TO STACK OR NOT TO STACK: SPECTRAL ENERGY DISTRIBUTION PROPERTIES OF Lyα-EMITTING GALAXIES AT z = 2.1

The nature of radiation emitted from astrophysical sources offers insights into their physical properties. Lyα photons are emitted when a neutral hydrogen atom undergoes a transition from the first excited state to the ground state. An immense number of Lyα photons are emitted at 1216 Å rest frame from H ii regions surrounding young, massive stars. The large luminosity of the Lyα emission line makes it easily identified even over vast distances, and thus it is widely used in research in extragalactic astronomy. Lyα emitting (LAE) galaxies are typically discovered using the narrowband (NB) filter technique (Cowie & Hu 1998; Rhoads et al. 2000), in which a detection of an LAE is made by finding greater signal-to-noise ratios (S/Ns) in an NB filter image than in images obtained with adjacent broadband filters.

LAE galaxies at 2 < z < 3 are thought to be building blocks of present-day galaxies like our own Milky Way (Gawiser et al. 2007; Guaita et al. 2011). These galaxies are easy to detect thanks to the strength of the Lyα line, but they usually have low stellar masses and are therefore dim in the continuum. As a result, it has been a common procedure to stack multiple photometric data points of LAEs in order to enhance their S/N (Gawiser et al. 2006a, 2007; Nilsson et al. 2007; Pirzkal et al. 2007; Finkelstein et al. 2007; Lai et al. 2008; Pentericci et al. 2009; Ono et al. 2010a, 2010b; Ouchi et al. 2010; Nilsson et al. 2011; Guaita et al. 2011; Finkelstein et al. 2011a; Acquaviva et al. 2011, 2012b).

Inconsistencies in the literature, particularly in estimating ages and sample diversity, motivate a study of the validity of the stacking method for analyzing the spectral energy distributions (SEDs) of dim LAEs. To date, most SED studies of high-redshift LAEs have been done by stacking the catalog fluxes or ground-based postage-stamp images of galaxies too faint to study individually. By stacking numerous faint objects (via average or median), we increase the S/N significantly. It is believed that a study of these images will represent the characteristics of a sample of galaxies on the average. However, this interpretation of median as typical relies on the assumption that the best-fit SED parameters of the stacked galaxies have similar physical properties. An additional key assumption is that the median (average) SEDs will match the median (average) physical properties of individual galaxy SEDs. This assumption appears reasonable for stellar mass if the mass-to-light ratio of the stellar populations of different LAEs is similar, but it is not guaranteed for other parameters like age or E(B − V).

Results from the analysis of stacked samples of LAEs indicate a large spread in the properties of these galaxies and introduce further concern about the validity of the stacking technique. A recent study of median image-stacked LAEs at z = 3.1 found them to be older than median flux-stacked LAEs at z = 2.1 (Acquaviva et al. 2012a); this puzzling result would imply that LAEs grow younger as cosmic time progresses. However, it might instead result from the failure of image and/or flux stacking to accurately measure the SED characteristics of individual galaxies, and from systematic differences between image stacking and flux stacking. Additionally, a previous study of individual LAEs at z = 4.5 (Finkelstein et al. 2009) found a wide range of physical properties, including many LAEs that appeared dusty, something that would not have been expected given the earlier results at z = 3.1 from Gawiser et al. (2006b), which implied, via a stacking analysis, that LAEs were dust free.

In this paper, we perform SED fitting on 20 LAEs at z = 2.1, as well as on their flux- and image-stacked SEDs, in order to clarify whether these discrepancies are attributable to the failure of one or all stacking methods. The use of deep Hubble Space Telescope (HST) and Spitzer Infrared Array Camera (IRAC) data gives us the opportunity, for the first time, to study rest-frame ultraviolet (UV) and optical properties of individual objects, which is crucial for accurate constraints on age and stellar mass. We use the Markov Chain Monte Carlo (MCMC) algorithm SpeedyMC (Acquaviva et al. 2012a) to analyze the SED properties of our sample. SpeedyMC is a streamlined implementation of GalMC (Acquaviva et al. 2011) designed to handle large samples of SEDs. This code utilizes Bayesian statistics to determine the expectation values of SED parameters and provide estimates of the uncertainties associated with their measurements. All magnitudes referenced in the text hereafter are in the AB magnitude system. All SED fitting calculations assume H₀ = 73 km s⁻¹ Mpc⁻¹, Ω_M = 0.258, and Ω_Λ = 0.742.

