UV CONTINUUM SLOPE AND DUST OBSCURATION FROM z ∼ 6 TO z ∼ 2: THE STAR FORMATION RATE DENSITY AT HIGH REDSHIFT^*

We used SExtractor (Bertin & Arnouts 1996) to perform object detection and photometry. Object detection was performed using the square root of the χ² images (Szalay et al. 1999; essentially a co-addition of the relevant images) constructed from all ACS images with coverage redward of the dropout band (e.g., V₆₀₆i₇₇₅z₈₅₀ in the case of a B₄₃₅-dropout selection) or the WFPC2 images redward of the dropout band in the case of our U₃₀₀ dropout selection (where the B₄₅₀V₆₀₆I₈₁₄ bands are used). For clarity, the χ² image is equal to

$$\begin{eqnarray*} &&\Sigma _{k} (I_k (x,y) / N_k)^2, \end{eqnarray*}$$

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

Our procedure for measuring colors depends upon the point-spread function (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 H₁₆₀-band. Then, we use the same apertures described above if the area in those apertures is ⩾0.38 arcsec² (i.e., ⩾07-diameter apertures). This only applies for the largest sources in our selection. Otherwise (and in most cases), the colors are measured in fixed 07-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 χ² 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 ∼0.6 mag for the first step and ∼0.1–0.2 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.

We select star-forming galaxies over the redshift range z ∼ 6 to z ∼ 2 using the well-established dropout technique (e.g., Steidel et al. 1996; Bunker et al. 2003; Vanzella et al. 2006, 2009; Dow-Hygelund et al. 2007; Stanway et al. 2007). Our U-dropout criterion is similar to those used by Bouwens et al. (2003) while our B, V, and i dropout criteria are almost identical to those adopted by Bouwens et al. (2007) except that our V-dropout criterion uses a sharper (i₇₇₅ − z₈₅₀) < 0.6 cut to exclude dusty z ∼ 1 interlopers. These criteria are

$$\begin{eqnarray*} &&(U_{300}-B_{450} > 1.1)\wedge (B_{450}-I_{814} < 1.5) \\ &&\wedge\, (U_{300}-B_{450} > 0.66 (B_{450} - I_{814}) + 1.1) \end{eqnarray*}$$

for our U-dropout sample

$$\begin{eqnarray*} (B_{435}-V_{606} > 1.1)& \wedge &(B_{435}-V_{606} > (V_{606}-z_{850})+1.1) \\ &\wedge &\, (V_{606}-z_{850}<1.6) \end{eqnarray*}$$

for our B-dropout sample,

$$\begin{eqnarray*} &&[(V_{606}-i_{775} > 0.9(i_{775}-z_{850})) \vee (V_{606}-i_{775} > 2)] \\ &&\wedge \,(V_{606}-i_{775}>1.2) \wedge (i_{775}-z_{850}<0.6) \end{eqnarray*}$$

for our V-dropout sample, and

$$\begin{eqnarray*} &&(i_{775}-z_{850}>1.3) \wedge ((V_{606}-i_{775} > 2.8) \vee ({\rm S}/{\rm N}(V_{606})<2)) \end{eqnarray*}$$

for our i-dropout sample, where ∧ and ∨ represent the logical AND and OR symbols, respectively, and S/N represents the signal-to-noise ratio. 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).

Zoom In Zoom Out Reset image size

To ensure that our selections did not contain any spurious sources, we required that sources in our U, B, V, and i dropout selections be ⩾5.5σ detections in the I₈₁₄, z₈₅₀, z₈₅₀, and z₈₅₀ 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 (≲2% of the relevant sources).

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

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 nonlinearity in its count rate. While there are now procedures (e.g., de Jong et al. 2006) to correct for these nonlinearities, 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. Second, the colors we measure 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 (22 < z_850,AB < 26) point-like stars (SExtractor stellarity >0.8) over the HUDF/GOODS fields, and then compared their observed colors with that expected from the Pickles (1998) atlas of stellar spectra. Given the very limited range in spectral energy distribution (SED) shape found in real stars, these comparisons should allow us to make reasonable corrections to the J- and H-band photometries. We determined the average differences between the measured and best-fit near-IR fluxes (both in the J₁₁₀ and H₁₆₀ bands) and found corrections of −0.13 mag and −0.07 mag for the J₁₁₀ and H₁₆₀ bands, respectively.

To ensure that these corrections were as accurate as possible, we also identified a sample of ∼60 highest S/N B-dropouts from our search fields (GOODS and HUDF) and then fit their observed SEDs with Bruzual & Charlot (2003) τ models. Since the 1.1 μm J- and 1.6μm H-band fluxes of these galaxies probe the light blueward of the age sensitive break at 3600 Å (probing 2300 Å and 3300 Å rest frame, respectively), it should be reasonable to use the available i (1600 Å rest frame) and z-band (1900 Å rest frame) fluxes to establish a UV-continuum slope β and extrapolate to these redder wavelengths. For our fiducial model-fit parameters, we assumed a e^−t/τ star formation history (with t = 70 Myr, τ = 10 Myr), a Salpeter initial mass function (IMF), and [Z/Z_☉] = −0.7, and let the dust extinction and redshift be free parameters (the fiducial model considered by Papovich et al. 2001 in their stellar population modeling of z ∼ 3 U-dropouts). Comparing the predicted near-IR fluxes (from the models) with the measured fluxes, we computed corrections to the J- and H-band fluxes. The corrections were −0.08 ± 0.02 mag and 0.04 ± 0.2 mag, respectively (the uncertainties were derived using other SED models (the Appendix) to determine these corrections). Because of the size of the uncertainties in the H-band corrections (from the z ∼ 4 B-dropouts), we gave those results very low weight overall for the H-band correction.

Combining the constraints we derived from our sample of stars and z ∼ 4 Lyman Break Galaxies (LBGs)—which were consistent within 0.05 mag—we derived a −0.10 mag correction to the measured J₁₁₀-band magnitudes. For the NICMOS H₁₆₀-band, we derived a −0.05 mag correction from the above fits—using the SED fit results to stars as our primary constraint.⁸ The measured J₁₁₀- and H₁₆₀-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 UV-continuum slopes β is given in Section 3.8.

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

We estimate the UV-continuum slopes β 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 UV wavelengths were as very close to 1650 Å and 2300 Å as 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 1650–2300 Å is a good match to the wavelength range used for studies of the dust extinction in galaxies at z ∼ 0 (e.g., Meurer et al. 1995, 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 B-dropout sample, for example, we use the i₇₇₅ − z₈₅₀ colors for these estimates to take advantage of the much higher quality (deeper, wider area) ACS data available. For our V-dropout sample, we use z₈₅₀ − (J₁₁₀ + H₁₆₀)/2 colors predominantly for these estimates, but we also make use of z₈₅₀ − H₁₆₀ colors for these estimates to take advantage of the large areas within the GOODS fields which have only NICMOS H₁₆₀-band coverage (primarily from the C. Conselice et al. 2009, in preparation, NICMOS program; see the light orange squares in Figure 1).

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

To correct for these passband and wavelength-dependent effects, we base our estimates of β on a small number of more realistic SEDs calculated from stellar population models (Bruzual & Charlot 2003). These model SEDs are calibrated to have UV-continuum slopes β of −2.2, −1.5, and −0.8 over the wavelength range and assume a e^−t/τ star formation history (t = 70 Myr, τ = 10 Myr, [Z/Z_☉] = −0.7) 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 UV-continuum slope β are presented in the Appendix. 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 UV-continuum slopes β of ∼0.1–0.2.

