Lyman-break Galaxies at z ∼ 3 in the Subaru Deep Field: Luminosity Function, Clustering, and [O iii] Emission

The SDF has a set of very wide and deep multi-waveband optical imaging data from the Subaru Deep Survey project.⁶ These data cover one Subaru/Suprime-Cam field (Miyazaki et al. 2002) $34^{\prime} \times 27^{\prime}$ in size, with $0\buildrel{\prime\prime}\over{.} 202$ pixels, in five standard broad-band filters, B, V, R_c, $i^{\prime}$, and $z^{\prime}$. The observations, data processing, and the source detections are described in Kashikawa et al. (2004). Basic parameters are summarized in Table 1. The $3\sigma$ limiting magnitudes of the Suprime-Cam imaging, measured in $2^{\prime\prime}$-diameter apertures, are 28.45, 27.74, 27.80, 27.43, 26.62 in B, V, R, $i^{\prime}$, and $z^{\prime}$, respectively. The final co-added image has an effective area of 876 arcmin², with a seeing FWHM of 098 in each band. The wide FOV enables us to probe large comoving volumes exceeding 10⁶ Mpc³ for our LBG sample at $\langle z\rangle \simeq 3$. The absolute error of astrometry was found to be $0\buildrel{\prime\prime}\over{.} 21$–$0\buildrel{\prime\prime}\over{.} 27$.

Table 1.  Photometry in Subaru Deep Field

Band	${\lambda }_{\mathrm{cent}}$ a	${\rm{\Delta }}\lambda$ b	3σ Limitc	Seeing FWHM	SDF Coveraged
	(Å)	(Å)	[AB-mag, $2^{\prime\prime}$]	(arcsec)	%
U	3630	750	27.19	1.49	100
B	4440	690	28.45	0.93	100
V	5460	890	27.74	0.940	100
Rc	6520	1100	27.80	0.97	100
$i^{\prime}$	7660	1420	27.43	0.98	100
$z^{\prime}$	9020	960	26.62	0.98	100
J	12560	1600	24.75	1.10	82
H(WFCAM)	16510	2900	24.05	1.10	67
H(NEWFIRM)	16190	3000	24.20	0.90	95
K	22360	3400	24.15	1.10	82
[3.6]	35500	7500	24.29	1.92	81
[4.5]	44930	10150	23.97	1.79	75
[5.8]	57310	14250	22.86	2.00	81
[8.0]	78720	29050	22.89	2.23	75

Notes.

^aCentroid wavelength for the filters in Å. ^bFilter bandwidths in Å, not counting atmospheric transmission. ^cLimiting magnitudes measured in $2^{\prime\prime}$ diameter apertures across all portions of the image utilized in the study (across regions of non-uniform depth). ^dCoverage of the SDF in each image that was utilized for this study. Full coverage corresponds to 876 arcmin².

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Below, we describe the U-band, NIR (J-, H-, and K-band), and mid-IR data of the SDF utilized for this study. The photometric sensitivity, seeing, and image scales and sizes for these data are summarized in Table 1.

2.1.1. U-band Imaging

2.1.2. Near-infrared Imaging

2.1.3. Mid-infrared Imaging

Object detection and photometry were performed using SExtractor (version 2.3; Bertin & Arnouts 1996). We masked out regions of low S/N at the edges of the image, near bright stellar halos, and saturated CCD blooming, as well as bad pixels of abnormally high or low count spikes. We first detected objects in the (extremely deep) R_c-band image. We considered an object detected when if had more than five contiguous pixels detected at greater than the $2\times \mathrm{rms}$ threshold.

For each detected object, photometry was performed in the other waveband images at exactly the same positions by running SExtractor in dual-image mode. We adopted Kron aperture magnitudes (MAG_AUTO) in SExtractor for total magnitudes, and used magnitudes within a $2^{\prime\prime}$ diameter aperture to derive colors.⁹ We applied a differential aperture correction of −0.2 mag to the U magnitudes to compensate for the larger PSF size in the U-band image. The value −0.2 was determined so that the difference between the aperture magnitudes and the total magnitudes of the U band became equal to those of the other bands for LBG candidates.

The magnitudes of objects were corrected for a small amount of foreground Galactic extinction using the dust map of Schlegel et al. (1998). The value of $E(B-V)=0.017$ corresponds to an extinction of A_U = 0.08, A_B = 0.07, A_V = 0.05, ${A}_{{R}_{c}}=0.04$, ${A}_{i^{\prime} }=0.03$, ${A}_{z^{\prime} }=0.02$.

We find that a combination of U, V, and R_c bands works best to select LBGs at $z\sim 3$ among our bandpass set. The V and R_c filters are redward of Lyα at z = 2.9, while half of the U-band is entirely below the Lyman limit, causing a substantial jump in the U − V color. We note that comparing the V filter (rather than the B) to find U-band "drop-outs" mostly eliminates the ambiguity from galaxies at somewhat lower redshifts, where a red color could be produced by partial absorption due to the Lyα forest and intervening Lyman limit absorbers, rather than a true Lyman limit cutoff. The Lyman limit absorption is not shifted into the V filter until redshifts above $z\simeq 4.5$. The idea of using the partial Lyα forest absorption to identify galaxies at $z\sim 2$ is the foundation of the so-called "BX" method (Adelberger et al. 2004). L11 has previously used the U, B and V photometry in SDF to identify these BX galaxies.

In Figure 1, we show the distribution of detected objects in the U − V versus $V-{R}_{c}$ diagram, as well as predicted positions of high-redshift galaxies (described below) and foreground objects (lower-redshift galaxies and Galactic stars). We assigned the 1σ limiting-magnitude for objects with U and/or V magnitudes fainter than this level.

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We set the selection criteria for LBGs at $z\sim 3$ as:

$$\begin{eqnarray}&&23.0\leqslant {R}_{c}\leqslant 27.8,\end{eqnarray} \tag{ 1 }$$

$$\begin{eqnarray}&&U-V\geqslant 0.12,\end{eqnarray} \tag{ 2 }$$

$$\begin{eqnarray}&&U-V\geqslant 4.1\times (V-{R}_{c})+1.8,\end{eqnarray} \tag{ 3 }$$

$$\begin{eqnarray}&&\ \mathrm{and}\ V-{R}_{c}\leqslant 0.41.\end{eqnarray} \tag{ 4 }$$

The selection boundaries in the U − V versus $V-{R}_{c}$ diagram defined by these color criteria are outlined with the thick black line in Figure 1. The small dots represent all objects we measured in SDF brighter than R_c = 27.8. Gray symbols represent galaxies detected in the U band; the more numerous (70% of the sample) blue symbols represent LBGs with upper limits in the U band, meaning that their true locations in the diagram could be higher than what is shown.

To include extremely faint LBGs with low photometric SNRs, we accepted galaxies with formal U − V colors down to 0.12 mag. This is bluer than allowed by some previous searches for "U dropouts," but it turned out that this made hardly any difference. If instead we had set the U − V cut at ≥0.50 mag, we would only have lost 48 LBG candidates—less than 1% of our final sample. This would not have significantly changed any of our results.

In general, our color selection is deliberately conservative. We have excluded relatively red LBGs from our sample because we wanted to minimize the possible contamination from foreground interlopers. We have thus aimed to construct a z = 3 LBG sample of high purity. Although its completeness may be lower than some previous studies, we are able to estimate and correct for this effect (described in Section 3.3). We also excluded candidates brighter than R_c = 23.0 because, at that high brightness level, extremely rare, very luminous LBGs would likely be overwhelmed by spurious contaminants.

The data were compared with the predicted colors of model galaxies that are expected to span the range of properties found in LBGs, shown as the red circles. These models were produced using the stellar synthesis code of Bruzual & Charlot (2003) (hereafter BC03), assuming a Salpeter IMF. We considered both constant and exponentially decaying SFR histories. In our modeling of the stellar populations of LBGs, we have considered three metallicities, $Z=0.004,0.008$ and 0.02 (solar), and ages of 0.01, 0.1, 0.5, and 1 Gyr. We assumed reddenings of $E(B-V)=0.0$, 0.1, 0.2, 0.3, and 0.4, and the Calzetti et al. (2000) attenuation curve. Although we have no particular reason to believe that this law is more applicable to LBGs than other possibilities, we adopt it to facilitate comparison with many previous studies that also assume the Calzetti et al. (2000) curve. The absorption due to the intergalactic medium was applied following the prescription by Madau (1995). The colors were calculated by convolving the constructed model spectra with the response functions of the Suprime-Cam and MOSAIC broadband filters. These models were computed at redshift intervals of ${\rm{\Delta }}z=0.1$. These values reproduce the average rest-frame ultraviolet-optical SED of LBGs observed at $z\sim 3$ (e.g., Papovich et al. 2001; Shapley et al. 2001; Ly et al. 2011b).

