1 Introduction

Continuum Slope and Dust Obscuration from to : The Star Formation Rate Density at High Redshift

Abstract

We provide a systematic measurement of the rest-frame UV continuum slope over a wide range in redshift (2-6) and rest-frame luminosity (0.1 to 2) to improve estimates of the SFR density at high redshift. We utilize the deep optical and infrared data (ACS/NICMOS) over the CDF-South and HDF-North GOODS fields, as well as the UDF for our primary “dropout” Lyman Break Galaxy sample. We also use strong lensing clusters to identify a population of very low luminosity, high-redshift dropout galaxies. We correct the observed distributions for both selection biases and photometric scatter. We find that the -continuum slope of the most luminous galaxies is substantially redder at 2-4 than it is at 5-6 (from at to at ). Lower luminosity galaxies are also found to be bluer than higher luminosity galaxies at and . We do not find a large number of galaxies with ’s as red as in our dropout selections at , and particularly at , even though such sources could be readily selected from our data (and also from Balmer Break Galaxy searches at ). This suggests that star-forming galaxies at almost universally have very blue -continuum slopes, and that there are not likely to be a substantial number of dust-obscured galaxies at that are missed in “dropout” searches. Using the same relation between -continuum slope and dust extinction as has been found to be appropriate at both and , we estimate the average dust extinction of galaxies as a function of redshift and luminosity in a consistent way. As expected, we find that the estimated dust extinction increases substantially with cosmic time for the most luminous galaxies, but remains small () at all times for lower luminosity galaxies. Because these same lower luminosity galaxies dominate the luminosity density in the continuum, the overall dust extinction correction remains modest at all redshifts and the evolution of this correction with redshift is only modest. We include the contribution from ULIRGs in our star formation rate density estimates at 2-6, but find that they contribute only 20% of the total at 2.5 and 10% at .

Subject headings:
galaxies: evolution — galaxies: high-redshift

1. Introduction

Optical -band Near-IR -band
Field Passbands Area1 limit2 Passbands Area3 limit4
HDF-N 5 27.6 5 27.3
HDF-S 4 28.0
GOODS 316 27.4 18 26.4
GOODS 36 26.4
HUDF05 22 28.4 2.5 28.6
HUDF 11 28.8 6 27.2
Abell 2218 11 27.2 1.5 26.8
MS1358 11 27.2 1.5 26.8
CL0024 11 27.2 3.0 26.8
Abell 1689 11 27.2 5
Abell 1703 11 27.2 6
Table 1Observational data used to select , , , and dropouts and to measure the -continuum slope.

Quantifying the star-formation rate (SFR) density at is a challenging endeavor. While there are a wide variety of techniques to estimate this rate at from light at different wavelengths (Condon 1992; Kennicutt 1998; Madau et al. 1998; Ranalli et al. 2003; Yüksel et al. 2008; Li 2008; Reddy et al. 2006; Yun et al. 2001; Reddy & Steidel 2004; Reddy et al. 2006; Erb et al. 2006b), the situation becomes considerably more uncertain at redshift . This is because most of the techniques effective at low redshift rely upon light at wavelengths that simply cannot be detected at high redshifts for all but the most exceptional sources. Determinations of the SFR density from x-ray or radio emission appear to be just possible at 2-4 by stacking a large number of sources in very deep integrations (e.g., Seibert et al. 2002; Nandra et al. 2002; Reddy & Steidel 2004), but do not work at (Lehmer et al. 2005; Carilli et al. 2008). Similarly, while deep mid-IR data would seem to be quite effective in estimating the SFR at (e.g., Reddy et al. 2006; Caputi et al. 2007), at many of the key spectral features redshift to wavelengths not very amenable to study with current instrumentation.

This leaves us with sparingly few techniques for estimating the SFR density at high redshift, most of which have only recently been developed. These techniques include traditional approaches like a consideration of the light to estimate the SFR densities to newer approaches that use gamma-ray burst (GRB) rate densities (e.g., Yüksel et al. 2008; Li 2008) or possible detections of redshifted H (e.g., Chary et al. 2005) to make these estimates.

Perhaps, the simplest and most practical of these techniques is to rely on the light emanating from the high-redshift galaxies themselves. Young stars emit a large fraction of their energy at these wavelengths, and it is very easy using current instrumentation to measure this energy to very low SFR rates (i.e., 0.1 yr at z4 in the Hubble Ultra Deep Field [HUDF: Beckwith et al. 2006]). One of the most significant challenges in establishing the SFR from the UV data is estimating the absorption by dust. Since light is subject to substantial attenuation by dust, this entire issue is quite important. One of the most practical methods for determining the dust attenuation is through the -continuum slopes (). Since these slopes have been shown to be well correlated with the dust extinction at (Meurer et al. 1995; Meurer et al. 1997; Meurer et al. 1999; Burgarella et al. 2005; Dale et al. 2007; Treyer et al. 2007; Salim et al. 2007), (Laird et al. 2005), and (Reddy et al. 2006; Daddi et al. 2004), we can use these slopes to estimate the dust obscuration at even higher redshift.

Not surprisingly, there have already been a number of attempts to determine these slopes at high redshift (2-6) from the available imaging data. Adelberger & Steidel (2000) and Meurer et al. (1999) examined the UV slopes for a sample of -dropouts and inferred modest (factor of 5) dust extinctions. Other groups (Lehnert & Bremer 2003; Ouchi et al. 2004; Stanway et al. 2005; Yan et al. 2005; Bouwens et al. 2006) have looked at these slopes at higher redshifts (), again on a dropout selection. Broadly, it was found that higher redshift galaxies had bluer -continuum slopes than lower redshift galaxies have, suggesting that the typical dust extinction at high redshift is significantly less than it is at lower redshift. There was also some evidence from previous analyses (e.g., Meurer et al. 1999 at ) that lower luminosity galaxies had bluer -continuum slopes and thus lower dust extinctions (though the emphasis in Meurer et al. (1999) was the correlation between and the dust-corrected luminosity, not the observed luminosity).

Regrettably, apart from these two trends (higher redshift galaxies are bluer and lower luminosity galaxies are bluer), it has been difficult to make quantitative statements about the distribution of -continuum slopes or dust extinctions for galaxies at high redshift. In part, this can be attributed to the piecemeal way the -continuum slopes and dust extinction have been derived at high redshift, as different analyses have derived -continuum slopes based upon a wide variety of disparate high-redshift galaxy samples, using different techniques (not all of which are consistent). In part, this can also be attributed to the limited amounts of data available to each particular study. Fortunately, the situation has begun to change, and there is now a wide variety of HST data that are available to select high redshift galaxies and determine their slopes over a wide range in redshift and luminosity. Determining how these slopes also depend upon luminosity seems particularly relevant given the fact that these sources likely dominate the luminosity density and perhaps star formation rate density at (e.g., Bouwens et al. 2006; Sawicki & Thompson 2006b; Bouwens et al. 2007; Reddy et al. 2008; Yan & Windhorst 2004; Reddy & Steidel 2009).

It seems clear that a comprehensive, systematic determination of the distribution of -continuum slopes is needed, both as a function of redshift and luminosity. The goal of the present paper is to provide just such an analysis, taking advantage of the considerable quantity of deep HST data to clarify our understanding of dust obscuration and, by consequence, the star formation rate density at high redshift. We select galaxies for -continuum slope measurements over a wide range in redshift using well-tested , , , and dropout selections. We use the wide-area ACS Great Observatories Origins Deep Survey (GOODS) fields (Giavalisco et al. 2004a) to quantify these slopes at bright magnitudes and extend our analyses to very low luminosities by examining very deep HST data sets like the HUDF, and also by examining dropouts behind massive galaxy clusters where the substantial magnification factors permit us to consider very faint sources.

The plan for this manuscript is as follows. In §2, we present the observational data set. In §3, we describe our procedure for generating source catalogs from these data, selecting the dropout samples, and estimating the -continuum slopes while correcting for the effect of object selection and photometric scatter. Finally, we discuss our results (§4), examine the likely implications of these results for the effective dust extinction and SFR density at 2-6 (§5), and then include a summary (§6). Throughout this work, we will 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. Where necessary, we assume , , . Although these parameters are slightly different from those determined from the WMAP five-year results (Dunkley et al. 2009), they allow for convenient comparison with other recent results expressed in a similar manner. Unless otherwise stated, the -continuum continuum slope presented here all assume a 1600Å-2300Åbaseline. We express all magnitudes in the AB system (Oke & Gunn 1983).

Figure 1.— Illustration of the deep near-IR data available over the two GOODS fields. Deep near-IR data are essential for measurements of the UV-continuum slope of galaxies at and valuable for a measurement of these slopes at . The red and dark red regions correspond to regions with deep and very deep NICMOS and coverage, respectively ( depths and AB mag). The light orange regions have deep NICMOS -band coverage ( depths of : H. Teplitz et al. 2009, in prep; C. Conselice et al. 2009, in prep) while the magenta regions have deep NICMOS -band coverage ( depths of : R.J. Bouwens et al. 2009, in prep). The blue and dark blue regions correspond to regions with deep and very deep optical ACS coverage, respectively ( depths of and AB mag). For more details on the NICMOS data over the two GOODS fields, see Bouwens et al. (2008) and R.J. Bouwens et al. (2009, in prep).