In order to accurately quantify the distribution of UV-continuum slopes β 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 UV-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 allowed us to find a sufficient number of sources to map out the distribution of UV-continuum slopes (which is helpful for determining the mean and 1σ scatter). In Table 3, we provide a list of the fields we use to quantify the distribution of UV-continuum slopes at a given luminosity and redshift.

In Figure 3, we plot the UV-continuum slopes β observed for galaxies in our four dropout samples versus their UV luminosities. The mean UV-continuum slope β and 1σ for galaxies is also included in this figure (blue squares) as a function of luminosity for each of the dropout samples. We adopt finer 0.5 mag bins for our B-dropout selections than the 1.0 mag bins we adopt for our U and V dropout selections or 2.0 mag bins we adopt for our i dropout selection. The width of the bins depends upon the number of sources present in each of our dropout selections. The UV luminosity as shown on this diagram is the geometric mean of the two UV luminosities used to establish the UV slope (see Table 2). Use of the geometric mean of the two UV luminosities is preferred here for examining the correlation of β 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 UV-continuum slope β 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 UV-continuum slope β as a function of the redder band.

Zoom In Zoom Out Reset image size

A clear trend is seen toward bluer UV-continuum slopes β at lower luminosities. This trend is similar to that found by Meurer et al. (1999) in their analyses of z ∼ 2.5 U-dropout galaxies in the HDF North and also noted by Overzier et al. (2008) in their analysis of z ∼ 4 g-dropouts in the TN1338 field (Figure 4 from that work).

In Section 3.5, we considered galaxies from both wide-area and deep surveys to establish the distribution of UV-continuum slopes β over a wide range in luminosity. We can increase the number of sources in our lower luminosity z ∼ 4–6 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 z∼ 4–6 from the available HST ACS data.

These clusters substantially amplify the flux from distant galaxies, making it possible to measure the UV-continuum slope of galaxies at lower luminosities than would otherwise be possible. Of course, in the case of B-dropout selections from very deep optical data like HUDF, we are able to reach to the same intrinsically luminosities as we reach in the B-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 z∼ 4–6 galaxies we find behind galaxy clusters are ∼ 5–10. We do not include the UV-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 preparation). 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 preparation).

We include the UV-continuum slope determinations of these galaxies with those estimated from our field samples in Figure 3 (black points on the z ∼ 4 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 UV-continuum slopes β derived from our field samples and those inferred from our selections around clusters.

The distribution of UV-continuum slopes that we derive is quite clearly affected by the manner in which sources are selected. This is because dropout criteria include galaxies with bluer UV-continuum slopes more efficiently (i.e., over a larger range in redshift) than they do for galaxies with other colors. This is because 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., UV-continuum slopes β larger than 0.5), it is essentially impossible to robustly select a 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.

Zoom In Zoom Out Reset image size

To control for this effect, we construct models of the UV-continuum slope β distribution, use these models to add artificial galaxies to real data, select sources from these data, and measure their UV-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 UV-continuum slopes β 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 z ∼ 4 B-dropout sample from the HUDF (Bouwens et al. 2007) and scale their sizes as (1 + z)⁻¹ (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, 1998b, 2003, 2006, 2007) to perform these simulations. For the model UV LF at z ∼ 2–6, we adopt the Schechter parameterizations determined by Reddy & Steidel (2009) at z ∼ 2.5 and by Bouwens et al. (2007) at z∼ 4–6.

In performing these simulations, we account for the modest correlation between the UV-continuum slopes β of galaxies and their surface brightnesses. We determined the approximation correlation by examining 192 luminous (∼1–2 L*_UV) galaxies in our GOODS B-dropout selection and comparing their observed β'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 UV-continuum slopes β (e.g., Figure 4). We find that sources with β ≳ −1 are ∼15% larger than galaxies with −2 ≲ β ≲ −1 and ∼40% larger than galaxies β ≲ −2—though there is significant object-to-object scatter (the correlation coefficient between size (half-light radius) and β is just ∼0.3). Fortunately, this correlation seems to only have a modest effect on these selection volumes, decreasing it by only ∼10% for the red galaxies and increasing it by only ∼10% for the blue galaxies.

We experimented with a range of model UV-continuum slope β distributions to determine the effect of object selection and photometric scatter on the observed distribution of UV-continuum slopes β. These experiments were performed as a function of magnitude and the input color distribution (with mean UV-continuum slopes ranging from −1.5 and −2.2 and the input color distribution taken to be Gaussian). In general, we found that the recovered distribution of UV-continuum slopes β (after selection) is bluer than the input distributions (by Δβ ∼ 0.1) at lower luminosities for all of our dropout samples. As expected, we found that noise in the observations broadened the distribution of UV-continuum slopes β somewhat over that present in the input distribution. Figure 4 illustrates how observational selection and noise modifies the input distribution of slopes for a B-dropout selection over the HUDF.

We used the simulations described above to quantify changes in this distribution as simple shifts in the mean UV-continuum slope and 1σ scatter. We used the estimated shifts to correct the observed distribution of UV-continuum slopes β for these effects. This corrected distribution of UV-continuum slopes is plotted in Figure 3 as open red squares, with the 1σ scatter shown with the red error bars.

To verify that the corrections we apply in this section are accurate, we compared the mean UV-continuum slope β we estimate near the faint end (z_850,AB∼26.5-27) of our shallower GOODS B-dropout selections (after correction) with those derived from our much higher S/N HUDF B-dropout selections in the same magnitude range. We find that they are in excellent agreement, i.e., −1.81 ± 0.04 at −19.2 AB mag for the GOODS fields versus −1.73 ± 0.07 for the HUDF. This suggests that the corrections we apply in this section are reasonably accurate (Δβ ≲ 0.1).

Before presenting the distribution of UV-continuum slopes β 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 zero points) would have on the UV-continuum slope determinations (Section 3.3). A ∼0.05 mag error in the derived colors would result in a ∼0.15 change in the derived UV-continuum slope β. 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 UV-continuum slope β from the fiducial SEDs we use to convert from the observed colors to UV-continuum slope β. The uncertainties on the derived slope from this conversion are ≲0.1 for our z ∼ 2.5, z ∼ 5, and z ∼ 6 and ≲0.2 for our z ∼ 4 sample (see the Appendix).

The UV-continuum slopes we derive for our selections also show some dependence upon the model redshift distributions. As discussed in the Appendix, small differences (Δz ∼ 0.1) between the mean redshifts of dropout galaxies in our models and that present in reality have a small effect on the derived UV-continuum slopes β. Errors of Δz ∼ 0.1 in the mean redshifts of these selections shift the derived β's by 0.02, 0.02, 0.01, and 0.05 for our z ∼ 2.5 U, z ∼ 4 B, z ∼ 5 V, and z ∼ 6 i dropout selections, respectively. These are much smaller uncertainties than the Δβ ∼ 0.15 errors that would result from small systematics in the photometry (or uncertainties in the conversion from observed colors to UV-continuum slopes).

