-0.04}^{+0.04}$	119	83
	11.125	${10.15}_{-0.06}^{+0.05}$	${11.63}_{-0.08}^{+0.07}$	${1.70}_{-0.07}^{+0.07}$	${10.37}_{-0.23}^{+0.02}$	${11.91}_{-0.06}^{+0.06}$	${1.98}_{-0.06}^{+0.06}$	30	19
$2.00\lt z\lt 2.50$	9.125	${10.04}_{-0.01}^{+0.01}$	${9.76}_{-0.46}^{+0.23}$	${0.51}_{-0.06}^{+0.06}$	${10.04}_{-0.01}^{+0.01}$	${9.76}_{-0.50}^{+0.29}$	${0.51}_{-0.06}^{+0.07}$	475	475
	9.375	${10.14}_{-0.01}^{+0.01}$	${10.42}_{-0.11}^{+0.09}$	${0.79}_{-0.05}^{+0.05}$	${10.14}_{-0.01}^{+0.01}$	${10.41}_{-0.15}^{+0.11}$	${0.79}_{-0.07}^{+0.06}$	514	513
	9.625	${10.27}_{-0.01}^{+0.02}$	${10.79}_{-0.07}^{+0.05}$	${1.05}_{-0.04}^{+0.03}$	${10.27}_{-0.01}^{+0.02}$	${10.79}_{-0.05}^{+0.04}$	${1.05}_{-0.03}^{+0.03}$	403	400
	9.875	${10.36}_{-0.02}^{+0.02}$	${11.19}_{-0.04}^{+0.03}$	${1.35}_{-0.03}^{+0.03}$	${10.38}_{-0.02}^{+0.04}$	${11.20}_{-0.04}^{+0.04}$	${1.36}_{-0.03}^{+0.03}$	250	236
	10.125	${10.28}_{-0.06}^{+0.08}$	${11.28}_{-0.03}^{+0.03}$	${1.40}_{-0.04}^{+0.03}$	${10.37}_{-0.02}^{+0.05}$	${11.35}_{-0.03}^{+0.03}$	${1.47}_{-0.03}^{+0.02}$	197	173
	10.375	${10.17}_{-0.07}^{+0.06}$	${11.54}_{-0.05}^{+0.05}$	${1.61}_{-0.05}^{+0.05}$	${10.25}_{-0.04}^{+0.05}$	${11.64}_{-0.05}^{+0.04}$	${1.71}_{-0.04}^{+0.04}$	124	103
	10.625	${9.95}_{-0.06}^{+0.04}$	${11.58}_{-0.06}^{+0.04}$	${1.64}_{-0.05}^{+0.04}$	${9.95}_{-0.07}^{+0.09}$	${11.72}_{-0.04}^{+0.04}$	${1.78}_{-0.04}^{+0.04}$	107	81
	10.875	${10.12}_{-0.05}^{+0.05}$	${11.78}_{-0.03}^{+0.04}$	${1.83}_{-0.03}^{+0.03}$	${10.13}_{-0.06}^{+0.05}$	${11.94}_{-0.03}^{+0.03}$	${1.99}_{-0.03}^{+0.03}$	79	55
	11.125	${10.16}_{-0.06}^{+0.03}$	${11.93}_{-0.04}^{+0.07}$	${1.98}_{-0.05}^{+0.06}$	${10.12}_{-0.04}^{+0.06}$	${12.04}_{-0.04}^{+0.04}$	${2.09}_{-0.04}^{+0.04}$	36	29
$2.50\lt z\lt 3.00$	9.125	${10.23}_{-0.02}^{+0.01}$	${9.35}_{-0.21}^{+0.41}$	${0.57}_{-0.06}^{+0.07}$	${10.23}_{-0.01}^{+0.02}$	${9.35}_{-0.73}^{+0.55}$	${0.58}_{-0.06}^{+0.06}$	242	240
	9.375	${10.29}_{-0.01}^{+0.01}$	${9.35}_{-0.59}^{+0.41}$	${0.69}_{-0.03}^{+0.04}$	${10.29}_{-0.01}^{+0.01}$	${9.36}_{-0.56}^{+0.41}$	${0.69}_{-0.06}^{+0.05}$	327	323
	9.625	${10.41}_{-0.02}^{+0.02}$	${10.99}_{-0.12}^{+0.09}$	${1.22}_{-0.08}^{+0.05}$	${10.41}_{-0.02}^{+0.02}$	${10.98}_{-0.14}^{+0.07}$	${1.22}_{-0.08}^{+0.06}$	269	265