The starting point for our analysis was the sample of 216 LAE candidates at z = 2.1 discovered in deep 3727 Å NB images of the 30'  × 30' Extended Chandra Deep Field-South (ECDF-S) by Guaita et al. (2010) as refined by Bond et al. (2012), who excluded 34 additional LAE candidates that appear to be low-redshift contaminants due to extended morphology in HST V-band optical images. These objects typically have low S/Ns in ground-based broadband images, with a median magnitude of R ∼ 25.5, and are unresolved at 01. Thus, the addition of deep HST imaging is crucial.

The Cosmic Assembly Near-Infrared Deep Extragalactic Legacy Survey (CANDELS; Grogin et al. 2011; Koekemoer et al. 2011) has produced a deep H-band-selected multiwavelength catalog of the GOODS-S field, which encompasses the central 16'  ×  10' of ECDF-S (Guo et al. 2013). The observations are 10 epochs deep in H band (limiting magnitude of 27.2), which corresponds to roughly six HST orbits. The catalog contains photometry from U-band images from CTIO and VIMOS, Advanced Camera for Surveys (BViz) images from the GOODS survey (Giavalisco et al. 2004), CANDELS WFC3 images (F098M, F105W, F125W, and F160W), K-band images from Hawk-I and ISAAC, and Spitzer Space Telescope IRAC (Fazio et al. 2004) 3.6, 4.5 μm images from SEDs (Ashby et al. 2013) and 5.8, 8.0 μm images from GOODS (Dickinson et al. 2003). For SED fitting, we excluded both U bands due to expected Lyα contamination at this redshift, F098M due to having coverage in a minority of our sample that might have different median physical properties than the rest, and both K bands due to incomplete coverage and shallower photometry. All of our analysis made use of imaging and photometry from the remaining 11 space-based bands.

We match the positions of our catalog of known LAEs at z = 2.1 (Guaita et al. 2010) to the positions of objects contained within the version of the CANDELS Multi-wavelength catalog presented in Guo et al. (2013). The positions within the z = 2.1 LAE catalog are accurate to within 01. Objects within a distance of 05 were recorded as a match, and their CANDELS SEDs were used in our analysis. Only two objects showed multiple possible counterparts within this tolerance, and they were discarded due to the strong likelihood of neighboring object contamination. In total, we found 20 LAE counterparts in the CANDELS Multi-wavelength catalog. We note that only 30 of the original 216 LAEs are within the GOODS-S field, and the 10 "missing" objects were probably too dim to have been detected at the 5σ level in H band (F160W) and were thus excluded from the CANDELS catalog. In our sample, 9 of 20 LAEs were observed with VLT+FORS (Berry et al. 2012), yielding five confirmed redshifts over 2.05 < z < 2.08, consistent with the expected scatter in redshifts from NB selection. The other four objects showed insufficient S/N to yield a spectroscopic redshift.

The main objective of this paper is the comparison of the SED fitting properties of individual objects and stacks. The SEDs of the stacks can be obtained via two different procedures: by combining the cataloged fluxes of the individual objects, or by stacking the postage-stamp images of the 20 individual objects at each wavelength and then performing photometry on the combined images. We refer to the SEDs obtained via these two methods as "flux-stacked" and "image-stacked," respectively. We took the median and average of the flux densities in each band of all of the z = 2.1 LAEs with CANDELS photometry in that band and will henceforth refer to these as the median and average flux stacks, respectively.

To create the image-stacked SED, we performed both average and median stacking on cutout images in each band centered around each of our LAEs with CANDELS imaging in that band. The extracted HST cutouts measured 211 pixels on a side, while the IRAC cutouts measured 21 pixels. As the processed HST image pixels measure 006 and the processed IRAC image pixels measure 06, the cutouts in all bands had approximately the same size of 126 on a side. We note that of the 20 objects, 6 of them were located in the northern ∼25% of the GOODS-S field, which has WFC3 data from the WFC3 Early Release Science program (Windhorst et al. 2011). While these images were obtained prior to the CANDELS data, they were reprocessed alongside the CANDELS data to make a single mosaic which was used to extract the cutouts. Along with the science images, we also extracted cutouts from the rms images for use with Source Extractor (Bertin & Arnouts 1996; see Koekemoer et al. 2011 for details on the science and rms mosaic construction).

