ULTRAVIOLET LUMINOSITY FUNCTIONS FROM 132 z ∼ 7 AND z ∼ 8 LYMAN-BREAK GALAXIES IN THE ULTRA-DEEP HUDF09 AND WIDE-AREA EARLY RELEASE SCIENCE WFC3/IR OBSERVATIONS^*

The full two-year WFC3/IR observations from the HUDF09 program consist of 192 orbits of ultra-deep WFC3/IR data over the HUDF09 (111 orbits), HUDF09-1 (33 orbits), and HUDF09-2 (48 orbits) fields. The observations from this program are now complete.

All 111 orbits of imaging data over the HUDF were obtained on a single ∼4.7 arcmin² WFC3/IR pointing and were distributed over three bands: Y₁₀₅ (24 orbits), J₁₂₅ (34 orbits), and H₁₆₀ (53 orbits). The reductions of those data are similar to those already described in Oesch et al. (2010a) and Bouwens et al. (2010b), but now include the full two years of observations of the HUDF. Great care was exercised to ensure the image registration to the optical HUDF was as accurate as possible. Not only were latest distortion solutions (10/11/2010) utilized, but each source in our WFC3/IR reductions was cross-correlated with the corresponding source in the v1.0 HUDF ACS z₈₅₀-band observations (Beckwith et al. 2006). Small corrections to the distortion solution were required to obtain excellent registration (i.e., rms differences of <001) for the Y₁₀₅-band data (no WFC3/IR distortion solution was explicitly derived for this filter by STScI). Pixels affected by source persistence were explicitly masked out. This masking was performed by remapping our initial reductions of the data back to the original frames, subtracting them from the original frames, co-adding these subtracted frames for all exposures within a visit, smoothing, and then flagging all pixels above a 3σ threshold. The final reduced frames were drizzled onto the v1.0 HUDF ACS reductions rebinned on a 006 pixel frame.

Our HUDF09 WFC3/IR reductions reach to 29.6, 29.9, and 29.9 AB mag (5σ: 035 diameter apertures) in the Y₁₀₅, J₁₂₅, and H₁₆₀ bands, respectively. The optical ACS imaging over the HUDF reach to 29.7, 30.1, 29.9, and 29.4 AB mag (5σ: 035 diameter apertures) in the B₄₃₅, V₆₀₆, i₇₇₅, and z₈₅₀ bands, respectively. These depths do not include any correction for the light outside of the 035 diameter aperture and on the wings of the point-spread function (PSF), so that the depths can be readily reproduced.

The availability of the WFC3/IR observations over the HUDF09-1 (33 orbits) and HUDF09-2 (48 orbits) fields allows us to substantially extend our z ∼ 7 and z ∼ 8 samples. In both fields, the WFC3/IR observations were concentrated in single ∼4.7 arcmin² WFC3/IR pointings and distributed across the three bands Y₁₀₅, J₁₂₅, and H₁₆₀. In the HUDF09-1 field, 8 orbits in the Y₁₀₅ band, 12 orbits in the J₁₂₅ band, and 13 orbits in the H₁₆₀ band were acquired. In the HUDF09-2 field, 11 orbits in the Y₁₀₅ band, 18 orbits in the J₁₂₅ band, and 19 orbits in the H₁₆₀ band were acquired. These new observations were reduced using the same procedures as for the HUDF. The WFC3/IR observations were registered and drizzled onto the same frame as our reductions of the ACS data on the two fields rebinned on a 006 pixel frame. As with our UDF reductions, care was taken to ensure that the registration to the optical data was as accurate as possible (<001) and that even issues such as velocity aberration were properly treated. Our reductions of the WFC3/IR over the HUDF09-1 field reach to 29.0, 29.3, and 29.1 AB mag in the Y₁₀₅, J₁₂₅, and H₁₆₀ bands, respectively. These depths are 29.1, 29.4, and 29.3 AB mag, respectively, over the HUDF09-2 field (see Table 1).

For the ACS observations over the HUDF09-1 field, we could only take advantage of those from the HUDF05 (GO10632; PI: Stiavelli) program, and so we used the publicly available v1.0 reductions of Oesch et al. (2007). For the HUDF09-2 field, on the other hand, deep ACS observations are available as a result of two programs: the HUDF05 and HUDF09 programs. The ACS observations over this field from the HUDF05 program (102 orbits in total: 9 orbits V₆₀₆ band, 23 orbits i₇₇₅ band, and 70 orbits z₈₅₀ band) were obtained over an ∼18 month period in 2005 and 2006 at a number of different orientations, as well as a 1' shift for 20 orbits from the primary pointing. One hundred eleven additional orbits of ACS data (10 orbits B₄₃₅ band, 23 orbits V₆₀₆ band, 23 orbits i₇₇₅ band, 16 orbits I₈₁₄ band, and 39 orbits z₈₅₀ band) were acquired over this field in 2009 and 2010 from our HUDF09 program.

Reductions of the recent ACS observations were conducted using the ACS GTO apsis pipeline (Blakeslee et al. 2003). Special care was required to cope with the significantly reduced charge transfer efficiency in the new post-SM4 ACS observations and to correct for row-by-row banding artifacts. Our recent ACS observations (from HUDF09) cover ∼70% of the HUDF09-2 footprint and add ∼0.15–0.5 mag to the depth of the earlier HUDF05 observations (after co-addition). Reductions of the HUDF05 ACS observations are described in Bouwens et al. (2007). The depths of the WFC3/IR and ACS observations over these two fields are given in Table 1 and reach to ≳29 AB mag (5σ).

The WFC3/IR ERS observations cover ∼40 arcmin² in the upper region of the CDF-South GOODS field (Windhorst et al. 2011). These observations include 10 separate ∼4.7 arcmin² WFC3/IR pointings (see Figure 1). Two orbits of near-IR data are obtained in the F098M, F125W, and F160W bands, for a total of six orbits per field (60 orbits in total). These near-IR observations are reduced in a very similar way to the procedure used for the WFC3/IR observations from our HUDF09 program (Oesch et al. 2010a; Bouwens et al. 2010b). To keep the size of the drizzled WFC3/IR frames manageable, we split the output mosaic into eight discrete pieces—corresponding to different segments in the ACS GOODS mosaic (i.e., S14, S24, S25, S34, S35, S43, S44, S45).⁹ For each segment, we first aligned the observations against the ACS data binned on a 006 pixel scale and then drizzled the data onto that frame. For the ACS data, we made use of our own reductions (Bouwens et al. 2006, 2007) of the deep GOODS ACS/WFC data over the GOODS fields (Giavalisco et al. 2004). These reductions are similar to the GOODS v2.0 reductions, but reach ∼0.1–0.3 mag deeper in the z₈₅₀ band due to our inclusion of the supernovae (SNe) follow-up data (e.g., Riess et al. 2007). Our reduced WFC3/IR data reach to 27.9, 28.4, and 28.0 in the Y₀₉₈, J₁₂₅ and H₁₆₀ bands, respectively (5σ: 035 apertures). The ACS observations reach to 28.2, 28.5, 28.0, and 28.0 in the B₄₃₅, V₆₀₆, i₇₇₅, and z₈₅₀ bands, respectively (5σ: 035 apertures). The FWHMs of the PSFs are ∼016 for the WFC3/IR data and ∼010 for the ACS/WFC data.

We generate separate catalogs for each of our Lyman-break selections, to obtain more optimal photometry for each selection. This is done through the use of square root of χ² detection images (Szalay et al. 1999: similar to a co-added inverse noise-weighted image) constructed from only those bands that are expected to have flux for a given Lyman-break selection. Specifically, this image is constructed from the Y₁₀₅-, J₁₂₅-, and H₁₆₀-band images for our z ∼ 7 HUDF09 z₈₅₀-dropout selection, the J₁₂₅- and H₁₆₀-band images for our z ∼ 8 HUDF09 Y₁₀₅-dropout selection, and from the J₁₂₅- and H₁₆₀-band images for our ERS z₈₅₀- and Y₀₉₈-dropout selections; the square root of χ² image therefore includes all deep WFC3/IR observations redward of the break.

Object detection and photometry are performed using the SExtractor (Bertin & Arnouts 1996) software run in dual image mode. Object detection is performed off the square root of χ² image. Colors are measured in small scalable apertures (MAG_AUTO) defined using a Kron (1980) factor of 1.2 (where the Kron radius is established from the square root of χ² image). All of our imaging data (optical/ACS and near-IR/WFC3) are PSF-matched to the WFC3/IR F160W imaging data before making these color measurements. Fluxes in these small scalable apertures are then corrected to total magnitudes in two steps. First a correction is made for the additional flux in a larger scalable aperture (with Kron factor of 2.5). Then a correction is made for the light outside this larger scalable aperture (a 07 diameter aperture is typical) and on the wings of the PSF. The latter correction (typically 0.2 mag) is made based upon the encircled energy measured outside this aperture (for stars).¹⁰ The typical size of the total correction (including both steps) is ∼0.4–0.8 mag.

