ALMA SPECTROSCOPIC SURVEY IN THE HUBBLE ULTRA DEEP FIELD: THE INFRARED EXCESS OF UV-SELECTED z = 2–10 GALAXIES AS A FUNCTION OF UV-CONTINUUM SLOPE AND STELLAR MASS

The principal data used are the ALMA observations from the 2013.1.00718.S program (PI: Aravena) over the HUDF. Those observations were obtained through a full frequency scan in band 6 (212–272 GHz) with ALMA in its most compact configuration. The observations are distributed over seven pointings and cover an approximate area of ∼1 arcmin² to near uniform depth. As described in Paper II (Aravena et al. 2016a), we collapsed our spectral data cube along the frequency axis in the uv-plane, inverting the visibilities using the CASA task CLEAN using natural weighting and mosaic mode, to produce the continuum image. The peak sensitivity we measure in these continuum observations is 12.7 μJy (1σ) per primary beam. Our observations fall within the region of the HUDF that possesses the deepest UV, optical, and near-infrared observations (see Illingworth et al. 2013; Teplitz et al. 2013).

The sensitivity of our ALMA observations allows us to provide useful individual constraints on the far-IR dust emission from normal sub-L* galaxies. If we adopt a modified blackbody form for the shape of the spectral energy distribution (SED) with a dust temperature of 35 K and a power-law spectral index for the dust emissivity of β_d = 1.6 (Eales et al. 1989; Klaas et al. 1997), and account for the impact of the cosmic microwave background (CMB) (e.g., da Cunha et al. 2013b: Section 3.1.1), we estimate that we should be able to tentatively detect at 2σ any star-forming galaxy at z > 3 with an IR luminosity (8–1000 μm rest-frame) in excess of 4 × 10¹⁰ L_⊙ (see Table 1 and Figure 1). For comparison, the characteristic luminosity of galaxies in the rest-frame UV is approximately equal to 4 × 10¹⁰ L_⊙ from z ∼ 3 to z ∼ 8. The implication is that the typical L^∗ galaxy should be tentatively detected at ≳2σ in our data set if it were outputting equal amounts of energy in the far-IR and rest-frame UV.

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Table 1.  2σ Sensitivity Limits for our Probe of Obscured Star Formation from Individual z ≳ 2 Galaxies and the Dependence on SED

Far-Infrared	2σ Sensitivity Limits (1010 L⊙)
SED Model	z ∼ 2	z ∼ 3	z ∼ 4	z ∼ 5	z ∼ 6	z ∼ 7	z ∼ 8	z ∼ 9	z ∼ 10
35 K modified blackbodya (fiducial)	5.0	4.4	4.0	3.7	3.7	3.8	4.2	4.8	6.0
Modified blackbody with evolving Tdb	6.3	7.8	8.8	9.5	10.1	10.2	9.5	9.2	9.1
25 K modified blackbodya	1.4	1.4	1.4	1.6	1.9	2.7	4.1	6.9	12.3
30 K modified blackbodya	2.7	2.5	2.4	2.4	2.5	2.9	3.6	4.9	7.0
40 K modified blackbodya	8.5	7.3	6.3	5.7	5.3	5.2	5.3	5.7	6.4
45 K modified blackbodya	13.9	11.6	9.8	8.5	7.7	7.3	7.1	7.2	7.5
50 K modified blackbodya	21.6	17.6	14.6	12.4	11.0	10.1	9.5	9.3	9.3
NGC 6946c	1.3	1.3	1.4	1.6	1.9	2.5	3.5	5.4	8.7
M51c	1.4	1.4	1.5	1.6	1.9	2.5	3.4	5.2	8.2
Arp 220c	7.4	6.6	6.1	5.6	5.4	5.5	5.7	5.9	6.6
M82c	11.8	10.4	9.4	8.7	8.4	8.4	8.6	9.1	9.9
	2σ Limit for Probes of the Obscured SFR (M⊙ yr−1)d
SED Model	z ∼ 2	z ∼ 3	z ∼ 4	z ∼ 5	z ∼ 6	z ∼ 7	z ∼ 8	z ∼ 9	z∼ 10
35 K modified blackbodya (fiducial)	5.0	4.4	4.0	3.7	3.7	3.8	4.2	4.8	6.0
Modified blackbody with evolving Tdb	6.3	7.8	8.8	9.5	10.1	10.0	9.4	9.0	8.9
25 K modified blackbodya	1.4	1.4	1.4	1.6	1.9	2.7	4.1	6.9	12.3
30 K modified blackbodya	2.7	2.5	2.4	2.4	2.5	2.9	3.6	4.9	7.0
40 K modified blackbodya	8.5	7.3	6.3	5.7	5.3	5.2	5.3	5.7	6.4
45 K modified blackbodya	13.9	11.6	9.8	8.5	7.7	7.3	7.1	7.2	7.5
50 K modified blackbodya	21.6	17.6	14.6	12.4	11.0	10.1	9.5	9.3	9.3
NGC 6946c	1.3	1.3	1.4	1.6	1.9	2.5	3.5	5.4	8.7
M51c	1.4	1.4	1.5	1.6	1.9	2.5	3.4	5.2	8.2
Arp 220c	7.4	6.6	6.1	5.6	5.4	5.5	5.7	5.9	6.6
M82c	11.8	10.4	9.4	8.7	8.4	8.4	8.6	9.1	9.9

Notes.

^aStandard modified blackbody form (e.g., Casey 2012) with a power-law spectral index for the dust emissivity of β_d = 1.6 (Eales et al. 1989; Klaas et al. 1997). ^bAssuming dust temperature T_d evolves as (35 K)${((1+z)/2.5)}^{0.32}$ (Béthermin et al. 2015) such that T_d ∼ 44–50 K at z ∼ 4–6. See Section 3.1.3. ^cEmpirical SED template fits to specific galaxies in the nearby universe (Silva et al. 1998). ^dUsing the conversion $\mathrm{SFR}={L}_{\mathrm{IR}}/({10}^{10}\,{L}_{\odot })$ appropriate for a Chabrier IMF (Kennicutt 1998; Carilli & Walter 2013).

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We consider tentative 2σ detections in our examination of our ASPECS field, instead of the usual 3σ or 3.5σ limit, to push as faint as possible in looking for evidence of obscured star formation. We can use this aggressive limit because of the relatively modest number of z = 2–10 sources over ASPECS and the availability of sensitive MIPS 24 μm observations and photometrically inferred physical properties to evaluate any tentative detections.

To ensure accurate far-IR flux measurements for sources over our HUDF mosaic, care was taken in determining the offset between the nominal sky coordinates for sources in our deep ALMA continuum observations and the positions in the ultraviolet, optical, and near-IR observations using the six best-continuum-detected sources over the HUDF (Aravena et al. 2016a). The positional offset between the images was found to be such that sources in our ALMA continuum image were positioned ∼03 to the south of sources in the HST mosaic, with ∼02 source-to-source scatter in the derived offset. Such source-to-source offsets from UV to far-IR are not surprising for bright sources, but are expected to be smaller for most of the fainter sources we are stacking, as the results we present in Section 3.2 indicate. Overall, on the basis of the source-to-source scatter, we estimate that we can register the HST and ALMA mosaics to better than 01 on average.

Flux measurements themselves were made using the nominal flux at the position of the source in the continuum map divided by the primary beam. We checked those flux measurements against those we derive after convolving the maps by the primary beam and looking at the flux at source center. For this latter procedure, we found that we recovered a flux that was less than 5% higher than using the flux at the position of the source.

For a more detailed summary of the ASPECS data set and the basic results, we refer the reader to Papers I and II in this series (Walter et al. 2016; Aravena et al. 2016a). Paper V in this series (Aravena et al. 2016b) provides a comprehensive discussion of the candidate [C ii] 158 μm lines identified in the band-6 data.

The high-redshift star-forming galaxies that we analyzed in this study were selected for this study using the best existing HST observations over the HUDF.

We briefly describe the z = 1.5–3.5 samples that we constructed over the HUDF using the available WFC3/UVIS, ACS/WFC, and WFC3/IR observations. For HST optical ACS/WFC and near-infrared WFC3/IR observations, we make use of the XDF reductions (Illingworth et al. 2013), which incorporated all ACS+WFC3/IR data available over the HUDF in 2013. The XDF reductions are ∼0.1–0.2 mag deeper than original reductions of Beckwith et al. (2006) at optical wavelengths and also provide coverage in the F814W band. The WFC3/IR reductions made available as part of the XDF release include all data from the original HUDF09 (Bouwens et al. 2011), CANDELS (Grogin et al. 2011; Koekemoer et al. 2011), and HUDF12 (Ellis et al. 2013) programs.

In the process of assembling our z = 1.5–3.5 samples, we derived our initial source catalogs and performed photometry on sources using our own modified version of the SExtractor (Bertin & Arnouts 1996) software. Source detection was performed on the square root of χ² image (Szalay et al. 1999: similar to a coadded image) constructed from the V₆₀₆, i₇₇₅, Y₁₀₅, J₁₂₅, JH₁₄₀, and H₁₆₀ images. After correcting fluxes for point-spread function (PSF) to match the H₁₆₀-band image, color measurements were made in scalable apertures in the style of Kron (1980) with a Kron factor of 1.6. "Total magnitude" fluxes were derived from the smaller scalable apertures by (1) correcting the flux to account for the additional flux seen in a larger scalable aperture (Kron factor of 2.5) as seen on the square root of χ² image and (2) correcting for the flux outside these larger scalable apertures and on the wings of the PSF using tabulations of the encircled energy (Dressel et al. 2012).

Galaxies at intermediate redshift z ∼ 2–3 were selected by applying simple two-color criteria using the Lyman-break galaxy (LBG) strategy:

$$\begin{eqnarray*}&&\begin{array}{l}({{UV}}_{275}-{U}_{336}\gt 1)\wedge ({U}_{336}-{B}_{435}\lt 1)\wedge \\ ({V}_{606}-{Y}_{105}\lt 0.7)\wedge (\text{S/N}({{UV}}_{225})\lt 1.5)\end{array}\end{eqnarray*}$$

for z ∼ 2 galaxies and

$$\begin{eqnarray*}&&({U}_{336}-{B}_{435}\gt 1)\wedge ({B}_{435}-{V}_{606}\lt 1.2)\wedge \\ &&({i}_{775}-{Y}_{105}\lt 0.7)\wedge ({\chi }_{{{UV}}_{225},{{UV}}_{275}}^{2}\lt 2)\end{eqnarray*}$$

for z ∼ 3 galaxies, where ∧, ∨, and S/N represent the logical AND, OR symbols, and signal-to-noise in our smaller scalable apertures, respectively. We define ${\chi }_{{{UV}}_{225},{{UV}}_{275}}^{2}$ as ${{\rm{\Sigma }}}_{i}\mathrm{SGN}{({{f}}_{i})({{f}}_{i}/{\sigma }_{i})}^{2}$ where f_i is the flux in bands UV₂₂₅ and UV₂₇₅ in a small scalable aperture, σ_i is the uncertainty in this flux, and SGN(f_i) is equal to 1 if f_i > 0 and −1 if f_i < 0. These criteria are similar to the two-color criteria previously utilized in Oesch et al. (2010) and Hathi et al. (2010).

