-continuum slopes at from the HUDF09+ERS+CANDELS observations: Discovery of a well-defined -color magnitude relationship for star-forming galaxies11affiliation: Based on observations made with the NASA/ESA Hubble Space Telescope, which is operated by the Association of Universities for Research in Astronomy, Inc., under NASA contract NAS 5-26555. These observations are associated with programs #11563, 9797.
Ultra-deep ACS and WFC3/IR HUDF+HUDF09 data, along with the wide-area GOODS+ERS+CANDELS data over the CDF-S GOODS field, are used to measure UV colors, expressed as the UV-continuum slope , of star-forming galaxies over a wide range in luminosity ( to ) at high redshift ( to ). is measured using all ACS and WFC3/IR passbands uncontaminated by Ly and spectral breaks. Extensive tests show that our measurements are only subject to minimal biases. Using a different selection procedure, Dunlop et al. recently found large biases in their measurements. To reconcile these different results, we simulated both approaches and found that measurements for faint sources are subject to large biases if the same passbands are used both to select the sources and to measure . High-redshift galaxies show a well-defined rest-frame UV color-magnitude (CM) relationship that becomes systematically bluer towards fainter UV luminosities. No evolution is seen in the slope of the UV CM relationship in the first 1.5 Gyr, though there is a small evolution in the zero-point to redder colors from to . This suggests that galaxies are evolving along a well-defined sequence in the -color () plane (a “star-forming sequence”?). Dust appears to be the principal factor driving changes in the UV color with luminosity. These new larger samples lead to improved dust extinction estimates at -7 and confirm that the extinction is essentially zero at low luminosities and high redshifts. Inclusion of the new dust extinction results leads to (i) excellent agreement between the SFR density at -8 and that inferred from the stellar mass density, and (ii) to higher SSFRs at , suggesting the SSFR may evolve modestly (by factors of 2) from -7 to .
Subject headings:galaxies: evolution — galaxies: high-redshift
One of the biggest frontiers in extragalactic cosmology is to characterize the early build-up and evolution of galaxies. It is important for our understanding of early gas accretion and star formation in the universe, estimating the rate of early metal injection into the IGM, and assessing the impact of galaxies on reionization. From the observations, we already have a good measure of how fast galaxies build up through Lyman-Break selections and luminosity function (LF) studies reaching all the way to (e.g., Bouwens et al. 2010b; McLure et al. 2010; Bunker et al. 2010; Yan et al. 2010). Galaxies show a remarkably uniform brightening in their LFs from to (e.g., Bouwens et al. 2011b) and plausibly from (Bouwens et al. 2011a). There is even some evidence for very rapid evolution at (Bouwens et al. 2011a; Oesch et al. 2012a).
Despite general constraints on how the galaxy population builds up with cosmic time, much less is known about how individual galaxies grow. Qualitatively, we would expect galaxies to build up monotonically in mass, metallicity, and dust content as they form stars. Quantifying how this build up occurs and with what star formation history is very challenging however. The effects of dust, metal, and age on the colors are very similar and make it very difficult to disentangle one factor from the others. Nonetheless, there is enough information available, i.e., the -to-optical colors (Stark et al. 2009; González et al. 2010; Labbé et al. 2010b), colors (e.g., Bouwens et al. 2009; Hathi et al. 2008), and high-resolution spectra of high redshift galaxies (e.g., Stark et al. 2010; Vanzella et al. 2009), that significant progress should be made in better characterizing galaxies throughout the build-up process.
An important piece of the puzzle in deciphering how galaxies build up is provided by the rest-frame spectrum and in particular the colors. The rest-frame color provides us with perhaps our best means for estimating the dust extinction and star formation rate (SFR) for faint galaxies, given that other techniques for probing the SFR tend to only detect the most bolometrically luminous galaies (e.g., Bouwens et al. 2009; Smit et al. 2012). The colors also show a systematic dependence on the luminosities of star-forming galaxies (e.g., Bouwens et al. 2009; Bouwens et al. 2010a) and are much more amenable to direct measurement than rest-frame UV-optical colors where mid-IR (IRAC) photometry is necessary.
A significant amount of effort has gone into establishing the -continuum slope distribution at high redshift and determining its dependence upon redshift and luminosity. The earliest analyses were at -3 using either ground-based observations (Steidel et al. 1999; Adelberger & Steidel 2000) or the deep WFPC2 observations over the HDF North (Meurer et al. 1999). Subsequent analyses pushed -continuum slope measurements to -6 using Subaru Suprime-Cam, HST ACS, or HST NICMOS observations (Ouchi et al. 2004a; Papovich et al. 2004; Stanway et al. 2005; Bouwens et al. 2006; Hathi et al. 2008). Bouwens et al. (2009) extended these previous works by examining the slopes as a function of luminosity over the entire redshift range -6 – establishing a coherent framework for understanding the observational results at that time.
The availability of both deep and wide WFC3/IR imaging has made it possible to substantially improve these early measurements of the -continuum slope. These imaging observations allow for accurate measurements of the -continuum slopes for large numbers of -7 galaxies. Bouwens et al. (2010a) made use of the early WFC3/IR observations to examine the -continuum slope distribution out to (see also Oesch et al. 2010a; Bouwens et al. 2010b; Bunker et al. 2010; Finkelstein et al. 2010; Robertson et al. 2010; Dunlop et al. 2012; Wilkins et al. 2011). Bouwens et al. (2010a) found that very low luminosity -8 galaxies in the ultra-deep HUDF09 WFC3/IR field had -continuum slopes as steep as and plausibly consistent with but with uncertainties of 0.2-0.3. Finkelstein et al. (2010) also reported on the colors of ultra-faint -8 galaxies in the HUDF finding similar steep values, but with somewhat larger uncertainties.
Since the early WFC3/IR campaign over the HUDF (Bouwens et al. 2010b; Oesch et al. 2010a) from the HUDF09 program (GO 11563: PI Illingworth), the amount of deep, wide-area WFC3/IR observations over well-known legacy fields has increased dramatically. At present, we have ultra-deep WFC3/IR observations over the two HUDF09/HUDF05 fields (Bouwens et al. 2011b; Oesch et al. 2007, 2010a), wide-area (145 arcmin) data over the CDF-South GOODS field as a result of the Early Release Science (Windhorst et al. 2011) and CANDELS (Grogin et al. 2011; Koekemoer et al. 2011) programs, and even deeper WFC3/IR observations over the HUDF and HUDF05 fields (Bouwens et al. 2011b). These observations greatly improve the luminosity baseline, redshift range, and precision with which we can define the -continuum slope distribution for -7 galaxies.