To create our image stacks, we created median and average images by taking the median/average value for each pixel from all of the cutout images for a given band, excluding pixels with rms values >10—a very high value, indicative of a problem with the given pixel in that image. We then measured photometry on the image stacks using Source Extractor in dual-image mode, with the F160W stack as the detection image for the HST bands and the 3.6 μm image as the detection band for the IRAC images. We measured photometry using Kron-style elliptical apertures, denoted by MAG_AUTO within Source Extractor. For the HST bands, we measured colors in small Kron apertures (with the parameter PHOT_AUTOPARAMS set to 1.2, 1.7), with the fluxes corrected to total using an aperture correction derived by the ratio of the default MAG_AUTO flux value to our smaller Kron flux (this procedure has been optimized for faint, high-redshift sources; see Finkelstein et al. 2010 for details). For the IRAC stacks, we measured photometry directly from MAG_AUTO, which is designed to approximate a total flux. The photometry for the individual objects was also performed in the same way. For reference, typical S/N values in H band of the individual objects are ∼8, while that of the average flux stack is ∼16.

To further our analysis, we also introduce an additional set of image stacks centered on the object positions found in the MUSYC NB catalog of LAEs. Comparison of the primary HST-centered image stacks with these NB-centered stacked images will illustrate any improvement that results from eliminating the ∼01 NB centroiding errors (Guaita et al. 2010) and any astrometric offsets between ground-based and space-based coordinates. Figure 1 shows F160W filter postage-stamp images of each galaxy in our sample and those of the HST and NB-centered image stacks. Typically, LAEs have half-light radii of 0.5–2 kpc in HST images (Bond et al. 2012; Gronwall et al. 2007; Malhotra & Rhoads 2002). While some of the galaxies seen in Figure 1 are mildly resolved, their light distribution is accounted for by the TFIT (Laidler et al. 2007) process used to make the CANDELS catalogs, and the majority of their light will fall inside the apertures used to obtain photometry from image stacks. In order to explore the effects of potential IRAC contamination, we introduce IRAC-clean image stacks. These stacks were composed of 11 individual objects determined by visual inspection to be free of significant flux contamination from neighboring objects in the IRAC 3.6 μm band. The IRAC-clean stack postage-stamp images are compared to those of the entire sample in Figure 2. A slight loss of S/N in the clean sample is apparent due to the reduced sample size.

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2.1. Error Estimation

The uncertainty in the fluxes for individual objects' SEDs includes an effective zero-point calibration error, which is added in quadrature to the photometric uncertainty. We assume that these zero-point errors are 3% of the flux in HST bands and 8% in the IRAC bands (CANDELS team 2012, private communication).

Errors on the flux-stacked photometry are determined by a bootstrap procedure, as in Guaita et al. (2011) and Acquaviva et al. (2011). This accounts for both the photometric error and the sample variance resulting from the spread in the SEDs of different galaxies; for our sample, the latter is the dominant source of uncertainty. The same calibration uncertainty used for the individual objects' SED is added in quadrature to the bootstrap error in both the average and median flux stacked SED. For image-stacked SEDs, we adopt a conservative error estimate in each band by taking the larger of the photometric error indicated by SExtractor (Bertin & Arnouts 1996) and the bootstrap error and adding it in quadrature to the calibration error. Figure 3 shows the SEDs of all individual objects and all stacks overlaid atop one another for comparison.

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2.2. Stacking Simulations

Information about the physical properties of galaxies, including redshift, stellar mass, age, dust content, and metallicity, can be determined by fitting their SEDs. The algorithm that was used for this analysis, SpeedyMC, is a faster version of GalMC introduced in Acquaviva et al. (2012a). Rather than using GALAXEV to generate the model SED at each point, a template library on a grid of locations encompassing the entire parameter space is generated beforehand. The exploration of parameter space is then carried out using the same MCMC algorithm, but at each location, multi-linear interpolation between the pre-computed spectra is used to calculate the model SED and its corresponding χ² value. The use of the SpeedyMC method allows us to fit the SEDs of galaxies at a rate 20,000 times faster than GalMC, and corresponding to about 1 s per galaxy on a 2.2 GHz MacBook Pro laptop.