Our z = 4–8 samples were drawn from the samples of Bouwens et al. (2015a) and include all z = 3.5–8.5 galaxies located over the 1 arcmin² ASPECS region. The samples of Bouwens et al. (2015a) were based on the deep optical ACS and WFC3/IR observations within the HUDF. z = 4–8 samples were constructed by applying Lyman-break-like color criteria to the XDF reduction (Illingworth et al. 2013) of the HUDF.

The z = 9–10 samples of R. J. Bouwens et al. (2016, in preparation) were constructed applying a Y₁₀₅ or J₁₂₅-dropout Lyman-break color criterion to the available HST data and then splitting the selected sources into z ∼ 9 and z ∼ 10 subsamples. Two sources from that z = 9–10 sample lie within the ASPECS region (see also Bouwens et al. 2011; Ellis et al. 2013; Oesch et al. 2013b).

To maximize the total number of star-forming galaxies at z ∼ 2–10 considered in this study, we also applied the EAZY photometric redshift code to the HST WFC3/UVIS, ACS, and WFC3/IR photometric catalogs we had available over our deep ALMA field and included all sources with a best-fit redshift solution between z ∼ 1.5 and z ∼ 8.5, which were not in our Lyman-break catalogs, and which utilized star-forming or dusty SED templates to reproduce the observed SED. We also made use of the photometric catalog of Rafelski et al. (2015) and included those sources in our samples, if not present in our primary two selections.

The star-forming galaxies selected by photometric redshift added 64, 31, 2, and 1 z ∼ 2, z ∼ 3, z ∼ 4, and z ∼ 7 galaxies to our study, respectively. Sources in our photometric-redshift selections showed almost identical distributions of properties to our LBG selections at z ∼ 2–3 (where our photometric-redshift selections add sources), with a median β and stellar mass of −1.84 and 10^8.41 M_⊙ for the photometric-redshift selections versus −1.82 and 10^8.37 M_⊙ for the z ∼ 2–3 LBG selections. 9% (7/79) and 3% (3/96) of the sources in our LBG and photometric-redshift selections, respectively, have measured β's redder than −1.

Eighty-one z ∼ 2 and 81 z ∼ 3 sources in total were identified using our dropout + photometric-redshift criteria over the 1 arcmin² region of the HUDF where we have deep ALMA observations. Our higher-redshift z ∼ 4, z ∼ 5, z ∼ 6, z ∼ 7, z ∼ 8, z ∼ 9, and z ∼ 10 samples (Bouwens et al. 2015a, 2016, in preparation) contain 80, 34, 30, 16, 6, 1, and 1 sources, respectively (Table 2) over this same region. The expected contamination rate in these color-selected samples by lower-redshift galaxies (or stars) is estimated to be of the order of 3%–8% (e.g., Bouwens et al. 2015a). In terms of apparent magnitude in the UV continuum, these sources extend from 21.7 mag to 30.8 mag (Figure 2: left panel).

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We provide a brief description of our estimates of the stellar mass for z = 2–10 sources over the HUDF. As in other work (e.g., Sawicki & Yee 1998; Brinchmann & Ellis 2000; Papovich et al. 2001; Labbé et al. 2005; Gonzalez et al. 2014), we estimate stellar masses for individual sources in our samples by modeling the observed photometry using stellar population libraries and considering variable (or fixed) star formation histories, metallicities, and dust content.

For convenience, we make use of the publicly available code FAST (Kriek et al. 2009) to perform this fitting. We assume a Chabrier (2003) IMF, a metallicity of 0.2 Z_⊙, an approximately constant SFR in modeling the star formation history while performing the fits (keeping the parameter τ equal to 100 Gyr where the star formation history is proportional to e^−t/τ), and we allow the dust extinction in the rest-frame V band to range from zero to 2 mag. Our fixing the fiducial metallicity to 0.2 Z_⊙ is well motivated based on studies of the metallicity of individual z ∼ 2–4 galaxies (Pettini et al. 2000) or as predicted from cosmological hydrodynamical simulations (Finlator et al. 2011; Wise et al. 2012). While the current choice of parameters can have a sizeable impact on inferred quantities such as the age of a stellar population (changing by >0.3–0.5 dex), these choices typically do not have a major impact (≳0.2 dex) on the inferred stellar masses.

In deriving the stellar masses for individual sources, we made use of flux measurements from 11 HST bands (UV₂₂₅, UV₂₇₅, U₃₃₆, B₄₃₅, V₆₀₆, i₇₇₅, z₈₅₀, Y₁₀₅, J₁₂₅, JH₁₄₀, H₁₆₀), one band in the near-IR from the ground (K_s), and four Spitzer/IRAC bands (3.6 μm, 4.5 μm, 5.8 μm, and 8.0 μm). The HST photometry we use for estimating stellar masses was derived by applying the same procedure as used for selecting our z ∼ 2–3 LBG samples (see Section 2.2).

Our Spitzer/IRAC flux measurements were derived for individual sources from ∼100–200 hr stacks of the IRAC observations over the HUDF (Labbé et al. 2015) from the IUDF program (PI: Labbé) and the Spitzer/IRAC program of Oesch et al. (2013a). As has become standard procedure (e.g., Labbé et al. 2005, 2015; Shapley et al. 2005; Grazian et al. 2006; Laidler et al. 2007; Merlin et al. 2015), we use the HST observations as a template to model the fluxes of sources in the Spitzer/IRAC observations and thus perform photometry below the nominal confusion limit. The model flux from neighboring sources is subtracted before attempting to measure fluxes for the sources of interest. Source photometry is performed in 18 diameter circular apertures for the Spitzer/IRAC 3.6 μm and 4.5 μm bands and 20 diameter circular apertures for the 5.8 μm and 8.0 μm bands. The observed fluxes are corrected to the total based on the inferred growth curve for sources after PSF correction to the Spitzer/IRAC PSF. We utilize a similar procedure to derive fluxes for sources based on the deep ground-based K-band observations available from VLT/HAWK-I, VLT/ISAAC, and PANIC observations (Fontana et al. 2014) over the HUDF (5σ depths of 26.5 mag).

A modest correction is made to the IRAC 3.6 μm and 4.5 μm photometry to account for the impact of nebular emission lines on the observed IRAC fluxes, decreasing the brightness of the 3.6 μm and 4.5 μm band fluxes by 0.32 mag to account for the presence of Hα and by 0.4 mag to account for the presence of [O iii]+Hβ emission where present. These corrections are well motivated based on observations of z ∼ 4–8 galaxies (Labbé et al. 2013; Stark et al. 2013; Smit et al. 2014, 2015; Marmol-Queralto et al. 2016; Rasappu et al. 2016) and lower the median inferred stellar mass for z > 3.8 galaxies in our sample by ∼0.1 dex.

The stellar masses we estimate for the highest-redshift sources in our selection, z > 5, are not as well constrained as at lower redshifts, where our sensitive photometry extends to rest-frame 1 μm. To guard against noise in the modeling process scattering lower-mass galaxies into higher-mass bins, we also model the photometry of galaxies and force the dust extinction to be zero in fitting the observed SEDs with FAST. For sources where the stellar-mass estimates exceed the dust-free stellar-mass estimates by more than 0.9 dex and the photometric evidence for a particularly dusty SED was weak (applicable to only six sources from our total sample of 330 sources), we made use of the dust-free stellar mass estimates instead.

We also estimated stellar masses for our sources by using the MAGPHYS software (da Cunha et al. 2008) to model the photometry for the 330 z = 2–10 sources that make up our samples. For sources with redshifts z < 3.8, the stellar masses we estimated were in excellent agreement with our fiducial results, with the median and mean stellar mass derived by MAGPHYS being 0.02 dex and 0.04 dex lower, respectively. This points toward no major systematic biases in the results from the present study—which rely on FAST-estimated masses—and from the other papers in the ASPECS series—where the reliance is on MAGPHYS-estimated masses.

The middle panel of Figure 2 illustrates the effective range in stellar mass probed by our z = 2–10 sample. Most sources from our HUDF z = 2–10 sample have stellar masses in the range 10^7.5 M_⊙ to 10^9.5 M_⊙. The most massive sources probed by our program extend from 10¹⁰ to 10^11.2 M_⊙. Beyond the stellar mass itself, Figure 2 also illustrates the range in UV-continuum slope β probed by our samples (see Section 3.1 for details on how β is derived). Since the measured β has been demonstrated to be quite effective in estimating the infrared excess for lower-redshift UV-selected samples (e.g., M99; Reddy et al. 2006; Daddi et al. 2007), it is useful for us to probe a broad range in β. As can be seen from Figure 2, our samples probe the range β ∼ −1.5 to ∼−2.5 quite effectively.

3.1.1. Expectations Using the z ∼ 0 IRX–β Relations

We commence our analysis of our ALMA HUDF observations by first asking ourselves which sources we might expect to detect, given various results at lower redshift. Such an exercise will help us to interpret the results that follow and also to evaluate whether or not the number of sources we detect and the rest-frame far-IR flux density we measure for z > 2 galaxies are similar to those found for galaxies at z ∼ 0.

We adopt as our z ∼ 0 baseline the now canonical IRX–β relationship of M99, where a connection was found between the infrared excess (IRX) of galaxies and the spectral slope of the UV continuum:

$$\begin{eqnarray}&&{M}99:{\mathrm{IRX}}_{{\rm{M}}99}=1.75({10}^{0.4(1.99(\beta +2.23))}-1)\end{eqnarray} \tag{ 1 }$$

The factor of 1.75 in the above relationship is needed to express the M99 relation in terms of the IR luminosity, rather than the far-IR luminosity utilized by M99. See the discussion in Section 5.1 of Reddy et al. (2006). This relationship implicitly includes the slope of the dust law of Calzetti et al. (2000). Despite modest scatter (∼0.3 dex), redder galaxies were systematically found to show higher infrared excesses than blue galaxies. For simplicity, the factor $B={BC}{(1600)}_{* }/{BC}(\mathrm{FIR})$ from M99 is taken to equal one, consistent with the measurements made in that study.

Importantly, the M99 IRX–β relationship was shown to have a basic utility that went beyond the z ∼ 0 universe for UV-selected samples. A series of intermediate-redshift studies (Reddy et al. 2006, 2010; Daddi et al. 2007; Pannella et al. 2009) found this relationship to be approximately valid when comparing the observed IR luminosities of galaxies to the predictions from the M99 IRX–β relationship.

As an alternative baseline, we also consider the expectations when adopting the so-called SMC IRX–β relationship, where a connection is again assumed between the infrared excess of a galaxy and its spectral slope in the UV continuum. However, since the SMC dust curve is steeper in the near-UV than dust laws like that of Calzetti et al. (2000), a small optical depth in dust extinction can have a large impact on the observed color of a galaxy in the UV continuum. The infrared excess, given an SMC extinction, can be expressed as follows:

$$\begin{eqnarray}&&\mathrm{SMC}:{\mathrm{IRX}}_{\mathrm{SMC}}={10}^{0.4(1.1(\beta +2.23))}-1\end{eqnarray} \tag{ 2 }$$

This relationship is derived based on the observational results of Lequeux et al. (1982), Prevot et al. (1984), and Bouchet et al. (1985: see also Pei 1992; Pettini et al. 1998; Smit et al. 2015).