In this paper, we take advantage of these new observations to establish the distribution of -continuum slopes over a wide range in luminosity and redshift. These new observations allow us to determine with great precision how the -continuum slope distribution depends upon luminosity, in four distinct redshift intervals. This new information puts us in a position to look for a possible star-forming sequence of galaxies at high redshift and to characterize its evolution with cosmic time. The evolution of such a sequence provides useful information for better understanding early galaxy build-up. For example, the slope of a -luminosity relationship constrains how the dust and age of the galaxy population vary as a function of luminosity. Scatter in the -luminosity relationship constrains the overall scatter in the stellar populations or dust extinction of individual galaxies. Finally, evolution in the slope and offset of the relation with cosmic time gives us clues as to possible changes in how galaxies evolve, either in age or dust extinction. Independent analyses of the -continuum slope in -7 galaxies are provided by Dunlop et al. (2012) and Wilkins et al. (2011), but both are based on a much smaller, shallower set of observations. A somewhat complementary analysis to the one described here is given by González et al. (2012) who quantify the changes in -optical colors as a function of luminosity and redshift.
We provide a brief overview of the paper here. We begin with a brief summary of the observational data (§2). In §3, we describe the manner in which we construct our high-redshift samples from the observational data, measure the -continuum slope distribution, correct for measurement and selection biases, and present evidence for well-defined color-magnitude relation for galaxies in the rest-fame . In §4, we compare the present -continuum slope determinations with previous determinations. In §5, we explore the implications of such a color-magnitude relation for galaxy growth – using the -continuum slopes to infer a luminosity-dependent dust correction for galaxies. In §6, we use these extinction estimates to rederive the SFR density at and then compare these results with what one infers from the stellar mass density. Finally, in §7, we conclude and provide a summary of our primary results. The appendices include a detailed description of many quantitative results and simulations essential for accurate measurements of the -continuum slopes.
Throughout this work, we find it convenient to quote results in terms of the luminosity Steidel et al. (1999) derived at , i.e., , for consistency with previous work – though we note that the Steidel et al. (1999) LF results are now updated (: Reddy & Steidel 2009) but still consistent with the previous determination. We present our dust extinction estimates as the ratio of the bolometric luminosity ( + luminosity: + ) to the luminosity (), i.e., . We refer to the HST F435W, F606W, F775W, F814W, F850LP, F098M, F105W, F125W, and F160W bands as , , , , , , , , and , respectively. Where necessary, we assume , , . We quote all star formation rates and stellar masses assuming a Salpeter (1955) IMF. All magnitudes are in the AB system (Oke & Gunn 1983).
|Detectionaa detection limits for our -7 ACS+WFC3/IR selections were measured in a -diameter aperture.||PSF FWHM||Areal Coverage|
|HUDF09 (WFC3/IR HUDF)|
|HUDF09-1 (WFC3/IR P12)|
|HUDF09-2 (WFC3/IR P34)|
|bbOur reductions of the ACS data over the HUDF09-2 field include both those observations taken as part of the HUDF05 (82 orbits) and HUDF09 (111 orbits) programs (see Figure 1 from Bouwens et al. 2011b). The latter observations add 0.15-0.4 mag to the total optical depths.||28.8||0.09||3.3|
|bbOur reductions of the ACS data over the HUDF09-2 field include both those observations taken as part of the HUDF05 (82 orbits) and HUDF09 (111 orbits) programs (see Figure 1 from Bouwens et al. 2011b). The latter observations add 0.15-0.4 mag to the total optical depths.||29.9||0.09||4.7|
|bbOur reductions of the ACS data over the HUDF09-2 field include both those observations taken as part of the HUDF05 (82 orbits) and HUDF09 (111 orbits) programs (see Figure 1 from Bouwens et al. 2011b). The latter observations add 0.15-0.4 mag to the total optical depths.||29.3||0.09||4.7|
|ERS (CDF-S GOODS)|
|28.0ccThe depth of the F814W observations vary considerably over the ERS and CDF-South CANDELS Wide field, from very deep coverage (28.8 mag) in some regions to essentially no coverage over a small fraction of the area. Typical depths are 28.0 AB mag (5).||0.09||39|
|CDF-S CANDELS Deep|
|CDF-S CANDELS Wide|
|28.1ccThe depth of the F814W observations vary considerably over the ERS and CDF-South CANDELS Wide field, from very deep coverage (28.8 mag) in some regions to essentially no coverage over a small fraction of the area. Typical depths are 28.0 AB mag (5).||0.09||40|
2. Observational Data
Here we utilize two primary data sets to examine the -continuum slope distribution of galaxies from to : (1) the ultra-deep ACS+WFC3/IR observations over the three HUDF09 fields (Bouwens et al. 2010b; Bouwens et al. 2011b; Oesch et al. 2010a) and (2) the wide-area ACS+WFC3/IR observations taken over the CDF-South GOODS field as a result of Early Release Science and CANDELS programs (Windhorst et al. 2011; Giavalisco et al. 2004; Grogin et al. 2011; Koekemoer et al. 2011). A brief summary of the properties of these observations is provided in Table 1. Figure 1 shows the layout of these observations over the CDF-South GOODS area.
2.1. HUDF09 Observations
The first set of observations we utilize is from the HUDF09 program (GO11563: PI Illingworth) and involves ultra-deep WFC3/IR observations over three fields that already have ultra-deep ACS observations. These fields include the HUDF (Beckwith et al. 2006) and two HUDF05 flanking fields (Oesch et al. 2007). These observations are primarily of use in establishing the distribution of -continuum slopes for very faint -7 galaxies.
At present, the full two years of WFC3/IR observations (all 192 orbits from the GO11563 program) have been obtained over the three HUDF09 fields. The WFC3/IR observations over the HUDF include some 111 orbits of observations – while the HUDF09-1 and HUDF09-2 fields include 33 orbits and 48 orbits of observations, respectively. Our reductions of the WFC3/IR observations over these fields are already described in Bouwens et al. (2011b) and are conducted using standard procedures. Particularly important to this process was ensuring that the geometric distortion and velocity abberation were treated properly so that the registration between bands and with the ACS observations was very accurate (0.01). Accurate registration is absolutely essential not only for maximizing the accuracy of our color measurements, but also for minimizing the effect that misregistration might have on scatter in these same color measurements. The depths of our WFC3/IR observations reach to 29 AB mag at and are presented in detail in Table 1. The FWHM of the PSF in the WFC3/IR observations is 0.16.
Our reductions of the ACS observations over the HUDF09-2 area include all available data – including the 82 orbits that were taken during the execution of the original HUDF05 program and an additional 111 orbits that were taken in parallel with the HUDF09 WFC3/IR observations over the HUDF. The total number of orbits per filter over the HUDF09-2 are 10 orbits F435W, 32 orbits F606W, 46 orbits F775W, 16 orbits F814W, and 89 orbits F850LP. The total integration time is 50% of that available for the HUDF, allowing us to reach within 0.4 mag of the HUDF. This provides us with sufficiently high S/N levels required to keep contamination in selections to a minimum. A more detailed description of this data set is provided by Bouwens et al. (2011b).