Similarly, the presence or absence of Lyα emission (at 1216 Å) in the spectra of sources in our selections is not expected to have a large effect on the measured UV-continuum slopes β themselves. This is because the measurements are made in passbands which are only sensitive to light redward of 1216 Å. The only exception to this is for our z ∼ 6 i-dropout selection, but even there for typical Lyα equivalent widths (EWs; ∼50 Å rest frame; Dow-Hygelund et al. 2007; Vanzella et al. 2006, 2009; Stanway et al. 2007), the J₁₁₀ − H₁₆₀ colors would only change by ∼0.07—which would make the derived β's ∼0.17 bluer. Nonetheless, the amount of flux in Lyα has an effect on the redshift distribution of the sources selected with our dropout criteria. For galaxies with Lyα EWs toward the upper end of the observed range (∼50 Å rest frame; Shapley et al. 2003; Dow-Hygelund et al. 2007; Vanzella et al. 2006, 2009; Stanway et al. 2007), the mean redshift of our dropout selections increases by Δz ∼ 0.3. This changes the derived β's by 0.06, 0.06, 0.03, and 0.15, respectively, for our four selections (the Appendix). However, since only a small fraction (≲50%) of the star-forming galaxies at z ∼ 2–6 appear to show Lyα 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 β 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 UV-continuum slopes β).

Lastly, one might ask whether interstellar absorption features may have an effect on the UV-continuum slopes β estimated from the broadband photometry. Fortunately, these absorption lines appear not to have a big effect on the measured slopes (i.e., Δβ ≲ 0.1) as demonstrated by Meurer et al. (1999; see Section 3.4 and Figure 3 from Meurer et al. 1999) 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 ≲1700 Å.

In Table 4, we present the mean UV-continuum slopes and 1σ 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 UV-continuum slope β of 0.15 as a result of possible (∼0.05 mag) systematics in the measured colors (see Section 3.3) and small model dependencies in the conversion from the observed colors to UV-continuum slope β (see the Appendix).

There are clear trends that seem to be present in the distributions of UV-continuum slopes β as a function of redshift and UV luminosity. The first trend is a correlation between the mean UV-continuum slope and UV luminosity, in the sense that galaxies become bluer at lower UV luminosities. This trend is particularly significant for our z ∼ 2.5 U-dropout and z ∼ 4 B-dropout selections, and fitting the mean β versus UV luminosity to a line, we find β = (−0.20 ± 0.04)(M_UV,AB + 21) − (1.40 ± 0.07 ± 0.15) for our z ∼ 2.5 U-dropout sample and β = (−0.15 ± 0.01)(M_UV,AB + 21) − (1.48 ± 0.02 ± 0.15) for our z ∼ 4 B-dropout sample (the fit is shown in Figure 3 with the red lines).⁹ The Δβ errors of ±0.15 given in the equations above are our estimates of the systematic errors (see discussion in the previous section). We find no significant trend in the width of the UV-continuum slope β distribution as a function of luminosity.

We also find a clear correlation between the mean UV-continuum slope β and redshift, in the sense that higher redshift galaxies are bluer. This is most evident in Figure 5 where the mean UV-continuum slope β is shown at a luminosity of L*_{z = 3} (i.e., M*_UV ∼ −21) and 0.15L*_{z = 3} (i.e., M*_UV ∼ −19) for each of our dropout samples. At z ∼ 2–4, the mean UV-continuum slope is ∼−1.5 while at z ≳ 5, it is ≲−1.8. No change is evident in the width of the UV-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.

Zoom In Zoom Out Reset image size

In Section 3, we derived the distribution of UV-continuum slopes β over a wide range in redshift and UV 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 UV-continuum slope β is bluer at z ∼ 5–6 than it is at z ∼ 2–4 and that this slope is also bluer at lower UV luminosities at z ∼ 2–4.

Previously, there had been a variety of different attempts to measure these slopes at specific redshifts or luminosities (typically ∼L*_{z = 3}), 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 z ∼ 2–3 and was based upon U-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 these papers, the UV-continuum slope β was found to have a mean value of ∼−1.4 and 1σ dispersion of ∼0.5–0.6 at ∼L*_UV. No significant correlation of UV-continuum slope β with UV magnitude was found to −20 AB mag (Adelberger & Steidel 2000; Reddy et al. 2008), though there would appear to be some evidence in the β distributions presented by Meurer et al. (1999, e.g., Figure 5 from that paper) that the β distribution becomes a little bluer at lower UV luminosities (i.e., −18.5 AB mag).¹⁰

Ouchi et al. (2004) extended these studies to higher redshift by presenting a determination of the UV-continuum slope distribution at z ∼ 4 based on a large selection of B-dropout galaxies from the Subaru Deep Field and Subaru XMM/Newton Deep Field. Ouchi et al. (2004) found that the UV-continuum slope distribution at z ∼ 4 was consistent with that determined at z ∼ 3 and that there was only a marginal trend (not statistically significant) toward bluer slopes at lower luminosities. Papovich et al. (2004), by contrast, working with a selection of z ∼ 4 B-dropouts from the GOODS fields, found that galaxies at z ∼ 4 were slightly bluer in their UV colors than at z ∼ 2.5 in the HDF North and South. At somewhat fainter magnitudes, Beckwith et al. (2006) remarked that the UV colors of z ∼ 4 B-dropouts in the HUDF were very blue in general (with β's of ∼−2) and hence suggested very little dust extinction. The somewhat bluer UV-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 toward bluer UV-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 z ∼ 5 V dropouts had UV-continuum slopes β very close to −2 while later Stanway et al. (2005) and Bouwens et al. (2006) found UV-continuum slopes of −2.2 ± 0.2 and −2.0 ± 0.3, respectively, at z ∼ 6 from a selection of i-dropouts in the HUDF. Hathi et al. (2008) used a small sample of galaxies at z∼4–6 from the HUDF and larger sample of U-dropouts at z ∼ 3 in an attempt to quantify the change in the UV-continuum slope β as a function of redshift. Hathi et al. (2008) found that the UV-continuum slope β showed only a slight change from z ∼ 3 (where β ∼ −1.5) to z ∼ 5–6 (where β ∼ −1.8).

Broadly, the picture that has emerged from these studies is that star-forming galaxies become bluer toward 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 UV-continuum slope is ∼1.0 ± 0.4 bluer at z ∼ 6 than it is at z ∼ 2.5 and ∼0.5 ± 0.1 bluer at −18 AB mag (∼0.1L*_{z = 3}) than it is at −21 AB mag (∼L*_{z = 3}). 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 UV-continuum slope β 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 UV-continuum slope β at z ∼ 6 where a mean β of −1.73 ± 0.35 was found. This appears to result from the relation Hathi et al. (2008) use to compute the UV-continuum slope β at z ∼ 6 (i.e., β = 2.56(J − H) − 2). This relation does not account for the fact that the NICMOS J₁₁₀ band extends down to 8000 Å and therefore z ∼ 6 i-dropouts partially drop out in the J₁₁₀ band (i.e., are fainter in the J₁₁₀ band and hence have redder J₁₁₀ − H₁₆₀ colors). Comparing β = 2.56(J − H) − 2 with Equation (A5) from the Appendix, we can readily see why the mean UV-continuum slopes we derive are much steeper (by ∼0.3). We note that an additional −0.1 mag shift in β relative to previous z ∼ 6 measurements of β (e.g., Bouwens et al. 2006; Hathi et al. 2008) comes from small offsets we make to the J₁₁₀-band photometry (Section 3.3).