2. Observational Data

The purpose of this study is to measure the -continuum slope over as wide a range in luminosity and redshift as is practical from current observations. Of course, establishing this distribution even over a limited range in redshift or luminosity can be a challenge, as it requires very deep imaging over a wide wavelength range, both to select galaxies through the Lyman break technique and to measure their UV-continuum slope redward of this break. It also requires sufficient search area to provide a sufficiently large samples of sources to effectively define the distribution of -continuum slopes.

At , we use the deep -band coverage available over the WFPC2 Hubble Deep Field (HDF) North (Williams et al. 1996) and HDF South (Casertano et al. 2000) to select -dropouts. Those two deep WFPC2 fields are invaluable for defining the distribution of -continuum slopes at lower luminosities, but their small area limits their value for establishing the distribution at the higher luminosities. However, as we will see, the -continuum slope distribution for bright galaxies is similar to what Adelberger & Steidel (2000) and Reddy et al. (2008) derived previously. For our , , and -dropout samples, we utilize the ACS data available over the HUDF, HUDF05 (Oesch et al. 2007; Bouwens et al. 2007), and GOODS fields. The wide-area GOODS fields are used for establishing the -continuum slope distribution for luminous galaxies while the deeper HUDF and HUDF05 fields are used for establishing the -continuum slope distribution for lower luminosity galaxies. The reduction of these latter data was performed with “apsis” (Blakeslee et al. 2003) and is described in Bouwens et al. (2006, 2007). A summary of the properties of these fields is given in Table 1.

Importantly, the HUDF, HUDF05, and GOODS fields also have arcmin of very deep near-IR coverage with NICMOS – reaching limits of (-diameter apertures). This coverage is important for measurements of the -continuum slope for galaxies in our -dropout and -dropout selections. The NICMOS data are described in more detail in R.J. Bouwens et al. (2009, in prep) and are illustrated in Figure 1 (see also Bouwens et al. 2008). These data were obtained as a result of the cumulative observations of many different HST programs (e.g., Dickinson 1999; Thompson et al. 1999; Thompson et al. 2005; Bouwens et al. 2005; Oesch et al. 2007; Riess et al. 2007; C. Conselice et al. 2009, in prep). All the NICMOS data taken since 2002 (when the NICMOS cryocooler went into operation) were reduced with our NICMOS pipeline “nicred.py” (Magee et al. 2007).

Finally, we take advantage of the deep HST data over three massive galaxy clusters to identify a small number of lower luminosity star-forming galaxies at behind these clusters. Because of lensing by the foreground cluster (magnifying faint 5-6 galaxies by factors of ), it is possible to select and to measure -continuum slopes for a small number of extremely low luminosity galaxies. The five galaxy clusters (MS1358, CL0024, Abell 2218, Abell 1689, Abell 1703) all have deep ACS data and cover 55 arcmin in total. These clusters also have 6 arcmin of deep NICMOS data necessary to measure the -continuum slope for dropout and -dropout galaxies behind these clusters. The ACS data available over these clusters were also reduced with “apsis” while the NICMOS data were reduced with “nicred.py”. These reductions are described in more detail in Bouwens et al. (2009a) and Zheng et al. (2009).

3. Analysis

In this section, we describe our procedure for determining the distribution of -continuum slopes from our observational data as a function of redshift and luminosity. We begin by discussing how we do photometry on our observational data and then select 2-6 galaxies using various dropout criteria (§3.1-3.2). In §3.3, we calibrate our optical/near-IR photometry and then derive -continuum slopes from the measured colors (§3.4). By combining measurements of the -continuum slope determinations from both very deep and wide-area data sets, we derive the -continuum slope distribution over a wide range in luminosity (§3.5). In §3.6, we take advantage of lensing from massive galaxy clusters to examine the -continuum slope distribution for a set of lower luminosity galaxies. We correct the observed distributions of -continuum slopes for the effect of object selection and photometric scatter (§3.7). Finally, we discuss the uncertainties that likely exist in our determinations (§3.8) and then give our final estimates of the -continuum slope distributions in §3.9.

3.1. Photometry

We used SExtractor (Bertin & Arnouts 1996) to perform object detection and photometry. Object detection was performed using the square root of the images (Szalay et al. 1999: essentially a coaddition of the relevant images) constructed from all ACS images with coverage redward of the dropout band (e.g., in the case of a -dropout selection) or the WFPC2 images redward of the dropout band in the case of our dropout selection (where the bands are used). For clarity, the image is equal to

where is the intensity of image at pixel , where is the RMS noise on that image, and where the index runs over all the relevant images (i.e., those redward of the dropout band).

Our procedure for measuring colors depends upon the PSF of the data we are using. For colors that include only the ACS or WFPC2 bands (where the PSF is much sharper than for NICMOS), we measure these colors in scalable apertures determined using a Kron (1980) factor of 1.2. When measuring colors that include the NICMOS near-IR bands, we first PSF match the ACS/WFPC2 data to the NICMOS -band. Then, we use the same apertures described above if the area in those apertures is arcsec (i.e., 0.7-diameter apertures). This only applies for the largest sources in our selection. Otherwise (and in most cases), the colors are measured in fixed -diameter apertures.

We correct the fluxes measured in the above apertures (used for color measurements) to total fluxes using a two step procedure. First, the individual flux measurements are corrected to a much larger scalable aperture (using a Kron factor of 2.5). These corrections are made on a source-by-source basis using the square root of the image (Szalay et al. 1999). Second, a correction is made to account for the light outside of these larger apertures and on the wings of the ACS Wide Field Camera (WFC) PSF (Sirianni et al. 2005). Typical corrections are mag for the first step and mag for the second step. While there are modest uncertainties in these corrections (typically 0.2 mag), these corrections only affect the total magnitude measurements and do not affect the colors.

Figure 2.— Two-color diagrams showing the selection criteria (see §3.2) we use to find , , , and dropouts in our search fields (upper left, upper right, lower left, and lower right panel, respectively). Sources that fall within the shaded regions would be included in our selections. The blue tracks show the expected colors of starbursts with different -continuum slopes as a function of redshift, while the red lines show the colors of low-redshift interlopers (Coleman et al. 1980). The black points illustrate the range of colors that real sources in our search fields possess. To ensure that these plotted distributions of colors are as realistic as possible, we only include sources from our deepest fields (the HDFs for our -dropout selection and HUDF for our , , and -dropout selection) and at relatively bright magnitudes ( for our -dropout selection, for our -dropout selection, and for our ,-dropout selections). The red arrows show the Calzetti et al. (2000) reddening vectors. These diagrams show that our selection criteria are effective in identifying high-redshift galaxies with -continuum slopes bluer than 0.5, but less effective in identifying galaxies redward of this limit (Figures 4 and 6 show the effective selection volume calculated to be available to galaxies with various -continuum slopes).

3.2. Sample Selection

We select star-forming galaxies over the redshift range to using the well-established dropout technique (e.g., Steidel et al. 1996; Bunker et al. 2003; Vanzella et al. 2006; Dow-Hygelund et al. 2007; Stanway et al. 2007; Vanzella et al. 2009). Our -dropout criterion is similar to those used by Bouwens et al. (2003) while our , , and dropout criteria are almost identical to those adopted by Bouwens et al. (2007) except that our -dropout criterion uses a sharper cut to exclude dusty interlopers. These criteria are

for our -dropout sample

for our -dropout sample,

for our -dropout sample, and

for our -dropout sample, where and represent the logical AND and OR symbols, respectively, and S/N represents the signal-to-noise. These criteria are illustrated in two color diagrams in Figure 2 and are very similar to those considered by Giavalisco et al. (2004b) and Beckwith et al. (2006).

To ensure that our selections did not contain any spurious sources, we required that sources in our , , , and dropout selections be 5.5 detections in the , , , and bands, respectively. We inspected all sources in our dropout samples by eye to purge them of any obvious contaminants (diffraction spikes, irregularities in the background). Pointlike sources (SExtractor stellarity S/G 0.8) were also removed (% of the relevant sources).

Dropout Mean Filters used for determination
Sample Redshift Lower Upper
-dropout 2.5 (1730Å) (2330Å)
-dropout 3.8 (1610Å) (1940Å)
-dropout 5.0 (1550Å) (2250Å)
-dropout 5.0 (1550Å) (2660Å)
-dropout 5.9 (1570Å) (2290Å)
78
Table 2Wavebands used to derive the continuum slope for individual galaxies in high-redshift , , , and dropout samples.9

3.3. Calibration of the NICMOS Magnitudes

To accurately estimate the -continuum slopes for galaxies in our sample, it is absolutely essential that the colors that we measure are accurate. For example, for a source with colors measured across the baseline 1600Åto 2300Å, a 0.1 mag error in this color translates into an error of 0.3 in the estimated slope .