2.2. ERS/CANDELS Observations
Wide-area WFC3/IR observations allow us to establish the -continuum slope distribution for the rarer, higher luminosity sources. These observations are available over the CDF-South GOODS field from the Early Release Science (Windhorst et al. 2011) and CANDELS (Grogin et al. 2011; Koekemoer et al. 2011) programs.
The WFC3/IR observations from the ERS program are distributed over the northern part of the CDF South GOODS field (Windhorst et al. 2011: Figure 1). These observations consist of 60 orbits of F098M, F125W, and F160W observations distributed over 10 distinct pointings, with 2 orbits F098M, 2 orbits F125W, and 2 orbits F160W per pointing. The WFC3/IR observations extend over 43 arcmin in total, although only 39 arcmin of those observations overlap with the ACS GOODS data and can be used here.
WFC3/IR observations from the CANDELS program are distributed over the southern two-thirds of the CDF-South GOODS field (106 arcmin), with the deepest section planned for the central 66 arcmin portion. 3 orbits of F105W, 4 orbits of F125W, and 4 orbits of F160W observations are available over the central sections of the CDF-South (representing all 10 SNe search epochs to February 18, 2012) while only 1 orbit of F105W, F125W and F160W observations are available over the southern portion. The layout of the CANDELS and ERS observations with the CDF-South GOODS field is shown in Figure 1.
Our reductions of the WFC3/IR observations over the CDF-South are performed in exactly the same manner as for the HUDF09 observations. See Bouwens et al. (2011b) for a detailed description. As in our HUDF09 reductions, a pixel size was used. For the ACS observations over the CDF-South, we used the Bouwens et al. (2007) reductions for the F435W, F606W, F775W, and F850LP bands. These reductions are comparable to the GOODS v2.0 reduction (Giavalisco et al. 2004) but take advantage of the substantial SNe follow-up observations over the CDF-South (Riess et al. 2007) which add 0.1-0.3 mag of depth to the band.
We also reduced the new ACS -band observations over the CDF-South GOODS fields from the CANDELS+ERS programs to take advantage of the superb depth available with these data at 8000Å (typically 13 orbits over the CANDELS-DEEP region, or 0.8 mags deeper than the GOODS band exposures). The F814W observations are valuable for controlling for contamination in our selections and for minimizing the uncertainties in our determinations at . We included all F814W observations from the ERS+CANDELS programs over the CDF-South. After using public codes (e.g., Anderson & Bedin 2010) to correct the raw frames for charge transfer efficiency defects and row-by-row banding artifacts, we performed the alignment, cosmic-ray rejection, and drizzling with the ACS GTO apsis pipeline (Blakeslee et al. 2003).
|SampleaaThe mean redshift we estimate for these samples is 3.8, 4.9, 5.9, and 7.0, respectively (Figure 3).||Field||RangebbThe faint magnitude limit is set by the 5 depth of the WFC3/IR near-IR observations over our search fields. See §3.2.||Sources|
In this section, we describe the procedure we use to establish the distribution of -continuum slopes versus luminosity for -7 galaxies. We begin with a description of the technique we use for generating the source catalogs needed for sample selection and -continuum slope measurements (§3.1). We then describe our procedure for selecting our high-redshift samples (§3.2) and for deriving the -continuum slope for individual sources (§3.3). In §3.4, we detail the methods we use to correct the observed distribution of -continuum slopes for selection and measurement biases. In §3.5 we combine our -continuum slope determinations in the ultra-deep + wide-area data sets to establish -continuum slope over a wide-range in redshift and luminosity. We then conclude with a discussion of the correlation we find between the -slope and the luminosity (§3.6).
3.1. Source Catalogs
The procedure for source detection and photometry are discussed in many of our previous papers (e.g., Bouwens et al. 2007); we just summarize the key steps here. We generate catalogs for our samples using the SExtractor (Bertin & Arnouts 1996) software run in double image mode, with the detection image taken to be the square root of image (Szalay et al. 1999: similar to a coadded image of the WFC3/IR observations) and the measurement image to be a PSF-matched image from our ACS+WFC3/IR image sets (see below).
We select each of our high-redshift samples (i.e., , , , and ) from separately-generated source catalogs. This allows us to take advantage of the fact that -6 samples can be selected entirely (or almost entirely) based on the ACS data (involving generally smaller apertures and therefore reducing the uncertainties in our color measurements) while selections require PSF-matching to the WFC3/IR data.111For our selections, some use of the available WFC3/IR data is made. In addition to satisfying a criterion, sources in our selection must also satisfy a color criterion (§3.2). For our -6 catalogs, the first step is to match PSFs by smoothing the shorter wavelength images to the resolution of the ACS -band image. The square root of images we use for the SExtractor detection images are constructed from the , , and images for our , , and selections, respectively. For our catalogs, all the observations are PSF-matched to the WFC3/IR band. The square root of images are constructed from the available WFC3/IR imaging observations, e.g., for our HUDF09 fields, images for the ERS observations, and for the CANDELS observations.
Colors are measured within small scalable apertures using a Kron (1980) factor of 1.2. These small-aperture color measurements are then corrected to total magnitudes in two steps. First, a correction is made for the light in a larger scalable aperture, with Kron factor equal to 2.5. This correction is made from the square root of image to optimize the S/N. Secondly, a correction is made for light outside this large scalable aperture – using the encircled energies measured for point sources (typically a 0.1-0.15 mag correction).
Photometry was done using the latest WFC3/IR zeropoint calibrations (March 6, 2012) which differ by 0.01-0.02 mag from the earlier zeropoints. In addition, a correction to the photometry (i.e., ) was performed to account for foreground dust extinction from our own galaxy. This correction was based on the Schlegel et al. (1998) dust map.
3.2. Sample Selection
To select star-forming galaxies at high-redshift, we will use the well-established Lyman-Break Galaxy (LBG) technique. This technique takes advantage of the unique spectral characteristics of high-redshift star-forming galaxies, which show a very blue spectrum overall but a sharp cut-off blueward of Ly. It has been shown to be very robust through extensive spectroscopic follow-up (Steidel et al. 1996; Steidel et al. 2003; Bunker et al. 2003; Dow-Hygelund et al. 2007; Popesso et al. 2009; Vanzella et al. 2009; Stark et al. 2010).
|Filters used||Mean rest-frame|
|Sample||to derive||WavelengthbbGeometric mean|
|CDF-S CANDELS Observations|
We will base our high-redshift samples on selection criteria from previous work on galaxies. For our and dropout samples, we will apply almost the same criteria as Bouwens et al. (2007), namely,
for our -dropout sample and
for our -dropout selection (but note we use a color selection instead of the selection used by Bouwens et al. 2007 to minimize contamination). For our -dropout selection, we expanded the criteria used in Bouwens et al. (2007) to take advantage of the deep near-IR observations from WFC3/IR to set limits on the color redward of the break. Our -dropout criterion is
Finally, our -dropout criterion is
depending upon whether our search field is from the HUDF09/CANDELS datasets or the ERS dataset, respectively. The above selection criteria closely match those used in our previous study of the -continuum slope at (Bouwens et al. 2010a). These criteria are slightly more inclusive than those considered by Bouwens et al. (2011b), but this is to allow us to maximize the size of our samples and to extend our selection to the highest redshift possible without suffering significant contamination from Ly emission or IGM absorption. In cases of a non-detection in the dropout band, we set the flux in the dropout band to be equal to the upper limit. To take advantage of the deep -band observations over the CDF-South to keep contamination in our samples to a minimum, we required that all -dropouts in our selection to have colors greater than 2.0 or to be undetected in the band (at ). No attempt is made to select or measure -continuum slopes for star-forming galaxies at due to the fact that the color (the only available color providing high S/N information on the rest-frame SED at ) is affected by Lyman-series absorption and Ly emission at (but see Taniguchi et al. 2011).