Our primary z ∼ 7–8 Lyman-break samples are based upon the ultra-deep WFC3/IR observations over the three HUDF09 fields. The availability of ultra-deep three-band (Y₁₀₅, J₁₂₅, and H₁₆₀) observations over these fields allows us to use traditional two-color LBG selection criteria to identify star-forming galaxies at z ≳ 7. Spectroscopic follow-up has shown that these criteria are generally quite robust in identifying star-forming galaxies at z ∼ 3–7 (e.g., Steidel et al. 1996, 2003; Popesso et al. 2009; Vanzella et al. 2009; Stark et al. 2010) though admittedly spectroscopic follow-up of the very faintest sources at z ⩾ 4 has been ambiguous since only those sources with Lyα emission can be confirmed.

We favor the use of LBG "dropout" selection criteria over the use of photometric redshifts for the identification of star-forming galaxies at z ∼ 7 and z ∼ 8 due to (1) the simplicity and transparency of LBG criteria, (2) the ease with which optical non-detection criteria can be engineered to optimally remove contaminants (e.g., see Appendix D), (3) the economy with which these criteria capture the available redshift information (while minimizing redshift aliasing effects), and (4) the accuracy with which it is possible to model the effective selection volumes.

The color criteria we utilize for our selection are similar to those already used by Oesch et al. (2010a) and Bouwens et al. (2010b). In detail, these criteria are

$$\begin{displaymath} (z_{850} - Y_{105} > 0.7)\wedge (Y_{105}-J_{125} < 0.45) \end{displaymath}$$

$$\begin{displaymath} \quad\wedge(z_{850}-Y_{105}>1.4(Y_{105}-J_{125})+0.42) \end{displaymath}$$

for our z ∼ 7 z₈₅₀-dropout sample and

$$\begin{displaymath} (Y_{105} - J_{125} > 0.45)\wedge (J_{125}-H_{160} < 0.5) \end{displaymath}$$

for our z ∼ 8 Y₁₀₅-dropout selection, where ∧ represents the logical AND symbol. These two-color criteria are illustrated in Figures 2 and 3, respectively. The criteria were chosen to allow for a fairly complete selection of star-forming galaxies at z ∼ 7 and z ∼ 8 and so that our HUDF09 samples have a very similar redshift distribution to those galaxies selected with the ERS Y₀₉₈, J₁₂₅, and H₁₆₀ bands (see Figure 4 and Section 3.4). They are also carefully crafted so that all star-forming galaxies at z ≳ 6.5 are included and do not fall between our z ∼ 7 and z ∼ 8 selections. To ensure that sources are real, we require that sources be detected at ≳3.5σ in the J₁₂₅ band and 3σ in the Y₁₀₅ or H₁₆₀ bands. This is equivalent to a detection significance of 4.5σ (when combining the J₁₂₅ and Y₁₀₅/H₁₆₀ detections).

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For both of our z ∼ 7 z₈₅₀-dropout and z ∼ 8 Y₁₀₅-dropout samples, we enforce very stringent optical non-detection criteria. Not only do we reject sources detected at 2σ in a single passband or 1.5σ in more than one band, but we also compute a collective χ²_opt value from all of the optical data together and eliminate sources above specific thresholds. We outline this procedure in Section 3.3.

To ensure that our selections do not suffer from significant contamination from SNe or low-mass stars (both of which have point-like profiles), we examined each of our candidate z ∼ 7–8 galaxies with the SExtractor stellarity parameter to identify those sources consistent with being point-like. The only source that we identified in our fields that was point-like and satisfied our dropout criteria was the probable SNe (03:32:34.53, −27:47:36.0) previously identified over the HUDF (Oesch et al. 2010a: see also McLure et al. 2010; Bunker et al. 2010; Yan et al. 2010). No other point-like sources were found (or removed).

We use the above criteria to search for candidate z ≳ 7 galaxies over all areas of our HUDF09 fields where the WFC3/IR observations are at least half of their maximum depth in each field (or ∼4.7 arcmin² per field). In total, 29 z ∼ 7 z₈₅₀-dropout sources and 24 z ∼ 8 Y₁₀₅ dropouts were identified over our ultra-deep WFC3/IR HUDF pointing. Seventeen z ∼ 7 z₈₅₀-dropout sources and 14 z ∼ 8 Y₁₀₅ dropouts were found over our ultra-deep WFC3/IR HUDF09-1 pointing, while 14 z ∼ 7z₈₅₀-dropout sources and 15 z ∼ 8 Y₁₀₅ dropouts were found over our ultra-deep WFC3/IR HUDF09-2 pointing. As in the Oesch et al. (2010a) and Bouwens et al. (2010b) selections (see also McLure et al. 2010; Bunker et al. 2010; Yan et al. 2010; Finkelstein et al. 2010), the sources we identified have H₁₆₀-band magnitudes ranging from ∼26 mag to ∼29 mag. A catalog of the z ∼ 7 z₈₅₀ dropouts in the HUDF09 fields is provided in Tables 10–12 of Appendix E. A similar catalog of Y₁₀₅ dropouts is provided in Tables 14–16 of Appendix E. Figures 20 and 21 of Appendix E show image cutouts of all of these candidates. Figure 5 shows the approximate surface density of these candidates as a function of magnitude in our HUDF09 search fields.

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As a result of the very limited depths of the ACS data over our fields, it was necessary to exercise considerable caution in selecting z ≳ 7 galaxies over our fields. Our conservative selection procedure resulted in our eliminating a modest number of sources from our still very large z ∼ 7 and z ∼ 8 samples. For comparison with other studies, we have included a list of these likely contaminants in Tables 18 and 19 of Appendix E.

Identifying lower redshift contaminants in our HUDF09 selections is very challenging due to the limited depths of the available optical data. This is especially true at the faintest magnitudes. We have found that the only truly effective way to identify these contaminants is to make full use of the optical data.

Accordingly, we have developed a rather sophisticated multi-step procedure to obtain tight controls on contamination. The first step is rather standard for high-redshift selections: remove all the sources detected at 2σ in a single optical band or at 1.5σ in more than one optical band. The second step is more complex, but very effective. In this second step we compute a collective χ²_opt from all of the optical data as

$$\begin{equation} \chi _{{\rm opt}} ^2 = \Sigma _{i} \hbox{SGN}(f_{i}) (f_{i}/\sigma _{i})^2. \end{equation} \tag{ 1 }$$

We reject all sources with χ²_opt greater than a specific value χ²_lim that we establish through simulations (Appendix D). At brighter 0.5(J_{125, AB} + H_{160, AB}) ≲ 28.5 magnitudes, we take χ²_lim equal to 5, 3, and 5 for our HUDF09, HUDF09-1, and HUDF09-2 selections, respectively, while at fainter 0.5(J_{125, AB} + H_{160, AB}) > 29.2 magnitudes, χ²_lim is taken to be half that. At magnitudes between those limits, the limiting χ²_lim is an interpolation between the two extremes. More stringent limiting values on χ²_lim are adopted for the faintest sources or where the optical data are shallower; more permissive limiting values on χ²_lim are adopted for brighter sources or where the optical data are deeper. Note that f_i is the flux in band i in our smaller scalable apertures, σ_i is the uncertainty in this flux, and SGN(f_i) is equal to 1 if f_i > 0 and −1 if f_i < 0. The filters included in the χ²_opt summation are B₄₃₅, V₆₀₆, and i₇₇₅ for the z₈₅₀-dropout selection and B₄₃₅, V₆₀₆, i₇₇₅, and z₈₅₀ for the Y₁₀₅-dropout selection.

These particular limits on χ²_opt were chosen to minimize contamination for our Lyman-break selections while maximizing the completeness of the z ∼ 7 and z ∼ 8 sources in our samples (see Appendix D for the relevant simulations). We determine χ²_opt in three different sets of apertures (035 diameter apertures, 018 diameter apertures, and scalable Kron apertures with typical and maximum radii of 02 and 04) to maximize the information we have on possible optical flux in our candidates. Larger aperture measurements are useful for flagging more extended low-redshift galaxies in the selection. Meanwhile, the smaller aperture measurements provide higher signal-to-noise ratio (S/N) measurements from the higher spatial resolution ACS data. Making measurements in such small apertures is meaningful, given that the alignment we achieve between sources in the WFC3/IR and ACS images is better than 001.

Overall, we have found that a χ²_opt criterion is extremely effective at reducing the contamination rate from low-redshift galaxies that scatter into our selection. The reductions in the contamination rate are very substantial, i.e., factors of ≳2–3 over what it would be excluding only those sources detected at 2σ in one optical band or >1.5σ in ⩾2 optical bands. Finally, we emphasize that use of the χ²_opt statistic and an examination of the χ²_opt distribution allows us to verify that our samples are not subject to significant contamination (since contaminating sources would show up as a tail to positive χ²_opt values; Section 4.2 and Appendix D.4).