The LBG 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 z ∼ 2–6 (Steidel et al. 1996; Williams et al. 1996; Bunker et al. 2003; Vanzella et al. 2006, 2009; Dow-Hygelund et al. 2007). This technique provides a very complete census of UV 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 UV 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 UV-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 Lyα forest. However, we might expect criterion (2) to fail if the UV-continuum slopes β of the sources were sufficiently red.

We can attempt to address this question by looking at how many galaxies have UV-continuum slopes β 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 (Section 3.7) for galaxies (dashed black lines on the left-hand panels) as a function of UV-continuum slope β (see also Figure 4). It is quite clear from these figures that our selection criteria are effective in identifying star-forming galaxies at z ∼ 2–6 with β's bluer than ∼0.5, and it is striking how much redder this limit is than the UV-continuum slopes derived for galaxies in our dropout samples at z ∼ 2–6 (solid blue histograms in Figure 6). The situation is particularly conspicuous for galaxies with UV-continuum slopes β redder than −1 at z ≳ 5. While our simulations (Section 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 UV-continuum slopes β are extremely rare.

Zoom In Zoom Out Reset image size

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 ∼3700 Å rest 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 dustier. 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 2 < z < 3 and 3 < z < 4.5 (see right-hand panels in Figure 6) based upon the Faint Infrared Extragalactic Survey (FIRES) data (Labbé et al. 2003; Förster Schreiber et al. 2006) and derived UV-continuum slopes for sources in their selections from SED fits to the photometry. They found that ∼67% of the galaxies in their z ∼ 2–3 selection had UV-continuum slopes bluer than 0.5, but almost all (≳90%) of the galaxies in their z∼3–4.5 sample had such blue slopes. This is highly encouraging for z ∼ 4 LBG selections given the independent nature of Balmer Break selections—suggesting that z ∼ 4 LBG selections may be largely complete.

By contrast, the fact that ∼33% of the galaxies in the Brammer & van Dokkum (2007) z ∼ 2–3 BBG sample have β's redder than 0.5 suggests that completeness could be somewhat of a concern for LBG selections at z ∼ 2.5, and in fact it is well known that at z ∼ 2–3, there is a substantial population of dust-obscured galaxies undergoing vigorous star formation (e.g., Hughes et al. 1998; Barger et al. 1998; Chapman et al. 2005; Labbé et al. 2005; Papovich et al. 2006; Pope et al. 2006).

Searches for BBGs at z ≳ 5 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 BBGs at z ∼ 5–7 over the Chandra Deep Field-South GOODS field, 7 of which are detected at 24 μm with MIPS, while Dunlop et al. (2007) find no BBGs over the same redshift range, in their analysis of the same field. While the MIPS detections for 7 of the 11 z ∼ 5–7 Wiklind et al. (2008) BBG candidates may be interpreted as due to an obscured active galactic nucleus (AGN), we believe that 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 z ∼ 5–7, but may also have lower redshifts.

Implications. Putting together the present LBG selection results at z ∼ 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 z ∼ 2–6, and since there is no reason to suppose that these trends come from various selection biases (not only do the sources have β'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; Section 4.1) have discussed this in the context of z ∼ 4 B-dropout selections.

Obviously we would expect galaxies at very high redshifts (z ∼ 5–6) to have much bluer UV-continuum slopes than at z∼2–3, because of the much smaller time baseline over which to produce metals and dust, as well as the shorter dynamical timescales of galaxies at z ≳ 5 (and hence much reduced ages). We would also expect this result based upon the evolution of the UV 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, 2007; Yoshida et al. 2006; 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 z ≳ 5 with substantial dust extinction. Finally there is the evolution seen in the relationship between SFR and dust extinction. From z ∼ 0 to z ∼ 0.7 to z ∼ 2, it has been found that the effective dust extinction for a given SFR 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 uniformly very blue and LBG selections very complete.

Nonetheless, we know there is a substantial population of luminous, dust-obscured galaxies at z∼2–3 (e.g., Hughes et al. 1998; Barger et al. 1998; Chapman et al. 2005; Labbé et al. 2005; Papovich et al. 2006; Pope et al. 2006) and extending out to redshifts as high as z ∼ 4.5 (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 z ≳ 4, 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 z ≳ 4. We discuss and quantify this point in Section 6.2 (see also Tables 5, 7, and Figures 13 and 14).

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

Given this impact, it has been somewhat surprising that different teams have made very different assumptions about the distribution of UV-continuum slopes in their determinations of the UV LFs. Some teams have adopted UV-continuum slopes of β ∼ −1.4 (Sawicki & Thompson 2006a; Yoshida et al. 2006), other teams have assumed UV-continuum slopes of β ∼ −2.0 (Beckwith et al. 2006), and yet other teams have adopted luminosity-dependent UV-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 UV-continuum slopes β at high redshift (versus redshift and luminosity), we now have a relatively uniform set of assumptions that can be used for quantifying the UV LFs at z ∼ 2–6 and in extending such determinations to z ≳ 7.

Having used the available observations to derive the distribution of UV-continuum slopes β for high-redshift galaxies as a function of UV 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 UV-continuum slope β 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 β is through changes in the dust content of galaxies. While there are certainly 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 UV-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 z ∼ 2–4 is just ∼3% for moderately bright galaxies at z∼2–3 (e.g., Nandra et al. 2002) and there is little evidence they are more frequent or important at higher redshifts 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 UV-continuum slopes β, we consider the effect that changes in many of the above quantities would have on the UV-continuum slope β. For simplicity, we model the star formation history of galaxies with a simple τ model e^−t/τ with τ = 10 Myr (e.g., as in Papovich et al. 2001) and t 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) modeling of z ∼ 2.5U-dropouts in the HDF North as a guide, we adopt t = 70 Myr, Salpeter IMF, [Z/Z_☉] = −0.7, and E(B − V) = 0.15 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 UV-continuum slope β predicted for a galaxy, as measured over the baseline 1600–2300 Å. The results are shown in Figure 7. From this figure, it is clear that the largest changes in the UV-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 UV-continuum slope β change by ∼0.05, ∼0.1, ∼0.35, respectively.

Zoom In Zoom Out Reset image size

Of course, we also see that age has a modest effect on the observed UV-continuum slope β. 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 β to age, no significant change is found in the median ages of star-forming galaxies from z ∼ 6 to z ∼ 4 (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 times 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 β is at most modest (≲0.2). 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 timescale as ρ^−1/2 ∼ (1 + z)^−3/2—which is a factor of ∼3 (0.5 dex) from z ∼ 6 to z ∼ 2.5. For such a change in age, the change in β would be Δβ = 0.2, which is small compared to the observed differences—where Δβ is 0.5–1.0.

Metallicity has an even smaller effect (by a factor of ∼8) on the observed UV-continuum slopes β than either dust or age do, so we would require very large variations in the metallicity to affect the UV-continuum slope β 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 β resulting from metallicity would be completely overwhelmed by changes in β resulting from dust.

Changing the IMF of galaxies only appears to have a modest effect on the UV-continuum slope β. For example, changing the slope of the IMF by ∼0.5 only changes β by ≲0.1. Moreover, for a steep enough UV-continuum slope and a top heavy enough IMF, there is very red nebular continuum emission (resulting from the ionizing radiation; Schaerer 2002, 2003; Venkatesan et al. 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 & Pelló 2005).