Figure 2 provides a convenient illustration of the approximate range in -continuum slopes and redshifts selected by the above two-colour criteria.
We utilize several additional selection criteria in defining our final samples. Sources are required to have SExtractor stellarity parameters less than 0.8 (i.e., they show evidence of being extended) to ensure our samples are free of contamination from AGN or low-mass stars. Given that 97% of bright sources in Lyman-Break selections show clear evidence of having extended profiles (i.e., are not pointlike), this criterion has almost no effect on the overall composition of our samples.
We also require that sources be undetected in all bands blueward of the dropout bands. Sources are rejected if they are detected at 2 in a single band, 1.5 in 2 bands, or have a in the optical bands. We take where is the flux in band in our smaller scalable apertures, is the uncertainty in this flux, and SGN() is equal to 1 if and if (Bouwens et al. 2011b). As Bouwens et al. (2011b) illustrate, a criterion can be particularly useful for minimizing contamination in high redshift samples (see also Oesch et al. 2012a,b).
Table 2 summarizes the properties of the , , , and samples we derive from the HUDF09+ERS+CANDELS observations. In total, 308, 137, 70, and 57 , , , and galaxies are found in the HUDF09 fields and 1524, 277, 101, and 44 , , , and galaxies are found in the ERS+CANDELS fields. The approximate redshift distributions for our , , , and selections are shown more explicitly in Figure 3. These redshift distributions are as calculated in Bouwens et al. (2007) and Bouwens et al. (2011b).
3.3. Estimating the -Continuum Slope
Establishing the -continuum slope for a source (where is defined such that ) requires that we have at least two flux measurements of a source in the -continuum not significantly affected by Ly or IGM absorption (unless both the redshift and Ly flux are already known). In practice, this means that we should not make use of the flux information in the band immediately redward of the break – since it is frequently contaminated by Ly emission or absorption from the IGM. Assuming we have wavelength coverage in the bands, we can obtain good estimates of the -continuum slope for high-redshift sources from to at least .
There are a few different approaches we could adopt in using the available flux measurements in the -continuum to estimate these slopes. One approach is to derive the -continuum slopes from a fit to all available flux measurements in the rest-frame . An alternate approach is to derive a slope from the flux measurements at both ends of a fixed rest-frame wavelength range (e.g., as utilized by Bouwens et al. 2009).
Each approach has its advantages. The first approach uses the full flux information available on each source and also takes advantage of a much more extended wavelength baseline. This results in much smaller uncertainties in our determinations, particularly at and where fluxes in four separate bandpasses are available. The simulations we perform in Appendix B.3 suggest factors of 1.5 improvement in the uncertainties on at and . On the other hand, the second approach has the advantage that all -continuum slope determinations are made using a similar wavelength baseline at all redshifts. As a result, even if the SED of galaxies in the rest-frame is not a perfect power law (e.g., from the well-known bump in dust extinction law at 2175Å: Stecher 1965), we would expect determinations made at one redshift to show no systematic offset relative to those made at another, allowing for more robust measurements of evolution across cosmic time.
After some testing, we adopted the approach that utilizes all the available flux information to determine the -continuum slopes . Figure 4 provides an example of such a determination for a galaxy in our HUDF sample and an example of such a determination for a galaxy. The passbands we will consider in deriving the slopes include the bands for galaxies, the bands for galaxies, the bands for galaxies, and the bands for galaxies. We include the band in these fits for galaxies over the ERS/CANDELS fields. We replace the band with the band in these fits for galaxies over the ERS field. In selecting these passbands, we explicitly excluded passbands which could be contaminated by Ly emission, the Lyman-continuum break, or flux redward of the Balmer break. These choices are motivated by our expected redshift distributions for these samples (Figure 3). Table 3 includes a list of all the bands we use to perform these fits. The mean rest-frame wavelengths for our derived -continuum slopes at , , , and are 2041Å, 1997 Å, 1846Å, and 1731Å, respectively.
In determining the -continuum slopes from the flux information just discussed, we find the -continuum slope that minimizes the value of :
where and are the observed fluxes and uncertainties, respectively, is the best-fit -continuum slope, and is the best-fit normalization factor. The fluxes in the above fits are from photometry of the full ACS+WFC3/IR data set PSF-matched to the WFC3/IR -band, using typical Kron-style apertures.
|MeanccThese samples include sources from the Bouwens et al. (2011b) HUDF09 samples and the lowest redshift galaxies in the Bouwens et al. (2011b) HUDF09 selection.||ScatterddTo keep contamination in our faint selection to a minimum, we only include faint sources from the HUDF09 fields with the deepest optical data. We therefore only consider sources from the HUDF (HUDF09) and the 3 arcmin (70%) area in the HUDF09-2 field with deep optical coverage from the HUDF09 program (see Figure 1 from Bouwens et al. 2011b).|
|18.50||2.160.030.10||—eeIt is challenging to establish the intrinsic scatter in the distribution in this magnitude interval to 0.1 accuracy in either because of a large photometric scatter in the individual measurements or because of a limited number of sources.|
|17.50||2.240.060.10||—eeIt is challenging to establish the intrinsic scatter in the distribution in this magnitude interval to 0.1 accuracy in either because of a large photometric scatter in the individual measurements or because of a limited number of sources.|
|16.50||2.200.070.10||—eeIt is challenging to establish the intrinsic scatter in the distribution in this magnitude interval to 0.1 accuracy in either because of a large photometric scatter in the individual measurements or because of a limited number of sources.|
This photometry is therefore distinct from that used to select our , , and samples – since sources in our , , and Lyman-break samples were selected based on catalogs where the PSF matching was done to the ACS band. This approach offers two clear advantages. (1) By selecting sources based on catalogs where the PSF-matching is done to the -band and therefore our photometric apertures are smaller, we utilise much higher S/N photometry for the selection process, improving the robustness of our , 5, and 6 selections. (2) By using different photometry (and smaller apertures) to select sources than we use to estimate (typically involving larger apertures), we ensure that our measurements are not subject to exactly the same noise as affects source selection (because the photometric apertures are different and will have somewhat different noise characteristics). This makes our -6 measurements less susceptible (by 50%) to the biases that arise from a coupling of errors in our measurements with similar errors in our photometric selection (“photometric error coupling bias”: Appendix B.1.2). For our Lyman-break sample, however, the photometry used to estimate is the same as what we use to select the sources. See §3.1 for a description of the photometry.