For each of the z = 2–10 sources in our ALMA field, we fit the HST photometry in various bands probing the UV continuum to a power law ${f}_{1600}{(\lambda /1600\mathring{\rm A} )}^{\beta }$ to derive a mean flux at ∼1600 Å and also a spectral slope β. We derive a nominal luminosity for the source in the rest-frame UV by multiplying the flux density of the source at 1600 Å by the frequency at that wavelength (ν₁₆₀₀f₁₆₀₀) and convert that to an expected IR luminosity for the source (considered to extend from 8 μm to 1000 μm).²⁵

The equivalent flux at an observed wavelength of 1.2 mm is then computed by adopting a modified blackbody form with a dust temperature of 35 K and a power-law spectral index for the dust emissivity of β_d = 1.6. A value of T_d = 35 K is intermediate between the temperatures found for main-sequence galaxies by Elbaz et al. (2011) and Genzel et al. (2015), i.e., ∼30 K at z ∼ 2, and ∼37–38 K found for stacked sources in other studies (Coppin et al. 2015).

Since the flux density we would measure with ALMA is reduced somewhat by the effective temperature of the CMB at z ∼ 2–10, we multiply the measured flux by C_ν,

$$\begin{eqnarray}&&{C}_{\nu }=1-\displaystyle \frac{{B}_{\nu }({T}_{{\rm{CMB}}}(z))}{{B}_{\nu }(35\,{\rm{K}})},\end{eqnarray} \tag{ 3 }$$

to compute the expected signal. This treatment follows prescriptions given in da Cunha et al. (2013b).

Performing this exercise over all 330 z = 2–10 galaxies with coverage from our ALMA mosaic, we calculated expected fluxes for these sources at 1.2 mm assuming that sources follow the M99 and SMC IRX–β relations. These calculations suggested that 35 and 26 of these galaxies should be detected at ≥2σ and ≥3σ significance, respectively, in our observations if the M99 IRX–β relation applied, while eight and five of these sources would be detected at ≥2σ and ≥3σ significance, respectively, if the SMC IRX–β relationship applied to z = 2–10 galaxies in our samples.

To illustrate these expectations for our z ≥ 2 study, we present the predicted IR luminosities for our z ∼ 2–10 sample versus redshift in the left panel of Figure 3. The red solid and open circles indicate those sources for which a 3σ and 2σ detection, respectively, is expected in our 1.2 mm continuum observations, while the solid black circles indicate those sources for which a detection is not expected. The solid and dotted red lines indicate the lowest IR luminosities at which we would detect sources at 3σ and 2σ, respectively, over the ∼1 arcmin² ASPECS region.

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To help guide the discussion that follows, we provide a complete list of the sources with expected detections in Table 3. Comparisons of the actual flux measurements with estimates based upon various z ∼ 0 IRX–β relations provide us with a quantitative sense of how much these relations have evolved from z ≥ 3, while also illustrating the source-to-source scatter.

Table 3.  z ≳ 2 UV-Selected Sources Expected to Show Tentative 2σ Detections Adopting the M99 IRX–β Relationship and Assuming a 35 K Modified Blackbody SED and β_d = 1.6

							Predicted			Measured	Inferred
			${m}_{\mathrm{UV},0}$				${f}_{1.2\mathrm{mm}}$ ([μJy)			${f}_{1.2\mathrm{mm}}$
IDa	R.A.	Decl.	(mag)	zph	log10(M/M⊙)	βb	Calz.c	SMCc	Massc,d	(μJy)	LIR/(1010 L⊙)
z ∼ 2–3 Sample
XDFU-2397246112(C2)	03:32:39.72	−27:46:11.2	24.4	1.55e	11.21	0.3 ± 0.1	1426	99	946f	261 ± 25	50 ± 5
XDFU-2373546453(C5)	03:32:37.35	−27:46:45.3	23.7	1.85e	10.52	−0.5 ± 0.1	1028	121	552f	71 ± 14	14 ± 3
XDFU-2393346236	03:32:39.33	−27:46:23.6	25.5	2.59e	10.18	−0.5 ± 0.1	369	44	93f	−12 ± 13	−2 ± 2
XDFU-2370746171	03:32:37.07	−27:46:17.1	23.7	2.24e	10.09	−1.2 ± 0.1	350	67	306f	34 ± 14	6 ± 2
XDFU-2358146436	03:32:35.81	−27:46:43.6	24.6	1.90e	9.98	−0.9 ± 0.1	194	31	75	16 ± 52	3 ± 10
XDFU-2356746283	03:32:35.67	−27:46:28.3	25.2	3.17	9.93	−1.3 ± 0.1	153	30	108f	−13 ± 21	−2 ± 4
XDFU-2385446340(C1)	03:32:38.54	−27:46:34.0	24.3	2.54e	9.9	−1.2 ± 0.1	242	47	143f	571 ± 14	97 ± 2
XDFU-2388246143	03:32:38.82	−27:46:14.3	26.3	3.38	9.85	−0.5 ± 0.1	321	38	37f	10 ± 14	2 ± 2
XDFU-2387446541	03:32:38.74	−27:46:54.1	25.7	2.74	9.85	−1.2 ± 0.1	81	15	40	18 ± 26	3 ± 4
XDFU-2365446123	03:32:36.54	−27:46:12.3	24.1	1.87e	9.77	−1.5 ± 0.1	85	19	68f	38 ± 16	7 ± 3
XDFU-2384246348	03:32:38.42	−27:46:34.8	23.8	2.70	9.75	−2.0 ± 0.1	46	13	181f	36 ± 14	6 ± 2
XDFU-2369146023	03:32:36.91	−27:46:02.3	24.1	2.33e	9.47	−1.8 ± 0.1	65	17	56f	13 ± 16	3 ± 3
XDFU-2395845544	03:32:39.58	−27:45:54.4	24.4	3.21e	9.46	−1.5 ± 0.1	215	47	78	14 ± 64	2 ± 11
XDFU-2369146348	03:32:36.91	−27:46:34.8	24.7	1.76	9.46	−1.3 ± 0.1	75	15	17	9 ± 14	2 ± 3
XDFU-2370846470	03:32:37.08	−27:46:47.0	24.4	1.85e	9.44	−1.3 ± 0.1	102	20	23	0 ± 15	0 ± 3
XDFU-2363346155	03:32:36.33	−27:46:15.5	25.3	2.34	9.32	−1.6 ± 0.1	35	8	13	1 ± 16	0 ± 3
XDFU-2366846484	03:32:36.68	−27:46:48.4	25.3	1.88e	9.27	−1.0 ± 0.1	84	14	8	−10 ± 19	−2 ± 4
XDFU-2366946210	03:32:36.69	−27:46:21.0	24.9	1.96	9.26	−1.7 ± 0.1	28	7	11	2 ± 13	0 ± 3
XDFU-2382946284	03:32:38.29	−27:46:28.4	26.2	1.76	9.22	−1.0 ± 0.1	32	5	3	−2 ± 14	−0 ± 3
XDFU-2378846451	03:32:37.88	−27:46:45.1	26.4	1.89	8.9	−0.9 ± 0.1	37	6	1	3 ± 15	0 ± 3
XDFU-2372446294	03:32:37.24	−27:46:29.4	27.2	3.25	8.77	−1.2 ± 0.1	28	5	1	−34 ± 14	−6 ± 2
XDFU-2379146261	03:32:37.91	−27:46:26.1	27.1	2.48	8.41	−0.4 ± 0.1	85	10	0	0 ± 13	0 ± 2
XDFU-2379046328	03:32:37.90	−27:46:32.8	26.8	3.38	8.25	−1.4 ± 0.1	30	6	1	−3 ± 14	−1 ± 2
z ∼ 4 Sample
XDFB-2394046224	03:32:39.40	−27:46:22.4	25.5	2.94g	9.76	−1.2 ± 0.1	119	22	45f	−22 ± 13	−3 ± 2
XDFB-2368245580	03:32:36.82	−27:45:58.0	24.4	3.87e	9.45	−1.5 ± 0.1	291	64	105f	−6 ± 22	−1 ± 3
XDFB-2375446199	03:32:37.54	−27:46:19.9	25.9	4.21	9.14	−1.3 ± 0.1	139	27	16	4 ± 14	1 ± 2
XDFB-2394246267	03:32:39.42	−27:46:26.7	27.1	4.99	8.73	−0.7 ± 0.2	200	28	3	−8 ± 13	−1 ± 2
XDFB-2381646267h	03:32:38.16	−27:46:26.7	28.5	4.16	8.57	−0.6 ± 0.4	50	7	0	−21 ± 13	−3 ± 2
z ∼ 5 Sample
XDFV-2372946175	03:32:37.29	−27:46:17.5	28.1	5.00	8.92	−1.1 ± 0.3	38	7	2	−1 ± 14	−0 ± 2
z ∼ 6 Sample
GSDI-2374046045	03:32:37.40	−27:46:04.5	26.7	5.85	9.52	−1.5 ± 0.8	71	16	34f	4 ± 14	1 ± 2
XDFI-2374646327	03:32:37.46	−27:46:32.7	26.4	6.49	9.35	−1.5 ± 0.2	111	25	34f	12 ± 14	2 ± 2
XDFI-2364964171	03:32:36.49	−27:46:41.71	25.5	6.12	8.97	−2.0 ± 0.2	50	14	31	−11 ± 18	−2 ± 2
GSDI-2382846172	03:32:38.28	−27:46:17.2	26.2	6.12	8.66	−1.9 ± 0.3	34	9	8	−4 ± 14	−1 ± 2
XDFI-2378346180h	03:32:37.83	−27:46:18.0	29.3	6.20	8.54	−0.3 ± 0.6	87	9	0	1 ± 14	0 ± 2
z ∼ 7 Sample
XDFZ-2381446048h	03:32:38.14	−27:46:04.8	29.5	6.66	7.98	0.2 ± 1.7	196	14	0	−11 ± 15	−2 ± 2

Notes.

^aSource ID from Bouwens et al. (2015a). Otherwise selected from either a new catalog constructed here or the catalog of Rafelski et al. (2015) based on the HST WFC3/UVIS, ACS, and WFC3/IR observations over the HUDF. C1, C2, and C5 correspond to the continuum detections identified in our blind search of our ALMA 1.2 mm observations (Paper II from this series: Aravena et al. 2016a). ^bUV-continuum slope β estimated by fitting the UV-continuum fluxes to a power law (Bouwens et al. 2012; Castellano et al. 2012; Rogers et al. 2013). ^cAssuming a standard modified blackbody SED with dust temperature of 35 K and accounting for the impact of the CMB on the measured flux (da Cunha et al. 2013b). ^dAssuming the consensus z ∼ 2–3 relationship between the infrared excess and the inferred stellar mass of the galaxy (Appendix A). ^eSpectroscopic redshift from 3D-HST (Momcheva et al. 2015). ^fTentative ≳2σ detection of the source is expected. ^gThe B₄₃₅-dropout color selection criterion from Bouwens et al. (2015a) identifies galaxies with photometric redshifts as low as z ∼ 3. We retain this source in our z ∼ 4 sample, consistent with the B₄₃₅-dropout selection function of Bouwens et al. (2015a). ^hThese sources are nominally expected to be detected in our ALMA mosaic based on their very red measured β's. However, the β measurements for these sources are quite uncertain. It is anticipated that a few of the reddest z = 6–8 galaxies over our small field would have these colors due to the impact of noise.