The red lines in Figure 1 show some of our BC03 stellar synthesis model tracks. Regardless of the stellar population, our selection box is sensitive to LBGs having redshifts of $z\simeq 2.9\mbox{--}3.5$ (the squares on the model tracks bracket this range). The yellow stars show observed colors of Gunn & Stryker (1983) stars. For a given $V-{R}_{c}$ color, the stars are always bluer in U − V, because they do not have so steep a "U-drop" as our LBGs. This is not the case for the ${{ \mathcal U }}_{n}{ \mathcal G }{ \mathcal R }$ selection criterion, which is affected by contamination from main-sequence stars L11. The total number of LBGs selected is 5161, with a raw detected surface density of $\approx 6$ arcmin⁻².

A small fraction of our LBG candidates (roughly 15%) have clear U dropouts (U − V ∼ 1) while showing extremely blue V − R colors (of −0.3 to −0.4). Their V − R colors appear to be 0.1–0.2 mag bluer than our most extreme young starburst model. Closer examination reveals that these blue objects are entirely confined to our faintest LBG candidates (R ≥ 27), of which they constitute the majority. They are, by definition, required to have solid detections in the V band, but with otherwise quite noisy photometry. Thus, given their large photometric uncertainties, their true V − R colors are probably somewhat redder than shown in the diagram, in many cases. As discussed below, we have modeled these photometric uncertainties and find that our selection of LBGs fainter than R = 27.0 is, indeed, extremely incomplete. However, even at these faint magnitudes, we subsequently find that the contamination fraction of non-LBG interlopers must be small. This is demonstrated by the extremely strong K-band excess that the stack of our 1294 faintest LBG candidates show, attributable to [O iii]5007 emission at z ∼ 3, discussed in Section 5.

Beyond photometry errors, there is an additional factor that may cause the extremely blue V − R colors in our faintest LBG candidates. As discussed in Section 5.3.1, the strong nebular emission we find in all faint LBGs is also seen in substantial Lyα emission, which can be strong enough to increase the observed broadband V flux. The blue track (left-most curve) shows the model of the youngest stellar population with the addition of Lyα emission, ${W}_{0}=20$ Å. This Lyα-enhanced track matches the colors of our faintest, bluest LBG candidates.

Table 2 gives the distribution of R_c-band magnitudes along with surface density counts of our 5161 LBGs. Our sample includes ∼2000 LBGs fainter than R_c = 27.0. This gives us a significantly better statistical measure of the faint end of the LBG LF than previous studies, which covered smaller areas or were less sensitive.

Table 2.  Magnitude Distribution, Number Counts, and Effective Survey Volumes of the LBGs

Rc	${N}_{\mathrm{LBG}}$	Σ (num arcmin−2)a	${V}_{\mathrm{eff}}$ $({\mathrm{Mpc}}^{3})$
23.0–23.5	11	0.025	$1.32\times {10}^{6}$
23.5–24.0	49	0.112	$1.26\times {10}^{6}$
24.0–24.5	146	0.333	$1.19\times {10}^{6}$
24.5–25.0	334	0.762	$0.94\times {10}^{6}$
25.0–25.5	482	1.100	$7.08\times {10}^{5}$
25.5–26.0	590	1.347	$4.81\times {10}^{5}$
26.0–26.5	757	1.728	$3.40\times {10}^{5}$
26.5–27.0	929	2.121	$2.52\times {10}^{5}$
27.0–27.5	1132	2.584	$2.22\times {10}^{5}$
27.5–28.0	731	1.669	$1.57\times {10}^{5}$
Total	5161	11.783

Note.

^aSurface density of LBGs in each R_c-band magnitude bin.

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The surface density counts of our LBG candidates are consistent with previous studies, except that we have a lower surface density of the brightest LBGs in the SDF. Given the low surface density of such rare luminous galaxies, this apparent discrepancy could be the result of cosmic variance. Another possibility is that these previous surveys might have suffered some contamination from interlopers at the bright end (Ly et al. 2011b).

The redshift distribution function of the LBG sample and its completeness were estimated as a function of magnitude through Monte Carlo simulation. We generated artificial LBGs over an apparent magnitude range of $23.0\leqslant z^{\prime} \leqslant 27.5$, in intervals of ${\rm{\Delta }}m=0.5$, and over a redshift range of $2\leqslant z\leqslant 4$, in intervals of ${\rm{\Delta }}z=0.1$. Model spectra of the artificial LBGs are constructed using the BC03 stellar population synthesis code, as described above.

The median size of our LBG candidates is FWHM $\approx 1\buildrel{\prime\prime}\over{.} 1$. Their unresolved or marginally resolved morphology is consistent with previous expectations that the intrinsic half-light diameters of LBGs at $z\sim 3$ should be about 2 kpc, or only a few tenths of an arcsecond (Shibuya et al. 2015). There is a significant tail of resolved LBGs—about 10% of them have FWHM $\gt \,2^{\prime\prime}$. We assumed that the shape of LBGs is Gaussian. We assigned apparent sizes to the artificial LBGs in our Monte Carlo simulation matching the size distribution of observed LBG candidates measured by SExtractor.

The artificial LBGs were then distributed randomly on the original images after adding Poisson noise appropriate to their magnitudes, and object detection and photometry were performed in the same manner as done for real objects. We generated 30,000 artificial galaxies on the image, and repeated this five times, to obtain statistically accurate values of completeness. In the simulations, the completeness for a given apparent magnitude, redshift, and $E(B-V)$ value is defined as the ratio in number of the simulated LBGs that are detected and also satisfy the selection criteria to all the simulated objects with the given magnitude, redshift, and $E(B-V)$ value. We calculated the completeness of the LBG sample by taking a weighted average of the completeness for each of the five $E(B-V)$ values. The weight is taken using the $E(B-V)$ distribution function of $z\sim 4$ LBGs derived by Ouchi et al. (2004a) (the open histogram in the bottom panel of their Figure 20), which was corrected for incompleteness due to selection biases.

This Monte Carlo simulation ignores the possible systematic positive correlation of $E(B-V)$ with galaxy luminosity, which is seen in observations by Labbé et al. (2007), Bouwens et al. (2009), and Sawicki (2012). Our simple approach is the same as used by Yoshida et al. (2006), Bian et al. (2013), and also, judging from the descriptions they give, by Reddy & Steidel (2009) and Hathi et al. (2010). We and other groups have not included a correlation between $E(B-V)$ and magnitude in our simulations, mainly because determining the true intrinsic correlation is difficult, given the selection effects operating, and any correlation has such a large cosmic scatter that it does not describe many LBGs. The effects of different reddening assumptions have been considered. Reddy et al. (2008) found that their "results are also insensitive to small variations in the assumed $E(B-V)$ distribution as long as the range of $E(B-V)$ chosen reflects that expected for the galaxies." Similarly, Sawicki & Thompson (2006) concluded that "the dependence of the LF on these assumed dust and age values is negligibly small at z ∼ 4 and 3 but becomes more significant at the lower redshifts."

As shown in Figure 1, the diagonal color cut we used brings in LBGs of all reddenings at nearly the identical redshift, z = 2.9, shown by the red squares. This results in a sharp redshift cut-on for our LBG sample. The high-redshift cut-off is, at least in principle, not so sharp. Theoretically, the bluest galaxies could still fall within the selection box up to a redshift of $z\leqslant 3.7$, whereas the reddest galaxies begin to leave the box at $z\gt 3.3$. This is a common feature of U dropout selections: toward the flux limits of the surveys, redder galaxies have a somewhat smaller chance of being included. Our simple Monte Carlo simulation selected galaxy reddenings independent of their magnitude. A more sophisticated simulation might have included a tendency for brighter LBGs to be redder, presumably because of higher dust content (Steidel et al. 2003). This could reveal a weak bias in favor of discovering bluer LBGs at lower redshifts. We, along with others using two-color selections, have not included such a correlation into our completeness simulations. To the extent that this correlation is not merely a selection artifact of magnitude-limited surveys, its form is quite uncertain, with a great deal of intrinsic scatter. In any case, the reality, shown in the figures, is that our selection finds very few LBGs—of any type—at $z\geqslant 3.5$. There is therefore little room for a possible (highly uncertain) color bias to undermine our completeness estimates.