In Appendix A, we show that the present approach produces similar results to that based on colors spanning the wavelength range 1600Å to 2200Å(but with smaller errors). For typical sources, the derived ’s agree to within 0.1 which is about as well as we can determine using colors (where possible systematics in estimates are on the order of 0.1). The present -continuum slope estimates should therefore be directly comparable with others in the literature (Bouwens et al. 2009; Meurer et al. 1999; Ouchi et al. 2004a; Stanway et al. 2005; Bouwens et al. 2006; Bouwens et al. 2010a; Finkelstein et al. 2010) where the wavelength baseline 1600Åto 2200Åwas typical for determinations.
In deriving ’s for sources in our samples from the available photometry, we tested for significant deviations from a pure power-law shape. This is important for ensuring that the model we use to characterize the -continuum SEDs of sources is meaningful. For the average source, we found that the observed photometry and the best-fit power-law SED differed by less than mag. Such deviations are not large enough to have a sizeable impact on our determinations, shifting the observed determinations by . Such changes are well within the uncertainties we quote on our determinations (§3.5).
3.4. Possible Selection and Measurement Biases
To establish the actual distribution of -continuum slopes from the observations, we must account for the effect that object selection and measurement have on the observed distribution. These effects can cause the observed distribution to look very different from what it is in reality. Examples of such effects include (1) our LBG selection criteria preferentially including those galaxies in our samples with the bluest -continuum slopes and (2) photometric noise increasing the spread in the measured -continuum slopes .
To correct for these selection and measurement biases, we follow a very similar procedure to what we used in our first major study of the -continuum slope distribution at high redshift (Bouwens et al. 2009). We create mock catalogs of galaxies, generate artificial images for each source in our catalogs, add these sources to the real observations, and then reselect the sources and measure their properties in the same way as with the real observations. We then determine the approximate biases that we would expect in the mean -continuum slope and scatter due to source selection or photometric scatter. We compute these biases as a function of magnitude for each of our data sets. To ensure that we are able to determine the relevant biases to high precision, we repeat these experiments for 10 artificial galaxies.
Appendix B provides a detailed discussion of the simulations we use to estimate the likely biases in the -continuum slope distribution and establish the needed corrections.
Figure 5 shows the results of these simulations for selections. The effect of biases related to the LBG selection itself (Figure 21) and due to a possible coupling between the selection and measurement processes (“photometric error coupling bias”) is shown explicitly. It is immediately clear from this figure that we can successfully recover the input values of the -continuum slopes and do not expect especially large biases in the derived slopes. We include similar panels for our , , and selections in Figure 22 of Appendix B.
An important check to perform in assessing the overall quality of our -continuum slope determinations here is to compare the results from our deep data with the results from our wide data. Such a check is provided in Figure 25 of Appendix B, and it seems clear that the results from the HUDF09 and ERS+CANDELS data sets are in good agreement (-0.2) over the luminosity range where they overlap and have good statistics. This suggests that the -continuum slope distributions we recover are accurate and free of any sizeable biases.
3.5. Establishing the Distribution over a Wide Luminosity Baseline
We can utilize the -continuum slopes measured for individual sources (§3.3) to establish the distribution of -continuum slopes as a function of luminosity for each of our -7 Lyman-Break Galaxy samples.
To establish this distribution while properly accounting for the relevant selection and measurement biases, we analyze the -continuum slope distribution for each data set and Lyman-Break sample separately. We determine the mean -continuum slope and scatter. We correct the mean and scatter for the relevant biases (Appendix B and §3.4). Typical corrections in the mean -continuum slope are 0.1 in general; the corrections do however reach values of near the selection limit of our shallow and deep probes.
When determining the mean and scatter for each distribution, we use the biweight mean
and biweight scale
where , , is the median of the ’s, is the data, is the number of sources summed over, and is the “tuning constant” (Beers et al. 1990). is taken to be equal to 6 in computing the biweight mean and 9 in computing the biweight scale. Use of robust statistics like the biweight mean and scale is valuable, given that a small fraction of the sources in our samples may be contaminated by light from nearby sources or may lie outside the target redshift range. Biweight mean determinations of are similar to median determinations of (median difference with 0.04 scatter), but somewhat bluer () than mean determinations (due to the distribution having a somewhat extended tail towards red ’s and the biweight mean de-weighting the tail).
For the absolute magnitude of each galaxy in the , we use the mean absolute magnitude of that source in all the HST bands that contribute to its -continuum slope determinations (Table 3). By using the geometric mean for this luminosity, we aim to avoid giving too much weight to the bluer or redder bands in defining this luminosity and hence artificially creating a correlation between the derived -continuum slope and luminosity.
Figure 6 shows the -continuum slope distribution versus the magnitude for our , , , and samples. Scatter in the distribution is relatively modest for our and samples, but is larger for our and samples. This is the result of the fact that our -5 samples use a larger number of passbands and larger wavelength baseline to measure the -continuum slope .
In Table 4, we present our corrected determinations of the biweight mean -continuum slope and scatter as a function of luminosity, for our , , , and samples. We also include an estimate of the uncertainties in the biweight mean -continuum slopes that result from the measurement uncertainties, intrinsic scatter in the distribution, and small number statistics.
For the systematic error on our -continuum slope measurements, we conservatively adopt values of 0.10-0.28. The dominant component of this error comes from uncertainties in our photometry. Allowing for small errors in the photometric zeropoints, aperture corrections, and PSF-matching, we estimate a maximum error of 0.05 mag in our flux measurements. This translates into possible systematic errors of 0.10, 0.10, 0.14, and 0.28 in the measurements at , , , and .
We also allow for small systematic errors in our -continuum slope measurements as a result of differences in the wavelength baseline we use to derive (Table 3). These differences would have an effect on the ’s we derive from the available photometry, if the SEDs differed substantially from a pure power-law form (i.e., ). The tests we perform in Appendix A suggest the relevant systematic errors are small . Similar ’s are found from fits using the full -continuum (§3.3 and Figure 4) as are found using a smaller wavelength range.