Measuring colors to this level of accuracy is challenging for a number of reasons, particularly in the near-IR. First and foremost, the NICMOS detector (essential for measuring the fluxes of faint sources in the near-IR) has been shown to suffer from significant non-linearity in its count rate. While there are now procedures (e.g., de Jong et al. 2006) to correct for these non-linearities, as utilized here, it is unlikely that the corrections are perfect in the magnitude range we are considering. This is because there are comparatively few deep fields available where the near-IR fluxes can be measured and calibrated against other data. Secondly, the colors we are measuring are derived from both optical (ACS) and near-IR (NICMOS) data where the PSF is very different. While such differences can be effectively addressed by carefully matching the PSFs of the two data sets before measuring fluxes (as we do here), PSF matching can be challenging to perform perfectly and so it is easy to introduce small systematic errors at this stage.

As a result of these issues, we took great care in deriving correction factors that we could apply to the measured NICMOS near-IR fluxes. For simplicity, we assumed that these correction factors did not depend on the luminosity of the sources we were considering – but rather were simple offsets we could add to the measured magnitudes. We established these correction factors by examining the photometry of a sample (50) of relatively faint () point-like stars (SExtractor stellarity 0.8) over the HUDF/GOODS fields, and then compared their observed colors with that expected from the Pickles et al. (1998) atlas of stellar spectra. Given the very limited range in SED shape found in real stars, these comparisons should allow us to make reasonable corrections to the and -band photometry. We determined the average differences between the measured and best-fit near-IR fluxes (both in the and bands) and found corrections of 0.13 mag and 0.07 mag for the and band, respectively.

To ensure that these corrections were as accurate as possible, we also identified a sample of highest S/N dropouts from our search fields (GOODS and HUDF) and then fit their observed SEDs with Bruzual & Charlot (2003) models. Since the and band fluxes of these galaxies probe the light blueward of the age sensitive break at 3600Å(probing 2300Åand 3300Årest-frame, respectively), it should be reasonable to use the available (1600Å  rest-frame) and -band (1900Årest-frame) fluxes to establish a -continuum slope and extrapolate to these redder wavelengths. For our fiducial model fit parameters, we assumed a star formation history (with Myr, Myr), a Salpeter IMF, and , and let the dust extinction and redshift be free parameters (the fiducial model considered by Papovich et al. 2001 in their stellar population modelling of -dropouts). Comparing the predicted near-IR fluxes (from the models) with the measured fluxes, we computed corrections to the and -band fluxes. The corrections were 0.080.02 mag and 0.040.2 mag, respectively (the uncertainties were derived using other SED models [Appendix A] to determine these corrections). Because of the size of the uncertainties in the -band corrections (from the -dropouts), we gave those results very low weight overall for the -band correction.

Combining the constraints we derived from our sample of stars and LBGs – which were consistent within mag – we derived a 0.10 mag correction to the measured -band magnitudes. For the NICMOS -band, we derived a 0.05 mag correction from the above fits – using the SED fit results to stars as our primary constraint.10 The measured and -band fluxes were therefore somewhat too faint (before correction). A brief discussion of the effect that uncertainties in the photometry/zero points could have on our derived -continuum slopes is given in §3.8.

3.4. Measuring the -continuum slope

Our principal interest is to determine the rest-frame -continuum slope for the galaxies in our dropout samples. As noted in the introduction, the -continuum slope specifies how the flux density of a galaxy varies with wavelength (i.e., ) in the -continuum (i.e., from 1300Åto 3500Å).

We estimate the -continuum slopes from the broadband colors available for each galaxy in our sample. To make our estimates of these slopes as uniform as possible, we selected filters at each redshift where the rest-frame wavelengths were as very close to 1650Åand 2300Åas possible. Of course, the actual effective wavelengths of these filters will vary somewhat from sample to sample – depending on the passbands in which broadband imaging is available for our different samples or the precise redshift of specific galaxies in our samples. Fortunately, the wavelength range 1650Åto 2300Åis a good match to the wavelength range used for studies of the dust extinction in galaxies at (e.g., Meurer et al. 1995; Meurer et al. 1999; Burgarella et al. 2005). Table 2 summarizes the passbands we use to probe these wavelengths for each of our dropout samples.

Some additional clarification may be helpful regarding specific filter choices. For our dropout sample, for example, we use the colors for these estimates to take advantage of the much higher quality (deeper, wider area) ACS data available. For our -dropout sample, we use colors predominantly for these estimates, but we also make use of colors for these estimates to take advantage of the large areas within the GOODS fields which have only NICMOS -band coverage (primarily from the Conselice et al. 2009, in prep, NICMOS program: see the light orange squares in Figure 1).

In principle, deriving the -continuum slopes from the measured colors is straightforward. One simply takes power-law spectra , calculates their colors at the mean redshift of each of our dropout samples, and then determines for which the model spectrum reproduces the observed colors. In practice, however, since the spectrum of star-forming galaxies in the -continuum is not expected to be a perfect power law (as noted above), small differences in the wavelength range over which the -continuum slope is measured will have an effect of the derived results.

To correct for these passband and wavelength-dependent effects, we base our estimates of on a small number of more realistic SEDs calculated from stellar population models (Bruzual & Charlot 2003). These model SEDs are calibrated to have -continuum slopes of 2.2, 1.5, and 0.8 over the wavelength range and assume a star formation history ( Myr, Myr, ) with varying amounts of dust extinction calculated according to the Calzetti et al. (2000) law. The formulae we use to convert between the measured colors and the -continuum slope are presented in Appendix A. The formulae we use to do the conversion depend somewhat on the star formation history assumed and likely result in an additional uncertainty in the derived -continuum slopes of .

Dropout Luminosity # of
Sample Field Range11 Sources
-dropout HDF-North 97
HDF-South 71
-dropout GOODS-North 218
GOODS-South 251
HUDF 470
Cluster 195
-dropout GOODS-North 22
GOODS-South 19
HUDF 10
HUDF05 16
Cluster 12
-dropout GOODS-North 2
GOODS-South 2
HUDF 5
HUDF05 3
Cluster 2
Table 3Dropout samples used to measure the distribution of -continuum slopes as a function of redshift and luminosity.

3.5. Constructing the -continuum Slope Distribution from Multiple (Deep or Wide-Area) Dropout Selections

In order to accurately quantify the distribution of -continuum slopes over a wide range in luminosity, we alternatively used our deepest selections and our wide-area selections. We used the deepest selections to define the -continuum slope distribution at the lowest luminosities to maximize the S/N on the derived colors (while minimizing the importance of selection effects). Meanwhile, we used our widest-area selections to define this distribution at higher luminosities. This allows us to find a sufficient number of sources to map out the distribution of -continuum slopes (which is helpful for determining the mean and scatter). In Table 3, we provide a list of the fields we use to quantify the distribution of -continuum slopes at a given luminosity and redshift.

In Figure 3, we plot the -continuum slopes observed for galaxies in our four dropout samples versus their luminosities. The mean -continuum slope and for galaxies is also included on this figure (blue squares) as a function of luminosity for each of the dropout samples. We adopt finer 0.5 mag bins for our dropout selections than the 1.0 mag bins we adopt for our and dropout selections or 2.0 mag bins we adopt for our dropout selection. The width of the bins depends upon the number of sources present in each of our dropout selections. The luminosity as shown on this diagram is the geometric mean of the two luminosities used to establish the slope (see Table 2). Use of the geometric mean of the two luminosities is preferred here for examining the correlation of with luminosity since it allows us to evaluate the correlation without introducing any artificial correlations. Had we, for example, elected to use the bluer bands for examining this correlation, the -continuum slope would be bluer as a function of the bluer-band luminosity, simply as a result of our examining the relationship as a function of this quantity. A similar (but opposite) bias would be introduced, in examining the -continuum slope as a function of the redder band.

A clear trend is seen towards bluer -continuum slopes at lower luminosities. This trend is similar to that found by Meurer et al. (1999) in their analyses of -dropout galaxies in the HDF North and also noted by Overzier et al. (2008) in their analysis of -dropouts in the TN1338 field (Figure 4 from that work).

3.6. -continuum Slopes Derived for Galaxies Gravitationally Lensed By Galaxy Clusters

In §3.5, we considered galaxies from both wide-area and deep surveys to establish the distribution of -continuum slopes over a wide range in luminosity. We can increase the number of sources in our lower luminosity samples, by considering gravitationally lensed sources behind high-redshift galaxy clusters. The clusters under consideration include Abell 2218, MS1358, CL0024, Abell 1689, and Abell 1703, and all allow us to select very faint star-forming galaxies at from the available HST ACS data.