To complement our ultra-deep searches for z ∼ 7–8 galaxies in the HUDF09 observations, we also take advantage of the wide-area (∼40 arcmin²) ERS observations over the upper portion of the CDF-South GOODS field (Figure 1).

Ideally we would use the same selection criteria for identifying z ∼ 7–8 galaxies over the ERS observations as we use over the three ultra-deep HUDF09 fields. This is not possible, however, since the ∼40 arcmin² WFC3/IR ERS observations use a different Y-band filter (Y₀₉₈) than in the WFC3/IR HUDF09 observations (Y₁₀₅). The J₁₂₅ and H₁₆₀ filters remain the same. We can nevertheless do quite well in selecting galaxies over a similar redshift range and with similar properties by judiciously choosing our selection criteria (as can be seen in Figure 4).

The multi-color selection criteria we use to select galaxies at z ∼ 7 and z ∼ 8 are illustrated in Figures 6 and 7, respectively, and nominally correspond to z₈₅₀-dropout and Y₀₉₈-dropout selections. We base our z ∼ 7 z₈₅₀-dropout selection on a z₈₅₀ − J₁₂₅-dropout criterion rather than a z₈₅₀ − Y₀₉₈-dropout criterion to extend our selection over a wider redshift range than is possible using a z₈₅₀ − Y₀₉₈ criterion. Dropout selections based upon the latter color criterion (e.g., as used by Wilkins et al. 2010) result in galaxies starting to drop out of the Y₀₉₈ band at z ≳ 6.7. This makes it more difficult to identify the Lyman break at z ≳ 7, resulting in a redshift selection window that is artificially narrow.

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In detail, we require that galaxies show a strong z₈₅₀ − J₁₂₅ > 0.9 break and satisfy z₈₅₀ − J₁₂₅ > 0.8 + 1.1(J₁₂₅ − H₁₆₀) and z₈₅₀ − J₁₂₅ > 0.4 + 1.1(Y₀₉₈ − J₁₂₅). The latter two criteria exclude sources with redshifts z ≲ 6.5. To bound our redshift window selection on the high end to redshifts z ≲ 7.4, we require that their Y₀₉₈ − J₁₂₅ colors be bluer than 1.25. In addition, sources are required to have J₁₂₅ − H₁₆₀ colors (redward of the break) bluer than 0.5 to exclude intrinsically red sources (which otherwise might contaminate our selection). Candidates are required to be detected at ⩾4.5σ in the J₁₂₅ band and at ⩾3.5σ in the H₁₆₀ band (to ensure that they correspond to real sources). A higher detection significance is required for our ERS selections than our HUDF09 selections (Section 3.2) to compensate for the fewer exposures that go into each ERS field (and hence less Gaussian noise).

For our higher z ∼ 8 Y₀₉₈-dropout selections, we require that galaxies show a strong Y₀₉₈ − J₁₂₅ > 1.25 break and, similar to the previous z ∼ 7 z₈₅₀-dropout selection, have J₁₂₅ − H₁₆₀ colors bluer than 0.5. As we will see later, our use of a Y₀₉₈ − J₁₂₅ > 1.25 criterion selects for star-forming galaxies at z ≳ 7.4 and is convenient for combining the ERS dropout samples with the HUDF09 samples (selected using a slightly different Y₁₀₅ filter). Candidates are again required to be detected at ⩾4.5σ in the J₁₂₅ band and 3σ in the H₁₆₀ band.

To minimize the contamination from low-redshift galaxies that scatter into our dropout color windows, we utilize very stringent criteria to exclude sources that show any evidence of detection in the optical. This includes being detected at 2σ in a single optical band, 1.5σ in more than one optical band, or having an optical χ²_opt value >2.5 (see Section 3.3 and Appendix D for our definition of χ²_opt).

To control for potential contamination from low-mass stars or SNe—both of which have point-like profiles—we examined each of our candidate z ∼ 7 z₈₅₀- or z ∼ 8 Y₀₉₈-dropout galaxies with the SExtractor stellarity parameter to identify those sources consistent with being point-like. One of the sources (03:32:27.91, −27:41:04.2) in our selection had measured stellarity parameters consistent with being point-like (with stellarities >0.8 in the J₁₂₅ and H₁₆₀ bands). This source had a J − H∼−0.6 color—which is much bluer than typical L* galaxies at z ∼ 7 (Bouwens et al. 2010a), but consistent with the colors of a T dwarf (e.g., Knapp et al. 2004). We therefore identified this source as a probable T dwarf and removed it from our z₈₅₀-dropout selection. For all other sources in our selection, the measured stellarity parameters were substantially less, with typical values ∼0.05–0.3 indicative of extended sources. From a quick inspection of Figures 20 and 21, it is clear that essentially all the sources in our samples are extended.

The above selection criteria are only applied to those regions of the WFC3/IR ERS observations over the CDF-South GOODS field where the optical and near-IR observations reach within 0.2 mag of the typical ACS GOODS field depth and ERS WFC3/IR depth, i.e., ≳3 orbits (ACS B₄₃₅ band), ≳2 orbits (ACS V₆₀₆ band), ≳3 orbits (ACS i₇₇₅ band), ≳7 orbits (ACS z₈₅₀ band), ≳2 orbits (Y₁₀₅), ≳2 orbits (J₁₂₅), and ≳2 orbits (H₁₆₀). The total effective area is 39.2 arcmin² (∼3 arcmin² of the WFC3/IR ERS fields extend outside of the deep ACS GOODS data; see Figure 1).

With the above criteria, we selected 13 z ∼ 7 z₈₅₀-dropout galaxies and 6 z ∼ 8 Y₀₉₈-dropout galaxies. The z ∼ 7 z₈₅₀-dropout candidates have magnitudes ranging from 25.8 to 27.6 AB mag, while the z ∼ 8 Y₀₉₈-dropout candidates have magnitudes ranging from 26.0 to 27.2 AB mag. The properties of the sources are given in Tables 13 and 17. The surface densities of our z ∼ 7 z₈₅₀ and z ∼ 8 Y₀₉₈ dropouts are shown in Figure 5.

As with our HUDF09 dropout selections, we have compiled a list of other possible z ≳ 7 candidates that narrowly missed our ERS selection. These sources are given in Table 20 along with our reasons for excluding them from our samples.

We have carefully crafted our selection criteria to optimally identify star-forming galaxies at z ∼ 7 and z ∼ 8 while minimizing contamination. However, some amount of contamination is almost inevitable, and so it is important to try to quantify the contamination rate. The five most important sources of contamination for our z ∼ 7 and z ∼ 8 selections are (1) low-mass stars, (2) SNe and other transient sources, (3) active galactic nuclei (AGNs), (4) lower redshift sources and photometric scatter, and (5) spurious sources. We consider each source of contamination in the paragraphs that follow.

3.5.1. Low-mass Stars

3.5.2. Transient Sources

3.5.3. Active Galactic Nuclei (AGNs)

3.5.4. Lower Redshift Sources

3.5.5. Lower Redshift Sources and Photometric Scatter

3.5.6. Spurious Sources

3.5.7. Summary

In total, 73 z ∼ 7 z₈₅₀-dropout candidates are identified over all of our search fields and 59 z ∼ 8 Y₁₀₅/Y₀₉₈-dropout candidates are found. A complete catalog of the dropouts in those samples is provided in Tables 10–17. Postage stamps of these dropout candidates are given in Figures 20 and 21.

A convenient summary of the properties of our z ∼ 7 z₈₅₀- and z ∼ 8 Y₁₀₅/Y₀₉₈-dropout samples and our search fields is provided in Table 2. The redshift distributions derived for the samples are given in Figure 4. The mean redshift for our z₈₅₀-dropout selections is 6.8 and 6.7 for our HUDF09 and ERS selections, respectively. Meanwhile, the mean redshift for our Y₁₀₅/Y₀₉₈-dropout selections is 8.0 and 7.8 for our HUDF09 and ERS selections, respectively.

Before using the present LBG selections to make inferences about the rest-frame UV LF at z ≳ 7 and its evolution across cosmic time, it is instructive to compare these samples with previous samples over the same fields.

4.1.1. Previous z ≳ 6.5 Galaxy Selections within the HUDF09 Observations Over the HUDF

4.1.2. Previous z ≳ 6.5 Galaxy Selections within the HUDF09-1/HUDF09-2 Observations

4.1.3. Previous z ≳ 6.5 Galaxy Selections in the ERS Observations

One of the most significant challenges in constructing large samples of z ⩾ 7 galaxies to very faint magnitudes is the limited depth of the optical data. Without deep optical data, it is very difficult to discriminate between bona fide high-redshift galaxies and z ∼ 1–2 galaxies that are simply faint in the optical. How confident are we that we have identified all the contaminants in our selections (or conversely that we have not discarded a substantial number of bona fide z ⩾ 7 galaxies)?

