CONFIRMING THE EXISTENCE OF A QUIESCENT GALAXY POPULATION OUT TO z = 3: A STACKING ANALYSIS OF MID-, FAR-INFRARED, AND RADIO DATA

Studies suggest that when the universe was only 4 Gyr old (z ∼ 1.5), about half of the most massive (${M}_{\star }\geqslant {10}^{11}\;{M}_{\odot }$) galaxies already had evolved stellar populations and unobscured SFRs of only a few M_⊙ yr⁻¹ (e.g., Ilbert et al. 2013; Muzzin et al. 2013, and references therein). They are thought to have undergone a rapid build-up of stellar mass followed by an effective phase of star formation quenching, possibly via AGN feedback (e.g., Bower et al. 2006; Croton et al. 2006). However, these claims rest on the assumption of a universal dust attenuation law (Calzetti et al. 2000), which may in fact vary with galaxy spectral types (Kriek & Conroy 2013). If a significant amount of heated dust is present in these galaxies, however, it would imply that the SFRs inferred from the rest-frame ultraviolet (UV) are severely underestimated, and that their red colors are due to strong dust reddening, rather than evolved stellar populations. Direct FIR measurement of the cold dust is essential to unambiguously assess the level of obscured SF. A recent Herschel¹³ stacking analysis by Viero et al. (2013) found that massive QGs at z > 2 have IR luminosities comparable to local starbursts (i.e., ultra-luminous IR galaxies or ULIRGs, ${L}_{{\rm{IR}}}\geqslant {10}^{12}\;{L}_{\odot }$), inconsistent with the quiescence inferred from the UV continua (e.g., Ilbert et al. 2013) as well as their low 24 μm stacked flux densities (Fumagalli et al. 2014; Utomo et al. 2014). If QGs harbor significant dust-obscured SF, it would bring into question the usefulness of the rest-frame color selections commonly used for identifying QGs (e.g., Williams et al. 2009; Ilbert et al. 2013), therefore challenging the need for powerful quenching mechanisms.

Here, we analyze a sample of ∼14,200 QG candidates with ${M}_{\star }\;=\;{10}^{9.8{\rm{\mbox{--}}}12.2}\;{M}_{\odot }$ out to z = 3, selected over an area of 1.48 deg² in the COSMOS field (Scoville et al. 2007). The large sample size enables us to use stacking to enhance the sensitivity to study otherwise undetected sources. A comprehensive stacking analysis across MIR, FIR, and radio wavelengths using the same sample is necessary for interpreting the inconsistent dust-obscured SFRs in the literature obtained by stacking 24 μm and Herschel, as mentioned above. Taking advantage of the available deep multi-wavelength data, we constrain their dust-obscured SFRs through stacking in the Spitzer Multiband Imaging Photometer (MIPS; Rieke et al. 2004), the Herschel Photodetector Array Camera, and Spectrometer (PACS; Poglitsch et al. 2010), and the Spectral and Photometric Imaging Receiver (SPIRE¹⁴ ; Griffin et al. 2010) maps. These are compared with stacks in deep Very Large Array (VLA) radio maps. Magnitudes are quoted in the AB system. We adopt a Chabrier (2003) initial mass function (IMF), and H₀ = 70 km s⁻¹ Mpc⁻¹, Ω_M = 0.3, and Ω_Λ = 0.7. All the infrared luminosities (L_IR) used in this work are integrated from the rest-frame wavelength range of 8–1000 μm.

We select galaxies brighter than K_s = 24 from the UltraVISTA survey (McCracken et al. 2012) which have ${M}_{\star }\geqslant {10}^{9.8}\;{M}_{\odot }$ and photometric redshifts z_phot = 0.1–3.0. Both M_⋆ and z_phot are from the catalog of Ilbert et al. (2013), derived from spectral energy distribution (SED) fits to the broadband UV-to-IRAC photometry (Capak et al. 2007). A small fraction of the UltraVISTA galaxies appears to host AGNs (<5%), as indicated via their emission in X-rays (Brusa et al. 2010; Civano et al. 2012), IRAC bands (Donley et al. 2012), or the radio (Schinnerer et al. 2007, 2010). We have included them in the analysis presented below, but we note that excluding them does not change the 24 μm and radio (the two bands expected to be most sensitive to the presence of an AGN) stacked flux densities within the stated uncertainties, nor the conclusions of this work.

Multiple rest-frame color selection techniques have been devised to separate QGs from SFGs, such as U − V versus V − J ("UVJ-selection"; Williams et al. 2009), NUV − r versus r − J ("NUVrJ-selection"; Ilbert et al. 2013), and NUV − r versus r − K (Arnouts et al. 2013). Here, we have adopted the NUVrJ selection technique for the reasons explained in Ilbert et al. (2013, Section 3.3), although we note that the stacking results from our selection are comparable to those using the UVJ-selection (see Section 4.7). Each galaxy is classified as a QG or a SFG candidate¹⁵ based on its rest-frame NUV − r and r − J colors, where the criterion is shown on Figure 1. NUV − r is a measure of the amount of UV light from young stars (i.e., recent SF) relative to the red optical light from evolved stellar populations, and when compared to r − J we obtain constraints on the degree of dust attenuation. The QG candidates are divided into six bins of z_phot, each of which is split into four M_⋆-bins (see Table 2); however, only M_⋆-bins which are >90% mass-complete (according to the limits presented in Ilbert et al. 2013) are considered and stacked,¹⁶ as listed in Table 2.

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To weed out dusty galaxies erroneously classified as QGs, we cross-correlate our sample, using a radius of $2^{\prime\prime}$, with the MIPS 24 μm catalog of Le Floc'h et al. (2009), which is 90% complete for sources with ${S}_{24}\;\geqslant$ 80 μJy. A redshift-dependent 24 μm flux density (S₂₄) cut-off¹⁷ is then applied to separate the IR-bright QG candidates with SFR${}_{24}\geqslant 100\;{M}_{\odot }$ yr⁻¹, where SFR₂₄ is the SFR inferred from their 24 μm flux density (S₂₄; see Section 4.4 for derivation). The fraction of QG candidates with SFR${}_{24}\geqslant 100\;{M}_{\odot }$yr⁻¹ (${f}_{{\rm{QG,24}}}$) increases with z and peaks at 15% for the most massive ones at z ≳ 2 (see Table 1), qualitatively similar to the trend found in Marchesini et al. (2014). This suggests a higher fraction of misclassified QG candidates at z ≳ 2, which results from a higher fraction of dusty galaxies at higher z (e.g., Greve et al. 2010), and more uncertain rest-frame colors (Williams et al. 2009). Overall, however, the fractions are reassuringly small. A similar conclusion is reached for the fraction (${f}_{{\rm{QG,H}}}\lt 6\%;$ Table 1) of QG candidates that are further detected by Herschel (i.e., S/N $\geqslant \;5$ in at least two PACS and/or SPIRE bands), determined using the catalog of Lee et al. (2013) in which the 24 μm sources presented in Le Floc'h et al. (2009) are cross-identified to the Herschel detections. The linear inversion technique of cross-identification being used is described in Roseboom et al. (2010, 2012).