	9.875	${10.52}_{-0.06}^{+0.02}$	${11.27}_{-0.10}^{+0.08}$	${1.45}_{-0.07}^{+0.06}$	${10.52}_{-0.03}^{+0.04}$	${11.27}_{-0.07}^{+0.07}$	${1.45}_{-0.05}^{+0.06}$	151	150
	10.125	${10.39}_{-0.03}^{+0.04}$	${11.67}_{-0.06}^{+0.05}$	${1.76}_{-0.06}^{+0.05}$	${10.42}_{-0.04}^{+0.04}$	${11.68}_{-0.05}^{+0.04}$	${1.76}_{-0.04}^{+0.04}$	104	97
	10.375	${10.14}_{-0.06}^{+0.03}$	${11.66}_{-0.12}^{+0.07}$	${1.73}_{-0.11}^{+0.07}$	${10.29}_{-0.11}^{+0.11}$	${11.79}_{-0.10}^{+0.05}$	${1.86}_{-0.08}^{+0.06}$	86	67
	10.625	${9.92}_{-0.04}^{+0.16}$	${11.86}_{-0.05}^{+0.05}$	${1.91}_{-0.05}^{+0.05}$	${10.03}_{-0.15}^{+0.05}$	${11.95}_{-0.04}^{+0.04}$	${2.00}_{-0.04}^{+0.04}$	66	56
	10.875	${9.95}_{-0.05}^{+0.18}$	${11.98}_{-0.06}^{+0.06}$	${2.03}_{-0.06}^{+0.06}$	${9.89}_{-0.03}^{+0.06}$	${12.09}_{-0.06}^{+0.05}$	${2.13}_{-0.05}^{+0.05}$	29	24
	11.125	${10.23}_{-0.08}^{+0.22}$	${12.32}_{-0.10}^{+0.08}$	${2.37}_{-0.10}^{+0.08}$	${10.32}_{-0.14}^{+0.13}$	${12.35}_{-0.09}^{+0.08}$	${2.40}_{-0.09}^{+0.07}$	15	14
$3.00\lt z\lt 4.00$	9.625	${10.42}_{-0.01}^{+0.01}$	${10.64}_{-0.25}^{+0.17}$	${1.04}_{-0.10}^{+0.08}$	${10.43}_{-0.01}^{+0.01}$	${10.64}_{-0.26}^{+0.11}$	${1.05}_{-0.09}^{+0.07}$	256	253
	9.875	${10.53}_{-0.01}^{+0.04}$	${11.48}_{-0.09}^{+0.08}$	${1.61}_{-0.07}^{+0.06}$	${10.54}_{-0.03}^{+0.03}$	${11.47}_{-0.09}^{+0.06}$	${1.61}_{-0.07}^{+0.06}$	166	161
	10.125	${10.55}_{-0.05}^{+0.03}$	${11.63}_{-0.06}^{+0.07}$	${1.74}_{-0.05}^{+0.06}$	${10.58}_{-0.02}^{+0.04}$	${11.67}_{-0.08}^{+0.06}$	${1.78}_{-0.06}^{+0.06}$	122	111
	10.375	${10.41}_{-0.07}^{+0.08}$	${11.83}_{-0.07}^{+0.06}$	${1.90}_{-0.06}^{+0.06}$	${10.49}_{-0.10}^{+0.08}$	${11.86}_{-0.06}^{+0.05}$	${1.94}_{-0.06}^{+0.05}$	65	58
	10.625	${10.34}_{-0.16}^{+0.21}$	${12.06}_{-0.12}^{+0.12}$	${2.11}_{-0.12}^{+0.11}$	${10.54}_{-0.30}^{+0.10}$	${12.14}_{-0.11}^{+0.11}$	${2.20}_{-0.11}^{+0.09}$	34	28
	10.875	${10.14}_{-0.10}^{+0.17}$	${12.20}_{-0.07}^{+0.07}$	${2.24}_{-0.07}^{+0.07}$	${10.07}_{-0.06}^{+0.18}$	${12.26}_{-0.08}^{+0.06}$	${2.31}_{-0.07}^{+0.06}$	30	24
	11.125	${10.22}_{-0.13}^{+0.21}$	${12.22}_{-0.13}^{+0.07}$	${2.27}_{-0.12}^{+0.07}$	${10.22}_{-0.12}^{+0.34}$	${12.33}_{-0.09}^{+0.09}$	${2.37}_{-0.09}^{+0.08}$	14	12
	11.375	${10.16}_{-0.07}^{+0.03}$	${12.43}_{-0.05}^{+0.04}$	${2.47}_{-0.05}^{+0.04}$	${10.16}_{-0.07}^{+0.03}$	${12.48}_{-0.07}^{+0.05}$	${2.52}_{-0.07}^{+0.05}$	7	7