The resulting completeness, $p(m,z)$, is shown in Figure 2. Because we have been careful to make generous allowances for the number of LBGs that could have been missed due to reddening, our total completeness is not extremely high, dropping to 50% at R_c = 25.0 and 20% at R_c = 26.0. This means that a large number of our fainter LBGs are in a regime where our survey (and nearly all others) is not very complete. The magnitude-weighted redshift distribution function of our LBG sample was derived from $p(m,z)$ by averaging the magnitude-dependent completeness weighted by the number of LBGs in each magnitude bin. The average redshift and its standard deviation are calculated to be $\langle z\rangle =3.3$ and ${\sigma }_{z}=0.3$.

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A small minority (140) of the 5161 LBGs are individually detected ($\gt 3\sigma$) in both of the WFCAM H and K band images (hereafter referred to as our NIR subsample). Of these, slightly over half (77) are also detected in IRAC bands 1 and 2. These are naturally the reddest LBGs in the parent sample and are expected to be the most massive (containing older stellar populations) or dust-obscured. So, although they are not representative of the full LBG population as a whole, they do have enough accurate multi-band photometry over a wide wavelength range to allow us to measure their photometric redshift distribution. In particular, the H and K band photometry spans the Balmer break at these redshifts. We input their optical-to-infrared SEDs into the Easy and Accurate Zphot from Yale (EAZY; Brammer et al. 2008). The resulting distribution of z_phot for the NIR subsample is shown in Figure 3. The NIR subsample has a median redshift ${\langle {z}_{\mathrm{phot}}\rangle }_{\mathrm{med}}=3.03$, with ∼90% of the LBG candidates having photometric redshifts in the range ${z}_{\mathrm{phot}}=2.5\mbox{--}3.5$. Thus, robust photometric redshifts derived for a subset of massive LBGs yield confidence in the effectiveness of our color selection (described above) to select $z\sim 3$ star-forming galaxies. We further discuss properties of the NIR subsample in Section 5.

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The same ${{UBVR}}_{c}i^{\prime} z^{\prime}$ photometry in SDF was previously used by L11 to identify LBGs. That study aimed to duplicate exactly the U-dropout method used by Steidel and collaborators, which selects objects with very red ${{ \mathcal U }}_{n}-{ \mathcal G }$ colors (e.g., Steidel et al. 1999). L11 approximated the ${ \mathcal G }$ filter with a linear combination of B and V photometry. Because the G band is about 500 Å bluer than the V band used in this paper, their LBG sample starts at z = 2.5 rather than our z = 2.8. This results in their finding a higher surface density of bright U-dropouts than we did with our U − V selection. However, when we account for the different selection functions and cosmic volumes surveyed, our resulting LF is very similar to that of ${{ \mathcal U }}_{n}-{ \mathcal G }$ dropouts (see Section 4 below). In contrast to our selection down to R_c = 27.8, L11 cut their LBG selection at relatively bright U-dropouts, having $R\leqslant 25.5$. They also used a combination of photometric and image size information to exclude stars, which removed about 10% of the LBG candidates at $R\lt 24$, and a smaller, negligible fraction fainter than that. Overall, from cross-matching our U − V dropout catalog with the ${{ \mathcal U }}_{n}-{ \mathcal G }$ dropout catalog in L11, we find only 651 sources in common (13% of our sample and 27% of the L11 sample). This lack of overlap is due to the effects mentioned above—primarily the differing color selections. Our sample occupies a bluer, fainter locus on the U − V versus $V-{R}_{c}$ diagram than the ${{ \mathcal U }}_{n}-{ \mathcal G }$ dropouts from L11. For comparison, we included the L11 sample in our analysis of average UV-to-IR SEDs and inferred physical properties discussed in Section 5.

To stack LBGs in the infrared, we divided our sample into bins of $i^{\prime}$-magnitude, a proxy of SFR at $z\sim 3$ (probing rest-frame ∼1900 Å). Specifically, we stacked our sample of LBGs in bins of $i^{\prime} =23.5\mbox{--}24.5$, 24.5–25.5, 25.5–26.5 and 26.5–27.5. These magnitude bins contain 249, 806, 1256, and 1588 LBGs, respectively. The ∼1200 LBGs with $i^{\prime} \gt 27.5$ did not yield stacked detections, and are thus excluded from the following analysis. An image of a representative galaxy in each magnitude bin was formed by finding the median of each pixel in stacks of roughly $30^{\prime\prime} \times 30^{\prime\prime}$ cut-outs, centered around the optical position of each LBG in the bin. Here, we use the median of pixels rather than the mean, as the latter is significantly more prone to outliers.

Stacking is performed in the WFCAM J, H, and K bands, along with the NEWFIRM H band, and all four IRAC mosaics ([3.6], [4.5], [5.8], and [8.0]). Stacked fluxes were obtained from aperture photometry on the stacked images. We applied aperture corrections that were determined for the WFCAM/NEWFIRM images of the SDF by identifying bright/isolated point sources and computing the median curve of growth. For the IRAC mosaics, we used the aperture corrections given in the IRAC Instrument Handbook.¹⁰ As a check, we calculated our own aperture corrections and found that they are within 0.1 mag of those quoted in the handbook. Examples of stacked images of our LBGs in the K-band and IRAC 3.6 μm mosaics are shown in Figure 5.

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For each image, we computed the statistical significance of stacked flux as follows. First, we generated a set of N random image coordinates (in the same regions used for stacking), formed a median-stacked image for these N random coordinates, and found the aperture flux (with the same aperture size used to measure our LBG stacked flux). We then repeated this process for 1000 iterations to form a distribution of aperture flux from stacks of random image positions. The width of this distribution (which is Gaussian) gives the 1σ flux level corresponding to aperture photometry for stacks of N coordinates on a given image of the SDF. Thus, we derived curves of 1σ flux versus number of coordinates in a stack N for each image. As expected, we find that every curve follows a $1/\sqrt{N}$ law. For each band, the 3σ limiting-magnitudes from stacking N = 1000 sources, after aperture correction, are: J = 27.44, $H(\mathrm{WFCAM})=26.86$, $H(\mathrm{NEWFIRM})=26.74$, K = 27.12, $[3.6]=26.34$, $[4.5]\,=26.35$, $[5.8]=24.82$, and $[8.0]=24.85$.

5.1.1. Faint-stack Defect and Corrections

In the stacks of fainter bins, such as in those displayed in Figure 5, we encounter perhaps the largest source of uncertainty in stacked flux: a depression in the background within the immediate vicinity ($\lesssim 10^{\prime\prime}$) of the detected source. To understand the cause of this defect and determine an appropriate method to correct for it, we performed the following analysis. First, we used publicly available SDF SExtractor detection catalogs¹¹ and stacked on random source positions listed in the catalog, binning by $i^{\prime}$ magnitude (down to total mag of $i^{\prime} =29$). The defect remains present in these test stacks of a very heterogeneous sample, so we rule out the defect as being a result of our LBG selection technique. We find that the defect is present in the faint $i^{\prime}$-mag stacks of SDF images in all wavebands, including the publicly available ${{BVR}}_{c}i^{\prime} z^{\prime}$ images. The deficit gets more severe with fainter source magnitude, and is perhaps most significant in the IRAC [3.6] and [4.5] bands.

From this test, we posit that the defect is caused by background counts around faint sources in the detection catalogs being systematically lower than the background counts around brighter sources. In other words, faint objects tend to enter these catalogs when they are found in regions of the sky with relatively low surface density of faint, blended objects (i.e., low confusion). This bias is due to SExtractor requiring a lower local background level (higher S/N) in order to include a faint source as a detection. To test this hypothesis, we performed a test similar to the calculation of the completeness of our sample. We created 50,000 fake galaxies (Gaussian-shaped) with chosen total magnitude and size, and added them to random positions in the Suprime-Cam R_c-band image (the detection image), cataloging their random positions. Next, we ran SExtractor on the new detection image (with all the synthetic galaxies added), with parameters identical to those used for LBG detection, and recovered the coordinates of all detected synthetic objects.¹² According to our hypothesis, these coordinates should be positions with relatively lower background for fainter objects in the detection image. Finally, we stacked on those positions in the IR and obtained a "hole" representing the pure background deficiency (because the synthetic galaxies were added only to the detection image). This effect is shown in Figure 6. Stacking on all fake object input positions (which are just random image coordinates) results in a uniform stack with no hole apparent. As expected, the amplitude (depth) of the hole gets larger for stacks on recovered positions of fainter synthetic objects. Furthermore, we find that the profile gets deeper as the fake galaxies added are increased in size.

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In summary, we confirm that faint objects in SExtractor detection catalogs are located in regions of lower local background (due to a lack of faint, blended sources) than the brighter sources in the catalog. This effect is also likely present in catalogs generated with other detection algorithms. Thus, in any study utilizing such catalogs for stacking analysis of faint objects, the stacked flux of these objects will be systematically underestimated if this defect is not accounted for. We also note that low-resolution images that are deep enough to begin to approach the confusion limit are particularly affected by this bias.