Table 1.  24 μm- and Herschel-bright Fractions for QG Candidates

	log(M⋆/M⊙)
Redshift	11–12.2	10.6–11	10.2–10.6	9.8–10.2
fQG, 24
0.1–0.5	0%	0%	0%	0%
0.5–1.0	0.2%	0.1%	0%	0%
1.0–1.5	2.1%	0.8%	0.2%	0%
1.5–2.0	2.4%	4.7%	2.6%	⋯
2.0–2.5	14.7%	9.9%	⋯	⋯
2.5–3.0	13.3%	⋯	⋯	⋯
${f}_{{\rm{QG,H}}}$
0.1–0.5	0%	0%	0%	0%
0.5–1.0	0.2%	0.1%	0%	0%
1.0–1.5	0.8%	0.1%	0%	0%
1.5–2.0	1.2%	1.0%	0.6%	⋯
2.0–2.5	6.0%	2.5%	⋯	⋯
2.5–3.0	2.7%	⋯	⋯	⋯

Note. ${f}_{{\rm{QG,24}}}$ is the fraction of QG candidates (classified by their NUV − r and r − J colors) with SFR₂₄ ≥ 100 M_⊙ yr⁻¹, where SFR₂₄ is the SFR as inferred from the 24 μm flux density. ${f}_{{\rm{QG,H}}}$ is the fraction of QG canididates fulfilling the above criterion that are also detected in at least two Herschel PACS+SPIRE bands (S/N ≥ 5).

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For the stacking analysis (Section 3) we use the MIPS 24 μm imaging (FWHM ≃ 6'') from Sanders et al. (2007), while the Herschel PACS and SPIRE maps are from the PACS Evolutionary Probe survey (PEP; Lutz et al. 2011) and the Herschel Multi-tiered Extragalactic Survey (HerMES; Oliver et al. 2012), respectively. The PACS maps reach depths of 5 and 10.3 mJy beam⁻¹ (3σ) at 100 and 160 μm, respectively (FWHM ≃ 68 and 11''), and SPIRE 250, 350, and 500 μm depths are 8, 11, and 13 mJy beam⁻¹ (3σ), respectively (FWHM ≃ 182, 249, and 363). The Herschel maps are made assuming a flat spectrum. For the radio stacking we use the 1.4 GHz VLA-COSMOS large survey (Schinnerer et al. 2007, 2010), which reaches a root-mean-square noise (rms) of 15 μJy beam⁻¹ at an angular resolution of ∼15 (FWHM).

Our Herschel maps are characterized by a high level of source confusion (multiple sources blended in a beam due to coarse angular resolution; Nguyen et al. 2010) which, if unaccounted for, will bias a stacked signal of clustered sources (Marsden et al. 2009; Béthermin et al. 2010; Kurczynski & Gawiser 2010; Viero et al. 2013). Here, we apply the global stacking and deblending method presented in Viero et al. (2013), which is publicly distributed as an IDL code called SIMSTACK.¹⁸ In short, we stacked and deblended multiple galaxy samples simultaneously, including a separate list of 24 μm sources from Le Floc'h et al. (2009) that are not already included in the UltraVISTA sample. The median M_⋆ and z for IR-faint QGs, IR-bright QGs, and SFGs are listed in Tables 2–4, respectively. We estimated the errors of the Herschel stacking following the extended bootstrap technique presented in Viero et al. (2013, Section 3.4). Having created 100 fake UltraVISTA catalogs by perturbing M_⋆ and z of all sources within their uncertainties, we re-did the sample selection and re-ran SIMSTACK 100 times. The stacked flux densities and errors are taken to be the mean and standard deviation of the 100 runs.

Table 2.  Stacked Flux Densities of Spitzer/MIPS 24 μm, Herschel (PACS+SPIRE), and VLA 1.4 GHz and the Inferred SFRs for IR-faint QG Candidates