These clusters substantially amplify the flux from distant galaxies, making it possible to measure the -continuum slope of galaxies at lower luminosities than would otherwise be possible. Of course, in the case of -dropout selections from very deep optical data like HUDF, we are able to reach to the same intrinsically luminosities as we reach in the -dropout samples we compile from searches behind lensing clusters. We correct the luminosities of the dropout galaxies identified behind these clusters by applying gravitational lensing models from the literature. For Abell 2218, MS1358, CL0024 (sometimes known as CL0024+1652), Abell 1689, and Abell 1703, we adopt the models constructed by Elíasdóttir et al. (2007), Franx et al. (1997), Jee et al. (2007), Limousin et al. (2007), and Limousin et al. (2008), respectively. Typical magnification factors for the galaxies we find behind galaxy clusters are 5-10. We do not include the -continuum slope results for sources where the model magnification factors are substantially greater than 10 – due to sizeable model-dependent uncertainties in the actual magnification and hence their luminosities (the uncertainties in the magnification factor for sources with such high predicted magnifications can be very large, i.e., greater than factors of 3-4: see Appendix A of R.J. Bouwens et al. 2009, in prep). Even for sources where the predicted magnification factors are smaller, the model magnification factors (and hence the luminosities corrected for lensing) are likely uncertain at the factor of 2 level (R.J. Bouwens et al. 2009, in prep).

Figure 3.— Determinations of the -continuum slope (1600Åto 2300Å) versus rest-frame luminosity for the present selection of , , , and dropouts. The luminosity here is the geometric mean of the luminosities measured in the bands used to establish the slope (see Table 2). On the right axes are the equivalent extinctions (Calzetti et al. 2000 law) to a given if the base spectrum is a young star-forming galaxy (100 Myr of constant star formation) – which is only bluer than our fiducial model (i.e., the one referred to in footnote b of Table 2). The small points represent the -continuum slopes measured for individual galaxies in our samples while the open blue squares and vertical bars show the mean -continuum slope and scatter (shown in a darker blue in the panel for clarity). Blue upper limits are shown for a few and sources with ’s bluer than 3.5. The green points on the panel show the -continuum slope determinations derived from our -dropout samples where only NICMOS -band data are available (i.e., with no deep NICMOS -band coverage). The black points on the panel show the -continuum slope determinations derived from our -dropout samples behind lensing clusters (where we have corrected their luminosities for model lensing magnification factors: see §3.6). It is encouraging that the -continuum slopes we derive from both of the latter and dropout selections agree with the determinations derived from our primary selections. The open red squares and vertical bars show the mean -continuum slope and error in this slope, after correcting for the effect of object selection and photometric scatter (§3.7). See Table 4 for a tabulation of these slopes. The red lines show the best fit relationship between continuum slope and the observed magnitude of a galaxy in the (see §3.9 and §4.5). It is also clear that the -continuum slope of star-forming galaxies is much redder at higher luminosities than it is at lower luminosities (particularly at and ).

We include the -continuum slope determinations of these galaxies with those estimated from our field samples on Figure 3 (black points on panel). As a result of a substantial magnification from gravitational lensing (typical factors of 5-10), the intrinsic luminosities of galaxies found behind clusters are much lower on average than those found in our field samples. In general, we observe good agreement between the distribution of -continuum slopes derived from our field samples and those inferred from our selections around clusters.

Figure 4.— (left) Comparison between the -continuum slope distribution input into the simulations (black histogram) and that directly recovered from the simulations (red histogram: which included the effect of object selection and measurement error). This panel is for our faintest, lowest S/N -dropout selection over the HUDF (-band magnitudes of 28-29). Both the input (and recovered) distributions are normalized to have the same area. On the top axes are the extinctions (Calzetti et al. 2000 law) that are equivalent to a given if the base spectrum is a young star-forming galaxy (100 Myr of constant star formation). The median -continuum slope input into the simulations and that recovered are shown with the black and red arrows, respectively. The recovered distribution of -continuum slopes is broader and peaked to somewhat bluer values than the input distribution. For our higher S/N selections, the differences between the input and recovered distributions are much smaller (the offsets between the red and blue squares in Figure 3 show the size of the corrections). For our faintest selections (as shown here) the differences can be larger. It is nonetheless reassuring that when these corrections are applied there is agreement between the mean -continuum slopes estimated in our shallower selections (where the corrections are larger) and in our deeper selections (where the corrections are smaller). For example, the mean -continuum slope derived at 26.5-27 for our GOODS -dropout selections (i.e., at AB mag) agrees very well with that derived from our much higher S/N HUDF selections for the GOODS fields (i.e., for the HUDF). (right) The selection volumes (arbitrary units) available to galaxies for our -dropout selection over the HUDF versus -continuum slope . Shown are the relative selection volumes for galaxies with 24.7-26.7 (red line), 26.7-27.7 (green line), and 27.7-29.0 (blue line). We emphasize that the same normalization for the selection volume is used for all the three magnitude selections, demonstrating that it is much easier to select galaxies with red -continuum slopes at bright magnitudes than it is at faint magnitudes. The reason bright galaxies are easier to select is that one needs to confirm that a source has a large Lyman break at 1, and this is easier to do at bright magnitudes. Selection biases and measurement errors are less for brighter samples (where the S/N is higher) and greater for fainter samples. The two panels here demonstrate how important it is for us to correct for selection biases and measurement uncertainties as we do in §3.7.

3.7. Selection Biases and Measurement Uncertainties

The distribution of -continuum slopes that we derive is quite clearly affected by the manner in which sources are selected. Dropout criteria include galaxies with bluer -continuum slopes more efficiently (i.e., over a larger range in redshift) than they do for galaxies with other colors. It is much easier to identify a sharp break in the SED of a blue galaxy than it is for a galaxy that is somewhat redder. For galaxies with red enough colors (i.e., -continuum slopes larger than 0.5), it is essentially impossible to robustly select high-redshift galaxy using the dropout criteria, and in fact the only sources that would satisfy the selection criteria would do so because of photometric scatter. This is illustrated in Figures 2 and 4.

To control for this effect, we construct models of the -continuum slope distribution, use these models to add artificial galaxies to real data, select sources from these data, and measure their -continuum slopes in the same way as from the real data. The goal is to construct a model that when “observed” reproduces the distribution of -continuum slopes measured from the data. For the sizes and morphologies of the model galaxies used in the simulations, we start with the pixel-by-pixel profiles of the -dropout sample from the HUDF (Bouwens et al. 2007) and scale their sizes as (for fixed luminosity) to match the observed size-redshift trends (Bouwens et al. 2006; see also Ferguson et al. 2004 and Bouwens et al. 2004). We use our well-tested “cloning” software (Bouwens et al. 1998a,b; Bouwens et al. 2003; Bouwens et al. 2006, 2007) to perform these simulations. For the model LF at 2-6, we adopt the Schechter parameterizations determined by Reddy & Steidel (2009) at and by Bouwens et al. (2007) at .

In performing these simulations, we account for the modest correlation between the -continuum slopes of galaxies and their surface brightnesses. We determined the approximation correlation by examining 192 luminous (1-2 ) galaxies in our GOODS -dropout selection and comparing their observed ’s with their sizes (half-light radii). Any correlation is potentially important, since it could substantially lower the efficiency with which we can select galaxies with very red -continuum slopes (e.g., Figure 4). We find that sources with are 15% larger than galaxies with and 40% larger than galaxies – though there is significant object-to-object scatter (the correlation coefficient between size (half-light radius) and is just 0.3). Fortunately, this correlation seems to only have a modest effect on these selection volumes, decreasing it by only % for the red galaxies and increasing it by only 10% for the blue galaxies.

We experimented with a range of model -continuum slope distributions to determine the effect of object selection and photometric scatter on the observed distribution of -continuum slopes . These experiments were perfomed as a function of magnitude and the input color distribution (with mean -continuum slopes ranging from and and the input color distribution taken to be Gaussian). In general, we found that the recovered distribution of -continuum slopes (after selection) is bluer than the input distributions (by ) at lower luminosities for all of our dropout samples. As expected, we found that noise in the observations broadened the distribution of -continuum slopes somewhat over that present in the input distribution. Figure 4 illustrates how observational selection and noise modifies the input distribution of slopes for a -dropout selection over the HUDF.

We used the simulations described above to quantify changes in this distribution as simple shifts in the mean -continuum slope and scatter. We used the estimated shifts to correct the observed distribution of -continuum slopes for these effects. This corrected distribution of -continuum slopes is plotted in Figure 3 as open red squares, with the scatter shown with the red error bars.

To verify that the corrections we apply in this section are accurate, we compared the mean -continuum slope we estimate near the faint-end (26.5-27) of our shallower GOODS -dropout selections (after correction) with those derived from our much higher S/N HUDF -dropout selections in the same magnitude range. We find that they are in excellent agreement, i.e., at AB mag for the GOODS fields vs. for the HUDF. This suggests that the corrections we apply in this section are reasonably accurate ().