In our analysis, the parameter space consisted of stellar mass, age, and dust reddening defined by excess color E(B − V). We assumed a constant star formation history and fixed the redshifts at z = 2.1 and the metallicity, Z, at the value of 0.2 Z_☉ as indicated by both spectroscopic analysis and SED fitting results (Acquaviva et al. 2011; Finkelstein et al. 2011b). We used the stellar population synthesis models of Bruzual & Charlot (2003, hereafter BC03) and S. Charlot & G. Bruzual (2007, private communication, hereafter CB07) and included nebular emission according to the procedure described in Acquaviva et al. (2011). Galactic absorption was taken into account using the Calzetti law (Calzetti et al. 1994), with a value R_V = 4.05, and starlight absorption by the intergalactic medium using the prescription from Madau (1995). We used the Salpeter initial mass function (IMF) for consistency with the previous literature on this subject. Our reference grid contained 100 values of both age and E(B − V) (the stellar mass is a free normalization parameter that can be varied without the need to build extra models in the grid). The number of values in the grid was chosen by refining the resolution of the grid until the resulting probability distributions obtained from GalMC and SpeedyMC were virtually identical, a sign that the linear interpolation between reference SEDs was working correctly. We limit the size of the grid in age to be between 10⁶ and 3.2 × 10⁹ yr, as the universe at z = 2.1 is 3.2 × 10⁹ yr old. In mass, we limit the grid to range 10⁴–10¹⁵ M_☉, and in E(B − V) we range 0–1. We ran six chains on each object, using 100,000 steps for each and starting from six different locations to ensure sampling of the entirety of parameter space and minimize "chain locking" in local minima. The "GetDist" software from Lewis & Bridle (2002) was used to analyze the chains and to make sure that the distribution inferred from the chains had converged to the true one. In the left panel of Figure 4, we plot the observed SED of CANDELS object 3826 overlaid with its best-fitting model produced by SpeedyMC. A similar plot for the median flux stacked SED is included as the right panel of Figure 4. We plot the observed U-band data points in this figure as an illustration of possible Lyα contamination in that band.

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The BC03 stellar population synthesis models performed better than the CB07 ones, resulting in smaller χ² values on average. This behavior is in agreement with results in the recent literature that seemed to favor a low contribution of thermally pulsating asymptotic giant branch stars (e.g., Kriek et al. 2010; Meidt et al. 2012; Zibetti et al. 2013). For these reasons, we show the parameters obtained by using the BC03 models in Table 1 and the figures. Individual LAEs at z = 2.1 are found to have stellar masses ranging from 2.3 × 10⁷ M_☉ to 8.5 × 10⁹ M_☉, ages ranging from 0.004 Gyr to 0.47 Gyr, and E(B − V) between 0.02 and 0.24.