This systematic loss of faint galaxies in the near vicinity of other galaxies was already corrected for in our LF calculations in Section 4, using our detailed simulations of incompleteness. However, corrections were still required for our stacked images in order to obtain accurate photometric fluxes.

We performed a simple least-squares fit to each image with a two-component model consisting of a positive Gaussian function for the stacked emission and a negative Lorentzian component for the background depression. We fixed both components to be circularly symmetric with centroids on the center of the image. After the fit was performed, the corrected image was generated by subtracting the negative Lorentzian component off of the original stacked image. An example of this correction procedure is shown in Figure 7 for the IRAC 3.6 μm stack in the $\langle i^{\prime} \rangle =26$ bin, showing the raw image, the model, and the corrected image along with corresponding radial profiles. Although the fit is good for all stacked images, this correction is considered the primary source of uncertainty in measuring the stacked flux of faint bins.

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We compute representative LBG SEDs using the median of ${{UBVR}}_{c}i^{\prime} z^{\prime}$ photometry and photometry from the median stacks of J, H, K, [3.6], [4.5], [5.8], and [8.0] images in each $i^{\prime}$-mag bin. Prior to measuring fluxes from stacked images, we implemented corrections to all of the NIR through mid-infrared stacks in the $\langle i^{\prime} \rangle =25$ bin and fainter, in order to account for the faint stack background deficiency (described in Section 5.1.1). Error bars on stacked fluxes are calculated from the random stack analysis described above. The reported stacked H-band fluxes are formed from a weighted mean of the flux from the WFCAM and NEWFIRM stacked images. The average SEDs for our U − V selected LBGs are plotted in the left panel of Figure 8. For comparison, we additionally performed stacking to form SEDs of the sample of ${{ \mathcal U }}_{n}-{ \mathcal G }$ selected LBGs from L11. As explained in Section 3.4, this sample is redder and includes a higher proportion of galaxies on the lower-redshift end of our redshift distribution function. We stacked this galaxy sample in three bins: $i^{\prime} =23\mbox{--}24$, 24–25, and $\gt 25$ (the sample is selected with $R\lt 25.5$, yielding sources down to $i^{\prime} \simeq 26$). The faint stack defect is only non-negligible in the faintest bin $\langle i^{\prime} \rangle =25.5;$ in this bin, all reported stacked data were corrected using the method described above. The average SEDs for these ${{ \mathcal U }}_{n}-{ \mathcal G }$ dropouts are shown in the right panel of Figure 8. We also form the stacked SED of our massive, NIR-detected subset of 140 LBGs.

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Average LBG properties as a function of $i^{\prime}$ magnitude are derived using the Fitting and Assessment of Synthetic Templates (FAST) code (Kriek et al. 2009) to fit stellar population synthesis templates to each stacked LBG SED. We use the BC03 stellar population template library, with a Salpeter IMF, and assume the Calzetti et al. (2000) dust extinction curve. We fit an exponentially decaying star formation history of the form $\mathrm{SFR}\sim \exp (-t/\tau )$. The ranges of parameter values for fitting are set to: age $t={10}^{6}\mbox{--}{10}^{10}$ years, star formation history $\tau ={10}^{7}\mbox{--}{10}^{9}\,\mathrm{years}$ (in steps of 0.5 dex), and extinction ${A}_{V}=0\mbox{--}3$. Metallicity is fixed at solar Z = 0.02. For our LBG sample, we fix the redshift of the fit to z = 3.1, which is the approximate peak of the completeness distribution for each magnitude bin. For the L11 sample, we fix the redshift to the median photometric redshift in each bin. The resulting best-fit age, τ, A_V, stellar mass M_*, and SFR corresponding to each average LBG are given in Table 4. Uncertainties for these physical quantities are determined by FAST, using Monte Carlo simulations that redistribute the photometric data according to their error bars, and include additional uncertainty based on the uncertainties in the stellar continuum models (see the appendix of Kriek et al. 2009 for more information).

Table 4.  Stellar Properties of Average $z\sim 3$ LBGs, Derived from Stellar-continuum Fits to the Stacked SEDs

$\langle i^{\prime} \rangle$ a	${N}_{\mathrm{stack}}$ b	${\mathrm{log}}_{10}({M}_{* }/{M}_{\odot })$	SFR $({M}_{\odot }{\mathrm{yr}}^{-1})$	Age (Myr)	τ (Myr)	AV
NIR-detected U − V dropouts
24.3	113–140	10.47 ± 0.06	45 $(+3,-17)$	398 $(+233,-194)$	316	0.6
U − V dropouts
24.2	169–217	10.09 $(+0.16,-0.33)$	51 $(+48,-49)$	251 $(+456,-208)$	1000	0.5
25.1	589–724	9.58 $(+0.00,-0.26)$	12 $(+0,-11)$	251 $(+37,-192)$	316	0.2
26.1	878–1078	9.02 $(+0.07,-0.69)$	3 $(+22,-1)$	251 $(+165,-242)$	316	0.0
26.9	1138–1390	7.79 $(+0.74,-0.10)$	4 $(+12,-3)$	10 $(+253,-6)$	10	0.3
${{ \mathcal U }}_{n}-{ \mathcal G }$ dropouts (L11)
23.8	104–130	10.22 $(+0.12,-0.10)$	27 $(+38,-25)$	100 $(+157,-45)$	31	0.4
24.7	668–867	10.10 $(+0.00,-0.30)$	13 $(+0,-12)$	251 $+0,-188)$	100	0.3
25.3	828–1048	9.54 $(+0.14,-0.44)$	6 $(+139,-6)$	100 $(+175,-90)$	31	0.1

Notes.

^aMedian $i^{\prime}$ mag in bin. ^bRange in number of objects in each stack. The minimum number of objects are in the WFCAM H-band stacks due to its incomplete coverage of the SDF.

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The SEDs show a K-band excess that is more pronounced for fainter bins. We attribute this excess to contamination from redshifted [O iii]λλ4959,5007+Hβ nebular line emission. As such, we perform fits both including and excluding the K-band photometry. The best-fit models are shown in Figure 8 with the data; the blue and red curves exclude and include the K-band data, respectively. The quoted stellar parameters in Table 4 correspond to the blue curves. Note that these fits only consist of a stellar continuum; we discuss the implications of nebular emission in Section 5.3.

In general, we find that the properties derived for the LBGs studied here, given in Table 4, are consistent with those derived in other studies of $z\sim 3$ star-forming galaxies. We compare our LBG sample to the "star-forming main sequence" (Speagle et al. 2014, and references therein) and plot their ${M}_{* }\mbox{--}\mathrm{SFR}$ correlation in Figure 9. The dashed line in Figure 9 is a linear fit to a sample of massive (average stellar mass of $\simeq 5\times {10}^{10}$ ${M}_{\odot }$) LBGs at $z\simeq 3$ studied in Magdis et al. (2010). Our average LBGs have stellar masses between ${10}^{7.8}$ and ${10}^{10.1}$ ${M}_{\odot }$ and SFRs between 3 and 51 ${M}_{\odot }$ yr⁻¹. As expected, the stack of the NIR-detected sample is more massive, with ${M}_{* }\simeq {10}^{10.5}$ ${M}_{\odot }$ and $\mathrm{SFR}=45$ ${M}_{\odot }$ yr⁻¹. The ${{ \mathcal U }}_{n}-{ \mathcal G }$ dropouts in L11 are found to be slightly more massive for a given $i^{\prime}$ magnitude (which is also expected based on their redder colors). The average LBG in the faintest bin $\langle i^{\prime} \rangle =27$ has a best-fit model SED with low stellar mass ${M}_{* }\simeq {10}^{8}$ ${M}_{\odot }$, very young age $t\simeq 10\,\mathrm{Myr}$, and high specific star formation rate (sSFR) of ${10}^{-7.2}$ yr⁻¹ or 60 Gyr⁻¹, placing it nearly ∼1 dex above the fit to the Magdis et al. (2010) sample. Although the fitting parameters are highly uncertain, the properties are indicative of dwarf galaxies forming stars at a prolific rate. Their presence in large numbers suggests that the "star-forming sequence" in reality contains significant dispersion. The smaller the mass of the galaxy, the more stochastic its star formation episodes are likely to be, and the greater scatter this should produce in a ${M}_{* }\mbox{--}\mathrm{SFR}$ correlation. This insight is entirely consistent with what surveys at somewhat lower redshifts have found when they select galaxies by their line emission rather than broadband continuum (Atek et al. 2011, 2014; Ly et al. 2014, 2015).