Figure 5.— (top) Determinations of the mean -continuum slope for galaxies with a luminosity of () as a function of redshift (solid red circles). The error bars in are and include possible systematic errors in both the measured colors and corrections for selection effects. The error bars in (with the wider horizontal bars) are for the random errors. The random errors are generally much smaller than those including the estimated systematic errors (shown with the narrower horizontal bars). The right axes give the equivalent extinctions (Calzetti et al. 2000 law) for a given if the input spectrum is a young star-forming galaxy (100 Myr of constant star formation). Published determinations of the mean -continuum slope at (black star: Adelberger & Steidel 2000), at (blue open pentagon: Ouchi et al. 2004), at (solid cyan square: Stanway et al. 2005 and solid magenta triangle: Bouwens et al. 2006), and at (green open squares: Hathi et al. 2008) are also plotted. galaxies are found to have bluer colors at 5-6 than they do at 2-4. (bottom) Determinations of the mean -continuum slope for galaxies with a luminosity of () as a function of redshift (solid red squares). For comparison, our determinations of the mean -continuum slope at (from top panel) are included in this panel as solid red circles (and offset somewhat from the lower luminosity determinations in redshift for clarity). For each of our dropout samples, the lower luminosity galaxies are found to be bluer.
-continuum slope
Mean12 Scatter13
-dropouts
21.73 1.180.170.15 0.32
20.73 1.580.100.15 0.27
19.73 1.540.060.15 0.34
18.73 1.880.050.15 0.36
-dropouts
22.22 1.320.120.15 0.51
21.72 1.330.070.15 0.31
21.22 1.440.040.15 0.44
20.72 1.510.030.15 0.40
20.22 1.580.100.15 0.34
19.72 1.680.090.15 0.35
19.22 1.730.070.15 0.55
18.72 1.930.060.15 0.31
18.22 1.980.050.15 0.5414
17.72 2.030.040.15 0.2915
17.22 1.910.050.15 0.2616
-dropouts
21.90 1.810.220.15 0.45
20.90 1.760.100.15 0.64
19.90 2.170.080.15 0.6617
18.90 2.640.160.15 1.0218
-dropouts
20.76 2.160.150.15 0.41
18.76 2.320.190.15 0.9119
2021
Table 4The mean -continuum slope and scatter of galaxies, as a function of luminosity.22

3.8. Uncertainties and Model Dependencies

Before presenting the distribution of -continuum slopes derived for each of the dropout selections, it is worthwhile to ask ourselves how our determinations may be affected by specific assumptions we make. We have already discussed the effect that errors in our photometry (and zeropoints) would have on the -continuum slope determinations (§3.3). A mag error in the derived colors would result in a change in the derived -continuum slope . In an attempt to minimize such errors, we have exercised great care in obtaining a consistent set of colors across the optical and near-IR passbands for which we have data.

We would expect a similar error in the -continuum slope from the fiducial SEDs we use to convert from the observed colors to -continuum slope . The uncertainties on the derived slope from this conversion are 0.1 for our , , and and 0.2 for our sample (see Appendix A).

The -continuum slopes we derive for our selections also show some dependence upon the model redshift distributions. As discussed in Appendix A, small differences () between the mean redshifts of dropout galaxies in our models and that present in reality have a small effect on the derived -continuum slopes . Errors of in the mean redshifts of these selections shift the derived ’s by 0.02, 0.02, 0.01, and 0.05 for our , , , and dropout selections, respectively. These are much smaller uncertainties than the errors that would result from small systematics in the photometry (or uncertainties in the conversion from observed colors to -continuum slopes).

Similarly, the presence or absence of Ly emission (at 1216Å) in the spectra of sources in our selections is not expected to have a large effect on the measured -continuum slopes themselves. This is because the measurements are made in passbands which are only sensitive to light redward of 1216Å. The only exception to this is for our -dropout selection, but even there for typical Ly EWs (50Årest-frame: Dow-Hygelund et al. 2007; Vanzella et al. 2006; Stanway et al. 2007; Vanzella et al. 2009), the colors would only change by 0.07 – which would make the derived ’s steeper.

Nonetheless, the amount of flux in Ly has an effect on the redshift distribution of the sources selected with our dropout criteria. For galaxies with Ly EWs towards the upper end of the observed range (50Årest-frame: Shapley et al. 2003; Dow-Hygelund et al. 2007; Vanzella et al. 2006; Stanway et al. 2007; Vanzella et al. 2009), the mean redshift of our dropout selections increases by . This changes the derived ’s by 0.06, 0.06, 0.03, and 0.15, respectively, for our four selections (Appendix A). However, since only a small fraction (%) of the star-forming galaxies at 2-6 appear to show Ly emission at these levels (Shapley et al. 2003; Dow-Hygelund et al. 2007; Vanzella et al. 2006; Stanway et al. 2007), the effect of the quoted uncertainties on is likely much smaller than this (i.e., 0.075). Again, this is much smaller than the errors we would expect to result from small systematics in the photometry (or conversions to -continuum slopes ).

Lastly, one might ask whether interstellar absorption features may have an effect on the -continuum slopes estimated from the broadband photometry. Fortunately, these absorption lines appear not to have a big effect on the measured slopes (i.e., ) as demonstrated by Meurer et al. (1999: see §3.4 and Figure 3 from that work) given the wide wavelength range of the broadband filters used to estimate the slopes and the fact that most of these absorption lines occur at 1700Å.

3.9. Results

In Table 4, we present the mean -continuum slopes and scatter observed for our four dropout samples after correction for selection effects and photometric scatter. This distribution is the same as that presented in Figure 3). We assume a minimum systematic error in the mean -continuum slope of 0.15 as a result of possible ( mag) systematics in the measured colors (see §3.3) and small model dependencies in the conversion from the observed colors to -continuum slope (see Appendix A).

There are clear trends that seem to be present in the distributions of -continuum slopes as a function of redshift and luminosity. The first trend is a correlation between the mean -continuum slope and luminosity, in the sense that galaxies become bluer at lower luminosities. This trend is particularly significant for our -dropout and -dropout selections, and fitting the mean vs. luminosity to a line, we find for our -dropout sample and for our -dropout sample (the fit is shown on Figure 3 with the red lines).23 The errors of given in the equations above are our estimates of the systematic errors (see discussion in previous paragraph). We find no significant trend in the width of the -continuum slope distribution as a function of luminosity.

We also find a clear correlation between the mean -continuum slope and redshift, in the sense that higher redshift galaxies are bluer. This is most evident in Figure 5 where the mean -continuum slope is shown at a luminosity of (i.e., ) and (i.e., ) for each of our dropout samples. At 2-4, the mean -continuum slope is 1.5 while at , it is . No change is evident in the width of the -continuum slope distribution – though this width is difficult to quantify for our highest redshift samples due to the small number of sources in these high redshift samples and the large photometric errors of the sources that are available.

4. The -continuum slope Distribution: Direct Implications

In §3, the -continuum slope distribution was derived as a function of both redshift and luminosity. Being able to establish this distribution as a function of these two quantities is an important empirical result and has a number of noteworthy implications, which we will detail in this section. However, before describing these implications, it is prudent to compare the present -continuum slope determinations with those in the literature to put them in context (see §4.1).

One of the most significant implications of these results is for the completeness of dropout selections at , which we discuss in §4.2. Being able to establish these distributions is also important for a determination of the LFs (§4.3). In §4.4, we consider the question of how variations in the -continuum slope likely arise, and we argue that changes in the dust extinction likely dominate the observed variations. Finally, in §4.5, we attempt to connect the sequence we find in the -continuum slope versus luminosity to similar trends found at lower redshift (and over other wavelength baselines).

4.1. Comparison to previous determinations of the -continuum slope

In §3, we derived the distribution of -continuum slopes over a wide range in redshift and luminosity using a very systematic approach while taking advantage of a wide-variety of both deep and wide-area HST data. We presented evidence that the mean -continuum slope is bluer at 5-6 than it is at 2-4 and that this slope is also bluer at lower luminosities at 2-4.

Previously, there had been a variety of different attempts to measure these slopes at specific redshifts or luminosities (typically ), e.g., Steidel et al. (1999), Meurer et al. (1999), Adelberger & Steidel (2000), Ouchi et al. (2004), Stanway et al. (2005), Bouwens et al. (2006), and Hathi et al. (2008). The top panel of Figure 5 provides a summary of many of these previous measurements. Most of the early high-redshift work focused on 2-3 and was based upon dropout selections from the HDF and large ground-based LBG searches (e.g., Steidel et al. 1999; Adelberger & Steidel 2000; Meurer et al. 1999). In those papers, the -continuum slope was found to have a mean value of 1.4 and dispersion of 0.5-0.6 at . No significant correlation of -continuum slope with magnitude was found to AB mag (Adelberger & Steidel 2000; Reddy et al. 2008), though there would appear to be some evidence in the distributions presented by Meurer et al. (1999: e.g., Figure 5 from that paper) that the distribution becomes a little bluer at lower luminosities (i.e., 18.5 AB mag).24

Ouchi et al. (2004) extended these studies to higher redshift by presenting a determination of the -continuum slope distribution at based on a large selection of dropout galaxies from the Subaru Deep Field and Subaru XMM/Newton Deep Field. Ouchi et al. (2004) found that the -continuum slope distribution at was consistent with that determined at and that there was only a marginal trend (not statistically significant) towards bluer slopes at lower luminosities. Papovich et al. (2004), by contrast, working with a selection of dropouts from the GOODS fields, found that galaxies at were slightly bluer in their colors than at in the Hubble Deep Field North and South. At somewhat fainter magnitudes, Beckwith et al. (2006) remarked that the colors of dropouts in the HUDF were very blue in general (with ’s of ) and hence suggested very little dust extinction. The somewhat bluer -continuum slopes found by Beckwith et al. (2006) than by e.g. Ouchi et al. (2004) is not particularly surprising given the substantial differences in the mean luminosities of the two samples. Incidentally, the trend towards bluer -continuum slopes at lower luminosities reported on here is evident in Figure 18 of Beckwith et al. (2006) although it was not specifically noted.