3.6. Dependence of on Luminosity
We observe a clear trend in the -continuum slope as a function of luminosity in all four LBG samples, such that the -continuum slope becomes progressively bluer at fainter luminosities (Figure 6 and 25). Such trends in had already been identified by Bouwens et al. (2009) in their analyses of and galaxy samples (see also Meurer et al. 1999; Labbé et al. 2007; Figure 8 of Overzier et al. 2008) and by Bouwens et al. (2010a) in their analyses of -7 galaxies (see also Wilkins et al. 2011). The observed luminosity dependence of is thought to be due to a change in the dust content and perhaps the age of galaxies as a function of luminosity (Labbé et al. 2007; Bouwens et al. 2009). It is likely to be a manifestation of the well-established mass-metallicity relationship seen at a wide variety of redshifts (e.g., Tremonti et al. 2004; Erb et al. 2006a; Maiolino et al. 2008).
|3.8||2.000.020.10||0.110.01aaThe biweight mean -continuum slopes presented here are also shown on Figure 6.|
|5.0||2.080.030.10||0.160.03aaWhile the slopes for our , 5, and 6 samples are quite similar overall, these slopes show even better agreement if we consider the correlations over the same luminosity range, i.e., excluding determinations faintward of AB mag. In this case, we find a slope of 0.130.02 for our sample.|
|5.9||2.200.050.14||0.150.04aaWhile the slopes for our , 5, and 6 samples are quite similar overall, these slopes show even better agreement if we consider the correlations over the same luminosity range, i.e., excluding determinations faintward of AB mag. In this case, we find a slope of 0.130.02 for our sample.|
|bbThe slopes presented here have been corrected for selection effects and measurement errors (see §3.4 and Appendix B). The tabulated luminosities are the geometric mean of the measured luminosities in the bands used to establish the slope (see Table 3).||2.5||1.700.070.15||0.200.04|
The -continuum slope shows an approximately linear relationship on the magnitudes of the sources (Figure 6), and therefore it makes sense for us to model this relationship using a first-order polynomial and determine the best-fit slopes and intercepts .
In fitting a line to our -continuum slopes, we use a finer binning (i.e., 0.5 mag) than shown in Figure 6 to minimize the impact of the binning scheme on the best-fit slopes and intercept. The results of our fits to the biweight means are shown as blue lines on Figure 6, and it is clear that the mean ’s are well fit by the lines.222There may nonetheless be weak evidence in the results from our and samples that the dependence of on luminosity is weaker faintward of 18 mag (see Figure 6). The fits exhibit a similar dependence on the magnitude at each redshift. The intercept to the lines is also similar at all redshifts but appears to evolve monotonically with cosmic time. The best-fit parameters – slope and intercept – we derive for these lines are presented in Table 5 and Figure 7. These determinations are in excellent agreement with previous work, as we discuss in §4.
4. Comparison to Previous Results
In the present section, we compare the present observational results on the -continuum slope with those previously obtained in the literature (Figure 8). The goal is to assess the robustness of the present observational results and to give some perspective on which trends are gaining widespread observational support.
4.1. Comparison with Bouwens et al. 2009 (-6)
Before the WFC3/IR camera on HST became operational, Bouwens et al. (2009) made use of the ACS+NICMOS observations to quantify the distribution of -continuum slopes for star-forming galaxies over range in redshift (-6) and luminosities. Bouwens et al. (2009) derive directly from the colors (e.g., as in Appendix A). How do the present -continuum slope determinations compare with those from Bouwens et al. (2009)? Both old and new results are shown in Figure 8. Comparing the -continuum slope measurements made on the identical sources in the two data sets (old ACS+NICMOS data vs. the new WFC3/IR observations), we find reasonable agreement, with mean offsets of only 0.10, 0.04, 0.09 in the derived ’s at , , and , respectively. The biweight mean -continuum slopes found here are also in good agreement with the slopes derived by Bouwens et al. (2009). At and , the present biweight mean ’s are just 0.26 bluer and 0.16 redder than the values found in Bouwens et al. (2009); the present biweight mean ’s at show no average shift at all relative to the values found by Bouwens et al. (2009). of the differences result from our use of the more robust biweight means to express the central ’s, so the agreement is quite good overall.
4.2. Comparison with Bouwens et al. 2010a (-7)
Bouwens et al. (2010a) took advantage of the first-year WFC3/IR observations over the HUDF and the ERS WFC3/IR observations to quantify the -continuum slope distribution to higher redshifts () and lower luminosities. How do the present determinations of the -continuum slope compare with Bouwens et al. (2010a)? At high luminosities, we find excellent agreement, with both studies preferring mean -continuum slopes of (see Figure 8). At lower luminosities, however, we now find somewhat redder values of the -continuum slope , i.e., than found by Bouwens et al. (2010a). The observed differences in the mean -continuum slope seem to have resulted from both the small number of sources in previous samples and uncertainties in the photometry of faint sources in the early HUDF09 data. We remark that the current WFC3/IR observations of the HUDF from the HUDF09 WFC3/IR program are approximately twice as deep as what Bouwens et al. (2010a) used.
4.3. Comparison with the literature: Does correlate with redshift?
In the present analysis, we find that the -continuum slope shows a correlation with the redshift of galaxies, with higher redshift galaxies being bluer. This evolution is evident both in Figure 6 (compare the solid lines in the , , panels with the dotted lines) and in Figure 7.
How does the correlation we find compare with other studies? A brief summary of the evidence for this redshift dependence is provided in Figure 8, and there is a clear trend from bluer -continuum slopes at -7 to redder slopes at -4. Essentially all studies of the -continuum slope over the range -7 (Lehnert & Bremer 2003; Stanway et al. 2005; Bouwens et al. 2006; Bouwens et al. 2009; Bouwens et al. 2010a; Wilkins et al. 2011; Finkelstein et al. 2012) find evidence for this evolution of with redshift. The only exception to this is the recent study of Dunlop et al. (2012) who find no dependence (but Dunlop et al. 2012 do note that a comparison of their results with those at does argue for evolution); we discuss the Dunlop et al. (2012) results in §4.5.
4.4. Comparison with the literature: Does correlate with luminosity?
As we discuss in §3.6, we find that the -continuum slope shows a clear correlation with the rest-frame luminosity of galaxies, with lower luminosity galaxies being bluer at all redshifts. A good illustration of this correlation is provided in Figure 6. The best-fit relationship (solid blue line) shows almost exactly the same dependence for each high redshift sample. The uniformity of the slope and modest variation as a function of redshift is also clearly illustrated in Figure 7. The uniformity extends even to our samples. While in these samples appear to show a somewhat weaker dependence on luminosity than our other samples, a slightly steeper dependence is found, i.e., , if we use the same luminosity baseline as our and samples (excluding sources faintward of AB mag).