Redshift	$\bar{{z}_{{\rm{phot}}}}$	Ngal	S24 μm	S100 μm	S160 μm	S250 μm	S350 μm	S500 μm	Sradio	log(${L}_{{\rm{IR,H}}}$)	log(${L}_{1.4\mathrm{GHz}}$)	qIR	SFRSED	SFR24	SFRH	SFRradio	log(sSFRH)
			(μJy)	(mJy)	(mJy)	(mJy)	(mJy)	(mJy)	(μJy)	log(L⊙)	log(W Hz−1)		(M⊙ yr−1)	(M⊙ yr−1)	(M⊙ yr−1)	(M⊙ yr−1)	log(yr−1)
log(M⋆/M⊙) = 11–12.2 (median = 11.2)
0.1–0.5	0.4	232	34.0 ± 1.5	0.5 ± 0.1	2.0 ± 0.2	0.6 ± 0.3	1.9 ± 0.2	1.1 ± 0.2	9.6 ± 1.1	9.8 ± 0.1	21.7 ± 0.0	2.2	${0.05}_{-0.05}^{+2.97}$	0.5 ± 0.2	${0.7}_{-0.1}^{+0.2}$	1.7 ± 0.1	−11.3
0.5–1.0	0.8	1298	24.1 ± 0.9	0.1 ± 0.0	1.2 ± 0.1	0.6 ± 0.1	2.3 ± 0.1	1.2 ± 0.1	7.8 ± 0.5	10.5 ± 0.1	22.5 ± 0.0	2.1	${0.20}_{-0.20}^{+12.39}$	1.8 ± 0.7	${3.4}_{-0.4}^{+0.4}$	9.2 ± 0.5	−10.6
1.0–1.5	1.2	827	21.3 ± 1.0	0.2 ± 0.0	1.0 ± 0.1	0.7 ± 0.2	2.5 ± 0.1	1.4 ± 0.1	6.4 ± 0.6	10.9 ± 0.1	22.8 ± 0.0	2.1	${0.59}_{-0.56}^{+13.21}$	4.6 ± 1.7	${8.1}_{-1.0}^{+1.2}$	19.0 ± 1.7	−10.2
1.5–2.0	1.6	320	13.0 ± 1.2	0.0 ± 0.1	0.6 ± 0.2	−0.1 ± 0.3	2.4 ± 0.3	2.0 ± 0.2	3.0 ± 0.9	11.2 ± 0.2	22.8 ± 0.1	2.4	${0.42}_{-0.39}^{+6.50}$	4.8 ± 1.8	${14.8}_{-5.0}^{+7.6}$	20.0 ± 6.0	−10.0
2.0–2.5	2.2	185	17.6 ± 1.6	−0.1 ± 0.1	1.3 ± 0.3	0.6 ± 0.4	3.1 ± 0.4	2.5 ± 0.3	5.7 ± 1.1	11.7 ± 0.1	23.4 ± 0.1	2.3	${0.78}_{-0.71}^{+7.74}$	9.4 ± 3.5	${50.1}_{-12.1}^{+16.0}$	82.7 ± 16.6	−9.4
2.5–3.0	2.6	65	11.7 ± 2.2	−0.2 ± 0.1	0.4 ± 0.5	−1.2 ± 0.6	1.4 ± 0.6	1.7 ± 0.6	6.0 ± 2.1	11.5 ± 0.4	23.6 ± 0.1	1.9	${1.20}_{-1.16}^{+34.28}$	12.8 ± 5.4	${35.5}_{-22.3}^{+60.0}$	141.2 ± 48.4	−9.5
log(M⋆/M⊙) = 10.6–11.0 (median = 10.8)
0.1–0.5	0.4	518	26.3 ± 1.0	0.7 ± 0.1	2.5 ± 0.2	0.4 ± 0.2	1.8 ± 0.2	1.0 ± 0.2	4.8 ± 0.7	9.9 ± 0.1	21.4 ± 0.1	2.5	${0.02}_{-0.02}^{+1.03}$	0.3 ± 0.1	${0.8}_{-0.1}^{+0.1}$	0.9 ± 0.1	−10.9
0.5–1.0	0.8	2296	16.5 ± 0.7	0.2 ± 0.0	1.2 ± 0.1	−0.1 ± 0.1	1.9 ± 0.1	1.3 ± 0.1	2.9 ± 0.3	10.5 ± 0.0	22.0 ± 0.1	2.5	${0.13}_{-0.13}^{+6.18}$	1.2 ± 0.4	${3.1}_{-0.3}^{+0.3}$	3.3 ± 0.4	−10.3
1.0–1.5	1.2	1764	10.7 ± 0.5	0.1 ± 0.0	0.7 ± 0.1	−0.9 ± 0.1	1.6 ± 0.1	0.9 ± 0.1	2.3 ± 0.4	10.8 ± 0.1	22.3 ± 0.1	2.4	${0.26}_{-0.25}^{+6.35}$	2.0 ± 0.7	${5.8}_{-0.9}^{+1.0}$	6.8 ± 1.1	−10.0
1.5–2.0	1.7	550	8.5 ± 0.8	−0.1 ± 0.1	0.4 ± 0.2	−1.5 ± 0.3	1.3 ± 0.2	1.1 ± 0.2	2.8 ± 0.7	11.0 ± 0.2	22.8 ± 0.1	2.2	${0.34}_{-0.31}^{+3.46}$	2.8 ± 1.1	${10.2}_{-4.1}^{+6.7}$	20.2 ± 4.8	−9.8
2.0–2.5	2.2	326	13.2 ± 1.2	0.1 ± 0.1	0.8 ± 0.3	−0.6 ± 0.4	2.6 ± 0.4	2.1 ± 0.3	3.0 ± 0.9	11.6 ± 0.2	23.2 ± 0.1	2.4	${0.66}_{-0.61}^{+7.28}$	7.5 ± 2.8	${36.3}_{-12.3}^{+18.6}$	45.6 ± 13.4	−9.2
log(M⋆/M⊙) = 10.2–10.6 (median = 10.4)
0.1–0.5	0.4	708	13.7 ± 0.8	0.5 ± 0.1	1.9 ± 0.2	−1.0 ± 0.1	0.7 ± 0.1	0.2 ± 0.1	3.5 ± 0.6	9.7 ± 0.1	21.2 ± 0.1	2.5	${0.02}_{-0.01}^{+0.57}$	0.2 ± 0.1	${0.5}_{-0.1}^{+0.1}$	0.8 ± 0.1	−10.7
0.5–1.0	0.8	2365	12.1 ± 0.5	0.2 ± 0.0	1.0 ± 0.1	−1.0 ± 0.1	1.1 ± 0.1	0.6 ± 0.1	1.5 ± 0.3	10.3 ± 0.1	21.7 ± 0.1	2.6	${0.08}_{-0.08}^{+3.31}$	0.8 ± 0.3	${2.1}_{-0.3}^{+0.4}$	1.8 ± 0.3	−10.1
1.0–1.5	1.2	1268	6.4 ± 0.5	0.0 ± 0.0	0.7 ± 0.1	−1.7 ± 0.1	1.1 ± 0.1	0.9 ± 0.1	1.5 ± 0.5	10.7 ± 0.1	22.1 ± 0.1	2.6	${0.16}_{-0.15}^{+2.79}$	1.1 ± 0.4	${4.9}_{-1.1}^{+1.4}$	4.3 ± 1.3	−9.7
1.5–2.0	1.7	478	7.0 ± 0.9	0.0 ± 0.1	0.7 ± 0.2	−1.6 ± 0.3	1.3 ± 0.3	1.1 ± 0.2	2.0 ± 0.7	11.1 ± 0.2	22.7 ± 0.1	2.4	${0.32}_{-0.28}^{+2.70}$	2.3 ± 0.9	${11.7}_{-4.0}^{+6.0}$	14.4 ± 5.1	−9.3
log(${M}_{\star }/{M}_{\odot }$) = 9.8–10.2 (median = 10.0)
0.1–0.5	0.4	590	14.1 ± 0.8	0.3 ± 0.1	1.0 ± 0.2	−2.1 ± 0.2	0.3 ± 0.1	−0.3 ± 0.2	2.2 ± 0.7	9.4 ± 0.3	21.0 ± 0.1	2.5	${0.02}_{-0.01}^{+0.47}$	0.1 ± 0.1	${0.3}_{-0.1}^{+0.2}$	0.5 ± 0.1	−10.5
0.5–1.0	0.8	1349	6.6 ± 0.6	0.1 ± 0.0	0.8 ± 0.1	−1.5 ± 0.1	1.0 ± 0.1	0.7 ± 0.1	1.1 ± 0.4	10.2 ± 0.1	21.6 ± 0.1	2.7	${0.07}_{-0.06}^{+2.07}$	0.4 ± 0.1	${1.7}_{-0.4}^{+0.5}$	1.3 ± 0.4	−9.8
1.0–1.5	1.2	709	3.6 ± 0.7	−0.0 ± 0.1	0.1 ± 0.2	−2.2 ± 0.2	0.8 ± 0.2	0.4 ± 0.2	1.4 ± 0.6	10.3 ± 0.6	22.1 ± 0.2	2.2	${0.21}_{-0.20}^{+3.25}$	0.5 ± 0.2	${2.0}_{-1.5}^{+5.9}$	4.0 ± 1.8	−9.7