There are two arguments that suggest the situation is largely under control. First, carefully modeling the selection efficiency for each of our dropout samples and fixing the shape of the LF, we can estimate ϕ* for the z ∼ 7 and z ∼ 8 LFs on a field-by-field basis (Section 6.2). If we were misidentifying too many low-redshift galaxies as high-redshift galaxies—or vice versa—we would expect to observe huge fluctuations in the value of ϕ* in precisely those fields with the shallowest optical data. However, we find essentially the same value of ϕ* (within 20%) for each of the three ultra-deep fields we consider (HUDF09, HUDF09-1, and HUDF09-2). This suggests that we have a reasonable handle on contamination in our samples.

Second, we can look at the extent to which our z₈₅₀-dropout and Y₁₀₅/Y₀₉₈-dropout samples show detections in the optical data (quantified here using a χ²_opt statistic). If our selections were largely composed of bona fide z ∼ 7–8 galaxies, then we would expect there to be no more sources with positive detections in the optical data than there are negative detections. Indeed, our samples show a similar number of sources with negative and positive detections. See Appendix D.4 for more details.

It is worthwhile noting that much of this discussion will likely be highly relevant to future z ⩾ 7 selections with the James Webb Space Telescope—where the lack of visible imaging of comparable depths may make the selection of z ⩾ 7 galaxies challenging.

We first consider a stepwise determination of the UV LF at z ∼ 7. Stepwise determinations of the UV LF are valuable since they allow for a relatively model-independent determination of the shape of the LF. This ensures that our determinations are not biased to adhere to specific functional forms such as a Schechter function or power law.

Here the stepwise LF ϕ_k is derived using a very similar procedure to the Efstathiou et al. (1988) stepwise maximum likelihood (SWML) method. With this method, the goal is to find the shape of the LF which is most likely given the observed distribution of magnitudes in our search fields. Since only the shape of the distribution is considered in this approach (and there is no dependence on the volume density of sources), we would expect the LF to be largely insensitive to large-scale structure effects and hence more robust. Previously, we employed this procedure to derive the LFs at z ∼ 4–6 (Bouwens et al. 2007).

In detail, one can write this likelihood as

$$\begin{equation} {\cal L}=\Pi _{{\rm field}} \Pi _i p(m_i), \end{equation} \tag{ 2 }$$

where

$$\begin{equation} p(m_i) = \left(\frac{n_{{\rm expected},i}}{\Sigma _j n_{{\rm expected},j}} \right)^{n_{{\rm observed},i}}; \end{equation} \tag{ 3 }$$

n_{observed, i} is the observed number of sources in the magnitude interval i, n_{expected, j} is the expected number of sources in the magnitude interval j, and Π is the product symbol. The above expression gives the likelihood that a given magnitude distribution of sources is observed given a model LF. The expected number of sources n_{expected, i} is computed from a model LF as

$$\begin{equation} \Sigma _{k} \phi _k V_{j,k} = n_{{\rm expected},j}, \end{equation} \tag{ 4 }$$

where n_{expected, j} is the surface density of galaxies in some search with magnitude j, ϕ_k is the volume density of galaxies with absolute magnitude k, and V_{i, j, k} is the effective volume over which galaxies in the absolute magnitude interval k are selected with a measured magnitude in the interval j. With the V_{i, j, k} factors, we implicitly account for the effects of photometric scatter (noise) in the measured magnitudes on the derived LFs—allowing us to account for effects like the Malmquist bias on our LF determinations. The Malmquist bias is known to boost the number of sources near the selection limit, simply as a result of the lower significance sources (≲5σ) scattering into the selection. See Appendix B of Bouwens et al. (2007; see also Bouwens et al. 2008) for a more detailed description of the above maximum likelihood procedure. We elected to use magnitude intervals of width 0.5 mag for our LF determination as a compromise between S/N (i.e., the number of sources in each interval) and resolution in luminosity.

We estimate the selection volumes V_{j, k} for our LBG selections using the same set of simulations described in Appendix C of this paper. This procedure is the same as used in our many previous LF studies (e.g., Bouwens et al. 2007, 2008, 2010b) and involves taking the pixel-by-pixel profiles of actual HUDF z ∼ 4 B dropouts, inserting them at random positions in the actual observations, and then attempting to select them using the same procedures as we use on the real data (including the χ²_opt non-detection criterion we describe in Section 3.3 and Appendix D). The UV-continuum slopes β adopted for the z ∼ 7–8 model galaxies are chosen to match the observed color distribution (Bouwens et al. 2010a). The selection volumes V_{i, j, k} we derived from these simulations have a very similar form to those shown in Figure 8 of Bouwens et al. (2006) or Figure A2 of Bouwens et al. (2007). We would expect these volumes to be reasonably accurate—since we reproduce both the observed sizes of z ∼ 7–8 galaxies (Appendix A) and the UV-continuum slopes.

Due to its relative insensitivity to large-scale structure effects, the SWML procedure described above only provides us with constraints on the shape of the LF. To provide a normalization for the LF, we require that the total number of z ∼ 7–8 galaxies we predict from Equation (4) in our search fields matches those we actually find (after correcting for contamination; see Table 9 and Appendix B for details on how the precise corrections depend on search field and the luminosity of the sources).

By applying the above maximum likelihood procedure to our ERS, HUDF09, HUDF09-1, and HUDF09-2 z ∼ 7 samples, we derive the stepwise LF at z ∼ 7. The resulting LF is presented in Table 3 and Figure 8 (red circles). The z ∼ 7 LFs inferred from our z ∼ 7 ERS samples alone are shown separately (black squares) to demonstrate the consistency of our derived LFs in both the wide-area and ultra-deep data. Also included in Figure 8 are the LF determinations at z ∼ 7 from two wide-area searches (Ouchi et al. 2009; Bouwens et al. 2010c; see Table 4). These searches are important in establishing the shape of the LF at very high luminosities (≲ −21 AB mag).

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In the present section, we consider representations of the z ∼ 7 LF with a Schechter parameterization ($\phi ^* (\ln (10)/2.5) 10^{-0.4(M-M^{*})\alpha } e^{-10^{-0.4(M-M^{*})}}$). This parameterization features an exponential cutoff at high luminosities and a power-law shape at fainter luminosities. Such a parameterization has been almost universal in characterizing the shape of the galaxy LF since it works so well. From the results of the previous section (e.g., Figure 8), it also appears to be relevant in describing the LF of z ∼ 7 galaxies.

As with our stepwise LF determinations, we use a maximum likelihood approach that considers only the shape of the LF in deriving the best-fit Schechter parameters. The approach is analogous to that developed by Sandage et al. (1979), but is formulated in terms of the apparent magnitude distribution (see Section 3.1 of Bouwens et al. 2007). The approach has the important advantage that it is almost entirely insensitive to large-scale structure effects (see Appendix C of Bouwens et al. 2007).

To perform this likelihood analysis, we start with various Schechter parameter combinations, calculate the equivalent stepwise LF ϕ_k's (adopting 0.1 mag bins), and then make use of Equations (2) and (4). This technique allows us to set constraints on the characteristic luminosity M* and faint-end slope α. The lower panel of Figure 8 (dotted black lines) shows the 68% and 95% confidence intervals we are able to obtain on the z ∼ 7 LF.

Inspecting this lower panel, we see that there is a large degree of freedom in the Schechter parameters M* and α allowed by our search results. The principal reason for this latitude (which occurs despite the large number of sources and large luminosity range) lies in the almost featureless power-law shape exhibited by the LF at z ∼ 7 (Figure 8, top panel). This restricts us to characteristic luminosities M* brighter than ∼ − 19.7 and approximate power-law slopes α of −2.0.

To obtain much tighter constraints on the Schechter parameters α and M*, we need to incorporate observations which allow us to resolve more distinct features in the LF (e.g., the expected break at brighter magnitudes that is not apparent in our small-area HUDF09+ERS data sets). Such constraints can be obtained by combining our WFC3/IR results with wide-area searches (e.g., Ouchi et al. 2009; Castellano et al. 2010a, 2010b; Bouwens et al. 2010c) which more effectively probe the rarer sources found at the bright end of the LF. Such searches have consistently found a distinct falloff in the volume density of z ∼ 7 sources at bright magnitudes (e.g., upper panel in Figure 8). Several of the most notable wide-area z ∼ 7 searches (Table 4) include a ∼85 + 248 arcmin² search with NICMOS and ISAAC/MOIRCS by Bouwens et al. (2010c; see also Mannucci et al. 2007; Stanway et al. 2008; Henry et al. 2009), a ∼150 arcmin² search with VLT HAWK-I by Castellano et al. (2010b; but see also Castellano et al. 2010a and Hickey et al. 2010), and a 1568 arcmin² search with Subaru Suprime-Cam by Ouchi et al. (2009). The mean redshift expected for these samples is typically ∼6.8–7.0, which is comparable to that for the present WFC3/IR z₈₅₀-dropout selections.