Note. A downloadable ascii version of this table will be hosted at http://zfourge.tamu.edu/

A machine-readable version of the table is available.

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We test the W08 template against the present data set using a sample of 1050 well-detected galaxies (S/N > 3 in all FIR bands). For each galaxy we fit the W08 template separately to the MIPS (24 μm) and the PACS (100+160μm) photometry. In Figure 2 we show a comparison between the MIPS-only and PACS-only cases. Although we observe a general scatter of ∼0.2 dex we find an overall consistency with no dominant systematic trends. Even when subsampling in redshift and stellar mass the mean offset is nearly always within the scatter. We do note, however, the presence of a weak systematic trend with redshift in the middle panel of Figure 2 which is likely caused by PAH features shifting through the MIPS 24 μm passband. Nevertheless, the consistency between the MIPS-only and the PACS-only estimates of L_IR suggests that the W08 template effectively describes the average IR SED of galaxies. It further suggests that reasonably reliable SFR estimates can be made with just a single IR band. In this section we have focused only on galaxies that are individually detected in the FIR bands. In Section 3.1 we present evidence that the systematic errors become slightly larger for our fainter stacked samples.

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In Figure 4 we compare our SFR–M_* relations to recent results from the literature. The chosen SFR–M_* relations come from Rodighiero et al. (2010), Karim et al. (2011), Whitaker et al. (2014), Tasca et al. (2015), Speagle et al. (2014), and Schreiber et al. (2015). All works have been scaled to a Chabrier (2003) IMF for consistency. Overall there is good agreement among all of the measurements presented in Figure 4. For the full sample of galaxies (star-forming plus quiescent), the median inter-survey discrepancy at fixed stellar mass is 0.2, 0.17, 0.15, 0.16, 0.21, and 0.13 dex in the redshift bins (0.5 < z < 1), (1 < z < 1.5), (1.5 < z < 2), (2 < z < 2.5), (2.5 < z < 3), and (3 < z < 4) respectively. For the star-forming sample these median discrepancies are 0.24, 0.17, 0.32, 0.17, 0.30, and 0.17 dex respectively. These differences are consistent with the inter-publication scatter found by Speagle et al. (2014) which draws on a larger sample of published SFR–M_* relations. Furthermore, differences of this order are comparable to variations in stellar mass estimates produced by SED fitting assuming different stellar population synthesis models (Conroy 2013).

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It is worth mentioning cosmic variance as a potential source for differences between SFR–M_* relations from different surveys. Whitaker et al. (2014) investigated this in detail by comparing the SFR–M_* relation as measured from each of the five CANDELS/3D-HST fields individually (see Appendix B of their paper). The field-to-field variation they find is comparable to the inter-survey variation that we find. Given that ZFOURGE covers less area (≈400 arcmin² versus ≈900 arcmin² for 3D-HST) we acknowledge that cosmic variance could explain the differences mentioned in the previous paragraph. Furthermore, Tomczak et al. (2014) quantify cosmic variance among the three ZFOURGE fields in their measurement of the SMF, finding it to range from 8% to 25% over roughly the same mass and redshift ranges as used in this work.

We do note, however, that for the full sample in Figure 4 at >10^10.5 M_⊙ and 1.5 < z < 2.5 our SFRs are systematically below the others. This offset goes away when we recalculate Ψ_UV+IR from our sample excluding the Herschel PACS 100 and 160 μm photometry but keeping the Spitzer MIPS 24 μm photometry. In Figure 5 we further investigate the impact that the Herschel PACS imaging has on our stacked SFRs. For this comparison we only consider star-forming galaxies. We first perform an internal comparison where we calculate Ψ_UV+IR both including and excluding the Herschel stacks. Interestingly we find a systematic trend wherein the SFRs of higher mass galaxies tend to be overestimated when relying on the MIPS 24 μm stacks alone for the IR contribution. This result contrasts what was found for galaxies that are individually detected in the far-IR images which show no strong evidence of a systematic trend (see Figure 2). This suggests that the fainter, non-detected galaxies have SEDs that are not consistent with their more luminous counterparts (Muzzin et al. 2010; Wuyts et al. 2011). Nevertheless, this discrepancy is not dominant, but comparable to the level of scatter between surveys noted earlier (≲0.2 dex).