At even higher redshifts, Lehnert & Bremer (2003) found that dropouts had -continuum slopes very close to while later Stanway et al. (2005) and Bouwens et al. (2006) found -continuum slopes of and , respectively, at from a selection of -dropouts in the HUDF. Hathi et al. (2008) used a small sample of galaxies at from the HUDF and larger sample of -dropouts at in an attempt to quantify the change in the -continuum slope as a function of redshift. Hathi et al. (2008) found that the -continuum slope showed only a slight change from (where ) to 5-6 (where ).

Broadly, the picture that has emerged from these studies is that star-forming galaxies become bluer towards higher redshifts and to lower luminosities, but it has been difficult, given possible systematics between the different studies, to quantify the size of the changes. We confirm this overall picture, finding that the mean -continuum slope is bluer at than it is at and bluer at AB mag () than it is at AB mag (). Because of the larger samples and use of a consistent approach in deriving these slopes at all redshifts and luminosities, the differential evolution we measure is more robust than in previous studies.

Looking more specifically at the -continuum slope measurements we have derived at various redshifts and comparing those measurements with those obtained in previous studies, we find very good agreement in general, in almost all cases within the quoted errors. The only significant exception to this is the Hathi et al. (2008) measurements of the -continuum slope at where a mean of was found. This appears to result from the relation Hathi et al. (2008) use to compute the -continuum slope at (i.e., ). This relation does not account for the fact that the NICMOS band extends down to 8000Åand therefore -dropouts partially drop out in the band (i.e., are fainter in the band and hence have redder colors). Comparing with Eq. A5 from Appendix A, we can readily see why the mean -continuum slopes we derive are much steeper (by 0.3). We note that an additional mag shift in relative to previous measurements of (e.g., Bouwens et al. 2006; Hathi et al. 2008) comes from small offsets we make to the -band photometry (§3.3).

Figure 6.— (left) Distribution of -continuum slopes observed at 2-6 for the present Lyman-Break Galaxy selections (solid blue histograms). We only include the observed values for Lyman-Break galaxies 1 mag brighter than the selection limit to avoid being dominated by selection effects. On the top axes are the extinctions (Calzetti et al. 2000 law) that are equivalent to a given if the base spectrum is a young star-forming galaxy ( Myr of constant star formation). The selection volume (§3.7) available for galaxies with -continuum slope and magnitudes mag brighter than the selection limit is shown with the thick dashed lines in each panel (the normalization is arbitrary). While it is more difficult to select galaxies with redder -continuum slopes with a Lyman Break criterion, our simulations (§3.7) indicate that our dropout selections should be capable of identifying modest numbers of galaxies with ’s as red as 0.5. Indeed, such galaxies (with ’s redder than ) are present in our 2-4 LBG selections albeit in small numbers. However, they are conspicuously absent in our selections. This suggests that such sources are exceedingly rare at . Conversely, the modest number of galaxies with red -continuum slopes ’s seen at seem consistent with the small samples of ULIRGs that have been found in the observations (e.g., Daddi et al. 2009). The lack of very blue galaxies is also consistent with expectations from models and suggest that the distributions derived here are reliable. (right) Distribution of -continuum slopes observed for Lyman Break Galaxy selections at 2-4 (solid blue histogram) and for Balmer-Break Galaxy selections at 2-4 (red histogram: Brammer & van Dokkum 2007). The extinctions given on the top axis are as in the left panel (but are not expected to be particularly representative for the somewhat older BBGs). Both distributions plotted are as observed, with no correction for selection or measurement effects. Balmer-Break Galaxy (BBG) selections provide us with an independent measure of the -continuum slope distribution since they do not depend upon their -continuum slope or the brightness of these sources in the -continuum. The fact that the distribution of -continuum slopes is becoming substantially bluer for both selections (and in particular the Brammer & van Dokkum 2007 BBG selection at ) suggests that Lyman-Break selections (sensitive to galaxies with ’s bluer than 0.5: see Figures 2 and 4) should be largely complete at (see §4.2).

4.2. Do LBG selections miss a substantial population of red star-forming galaxies at ?

The Lyman Break galaxy selection technique identifies galaxies at high redshift through simple color criteria. This technique is well established to be an efficient and robust method for identifying star-forming galaxies over a wide range in redshift 2-6 (Steidel et al. 1996; Williams et al. 1996; Bunker et al. 2003; Vanzella et al. 2006; Dow-Hygelund et al. 2007; Vanzella et al. 2009). This technique provides a very complete census of light at high redshift – simply by virtue of the selection wavelength itself. However, it is less efficient for probing the total stellar mass or even the total SFR at high redshift (e.g., van Dokkum et al. 2006). This is because galaxies with the highest stellar masses or SFRs are often either old or dust obscured – making them fainter in the and thus more difficult to select with the LBG technique.

Our interest here is in examining the systemic completeness of LBG selections. We want to determine whether there is a set of galaxies at high redshift that we miss altogether by virtue of the LBG selection technique. Note that this is a very different question from determining whether there is a class of galaxy at high redshift that we select less efficiently because they are faint in the UV (e.g., because of dust or age). In general, we would expect sources to miss our LBG selections if one of the two LBG color criteria failed to hold, i.e., if (1) the sources did not show a strong Lyman break or (2) the sources were too red in their -continuum slope to satisfy the LBG selection. We would expect criterion (1) to always hold for a sufficiently high redshift source, as a result of line blanketing by the Ly forest. However, we might expect criterion (2) to fail if the -continuum slopes of the sources were sufficiently red.

We can attempt to address this question by looking at how many galaxies have -continuum slopes that lie close to the selection limits. If our samples contain a large number of such galaxies, it would suggest that our LBG selections suffer from significant incompleteness near these limits. Figure 2 shows the selection criteria for each of LBG selections and Figure 6 presents the effective search volume (§3.7) for galaxies (dashed black lines on the left-hand panels) as a function of -continuum slope (see also Figure 4). It is quite clear from these figures that our selection criteria is effective in identifying star-forming galaxies at 2-6 with ’s bluer than , and it is striking how much redder this limit is than the -continuum slopes derived for galaxies in our dropout samples at 2-6 (solid blue histograms in Figure 6). The situation is particularly conspicuous for galaxies with -continuum slopes redder than at . While our simulations (§3.7) suggest that galaxies with these colors should show up in our selections if they existed (dashed lines in Figure 6), essentially none are found. This suggests that star-forming galaxies with red -continuum slopes are extremely rare.

Complementary Balmer Break Selections: Another way we can investigate the issue of possible systemic incompleteness in LBG selections is through other selection techniques. One such technique identifies galaxies based upon their rest-frame optical light and searches for a prominent Balmer Break at 3700Årest-frame. Such selections are not surprisingly called Balmer Break Galaxy (BBG) selections and typically identify the oldest and most massive galaxies. Since such galaxies tend to be more chemically evolved, they are frequently more dusty. Considering such selections therefore permit us to evaluate the extent to which our LBG selections may miss redder and more dust-obscured starbursts. Brammer & van Dokkum (2007) performed such a selection at 23 and 34.5 (see right-hand panels on Figure 6) based upon the Faint Infrared Extragalactic Survey (FIRES) data (Labbé et al. 2003; Förster Schreiber et al. 2006) and derived -continuum slopes for sources in their selections from SED fits to the photometry. They found that 67% of the galaxies in their 2-3 selection had -continuum slopes bluer than 0.5, but almost all (%) of the galaxies in their 3-4.5 sample had such blue slopes. This is highly encouraging for LBG selections given the independent nature of Balmer Break selections – suggesting that LBG selections may be largely complete.

By contrast, the fact that 33% of the galaxies in the Brammer & van Dokkum (2007) 2-3 BBG sample have ’s redder than 0.5 suggests that completeness could be somewhat of a concern for LBG selections at , and in fact it is well known that at 2-3, there is a substantial population of dust-obscured galaxies undergoing vigorous star formation (e.g., Hughes et al. 1998; Barger et al. 1999; Chapman et al. 2005; Labbé et al. 2005; Papovich et al. 2006; Pope et al. 2006).