Combining the wide-area z ∼ 7 search results with those available from the ultra-deep WFC3/IR observations, we arrive at much tighter constraints on M* and α. The best results are M* = −20.14  ±  0.26 and α = −2.01  ±  0.21 and are presented in the lower panel to Figure 8 with the solid red lines. In combining the wide-area search results with the WFC3/IR results (where the overlap in identified z ∼ 7 candidates is minimal), we find it convenient to modify our LF estimation methodology so that the likelihood incorporates information on the volume density of the LF (and not simply its shape)—comparing the volume density of the sources found with what is expected from a model LF. As in our Bouwens et al. (2008) LF analysis, likelihoods are computed assuming Poissonian statistics. We verified that the 68% and 95% likelihood contours we compute for the WFC3/IR search results reproduced those calculated using the STY method.

It is trivial for us to extend the present analysis to constrain the normalization ϕ* of the Schechter function. Utilizing the wide-area and ultra-deep WFC3 search results, we find a ϕ* of 0.86^+0.70_−0.39 × 10⁻³ Mpc⁻³. Together with the best-fit values of α and M*, we conveniently include these parameters in Table 5. The effects of large-scale structure add somewhat to the uncertainties on ϕ*. Given that most of the sources which contribute to the faint counts in the LF are from the CDF-South, we estimate that the added uncertainty is 25% on ϕ*—which corresponds to the sample variance expected assuming a ∼100 arcmin² search area (the approximate area within the CDF-South where our fields are found), a redshift selection volume of width Δz ∼ 0.8 (from Figure 4), and a bias factor of ∼5 (corresponding to halo volume densities of ∼10⁻³ Mpc⁻³). These estimates were made using the Trenti & Stiavelli (2008) cosmic variance calculator (see also Section 6.2).

Of course, the volume density ϕ* in the Schechter parameterization shows a significant covariance with both the characteristic luminosity and faint-end slope α, and so there is considerable freedom in the Schechter parameters that can fit current search results. We present the likelihood contours on ϕ* and the other two Schechter parameters in Section 7.1.

We now derive the LF at z ∼ 8 based upon the z ∼ 8 selections presented in Sections 3.2 and 3.4. As expected, we will find that the z ∼ 8 LF is less well determined than that at z ∼ 7, but it is still striking to find very interesting constraints from the combined HUDF09+ERS data sets.

We use the same procedure for deriving the stepwise LF and Schechter parameters at z ∼ 8 as we had earlier used at z ∼ 7. We base our LF determinations on our combined HUDF09 + ERS z ∼ 8 selections, but also fold in the null results from a shallow, ∼150 arcmin² Y-dropout search over the CDF-South, Bremer Deep Field (BDF), and NTTDF (Castellano et al. 2010b). Finally, to account for the fact that the mean redshift of our wide-area ERS z ∼ 8 selection is Δz ∼ 0.2 lower than for our HUDF09 z ∼ 8 selection, we adjust the volume densities implied by the ERS search downward by a factor of 1.15 (which we derive from a fit to the stepwise UV LF evolution from z ∼ 6 to z ∼ 4). We use 0.6 mag intervals for the stepwise LFs at z ∼ 8 because of the fewer sources in z ∼ 8 samples.

Our results are presented in Table 3 and Figure 9 (red points and upper limits; both the HUDF09 and the ERS observations are incorporated into the red circles). Also included in this figure are our z ∼ 8 LF using only the ERS observations. Not surprisingly, given the general form of the LFs at z ∼ 4–7, the z ∼ 8 LF maintains an approximate power-law form over a ∼3 mag range (red points), with a steep faint-end slope. Brightward of −21 AB mag, the LF appears to cut off somewhat. This cutoff is clear not only in the upper limits we derive for the volume density of luminous sources (red and green downward arrows from WFC3/IR and HAWK-I searches, respectively), but also in the z ∼ 8 ERS results (black squares).

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Nonetheless, sufficiently small area has been probed at present that the bright end of the z ∼ 8 LF remains fairly uncertain, and indeed there is significant field-to-field variation across our fields. The most notable case is over the HUDF09-2 field where four bright H_{160, AB} ⩽ 27 z ∼ 8 galaxy candidates are found, two lying within 3'' of each other (UDF092y-03781204, UDF092y-03751196) and a third H₁₆₀ ∼ 27 candidate within 30'' (UDF092y-04640529; see Table 16), each of which have similar Y₁₀₅ − J₁₂₅ colors and presumably redshifts. As a result of such variations, the bright end of the z ∼ 8 LF is understandably still quite uncertain, with some differences between that derived from the ERS observations alone (black squares) and from the combined HUDF09 + ERS results (red circles).

Despite the still somewhat sizeable uncertainties, we can model the z ∼ 8 LF using a Schechter function and derive constraints on the relevant parameters. Formal fits yield best-fit Schechter parameters of α = −1.91 ± 0.32, M* = −20.10 ± 0.52, and ϕ* = 0.59^+1.01_−0.37 × 10⁻³ Mpc⁻³ for the z ∼ 8 LF.¹¹ Though the significance is only modest, Schechter fits to the z ∼ 8 results (red line in Figure 9) are preferred at 2σ over fits to a power law (dashed line). The implications of such steep faint-end slopes are discussed below, but we caution against giving too much weight to these slopes at this time, given the current large uncertainties. Table 5 gives a compilation of these parameters, along with a comparison with Schechter determinations at other redshifts. The trends of the Schechter parameters with redshift are also discussed more broadly below.

6.1.1. z ∼ 7 LF

A large number of different determinations of the UV LFs at z ∼ 7 now exist (Bouwens et al. 2008; Oesch et al. 2009, 2010a; Ouchi et al. 2009; McLure et al. 2010; Castellano et al. 2010a, 2010b; Capak et al. 2011; Wilkins et al. 2011). Stepwise determinations of the LF from these studies are presented in Figure 10. The corresponding Schechter parameters are given in Table 6.

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Table 6. Determinations of the Best-fit Schechter Parameters for the Rest-frame UV LFs at z ∼ 7 (Section 6.1.1; see also Figure 10)

Reference	M*UV	ϕ* (10−3 Mpc−3)	α
This work	−20.14 ± 0.26	0.86+0.70−0.39	−2.01 ± 0.21
Castellano et al. (2010a)	−20.24 ± 0.45	0.35+0.16−0.11	−1.71 (fixed)
McLure et al. (2010)	−20.04	0.7	−1.71 (fixed)
Oesch et al. (2010a)	−19.91 ± 0.09	1.4 (fixed)	−1.77 ± 0.20
Ouchi et al. (2009)	−20.1 ± 0.76	0.69+2.62−0.55	−1.72 ± 0.65
Oesch et al. (2009)	−19.77 ± 0.30	1.4 (fixed)	−1.74 (fixed)
Bouwens et al. (2008)	−19.8 ± 0.4	1.1+1.7−0.7	−1.74 (fixed)

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It is instructive to compare the present LF determination at z ∼ 7 with those previously available. We begin with those studies where the primary focus was on the search for luminous, rarer z ∼ 7 galaxies (e.g., Ouchi et al. 2009; Castellano et al. 2010a, 2010b; Wilkins et al. 2010; Bouwens et al. 2010c). Perhaps the simplest point of comparison is the volume density of luminous (∼ − 21 AB mag) galaxies at z ∼ 7 relative to that at z ∼ 6. Ouchi et al. (2009), Bouwens et al. (2010c), and Wilkins et al. (2010) find a factor of two decrease in the volume density of luminous (−21 AB mag) galaxies over this interval, while Castellano et al. (2010a, 2010b) find a factor of ∼3.5 decrease.¹² In the present search, we find a similar factor of ∼1.8 decrease at the bright end of the LF (i.e., −20.5 AB mag).

At fainter magnitudes, our HUDF09 search fields are essentially unique in the constraints they provide on the UV LF at z ∼ 7. Nonetheless, it is useful to compare against previous work, particularly from the early ultra-deep WFC3/IR pointing. Our results at −19 AB mag are in good agreement with the UV LF of Oesch et al. (2010a), but are somewhat higher than those found by McLure et al. (2010).

Most studies find best-fit values of M* and ϕ* at z ∼ 7 that are slightly fainter and of lower density than is found at z ∼ 4–6 (Bouwens et al. 2008; Oesch et al. 2009; Ouchi et al. 2009; Oesch et al. 2010a; McLure et al. 2010; Castellano et al. 2010a, 2010b). In the present study, we find something similar. The M* we find at z ∼ 7 is fainter than the z ∼ 6 value (Bouwens et al. 2007) by ∼0.1 mag and the ϕ* we find at z ∼ 7 is lower by a factor of ∼1.8. Of course, ϕ* and M* are sufficiently degenerate at z ∼ 7 that there is a modest tradeoff between the evolution found in either parameter.