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5.3.1. Nebular Emission in High sSFR LBGs

Galaxies at the faint end of the LF, especially at high redshifts, have SEDs showing significant contributions from nebular emission (e.g., Yabe et al. 2009; Atek et al. 2011; Stark et al. 2013; de Barros et al. 2014). Indeed, the observed average LBG SEDs in Figure 8 contain some clear features that are not explained by the best-fit stellar continuum models.

Compared with filters on either side, the K-band magnitude is brighter by roughly 0.2, 0.4, and 0.9 mag for the $\langle i^{\prime} \rangle =25$, 26, and 27 LBGs SEDs, respectively. The most likely explanation is that the stellar continuum in the K-band is boosted by a strong contribution from [O iii]λλ4959,5007 emission at $z\simeq 3.1\mbox{--}3.8$ (with some contribution from ${\rm{H}}\beta$). Assuming this interpretation is correct, we estimate the equivalent width $\mathrm{EW}([{\rm{O}}\,{\rm{III}}]\lambda \lambda 4959,5007+{\rm{H}}\beta )$ for each average LBG. Specifically, we calculate the expected stellar continuum at the K-band wavelength, ${m}_{2.2\mu {\rm{m}},\mathrm{cont}}$, by convolving the best-fit stellar continuum models with the WFCAM K-band response curve. Knowing the K-band magnitude m_K and the filter FWHM ${\rm{\Delta }}K=0.34$ μm, we then calculate the observed equivalent widths as:

$$\begin{eqnarray}&&\mathrm{EW}([{\rm{O}}\,{\rm{III}}]+{\rm{H}}\beta )\simeq (1-{10}^{-0.4[{m}_{K}-{m}_{2.2\mu {\rm{m}},\mathrm{cont}}]})\cdot {\rm{\Delta }}K.\end{eqnarray} \tag{ 9 }$$

Error bars on the equivalent widths are calculated by re-sampling the H, K, and [3.6] filter fluxes with random noise added based on their photometric errors. Assuming a redshift of z = 3.1, the implied rest-frame equivalent widths of the $\langle i^{\prime} \rangle =25$, 26, and 27 stacked LBGs are ${\mathrm{EW}}_{0}([{\rm{O}}\,{\rm{III}}]+{\rm{H}}\beta )\simeq 247\pm 105$, 443 ± 143, and 1743 ± 360 Å, respectively. The K-band excess is also observed in the stacked SEDs for the ${{ \mathcal U }}_{n}-{ \mathcal G }$ dropouts from L11, implying ${\mathrm{EW}}_{0}([{\rm{O}}\,{\rm{III}}]+{\rm{H}}\beta )\simeq 79\pm 86$ and 303 ± 110 Å for the $\langle i^{\prime} \rangle =24.5$ and 25.5 LBGs, respectively.

In order to facilitate comparison with other samples, we remove the predicted contribution of Hβ by assuming ratios for [O iii]λλ4959,5007/Hβ. The [O iii]/Hβ ratios were taken from mass-stacked spectra of galaxies in the WFC3 Infrared Spectroscopic Parallel survey (WISPS) in Henry et al. (2013) and Domínguez et al. (2013). Comparing the mean mass and resulting stacked [O iii]/Hβ in these studies with the stellar mass of our stacked LBGs, we use ratios of [O iii]/H $\beta =3.0$, 4.5, and 5.0 for our three faintest bins of LBGs, and [O iii]/H $\beta =2.5$ and 3.0 for the two faintest bins of the L11 ${{ \mathcal U }}_{n}-{ \mathcal G }$ dropouts.

Figure 10 shows ${\mathrm{EW}}_{0}([{\rm{O}}\,{\rm{III}}]\lambda \lambda \mathrm{4959,5007})$ versus stellar mass for the LBGs studied here, compared with emission-line galaxy (ELG) samples at different redshifts. We find a steep dependence of ${\mathrm{EW}}_{0}([{\rm{O}}\,{\rm{III}}])$ on stellar mass, which we quantify by fitting a simple power-law with amplitude C and slope p:

$$\begin{eqnarray}&&{\mathrm{EW}}_{0}([{\rm{O}}\,{\rm{III}}]\lambda \lambda \mathrm{4959,5007})=C{[{\mathrm{log}}_{10}({M}_{* }/{M}_{\odot })]}^{-p}.\end{eqnarray} \tag{ 10 }$$

The least-squares fit, shown as the black dashed line in Figure 10, is performed for both the stacks of U − V dropouts selected in this study and the stacks of ${ \mathcal U }-{G}_{n}$ dropouts selected in L11. The LBGs have a best-fit slope of $p=10.2\pm 1.5$. Although highly uncertain, the fit implies that the average $z\sim 3$ LBG with a stellar mass of ${M}_{* }={10}^{9}{M}_{\odot }$ has a rest-frame [O iii]λλ4959,5007 equivalent-width of $\simeq 340\,\mathring{\rm A}$, while a dwarf LBG with ${M}_{* }={10}^{8}{M}_{\odot }$ has ${\mathrm{EW}}_{0}([{\rm{O}}\,{\rm{III}}])\simeq 1130\,\mathring{\rm A}$. To emphasize how extraordinarily strong these [O iii] emission lines are in the faint LBGs, we note that this doublet is carrying several percent of the entire luminosity of the galaxy. This makes it one of the most distinctive observable features—after the Lyman break—in the entire galaxy spectrum.

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Local analogs to the faint LBGs are the "green pea" galaxies that were identified, in the SDSS, by their distinctive green color originating from strong [O iii]λλ4959,5007 emission (Cardamone et al. 2009). The SDSS green peas have masses in the range ${\mathrm{log}}_{10}({M}_{* }/{M}_{\odot })=8.5\mbox{--}10.5$ and typical sSFRs of $\simeq 4$ Gyr⁻¹, comparable to those derived for our average LBGs (excluding the faintest bin with extremely high implied sSFR). These local [O iii]-emitters are quite rare, with a rough surface density of $\sim {10}^{-3}$ arcmin⁻².

Less-massive ELGs at $z\lesssim 1$ are presented in Ly et al. (2015, 2016). In particular, Ly et al. (2015) presents a sample of metal-poor, [O iii]λ4363-detected $z\simeq 0.8$ galaxy sample from the DEEP2 Galaxy Redshift Survey (Newman et al. 2013). These galaxies have high sSFR near 10 Gyr⁻¹ and lie about 1 dex above the typical ${M}_{* }\mbox{--}\mathrm{SFR}$ correlation at $z\simeq 0.8$, similar to what is observed for the $\langle i^{\prime} \rangle =27$ stacked LBG in this study. This sample, shown on Figure 10 as cyan circles, has a median ${\mathrm{EW}}_{0}([{\rm{O}}\,{\rm{III}}]\lambda \lambda \mathrm{4959,5007})\sim 400\,\mathring{\rm A}$ and extends down to ${M}_{* }\simeq {10}^{8}$ ${M}_{\odot }$. The galaxies at $z=0.1\mbox{--}1$ from Ly et al. (2016), selected by narrow-band emission in the SDF, are shown in Figure 10 as red circles. The filled circles represent galaxies with detectable [O iii]λ4363 emission (≃36% of their sample), while those without detection of this line are shown as open circles. Taken together, these ELGs have a median ${\mathrm{EW}}_{0}([{\rm{O}}\,{\rm{III}}]\lambda \lambda 4959,5007)\simeq 160\,\mathring{\rm A}$.

WISPS has produced larger samples of emission-line galaxies at $z\sim 1\mbox{--}2$. The properties of our fainter LBGs overlap with some of the more extreme emission-line galaxies in WISP. For example, Atek et al. (2011) selected [O iii] and Hα emitters with $\mathrm{EW}\gt 200\,\mathring{\rm A}$ (rest-frame). The $\langle i^{\prime} \rangle =26$ average LBG in our study (with $\ {\mathrm{EW}}_{0}([{\rm{O}}\,{\rm{III}}])\simeq 360$ Å) is roughly within the top half of the [O iii] EW distribution in Atek et al. (2011), while our faintest average LBG ($\langle i^{\prime} \rangle =27$) has ${\mathrm{EW}}_{0}([{\rm{O}}\,{\rm{III}}])\simeq 1450\,\mathring{\rm A}$, which is in the top ∼15%. A larger WISP sample, presented in Atek et al. (2014), shows that our average $\langle i^{\prime} \rangle =24,25$, and 26 LBGs have sSFRs in the top ∼40% of these emission-line galaxies. Our faintest average LBG is near the top 5% of their sSFR distribution.