Table 1. Main SED Fitting Results

CANDELS ID	M*	Age	E (B − V)	Best-fit
	(109 M/M☉)	(Gyr)		χ2/dof
205 (Clean)	$2.474^{+0.328}_{-0.446}$	$0.115^{+0.032}_{-0.035}$	$0.169^{+0.014}_{-0.012}$	81.9/8
991 (Clean)	$0.407^{+0.353}_{-0.118}$	$0.095^{+0.259}_{-0.041}$	$0.137^{+0.018}_{-0.060}$	18.2/8
7097 (Clean)	$0.971^{+0.250}_{-0.158}$	$0.116^{+0.070}_{-0.032}$	$0.156^{+0.013}_{-0.022}$	111.3/8
9125 (Clean)	$0.280^{+0.052}_{-0.176}$	$0.473^{+0.138}_{-0.394}$	$0.029^{+0.059}_{-0.011}$	20.1/8
10220 (Clean)	$0.717^{+0.154}_{-0.164}$	$0.211^{+0.100}_{-0.083}$	$0.094^{+0.020}_{-0.019}$	151.0/8
11824 (Clean)	$0.023^{+0.050}_{-0.005}$	$0.013^{+0.052}_{-0.004}$	$0.031^{+0.022}_{-0.018}$	14.0/8
12369 (Clean)	$0.064^{+0.030}_{-0.020}$	$0.057^{+0.051}_{-0.024}$	$0.058^{+0.013}_{-0.022}$	13.4/8
14889 (Clean)	$0.101^{+0.078}_{-0.016}$	$0.015^{+0.017}_{-0.003}$	$0.156^{+0.011}_{-0.010}$	13.8/8
17254 (Clean)	$8.518^{+2.021}_{-0.893}$	$0.280^{+0.182}_{-0.063}$	$0.240^{+0.016}_{-0.029}$	189.3/8
22689 (Clean)	$0.286^{+0.118}_{-0.073}$	$0.092^{+0.093}_{-0.036}$	$0.064^{+0.015}_{-0.029}$	5.0/8
24168 (Clean)	$1.116^{+0.229}_{-0.219}$	$0.147^{+0.070}_{-0.051}$	$0.121^{+0.019}_{-0.019}$	61.0/8
2210	$1.533^{+0.289}_{-0.281}$	$0.160^{+0.069}_{-0.052}$	$0.168^{+0.018}_{-0.019}$	28.0/8
3826	$0.024^{+0.014}_{-0.003}$	$0.004^{+0.004}_{-0.002}$	$0.083^{+0.034}_{-0.026}$	4.2/8
6696	$0.231^{+0.415}_{-0.095}$	$0.078^{+0.581}_{-0.042}$	$0.192^{+0.025}_{-0.094}$	11.3/8
7408	$0.053^{+0.006}_{-0.002}$	$0.004^{+0.001}_{-0.001}$	$0.017^{+0.011}_{-0.010}$	40.4/8
11292	$0.154^{+0.080}_{-0.038}$	$0.031^{+0.027}_{-0.010}$	$0.113^{+0.007}_{-0.014}$	14.1/8
18836	$0.124^{+0.089}_{-0.035}$	$0.013^{+0.013}_{-0.004}$	$0.155^{+0.017}_{-0.019}$	24.0/8
21026	$0.060^{+0.086}_{-0.023}$	$0.017^{+0.041}_{-0.007}$	$0.072^{+0.016}_{-0.029}$	4.9/8
21553	$0.572^{+0.184}_{-0.169}$	$0.258^{+0.203}_{-0.134}$	$0.056^{+0.035}_{-0.031}$	6.1/8
25902	$2.966^{+0.807}_{-0.454}$	$0.290^{+0.212}_{-0.090}$	$0.150^{+0.020}_{-0.030}$	94.0/8
Average of individual results	1.03	0.123	0.113
Median of individual results	0.283	0.094	0.117
Scatter of individual results	1.896	0.121	0.059
Average flux stack	$1.050^{+0.796}_{-0.487}$	$0.184^{+0.471}_{-0.126}$	$0.146^{+0.042}_{-0.066}$	3.6/8
Median flux stack	$0.511^{+0.347}_{-0.237}$	$0.148^{+0.335}_{-0.104}$	$0.108^{+0.048}_{-0.062}$	1.7/8
Average HST image stack	$1.192^{+1.262}_{-0.448}$	$0.309^{+0.873}_{-0.214}$	$0.124^{+0.066}_{-0.059}$	22.5/8
Median HST image stack	$0.810^{+0.451}_{-0.428}$	$0.305^{+0.509}_{-0.245}$	$0.108^{+0.075}_{-0.053}$	11.8/8
Average NB image stack	$0.222^{+0.180}_{-0.106}$	$0.018^{+0.044}_{-0.010}$	$0.165^{+0.006}_{-0.057}$	43.5/8
Median NB image stack	$0.828^{+0.396}_{-0.489}$	$0.462^{+0.559}_{-0.401}$	$0.101^{+0.103}_{-0.042}$	16.9/8
Scaled average flux stack	$0.525^{+0.280}_{-0.043}$	$0.084^{+0.093}_{-0.017}$	$0.105^{+0.025}_{-0.014}$	48.2/8
Scaled median flux stack	$0.382^{+0.165}_{-0.074}$	$0.091^{+0.096}_{-0.028}$	$0.131^{+0.013}_{-0.030}$	3.31/8
Clean average flux stack	$1.314^{+1.124}_{-0.753}$	$0.206^{+0.635}_{-0.163}$	$0.160^{+0.057}_{-0.076}$	4.6/8
Clean median flux stack	$0.639^{+0.182}_{-0.526}$	$0.348^{+0.098}_{-0.331}$	$0.048^{+0.104}_{-0.016}$	3.7/8
Clean average HST image stack	$0.088^{+0.140}_{-0.031}$	$0.008^{+0.035}_{-0.004}$	$0.140^{+0.028}_{-0.069}$	20.9/8
Clean median HST image stack	$0.049^{+0.309}_{-0.009}$	$0.009^{+0.177}_{-0.004}$	$0.089^{+0.062}_{-0.058}$	23.2/8
Clean average NB image stack	$0.095^{+0.126}_{-0.035}$	$0.007^{+0.029}_{-0.004}$	$0.150^{+0.023}_{-0.072}$	15.8/8
Clean median NB image stack	$0.079^{+0.214}_{-0.049}$	$0.016^{+0.170}_{-0.012}$	$0.131^{+0.042}_{-0.098}$	10.4/8

Notes. Below the results for each individual object, we display the individual object sample average, mean, and standard deviation in each parameter, for comparison to stacking results.