The faint-end slope to our UV LF is steep with α = −2.01 ± 0.21. Previously, it was not possible to set tight constraints on the faint-end slope. Ouchi et al. (2009) estimated a slope α of −1.72 ± 0.65 while Oesch et al. (2010a)—fixing ϕ* to 0.0014 Mpc⁻³—found −1.77 ± 0.20. The present values of the faint-end slope are consistent with those measured at z ∼ 4–6 where this slope was also found to be steep at α ∼ −1.7 (Bouwens et al. 2007; Oesch et al. 2007; Yoshida et al. 2006; Yan & Windhorst 2004). This new result may suggest a possible steepening toward higher redshifts z ≳ 7, but the significance is weak, i.e., ∼1.5σ. Such a steepening is expected in many theoretical models (e.g., Salvaterra et al. 2011; Trenti et al. 2010; Jaacks et al. 2011). Clearly we will require deeper data to confirm this trend.

Capak et al. (2011) have suggested that the z ∼ 7 LF may extend to substantially brighter magnitudes than have been considered by other groups. Performing follow-up spectroscopy on three very bright (H ∼ 23 mag) z ∼ 7 candidates over the COSMOS field, Capak et al. (2011) claim to have uncovered statistically significant detections of line flux. These lines could represent possible Lyα emission given the existence of strong continuum breaks in the sources. If this is the explanation, these sources would imply that the UV LF at z ∼ 7 extends to very bright magnitudes indeed, i.e., −24 AB mag, and in fact the LF that such sources would suggest is essentially power law in shape.

What should we make of this intriguing possibility? First, such an LF would seem to violate the constraints on the bright end of the LF set by deep, wide-area searches for z ∼ 7 galaxies (Ouchi et al. 2009; Bouwens et al. 2010c; Castellano et al. 2010a, 2010b). Second, it is difficult to understand how one might create such bright −24 AB mag sources in the UV. The challenge is that any source forming stars rapidly enough to be so bright in the UV (i.e., ∼200 M_☉ yr⁻¹) would most likely have built up sufficient quantities of dust and metals to be quite optically thick in the UV. Under such circumstances, it is unlikely that the source would be brighter than ∼−22 AB mag. Such is the situation at significantly later times (z ∼ 2–3) where there are exceedingly few systems with UV magnitudes brighter than ∼ − 23 (e.g., Reddy et al. 2008), but many ULIRGs (e.g., Chapman et al. 2005). It is hard to imagine, therefore, that at z ∼ 7, just 500–600 Myr after the first stars, there is a population of galaxies that are substantially brighter in the UV than at z ∼ 3. Additional work utilizing deeper spectroscopic and photometric data is clearly required.

6.1.2. z ∼ 8 LF

There are also several published determinations of the UV LF at z ∼ 8. These LFs are presented in stepwise form in Figure 11. Schechter parameters for these LFs are given in Table 7. In general, the present z ∼ 8 LF determination is in good agreement with previous determinations, within the quoted errors—though the present z ∼ 8 LF is a little higher than the McLure et al. (2010) result at the lowest luminosities.

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Table 7. Determinations of the Best-fit Schechter Parameters for the Rest-frame UV LFs at z ∼ 8 (Section 6.1.2; see also Figure 11)

Reference	M*UV	ϕ* (10−3 Mpc−3)	α
This worka	−20.10 ± 0.52	0.59+1.01−0.37	−1.91 ± 0.32
McLure et al. (2010)	−20.04	0.35	−1.71 (fixed)
Bouwens et al. (2010b)	−19.5 ± 0.3	1.1 (fixed)	−1.74 (fixed)
Lorenzoni et al. (2011)	−19.5	0.93	−1.7 (fixed)

Notes. ^aThe derived Schechter parameters depend significantly on whether we include or exclude the two brightest (∼26 AB mag) z ∼ 8 galaxies in the HUDF09-2 field. If we exclude the two brightest galaxies (assuming they represent a rare overdensity), we derive a significantly fainter characteristic luminosity M* and larger normalization ϕ*: M* = −19.54 ± 0.56, ϕ* = 1.5^+2.9_−1.0 × 10⁻³ Mpc⁻³, and α = −1.67 ± 0.40 (similar to the values found by Bouwens et al. 2010b and Lorenzoni et al. 2011).

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For the derived Schechter parameters (see Table 7), we find an M* of −20.10 ± 0.52, a ϕ* of 0.59^+1.01_−0.37 × 10⁻³ Mpc⁻³, and faint-end slope α = −1.91 ± 0.32. These M* and ϕ* values are somewhat brighter and lower, respectively, than what we find at z ∼ 6 and z ∼ 7. This might indicate that some of the evolution in the UV LF from z ∼ 8 to z ≲ 6 can be represented by a change in the volume density ϕ*. However, the uncertainties on ϕ* and M* are too large to be sure of these conclusions. In particular, the existence of a possible overdensity at z ∼ 8 within the HUDF09-2 field (see Section 5.3) illustrates the challenges of establishing reliable parameters from these initial data sets.

One of the most important challenges for LF determinations is large-scale structure. Variations in the volume density of galaxies as a function of position can have a significant impact on both the shape and overall normalization of the derived LFs. As a result of our use of the SWML and STY procedures (Efstathiou et al. 1988; Sandage et al. 1979) to derive the LF, the impact of large-scale structure on the shape of our LF should be minimal. On the other hand, we would expect this structure to have an effect on the overall normalization of our LFs. One can estimate the size of these uncertainties using the Trenti & Stiavelli (2008) cosmic variance calculator. Using the approximate volume densities ∼5 × 10⁻³ Mpc⁻⁴ we derived for z ∼ 7–8 sources, the width of our redshift selection windows Δz ∼ 1, and the 10' × 10' search geometry we use within the CDF-South, we find that the uncertainty in the normalization of the LF at z ∼ 7 and z ∼ 8 is a very modest ∼25% and ∼20%, respectively.

Large-scale structure variations can also have an effect on the faint-end slope α determinations as a result of mass-dependent (and hence likely luminosity-dependent) bias parameters (e.g., Robertson 2010). We estimate the luminosity-dependent bias parameters based upon the approximate halo masses that correspond to the volume density of sources we are probing at specific points in the LF (again using the Trenti & Stiavelli 2008 calculator). We then perturb the LFs using these bias parameters and the power spectrum at z ≳ 7 and determine the effect on the faint-end slope α. We find that large-scale structure introduces a 1σ uncertainty of ∼0.05 in our faint-end slope determinations α at z ∼ 7 and z ∼ 8.

We can also look at the issue of field-to-field variations from a more empirical standpoint. Fixing the shape of the UV LFs at z ∼ 7 and z ∼ 8 to the best-fit α and M*, we can look at the variations in the normalization ϕ* for our three HUDF09 fields. We find a 1σ variation in ϕ* of just ∼20%. This is similar to what we would expect from the Poissonian uncertainties. This suggests that large-scale structure uncertainties do not pose an especially huge problem for LF determinations in the current luminosity range (−21 AB to −18 AB mag).

Thus, the impact of large-scale structure on our z ∼ 7 and z ∼ 8 LFs appears to be quite modest.

In Section 5, we derived detailed constraints on the UV LFs at z ∼ 7 and z ∼ 8. By comparing these LFs with those derived at z ∼ 4–6, we can examine the evolution with cosmic time. In Figure 12, we compare the stepwise LFs derived at z ∼ 7 and z ∼ 8 with those found at z ∼ 4–6. The z ∼ 7 LF presented in Figure 12 incorporates the wide-area NICMOS + ISAAC + MOIRCS z ∼ 7 search results from Bouwens et al. (2010c; see the LFs in Figure 10). It is remarkable how uniform the evolution of the UV LF is with cosmic time. At higher luminosities (−20 AB mag), the UV LF evolves by a factor of ∼6–7 from z ∼ 8 to z ∼ 4 while at lower luminosities (−18 AB mag), the decrease is a factor of ∼5.

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Figure 13 presents the 68% and 95% confidence intervals derived here for the UV LF at z ∼ 7 and z ∼ 8. For comparison, the LFs at z ∼ 4, z ∼ 5, and z ∼ 6 are also shown so that the evolution can readily be inferred. A clear trend is evident in the M*/ϕ* LF fit results (left panel) from z ∼ 8 to z ∼ 4. The characteristic luminosity shows evidence for a consistent brightening with cosmic time from z ∼ 7 to z ∼ 4. Ascertaining how the ϕ* and M* evolves out to z ∼ 8 is not possible at present, given the large uncertainties and degeneracy between these two parameters (though there is a hint that some of the evolution may be in ϕ*; see also the discussion in McLure et al. 2010). Tighter constraints on the volume density of bright z ∼ 8 galaxies are required for progress on this question.

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As far as constraints on the shape of the LF (middle panel), again we see a clear trend toward fainter values of M* from z ∼ 7 to z ∼ 4. The faint-end slope α at z ∼ 7 and z ∼ 8 also is very steep, similar to the values α ∼ −1.7 found at z ∼ 2–6 (Figure 15) and possibly even steeper. This possible steepening of the LF with redshift is in qualitative agreement with that predicted from various theoretical models for the LF (e.g., Trenti et al. 2010; Salvaterra et al. 2011; Bouwens et al. 2008). Such steep faint-end slopes are potentially very important for reionization since it would suggest that low luminosity galaxies at z ≳ 7 are major contributors to the needed UV ionizing photon flux. We discuss this in detail in Bouwens et al. (2011b).