In general, the correlation we find agrees very well with most previous studies. Figure 9 shows different determinations as a function of luminosity at -5 (Ouchi et al. 2004; Overzier et al. 2008; Bouwens et al. 2009; Lee et al. 2011; Wilkins et al. 2011; Dunlop et al. 2012).333The slopes we derive from Labbé et al. (2007) are based upon their results at 1700Åand 3600Åin their Figure 3. Previously, Bouwens et al. (2009) had estimated a slope of in the -continuum slope vs. relationship at -2.7 from the Labbé et al. (2007) results based on a shorter 1700Åand 2200Åwavelength baseline (see Figure 7). The reason we use a more extended wavelength baseline to estimate than Bouwens et al. (2009) had used is to allow for a fair comparison with the present results (which use an extended wavelength baseline to estimate : see §3.3). Despite some dispersion in the precise values of the measurements and some small variance in the best-fit slopes , nearly all published determinations of the -continuum slope at -5 find bluer values for at lower luminosities, with the same luminosity dependence. Two recent analyses that found minimal or no correlation of with luminosity are those of Dunlop et al. (2012) at -7 and Finkelstein et al. (2012) at -5. We discuss the Dunlop et al. (2012) results in §4.5 and the Finkelstein et al. (2012) results in §4.7.
Similar correlations with luminosity are found in the -continuum slope results at -3 (Labbé et al. 2007; Bouwens et al. 2009; Sawicki 2012) and in the -optical colors (Papovich et al. 2001; Labbé et al. 2007; González et al. 2012). Again, two analyses did not find a correlation of with luminosity, and those are the Adelberger & Steidel (2000) and Reddy et al. (2008) analyses at -3. In these cases, not only was the luminosity baseline too short to provide much leverage for quantifying this correlation, but also the luminosity range probed was around which is where the dispersion in dust properties relative to UV luminosity is at a maximum (e.g., Figure 13 of Reddy et al. 2010). However, when one adds the Adelberger & Steidel (2000) samples to the faint (26-27 mag) samples observed by Bouwens et al. (2009), a strong correlation is present (Figure 3 of Bouwens et al. 2009: see also Figure 5 of Sawicki 2012). Taken together these results indicate that there is also a clear trend with luminosity at -3.
4.5. Comparison with Dunlop et al. 2012 (-7)
Dunlop et al. (2012) use the WFC3/IR observations over the HUDF09 and ERS observations to quantify the -continuum slope distribution at -7. They select sources using a photometric redshift procedure and then measure their -continuum slopes from the colors. In both respects, their procedure differs from that followed here (§3.2 and §3.3). Overall, the individual -continuum slope measurements of Dunlop et al. (2012) are in reasonable agreement with the present results (see Figure 8). The mean they derive for bright galaxies is bluer and the mean they derive for faint galaxies is redder. Given the quoted uncertainties on the measurements, the differences are not particularly significant.
The main differences arise when we look at the trends in the -continuum slope with redshift and luminosity. Dunlop et al. (2012) find a mean -continuum slope of galaxies equal to , with no significant dependence on luminosity or redshift. This is in contrast to the strong correlation we find of with both luminosity (Figure 6 and 7) and redshift (Figure 7). Their results are also inconsistent with the trends reported in the literature (§4.3-§4.4; Figure 7 and 9).
We have attempted to understand the source of this difference both by a qualitative assessment of some issues that we know are important for obtaining reliable results and by a quantitative assessment using simulations (see Appendix D and §4.6). First, we remark that Dunlop et al. (2012) make no attempt to correct their mean ’s for the fact that galaxies with bluer -continuum slopes are easier to select than galaxies with redder -continuum slopes . This effectively biases their -continuum slopes to bluer values. It is a fairly straightforward process to correct for this issue (e.g., we describe such a correction in Appendix B.1.1 and Figure 5 where we show its effect).444See also Figure 5 of Wilkins et al. 2011. Second, Dunlop et al. (2012) use an overlapping set of information both to select sources and to measure the -continuum slopes . This biases their results (see Appendix D), though the magnitude of this bias is mitigated by their consideration of only the brightest sources.
Third, one further limitation of the Dunlop et al. (2012) analysis is their exclusion of the lowest luminosity sources. This significantly reduces the leverage they have to quantify trends in the mean -continuum slope as a function of luminosity. Dunlop et al. (2012) exclude faint sources, because their simulations suggested to them that could not be measured in an unbiased way. However, as we show in Appendix B and D, we are able to obtain reliable measurements (see also §4.6). Not only can we recover the mean -continuum slope for our samples to very faint magnitudes with excellent accuracy, but we can recover these slopes for samples selected using a photometric redshift procedure, if the information used for source selection is clearly separated from that used to measure (see Figure 10).
As mentioned above, these three issues led us to consider a more quantitative evaluation of the Dunlop et al. (2012) procedure so that we could better understand what was happening with their measurement of . This is discussed in more detail in the next section.
4.6. Can be measured in a largely bias-free way for low S/N sources?
As discussed above, a key question that has arisen in recent papers is whether it is possible to determine the -continuum slope distribution to very low luminosities with small biases. Dunlop et al. (2012), in particular, have suggested that it is not possible and have supported this suggestion with a series of simulations where they add noise to model galaxies and reselect these galaxies with their photometric redshift code. Dunlop et al. (2012) argue that noise in the photometry combined with a preference for selecting sources with blue colors would result in highly biased estimates for the mean -continuum slope . Dunlop et al. (2012) show that such a bias towards bluer -continuum slopes at faint magnitudes is present in their mock data sets. Dunlop et al. (2012) argue that similar biases are likely present in Lyman-break selections, without further substantiating this claim.
We agree that selection biases can affect the distribution of -continuum slopes . However, the size of these selection biases is extremely dependent upon how one selects the galaxies and measures their -continuum slopes . As we show in Figure 10, our Lyman-Break selections yield much smaller selection biases overall. The reason we expect biases in our selections to be small is that we select galaxies using a different part of the rest-frame SED (the bluest two passbands redward of the break) than we use to measure -continuum slopes (the second bluest passband and redder). As a result, photometric scatter in the bands we use to measure the -continuum slope is largely independent of similar scatter in the bands we use for selection. Therefore, we would not expect our measurements to be significantly biased as a result of the object selection process.555This issue is of course in addition to the normal selection biases that Lyman-Break samples show against sources with red -continuum slopes , but as we show in Appendix B.1.1 these biases are small for all but the reddest ’s.
Achieving similarly small biases to faint magnitudes with a photometric redshift technique is also possible. However, one must again be careful to use different information to select sources from what one uses to measure the -continuum slope . To illustrate this, we consider the situation at in the current HST ACS + WFC3/IR data set. We have run simulations where we attempt to measure the mean -continuum slope for a set of galaxies selected using a photometric redshift procedure. Both the simulations and results are discussed in Appendix D. We consider (1) the case where 5 HST bands are used to select sources and determine redshifts (this excludes those bands used to measure ), (2) the case where 6 bands are used (and so now including one of the two bands used to measure ), and (3) the case where all 7 HST bands are used for selection and redshift determination (and so both bands used to measure are also included to measure redshifts and select the sources). This latter approach is basically what Dunlop et al. (2012) do.