Note. The stacking results for IR-faint (SFR₂₄ < 100 M_⊙ yr⁻¹) QG candidates. The median redshifts ($\bar{{z}_{{\rm{phot}}}}$), the number of IR-faint QG candidates (N_gal), and the stacked flux densities are listed. SFR_SED is the median SFR from the UV-to-IRAC SED fitting. We infer SFRs from the stacked flux densities (Section 4.4), assuming that the 24 μm and radio emissions originate from SF only. The IR luminosity and specific SFR (${L}_{{\rm{IR,H}}}$ and sSFR_H) inferred from the Herschel SIMSTACK results are shown in logarithmic units. The radio index q₂₄ is computed as log(${S}_{24\mu {\rm{m}}}/{S}_{{\rm{radio}}}$).

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Table 3.  Stacked Flux Densities of Spitzer/MIPS 24 μm, Herschel (PACS+SPIRE), and VLA 1.4 GHz and the Inferred SFRs for IR-bright QG Candidates

Redshift	$\bar{{z}_{{\rm{phot}}}}$	Ngal	S24 μm	S100 μm	S160 μm	S250 μm	S350 μm	S500 μm	Sradio	log(${L}_{{\rm{IR,H}}}$)	log(${L}_{1.4\mathrm{GHz}}$)	qIR	SFRSED	SFR24	SFRH	SFRradio	log(sSFRH)
			(μJy)	(mJy)	(mJy)	(mJy)	(mJy)	(mJy)	(μJy)	log(L⊙)	log(W Hz−1)		(M⊙ yr−1)	(M⊙ yr−1)	(M⊙ yr−1)	(M⊙ yr−1)	log(yr−1)
log(M⋆/M⊙) = 11–12.2 (median = 11.2)
0.5–1.0	0.9	3	755.6 ± 17.1	5.1 ± 0.6	16.7 ± 1.5	21.7 ± 2.9	16.0 ± 1.6	12.0 ± 1.5	34.3 ± 8.3	11.8 ± 0.1	23.2 ± 0.1	2.5	${2.51}_{-1.07}^{+1.85}$	119.0 ± 44.5	${58.9}_{-7.6}^{+8.7}$	55.8 ± 13.5	−9.3
1.0–1.5	1.4	18	226.0 ± 5.6	2.5 ± 0.5	7.2 ± 1.3	14.0 ± 1.6	12.3 ± 1.2	8.3 ± 1.2	35.4 ± 3.9	11.9 ± 0.1	23.7 ± 0.1	2.2	${9.12}_{-7.61}^{+45.83}$	148.0 ± 55.9	${89.1}_{-15.0}^{+18.0}$	166.8 ± 18.6	−9.2
1.5–2.0	1.6	8	281.6 ± 11.1	2.5 ± 0.6	8.1 ± 2.0	14.7 ± 3.1	13.5 ± 2.3	9.6 ± 2.0	49.0 ± 6.0	12.1 ± 0.1	24.0 ± 0.1	2.1	${1.95}_{-1.60}^{+8.77}$	190.5 ± 72.3	${134.9}_{-34.9}^{+47.1}$	335.5 ± 41.0	−9.1
2.0–2.5	2.3	32	172.7 ± 4.7	1.2 ± 0.3	7.2 ± 1.0	15.6 ± 2.3	15.4 ± 2.1	11.2 ± 1.7	20.1 ± 2.9	12.4 ± 0.1	24.0 ± 0.1	2.4	${5.13}_{-4.57}^{+41.64}$	162.4 ± 61.4	${269.2}_{-55.4}^{+69.7}$	314.6 ± 45.1	−8.7
2.5–3.0	2.6	10	137.4 ± 6.3	1.3 ± 0.4	6.0 ± 1.6	11.7 ± 3.5	12.2 ± 3.0	8.6 ± 2.5	37.7 ± 4.6	12.5 ± 0.2	24.5 ± 0.1	2.1	${2.95}_{-2.86}^{+94.77}$	235.2 ± 89.6	${331.1}_{-112.4}^{+170.1}$	915.6 ± 112.3	−8.6
log(M⋆/M⊙) = 10.6–11.0 (median = 10.8)
0.5–1.0	1.0	3	404.7 ± 18.1	6.3 ± 0.7	23.7 ± 2.5	40.2 ± 3.3	25.0 ± 3.1	14.7 ± 1.6	48.9 ± 9.5	11.9 ± 0.1	23.4 ± 0.1	2.5	${1.00}_{-0.87}^{+6.59}$	61.3 ± 22.1	${91.2}_{-11.8}^{+13.5}$	84.0 ± 16.4	−8.9
1.0–1.5	1.2	14	353.4 ± 7.4	3.9 ± 0.5	10.7 ± 1.5	14.6 ± 2.2	12.9 ± 1.7	6.2 ± 1.3	41.4 ± 4.1	11.9 ± 0.1	23.7 ± 0.0	2.3	${2.45}_{-1.66}^{+5.13}$	137.5 ± 51.9	${85.1}_{-14.3}^{+17.2}$	146.3 ± 14.5	−8.8
1.5–2.0	1.7	27	238.9 ± 6.0	2.1 ± 0.3	8.4 ± 1.1	14.6 ± 1.6	12.2 ± 1.2	7.4 ± 1.0	34.5 ± 3.0	12.1 ± 0.1	23.9 ± 0.0	2.2	${1.26}_{-1.10}^{+8.74}$	150.3 ± 56.8	${131.8}_{-19.6}^{+23.1}$	247.5 ± 21.4	−8.6
2.0–2.5	2.3	36	130.9 ± 3.9	1.1 ± 0.2	6.0 ± 1.1	11.1 ± 1.9	12.5 ± 1.9	9.7 ± 1.6	18.3 ± 2.9	12.3 ± 0.1	24.0 ± 0.1	2.4	${2.14}_{-1.88}^{+15.64}$	114.1 ± 42.4	${218.8}_{-49.0}^{+63.1}$	291.0 ± 45.3	−8.5
log(${M}_{\star }/{M}_{\odot }$) = 10.2–10.6 (median = 10.4)
1.0–1.5	1.5	3	206.7 ± 10.3	4.4 ± 1.4	7.8 ± 3.0	7.5 ± 4.0	9.1 ± 3.9	2.0 ± 3.1	29.4 ± 9.9	11.9 ± 0.3	23.6 ± 0.1	2.3	${0.58}_{-0.45}^{+2.12}$	133.1 ± 50.8	${89.1}_{-41.3}^{+76.8}$	141.4 ± 47.5	−8.6
1.5–2.0	1.8	13	201.6 ± 5.8	2.1 ± 0.7	4.7 ± 2.4	7.2 ± 4.0	6.9 ± 2.9	5.5 ± 2.2	25.5 ± 4.5	12.0 ± 0.2	23.8 ± 0.1	2.2	${0.68}_{-0.50}^{+1.95}$	105.1 ± 38.5	${95.5}_{-39.3}^{+66.7}$	188.5 ± 33.1	−8.6