We also compare our LBGs to extreme emission-line galaxies (EELGs) at higher redshift from van der Wel et al. (2011) and Maseda et al. (2014). The galaxies in these studies were selected to have very large ${\mathrm{EW}}_{0}([{\rm{O}}\,{\rm{III}}])$, strong enough to produce detectable excess in the J and H bands for their samples at $z\sim 1.7$ and $z\sim 2.2$, respectively. These EELGs have typical masses of $\sim {10}^{8}\mbox{--}{10}^{9}{M}_{\odot }$ and high sSFRs ($\simeq 10$ Gyr⁻¹ for the galaxies in Maseda et al. (2014) that are comparable to the best-fit sSFRs for our average LBGs. The $\langle i^{\prime} \rangle =27$ stacked LBG from our sample is within the locus of the van der Wel et al. (2011) EELGs in Figure 10. Interestingly, these galaxies have a space density nearly two orders of magnitude higher than the local green peas.

The enhancement of ${\mathrm{EW}}_{0}([{\rm{O}}\,{\rm{III}}])$ is not only associated with low-mass dwarf galaxies with extreme starbursts, it also appears to increase at higher redshifts across the board. It has long been suspected that [O III] emission is much stronger at higher redshifts ($z\gt 3$) than in star-forming galaxies at lower redshifts. This was the conclusion of the first narrow-band imaging search for [O iii]-line-emitting galaxies at z = 3.3 (Teplitz et al. 1999). Near-infrared spectroscopy further confirmed the unusual strength of [O iii] in Lyman break galaxies at $z\sim 3$ (Teplitz et al. 2000), and gravitationally lensed galaxies at these redshifts hinted that [O iii] was even stronger in the less-luminous galaxies (Teplitz et al. 2004).

The next step forward came with multi-object spectroscopy, using MOSFIRE on Keck, which showed that strong [O iii] emission is common among $z\sim 3\mbox{--}4$ LBGs. Schenker et al. (2013) found strong K-band emission lines in 20/28 targeted LBGs at $z=3.0\mbox{--}3.8$ (13 of which had prior spectroscopic confirmation, and 15 of which were photometrically selected). These galaxies, which overlap the mass range of our sample, have a median of ${\mathrm{EW}}_{0}([{\rm{O}}\,{\rm{III}}]\mathrm{4959,5007})\simeq 280\,\mathring{\rm A}$, with a few reaching ∼1000 Å or higher. More recently, Holden et al. (2016) present K-band spectra of 15 spectroscopically confirmed and 9 photometrically selected LBGs at $z=3.2\mbox{--}3.8$. Again, strong [O iii] emission is detected in 18/24 LBGs (15/15 and 3/9 for the spectroscopic and photometric samples, respectively), with a median ${\mathrm{EW}}_{0}([{\rm{O}}\,{\rm{III}}])\simeq 180\,\mathring{\rm A}$. As shown by Figure 10, our stacked LBGs are quite consistent with their EW distribution. Combined with our result, these studies suggest that the fraction of [O iii]-emitters in the LBG population is substantial at $3\lesssim z\lesssim 4$.

Also shown in Figure 10 (as black and brown open triangles) are the implied EWs from average SEDs of a sample of I-dropouts presented in Smit et al. (2014). These $z\sim 7$ galaxies (with masses of ${10}^{9}\mbox{--}{10}^{9.5}{M}_{\odot }$) show a very large excess in the IRAC [3.6] filter, which indicates rest-frame equivalent widths of ${\mathrm{EW}}_{0}([{\rm{O}}\,{\rm{III}}])+{\rm{H}}\beta )\gt 637\,\mathring{\rm A}$ for the average galaxy in the sample and ${\mathrm{EW}}_{0}([{\rm{O}}\,{\rm{III}}]+{\rm{H}}\beta )\simeq 1582\,\mathring{\rm A}$ for a subset of the bluest galaxies. Signatures of such strong [O iii]+Hβ contamination are also reported for a handful of $z\sim 8$ galaxies with ${M}_{* }\simeq {10}^{9}$ ${M}_{\odot }$ in Labbé et al. (2013), appearing as an excess in the IRAC [4.5] band. The implied equivalent width for their sample is ${\mathrm{EW}}_{0}([{\rm{O}}\,{\rm{III}}]+{\rm{H}}\beta )=670\,\mathring{\rm A}$. Thus, we conclude that although very strong [O iii]-emitting galaxies are rare in the local universe, they become increasingly common from redshifts $1\lesssim z\lesssim 2$, to $2\lesssim z\lesssim 3$. At higher redshifts, very strong [O iii] emitters may even become the norm.

Aside from the K-band excess, we observe two more subtle features in the faint average LBG SEDs in Figure 8 that are not explained by the best-fit stellar models. First, there is an excess in the J-band flux over the stellar continuum. An analogous feature has been found in stacked SEDs of $z\sim 4$ galaxies in González et al. (2012), in the form an H-band excess. These authors attribute the excess to a bias due to the Balmer break falling into the H-band for the low-redshift tail of their sample, such that the stacked H-band flux is increased by flux redward of the break for the lower-redshift tail of their sample. However, our LBG selection is not sensitive to the low redshifts required for this bias to explain the observed stacked J-band excess. An alternate contribution to the observed J-band excess in our faint LBG SEDs might come from bound–free and free–free nebular continuum emission at rest-frame ∼3100 Å. We predict the possible contribution from nebular continuum emission at these wavelengths by assuming a typical $[{\rm{O}}\,{\rm{}}\mathrm{III}]/{\rm{H}}\beta =3$ to estimate the Hβ flux from the [O iii] equivalent widths. Using the tabulated Hβ and continuum emission coefficients at an electron temperature of ${T}_{e}$ = 10,000 K from Osterbrock (1989), we determine that the nebular continuum can contribute ∼10% to the observed J-band flux. This would still leave about 20% of the observed flux in the faintest bin unaccounted for. We thus conclude that the J-band excess is largely due to uncertainty in forming the stacked images, such as an over-correction for the faint-stack defect described in Section 5.1.1. Another uncertainty comes from our assumption of a simple stellar population to describe the starlight.

There is also weak evidence of Lyα line emission increasing V-band flux in the faintest LBGs ($\langle i^{\prime} \rangle =27$). Comparing the observed V flux to the flux predicted by convolving the stellar continuum model with the V filter transmission profile yields an excess of $\simeq 2\sigma$ significance, and implies a Lyα equivalent width of ${\mathrm{EW}}_{0}(\mathrm{Ly}\alpha )\simeq 30\,\mathring{\rm A}$. If correct, this would mean that most of the LBGs in the faintest stack are Lyα emitters (LAEs). This would be consistent with the increasing LAE fraction found in the faintest LBGs at $3\lt z\lt 7$ (Stark et al. 2010, 2011; Pentericci et al. 2011; Ono et al. 2012; Schenker et al. 2012; Treu et al. 2012). However, those studies generally do not include galaxies fainter than ${M}_{\mathrm{UV}}=-18.75$, while the LBGs in our sample extend to lower luminosity. Nonetheless, if we extrapolate the result from Stark et al. (2010) down to absolute magnitude ${M}_{\mathrm{UV}}\simeq -18$, this would predict that ∼70% of LBGs at $3\lt z\lt 6.3$ should be LAEs with ${\mathrm{EW}}_{0}(\mathrm{Ly}\alpha )\gt 50\,\mathring{\rm A}$. Our observed average 30 Å equivalent width could be explained if about half of the faintest LBGs are strong LAEs with ${\mathrm{EW}}_{0}(\mathrm{Ly}\alpha )\gtrsim 50\,\mathring{\rm A}$. Alternatively, a few bright sources in the faint bin might have extremely strong Lyα emission, with the rest having weak or no emission (or perhaps even absorption). This would be consistent with what is observed for bright LBGs (Shapley et al. 2003, for example); however, we are in the regime of faint dwarf galaxies with little extinction from dust.