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

One of the most salient trends we found in the UV-continuum slopes β derived for z ∼ 2.5 U-dropout and z ∼ 4 B-dropout galaxies was the presence of a strong correlation between the UV-continuum slope β and UV luminosity (see also Meurer et al. 1999). As detailed in Section 3.9, the mean UV-continuum slope β for z ∼ 2.5 and z ∼ 4 galaxies decreases (becomes bluer) by 0.20 ± 0.04 and 0.15 ± 0.01, 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 z ∼ 2–4 samples. In fact, a strong correlation between UV-optical colors and rest-frame optical magnitude has been found for star-forming galaxies in the "blue" cloud (e.g., Papovich et al. 2001, 2004; Baldry 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 UV-continuum slope β and UV luminosity. Looking at the comparison more quantitatively, the slope of the color–magnitude relationship Labbé et al. (2007) measure, for example, is equivalent to dβ/dM_UV of ∼−0.27. This is similar to the −0.20 ± 0.04 slope we estimate at z ∼ 2.5 for our U-dropout selection (see Section 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 (Δβ ∼ 0.6) in the UV-continuum slope β with luminosity was through a change in the dust content of the galaxy population. Explaining the change in β 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 versus wavelength and through a detailed examination of the z ∼ 0 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 z∼0–3 (e.g., Tremonti et al. 2004; Erb et al. 2006a; Maiolino et al. 2008).

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 UV-continuum slope β at z ∼ 0 (Meurer et al. 1999):

$$\begin{equation} A_{1600} = 4.43 + 1.99\beta ^{\prime }, \end{equation} \tag{ 1 }$$

where the dust extinction A₁₆₀₀ here is specified at rest frame 1600 Å and where β' is the UV-continuum slope measured from 1300 Å to 2600 Å. This relationship has been shown to be a reasonable predictor of the actual dust extinction in galaxies at z ∼ 0, z ∼ 1, z ∼ 2 for all but the youngest (Reddy et al. 2006; Siana et al. 2008, 2009) and most obscured starburst galaxies (e.g., Meurer et al. 1995, 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 z ∼ 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 Equation (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 asymptotic giant branch (AGB) stars should not work at z ≳ 5 because the universe is not old enough at these redshifts to have produced AGB stars. Dust in z ≳ 5 galaxies has therefore been suggested to form by another mechanism, most frequently in supernovae (SNe) ejecta (e.g., Maiolino et al. 2004; Maiolino 2006). Equation (1) is also known to fail at z ∼ 3 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 z ∼ 2 (Reddy et al. 2006).

To use Equation (1), we need to convert the UV-continuum slopes β we have derived (which have a baseline 1600–2300 Å) to that appropriate for Equation (1) (where β assumes a slightly more extended baseline 1300–2600 Å). We therefore explored the value of the UV-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 Δβ < 0.2. While there is clearly latitude in the precise relationship we use to convert the measured UV-continuum slopes β to this new wavelength, perhaps the least arbitrary set of SEDs to use to perform this conversion is those we use in the Appendix to estimate β from the observed colors. For those SEDs (which assume a e^−t/τ star formation history, t = 10 Myr, τ = 70 Myr, [Z/Z_☉] = −0.7, the Calzetti et al. (2000) extinction law, and a Salpeter IMF), we derive the following relationship β' = −2.31 + 1.11(β + 2.3). While obviously there are some uncertainties in using this conversion, there are at least as many uncertainties in using Equation (1) to estimate the dust obscuration in high-redshift galaxies, and so we will ignore these uncertainties in the subsequent discussion.

We will first discuss the dependence of the UV-continuum slope and inferred dust extinction on the UV luminosity. We find that the UV-continuum slope β shows a strong dependence on the UV luminosity at both z ∼ 2.5 and z ∼ 4. This suggests that dust extinction is positively correlated with the observed UV 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 z ∼ 3 U-dropouts—where no correlation was found between the UV-continuum slope β and the observed UV luminosity over the range ${\cal R}=22$ (M_UV,AB ∼ −23.5) to ${\cal R}=25.5$ (M_UV,AB ∼ −20). This is a somewhat brighter range than we probe with good statistics, suggesting that dust extinction (and the UV-continuum slope β) may be a weaker function of the observed UV magnitude at M_UV,AB ≲ −20 (${\cal R}\lesssim 25.5$) than it is over the whole range probed here. Indeed, using this same magnitude baseline (V_606,AB < 25.5) and a t test, we only find the correlation between β and UV luminosities to be significant at 75% confidence from our own z ∼ 2.5 U-dropout sample. Thus, while the correlation would appear to be weaker for the more luminous sources, the existence of a correlation between β and observed UV magnitude over the entire magnitude range probed here is very significant (i.e., >4σ result).

In order to understand the differences between our results and those of Adelberger & Steidel (2000), we must appreciate that galaxies exhibit very different behaviors as a function of UV luminosity, depending upon whether we are dealing with galaxies at higher or lower luminosities. For the lower luminosity regime, a large fraction of the sources should have very low SFRs. Given the well-known correlation between the SFR and dust extinction (e.g., Wang & Heckman 1996; Hopkins et al. 2001; Martin et al. 2005; Reddy et al. 2006; Zheng et al. 2007), these galaxies would have low dust extinction (less than ≲0.2 mag), and therefore the UV luminosity should be approximately proportional to the SFR. Consequently, we might expect the dust extinction (and UV-continuum slope) to be approximately proportional to UV luminosity at low luminosities. This is what we observe.

In the higher luminosity regime, we might expect the SFR to be much larger in general. We would also expect the dust extinction in these galaxies to be larger, given the correlation of dust extinction with the SFR. As such, any increase in the SFR of a galaxy would be largely offset by a similar increase in the dust extinction—resulting in a very weak correlation between the UV light output from a galaxy and its SFR (or dust extinction). This is exactly what Adelberger & Steidel (2000) and Reddy et al. (2008) found in their sample of luminous (>0.3L*_{z = 3}) z ∼ 3 U-dropout galaxies. Of course, we would also expect galaxies with very large extinctions to be present in galaxy samples at lower UV luminosities, albeit as a small fraction. We can infer this latter fact from the volume density of such systems: >10¹²L_☉ IR ultra-luminous galaxies (e.g., from the z ∼ 2 IR LF of Caputi et al. 2007) have a much smaller volume density than UV faint sources (by factors of >50; see discussion in Section 8 of Reddy & Steidel 2009). Consequently, we would expect the contribution from such IR ultra-luminous galaxies to the overall SFR density in UV faint samples to be relatively small.

Very high-redshift (z ≳ 5) galaxies are also much bluer in general than z ∼ 2–4 galaxies of the same luminosity. This suggests that z ≳ 5 galaxies are less dust obscured as well. This smaller dust obscuration is probably a consequence of an evolution in the relationship between dust extinction and the SFR, such that for a given SFR the observed dust obscuration increases as a function of cosmic time. This would cause higher redshift galaxies of a given UV luminosity to have bluer UV-continuum slopes. Such an evolution in extinction-SFR relationship is observed from z ∼ 2 to z ∼ 0 (Reddy et al. 2006; Buat et al. 2007; Burgarella et al. 2007), and it is expected that this relationship continues to evolve from z ∼ 6. This is presumably a result of the gradual build-up in metals in the universe with cosmic time.