Note. The legend follows that of Table 2.

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Table 4.  Stacked Flux Densities of Spitzer/MIPS 24 μm, Herschel (PACS+SPIRE), and VLA 1.4 GHz and the Inferred SFRs for SFG Candidates

Redshift	$\bar{{z}_{{\rm{phot}}}}$	Ngal	S24 μm	S100 μm	S160 μm	S250 μm	S350 μm	S500 μm	Sradio	log(${L}_{{\rm{IR,H}}}$)	log(${L}_{1.4\mathrm{GHz}}$)	qIR	SFRSED	SFR24	SFRH	SFRradio	log(sSFRH)
			(μJy)	(mJy)	(mJy)	(mJy)	(mJy)	(mJy)	(μJy)	log(L⊙)	log(W Hz−1)		(M⊙ yr−1)	(M⊙ yr−1)	(M⊙ yr−1)	(M⊙ yr−1)	log(yr−1)
log(M⋆/M⊙) = 11–12.2 (median = 11.2)
0.1–0.5	0.4	157	249.7 ± 4.6	8.3 ± 0.4	23.4 ± 1.1	23.6 ± 0.8	10.3 ± 0.4	3.8 ± 0.3	17.3 ± 1.3	10.9 ± 0.0	22.0 ± 0.0	2.9	${2.51}_{-1.82}^{+6.61}$	4.0 ± 1.4	${8.1}_{-0.4}^{+0.4}$	2.9 ± 0.2	−10.2
0.5–1.0	0.8	446	151.3 ± 3.1	3.1 ± 0.1	10.3 ± 0.3	13.6 ± 0.3	9.2 ± 0.2	4.8 ± 0.2	14.9 ± 0.8	11.4 ± 0.0	22.7 ± 0.0	2.7	${6.31}_{-4.17}^{+12.31}$	12.7 ± 4.5	${23.4}_{-1.1}^{+1.1}$	16.1 ± 0.9	−9.8
1.0–1.5	1.2	668	93.9 ± 2.0	1.7 ± 0.1	7.2 ± 0.2	11.4 ± 0.3	9.9 ± 0.2	5.5 ± 0.2	16.2 ± 0.7	11.7 ± 0.0	23.2 ± 0.0	2.5	${12.59}_{-9.43}^{+37.53}$	31.5 ± 11.3	${55.0}_{-2.5}^{+2.6}$	56.3 ± 2.3	−9.4
1.5–2.0	1.8	795	101.4 ± 2.2	1.1 ± 0.1	4.7 ± 0.2	7.8 ± 0.3	8.2 ± 0.2	5.9 ± 0.2	13.1 ± 0.6	11.9 ± 0.0	23.5 ± 0.0	2.4	${34.67}_{-28.07}^{+147.30}$	47.6 ± 17.0	${87.1}_{-3.9}^{+4.1}$	106.1 ± 4.9	−9.2
2.0–2.5	2.2	560	114.7 ± 2.2	1.0 ± 0.1	3.7 ± 4.1	8.4 ± 0.3	9.3 ± 0.3	7.2 ± 0.3	14.6 ± 0.7	12.2 ± 0.0	23.8 ± 0.0	2.3	${53.70}_{-40.82}^{+170.17}$	87.2 ± 31.1	${147.9}_{-9.9}^{+10.6}$	213.5 ± 10.6	−9.0
2.5–3.0	2.7	322	83.8 ± 1.7	0.7 ± 0.1	3.0 ± 5.3	7.0 ± 0.4	8.7 ± 0.4	6.9 ± 0.4	11.5 ± 0.9	12.3 ± 0.1	23.9 ± 0.0	2.4	${79.43}_{-63.21}^{+309.61}$	169.2 ± 63.9	${204.2}_{-26.3}^{+30.2}$	284.7 ± 22.5	−8.8
log(M⋆/M⊙) = 10.6–11.0 (median = 10.8)
0.1–0.5	0.4	547	204.6 ± 4.1	7.5 ± 0.2	18.9 ± 0.4	16.3 ± 0.4	7.9 ± 0.2	3.1 ± 0.2	16.4 ± 0.8	10.8 ± 0.0	21.9 ± 0.0	2.9	${2.51}_{-1.87}^{+7.26}$	2.9 ± 1.0	${6.3}_{-0.3}^{+0.3}$	2.6 ± 0.1	−9.9
0.5–1.0	0.8	1823	127.0 ± 2.5	2.5 ± 0.0	8.1 ± 0.1	9.6 ± 0.1	6.8 ± 0.1	3.3 ± 0.1	13.0 ± 0.4	11.2 ± 0.0	22.6 ± 0.0	2.6	${8.13}_{-6.22}^{+26.55}$	11.0 ± 3.9	${17.8}_{-0.4}^{+0.4}$	14.0 ± 0.5	−9.5
1.0–1.5	1.3	2587	76.1 ± 1.4	1.4 ± 0.0	5.3 ± 0.1	6.9 ± 0.1	6.5 ± 0.1	3.7 ± 0.1	11.2 ± 0.4	11.6 ± 0.0	23.1 ± 0.0	2.5	${23.99}_{-18.98}^{+90.83}$	27.9 ± 10.0	${39.8}_{-0.9}^{+0.9}$	41.0 ± 1.3	−9.2
1.5–2.0	1.8	2632	69.3 ± 1.4	0.7 ± 0.0	3.3 ± 1.2	3.8 ± 0.1	5.0 ± 0.1	3.4 ± 0.1	8.1 ± 0.3	11.7 ± 0.0	23.3 ± 0.0	2.4	${31.62}_{-24.86}^{+116.29}$	31.5 ± 11.3	${47.9}_{-2.2}^{+2.3}$	65.0 ± 2.6	−9.1
2.0–2.5	2.2	1671	68.9 ± 1.2	0.5 ± 0.0	0.7 ± 3.2	2.9 ± 0.2	4.8 ± 0.2	3.8 ± 0.2	7.2 ± 0.4	11.8 ± 0.0	23.5 ± 0.0	2.3	${38.02}_{-26.80}^{+90.81}$	49.5 ± 17.7	${69.2}_{-6.1}^{+6.7}$	106.5 ± 6.0	−8.9
2.5–3.0	2.7	898	44.5 ± 0.9	0.4 ± 0.0	−1.4 ± 6.1	1.9 ± 0.2	4.2 ± 0.2	3.5 ± 0.2	6.9 ± 0.5	12.0 ± 0.1	23.7 ± 0.0	2.2	${48.98}_{-33.49}^{+105.90}$	75.9 ± 27.1	${91.2}_{-13.6}^{+16.0}$	170.3 ± 13.4	−8.8
log(M⋆/M⊙) = 10.2–10.6 (median = 10.4)
0.1–0.5	0.4	1014	164.0 ± 3.2	5.5 ± 0.1	13.1 ± 0.3	10.1 ± 0.2	5.3 ± 0.1	1.8 ± 0.1	13.8 ± 0.5	10.6 ± 0.0	21.8 ± 0.0	2.8	${2.51}_{-1.77}^{+6.00}$	2.3 ± 0.8	${4.0}_{-0.1}^{+0.1}$	2.1 ± 0.1	−9.8
0.5–1.0	0.8	3689	102.8 ± 1.9	1.9 ± 0.0	6.0 ± 0.1	5.8 ± 0.1	4.7 ± 0.1	2.2 ± 0.1	9.2 ± 0.3	11.1 ± 0.0	22.5 ± 0.0	2.6	${15.85}_{-12.05}^{+50.22}$	8.7 ± 3.1	${12.3}_{-0.3}^{+0.3}$	9.9 ± 0.3	−9.3
1.0–1.5	1.3	4832	51.1 ± 0.9	0.9 ± 0.0	3.2 ± 0.5	2.7 ± 0.1	3.7 ± 0.1	2.0 ± 0.1	6.3 ± 0.2	11.3 ± 0.0	22.9 ± 0.0	2.5	${26.30}_{-20.14}^{+85.90}$	16.9 ± 6.0	${20.9}_{-0.5}^{+0.5}$	22.6 ± 0.9	−9.0
1.5–2.0	1.7	4443	53.1 ± 1.0	0.4 ± 0.0	3.1 ± 0.8	1.3 ± 0.1	3.0 ± 0.1	2.0 ± 0.1	4.3 ± 0.2	11.4 ± 0.0	23.0 ± 0.0	2.4	${28.84}_{-21.76}^{+88.65}$	23.3 ± 8.3	${27.5}_{-1.8}^{+2.0}$	33.6 ± 1.9	−8.9
log(M⋆/M⊙) = 9.8–10.2 (median = 10.0)
0.1–0.5	0.4	1477	96.7 ± 1.9	3.1 ± 0.1	7.3 ± 0.1	4.1 ± 0.1	2.8 ± 0.1	1.2 ± 0.1	7.5 ± 0.4	10.3 ± 0.0	21.5 ± 0.0	2.8	${2.09}_{-1.46}^{+4.83}$	1.2 ± 0.4	${2.0}_{-0.1}^{+0.1}$	1.2 ± 0.1	−9.7
0.5–1.0	0.8	5737	59.7 ± 1.1	0.9 ± 0.0	3.1 ± 0.1	1.7 ± 0.1	2.6 ± 0.1	1.3 ± 0.1	4.8 ± 0.2	10.8 ± 0.0	22.2 ± 0.0	2.6	${15.49}_{-11.12}^{+39.47}$	4.7 ± 1.7	${5.8}_{-0.1}^{+0.1}$	5.2 ± 0.2	−9.2
1.0–1.5	1.3	7900	26.3 ± 0.5	0.4 ± 0.0	−0.1 ± 1.3	−0.1 ± 0.1	1.9 ± 0.1	1.1 ± 0.1	3.1 ± 0.2	11.0 ± 0.0	22.5 ± 0.0	2.5	${19.95}_{-14.06}^{+47.66}$	7.7 ± 2.7	${10.7}_{-0.7}^{+0.8}$	11.1 ± 0.6	−8.9