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Figure 5 shows the relationships between parameters of age, stellar mass, and E(B − V). In our analyses of parameters that may have statistically significant correlations with other parameters, we use both the Pearson product-moment correlation coefficient and the Spearman rank correlation coefficient. The Pearson and Spearman correlation coefficients (ρ) describe the strength and direction of the correlation and range from −1 to 1. We define 1 − p as the confidence with which one can reject the null hypothesis of no correlation. In order for two parameters to be correlated at 95% confidence, their p_Spearman value must be less than 0.05. Because the intrinsic distribution of parameters is not well known, we adopt the Spearman p values. We see a correlation of age with stellar mass, as expected. The assumption of a constant star formation history forces older galaxies to be more massive, and thus we see a correlation between age and stellar mass. No correlation is found between E(B − V) and stellar mass. Also, there is an observed correlation between age and E(B − V), implying that older LAEs generally host more dust.

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4.1. Correlations with Equivalent Width

Figure 6(a) shows stellar mass versus the rest-frame Lyα equivalent width (EW). There is a rough upper envelope on the points such that no objects exhibit rest-frame EW > 60 Å and stellar mass >10⁹ M_☉. Because EW was measured via NB 3727 Å and broadband U, B photometry (Guaita et al. 2010), there is no direct selection effect on stellar mass. Hence, the lack of high-stellar-mass, high-EW objects implies that galaxies with high stellar mass also have high rest-frame UV luminosity. This should be reflected via a related dearth of objects with high stellar mass and low star formation rate (SFR), i.e., low specific SFRs. Also, the correlation between stellar mass and EW is likely due to larger mass galaxies hosting more O- and B-type stars, producing the UV radiation that leads to Lyα emission.

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Figure 6(b) shows stellar population ages versus rest-frame EW. Here, one might expect to see a strong correlation, as the Lyα EW of a starburst is a strongly decreasing function of age, and for continuous SFR the correlation remains strong (Shapley et al. 2003). The observed lack of correlation implies that Lyα radiative transfer is not trivial; however, it could also be produced by a disconnect between our measured ages and the true age of the starbursting population caused by our assumption of a single stellar population with constant SFR.

Figure 6(c) shows dust reddening, E(B − V), versus rest-frame EW. Though statistically correlated, the correlation is weak. One could expect that dustier galaxies would more effectively quench Lyα photons to produce lower EWs. The lack of strong correlation therefore implies that Lyα radiative transfer is working to prevent Lyα photons from repeated resonant scatters and hence prevents E(B − V) from strongly affecting the resulting Lyα luminosity. This is a particularly interesting point due to the fact that this effect is not seen in other recent LAE studies, but is seen in some local group studies, such as Giavalisco et al. (1996).

4.2. The Lyα q Factor

As mentioned above, we excluded the U-band data from SED fitting due to this likelihood of Lyα emission making a significant contribution to this broadband photometry and the impossibility of properly accounting for Lyα radiative transfer in our SED templates. However, this makes the U-band data ideal for an estimation of Lyα q factors, which represent the extent to which Lyα emission from a galaxy has been enhanced (q > 1) or quenched (q < 1) along the line of sight to Earth. As was first done by Finkelstein et al. (2008), we define q = τ_Lyα/τ_{λ = 1216}, where τ_λ = k_λE(B − V)/1.086 and k_λ follows a Calzetti et al. (2000) dust attenuation law. The q factor has typically been measured from the same NB imaging used to select LAEs, leading to a selection against low values of q. The deep U-band data in the CANDELS catalog allow us to make a measurement of q from broadband imaging sensitive to Lyα that is uncorrelated with the original selection of the LAE sample. A q factor of unity implies that Lyα photons see the same dust column as UV continuum photons, whereas a q factor of zero points to Lyα photons seeing no dust extinction whatsoever. A negative q factor would imply Lyα enhancement by a top-heavy IMF, clumpy dust, or anisotropic radiative transfer, and q > 1 implies that Lyα photons see a greater dust column than UV continuum photons, as expected for resonant scattering.

Panel (a) of Figure 7 shows that the mode in q factors lies between 0 and 1. In a few of our objects, noise in the U-band photometry results in very low, even negative, values of q. There is no statistically significant correlation with E(B − V) in panel (b), but panel (c) shows a correlation with EW in the expected direction.

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We can compare this study's q factors with that of a previous study by Finkelstein et al. (2007). In that study, a sample of four LAEs at z = 4.4 were studied. The q factors for that sample were all between 0 and 1, which is the most common range for q factors in our study, although we find a broader distribution.