It is interesting to estimate what the observed distribution of UV-continuum slopes β would imply for the dust extinction, if Equation (1) holds. For this calculation, we assume that we can represent the distribution of UV-continuum slopes β as a simple Gaussian with mean and 1σ scatter as derived earlier in our paper (Table 4; see Section 3.7). We then fold this distribution through Equation (1) to calculate the effective dust extinction integrated down to various limiting luminosities. We adopt the UV LF of Bouwens et al. (2007) at z ∼ 4, z ∼ 5, and z ∼ 6 and that of Reddy & Steidel (2009) at z ∼ 3 for this calculation.

The effective dust extinction is presented in Table 5 and Figure 8. To a limiting luminosity of 0.3 L*_{z = 3}, we infer a dust extinction of 6.0 ± 2.5× (or 1.6 mag) at z ∼ 2.5, very close to the 5.4 ± 0.9× dust attention Meurer et al. (1999) estimated previously from the UV continuum slopes of HDF-North U-dropouts. It is encouraging that this extinction estimate is very similar to the values estimated at z ∼ 2 using a wide variety of multiwavelength data and SFR calibrators (e.g., Reddy & Steidel 2004; Reddy et al. 2006; Erb et al. 2006b). However, these extinction estimates are a few times higher than those inferred by Carilli et al. (2008) at z ∼ 3 by stacking the Very Large Array (VLA) radio observations of >0.2L*_{z = 3}U-dropouts in the COSMOS field (where a factor of 1.8 ± 0.4 extinction was inferred). This may indicate that the Meurer et al. (1999) relationship substantially overestimates the dust extinction at z > 2, or it may indicate that the radio luminosity for a given SFR is systematically lower at high redshift (Carilli et al. 2008). At this time, it is not obvious how this issue will be resolved, though it seems clear that dust extinction will be less of a concern as we reach back further in cosmic time (e.g., see Section 6.2).

Zoom In Zoom Out Reset image size

At higher redshifts and integrated down to lower luminosities, the effective dust extinction is much less, i.e., factors of ≲2–3 (≲1 mag). Qualitatively, we would expect such a change due to the much bluer UV-continuum slopes β found at these luminosities and redshifts. Nonetheless it is still quite striking how much smaller the estimated dust extinction is when including light from the lowest luminosity galaxies. In detail, this is because lower luminosity galaxies dominate the total luminosity density in the UV continuum (see Figure 9) and the average dust extinction in these lower luminosity galaxies is quite low. Bouwens et al. (2007) estimated that half of the light in the UV-continuum originated in galaxies fainter than 0.06 L*_{z = 3} (∼−18 AB mag; see Figure 9). At ∼−18 AB mag, we find that galaxies have an average UV-continuum slope β of ∼−2 to −2.5.

Zoom In Zoom Out Reset image size

These dust corrections are much more modest than have occasionally been assumed to calculate the total SFR density at early times, e.g., the factor of ∼8 adopted by Giavalisco et al. (2004b) or the factor of ∼3 adopted by Hopkins (2004) for their common obscuration correction. Previously, Bouwens et al. (2006) and Lehnert & Bremer (2005) have argued that the dust extinction was likely less than a factor of ≲2 at z ∼ 5–6 and Reddy & Steidel (2009) argued that when integrated to lower luminosities the dust extinction was less than a factor of ∼1.5–2 at z ∼ 2–3 (adopting their luminosity_dependent reddening (LDR) model). The Reddy & Steidel (2009) LDR dust extinctions are somewhat lower than what we estimate here, largely because they assume there is little dust extinction for galaxies with luminosities fainter than ∼−18.5 (0.1 L*_{z = 3}). By contrast, the dust extinction we adopt come from the distribution of UV-continuum slopes we find and using Equation (1). While it is unclear which set of dust corrections is most accurate, it seems quite clear that the dust extinction must be very low for galaxies at very low UV luminosities. This follows from two observational findings: (1) the much larger volume density (i.e., ∼10⁻² Mpc⁻³) of galaxies with low UV luminosities than ultra-luminous IR galaxy (ULIRG)-type sources (i.e., where 10^−3.5 Mpc⁻³) and (2) the correlation between dust extinction and bolometric luminosity (e.g., Wang & Heckman 1996; Hopkins et al. 2001; Martin et al. 2005; Reddy et al. 2006; Buat et al. 2007; Burgarella et al. 2007). As a result of these two findings, we can conclude that essentially all of the galaxies with lower UV luminosities (>99%) do not have very high bolometric luminosities, and therefore do not exhibit large dust extinctions.

The viability of the Meurer et al. (1999) IRX-β relationship (Equation (1)) we used to correct the observed UV luminosities for the dust extinction is a strong function of the bolometric luminosity of the sources. For example, Reddy et al. (2006) found that while the Meurer et al. (1999) relationship worked well between 10¹¹L_☉ and 10¹²L_☉ at z ∼ 2, it failed badly for galaxies with luminosities >10¹²L_☉ (see also Chapman et al. 2005).

To investigate the validity of the estimates we made in the previous section, it is interesting therefore to estimate the bolometric luminosities for galaxies with specific UV luminosities in our LBG selections. We again take advantage of Equation (1), as well as the relationships between the UV-continuum slope β and UV luminosity (Section 3.9) to make this estimate. This results in the following relationship between the absolute magnitude of a galaxy in the UV and the bolometric luminosity L_bol at z ∼ 2.5:

$$\begin{equation} L_{\rm bol} = 10^{11.67 - 0.58 (M_{UV,AB}+21)} L_{\odot }. \end{equation} \tag{ 2 }$$

The relationship at z ∼ 4 is similar. We emphasize that this relationship does not generally hold for galaxies where L_bol > 10¹²L_☉ (e.g., Reddy et al. 2006; Chapman et al. 2005) or for galaxies which are particularly young (Reddy et al. 2006), because of difficulties in using the UV-continuum slope β to correct for dust extinction in these cases.

The above result is shown in Figure 10 with the red lines for the z ∼ 2.5 and z ∼ 4 samples. The hatched region shows the approximate scatter expected in the bolometric luminosity for a given UV luminosity as a result of scatter in the UV-continuum slope β. Also shown are the bolometric luminosities for individual dropout galaxies in the two samples, after correcting their UV luminosities for dust extinction using their measured β's and Equation (1).

Zoom In Zoom Out Reset image size

From Figure 10 and Equation (2), it seems clear that we would expect a substantial fraction of our dropout sample to have bolometric luminosities less than 10¹²L_☉. As such, we can plausibly make use of the IRX-β relationship to predict the mean dust extinction in galaxies—as we have done in the previous section. Of course, we might expect a few >10¹²L_☉ dust-obscured galaxies to be present in the normal dropout samples, with typical UV luminosities and UV-continuum slopes β where Equation (2) may not apply (e.g., Chapman et al. 2005). However, given that the volume density of such galaxies is observed to be much lower (>10–100×) than that of typical dropout galaxies, we would expect Equation (2) to be reasonably accurate in most cases.