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M99 found that individual sources exhibited a 0.3 dex scatter in L_FIR around the IRX–β relationship preferred in that study. If we include a similar scatter in predicting L_IR for individual sources, we predict 36.9 2σ and 28.4 3σ detections instead of 35 and 26, respectively.

The UV-continuum slopes β we use in setting these expectations are not known precisely, especially for the faintest sources in our z = 6–8 samples. In particular, if a source is measured to have an especially red β due to the impact of noise, we would predict its detection in the ASPECS data even if this source is actually intrinsically blue. The impact of the scatter is asymmetric since faint blue sources—with β's in the range of ∼−2 to ∼−2.3 (e.g., Wilkins et al. 2011; Dunlop et al. 2013; Kurczynski et al. 2014)—are already predicted to show essentially no dust emission and so the expected emission can only be larger when adding noise to the photometry of faint sources.

To determine the impact that this would have on the expected number of detected sources, we perturbed the measured β's for individual sources by the estimated uncertainty, and we calculated the total number of sources we would expect to find. Repeating this exercise multiple times, we found that this would boost the expected number of detections by ∼3.8 sources to 38.8 in total. This simulation result suggests that noise in the HST photometry does boost the expected numbers above what they would be in the noise-free case (by ∼11%). If we suppose that a similar correction applies to our nominal expectations for tentative detections (35 sources), ∼31.6 may be a better estimate for the expected number of tentative detections of z ∼ 2–10 galaxies in ASPECS.

3.1.2. Expectations Using the $z\sim 2$ IRX–Stellar Mass Relation and Assuming ${L}_{{IR}}={L}_{\mathrm{UV}}$

3.1.3. Impact of the Dust Temperature

Here we look for possible individual detections of z = 2–10 galaxies over our deep ALMA continuum map at 1.2 mm. As results from the previous section illustrate, we could reasonably expect the number of detections to be modest if various IRX–β relations from the z ∼ 0 universe serve as a useful guide.

Table 4 provides a summary of the properties of the z ≳ 2 sources from our catalog of 330 z = 2–10 sources that are nominally tentatively detected at ≳2σ in our data. The measured flux density for the detected sources was derived by taking the value in our 1.2 mm continuum image at the nominal optical position of each source in our LBG samples (after correcting for the 03 positional offset between the ALMA and optical maps). We verified that we would retain all of our most significantly detected sources from this table if we derived flux densities for sources using other methods (e.g., by scaling the normalization of the primary beam to fit the pixels in a 3'' × 3'' aperture centered on a source).

Table 4.  z ≳ 2 UV-Selected Galaxies Showing Tentative 2σ Detections in Our Deep ALMA Continuum Observations^a

							Predicted			Measured	Inferred
			${m}_{\mathrm{UV},0}$				${f}_{1.2\mathrm{mm}}$ (μJy)			${f}_{1.2\mathrm{mm}}$
ID	R.A.	Decl.	(mag)	zph	log10(M/M⊙)	β	Calz.	SMC	Mass	(μJy)	LIR/(1010 L⊙)
Tentative >2σ Detections (Most Credible)b
XDFU-2397246112(C2)	03:32:39.72	−27:46:11.2	24.4	1.55c	11.21	0.3 ± 0.1	1426	99	946	261 ± 25	50 ± 5
XDFU-2373546453(C5)	03:32:37.35	−27:46:45.3	23.7	1.85c	10.52	−0.5 ± 0.1	1028	121	552	71 ± 14	14 ± 3
XDFU-2370746171	03:32:37.07	−27:46:17.1	23.7	2.24c	10.09	−1.2 ± 0.1	350	67	306	34 ± 14	6 ± 2
XDFU-2385446340(C1)	03:32:38.54	−27:46:34.0	24.3	2.54c	9.90	−1.2 ± 0.1	242	47	143	571 ± 14	97 ± 2
XDFU-2365446123	03:32:36.54	−27:46:12.3	24.1	1.87c	9.77	−1.5 ± 0.1	85	19	68	38 ± 16	7 ± 3
XDFU-2384246348	03:32:38.42	−27:46:34.8	23.8	2.70	9.75	−2.0 ± 0.1	46	13	181	36 ± 14	6 ± 2
Not Especially Credible >2σ Detectionsd
XDFU-2403146258	03:32:40.31	−27:46:25.8	27.6	1.55	8.21	−1.9 ± 0.1	1	0	0	54 ± 26	10 ± 5
XDFB-2355846304	03:32:35.58	−27:46:30.4	29.9	4.06	8.15	−1.9 ± 1.0	1	0	0	71 ± 26	11 ± 4
>2σ "Detections" in the Negative Continuum Image
XDFU-2372446294	03:32:37.24	−27:46:29.4	27.2	3.25	8.77	−1.2 ± 0.1	28	5	1	−34 ± 14	−6 ± 2
XDFU-2390646560	03:32:39.06	−27:46:56.0	26.7	2.42	8.75	−1.5 ± 0.1	12	3	1	−82 ± 38	−14 ± 7
XDFU-2390446358	03:32:39.04	−27:46:35.8	28.1	3.25	8.62	−2.0 ± 0.2	1	0	0	−33 ± 14	−6 ± 2
XDFU-2375346041	03:32:37.53	−27:46:04.1	26.4	1.96	8.26	−2.1 ± 0.1	1	0	0	−32 ± 14	−6 ± 3
XDFU-2369446426	03:32:36.94	−27:46:42.6	28.6	1.60	8.08	−1.3 ± 0.2	2	0	0	−32 ± 14	−6 ± 3
XDFB-2401746314	03:32:40.17	−27:46:31.4	28.3	4.01	8.21	−2.0 ± 0.3	1	0	0	−56 ± 26	−8 ± 4
XDFV-2385645553	03:32:38.56	−27:45:55.3	29.8	5.07	6.99	−1.6 ± 0.9	3	1	0	−72 ± 26	−10 ± 4
XDFZ-2375446018	03:32:37.54	−27:46:01.8	29.3	7.05	7.85	−2.3 ± 1.5	0	0	0	−40 ± 16	−6 ± 2

Notes.

^aColumns in this table are essentially identical to those in Table 3. ^bThe reality of each of these tentatively detected sources is supported by there not being any comparable detections of >10^9.75 M_⊙ sources in the negative continuum images and each of these sources also showing a detection in the MIPS $24\,\mu {\rm{m}}$ observations (see Table 10 from Appendix B). ^cSpectroscopic redshift from 3D-HST (Momcheva et al. 2015). ^dWhile fewer low-mass (<10⁹ M_⊙) galaxies are tentatively detected at >2σ in the positive continuum image than in the negative continuum image, this appears not to be statistically significant. The use of spatially offset positions (by 03) to measure the flux in sources typically results in an essentially equal number of tentative >2σ detections in the positive and negative continuum images.

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Only two sources from the entire catalog are detected at ≫3σ significance. They are the z = 2.54 source XDFU-2385446340, where the detection significance is indeed very high, i.e., 41σ, with ${f}_{1.2\mathrm{mm}}=571\pm 14\,\mu$Jy, and the z = 1.55 source XDFU-2397246112, where the detection significance is 10σ, with ${f}_{1.2\mathrm{mm}}=261\pm 25\,\mu$Jy. In Paper IV in this series (Decarli et al. 2016), we discuss the far-IR SED and molecular gas properties of both sources in more detail. Based on its X-ray flux in the deep Chandra observations over the Chandra Deep Field South (Xue et al. 2011), the latter source (XDFU-2397246112) is known to host an X-ray active galactic nucleus (AGN).

Six other sources from our catalogs show convincing >3.5σ or tentative >2σ detections in our ASPECS data. However, two of these detections appear to be noise spikes. This can been seen by looking for similar >2σ detections in the negative continuum image for sources with similar stellar masses (also presented in Table 4). For the sources with the highest masses (i.e., >10^9.75 M_⊙), only four positive 2σ detections are found and no 2σ "detections" in the negative images. The positive detections correspond to XDFU-2373546453, XDFU-2370746171, XDFU-2365446123, and XDFU-2384246348 with 5.1σ, 2.6σ, 2.4σ, and 2.3σ detections, respectively. Given that there are only 13 sources in our highest-mass sample and six of them show at least a tentative >2.3σ detection in our ALMA observations (expected only 1% of the time assuming Gaussian noise), each of these detections is likely real.²⁶

We remark that each of these four sources is also detected in the MIPS 24 μm observations at ≳2σ significance, providing further support for our conclusions here (see Table 10 from Appendix B). This also points toward MIPS 24 μm data being a valuable probe of the infrared excess to z ∼ 3, given its competitive sensitivity to long exposures with ALMA.

For sources with estimated stellar masses in the range 10⁷ to 10⁹ M_⊙, tentative 2σ detections are seen in both the positive and negative images. The excess numbers in the negative image appear not to be statistically significant, because small changes to the positions where the flux measurements are made (by ∼01) typically result in essentially identical numbers of tentative 2σ detections in the positive and negative continuum images.

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To illustrate the significance of this apparent dependence on the inferred stellar mass, we present in Figure 5 the fraction of detected z = 2–10 galaxies versus mass, after correcting the positive >2σ detections in a mass bin for the >2σ detection seen in the negative image. For this figure, we consider only those sources (172 out of 330) over the ASPECS field for which the 1.2 mm continuum senstivities are the highest, i.e., with 1σ rms noise <17 μJy. ${63}_{-17}^{+14}$% of the galaxies with stellar mass estimates >10^9.75 M_⊙ are detected; none of the sources with masses lower than 10^9.75 M_⊙ is detected. Dust-continuum emission shows a clear connection with the apparent stellar mass in galaxies—which is similar to a few prominent earlier predictions for the expected findings from a deep far-IR continuum survey over the HUDF with ALMA (da Cunha et al. 2013b).

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As a separate illustration of the predictive power of stellar mass in estimating the approximate luminosity of galaxies in the IR, we present in Figure 6 the range in stellar mass versus redshift probed by our HUDF sample and indicate the sources we identify as detected in red (open and solid circles detected at >3σ and 2σ–3σ, respectively) and those we identify as only found in a blind search in green (ASPECS Paper II: Aravena et al. 2016a). The sources we detected at >3σ also appear in the blind search of Aravena et al. (2016a). It is clear that stellar mass is a good predictor of which sources are IR-luminous, for galaxies with z > 1.5. The stellar masses used for constructing Figure 6 are taken from the 3D-HST catalogs (Skelton et al. 2014) if at z < 1.5 (if available); otherwise, they are inferred as in Section 2.3. For the most obscured systems, estimates of the redshift and stellar mass can be quite uncertain (given degeneracies between dust and age and the challenge in locating spectral breaks), so some caution is needed in interpreting this figure.