Searches for Balmer Break galaxies at have also been conducted (e.g., Dunlop et al. 2007; Wiklind et al. 2008: see also Rodighiero et al. 2007 and Mancini et al. 2009). As with the Brammer & van Dokkum (2007) study, these searches would seem to be relevant to the present discussion we are having about the completeness of LBG selections for star-forming galaxies at high redshift. Unfortunately, there is a wide diversity of search results in this area that make it difficult to draw clear conclusions. Wiklind et al. (2008) report 11 plausible Balmer Break galaxies at 5-7 over the CDF South GOODS field, 7 of which are detected at with MIPS, while Dunlop et al. (2007) find no Balmer Break galaxies over the same redshift range, in their analysis of the same field. While the MIPS detections for 7 of the 11 5-7 Wiklind et al. (2008) BBG candidates may be interpreted as due to obscured AGN, we believe a much more likely explanation is due to PAH emission, as Chary et al. (2007b) argue for HUDF-JD2 (Mobasher et al. 2005). The other BBG candidates from Wiklind et al. (2008) may have redshifts of 5-7, but may also have lower redshifts.

Implications: Putting together the present LBG selection results at 2-6 with those from Balmer Break selections over the same range and other results in the literature, we find clear evidence that the galaxy population at high redshift is increasingly blue as one moves out to high redshift, and therefore high-redshift LBG selections seem likely to be increasingly complete (and consequently suffer much less from systematic incompleteness). These trends are very clear in LBG selections at 2-6, and since there is no reason to suppose that these trends come from various selection biases (not only do the sources have ’s very far from the selection limits, but the effect of observational selection are corrected for), it seems reasonable to take the observed evolution at face value. Supporting evidence comes from a similar evolution seen in complementary Balmer Break selections (at least according to Brammer & van Dokkum 2007). Previously, Bouwens et al. (2007: §4.1) have discussed this in the context of -dropout selections.

Obviously we would expect galaxies at very high redshifts (5-6) to have much bluer -continuum slopes than at , because of the much smaller time baseline over which to produce metals and dust, as well as the shorter dynamical time scales of galaxies at (and hence much reduced ages). We would also expect this result based upon the evolution of the LF. Since the characteristic luminosity of galaxies in the UV becomes progressively smaller as we move to higher redshifts (Dickinson et al. 2004; Shimasaku et al. 2005; Bouwens et al. 2006; Yoshida et al. 2006; Bouwens et al. 2007; Oesch et al. 2009), we would expect to find fewer and fewer galaxies at high redshift with very large SFRs. Since dust extinction is well correlated with the SFR (e.g., Wang & Heckman 1996; Hopkins et al. 2001; Martin et al. 2005; Reddy et al. 2006; Buat et al. 2007; Zheng et al. 2007), we would not expect to find many galaxies at with substantial dust extinction. Finally there is the evolution seen in the relationship between SFR and dust extinction. From to to , it has been found that the effective dust extinction for a given star formation rate decreases quite strongly as we move out to higher redshifts (Reddy et al. 2006; Buat et al. 2007; Burgarella et al. 2007). Each of these considerations suggest that galaxies at high-redshift should be almost uniformally very blue and LBG selections very complete.

Nonetheless, we know there is a substantial population of luminous, dust obscured galaxies at (e.g., Hughes et al. 1998; Barger et al. 1999; Chapman et al. 2005; Labbé et al. 2005; Papovich et al. 2006; Pope et al. 2006) and extending out to redshifts as high as (e.g., Capak et al. 2008; Daddi et al. 2009; Wang et al. 2009; Dannerbauer et al. 2008). While some authors (e.g., Daddi et al. 2009) make the suggestion that these populations might be quite significant and contribute substantially to the overall SFR density at , we would argue that the role of these galaxies is likely much more modest in scope and they do not provide a large fraction of the SFR density, particularly at . We discuss and quantify this point in §6.2 (see also Table 5, Table 7, Figure 13, and Figure 14).

4.3. Relevance for LF Determinations

As shown in Figure 2, Figure 4 and Figure 6 (and as discussed in the previous section), the effective selection volume for star-forming galaxies at 2-6 is very sensitive to its -continuum slope and luminosity and can vary by factors of (see also discussion in Sawicki & Thompson 2006a or Beckwith et al. 2006). Quantifying the distribution of -continuum slopes for star-forming galaxies (as a function of both luminosity and redshift) is therefore an important first step in determining the LFs at high redshift.

Given this impact, it has been somewhat surprising that different teams have made very different assumptions about the distribution of -continuum slopes in their determinations of the LFs. Some teams have adopted -continuum slopes of (Sawicki & Thompson 2006a; Yoshida et al. 2006), other teams have assumed continuum slopes of (Beckwith et al. 2006), and yet other teams have adopted luminosity-dependent -continuum slopes (Bouwens et al. 2007). Such differences have only added to the large dispersion seen in LF determinations at high redshift (see Figures 10, 11, and 13 from Bouwens et al. 2007).

As a result of the present quantification of -continuum slopes at high redshift (vs. redshift and luminosity), we now have a relatively uniform set of assumptions they can be used for quantifying the LFs at 2-6 and in extending such determinations to .

Figure 7.— -continuum slope (1600Å to 2300Å rest-frame baseline) predicted for specific fractional changes in the mean age, metallicity, or dust extinction of high-redshift star-forming galaxies from our fiducial stellar population model. For our fiducial model (where ), we assume Myr, Myr, , , and a Salpeter IMF (where the star formation history is parametrized as ) from the Papovich et al. (2001) fits to -dropouts from the WFPC2 HDF North. In modifying our fiducial model to have younger ages, we make changes to both and . A factor of 2 (0.3 dex) change (dahsed line) in the mean dust content, age, or metallicity of the star-forming galaxy population at high redshift shifts the -continuum slope by , , and , respectively. The effect of the stellar IMF on is not shown here, but results in only modest changes in , i.e., for a shift in the slope of the IMF. It seems clear that changes in the mean dust content of galaxies at high-redshift has the biggest effect on the -continuum slope – though some change is also likely the result of differences in the average age. Nonetheless, since stellar population modelling of dropout galaxies (Stark et al. 2009) suggest that age shows little dependence on redshift or luminosity (at least in the median), it seems likely that age only plays a minor role () in driving the observed trends (§4.4).

4.4. Interpreting variations in the -continuum slope

Having used the available observations to derive the distribution of -continuum slopes for high-redshift galaxies as a function of luminosity and a range in redshift (Table 4 and Figure 3), we might ask ourselves what this teaches us about the physical properties of high-redshift galaxies. Since the -continuum slope is purely an observational measure of the shape of the spectrum of high-redshift galaxies and can be affected by a variety of different physical conditions (including dust, age, metallicity, etc.), we cannot use the above measurements to make any unambiguous observational inferences about the nature of high-redshift galaxies.

Nevertheless, we will argue that the simplest way to explain most of observed differences in is through changes in the dust content of galaxies. While there are undoubtably variations in other quantities such as the age, metallicity, or IMF of the stars that affect these slopes, we will argue that variations in the dust content of galaxies likely have the largest effect on the observed -continuum slopes and we can use the observed variations in these slopes to make inferences about changes in the effective dust extinction as a function of galaxy luminosity and redshift.

We note that we would not expect the presence or luminosity of an AGN to have a big effect on these slopes, given that the observed incidence of such sources at 2-4 is just % for moderately bright galaxies at (e.g., Nandra et al. 2002) and there is little evidence they are more frequent or important at higher redshift or lower luminosities (e.g., Lehmer et al. 2005; Ouchi et al. 2008).

To ascertain which of the aforementioned factors (e.g., the overall dust content and properties, the age, metallicity, or IMF of a stellar population) are likely to be the most important for interpreting the variations we see in the -continuum slopes , we consider the effect that changes in many of the above quantities would have on the -continuum slope . For simplicity, we model the star formation history of galaxies with a simple model with Myr (e.g., as in Papovich et al. 2001) and equal to the age. We also take the metallicity to be fixed and not evolve over this entire history. Finally, we implement the dust extinction by applying the Calzetti et al. (2000) law to the SED resulting from the stellar population models (Bruzual & Charlot 2003). Using the Papovich et al. (2001) modelling of -dropouts in the HDF North as a guide, we adopt Myr, Salpeter IMF, , and as our fiducial parameters.

We then change the age, metallicity, and dust content of this model by various factors and calculate the effect it would have on the -continuum slope predicted for a galaxy, as measured over the baseline 1600Åto 2300Å. The results are shown in Figure 7. From this figure, it is clear that the largest changes in the -continuum slope come from variations in the dust content of galaxies and the other parameters have a smaller effect. For factor of 2 changes in the age, metallicity, and dust, we estimate that the -continuum slope change by 0.05, 0.1, 0.35, respectively.