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No strong trends are evident as a function of redshift in ϕ* and α considered together (Figure 13, right panel). However, the uncertainties remain large at z ∼ 7 and z ∼ 8. Given these uncertainties, we can potentially gain more insight into the evolution of the LF at high redshift by fixing the faint-end slope α to −1.7. The faint-end slope α is found to be consistent with this value over a very wide redshift range z ∼ 1–8 (Bouwens et al. 2007; Reddy & Steidel 2009; Beckwith et al. 2006; McLure et al. 2009; Ouchi et al. 2009; Oesch et al. 2007, 2010a, 2010c), so it seems reasonable to fix α to −1.7 at z ∼ 7 and z ∼ 8. The constraints we derive on M* and ϕ* fixing α are shown in Figure 15. This figure provides further support to the suggestion that the primary mode of evolution in the LF from z ∼ 7 to z ∼ 4 is in the characteristic luminosity (see also Bouwens et al. 2006, 2007; Yoshida et al. 2006; McLure et al. 2009). For z ∼ 8 LF, the normalization is somewhat lower—suggesting that some of the evolution in the LF at z ⩾ 4 may be in volume density (e.g., McLure et al. 2009; Castellano et al. 2010b).

One potentially important open question about the UV LFs at z ⩾ 7 concerns their overall shape (e.g., see discussion in Section 5.5 of Bouwens et al. 2008). While we have generally attempted to represent the LFs here with Schechter functions (i.e., a power law with an exponential cutoff at the bright end; see Sections 5.2 and 5.3), it is possible that the LFs may have a shape closer to that of a power law, e.g., resembling the halo mass function (e.g., Douglas et al. 2009). This is simply because at the high redshifts we are probing the typical halo masses would be much lower (<10¹² M_☉) and therefore many of the physical mechanisms likely to be important for truncating the LFs at z ∼ 0 (e.g., transition to hot flows: Birnboim & Dekel 2003; Rees & Ostriker 1977; Binney 1977; Silk 1977 or AGN feedback: Croton et al. 2006) may not be as relevant at z ≳ 8. Of course, at sufficiently high redshifts (i.e., z ≳ 7), we would eventually expect some truncation in the LF at high masses (and luminosities) due to a similar exponential cutoff in the halo mass function (e.g., Trenti et al. 2010).

Overall, we do not find any evidence that the UV LF at z ∼ 7 or z ∼ 8 has a fundamentally different shape than the LF has at lower redshift. Both LFs seem to have roughly a power-law-like shape at the faint end and a modest cutoff at the bright end—consistent with that of a Schechter function (though the evidence for such a cutoff in the z ∼ 8 LF is weak). Ascertaining whether the LF really has a Schechter-like form or has a tail extending to bright magnitudes (e.g., Capak et al. 2011) will require significantly more statistics, particularly at bright magnitudes, to better establish the shape of the LF there. Fortunately, we should soon have such statistics as a result of observations being obtained over the GOODS fields as part of the CANDELS multi-cycle treasury program, the wide-area SEDS/CANDELS fields, and two pure parallel programs (Trenti et al. 2011; Yan et al. 2011).

With the completion of the wide-area ERS and ultra-deep HUDF09 observations, we have been able to significantly improve our constraints on the UV LFs at z ∼ 7 and z ∼ 8. These new LF determinations allow us to update our luminosity density and SFR density estimates at z ∼ 7–8 from those already given in Bouwens et al. (2010b) based on the initial HUDF09 data on the HUDF. These densities continue to be of great interest for discussions about the role of galaxies in reionization as well as the buildup of stellar mass.

For our luminosity density (and SFR density) estimates, we simply integrate up our stepwise LF determinations to the limiting depth of our LF determinations, i.e., to −17.7 AB mag (0.05 L*_{z = 3}). The resultant luminosity densities are tabulated in Table 8. They are also shown in Figure 16 along with luminosity density determinations at lower redshift.

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Table 8. UV Luminosity Densities and Star Formation Rate Densities to −17.7 AB mag^a

Dropout	〈z〉	$\hbox{log}_{10} \mathcal {L}$	log10 SFR density
Sample		(erg s−1	(M☉ Mpc−3 yr−1)
		Hz−1 Mpc−3)	Uncorrected	Correctedb
z	6.8	25.88 ± 0.10	−2.02 ± 0.10	−2.02 ± 0.10
Y	8.0	25.65 ± 0.11	−2.25 ± 0.11	−2.25 ± 0.11
Jd	10.3	24.29+0.51−0.76	−3.61+0.51−0.76	−3.61+0.51−0.76
Jd	10.3	<24.42c	<−3.48c	<−3.48c
B	3.8	26.38 ± 0.05	−1.52 ± 0.05	−0.90 ± 0.05
V	5.0	26.08 ± 0.06	−1.82 ± 0.06	−1.57 ± 0.06
i	5.9	26.02 ± 0.08	−1.88 ± 0.08	−1.73 ± 0.08

Notes. ^aIntegrated down to 0.05 L*_{z = 3}. Based upon LF parameters in Table 5 (see Section 7.3). The SFR density estimates assume ≳ 100 Myr constant SFR and a Salpeter IMF (e.g., Madau et al. 1998). Conversion to a Kroupa (2001) IMF would result in a factor of ∼1.7 (0.23 dex) decrease in the SFR density estimates given here. ^bThe dust correction is taken to be 0 for galaxies at z ∼ 7–8, given the very blue β's (see also Bouwens et al. 2009, 2010a; Oesch et al. 2010a). ^cUpper limits here are 1σ (68% confidence). ^dz ∼ 10 determinations and limits are from Bouwens et al. (2011) and assume 0.8 z ∼ 10 candidates in the first case and no z ∼ 10 candidates (i.e., an upper limit) in the second case.

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The luminosity density determinations are converted into SFR densities using the Madau et al. (1998) conversion factor—assuming a Salpeter initial mass function (IMF; with a stellar mass function ranging from 0.1 M_☉ and 125 M_☉). As often noted (e.g., Verma et al. 2007; Stanway et al. 2005; Bouwens et al. 2009), this conversion factor assumes that the SFRs in galaxies are relatively constant for ≳ 100 Myr prior to observation. Consequently, use of this conversion would result in an underestimate of the SFR (by factors of ∼2) for galaxies with particularly young ages.

Given the very blue UV-continuum slopes β of z ≳ 7 galaxies (e.g., Bouwens et al. 2010a, 2010b; Oesch et al. 2010a; Finkelstein et al. 2010; Bunker et al. 2010; see also Bouwens et al. 2009), the dust extinction in z ≳ 7 galaxies is likely small, and so no correction for dust is made in computing the SFR densities at z ≳ 7. At z ≲ 6, we adopt the dust corrections given in Schiminovich et al. (2005), Reddy & Steidel (2009), and Bouwens et al. (2009). Our SFR density estimates are included in Table 8 and in Figure 16.

The UV luminosity densities we find to −17.7 AB mag at z ∼ 7 and z ∼ 8 are just 32% and 18%, respectively, of that found to the same limiting luminosity at z ∼ 4. The SFR densities we find to these limits are just 6% and 4%, respectively, of that at z ∼ 4. This provides some measure of how substantial galaxy buildup has been in the ∼1 Gyr from z ∼ 7–8 to z ∼ 4 when the universe was nearing its peak SFR density at z ∼ 2–3.

It is interesting to compare the present SFR density determinations with that inferred from recent estimates of stellar mass density at z > 4 (Labbé et al. 2010a, 2010b; Stark et al. 2009; González et al. 2010). The comparison provides us with a very approximate test of the shape of the IMF at high redshift as well as allowing us to test the basic consistency of these very fundamental measurements.

In estimating the SFR densities from the stellar mass densities (Stark et al. 2009; Labbé et al. 2010a, 2010b; González et al. 2010), there are a few issues we need to consider. One issue is the stellar mass lost due to SNe recycling (Bruzual & Charlot 2003). To correct for this, we have multiplied the SFR densities we infer from the stellar mass densities by a factor of 1/0.72 ∼ 1.39 (appropriate for the Salpeter IMF assumed in the Labbé et al. 2010b mass estimates and a ∼300 Myr typical age).

A second issue we need to consider is that new galaxies will be entering magnitude-limited samples at all redshifts simply due to the growth of galaxies with cosmic time (e.g., galaxies not present in a z ∼ 7 sample would grow enough from z ∼ 7 to z ∼ 6 to enter magnitude-limited selections at z ∼ 6). Therefore, it is not quite possible to determine the SFR density by subtracting the stellar mass density determination in one redshift interval from that immediately below it. To account for those galaxies entering magnitude-limited samples due to luminosity growth, galaxies are assumed to brighten by 0.33 mag per unit redshift (the approximate M* evolution; see Section 7.5). This results in a typical 20% correction to the SFR density.