The results are shown in Figure 26 of Appendix D. While measurements show substantial biases when all 7 bands are used for the photometric redshift estimates (similar to the procedure of Dunlop et al. 2012) measurements made using 5 or 6 bands show much smaller biases. This demonstrates that the -continuum slope can be measured with very small biases to faint magnitudes.
4.7. Comparison with Finkelstein et al. 2012 (-7)
In an independent analysis, Finkelstein et al. (2012) also use the recent WFC3/IR observations over the HUDF and CDF-South GOODS field to quantify the -continuum slope distribution for star-forming galaxies at -8. Finkelstein et al. (2012) split sources into five different redshift samples using a photometric redshift procedure and then estimate by finding the model SED which best fits the photometry of a source and deriving from this model. Finkelstein et al. (2012) find that shows a clear dependence on redshift, but report only a limited dependence on the luminosity.
vs. Luminosity Trends: While the redshift dependence Finkelstein et al. (2012) find for is in excellent agreement with what we find (compare the solid black squares and large blue circles in the upper panel of Figure 8), the luminosity dependence Finkelstein et al. (2012) observe would appear to be considerably weaker. After all, Finkelstein et al. (2012) report no significant correlation of with luminosity in their baseline analyses of their five redshift samples – seemingly different than the clear correlation of with luminosity we report.
Despite these apparent differences, the overall results from the two studies are actually in fairly good agreement. For example, with regard to the and samples, the best-fit values Finkelstein et al. (2012) find, i.e., and , respectively, are strikingly similar to the values we find, i.e., and , respectively.
For the and samples, Finkelstein et al. (2012) do not report a significant correlation between and luminosity in what they identify as their baseline analysis (finding values of 0.010.03 and 0.000.06, respectively). However, in this analysis, Finkelstein et al. (2012) only make use of a 2-mag baseline in luminosity (due to their binning scheme), with their low luminosity anchor point largely coming from the relatively shallow 1.6 orbit CANDELS data (with only a small contribution from the HUDF data). The CANDELS observations are clearly poorly suited to determine the trend in to very low luminosities, given the very low S/N’s and potentially large biases expected for the faintest sources in the CANDELS fields. We might expect the situation to change taking advantage of the additional leverage in luminosity provided by the measurements they provide for faint sources in the ultra-deep HUDF observations.
We can check this by extracting the HUDF measurements from Figure 5 of their paper. By comparing their median measurements for brighter galaxies with their median measurements for fainter galaxies in the HUDF, we find evidence for a significant correlation with luminosity. We find trends of 0.060.02 and 0.130.04, respectively. In the final version of their paper, Finkelstein et al. (2012) also note a similar correlation with luminosity making use of the faintest HUDF sources, finding trends of 0.070.01 and 0.090.03, respectively. While not in exact agreement with the trends we derive based on our own measurements, i.e., 0.110.01 and 0.160.03, respectively, the agreement is much better. Use of the faint sources in the HUDF is important to take full advantage of the available leverage in luminosity to quantify the vs. trend.
In Figure 11, we show the vs. trend that Finkelstein et al. (2012) find in their baseline and analyses (magenta lines) and the trend we find from their measurements making exclusive use of sources from the HUDF to constrain to fainter magnitudes (dashed red lines). In addition, we show the median ’s Finkelstein et al. (2012) find for two fainter and subsamples within the HUDF (large solid red circles: we can extract measurements for individual sources within the HUDF from their Figure 5) and the bootstrap uncertainties on these medians. In both samples there is a clear trend in the median ’s towards bluer values at the lowest luminosities. It is striking how well the median ’s Finkelstein et al. (2012) derive from the HUDF agree with our own measurements, particularly in the luminosity interval [20 mag, 18 mag]. While it is true that the median ’s Finkelstein et al. (2012) derive for their entire ERS+CANDELS+HUDF09 sample are redder in general than what we find at these luminosities, these median ’s receive their largest weight from the shallower ERS+CANDELS samples and therefore may be subject to possible selection, measurement, or contamination biases (see the discussion at the end of this section).
Expected Trend in Luminosity: Finkelstein et al. (2012) defend the weak correlation of they report versus luminosity (particularly as derived in their baseline analysis), arguing that should show a stronger correlation with stellar mass than luminosity. We do not dispute this assertion; however, it would be most surprising if a correlation of with stellar mass did not also appear as a correlation with luminosity. Given the correlation found between SFR and stellar mass in high-redshift galaxies (e.g., Stark et al. 2009; González et al. 2011; McLure et al. 2011; Lee et al. 2012; Reddy et al. 2012b), we would expect the luminosity to be broadly correlated with stellar mass. A similar conclusion can be drawn from high-redshift angular correlation function results. Higher luminosity galaxies are consistently found to be more clustered than lower luminosity galaxies (e.g., Ouchi et al. 2004b; Lee et al. 2006). Such would not be the case if luminosity was not correlated with mass (in this case halo mass). Independent of these considerations, we remark that also shows a clear correlation with luminosity in various cosmological hydrodynamical simulations (e.g., Finlator et al. 2011; Dayal & Ferrara 2012), and the predicted trends (e.g., is expected in the Finlator et al. 2011 simulations) are comparable to what we find (Figure 7).
Possible Biases in the Finkelstein et al. measurements: Finkelstein et al. (2012) have suggested that the - trends we find may be stronger than what they find due to the fact that we measure the luminosity at a different rest-frame wavelength than they do, and vs. trends may depend on this rest-frame wavelength. For our samples, for example, we measure the rest-frame luminosity at 2041Å(see Table 3) while Finkelstein et al. (2012) measure it at 1500Å. We are in full agreement that will depend on the rest-frame wavelength where the luminosity is derived. However, the results from Figure 3 in Labbé et al. (2007) suggest one would find an even stronger vs. trend at bluer wavelengths than one would find at redder wavelengths, which is different from what Finkelstein et al. (2012) find. It is therefore not clear this explains the differences.
Instead, one might be concerned that the vs. trend
Finkelstein et al. (2012) find may be biased as a result of the
rest-frame wavelength Finkelstein et al. (2012) use to measure the
luminosity.666Finkelstein et al. (2012) also discuss
this issue at some length in their paper (as a source of differences
between the vs. trends we find) and would appear to
find a similar effect, but given its importance for understanding
differences between our results, we feel this discussion is worth
repeating. By determining luminosity at the blue end (at 1500Å)
of the wavelength baseline they use to derive , Finkelstein et
al. (2012) effectively introduce a coupling between the errors that
affect both their measurements and their determinations of the
luminosity . This could be problematic since any errors
in the flux measurements of sources would cause sources to be either
fainter and redder or brighter and bluer, causing the
trend derived by Finkelstein et al. (2012) to be biased towards too
high of values. Repeating the determination of at
-5 based on our own flux measurements but basing on
the flux measurement at the blue end of the wavelength baseline to