Examining Figure 10, it is certainly quite striking that z ∼ 2–4 LBGs start becoming extraordinarily rare brightward of ∼−21 or −22, at precisely the same luminosities (i.e., >10¹²L_☉) as we expect dust extinction to become increasingly important in galaxies. This suggests the abrupt cut-off in the UV LF at z ∼ 2–4, i.e., L*_UV, is set by dust extinction—or more precisely the luminosity at which dust obscuration becomes so significant as to offset the increase in energy output from stars. We can examine this by using the observed correlation between dust extinction and bolometric luminosity in the blue star-forming population at z ∼ 2 (Reddy et al. 2006) $\log L_{\rm bol} = (0.62\pm 0.06)\log \frac{L_{\rm IR}}{L_{1600}} + (10.95\pm 0.07)$ and applying it in Figure 10. Adopting the observed scatter of $\Delta \log \frac{F_{\rm IR}}{F_{UV}}\sim 0.4$, we show the envelope of M_UV and L_bol values expected for a wide range in L_bol. It seems clear from this exercise that we would not expect galaxies to have UV luminosities much brighter than −22 AB mag (∼2 L*_UV) and that dust effectively sets the cut-off in the UV LF. This may explain why the value of L*_UV (e.g., Steidel et al. 1999; Reddy et al. 2008) does not evolve much from z ∼ 4 to z ∼ 2.

The dust extinctions we infer are substantially less than have been inferred in some studies of specific high-redshift galaxies. One such example is the z = 6.56 galaxy behind Abell 370 (HCM6A; Hu et al. 2002). This source was found to have a significantly larger flux in the 4.5 μm band than in the 3.6 μm band—which may be due to substantial Hα emission falling within the 4.5 μm band (Chary et al. 2005). If Hα is in fact the explanation, the inferred SFR for this source from this Hα emission is ∼10 times higher than would be inferred from the UV or Lyα emission alone. Detailed stellar population modeling on the optical, near-IR, and Infrared Array Camera (IRAC) imaging data of HCM6A suggest an extinction factor of ∼11× (A_UV = 2.6 mag) which is much larger than what we infer on average for z ∼ 6 i-dropouts (Chary et al. 2005; Schaerer & Pelló 2005; see Table 5). However, in subsequent and much deeper IRAC observations of HCM6A (K. Lai et al. 2009, in preparation), no significant difference between the fluxes of HCM6A in the 3.6 μm and 4.5 μm bands is found, leaving us with no reason to believe this source shows significant Hα emission and hence no reason to suspect that its UV light is significantly obscured. Direct observations of this source with the MAMBO-2 bolometric array at 1.2 mm also set useful upper limits on its dust obscuration (Boone et al. 2007).

Other examples might include the many high-redshift QSOs discovered by the Sloan Digital Sky Survey (SDSS; Fan et al. 2006). The inferred dust masses of SDSS QSOs can be quite substantial (e.g., J1148+5251; Bertoldi et al. 2003), providing clear evidence that specific high-redshift sources can show modest to sizeable dust extinction. Nonetheless, we would not expect QSOs like J1148+5251 to be representative of z ∼ 6 galaxies, associated as they are with very massive halos and thus in very rich environments. Galaxies in such environments are known to show much greater evolution than galaxies in more typical environments (e.g., Tanaka et al. 2005) and hence much greater metal enrichment.

It makes sense for us to use the effective dust extinction inferred in the previous section (Table 5) to revise our estimates the SFR density at z ∼ 2–6. As per our calculation above, we adopt the UV LFs derived by Reddy & Steidel (2009) at z ∼ 3 and Bouwens et al. (2007) in calculating the SFR densities. In order to convert luminosities in the UV into estimates of the SFR we use the transformation presented by Madau et al. (1998; see also Kennicutt 1998):

$$\begin{equation} L_{UV} = \left(\frac{\hbox{SFR}}{M_{\odot } \hbox{yr}^{-1}} \right) 8.0 \times 10^{27} \hbox{erg}\, \hbox{s}^{-1}\, \hbox{Hz}^{-1}, \end{equation} \tag{ 3 }$$

where a 0.1–125 M_☉ Salpeter IMF and a constant SFR of ≳100 Myr are assumed. Conversion to a Kroupa (2001) IMF would result in factor of ∼1.7 smaller SFR estimates.

In view of the young ages (∼10–50 Myr) inferred for many star-forming galaxies at z ∼ 5–6 (e.g., Yan et al. 2006; Eyles et al. 2007; Verma et al. 2007), it is clear that adopting a simple formula like Equation (3) (which requires the assumption of ≳100 Myr ages) would result in a systematic underestimate of the SFR density of the universe at very early times (e.g., Verma et al. 2007; Reddy & Steidel 2009; Stanway et al. 2005). In any case, given our uncertain knowledge of how the age distribution of star-forming galaxies varies as a function of redshift and luminosity, it is difficult to know precisely how much we must revise our estimates upward. Working with a sample of bright z ∼ 5 galaxies, Verma et al. (2007) estimate that the actual SFRs may be twice as large as what Equation (3) would suggest. Of course, this effect may be partially offset by the fact that for these same young star-forming systems that standard dust corrections (e.g., following Meurer et al. 1999) seem to overpredict the SFRs by factors of ≳2 (Reddy et al. 2006; Siana et al. 2008). In any case, going forward, we will need to obtain tighter constraints on the age distribution of galaxies as a function of redshift and especially to very low luminosities (since faint galaxies provide the dominant contribution to the SFR density).

Our results are presented in Figure 11 and Table 6. These dust-corrected SFR densities are somewhat lower than presented by Bouwens et al. (2008). This is because the average dust extinction including the lowest luminosity galaxies is less than when only considering L* galaxies. We will discuss these results in more depth in Section 6.4 after including the contribution from ULIRGs.

Zoom In Zoom Out Reset image size

Of course, we would expect the extinction estimates above to underestimate the effective extinction, given that Equation (1) does not accurately describe galaxies with very high SFRs, dust extinctions, and bolometric luminosities (e.g., Elbaz et al. 2007), particularly those systems where L_bol > 10¹²L_☉. The SFR from such galaxies can be better accounted for by a direct consideration of IR LFs (e.g., Caputi et al. 2007).¹¹ Reddy & Steidel (2009) have already performed such a calculation at z ∼ 2, integrating the z ∼ 2 IR LF of Caputi et al. (2007) down to bolometric luminosities of 10¹²L_☉.¹² Reddy & Steidel (2009) infer an SFR density of 0.03 M_☉yr⁻¹Mpc⁻³ (Kroupa IMF) at z ∼ 2 from this population. Obviously, there are some modest uncertainties as to the exact bolometric luminosity down to which one should integrate the Caputi et al. (2007) LF (to correct for the ULIRG population not reasonably accounted for by the Meurer et al. 1999 relation), and so this SFR density could be larger. For example, assuming we integrated the Caputi et al. 2007 LF down to 10^11.8L_bol instead, the ULIRG contribution could increase by ∼50%.

Despite the existence of these possible uncertainties, we adopt as our baseline values at z ∼ 2.5 the contribution Reddy & Steidel (2009) derived from the z ∼ 2 Caputi et a. (2007) IR LF. We therefore correct our SFR density estimates at z ∼ 2.5 in Table 6 for this population. The results are given in the lower section of Table 7. The obscured population only has a modest effect on the dust extinction (and SFR density), increasing it by ∼20% (see also discussion in Reddy & Steidel 2009).