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We look into this difference in more detail by making use of the library of IR templates from Chary & Elbaz (2001, hereafter CE01). For each mass-redshift bin we find the individual best-fit CE01 template to the stacked 24–160 μm photometry. Comparing the IR luminosities derived from these best-fit templates to those derived from the W08 template we find a small scatter of ≈0.02 dex with a similar offset in log(L_IR/L_⊙), and a hint that this offset increases to ≈0.1 dex at z > 2. However, we do not correct for this effect because these discrepancies are poorly constrained given that the FIR data do not probe the peak of the dust emission. We also note that this difference is less than the intrinsic uncertainty in individual star formation rate calibrations (e.g., Bell 2003; Bell et al. 2005).

The differences between the 24 μm derived star formation rates for 3D-HST and ZFOURGE in Figure 5 are particularly interesting. These surveys cover the same fields (although ZFOURGE only covers half the area), rely on much of the same public imaging in the optical and IR bands, have had photometry performed using similar methods, and use the same conversions to calculate the star formation rates from the 2800 Å and 24 μm flux. Thus the systematic differences in star formation rates are indicative of the minimal differences that can be expected in inter-survey comparisons.

In Figure 6, we compare our measured SFR–M_* relations for the full sample of galaxies with results from the recent Illustris hydrodynamic simulation (Nelson et al. 2015) and the Munich semi-analytic galaxy formation model (Henriques et al. 2014). Gray lines correspond to the mean SFR in bins of stellar mass for each redshift interval indicated. Except at log(M_*/M_⊙) > 10.5, both simulations are consistently in good agreement with each other (for further discussion of this point see Weinmann et al. 2012). In general, the simulations reproduce the roughly constant slope at M_* ≲ 10¹⁰ M_⊙ albeit with an offset. This offset ranges between 0.17 and 0.45 dex at fixed stellar mass and decreases with redshift (Sparre et al. 2015). At higher masses, however, Illustris and the Munich galaxy formation model tend to under-predict and over-predict the strength of the turnover at z < 2 respectively.

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In Figure 7 we parameterize the SFR–M_* relation as a function of redshift. For this we adopt the same parameterization as Lee et al. (2015):

$$\begin{eqnarray}&&\mathrm{log}({\rm{\Psi }})={s}_{0}-\mathrm{log}[1+{(\displaystyle \frac{{M}_{*}}{{M}_{0}})}^{-\gamma }]\end{eqnarray} \tag{ 2 }$$

where s₀ and M₀ are in units of log(M_⊙/year) and M_⊙ respectively. This function behaves as a power law of slope γ at low masses which asymptotically approaches a peak value s₀ above a transitional stellar mass M₀. Originally this parameterization was defined for the SFR–M_* relation of star-forming galaxies, though we find it works similarly well for the SFR–M_* relation of all galaxies (star-forming plus quiescent) at the redshifts and stellar masses considered for this study. The righthand panels of Figure 7 show the best-fit parameters versus redshift. We consider two cases for fitting: "free γ" and "fixed γ." In the "free γ" case, we allow all three parameters to vary independently for each redshift bin. Noticing that γ does not show strong evidence for evolution, we perform the "fixed γ" case by refitting with γ fixed to its mean value from the "free γ" fits. We then parameterize the evolution of s₀ and M₀ with second-order polynomials.

$$\begin{eqnarray}\mathrm{All}\ \mathrm{Galaxies} & : & \\ {s}_{0}\ & = & \ \ 0.195+1.157z-0.143{z}^{2}\\ \mathrm{log}({M}_{0})\ & = & \ \ 9.244+0.753z-0.090{z}^{2}\\ \gamma \ & = & \ \ 1.118.\end{eqnarray} \tag{ 3 }$$