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We now return to discussing continuum-detected sources in ASPECS. 1.2 mm continuum images of the six sources showing meaningful detections are presented in Figure 4 together with their HST and Spitzer/IRAC images. We remark that the detected sources in the present samples are much fainter than those identified in many previous programs. For example, the typical flux measured by Scoville et al. (2016) for detected z ∼ 4.4 galaxies in their very high-mass (≳2 × 10¹⁰ M_⊙) sample is ∼200 μJy, which contrasts with the ∼35 μJy seen in the three faintest z ∼ 2–3 galaxies tentatively detected here. The observed differences in the typical fluxes of detected sources are a natural consequence of the relative sensitivities of the data sets, i.e., 12.7 μJy beam⁻¹ rms for ASPECS versus 65 μJy beam⁻¹ rms in the observations of Scoville et al. (2016).

In addition to looking at which z = 2–10 galaxies over the HUDF we can individually detect in our ALMA continuuum observations, we can gain powerful constraints on dust emission from high-redshift galaxies by stacking. For this, we subdivide our samples in terms of various physical properties and then make a weighted stack of the ALMA-continuum observations at the positions of the candidates.

For sources included in the stack, we map the ALMA continuum maps onto the same position and weight the contribution of each source to the stack according to its expected 1.2 mm continuum signal assuming L_IR ∝ L_UV and according to the inverse square of the noise (per beam). We derive a flux from the stack based on a convolution of the image stack (33 × 33 aperture) with the primary beam. No spatial extent is assumed in the stacked flux.

3.3.1. IRX versus Stellar Mass

We begin by looking at the average inferred infrared excesses of z = 2–10 galaxies as a function of the stellar mass. Segregating our samples in terms of stellar mass certainly is a logical place to start. Not only is there strong support in the literature for such a correlation at lower redshifts (e.g., Pannella et al. 2009, 2015; Reddy et al. 2010), but there is evidence for this correlation being present in our own limited samples (see Section 3.2).

In Figure 7, we show the stacked 1.2 mm continuum observations of z = 2–3 and z = 4–10 galaxies in four different bins of stellar mass: >10^9.75 M_⊙, 10^9.25–10^9.75 M_⊙, 10^8.75–10^9.25 M_⊙, and <10^8.75 M_⊙. For these stacks, we weight sources according to the square of the expected signal in the 1.2 mm continuum observations (assuming L_IR ∝ L_UV) and the inverse square of the noise (in μJy), i.e., ${({L}_{\mathrm{UV}}/\sigma ({f}_{1.2\mathrm{mm}}))}^{2}$.²⁷

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The implied constraints on IRX as a function of stellar mass are presented in Figure 8, Table 5, and Table 12 from Appendix D for both our z = 2–3 and z = 4–10 samples. Significantly enough, the only mass bin where we find a detection is for >10^9.75 M_⊙ galaxies at z = 2–3. This is not surprising since six of the 11 sources that compose this mass bin show at least tentative individual detections (≳2σ) in our ALMA observations. All the other mass bins are consistent with the infrared excess showing an approximate 2σ upper limit of IRX ∼ 0.4 for <10^9.75 M_⊙ galaxies.

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Table 5.  Inferred IRX versus Galaxy Stellar Mass and β from ASPECS (assuming T_d = 35 K and β_d = 1.6)^a

		# of
Stellar Mass	β	sources	IRXb
z = 2–3
>109.75 M⊙	All	11	${3.80}_{-2.40}^{+3.61}$ ± 0.19
<109.75 M⊙	All	151	${0.11}_{-0.42}^{+0.32}$ ± 0.34
z = 4–10
>109.75 M⊙	All	2	−0.49${}_{-1.13}^{+0.69}$ ±  0.71
<109.75 M⊙	All	166	${0.14}_{-0.14}^{+0.15}$ ± 0.18
z = 2–3
>109.75 M⊙	−4 < β < −1.75	1	${0.54}_{-0.00}^{+0.00}$ ± 0.29
	−1.75 < β < −1.25	2	${1.31}_{-0.94}^{+0.67}$ ± 0.72
	β < −1.25	8	${6.79}_{-4.51}^{+5.38}$ ± 0.26
<109.75 M⊙	−4 < β < −1.75	89	${0.19}_{-0.76}^{+0.40}$ ± 0.44
	−1.75 < β < −1.25	49	−0.01 ${}_{-0.35}^{+0.39}$ ±  0.58
	β < −1.25	13	−0.14 ${}_{-3.64}^{+4.65}$ ±  2.11
z = 4–10
<109.75 M⊙	−4 < β < −1.75	122	${0.03}_{-0.15}^{+0.22}$ ± 0.24
	−1.75 < β < −1.25	29	${0.33}_{-0.16}^{+0.11}$ ± 0.29
	β < −1.25	11	−1.03${}_{-1.23}^{+0.29}$ ±  1.46

Notes.

^aSee Tables 12–14 from Appendix D for a more detailed presentation of the stack results summarized here. ^bBoth the bootstrap and formal uncertainties are quoted on the result (presented in that order).

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Making use of the collective constraints across our z ∼ 2–10 sample, we find an approximate 2σ upper limit on the infrared excess of 0.4 for lower-mass (<10^9.75 M_⊙) galaxies. This suggests that dust emission from faint UV-selected sources is typically small.

In our stacking experiments, we also compute a constraint on the flux at 1.2 mm relative to the flux in the UV continuum. For these results, sources are weighted according to the square of their UV-continuum fluxes and inversely according to the noise in the ALMA 1.2 mm observations. Making use of all sources in our z ∼ 2–3 and z ∼ 4–10, <10^9.75 M_⊙ samples, we find 2σ upper limits of 20 and 44, respectively, on the ratio of fluxes at 1.2 mm and in the UV continuum.

The impact of this result is illustrated in Figure 9, by comparing current constraints against several possible SED templates at z ∼ 2–3 and z ∼ 4–10. The result provides information on the overall shape of the spectral energy distribution that is independent of the assumed SED template.

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3.3.2. IRX versus $\beta$

Next we subdivide our z = 2–10 samples in terms of their UV-continuum slopes. Given evidence that the infrared excess depends significantly on β at z ∼ 0 (M99) and also at z ∼ 2 (Reddy et al. 2006, 2010; Daddi et al. 2007; Pannella et al. 2009), we want to quantify this dependence in our own sample. We split our results by stellar mass (i.e., <10^9.75 M_⊙ and >10^9.75 M_⊙) motivated by the results of the previous section.

We examine the IRX–β relation for z = 2–3 sources with >10^9.75 M_⊙ in Figure 10, Table 5, and Table 13 from Appendix D using three different bins in β. The only source from the present ASPECS sample that shows a prominent X-ray detection (XDFU-2397246112) is excluded. Stacks of the ALMA continuum images at the positions of the sources are provided in Figure 11, after excluding those sources detected at >4σ.

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Given the small sample size, it is difficult to compute accurate uncertainties on the IRX–β relationship at z ∼ 2–3, but the large red solid circles give our best estimates. The present constraints appear most consistent with an SMC IRX–β relationship.

We formalize this analysis by calculating the region of the IRX–β plane preferred at 68% confidence. For this, we compare the derived IRX–β relationship with what would be predicted based on dust laws with various slopes ${{dA}}_{\mathrm{UV}}/d\beta$ (where ${\rm{IRX}}={10}^{0.4({{dA}}_{\mathrm{UV}}/d\beta )(\beta +2.23)}-1$). The result we obtain for dA_UV/dβ is ${1.26}_{-0.36}^{+0.27}$ (${1.26}_{-0.91}^{+0.49}$ at 95% confidence) and is presented in Figure 10 as a light-red shaded region. It is most consistent with an SMC dust law (where dA_UV/dβ ∼ 1.1).

We also quantify the IRX–β relationship for the lower-mass sources at z = 2–3 and present the stack results in Figure 10 as 2σ upper limits (downward green arrows) and also in Table 13 from Appendix D. The limits are much lower than the constraints we obtained for the highest-mass galaxies considered here and indicate that the IRX–β relationship depends on the stellar mass of the sources. As with our higher-mass sample, we use the stacked constraints to constrain the IRX–β relation (shown with the light-green shaded region in Figure 10), finding dA_UV/dβ < 1.22 at 95% confidence.

We also derive constraints on the IRX–β relationship for our z = 4–10 sample. This sample contains only two galaxies with stellar masses in excess of 10^9.75 M_⊙—neither of which is detected in our ALMA observations—so we do not consider a higher-mass subsample of galaxies. The image stamps showing the stack results are presented in Figure 11, while the infrared excess derived from these stack results is presented in Table 13 from Appendix D and in Figure 12. As in Figure 10, we determine the implications of these constraints for the IRX–β relationship and present the result in Figure 12 using a green-shaded contour. The 2σ upper limit we derive for dA_UV/dβ is 0.87. Again, our derived constraints on dA_UV/dβ suggest that dust emission from lower-mass <10^9.75 M_⊙ galaxies is generally quite small.

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Uncertainties in the measured UV-continuum slopes β also have an impact on these results, because scatter toward redder colors could cause us to include intrinsically blue sources in the reddest β bins. Since sources with the reddest colors are expected to have these colors due to dust extinction, these bins have significantly more leverage in determining the value that we derive for dA_UV/dβ. Noise has a particularly significant impact on the β's derived from the highest-redshift sources in our samples, i.e., particularly at z = 7–8.

To determine the impact of noise on the values we derive for dA_UV/dβ, we perturbed the best-estimate β values for individual sources by our uncertainty estimates on each β determination, rebinned the sources as for our fiducial results, and then rederived the dA_UV/dβ for the perturbed data set. We repeated this process 10 times and we found that the 2σ upper limit on dA_UV/dβ decreased on average by 0.1 for our low-mass, z = 4–10 sample, but did not have a noticeable impact on the dA_UV/dβ value we derived for our z = 2–3 sample.

We therefore correct the 1σ upper limit we derive on dA_UV/dβ by 0.0 and 0.1 for our low-mass z = 2–3 and z = 4–10 samples, respectively. This translates into 2σ upper limits on dA_UV/dβ of 1.22 and 0.97 for our low-mass samples at z = 2–3 and z = 4–10, respectively. All of the present constraints on dA_UV/dβ for sources at different redshifts and with different stellar masses are summarized in Table 6.

Table 6.  Present Constraints on the IRX–β Relationship

Sample	Mass Range	dAUV/dβ
Current Determinations
z ∼ 2–3	>109.75 M⊙	${1.26}_{-0.36}^{+0.27}$
z ∼ 2–3	<109.75 M⊙	<1.22a
z ∼ 4–10	<109.75 M⊙	<0.97a,b
Canonical IRX–β Relations
Meurer/Calzetti		1.99
SMC		∼1.10

Notes.