Accounting for the effect of such ULIRGs on the SFR density (and extinction estimates) at z ≳ 4 is a much less certain enterprise. Not only is it difficult to probe to moderate luminosities with current instrumentation, but conducting follow-up observations on specific z ≳ 4 sources (e.g., to determine redshifts) is also quite challenging at present. As a result, only a handful of high quality z ≳ 4 ULIRG candidates (e.g., Capak et al. 2008; Daddi et al. 2009; Wang et al. 2009; Schinnerer et al. 2008; Dunlop et al. 2004) currently exist, and essentially all of them are at z ∼ 4. Perhaps the deepest, most well-defined selection of such sources is found in the deep ∼50–100 arcmin² submillimeter SCUBA field lying within the HDF-North GOODS area (Pope et al. 2006). We will use those z ∼ 4 sources to set a lower limit on the contribution of highly obscured galaxies to the SFR density at z ∼ 4. Starting with the ∼5–8 ULIRGs within this field that are plausibly at z ∼ 4 (e.g., Daddi et al. 2009; Mancini et al. 2009) and then dividing by the comoving volume at z ∼ 4, one derives an SFR density of ∼0.01–0.02 M_☉yr⁻¹Mp⁻³ (depending upon what fraction of the candidates are actually at z ∼ 4). Again, we can correct our z ∼ 4 SFR density estimates for this population. Not surprisingly, this population only contributes ∼10% to the SFR density (see Figure 14).

Of course, we might expect there to be highly obscured galaxies below the 10¹² L_☉ luminosity limits that also contribute substantially to the SFR density relevant to these probes, so one might argue that 10% is just a lower limit. However, as we have already argued, we should be able to account for the star formation of these lower luminosity systems from LBG selections. The results of Reddy et al. (2006) suggest that the UV-corrected SFRs of galaxies can be accurately estimated at z ∼ 2 to 10¹²L_☉, and we would expect these corrections to be even easier to perform at z ∼ 4 than at z ∼ 2—given the trend of high-redshift galaxies to show less and less dust extinction at a fixed SFR at increasingly high redshifts (e.g., Reddy et al. 2006; Buat et al. 2007; Burgarella et al. 2007).

Given that >10¹²L_☉ ULIRGs only contribute ∼20% of the SFR density at z ∼ 2–3 and perhaps ∼10% at z ∼ 4, one might expect the contribution of such a population at z ∼ 5–6 also to be quite small. We can obtain a crude estimate of this fraction, if we assume that IR luminous galaxies have a bolometric LF similar to that derived by Caputi et al. (2007) at z∼2 (i.e., where L*_bol = 10^{11.8 ± 0.1}) and then evolve in a very similar way to the UV LF at z ⩾ 3. We therefore assume that L*_bol scales with redshift as 10^{−0.4(0.35(z − 3))} ∼ (1/1.38)^{z − 3} to match the observed M*_UV(z) = M*_UV(z = 4) + 0.35(z − 4) evolution in the UV LF from z ∼ 4 to z ∼ 7 (e.g., Bouwens et al. 2007, 2008).¹³ We then compare the fractional SFR density in L_bol > 10¹²L_☉ galaxies with that from the star-forming population as a whole. We find that the fractional SFR density in ULIRGs is 6% at z ∼ 5 and 2% at z ∼ 6. For these assumptions, the fractional SFR density in L_bol > 10¹²L_☉ galaxies is 23% and 10% at z ∼ 3 and z ∼ 4, respectively, which would seem to be a good match to the observations. We emphasize that this very low ULIRG estimate at z≳5 is consistent with the lack of very red UV-continuum slopes β at z ∼ 5–6 (Figure 6; which is in contrast to the tail toward redder (i.e., ∼−1 to 0) β's at z ∼ 4).

Recent work on GRBs provide strong support for the general conclusion that highly obscured ULIRGs contribute at most a small fraction of the SFR density at z ≳ 4. Since GRBs are perhaps a good tracer of massive star formation in the high-redshift universe (e.g., Yüksel et al. 2008; Li 2008; Kistler et al. 2009) and may not be biased to lower metallicity environments (Prochaska et al. 2007), we would expect GRBs to occur in galaxies roughly in proportion to their share of the star formation activity in the high-redshift universe. Follow-up work done on GRB host galaxies show that these systems are quite faint at both rest-frame UV (Chen et al. 2009) and optical wavelengths (Chary et al. 2007a). We can therefore conclude that most of the star formation at high redshift occurs in galaxies that are similarly faint.

Before using the extinction estimates for LBG samples (Table 5; Section 5.4) and the aforementioned estimates of the ULIRG contribution (Section 6.2) to estimate the SFR density, we first use them to assess the relative flux output from galaxies in the UV and that output in the mid-IR/far-IR, after dust reprocessing. This entire topic is important in thinking about how best to nail down the SFR density and energy output in the high-redshift universe and for our thinking about the contribution of the high-redshift universe to the extragalactic background light.

In calculating the relative flux output in the UV and mid-IR/far-IR, we assume that essentially all the light from the ULIRG population is emitted in the mid-IR/far-IR and adopting the extinction estimates from Table 5 for the LBG population. We include a plot of this fraction versus redshift in Figure 12. Not surprisingly the majority of the light output from star-forming galaxies at z∼2.5–4 occurs in the mid-IR/far-IR (after dust reprocessing) as has been emphasized repeatedly (e.g., Hughes et al. 1998; Pérez-González et al. 2005). However, as we will note in Section 6.4, it appears that we can account for most of this energy output using the LBG population. Again, the situation appears to be quite different at z ≳ 5, with most of the light output occurring in the UV.

Zoom In Zoom Out Reset image size

Including the contribution from highly obscured ULIRGs (Section 6.2) in with that derived from the UV LFs (Table 6 and Figure 11), our best estimates for the SFR densities are given in Table 7 and Figure 13. The ULIRG population increases the SFR density estimated at z ∼ 2.5 and z ∼ 4 (to faint limits) by ∼20% and 10%, respectively. We show the fractional contribution of ULIRGs to the SFR density at z ∼ 2–6 in Figure 14.

Zoom In Zoom Out Reset image size

Zoom In Zoom Out Reset image size

The present dust-corrected SFR densities are less than has been considered in many previous studies. This is largely because many previous studies corrected the SFR density by factors of ∼5–8 times (e.g., Giavalisco et al. 2004b)—which is the dust correction relevant for L* galaxies. This has important implications for studies (e.g., Hopkins & Beacom 2006) featuring comparisons of the stellar mass density with the SFR density.

The claim has been that the SFR density—after integration—exceeds the stellar mass density by factors of ∼3–4 (e.g., by Hopkins & Beacom 2006; Wilkins et al. 2008). Adopting a small dust correction (as is appropriate including the effect of the lower luminosity galaxies) substantially reduces the inferred SFR density at high redshift (by factors of ∼1.1, ∼1.1, ∼2.1, and ∼2.5 at z ∼ 2.5, z ∼ 4, z ∼ 5, and z ∼ 6, respectively relative to that used by Wilkins et al. 2008). This should help somewhat to resolve the apparent discrepancy in comparisons with the stellar mass density inferred at high redshift (see also discussion in Reddy & Steidel 2009). Another consideration that may help in this regard is the stellar mass in lower luminosity galaxies. Reddy & Steidel (2009) argue that a consideration of the SFR density and stellar mass densities to the same limiting luminosities largely resolves the apparent discrepancy between the inferred stellar mass density and that obtained by integrating SFR history (approximately doubling the inferred stellar mass densities at z ∼ 2–3).

What stands out in the above discussion is the role of lower luminosity galaxies and their apparently sizeable contribution to various volume-averaged quantities such as the SFR density and stellar mass density. Clearly, any reasonable examination of these quantities (or comparisons between the stellar mass density and the integral of the SFR density) must accurately account for this population or risk ignoring those galaxies that are most central to making sense of the cosmic evolution.