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Evolution in the transition mass is a new discovery, first reported quantitatively at 0.25 < z < 1.3 by Lee et al. (2015) from a study of the COSMOS 2 deg² field. Recent results from Gavazzi et al. (2015) show that this evolution extends to z ∼ 2.5 and that extrapolating to to z = 0 coincides with a break observed in the local SFR–M_* relation. Results from the VUDS spectroscopic survey have also found a redshift dependence for the turnover mass (Tasca et al. 2015). However, because we include all galaxies in the analysis presented here the observed evolution of M₀ may be a consequence of the increasing population of massive quenched galaxies at low-z. Therefore, we repeat this analysis for a sample of actively star-forming galaxies selected based on rest-frame (U − V) and (V − J) colors (see Section 2.4). Evolution in M₀ is still apparent; results are shown in Figure 8. In Figure 9 we compare our measured values of the turnover mass to those of Lee et al. (2015), Gavazzi et al (2015), and Tasca et al. (2015).

$$\begin{eqnarray}\mathrm{Star}-\mathrm{Forming} & : & \\ {s}_{0}\ & = & \ \ 0.448+1.220z-0.174{z}^{2}\\ \mathrm{log}({M}_{0})\ & = & \ \ 9.458+0.865z-0.132{z}^{2}\\ \gamma \ & = & \ \ 1.091.\end{eqnarray} \tag{ 4 }$$

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We remind the reader that these parameterizations may not extrapolate well outside of the redshift and/or stellar mass ranges used here. Nevertheless, our low-mass slope of γ ∼ 1 is consistent with the relative constancy of the low-mass slope α ∼ −1.5 of the SMF (Tomczak et al. 2014); if γ deviated significantly from unity then α would be expected to evolve strongly with redshift (Peng et al. 2010; Weinmann et al. 2012; Leja et al. 2015).

Figure 8 shows that (s₀, M₀, γ) follow similar trends with redshift between the star-forming and total samples indicating that the presence and evolution of the turnover mass is intrinsic to the star-forming population and is not simply due to a growing quiescent population diluting the star formation rates at the high mass end. We also point out that the apparent similarity in the fitted parameters does not suggest a redshift-independent quiescent fraction; in fact, the difference between the SFR–M* relations between the star-forming and the total sample are roughly consistent with the evolution of the quiescent fraction as derived from the Tomczak et al. (2014) mass functions, and assuming negligible star formation in the quiescent galaxies.

The growth of galaxies as predicted from the SFR–M_* relation can be compared directly to the evolution of the SMF. Leja et al. (2015) performed such an analysis using SMFs from Tomczak et al. (2014) and SFR–M_* relations from Whitaker et al. (2012) finding that the inferred growth from star formation greatly over-predicts the observed number densities of galaxies, even on short cosmic timescales (<1 Gyr). Those authors suggest that the SFR–M_* relation must have a steeper slope (α ≳ 0.9) at masses below 10^10.5 M_⊙ at z < 2.5, which is consistent with measurements from this work and recent literature (Whitaker et al. 2014; Lee et al. 2015; Schreiber et al. 2015).

Thus, we perform the same comparison using our updated SFR–M_* relation. At each stellar mass for a given SMF at a given redshift, SFRs are calculated from the parameterized SFR–M_* relation for all galaxies (Equations (2) and (3)). In times steps of 80 Myr each mass bin is shifted by the amount of star formation added to that bin. SFRs are recalculated at each new time step. Mass loss due to stellar evolution is accounted for according to Equation (16) of Moster et al. (2013). Using this technique, we evolve the observed SMF in each redshift bin forward and compare it to the observed SMF in the next redshift bin; results are shown in Figure 10.

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In general, we typically find reasonable agreement at intermediate stellar masses (10^10.5 < M_*/M_⊙ < 10¹¹). At lower masses, however, we find a consistent systematic offset in number density rising to ≈0.2–0.3 dex. It is important to note that this method does not incorporate the effect of galaxy–galaxy mergers whereas the observed evolution galaxy SMF necessarily does. Therefore, the disparity between these two curves inherently includes a signature of merging; in fact, Drory & Alvarez (2008) use this difference to constrain galaxy growth rates due to merging. Mergers will help to alleviate the discrepancy we observe if these low-mass galaxies are merging with more massive galaxies, thereby reducing their number density. However this would require between 25% and 65% of these galaxies to merge with a more massive galaxy per Gyr, which substantially exceeds current estimates of galaxy merger rates (e.g., Lotz et al. 2011; Williams et al. 2011; Leja et al. 2015). This disagreement thus implies that the SFRs are overestimated and/or the growth of the Tomczak et al. (2014) SMF is too slow. Similar issues were previously discussed by Weinmann et al. (2012) and Leja et al. (2015), but in contrast to the drastic discrepancies between star formation rates and stellar masses reported by those authors, here we show that using new data sets greatly reduces—but does not eliminate—the discrepancies.