^aUpper limits are 2σ. ^bUpper limit is corrected for the expected noise in the derived β values.

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The conclusions here can be impacted by our assumed dust temperatures. If we assume monotonically higher dust temperatures at z ≥ 3 such that T_d ∼ 44–50 K at z ∼ 4–6, then the 2σ upper limit on the IRX–β relation increases by 0.37 dex (shown as a dotted line), consistent with an SMC IRX–β relationship. Previously, Ouchi et al. (1999) had argued on the basis of SCUBA 850 μm observations over the Hubble Deep Field North (Hughes et al. 1998) that z ≳ 3 star-forming galaxies could be consistent with the z ∼ 0 M99 relation only if the dust temperature was ≳40 K.

The present results are not especially different from IRX versus β results found by Capak et al. (2015) for a small sample of z ∼ 5–6 galaxies, where most of the sources in their sample lie below the SMC relation, but are clearly lower than the results of Coppin et al. (2015) where IRX was found to be ∼8 for z ∼ 3 sources with β's of ∼−2 to −1.5. The results of Coppin et al. (2015) were based on a deep stack of SCUBA-2 (Holland et al. 2013) Cosmology Legacy Survey data (Geach et al. 2013) over the UKIDSS-UDS field (Lawrence et al. 2007). The explanation for differences relative to the results of Coppin et al. (2015) is not entirely clear.²⁸

3.3.3. IRX versus Apparent Magnitude in the Rest-frame UV

Essentially nothing is known about the typical dust temperature for sub-L* star-forming galaxies at z ∼ 2–10. While many star-forming galaxies have measured dust temperatures of ∼25–30 K (Elbaz et al. 2011; Magnelli et al. 2014; Genzel et al. 2015), there are many faint, individually detected sources that have much higher dust temperatures (Sklias et al. 2014), i.e., ∼40–50 K. Moreover, the dust temperature is known to depend on its specific SFR relative to that median value on the main sequence, taking values of ∼20 K, ∼30 K, or ∼40 K depending on whether a galaxy is below, on, or above the main sequence, respectively (Elbaz et al. 2011; Genzel et al. 2015).

Uncertainties in the dust temperature of lower-mass, z ≥ 2 galaxies are important since the results we derive depend significantly on the form of the far-IR SED we assume. To illustrate, if we assume that the dust temperatures are lower than 35 K, it would imply lower IR luminosities (and stronger upper limits on the luminosities). On the other hand, if we assume that the dust temperature of sub-L* galaxies at z ∼ 2–10 is higher than 35 K, it would imply higher IR luminosities (and weaker upper limits on the IR luminosity) for z ≥ 2 sources probed by the ASPECS program. The latter possibility would appear to be a particularly relevant one to consider in light of recent results from Magdis et al. (2012) and Béthermin et al. (2015), which have suggested that the mean intensity in the radiation field (and hence the typical dust temperature) of galaxies with moderate to high masses increases substantially toward higher redshifts, i.e., as (1 + z)^0.32 (Section 3.1.3), and therefore T_d is in the range 44–50 K at z = 4–10.

For convenience, we provide a table in Appendix C that indicates how the derived luminosities would change depending on the SED template or dust temperature assumed. Included in this table are dust temperatures ranging from 25 K to 45 K and also empirical SED templates for M51, M82, Arp 220, and NGC 6946 from Silva et al. (1998). The typical amplitude of these dependencies is a factor of 3 at z ∼ 2–3, a factor of 2 at z ∼ 6, and a factor of <1.5 at z ∼ 8–10.

While we would expect some uncertainties in the infrared excesses or IR luminosities we derive from the ALMA data, we have verified that the derived values are nonetheless plausible in the redshift range z ∼ 2–3 by comparing them with independent estimates made from the MIPS 24 μm observations and using a prescription from Reddy et al. (2010) to convert these 24 μm fluxes to IR luminosities (Appendix B). The IR luminosities we derive for the few detected sources agree to within 0.3 dex with the ALMA-estimated luminosities (Table 10 from Appendix B) if we adopt a fiducial dust temperature of 35 K. Even better agreement is obtained if we adopt higher values for the dust temperature.

Finally, we combine the current constraints on the infrared excess with those available in the literature to construct a more complete picture of the impact of dust obscuration on the overall energy output from star-forming galaxies at z = 4–10.

We focus in particular on constraints available from ALMA on luminous z = 4–10 galaxies due to the limited number of sources within the 1 arcmin² footprint of ASPECS. We focus on 12 fairly luminous z = 5–7 galaxies originally identified as part of wide-area surveys in the rest-frame UV (e.g., Willott et al. 2013) and recently examined with ALMA by Capak et al. (2015) and Willott et al. (2015). Results from those studies are particularly useful, since the 1.2 mm continuum fluxes, UV luminosities, and estimates of the stellar mass for individual sources are available. Six of the 12 galaxies from those two studies show tentative detections in the available ALMA data.

Combining our own results with those from Capak et al. (2015) and Willott et al. (2015)—self-consistently converting the observed ALMA fluxes to IR luminosities—we present our IRX versus stellar mass constraints in Figure 13 with the solid blue squares and large blue upper limits. For context, we compare these results with the consensus IRX–stellar mass relationship we derive from various results on IRX–mass found at z ∼ 2–3 (Reddy et al. 2010; Whitaker et al. 2014; Álvarez-Márquez et al. 2016: see Appendix A).

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For the case of a fixed dust temperature of 35 K, we find we can approximately match the current constraints at z = 4–10 (light thick dotted blue line in Figure 13) using the consensus IRX–stellar mass relationship at z ∼ 2–3 if sources of a given stellar mass exhibit IR luminosities at least ∼0.5 dex lower than at z ∼ 2–3. This would suggest lower dust extinctions at high redshift at a fixed stellar mass.

However, we should also look at how evolution in the dust temperature could impact the results. If the dust temperature exhibited a monotonic increase toward higher redshift, e.g., as found by Magdis et al. (2012) and Béthermin et al. (2015), we would infer much higher (by ∼0.4–0.5 dex) IR luminosities for the detected sources from the three samples considered here. This would translate into similarly higher infrared excesses at z = 4–10, which would be plausibly consistent with the IRX–stellar mass relationship at z = 0–3 (right panel of Figure 13), suggesting no significant evolution in this relationship from z ∼ 6 to $z\sim 0$.

Pannella et al. (2015) found no strong evidence for evolution in the IRX–stellar mass relation to z ∼ 3.5. Recent z ∼ 3–5 results on the average infrared excess for bright z ∼ 3–5 galaxies by Coppin et al. (2015), where IRX ∼ 5–6 (drawing values from their Table 2) for sources with UV SFRs of ∼18–33 (corresponding to a stellar mass of ${\mathrm{log}}_{10}\,(M/{M}_{\odot })\sim 9.8$: Duncan et al. 2014), are also consistent with no evolution in the IRX–stellar mass relation to z ∼ 5 (assuming minimal biases from stacking). While one might expect some evolution in this relationship due to the observed evolution in the mass–metallicity relation (e.g., Erb et al. 2006a), it is possible that higher amounts of gas and mass in the interstellar medium in z ≳ 2 galaxies could compensate for the lower metal content to produce a relatively unevolving IRX–stellar mass relation (Tan et al. 2014).

The results we obtained in the previous section imply that dust emission from lower-mass galaxies is not large, particularly relative to the emission from galaxies in the rest-frame UV and as apparent at 1.2 mm. The relative energy outputs in the IR from the "average" < 10^9.25 M_⊙ and <10^9.75 M_⊙ galaxies in our HUDF selection are estimated to be less than 42% and 32% (95% confidence), respectively, of what galaxies emit at rest-frame UV wavelengths (Table 12 from Appendix D).

4.1.1. Comparison with z ∼ 2 Spectroscopic Results

4.1.2. Do Examples of Low-mass but IR-luminous $z\gtrsim 2$ Galaxies Exist?

Synthesizing the results from our own program with those from other programs, we can derive an approximate expression for the average infrared excess in star-forming galaxies. For sources with stellar masses in excess of 10^9.75 M_⊙, we find that the IRX–β relationship is most consistent with an SMC IRX–β relation. While many previous studies (e.g., Reddy et al. 2006, 2010; Daddi et al. 2007) found evidence that the highest-mass sources followed an IRX–β relation such as that of Calzetti et al. (2000) or M99, implying more obscured star formation, here we are probing a smaller volume, and the highest-mass sources from the present study may not be especially dissimilar from the lowest-mass sources in many previous studies. For sources with stellar masses of 10¹⁰ M_⊙ and less, many previous studies also found evidence for an SMC IRX–β relation in z ∼ 2 galaxies, e.g., Baker et al. (2001), Reddy et al. (2006: their Figure 10), Reddy et al. (2010), and Siana et al. (2008, 2009).

Indeed, the present ALMA results are interesting in that they allow us to extend these analyses into an even lower mass regime than was generally studied before. Despite some dependence on the assumed SED template, our results suggest that dust emission from these lower-mass sources is much less significant than for even our high-mass subsample, i.e., with an IRX not larger than 0.40 (2σ) assuming T_d ∼ 35 K. Even if we conservatively adopt a modified blackbody SED with an evolving dust temperature similar to that found by Béthermin et al. (2015), our results imply that the infrared excess for lower-mass galaxies is not larger than 0.94 (2σ).

These results recommend to us a relatively simple recipe for the infrared excess of star-forming galaxies at z ≳ 2. For galaxies with stellar masses of >10^9.75 M_⊙, we make use of an IRX–β relationship intermediate between the dust laws of the SMC and Calzetti et al. (2000):²⁹

$$\begin{eqnarray}&&{A}_{\mathrm{UV}}=1.5(\beta +2.23).\end{eqnarray} \tag{ 4 }$$

For galaxies with stellar masses below 10^9.75 M_⊙, the dust extinction is much lower, as demonstrated, e.g., by our own results. For such systems, we postulate that IRX can be derived more reliably by using the correlation between IRX and stellar mass. Utilizing the IRX–stellar mass relationship from the previous section (keeping in mind the ±0.2 dex uncertainties), we propose that

$$\begin{eqnarray}&&{\mathrm{log}}_{10}\,\mathrm{IRX}={\mathrm{log}}_{10}[M/{M}_{\odot }]-9.67\end{eqnarray} \tag{ 5 }$$

where M is the inferred stellar mass (assuming a fixed dust temperature of 35 K). If the dust temperature evolves with cosmic time as found by Béthermin et al. (2015), i.e., as ${(1+z)}^{0.32}$ (Section 3.1.3), the latter expression could plausibly be replaced by ${\mathrm{log}}_{10}\,{\rm{IRX}}={\mathrm{log}}_{10}[M/{M}_{\odot }]-9.17$, i.e., approximately the same relationship as at z = 0–3 (see Section 3.5). Encouragingly enough, the IRX–stellar mass prescription we apply in the low-mass regime (Equation (5)) gives very similar estimates for the dust corrections in the high-mass regime (>10^9.75 M_⊙) as we find using our primary prescription (which relies on an IRX–β relationship). As such, there is a basic consistency to the present approach (despite some arbitrariness in how one parameterizes IRX in terms of various physical variables, i.e., stellar mass, β, or even the SFR itself).