Of course, we also see that age has a modest effect on the observed -continuum slope . This is well documented in the literature (e.g., Bell 2002; Kong et al. 2004; Cortese et al. 2006; Panuzzo et al. 2007). Despite this sensitivity of to age, no significant change are found in the median ages of star-forming galaxies from to (Stark et al. 2009), and little changes are found in the median ages as a function of luminosity (there is a hint that lower luminosity galaxies may be 1.5 younger). Galaxies are found to have median ages of 150 Myr. This suggests that the effect that systematic changes in the age of the stellar populations on the mean is at most modest (). Moreover, even if we ignore these observational results, it seems unlikely that the average ages of star-forming galaxies would increase more rapidly than some multiple of the dynamical time in a galaxy. From theory (e.g., Mo et al. 1998), the dynamical times scale as – which is a factor of (0.5 dex) from to . For such a change in age, the change in would be , which is small compared to the observed differences – where is 0.5-1.0.

Metallicity has an even smaller effect (by a factor of 8) on the observed -continuum slopes than either dust or age do, so we would require very large variations in the metallicity to affect the -continuum slope in any sizeable way. However, since any substantial change in metallicity would almost certainly be accompanied by a similar change in the dust content (given the correlation between these two quantities), we would again be left with a situation where the changes in resulting from metallicity would be completely overwhelmed by changes in resulting from dust.

Changing the IMF of galaxies only appears to have a modest effect on the -continuum slope . For example, changing the slope of the IMF by 0.5 only changes by 0.1. Moreover, for a steep enough -continuum slope and a top heavy enough IMF, there is very red nebular continuum emission (resulting from the ionizing radiation: Schaerer 2002; Venkatesan et al. 2003; Schaerer 2003; Zackrisson et al. 2008; Schaerer & de Barros 2009) that more than offsets the very blue spectrum from the stars themselves (Figure 1 from Schaerer & Pello 2005).

In summary, we expect that the observed variations in the mean -continuum slope to be largely the result of changes in the mean dust extinction.

4.5. Connection to trends found in star-forming galaxies at :

One of the most salient trends we found in the -continuum slopes derived for -dropout and -dropout galaxies was the presence of a strong correlation between the -continuum slope and luminosity (see also Meurer et al. 1999). As detailed in §3.9, the mean -continuum slope for and galaxies decreases (becomes bluer) by 0.200.04 and 0.150.01, respectively, for each magnitude we reach fainter in luminosity.

It seems reasonable to imagine that such trends might also be present in even lower redshift samples or in other colors for 2-4 samples. In fact, a strong correlation between -optical colors and rest-frame optical magnitude has been found for star-forming galaxies in the “blue” cloud (e.g., Papovich et al. 2001; Baldry et al. 2004; Papovich et al. 2004; Wyder et al. 2007; Labbé et al. 2007), in the sense that more luminous galaxies are redder. This is very similar to the relationship we find between the -continuum slope and luminosity. Looking at the comparison more quantitatively, the slope of the color-magnitude relationship Labbé et al. (2007) measure, for example, is equivalent to / of . This is similar to the slope we estimate at for our -dropout selection (see §3.9).

Finally, it is worthwhile to remark on the physical origin of this slope – which could plausibly be explained by changes in either the mean age or the dust content of galaxies as a function of luminosity. In the previous section, we argued that the simplest way of accommodating the sizeable change () in the -continuum slope with luminosity was through a change in the dust content of the galaxy population. Explaining the change in with age would require very large changes (factor of 10 changes in the mean galaxy age) – which seem contrary to the modest changes in age noted by Stark et al. (2009) as a function of luminosity.

Labbé et al. (2007) also argue that the slope in the color-magnitude relationship is predominantly the result of a variation in the dust content. Labbé et al. (2007) draw this conclusion based upon an examination of the slope of the color-magnitude relationship vs. wavelength and through a detailed examination of the Nearby Galaxy Field Sample (Jansen et al. 2000). This suggests that the color-magnitude relationship we observe for star-forming galaxies may simply be another manifestation of the well-known mass-metallicity relationship observed at (e.g., Tremonti et al. 2004; Erb et al. 2006a; Maiolino et al. 2008).

5. Inferred Dust Extinction

In §4.4-§4.5, we argued that the most likely interpretation of the systematic changes in the mean -continuum slope (as a function of redshift or luminosity) is a change in the dust content of galaxies. Given the sizeable (factor of 5) dust corrections inferred at for luminous galaxies (e.g., Reddy et al. 2006; Meurer et al. 1999; Erb et al. 2006b; Reddy & Steidel 2004), any changes in the estimated dust corrections would have a substantial effect on the SFR densities inferred at earlier redshifts and hence its evolution across cosmic time.

In this section, we use the measured -continuum slope distribution at 2-6 to estimate the dust corrections for -bright galaxies at 2-6. We begin by describing the formula we use to estimate dust extinction using the measured -continuum slopes (§5.1). Then, we discuss the likely significance of the observed trends in -continuum slope and dust extinction, versus luminosity (§5.2) and redshift (§5.3). In §5.4, we combine the derived trends in dust extinction with the observed LFs to calculate dust corrections, to various limiting luminosities. We then use the typical dust corrections for specific luminosity galaxies to estimate the bolometric luminosities of specific luminosity galaxies in the (§5.5). Finally, in §5.6, we discuss the few well-cited cases of dusty high-redshift galaxies and explain why they do not alter the simple picture we lay out in this section.

5.1. Inferred Dust Extinction

To estimate the effective dust extinction for galaxies in our various samples and as a function of redshift, we rely upon the correlation found between dust extinction and the -continuum slope at (Meurer et al. 1999):

(1)

where the dust extinction here is specified at rest-frame 1600Åand where is the -continuum slope measured from 1300Åto 2600Å. This relationship has been shown to be a reasonable predictor of the actual dust extinction in galaxies at , , for all but the youngest (Reddy et al. 2006; Siana et al. 2008; Siana et al. 2009) and most obscured starburst galaxies (e.g., Meurer et al. 1995; Meurer et al. 1999; Burgarella et al. 2005; Laird et al. 2005; Reddy et al. 2006), and therefore it is reasonable for us to use this equation to estimate extinction in each of our high-redshift dropout samples. This technique has already been employed in several previous studies estimating the SFR density at 2-6 (e.g., Adelberger & Steidel 2000; Meurer et al. 1999; Bouwens et al. 2006; Stark et al. 2007).

Of course, there are many reasons for suspecting that Eq 1 may not work very well at early times, both because of the very different ages and metallicities of stellar populations and because the dust itself may have very different characteristics. After all, the standard mechanism for forming dust in the envelopes of AGB stars should not work at because the universe is not old enough at these redshifts to have produced AGB stars. Dust in galaxies has therefore been suggested to form by another mechanism, most frequently in SNe ejecta (e.g., Maiolino et al. 2004; Maiolino 2006). Eq. 1 is also known to fail at for a few lensed sources (e.g., MS1512-cB58 [Siana et al. 2008] or the Cosmic Eye [Siana et al. 2009]) and for very young (100 Myr) star-forming galaxies at (Reddy et al. 2006).

To use Eq. 1, we need to convert the -continuum slopes we have derived (which have a baseline 1600Åto 2300Å) to that appropriate for Eq. 1 (where assumes a slightly more extended baseline 1300Åto 2600Å). We therefore explored the value of the -continuum slope over these two different baselines for a wide variety of star formation histories and values of the dust extinction. We found that they are very similar in general, with an absolute difference generally 0.2. While there is clearly latitude in the precise relationship we use to convert the measured -continuum slopes to this new wavelength, perhaps the least arbitrary set of SEDs to use to perform this conversion are those we use in Appendix A to estimate from the observed colors. For those SEDs (which assume a star formation history, Myr, Myr, , Calzetti et al. 2000 extinction law, and a Salpeter IMF), we derive the following relationship . While obviously there are some uncertainties in using this conversion, there are at least as many uncertainties in using Eq. 1 to estimate the dust obscuration in high redshift galaxies, and so we will ignore these uncertainties in the subsequent discussion.

Effective Extinction
Sample
Using Meurer et al. (1999) Relationship25
-dropouts 6.026 3.8 2.8
-dropouts 5.8 3.8 2.8
-dropouts 2.7 1.6 1.4
-dropouts 1.6 1.3 1.2
2728
Table 5The effective dust extinction (at 1600Å) estimated for the LBG population integrated down to various luminosities (see also Figure 8).

5.2. Dependence of Dust extinction on Luminosity

We will first discuss the dependence of the -continuum slope and inferred dust extinction on the luminosity. We find that the -continuum slope shows a strong dependence on the luminosity at both and . This suggests that dust extinction is positively correlated with the observed luminosity, in the sense that more luminous galaxies are more dust obscured.

At face value, this trend would appear to be contrary to what Adelberger & Steidel (2000) and Reddy et al. (2008) found in their examination of -dropouts – where no correlation was found between the -continuum slope and the observed luminosity over the range () to (). This is a somewhat brighter range than we probe with good statistics, suggesting that dust extinction (and the -continuum slope ) may be a weaker function of the observed magnitude at () than it is over the whole range probed here. Indeed, using this same magnitude baseline () and a t test, we only find the correlation between and luminosity to be significant at 75% confidence from our own -dropout sample. Thus, while the correlation would appear to be weaker for the more luminous sources, the existence of a correlation between and observed magnitude over the entire magnitude range probed here is very significant (i.e., 4 result).