Note. The legend follows that of Table 2.

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Source confusion is not an issue for our radio maps due to the higher angular resolution (see Section 2). The stacked signal was determined from the median-stacked images constructed from galaxy postage stamps, using the median value of each pixel after centering the galaxies with the UltraVISTA positions. MIPS 24 μm stacks were determined in a similar way, despite the larger beam size. The 24 μm flux densities were measured on the stacked images using an aperture radius of 35, with aperture corrections applied following Table 4.13 of the MIPS handbook, i.e., ${S}_{\mathrm{tot}}\;=\;2.80\times {S}_{3\buildrel{\prime\prime}\over{.} 5}$. For the radio fluxes we adopted the central pixel values. In both cases the errors were estimated from the rms of the background in the stacked images.

In summary, we derive three independent SFRs from stacking 24 μm, Herschel, and radio data (Sections 4.1 and 4.2). The L_IR inferred from the stacked 24 μm fluxes are a few factors below those from Herschel stacking (Section 4.3), which illustrates that SFG dust templates may be inadequate for QG candidates when converting 24 μm flux to L_IR. We find little SF from stacking Herschel data for IR-faint QG candidates at least out to z ∼ 2, thereby confirming their quiescent nature as indicated by their rest-frame NUV − r and r − J colors (Section 4.4). At z = 2–3, the obscured sSFRs of IR-faint QG candidates are somewhat higher but still below those of SFGs, hinting that the rest-frame color selection may be less robust beyond z ≳ 2. The radio emission of the most massive IR-faint QG candidates at z = 0.1–1.5 is in excess of that expected from the total (obscured & unobscured) SFR, suggesting contribution from low-luminosity AGNs (Section 4.5).