With future data—including both deeper and wider continuum mosaics with ALMA—it should be possible to improve on this prescription. Particularly valuable will be observations at bluer wavelengths (closer to the peak of the far-IR emission: see Figure 1) and complementary information from other probes, i.e., stacks of the PACS fluxes, X-ray, and near-IR spectra for even lower-mass sources. In addition, a measurement of the Balmer decrement out to z ∼ 6 should soon be possible with the James Webb Space Telescope.

The purpose of this section is to determine the approximate correction we should apply to the observed UV luminosity densities to correct for dust extinction and therefore obtain the SFR density.

We base our estimates of dust extinction on the large catalog of z = 4–10 galaxies from Bouwens et al. (2015a) in the CANDELS GOODS-North, GOODS-South, and Early Release Science (Windhorst et al. 2011) fields. Critically, each of the z ∼ 2–10 galaxies in these fields possesses individually estimated stellar masses and UV-continuum slopes β, all derived on the basis of the deep HST and Spitzer/IRAC photometry available over the GOODS-North and South (Labbé et al. 2015). Stellar mass, in particular, is an important variable to establish given its utility for predicting the IR luminosity and infrared excess for individual galaxies. In addition, as we saw in Section 3.3.2, the estimated stellar mass clearly impacts the dependence of IRX on β.

We estimate the dust correction in individual 0.5 mag bins of UV luminosity. For each bin, we first consider what fraction of galaxies have stellar masses in excess of 10^9.75 M_⊙. For galaxies in this mass range, we compute the estimated dust correction based on an IRX–β relation intermediate between those of the SMC and Calzetti et al. (2000) that utilizes the β distribution measured for such high-mass galaxies. In making use of the observed β's to derive the correction for galaxies in a given luminosity bin, we either make use of the full distribution of β's derived for individual sources (where the typical uncertainty in the β's measured for individual sources is <0.3) or make use of a model distribution (where the typical uncertainty in β is >0.3). The UV-continuum slope β we measure for an individual source is estimated based on the fit of a power-law slope to its UV-continuum fluxes (e.g., Castellano et al. 2012), avoiding those flux measurements that could be impacted by absorption by the intergalactic medium or rest-frame optical ≳3500 Å light.³⁰

However, for galaxies with estimated stellar masses <10^9.75 M_⊙, we derive the dust correction from the prescription given in Equation (5).

The top panel of Figure 14 shows the fraction of galaxies in our z ∼ 4, z ∼ 5, z ∼ 6, and z ∼ 7 samples whose estimated stellar masses exceed 10^9.75 M_⊙. As expected, the fraction of sources with masses >10^9.75 M_⊙ is relatively high for the brightest sources in the rest-frame UV, but decreases rapidly faintward of −21 mag, and is <3% at −19 mag.

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In the middle panel of Figure 14, we present the dust corrections we estimate for our samples based on our ALMA results and using the prescription that we describe above. The dust correction we estimate is only particularly significant at ${M}_{\mathrm{UV},{AB}}\lt -21$ mag and becomes negligible faintward of −20 mag. In the bottom panel of Figure 14, we present an alternative estimate of the dust correction assuming that IRX–stellar mass relation does not evolve from z ∼ 0–3 to z ∼ 7 (partially motivated by the findings in the right panel of Figure 13).

We can determine the approximate impact of dust corrections on the inferred SFR densities at z > 3 by multiplying the UV luminosity function (LF) by the inferred dust corrections and integrating to specific faint-end limits. For convenience, these dust corrections are presented in Table 7. For the brightest (<25.5 mag) galaxies in the rest-frame UV at z ∼ 3, we assume (e.g., Reddy & Steidel 2004) that the average dust correction for UV-bright (<25.5 mag) galaxies at z ∼ 3 is ∼5.

Table 7.  Estimated Dust Corrections to Apply the Results of UV Luminosity Density to Various Faint-End Limits

	log10(Dust Correction)
Sample	(>0.05 ${L}_{z=3}^{* }$)a	(>0.03 ${L}_{z=3}^{* }$)a
Assuming Td = 35 K (fixed)
z ∼ 3	0.37b	0.34b
z ∼ 4	0.15	0.14
z ∼ 5	0.16	0.14
z ∼ 6	0.09	0.07
z ∼ 7	0.04	0.03
z ∼ 8	0.04	0.03
Assuming Evolving Tdc
z ∼ 3	0.37b	0.34b
z ∼ 4	0.27	0.25
z ∼ 5	0.27	0.24
z ∼ 6	0.21	0.18
z ∼ 7	0.09	0.07
z ∼ 8	0.08	0.06

Notes.

^aThe specified limits 0.05 ${L}_{z=3}^{* }$ and 0.03 ${L}_{z=3}^{* }$ correspond to faint-end limits of −17.7 and −17.0, respectively, which is the limiting luminosity to which z ∼ 7 and z ∼ 10 galaxies can be found in current probes (Ellis et al. 2013; McLure et al. 2013; Oesch et al. 2013b; Schenker et al. 2013; Bouwens et al. 2015a). ^bFor uniquely the z ∼ 3 sample, we make use of the finding by, e.g., Reddy & Steidel (2004) and Reddy et al. (2010) that the average infrared excess for galaxies brighter than 25.5 mag at z ∼ 3 is a factor of ∼5. ^cWe adopt ${T}_{d}={((1+z)/2.5)}^{0.32}(35\,{\rm{K}})$ for the evolution following Béthermin et al. (2015). See Section 3.1.3.

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For the case of an evolving IRX–stellar mass relation, the corrections are smaller (by ∼0.2 dex) than earlier correction factors (e.g., Madau & Dickinson 2014, Bouwens et al. 2015a). This can be seen by comparing the dust extinction estimates in the middle set of rows in Table 8 (to be presented in Section 5.2) with the top set of rows.

Table 8.  UV Luminosity Densities and Star Formation Rate Densities to −17.0 AB mag (0.03 ${L}_{z=3}^{* }$)

Lyman		${\mathrm{log}}_{10}{ \mathcal L }$	Dust	log10 SFR density
Break		(erg s−1	Correction	(M⊙ Mpc−3 yr−1)
Sample	〈 z 〉	Hz−1 Mpc−3)a	(dex)b	Uncorrected	Corrected	Incl. ULIRGb
M99 IRX–β (as in Bouwens et al. 2015)
B	3.8	26.52 ± 0.06	0.42	−1.38 ± 0.06	−1.00 ± 0.13	−0.96 ± 0.13
V	4.9	26.30 ± 0.06	0.35	−1.60 ± 0.06	−1.26 ± 0.12	−1.25 ± 0.12
i	5.9	26.10 ± 0.06	0.25	−1.80 ± 0.06	−1.55 ± 0.13	−1.55 ± 0.13
z	6.8	25.98 ± 0.06	0.23	−1.92 ± 0.06	−1.69 ± 0.07	−1.69 ± 0.07
Y	7.9	25.67 ± 0.06	0.15	−2.23 ± 0.06	−2.08 ± 0.07	−2.08 ± 0.07
Fiducial Estimates: Assuming Td = 35 K (fixed)
U	3.0	26.55 ± 0.06	0.44	−1.35 ± 0.03	−1.01 ± 0.09	−0.91 ± 0.09
B	3.8	26.52 ± 0.06	0.21	−1.38 ± 0.06	−1.24 ± 0.13	−1.17 ± 0.13
V	4.9	26.30 ± 0.06	0.15	−1.60 ± 0.06	−1.46 ± 0.12	−1.45 ± 0.12
i	5.9	26.10 ± 0.06	0.08	−1.80 ± 0.06	−1.73 ± 0.13	−1.72 ± 0.13
z	6.8	25.98 ± 0.06	0.03	−1.92 ± 0.06	−1.89 ± 0.07	−1.89 ± 0.07
Y	7.9	25.67 ± 0.06	0.03	−2.23 ± 0.06	−2.20 ± 0.07	−2.20 ± 0.07
J	10.4	${24.62}_{-0.45}^{+0.36}$	0.00	−3.28${}_{-0.45}^{+0.36}$	−3.28${}_{-0.45}^{+0.36}$	−3.28 ${}_{-0.45}^{+0.36}$
Assuming Evolving Tdc
B	3.8	26.52 ± 0.06	0.30	−1.38 ± 0.06	−1.13 ± 0.13	−1.08 ± 0.13
V	4.9	26.30 ± 0.06	0.25	−1.60 ± 0.06	−1.36 ± 0.12	−1.35 ± 0.12
i	5.9	26.10 ± 0.06	0.20	−1.80 ± 0.06	−1.61 ± 0.13	−1.60 ± 0.13
z	6.8	25.98 ± 0.06	0.07	−1.92 ± 0.06	−1.85 ± 0.07	−1.85 ± 0.07
Y	7.9	25.67 ± 0.06	0.06	−2.23 ± 0.06	−2.17 ± 0.07	−2.17 ± 0.07

Notes.

^aIntegrated down to 0.03 ${L}_{z=3}^{* }$. Based upon LF parameters in Table 2 of Bouwens et al. (2015a, see Section 6.1). The SFR density estimates assume ≳100 Myr constant SFR and a Salpeter IMF (e.g., Madau et al. 1998). Conversion to a Chabrier (2003) IMF would result in a decrease by a factor of ∼1.8 (0.25 dex) in the SFR density estimates given here. ^bThis factor includes both the impact of dust on the z = 3–8 UV luminosity densities and also the contribution of far-IR bright (>10¹² L_⊙) galaxies, which might be missed in typical probes of Lyman-break galaxies or which might have their IR luminosities underestimated (Reddy et al. 2006; Reddy & Steidel 2009). ^cWe adopt ${T}_{d}={((1+z)/2.5)}^{0.32}(35\,{\rm{K}})$ for the evolution following Béthermin et al. (2015). See Section 3.1.3.

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However, if the IRX–stellar mass relation does not evolve significantly from z ∼ 0 to z ∼ 6 (i.e., as in the right panel of Figure 13), our estimated dust corrections are only slightly smaller (∼0.1 dex) than inferred in earlier work. Previously, Capak et al. (2015) had also considered the impact of new ALMA results for the implied SFR densities at z ∼ 4–7, finding suggestive evidence for smaller dust corrections than had previously been utilized.

We apply the dust corrections we derived in the previous sections to the UV luminosity densities and integrate the UV LF of Bouwens et al. (2015a) to 0.05 ${L}_{z=3}^{* }$ (−17.7 mag) and to 0.03 ${L}_{z=3}^{* }$ (−17.0 mag). Both the dust corrections and UV LFs were derived over a similar range in UV luminosity, so this process is self-consistent.