5.3.2. [O iii] Line Ratios and ISM Conditions

The physical conditions of star-forming regions at high redshift differ significantly from local systems. An observable consequence of these differences is higher [O iii]/Hβ and [O iii]/Hα ratios measured in distant versus local galaxies (e.g., Teplitz et al. 2004; Shapley et al. 2005; Erb et al. 2006; Ly et al. 2007a; Brinchmann et al. 2008; Liu et al. 2008; Schenker et al. 2013; Steidel et al. 2014; Sanders et al. 2015; Holden et al. 2016). To place the LBGs studied here in this context, we estimate [O iii]/Hα ratio using the implied $\mathrm{EW}([{\rm{O}}\,{\rm{III}}]\lambda \lambda \mathrm{4959,5007})$ and best-fit SFRs of the stacked LBGs. The [O iii] luminosity is calculated from

$$\begin{eqnarray}&&{L}_{[{\rm{O}}}\,{\rm{III}}]=4\pi {D}_{{\rm{L}}}^{2}{f}_{2.2\mu {\rm{m}},\mathrm{cont}}\times \mathrm{EW}([{\rm{O}}\,{\rm{III}}]),\end{eqnarray} \tag{ 11 }$$

where ${D}_{{\rm{L}}}(z=3.1)=26$ Gpc is the luminosity distance at z = 3.1, ${f}_{2.2\mu {\rm{m}},\mathrm{cont}}$ is the continuum flux density at 2.2 μm in ergs s⁻¹ cm⁻² Å⁻¹ (measured from the best-fit stellar continuum models as described above), and the equivalent widths are in the observed frame.

We estimate the ${L}_{{\rm{H}}\alpha }$ of the LBGs from their model SEDs with the same approach used in Ly et al. (2012). We use the Kennicutt (1998) relation to calculate intrinsic Hα luminosity from the best-fit SFRs: ${L}_{{\rm{H}}\alpha ,\mathrm{int}}=1.26\times {10}^{41}\cdot \mathrm{SFR}$ [${M}_{\odot }\,{\mathrm{yr}}^{-1}]$. Then, in order to compare our LBGs with other galaxy samples, we predict their reddened Hα luminosities by estimating the extinction for the Hα line, $A({\rm{H}}\alpha )$, from their best-fit visual extinctions A_V. The Hα extinction can be written in terms of an extinction curve evaluated at $\lambda =6563\,\mathring{\rm A}$, $k(6563\mathring{\rm A} )$, and the color excess for the gas $E{(B-V)}_{\mathrm{gas}}$:

$$\begin{eqnarray}&&A({\rm{H}}\alpha )=k(6563\mathring{\rm A} )\times E{(B-V)}_{\mathrm{gas}}.\end{eqnarray} \tag{ 12 }$$

For simplicity, we adopt $k(6563\,\mathring{\rm A} )=2.00$ from the SMC extinction curve (Gordon et al. 2003). We relate the color excess for the gas to that for the stellar continuum by assuming the Calzetti et al. (2000) relation: $E{(B-V)}_{\mathrm{gas}}=2.27E{(B-V)}_{\mathrm{stars}}$. We note that previous studies generally support the assumption that the extinction for the gas is roughly twice that of the stars (e.g., Ly et al. 2012; Reddy et al. 2015). Thus:

$$\begin{eqnarray}A({\rm{H}}\alpha ) & = & k(6563\,\mathring{\rm A} )\times 2.27\times E{(B-V)}_{\mathrm{stars}}\\ A({\rm{H}}\alpha ) & = & 4.54\times E{(B-V)}_{\mathrm{stars}}.\end{eqnarray} \tag{ 13 }$$

The color excess for the stars is calculated assuming the Calzetti attenuation law: $E{(B-V)}_{\mathrm{stars}}={A}_{V}/4.05$. We note that regardless of our assumptions for the reddening of Hα, the results discussed below are not significantly impacted ($\lesssim 0.1$ dex difference).

We plot estimated ${L}_{{\rm{H}}\alpha }$ versus ${L}_{[{\rm{O}}{\rm{III}}]}$ for the average LBG SEDs in Figure 11. The black dashed–dotted line indicates a slope of unity ([O iii]/Hα = 1). The average $\langle i^{\prime} \rangle =25$, 26, and 27 stacked LBGs have [O iii]/H$\alpha \simeq 1.6$, 3.1, and 4.1, respectively. The solid black line and gray band represent the mean relation and scatter for WISP galaxies in the redshift range $z=0.8\mbox{--}1.2$ reported in Mehta et al. (2015). Our two faintest stacked SEDs lie toward the bottom edge of the scatter, comparable to the more extreme WISP galaxies observed. Likewise, [O iii]/Hα in the faint LBGs is consistent with the ratios observed for a subset of WISP galaxies with Magellan/FIRE spectra in Masters et al. (2014), which reach [O iii]/H $\alpha \simeq 3$ (shown on the plot as yellow triangles). Domínguez et al. (2013) reports line ratios for stacked WISP spectra reaching an average $[{\rm{O}}\,{\rm{III}}]/{\rm{H}}\alpha \sim 2$ for their faintest bin of ${L}_{{\rm{H}}\alpha }$. At $z\sim 2.3$, galaxies from the MOSDEF survey have stacked spectra (binned by mass) indicating a maximum average [O iii]/Hα of $\simeq 1.8$ (Sanders et al. 2015). However, these stacks (shown as orange triangles in the plot) were normalized by ${L}_{{\rm{H}}\alpha }$, which prevents low-metallicity objects with high [O iii]/Hβ from dominating the stacked line ratios. The most extreme galaxies at $z\sim 1.95\mbox{--}2.65$ in the KBSS survey (Steidel et al. 2014) have individual spectra that reach up to [O iii]/H$\alpha \sim 4$. The $z\lesssim 1$ ELGs from Ly et al. (2015) and Ly et al. (2016), also included in Figure 11, form a locus with a shallower slope than the faint stacked LBGs. Perhaps most comparable to our sample is a subset of six "extreme" green peas (Jaskot & Oey 2013) that have $[{\rm{O}}\,{\rm{III}}]/{\rm{H}}\alpha \simeq 3.0$ on average (shown as the green circles in the plot).

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The average faint galaxy in our sample is clearly efficient at converting ionizing photons from starlight into [O iii] emission. By integrating the stellar continuum models fit to the faint average LBGs, we estimate the ratio of flux in [O iii] to observed (reddened) stellar luminosity. The ratio is $\simeq 1$% for $\langle i^{\prime} \rangle =26$ and $\simeq 3$% for $\langle i^{\prime} \rangle =27$ stacked LBGs. Spectroscopic surveys of high-redshift galaxies such as those described above support this trend of high-level [O iii] emission in the distant universe. Theoretical explanations include the possibility that the interstellar medium (ISM) has a larger ionization parameter (ionizing photon density to gas density), higher electron density, harder ionizing radiation spectrum, and/or very low metallicity (e.g., Kewley et al. 2013a, 2013b; Steidel et al. 2014; Nakajima & Ouchi 2014; Sanders et al. 2016). The very strong [O iii] emission implied for the average faint LBG in this study suggests that extreme ionization conditions (relative to those found in most local galaxies) are ubiquitous in low-mass galaxies at $z\sim 3$. Recent studies have shown that green peas (local strong [O iii]-emitters), which have ISM properties similar to LBGs, are good candidates for leaking a significant amount of ionizing radiation (e.g., Nakajima & Ouchi 2014; Izotov et al. 2016a, 2016b). Thus, galaxies at $z\gt 6$ that are directly analogous to the faint LBGs in this study might have been responsible for the bulk of the cosmic reionization of the intergalactic medium. Confirmation will require deep spectroscopy at wavelengths of 3.5 μm or longer. This will be possible with the James Webb Space Telescope in targeted surveys, or serendipitously with slitless spectroscopy.

We measure the clustering of both our LBG sample as a whole (all 5161 objects), and the brightest half of the sample (split by R-magnitude), in order to investigate mass-dependent clustering. We largely follow the methodology of Yoshida et al. (2008) to calculate the angular correlation function (ACF), $\omega (\theta )$, using the Landy & Szalay (1993) estimator:

$$\begin{eqnarray}&&\omega (\theta )=\displaystyle \frac{{dd}(\theta )-2{dr}(\theta )+{rr}(\theta )}{{rr}(\theta )},\end{eqnarray} \tag{ 14 }$$

where ${dd}(\theta )$, ${dr}(\theta )$, and ${rr}(\theta )$ are the numbers of galaxy–galaxy, galaxy–random, and random–random pairs, respectively, with angular separations θ. In creating the random catalog, we avoided regions where galaxies are not detected (for example, near bright stars). We generated a factor of 100 more random coordinates than the number of galaxies in the sample, and normalized dd, dr, and rr to the total number of pairs. Because the random pairs are subject to the same limitations as the real data, the deficiency of faint galaxies we found within several arcseconds of other galaxies due to confusion should not impact our estimated clustering.

We evaluate errors and covariance matrices of both the full sample and bright-half sample by the delete-one jackknife resampling method. The entire survey field is divided into 36 subfields and ACFs are calculated by omitting one subfield, repeating this procedure 36 times. The covariance matrix, C_ij, is derived as

$$\begin{eqnarray}&&{C}_{{ij}}=\displaystyle \frac{N-1}{N}\displaystyle \sum _{k=1}^{N}({\omega }_{k}({\theta }_{i})-\bar{\omega }({\theta }_{i}))\times ({\omega }_{k}({\theta }_{j})-\bar{\omega }({\theta }_{j})),\end{eqnarray} \tag{ 15 }$$

where $\bar{\omega }({\theta }_{i})$ is the averaged ACF of the ith angular bin.