4.1. Panchromatic UV-to-radio SED

Figure 2 summarizes our constraints on the average SEDs of IR-faint QG candidates at MIR, FIR, and radio wavelengths along with the median UV-to-NIR SEDs. All the stacked MIPS 24 μm, Herschel, and radio flux densities as a function of z and M_⋆ are listed for IR-faint QG candidates (Table 2), IR-bright QG candidates (Table 3), and SFG candidates (Table 4), respectively. The Herschel stacked flux densities have S/N > 5 for some but not all bands. The most massive (${M}_{\star }\geqslant {10}^{10.6}\;{M}_{\odot }$) IR-faint QG candidates are detected at all redshifts out to z = 3 in the 24 μm stacks (S/N ∼ 5–26) and out to z = 1.5 in the radio stacks (S/N ∼ 6–17). The lower mass IR-faint QG candidates (M_⋆ < 10^10.6 M_⊙) are detected at 24 μm (S/N ∼ 5–24) in all relevant (i.e., mass-complete) redshift bins, but only marginally detected in the radio (S/N ∼ 2–6). As expected, S₂₄ and S_radio generally decrease with z (cosmic dimming) and increase with M_⋆.

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4.2. Estimations of IR Luminosities and SFRs

4.3. Discrepant L_IR from Stacking $24\;\mu {\rm{m}}$ and Herschel

The ratio between the obscured SFRs derived from stacking $24\;\mu {\rm{m}}$ and Herschel, i.e., SFR₂₄/SFR${}_{{\rm{H}}}\equiv {L}_{{\rm{IR,24}}}/{L}_{{\rm{IR,H}}}$, is shown in Figure 3. In general, SFR₂₄ is lower than SFR_H for IR-faint QGs (red circles), a persistent trend that prevails for similar comparisons in the literature (see Sections 1 & 4.7). Notably, the two SFR estimates (SFR₂₄ and SFR_H) are more discrepant for IR-faint QGs than for SFGs. This implies that the discrepancy between SFR₂₄ and SFR_H for IR-faint QGs is, at least in part, driven by factors other than the systematic offsets in the conversion from S₂₄ to L_IR.

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The Rujopakarn et al. (2013) conversion used in this work is calibrated with SFGs and may not be applicable for QGs. The conversion of a single data point of observed flux density at 24 μm to a L_IR integrated over rest-frame 8–1000 μm involves systematic uncertainties depending on the choice of template (e.g., Lee et al. 2013). In particular, the intrinsic dust SED of QGs may differ from that of SFGs due to different interstellar medium properties, such as dust temperatures, opacities and geometry, ionization field strength, etc.

On the other hand, since mechanisms other than SF may contribute to the dust heating and are not accounted for in our conversion of ${L}_{{\rm{IR,H}}}$ to SFR_H, our finding implies that the SFR_H should be considered as an upper limit to the true obscured SFR. Evolved stellar populations have been shown to account for the low levels of L_IR in nearby quiescent early-type galaxies through the heating up of diffuse dust, and are a significant dust heating source contributing to emissions at rest-frame wavelengths longer than 160 μm (Bendo et al. 2012; see also Bell 2003; Salim et al. 2009; Fumagalli et al. 2014), which are most relevant for the Herschel SPIRE bands. We note in Figure 3 that the SFR₂₄/SFR_H ratio is less discrepant for massive IR-faint QGs candidates than less massive ones, which contradicts the expectations if old stellar populations are primarily responsible for the FIR emission. However, the temperature gradient is needed to determine the integrated dust SED shape (Bendo et al. 2012). The lack of its direct measurement in our QG sample restricts us from making further remarks on the mass-dependent trend of SFR₂₄/SFR_H for IR-faint QGs.

To quantify the contribution of the various dust heating sources in z ≳ 2 QGs, spatially resolved FIR maps are needed and will require hours of integration time even with ALMA, the most sensitive sub-millimeter interferometer existing to date. In the following, we quote the SFR_H as conservative upper limits for the obscured SFR, and discuss the implications in case of a significant FIR contribution of dust heating by old stellar populations.

4.4. How Quiescent are IR-faint QG Candidates Compared to SFGs?

Herschel stacking put stringent upper limits on the dust-obscured SFR of IR-faint QG candidates: [0.3–5, 2–15, $36{\rm{\mbox{--}}}50]\;{M}_{\odot }$ yr⁻¹ for z = [0.1–1, 1–2, 2–3], i.e., sSFR $\leqslant \quad {10}^{-(9.3{\rm{\mbox{--}}}11.3)}$ yr⁻¹ across all z and M_⋆ bins considered. IR-faint QGs form stars at a modest rate compared to SFGs (∼5–13×, Figure 4), although the difference is less obvious at z = 2–3 (∼2–5×). This may be partially due to the incomplete IR-bright QG selection at z = 2.5–3 (see Section 2), leading to higher contamination fraction of dusty sources in the IR-faint QG sample. Combined with the higher IR-bright fraction at high z (Table 1), we infer that the rest-frame color selection is less robust beyond z ∼ 2, as discussed in Section 2. As explained in Section 4.3, the SFR_H should be treated as an upper limit since old stellar populations may be responsible for part of the FIR emission, thus IR-faint QG candidates may in fact be more quiescent than the limits quoted above.

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As a consistency check, we derive SFRs for SFG candidates from stacking MIR, FIR, and radio, using the same methods and assumptions stated above for QG candidates. The stacked fluxes are shown in the Appendix. We find good agreement between our derived SFR-M_⋆ relation for the SFG candidates to that of Speagle et al. (2014; dark gray shade in Figure 4), with at most a factor of two difference (1σ scatter) for Herschel stacking (shown by blue circles in Figure 4). This demonstrates the reliability of the SIMSTACK code used for Herschel stacking in this work.

4.5. Do IR-faint QGs Host AGNs?

4.6. The Nature of IR-bright QG Candidates

Having separated the IR-bright QG candidates from the IR-faint ones in the stacking procedure, we list their stacked fluxes in Table 3 and plot their average SEDs in Figure 7 in the Appendix. This population of dusty galaxies with quiescent NUV − r and r − J colors could either be young dust-enshrouded SFGs without an evolved stellar population, or galaxies containing evolved stellar populations as well as dust heated by a rejuvenated SF episode (Lemaux et al. 2014). Both of these scenarios imply misclassification by the NUVrJ technique as they are dusty SFGs. They may also be post-starburst galaxies that no longer have current SF as traced by rest-frame UV, while the intermediate-type A-stars continue to heat the dust (Hayward et al. 2014). Alternatively, the IR emission could originate from a nearby SFG (in projection) despite being further away from the 24 μm counterpart used for cross-identification to the Herschel emission (see Section 2). Defining the nature of this subset of red galaxies requires spectroscopic and arc-second resolution FIR imaging, thus we defer further discussion until these data sets are obtained.