As in previous work, the UV luminosity densities are converted into SFR densities using canonical relationships of Madau et al. (1998) and Kennicutt (1998):

$$\begin{eqnarray}&&{L}_{\mathrm{UV}}=\left(\displaystyle \frac{\mathrm{SFR}}{{M}_{\odot }{\mathrm{yr}}^{-1}}\right)\,\times \,8.0\times {10}^{27}\,\mathrm{erg}\,{{\rm{s}}}^{-1}\,{\mathrm{Hz}}^{-1}.\end{eqnarray} \tag{ 6 }$$

This relationship assumes a Salpeter (1955) IMF and a constant SFR for 100 million years. We also apply these dust corrections to the LF results of Reddy & Steidel (2009) and McLure et al. (2013).

Our quantitative results for the corrected and uncorrected SFR densities at z ∼ 3–10 are presented in Table 8. In deriving the corrected SFR densities, we take the uncertainty in the dust correction to be equal to the difference in the estimated dust corrections found by adopting a fixed dust temperature of 35 K and allowing for an evolving dust temperature as implied by the results of Béthermin et al. (2015). We also present the derived SFR densities along with many previous estimates (Schiminovich et al. 2005; Reddy & Steidel 2009; McLure et al. 2013) in Figure 15.

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Of course, in computing the total SFR density from galaxies, we must account not only for the contribution from UV-selected galaxies, but also for the more massive, far-infrared bright sources where standard dust corrections are not effective or which are sufficiently faint in the UV to be entirely missed in standard LBG searches (estimated by Reddy et al. 2008 to occur when galaxies have IR luminosities >10¹² L_⊙). Such sources are known to contribute a substantial fraction of the SFR density at z ∼ 0–2 (Karim et al. 2011; Magnelli et al. 2013; Madau & Dickinson 2014). We account for this contribution by using published IR LFs to integrate those galaxies with IR luminosities >10¹² L_⊙. While one might be concerned this would "double count" the SFR coming from specific sources, it has been argued (e.g., Reddy et al. 2008) that the full SFR from such sources would not be properly accounted for in UV-selected samples (for reasons specified at the beginning of this paragraph).

We utilize the results of Magnelli et al. (2013) at z ∼ 0–2 (which build on the results of Caputi et al. 2007 and Magnelli et al. 2009, 2011), Reddy et al. (2008) at z ∼ 3, Daddi et al. (2009) and Mancini et al. (2009) at z ∼ 4, Dowell et al. (2014) at z ∼ 5, and Riechers et al. (2013) at z ∼ 6 (see also Wang et al. 2009; Boone et al. 2013; Asboth et al. 2016). All together these results suggest obscured SFR densities of 0.025, 0.01, 0.001, and < 0.0003 M_⊙ yr⁻¹ Mpc⁻³ at z ∼ 3, z ∼ 4, z ∼ 5, and z ∼ 6, respectively.

We combine these SFR densities with those we derived by correcting the UV LFs at z = 3–10 to present our best estimates for the SFR density at z = 3–10 in Table 8 and Figure 15, alternatively assuming a fixed dust temperature ${T}_{d}\sim 35$ K and supposing that the dust temperature increases monotonically toward high redshift as found by Béthermin et al. (2015). For context, we also present in Table 8 the SFR density that Bouwens et al. (2015a) estimated based on the M99 IRX–β relationship and making use of the observed β distribution at z ∼ 4–10.

Figure 16 shows the fraction of the SFR density that would be directly observable in the rest-frame UV and also in the IR. This figure is an update to Figure 12 from Bouwens et al. (2009). Similar to the findings from Bouwens et al. (2009), we find that most of the SFR density at z > 4.5 would be observable in the rest-frame UV.

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Another consequence of our new ALMA results is for the interpretation of the prominent Hα emission lines inferred in galaxies at z ∼ 4–5 based on the observed Spitzer/IRAC 3.6 μm and 4.5 μm fluxes (e.g., Schaerer & de Barros 2009; Shim et al. 2011; Stark et al. 2013).

As first noted by Shim et al. (2011), the total Hα luminosities of galaxies at z ∼ 4–5 are in excess of what one might expect based on their luminosities in the rest-frame UV. The excess is conservatively as large as a factor of 2 using the general z ∼ 4–5 LBG selections (Smit et al. 2015) but was earlier reported to be a factor of 6 (Shim et al. 2011) using samples selected based on their emission line properties. One explanation for these high luminosities would be if the selected galaxies predominantly had young stellar ages (given that the Lyα emission was prominent in the spectroscopic sample considered by Shim et al. 2011) or if dust extinction preferentially had a larger impact on the observed UV-continuum fluxes than it did on the observed Hα fluxes (Marmol-Queralto et al. 2016; Smit et al. 2015).

Neither mechanism appears to provide a fully satisfactory explanation for the high Hα fluxes in z ∼ 5 galaxies. Dust extinction falls off rapidly toward lower masses, but the Hα EWs in these sources remain essentially unchanged (or possibly increase toward lower mass: see Smit et al. 2015). Similarly, young ages for the z ∼ 4–5 galaxy population cannot provide an explanation, because a high ratio of Hα to UV continuuum is observed for typical star-forming galaxies at z ∼ 4–5 (Marmol-Queralto et al. 2016; Smit et al. 2015).³¹

This essentially forces us to conclude that the observed ratio of Hα to UV-continuum luminosity must be intrinsic and that star-forming galaxies indeed have very high Hα luminosities relative to their luminosities in the UV continuum, particularly in comparison with conventional stellar population models (i.e., Bruzual & Charlot 2003). It has been speculated that such could be achieved due to binarity or rapid rotation in massive stars, allowing them to output large amounts of ionizing radiation tens of million of years after an initial burst of star formation (e.g., Yoon et al. 2006; Eldridge & Stanway 2009, 2012; Levesque et al. 2012; de Mink et al. 2013; Kewley et al. 2013; Leitherer et al. 2014; Gräfener & Vink 2015; Szécsi et al. 2015). Changes to the IMF might also be a possibility (e.g., if there are a larger number of high-mass stars), but are disfavored given the general agreement between the observed stellar mass density and the integrated SFR densities (e.g., Stark et al. 2013).

These high ratios of Hα to UV continuum have implications for the Lyman-continuum photon production efficiencies ξ_ion used in reionization modeling. In Bouwens et al. (2015c), we pioneered the use of the relative luminosities in Hα and the UV continuum to estimate these efficiencies. Our results were consistent with the high-end assumed values for this efficiency in the literature, but were moderately sensitive to our assumptions about the dust extinction. Specifically, we derived intrinsic values of log₁₀ ξ_ion/(Hz erg s⁻¹) of ${25.27}_{-0.03}^{+0.03}$ and ${25.33}_{-0.03}^{+0.02}$ assuming a Calzetti dust law and an SMC one, respectively (where the systematic errors are likely 0.06 dex: see Section 3.3 of Bouwens et al. 2016). These compare to various canonical values ranging from 25.20 to 25.30 in the literature (e.g., Madau et al. 1999; Kuhlen & Faucher-Giguère 2012; Robertson et al. 2013).

The present ALMA results suggest that the dust extinction in z ∼ 4–5 galaxies is likely to be quite low. It is therefore interesting to estimate the mean Lyman-continuum photon production efficiency assuming that dust extinction is zero, repeating the estimates made in Bouwens et al. (2015c). This is almost certainly an extreme case, because even lower-mass galaxies likely show a small amount of dust extinction. Our new results for this efficiency factor ξ_ion are presented in Figure 17 as a function of UV luminosity. We compare the case of no dust extinction with the case in which all galaxies exhibit dust extinction according to Calzetti et al. (2000), and using similar dust extinction for nebular lines and the UV continuum.

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The impact is fairly dramatic. The Lyman-continuum photon production efficiency ${\mathrm{log}}_{10}\,{\xi }_{\mathrm{ion}}$/(Hz erg s⁻¹) that we derive is 25.51${}_{-0.03}^{+0.03}$, which is ∼1.6 times higher than the estimates of Bouwens et al. (2015c) assuming a dust law according to Calzetti et al. (2000) or an SMC one. This is very close to the efficiency of 25.53${}_{-0.06}^{+0.06}$ that Bouwens et al. (2015c) had previously derived for z ∼ 4–5 galaxies with the bluest UV continuum slopes β (β < −2.3) and very close to the values suggested by the stellar population models including the impact of binary stars on the evolution (e.g., Stanway et al. 2016; Wilkins et al. 2016b) or suggested by models including stellar rotation (e.g., Topping & Shull 2015).

Such a production efficiency is equivalent to z ∼ 5 star-forming galaxies producing ∼1.8 times as many ionizing photons per UV continuum photon as expected from standard stellar population models (which we take to be 10^25.2 to 10^25.3 (Hz erg s⁻¹) consistent with the use in the literature). In calculating this efficiency, we have assumed that the escape fraction is 0. If we assume that the escape fraction from galaxies is sufficient to reproduce observed constraints on the ionizing emissivity of the universe at z ∼ 4.4 (Becker & Bolton 2013), this would translate into a 0.02 dex higher mean ${\xi }_{\mathrm{ion}}$.

To determine the impact of this higher efficiency factor on reionization modeling, we take advantage of the modeling results from Bouwens et al. (2015c). Bouwens et al. (2015c) demonstrate that ionizing emissivity derived from observed z ≥ 6 galaxies matches that inferred from other observations (Planck, Lyα emission fractions, etc.) if the following condition applies:

$$\begin{eqnarray}&&{f}_{\mathrm{esc}}{\xi }_{\mathrm{ion}}{f}_{\mathrm{corr}}{({M}_{\mathrm{lim}})(C/3)}^{-0.3}={10}^{24.50\pm 0.10}\,{{\rm{s}}}^{-1}/({\mathrm{ergs}}^{-1}\,{\mathrm{Hz}}^{-1})\end{eqnarray} \tag{ 7 }$$

where f_esc is the escape fraction, $C=\langle {\rho }_{H}^{2}\rangle /\langle {\rho }_{H}{\rangle }^{2}$ is the clumping factor, and where ${{f}}_{{\rm{c}}{\rm{o}}{\rm{r}}{\rm{r}}}({M}_{{\rm{l}}{\rm{i}}{\rm{m}}})={10}^{0.02+0.078({M}_{{\rm{l}}{\rm{i}}{\rm{m}}}+13)-0.0088{({M}_{{\rm{l}}{\rm{i}}{\rm{m}}}+13)}^{2}}$ corrects the UV luminosity density ${\rho }_{\mathrm{UV}}(z=8)$ derived to a faint-end limit of ${M}_{\mathrm{lim}}=-13$ mag to account for different faint-end cut-offs. The above constraint is essentially identical to what Robertson et al. (2013) derive based on the available observations, but the above expression also shows how the result depends on the faint-end cut-off M_lim to the LF and the clumping factor C.

If we apply such an efficiency to the observed UV LFs and use fairly standard assumptions (integrating the observed LFs down to −13 mag and take the clumping factor C equal to 3: e.g., Robertson et al. 2013), these results would imply that galaxies can reionize the universe if the escape fraction f_esc is equal to (8 ± 2)%. An important corollary is that the escape fraction for z ≥ 6 galaxies cannot be significantly higher than 8%. If it were higher, it would imply an ionizing emissivity from galaxies that is higher than observed. Cosmic reionization would be complete at substantially earlier times than z ∼ 6 (i.e., z > 6.5).