4.3. Are LAEs on the SFR–M_* "Main Sequence"?

Lastly, in Figure 8, we explore the behavior of the individual object LAEs in the SFR–stellar mass relation. Our sample of LAEs are subject to a selection effect in both mass and SFR. The CANDELS catalog is mass selected, and the MUSYC catalog is SFR selected. SFRs below 1 M_☉ yr⁻¹ will not be detectable in the NB survey. Galaxies with stellar mass <3 × 10⁷ M_☉ would not be detectable in the CANDELS H-band catalog, even with the youngest ages observed in our sample. At stellar masses >10⁹ M_☉ where the SFR selection effect is unimportant, our LAEs appear to lie systematically above the galaxy main sequence (MS), implying larger SFRs than expected for galaxies of their mass.

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Interestingly, the scatter seems to increase toward low masses. This could happen because a star formation episode of a few M_☉ yr⁻¹ will disturb a lower mass galaxy much more than a higher mass galaxy. In the analysis done in Ibar et al. (2013) and seen in their Figure 8, Hα emitters at z = 1.47 also fall above the MS, though that study probes much higher masses. One should note that all other studies shown on this plot used the same (Salpeter) IMF, BC03 models, and constant star formation histories as used in this study, ruling out a systematic offset in SFR or M_* due to varying assumptions.

4.4. Stacking Results

4.5. Scaled Flux Stacks

4.6. Clean Stacks

Stacking techniques have been employed for studying low-S/N objects. However, some stacking methods may be more accurate at modeling the parameters of a sample than others. Through fitting individual SEDs of 20 LAEs at redshift 2.1 individually, and then fitting six types of stacked SEDs composed of the same 20 objects, we have tested the validity of these methods. Our choice of SED fitting under the assumption of a constant star formation history was made for two major reasons. First, our initial motivation for this paper came from inconsistencies in stacked LAE studies. Most of these studies used constant star formation histories in their SED fitting. Using other star formation histories in this study would make it more difficult to compare our results to other studies. Second, other functional forms of star formation histories, such as exponentially increasing or decreasing star formation histories and starburst star formation histories, do not significantly alter the SED fitting result for LAEs at z = 2.1 (Acquaviva et al. 2012b; Guaita et al. 2011).

We find that median flux stacking and average HST-centered image-stacking techniques correctly represent the age and stellar mass of a sample, while the median HST-centered image stack performs less adequately. This is a stark result, since our analysis serves as a best-case scenario for stacking. Stacking objects with CANDELS-defined coordinates maximizes the accuracy of our stacking sample. We further investigated this result by establishing a set of clean stacks, which are composed of objects free of nearby sources of flux, and with simulations. The clean stacking analysis sheds little light on the underlying reason for the discrepancies in image-stacked parameter estimates of a sample of individual objects. Future stacking analyses should be wary of disparities in analyses due to errors possibly brought on by the stacking method itself. The simulations show that median image stack photometry is biased considerably low when centroiding errors are significant compared to the PSF. Though this effect is not, in practice, catastrophic for either the NB- or HST-centered image stacks, it provides a note of caution. We caution the reader of inconsistencies any stacking analysis may bring forth, and we recommend the use of a scaled stacking method where stacking is absolutely necessary.

We can compare our results to other LAE stacking studies that also used SED fitting to derive galaxy parameters. There are three such studies at z ≃ 2.1: Guaita et al. (2011), Nilsson et al. (2011), and Acquaviva et al. (2012b). The SED fitting results from these studies are summarized in Table 2. Each of these studies at z ≃ 2.1 is in agreement within the uncertainties with the typical age, stellar mass, and E(B − V) found in this study with the exception of Acquaviva et al. (2012b), who report stellar masses higher by a factor of two, and Nilsson et al. (2011), who report stellar masses higher by a factor of 50 and ages higher by a factor of five. The latter discrepancy is likely caused by their higher Lyα flux detection limit and the sensitivity of the mean to outlying values, including interlopers. We also consider studies at larger redshifts, which are included in Table 2 as well: Ono et al. (2010b) at z = 3.1, Acquaviva et al. (2012b) at z = 3.1, Gawiser et al. (2007) at z = 3.1, and Finkelstein et al. (2007) at z = 4.4. Two of these higher redshift studies find LAEs to have significantly younger ages than in the present study; however, Acquaviva et al. (2012b) found significantly older ages at z = 3.1 than at z = 2.1. LAE stellar mass does not appear to evolve significantly with redshift, although it varies by a factor of 10 among the z = 3.1 studies, with our z = 2.1 result in the middle of this range. Finally, all three z = 3.1 LAE studies found typical LAEs to be consistent with no dust extinction, but this is not the case for any of the z ≃ 2.1 studies, which imply roughly a factor of 10 increase in reddening to still low values of E(B − V) ≃ 0.15.