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Could the IR-bright QGs be the bright end of a unimodal log(L_IR) distribution of all QGs? In Appendix we argue that the log(L_IR) distribution of QGs must be relatively wide (standard deviation $\gtrsim 0.4$ dex, compared to 0.2–0.3 dex for SFGs), if the IR-bright QGs belong to the bright tail of the log(L_IR) distribution of QGs modeled by a Gaussian distribution.

4.7. Comparison with the Literature

Our Herschel stacking results indicate that the NUVrJ selection successfully identify QGs over z = 0–3, with a maximum contamination fraction of 15% from dusty SFGs mostly at z ∼ 2–3, as estimated by cross-correlating to the 24 μm catalog with a sensitivity limit of 80 μJy. In other words, 24 μm fluxes are efficient for removing this small fraction of contaminants (IR-bright QG candidates) which host significant obscured SF. The IR-faint QGs have truly lower obscured SFRs compared to SFGs, as expected from their low unobscured SFRs measured from the UV continua. The average obscured sSFRs of IR-faint QGs with ${M}_{\star }\geqslant {10}^{10.6}{M}_{\odot }$ are 5–13× lower than those of SFGs out to z = 2, based on Herschel stacking. At z = 2–3, the sSFRs of IR-faint QGs are only 2–5× below that of SFGs, suggesting that the classification between the two galaxy types may be less robust due to more uncertain rest-frame colors. For the most massive (${M}_{\star }\geqslant {10}^{11}\;{M}_{\odot }$) IR-faint QGs at z = 0.1–1.5, the stacked radio emissions cannot be completely accounted for by the total level of SFRs derived from other indicators, suggesting the ubiquitous presence of low-luminosity AGNs at least out to z ∼ 1.5.

The ${L}_{{\rm{IR}}}$ (and the resulting obscured SFR) derived from stacking 24 μm is a few factors lower than that derived from stacking Herschel for IR-faint QG candidates. This indicates that QGs have intrinsically different dust SED shape compared to SFGs, leading to an underestimation of the L_IR from the stacked 24 μm flux. Alternatively, this suggests the presence of a cirrus dust component heated by old stellar populations in QGs. Spatially resolved FIR maps are needed to constrain the dust temperature gradient, and by comparing it to the distribution of old stars we can disentangle between these two scenarios. However, these resolved observations are currently only feasible for lensed galaxies. The James Webb Space Telescope will shed light on this matter in the near future.

We reaffirm that a population of truly quiescent galaxies is already in place by z = 3. This corroborates the need for powerful quenching mechanisms to terminate star formation in galaxies. While environmental quenching may be dominant for intermediate-mass QGs (Peng et al. 2010), stacking analyses at radio (this work) and X-ray (Olsen et al. 2013) wavelengths reveal that massive QGs harbor low-luminosity AGNs. AGNs provide a viable mechanism for quenching SF in galaxies, either through high-velocity outflows of gas (Tremonti et al. 2007; Cimatti et al. 2013; Cicone et al. 2014), or heating up the gas to prevent star formation. This is supported by the enhanced AGN fraction among transitory objects between SFGs and QGs (e.g., Barro et al. 2014). After galaxies are quenched, the AGNs may then proceed to "maintenance mode" suppressing further SF through a feedback cycle (Schawinski et al. 2009; Best & Heckman 2012). With upcoming surveys it will be possible to conduct a complete census of AGNs to sample the entire feedback duty cycle and constrain their energetics, in order to quantify their role in quenching star formation in galaxies.

We thank the COSMOS, UltraVISTA, PEP, and HerMES collaborations for providing the data used in this work. We are grateful to the anonymous referee for the invaluable comments that improved this manuscript. A.M. thanks Anna Gallazzi, Mark Sargent, Ryan Quadri, Brian Lemaux, and Julie Wardlow for helpful discussions. A.M. acknowledges the support of the Dark Cosmology Centre, which is funded by DNRF. T.R.G. acknowledges support from an STFC Advanced Fellowship. S.T. acknowledges support from the ERC Consolidator Grant funding scheme (project ConTExt, grant number 648179). A.K. acknowledges support by the Collaborative Research Council 956, sub-project A1, funded by the Deutsche Forschungsgemeinschaft (DFG). Support for B.M. was provided by the DFG priority program 1573 "The physics of the interstellar medium". C.M.C. acknowledges support from a McCue Fellowship through the University of California, Irvine's Center for Cosmology. V.S. is funded by the European Union's Seventh Frame-work program under grant agreement 337595 (ERC Starting Grant, "CoSMass"). A.Z. acknowledges support by the Instrument Centre for Danish Astrophysics (IDA). This work is based on data products from observations made with ESO Telescopes at the La Silla Paranal Observatory under ESO program ID 179.A-2005 and on data products produced by TERAPIX and the Cambridge Astronomy Survey Unit on behalf of the UltraVISTA consortium.

Are the IR-bright QGs merely the high-end tail of a broad unimodal distribution, or are they in fact misclassifications (i.e., the log(L_IR) is a bimodal distribution)?

Bimodal distributions are usually defined to have significant separation between the two means compared to the combined dispersions. The lack of individual IR detections for all QG candidates hinders us from deriving the underlying distributions and dispersions, therefore we are unable to answer this question directly. However, if their log(L_IR) distribution are unimodal, we can constrain the dispersion required to reproduce the population mean, as well as the mean of the bright subset, given the number of QGs and IR-bright QGs in each z and M_⋆ bin. Assuming the log(L_IR) distribution is Gaussian, the dispersion can be calculated as:

$$\begin{eqnarray}&&{\sigma }_{\mathrm{log}({L}_{{\rm{IR}}})}\;=\;\displaystyle \frac{\mathrm{log}({L}_{{\rm{IR,bright}}})-\mathrm{log}({L}_{{\rm{IR,all}}})}{\sqrt{2}\;{\rm{erfcinv}}({N}_{{\rm{bright}}}/{N}_{{\rm{all}}})}\end{eqnarray} \tag{ 1 }$$

where erfcinv(x) ≡ erfc⁻¹(x) is the inverse complementary error function of x.

Based on the L_IR,H and N_QG on Tables 2 and 3, we calculate the expected ${\sigma }_{\mathrm{log}({L}_{{\rm{IR}}})}$ and list them on Table 5. They are larger than the 0.3 dex of dispersion measured for SFGs (Speagle et al. 2014). While these numbers cannot be used to infer the intrinsic log(L_IR) distribution of QG candidates, we conclude that a broad distribution is required if the IR-bright QG canadidates are the brightest subset of all QG candidates. In another words, we rule out the possibility of a narrow (${\sigma }_{\mathrm{log}({L}_{{\rm{IR}}})}\lesssim 0.37$), unimodal Gaussian distribution of log(L_IR) of QG candidates.