The observed ACFs and associated error bars for the bright and full samples of LBGs are shown in Figure 13.

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At separations smaller than ≃10'', corresponding to about 80 kpc at z = 3, the observed ACFs in Figure 13 show a significant steepening. Such a steepening has also been found in previous LBG clustering studies, and is attributed to the additional clustering of $\gt 1$ galaxies residing in a single dark matter halo (Hildebrandt et al. 2007; Yoshida et al. 2008; Bielby et al. 2013; Bian et al. 2013). Thus, rather than fit a single power-law to the ACF, as is commonly done, we carried out a halo occupation distribution (HOD; e.g., Berlind & Weinberg 2002; Zheng et al. 2005; van den Bosch et al. 2007) analysis to interpret the relationship between the U-dropout galaxies and their host dark halos. We employ the standard halo occupation model proposed by Zheng et al. (2005). The occupation of central galaxies, ${N}_{{\rm{c}}}({M}_{{\rm{h}}})$, is formulated as

$$\begin{eqnarray}&&{N}_{{\rm{c}}}({M}_{{\rm{h}}})=\displaystyle \frac{1}{2}\left[1+\mathrm{erf}\left(\displaystyle \frac{\mathrm{log}({M}_{{\rm{h}}})-\mathrm{log}({M}_{\min })}{{\sigma }_{\mathrm{log}M}}\right)\right],\end{eqnarray} \tag{ 16 }$$

whereas the occupation of satellite galaxies, ${N}_{{\rm{s}}}({M}_{{\rm{h}}})$, can be described by:

$$\begin{eqnarray}&&{N}_{{\rm{s}}}({M}_{{\rm{h}}})={\left(\displaystyle \frac{{M}_{{\rm{h}}}-{M}_{0}}{{M}_{1}}\right)}^{\alpha }.\end{eqnarray} \tag{ 17 }$$

The number of total galaxies within dark halos with mass ${M}_{{\rm{h}}}$ is:

$$\begin{eqnarray}&&N({M}_{{\rm{h}}})={N}_{{\rm{c}}}({M}_{{\rm{h}}})[1+{N}_{{\rm{s}}}({M}_{{\rm{h}}})].\end{eqnarray} \tag{ 18 }$$

We vary all of the HOD free parameters, ${M}_{\min }$, ${M}_{1}$, ${M}_{0}$, ${\sigma }_{\mathrm{log}M}$, and α, and find the best-fit parameters to the observed ACFs.

The HOD analysis requires assuming some analytical models to characterize dark halos. We use the halo mass function of Sheth & Tormen (1999), the radial density profile of Navarro et al. (1997), the halo bias model of Tinker et al. (2010), and the halo mass-concentration parameter relation of Takada & Jain (2003). The redshift completeness functions $p(m,z)$ (Section 3.3) are employed as the redshift distributions of each subsample.

The HOD parameters are constrained through minimizing the ${\chi }^{2}$ as follows:

$$\begin{eqnarray}{\chi }^{2} & = & \displaystyle \sum _{i,j}({\omega }_{\mathrm{obs}}({\theta }_{i})-{\omega }_{\mathrm{HOD}}({\theta }_{i})){({C}^{-1})}_{{ij}}({\omega }_{\mathrm{obs}}({\theta }_{j})-{\omega }_{\mathrm{HOD}}({\theta }_{j}))\\ & & +\displaystyle \frac{{[{n}_{g}^{\mathrm{obs}}-{n}_{g}^{\mathrm{HOD}}]}^{2}}{{\sigma }_{{n}_{g}}^{2}},\end{eqnarray} \tag{ 19 }$$

where ${C}^{-1}$ is the inverse covariance matrix (Equation (15)), ${n}_{g}^{\mathrm{obs}}$ and ${n}_{g}^{\mathrm{HOD}}$ are the galaxy number densities from observation and the HOD model, and ${\sigma }_{{n}_{g}}$ is the statistical $1\sigma$ error of ${n}_{g}^{\mathrm{obs}}$, respectively. The best-fit parameter sets and those $1\sigma$ confidence intervals are estimated by Markov Chain Monte Carlo simulation. We note that effects of contaminations on observed ACFs are corrected by multiplying the factor of $1/{(1-{f}_{{\rm{c}}})}^{2}$, where ${f}_{{\rm{c}}}$ represents the fraction of contamination.

The best-fit ACFs for the full and bright LBG samples, computed by the HOD modeling, are shown with the observed ACFs in Figure 13. The best-fit HOD parameters and deduced parameters, i.e., mean halo masses and satellite fractions, are listed in Table 5. Mean halo masses, $\langle {M}_{{\rm{h}}}\rangle$, and satellite fractions, ${f}_{\mathrm{sat}}$, are defined as

$$\begin{eqnarray}&&\langle {M}_{{\rm{h}}}\rangle =\displaystyle \frac{\int {{dM}}_{{\rm{h}}}{M}_{{\rm{h}}}n({M}_{{\rm{h}}})N({M}_{{\rm{h}}})}{\int {{dM}}_{{\rm{h}}}n({M}_{{\rm{h}}})N({M}_{{\rm{h}}})},\end{eqnarray} \tag{ 20 }$$

and

$$\begin{eqnarray}&&{f}_{\mathrm{sat}}=1-\displaystyle \frac{\int {{dM}}_{{\rm{h}}}n({M}_{{\rm{h}}}){N}_{{\rm{c}}}({M}_{{\rm{h}}})}{\int {{dM}}_{{\rm{h}}}n({M}_{{\rm{h}}})N({M}_{{\rm{h}}})},\end{eqnarray} \tag{ 21 }$$

where $n({M}_{{\rm{h}}})$ is the halo mass function.

Table 5.  Best-fit HOD Parameters and Deduced Parameters with $1\sigma$ Confidence Intervals

N	${\mathrm{log}}_{10}({M}_{\min }/{h}^{-1}{M}_{\odot })$	${\mathrm{log}}_{10}({M}_{1}/{h}^{-1}{M}_{\odot })$	${\mathrm{log}}_{10}({M}_{0}/{h}^{-1}{M}_{\odot })$	${\sigma }_{\mathrm{log}M}$	α	${\mathrm{log}}_{10}(\langle {M}_{{\rm{h}}}\rangle /{h}^{-1}{M}_{\odot })$	${f}_{\mathrm{sat}}$	${\chi }^{2}/\mathrm{dof}$
$5,\,161$	${10.94}_{-0.28}^{+0.13}$	${12.75}_{-0.34}^{+0.48}$	${8.18}_{-2.16}^{+2.25}$	${0.51}_{-0.32}^{+0.18}$	${0.81}_{-0.20}^{+0.38}$	11.29 ± 0.12	0.05 ± 0.03	0.61
$2,\,581$	${11.21}_{-0.12}^{+0.11}$	${13.20}_{-0.42}^{+0.50}$	${8.18}_{-2.18}^{+2.22}$	${0.49}_{-0.29}^{+0.21}$	${0.83}_{-0.19}^{+0.38}$	11.49 ± 0.10	0.03 ± 0.02	1.40

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The mean halo masses implied for the total LBG sample and bright-half subsample are $\langle {M}_{{\rm{h}}}\rangle =({1.9}_{-0.5}^{+0.6})\times {10}^{11}$ ${h}^{-1}{M}_{\odot }$ and $\langle {M}_{{\rm{h}}}\rangle =({3.1}_{-0.6}^{+0.8})\times {10}^{11}$ ${h}^{-1}{M}_{\odot }$, respectively. Bian et al. (2013) have also carried out HOD analysis for their $z\sim 3$ LBG sample in two magnitude bins and found $\langle {M}_{{\rm{h}}}\rangle =(2.5\pm 0.3)\times {10}^{12}$ ${h}^{-1}{M}_{\odot }$ for LBGs with $24.0\lt R\lt 24.5$, and $\langle {M}_{{\rm{h}}}\rangle \,=({3.3}_{-0.4}^{+0.6})\times {10}^{12}$ ${h}^{-1}{M}_{\odot }$ for $23.5\lt R\lt 24.0$. The mean halo masses derived for our sample of LBGs are significantly smaller, consistent with the fact that we include much fainter galaxies in the sample (down to $R\simeq 28$ for the full sample). Overall, the HOD analysis of our LBG sample and comparison to that of brighter samples support the notion that more massive dark matter halos host more luminous LBGs.