Table 2. Comparison of Results

Study	Redshift	M*	Age	E (B − V)	Study Type
		(109 M/M☉)	(Gyr)
This study	2.1	$0.382^{+0.165}_{-0.074}$	$0.091^{+0.096}_{-0.028}$	$0.131^{+0.013}_{-0.030}$	Scaled median flux stack
This study	2.1	0.283 [0.024–8.518]	0.094 [0.004–0.290]	0.117 [0.029–0.240]	Median of individual objects
Guaita et al. (2011)	2.1	$0.398^{+0.602}_{-0.147}$	$0.012^{+0.149}_{-0.003}$	$0.22^{+0.13}_{-0.02}$	Median flux stack
Acquaviva et al. (2012b)	2.1	$0.64^{+0.13}_{-0.12}$	$0.09^{+0.04}_{-0.03}$	$0.14^{+0.02}_{-0.02}$	Median flux stack
Nilsson et al. (2011)	2.3	$16.98^{+3.91}_{-2.85}$	$0.44^{+0.40}_{-0.15}$	N/A	Average image stack
Nilsson et al. (2011)	2.3	24.55 [0.316–100]	1.02 [0.0–1.85]	N/A	Average of individual objects
Ono et al. (2010b)	3.1	$0.132^{+0.108}_{-0.076}$	$0.07^{+0.11}_{-0.06}$	$0.03^{+0.05}_{-0.03}$	Median image stack
Acquaviva et al. (2012b)	3.1	$1.30^{+0.50}_{-0.10}$	$0.70^{+0.17}_{-0.10}$	$0.015^{+0.005}_{-0.015}$	Median image stack
Gawiser et al. (2007)	3.1	$1.000^{+0.585}_{-0.369}$	$0.02^{+0.03}_{-0.01}$	$0.00^{+0.08}_{-0.00}$	Average image stack
Finkelstein et al. (2007)	4.4	0.548	0.001	N/A	Average image stack
Finkelstein et al. (2009)	4.5	[0.16–50.0]	[0.003–5.00]	N/A	Range of individual objects

Notes. We compare the results of various other LAE studies to the median of the individual object results of this study and the best-performing stack method of this study: the Scaled Median Flux Stack. Gawiser et al. (2007) were obtained using a two-population model, and the age listed is for the younger population. For individual object studies we list ranges in properties, and for stacked studies we list uncertainties.

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Our main conclusions are the following:

• 1.  
  A lack of correlation between Lyα EW and age implies either complicated radiative transfer mechanisms or an inappropriate assumption of a constant SFR in a starbursting sample.
• 2.  
  The narrow distribution of q values peaking near 1 and lack of correlation between q and E(B − V) imply that radiative transfer mechanisms seem to be working to prevent Lyα photons from resonantly scattering in dusty regions.
• 3.  
  Our sample of LAEs lies systematically above the SFR–stellar mass relation galaxy "MS" and shows an increase in scatter above this relation at low mass. This may be caused by ongoing starbursts in these galaxies causing a greater excursion in their specific SFRs.
• 4.  
  Though some types of stacking represent the average and median properties of a sample well, the large dispersion of individual object properties is obscured by stacking. We recommend a new approach using an H-band scaled median or average flux stack, which reduces uncertainties significantly.

We would like to thank both the CANDELS and MUSYC collaborations for making this work possible. We would also like to thank the anonymous referee for helpful comments that greatly improved the quality of the paper. This work is based on observations taken by the CANDELS Multi-Cycle Treasury Program with the NASA/ESA HST, which is operated by the Association of Universities for Research in Astronomy, Inc., under NASA contract NAS5-26555. These observations are associated with programs GO-9352, GO-9425, GO-9583, GO-9728, GO-10189, GO-10339, GO-10340, GO-11359, GO-12060, and GO-12061. Observations have been carried out using the Very Large Telescope at the ESO Paranal Observatory under Program ID(s): LP168.A-0485. This work is also based in part on observations made with the Spitzer Space Telescope, which is operated by the Jet Propulsion Laboratory, California Institute of Technology, under a contract with NASA. Support for Program numbers 12060.57 and 12445.56 was provided by NASA through a grant from the Space Telescope Science Institute, which is operated by the Association of Universities for Research in Astronomy, Incorporated, under NASA contract NAS5-26555. The Institute for Gravitation and the Cosmos is supported by the Eberly College of Science and the Office of the Senior Vice President for Research at the Pennsylvania State University. This material is based on work supported by the National Science Foundation under CAREER grant AST-1055919 awarded to Eric Gawiser. This research has made use of NASA's Astrophysics Data System.