The evolution of the SFR–M_* relation can be used to infer typical SFHs and stellar mass growth histories for individual galaxies. However, as seen in the previous subsection, there is tension between the growth of the galaxy population as inferred from the SFR–M_* relation when comparing to the observed SMF. Here we explore two different approaches for empirically deriving galaxy mass growth histories: (1) integrating differential star formation histories extracted from the SFR–M_* relation and (2) identifying descendents of high-z galaxies based on NDS samples (Figure 11). For the former, we start with the four sets of initial conditions (z₀, M₀, Ψ₀) indicated by the star symbols, where Ψ₀ is determined by our SFR parameterizations (Equations (2) and (3)). Stellar mass is then incrementally added assuming constant star formation over small time intervals of ≈80 Myr. At the end of each time step Ψ adjusted according to the SFR–M_* parameterization Ψ(z, M_*). At lower redshifts we interpolate between our lowest redshift SFR–M_* relation and Equation (13) of Salim et al. (2007). Mass loss due to stellar evolution is accounted for according to Equation (16) of Moster et al. (2013).

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In order to get a rough estimate of the scatter in these SFHs we run 1000 Monte Carlo simulations, resampling Ψ from a log-normal PDF with a mean value given by Ψ(z, M_*) and ±0.3 dex scatter. The approximate range of mass growth for each galaxy sample is calculated from the 16th and 84th percentiles of the distribution.

A number of studies have used this technique to estimate galaxy growth histories (e.g., Renzini 2009; Peng et al. 2010; Leitner 2012). An important point that should be kept in mind is that in this section we use the SFR–M_* relation for all galaxies—not just actively star-forming ones—as is appropriate for a comparison to NDS samples. Some other studies investigate the growth histories of galaxies that remain star-forming without ever quenching (Renzini 2009; Leitner 2012).

Also shown in the middle panel of Figure 11 are mass growth profiles predicted from the NDS approach. Using the same initial conditions shown by the star symbols, we calculate the corresponding cumulative co-moving number density from the SMFs of Tomczak et al. (2014) as parameterized by Leja et al. (2015). Note, this parameterization is limited to z ≤ 2.25, beyond which we interpolate between it and the best-fit Schechter function to the SMF at 2.5 < z < 3.

Using abundance matching with a dark matter simulation, whereby dark matter halos are assigned stellar masses from observations in a rank ordering fashion, Behroozi et al. (2013) have studied the number density evolution of galaxies. These authors demonstrate that galaxy descendants do not evolve in the same way as their progenitors, mainly due to scatter in dark matter accretion rates of halos. Thus, Behroozi et al. (2013) have provided a numerical recipe for estimating the number density evolution of galaxies, which we use to generate predictions for the number density evolution from the initial conditions given in Figure 11. These predictions include the median estimated number density as well as the 68th percentile range. The hatched regions in Figure 11 show these 68th percentile ranges converted to stellar masses by mapping number densities to the observed SMF.

On average we find that these two approaches agree within their combined 1σ confidence intervals. However there is a common systematic difference wherein the differential SFHs produce a steeper mass growth rate for the same progenitor galaxy at early times which provides us with a different view of the discrepancy shown in Figure 10. This disparity is illustrated in the rightmost panel of Figure 11 which shows the difference in the mass growth rates of both techniques from the middle panel. The differential SFHs build stellar mass more quickly at early times, but then slow down and are eventually overtaken by the NDS growth rates. We tested a wide range of initial conditions spanning 0.8 ≤ z₀ ≤ 2.75 and 8.8 ≤ log(M₀/M_⊙) ≤ 10.5, always finding this systematic trend. Papovich et al. (2015) find similar results in their analysis of the progenitors of galaxies with present-day masses of the Milky Way and M31 galaxies. Finally, we have repeated this comparison using the SFR–M_* parameterizations provided by Whitaker et al. (2014) and Schreiber et al. (2015) and in both cases we find similar disagreements. This suggests that the discrepancies between the star formation rates and mass evolution that were previously reported by Leja et al. (2015) have not been completely resolved using the more up-to-date data sets.