Based on these corrections and considerations and the stellar mass density at z ∼ 8 (Labbé et al. 2010b), we derive the SFR density down to z ∼ 4. The results are shown in Figure 17 (hatched red region).¹³ Interestingly, the SFR densities inferred from the stellar mass density are only ∼40% higher than that inferred from the UV LFs at z < 7 (where the stellar masses are more well determined).¹⁴ This suggests that (1) the IMF for SF galaxies at high redshift (z ∼ 4–7) may not be that different from what it is at later times (z ≲ 3; but see discussion in Chary 2008) and (2) the stellar mass densities at z > 4 inferred from the Spitzer Infrared Array Camera (IRAC) photometry are reasonably accurate. This latter conclusion is important since nebular emission line fluxes have been suggested to dominate the measured light in z ⩾ 7 galaxies with IRAC (Schaerer & de Barros 2010; but see Finlator et al. 2011). These comparisons suggest that nebular emission is not as dominant as has been suggested (Schaerer & de Barros 2010).

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The present LF determinations provide us with reasonably accurate measures of luminosity density and SFR density at z ∼ 7 and z ∼ 8—which are of considerable importance for assessing the impact of galaxies in the buildup of mass and metals in the universe, as well as understanding their role in reionization. However, knowing the LF results at z ∼ 7–8 only provides us with part of the information we want. To address some of the most outstanding of today's questions (i.e., whether galaxies reionize the universe, how stellar mass and metals are built up versus cosmic time), we really need to know what the UV LFs look like at even higher redshift.

For these questions, perhaps the best approach is simply to extrapolate the present LF results to even higher redshifts. Certainly, given the very uniform rate of evolution of the UV LF from z ∼ 8 to z ∼ 4, it seems quite plausible that such an extrapolation would be reasonably accurate. As in our previous work (Section 5.3 of Bouwens et al. 2008), we frame the evolution of the LF in terms of a Schechter function and assume that each Schechter parameter M*, α, and ϕ* can be expressed as a linear function of redshift. We then identify linear coefficients that maximize the likelihood of reproducing the observed contours in Figure 13 at z ∼ 4, z ∼ 5, z ∼ 6, z ∼ 7, and z ∼ 8. The result is

$$\begin{eqnarray*} M_{{\rm UV}} ^{*} =& (-21.02\pm 0.09) + (0.33\pm 0.06) (z - 3.8)\\ \phi ^* =& (1.14\pm 0.20) 10^{(0.003\pm 0.055)(z-3.8)}10^{-3} \,\hbox{Mpc}^{-3}\\ \alpha =& (-1.73\pm 0.05) + (-0.01\pm 0.04)(z-3.8). \end{eqnarray*}$$

We see that the evolution in M*_UV is significant at the 5σ level. However, the evolution in ϕ* and α is still not significant. The fitting formulas above are similar to those found in Bouwens et al. (2008) based upon a large z ∼ 4–6 ACS sample and z ∼ 7 NICMOS sample.

In the previous section, we combined the present constraints on the UV LFs at z ∼ 7 and z ∼ 8 with those available at z ∼ 4–6 to derive fitting formulas for the evolution of the UV LF with redshift. These formulas arguably provide us with our best means for extrapolating the UV LF to even higher redshifts and therefore estimating the form of the UV LF at z ∼ 10. This enterprise has important implications for z ∼ 10 galaxy searches in present and future WFC3/IR observations.

What surface density of z ∼ 10 J₁₂₅-dropout candidates do we expect to find in current ultra-deep WFC3/IR observations? We can estimate this by taking advantage of the fitting formulas we derive in the previous section and the detailed simulations carried out by Bouwens et al. (2011a) to estimate J₁₂₅-dropout selection efficiencies. We predict 1.0 and 2.3 J₁₂₅-dropouts arcmin⁻² to an H₁₆₀ magnitude of 28.5 and 29.5 AB mag, respectively. These numbers are similar albeit somewhat larger than that found by Bouwens et al. (2011a) over the HUDF to H₁₆₀ ∼ 29.5. Certainly, the fact that a few J₁₂₅-dropout galaxies are expected in the HUDF search suggests that the z ∼ 10 galaxy candidate identified by Bouwens et al. (2011a) is plausible. Of course, deeper observations are clearly needed for more definitive statements. We do not address the Yan et al. (2010) J₁₂₅-dropout numbers here, given the challenges in interpreting those numbers (both as a result of very low 3σ detection thresholds used in that study and the likely high contamination rates expected from the proximity of the many Yan et al. (2010) candidates to bright foreground galaxies (see discussion in Bouwens et al. 2011a).

To make maximal use of the optical imaging data that exist over the HUDF09 and GOODS fields, we clearly need to use a more sophisticated procedure. To this end, we define the quantity χ²_opt that effectively combines information in all the optical imaging data. We define χ²_opt as

$$\begin{equation} \chi _{{\rm opt}}^2 = \Sigma _{i} \hbox{SGN}(f_{i}) (f_{i}/\sigma _{i})^2, \end{equation} \tag{ D1 }$$

where f_i is the flux in band i in our smaller scalable apertures, σ_i is the uncertainty in this flux, and SGN(f_i) is equal to 1 if f_i > 0 and −1 if f_i < 0. Included in the computed χ²_opt values are the B₄₃₅-, V₆₀₆-, and i₇₇₅-band observations for the z ∼ 7 z₈₅₀-dropout selection and the B₄₃₅-, V₆₀₆-, i₇₇₅-, and z₈₅₀-band observations for the z ∼ 8 Y₁₀₅-dropout selection.

How shall we use the measured χ²_opt values for sources in our selection to exclude low-redshift contaminants? To answer this question, we must know the χ²_opt distribution expected for bona fide z ⩾ 7 galaxies and also that distribution expected to come from possible low-redshift contaminants. The χ²_opt distribution for z ∼ 7 candidates is calculated assuming no optical flux in z ∼ 7 candidates and ideal noise properties. Meanwhile, the χ²_opt distribution for the low-redshift contaminants is calculated in the same way as we calculate contamination from photometric scatter in Section 3.5, i.e., starting with the color distribution for an intermediate magnitude subsample of real sources in our data and then adding noise to match the fluxes of faint sources in our search fields. While discussing the use of χ²_opt to control for contamination in our z ≳ 6 selections, we will assume that we have already removed sources that are detected at ⩾2σ in individual images.

We show the model χ²_opt distributions in Figure 19 for the z ∼ 7 selections we perform over the ERS, HUDF09, and HUDF09-1 fields (which represent a good sampling of the selections considered here). For the HUDF09 and HUDF09-1 fields, these distributions are shown ∼0.7 mag from the approximate selection limit for each field. As Figure 19 shows, the χ²_opt values for z ≳ 7 galaxies are centered on 0 and have lower values than for low-redshift galaxies. However, due to the effects of noise, both z ≳ 7 galaxies and low-redshift contaminants are spread over a wide range in χ²_opt values.

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Indeed, it is striking how low the χ²_opt values are expected to become for some low-redshift galaxy contaminants that would appear in our selections. At relatively low χ²_opt values of 3, for example, the total number of contaminants in our ERS or HUDF09-1 selections (red lines) is expected to exceed the number of bona fide z ⩾ 7 galaxies (blue lines: Figure 19). A few contaminants are even expected to have observed χ²_opt values that are negative, particularly in those fields with the shallowest optical observations (i.e., the ERS or HUDF09-1 data). The prevalence of contaminants to even very low values of χ²_opt underlines the importance of adopting very conservative optical non-detection criteria in our selection.

We only select galaxies to a value of χ²_opt where we expect the number of z ∼ 7 galaxies to be greater than the number of contaminants. This critical value χ²_lim depends on both the depth of the optical data and the magnitude of the source—since we would expect our fainter selections to admit more contamination than our brighter selections. We have found that χ²_lim values of 5, 3, and 5, respectively, worked well for our HUDF09, HUDF09-1, and HUDF09-2 data sets, respectively, at brighter (J₁₂₅, H₁₆₀ ≲ 28.5) magnitudes; however, at fainter magnitudes (J₁₂₅, H₁₆₀ ∼ 29.2), χ²_lim needed to be 2.5, 1.5, and 2.5, respectively, for the same fields. For sources with magnitudes between 28.5 and 29.2, the upper limits on χ²_opt are an interpolation between these extremes. For the small fraction (∼30%) of sources in the HUDF09-2 field without additional optical/ACS coverage from our HUDF09 program (those not within the P1 field shown in Figure 1), it was appropriate to adopt the same χ²_opt limits as for the HUDF09-1 field.

We derive χ²_opt values for our z ⩾ 7 candidates in three different apertures: 035 diameter circular apertures, small 018 diameter circular apertures, and Kron-style scalable apertures (Kron factor of 1.2). The use of three different apertures allows us to be much more comprehensive in our identification of possible optical light in sources (which may be spatially very compact or extended) and therefore rejection of these sources as low-z contaminants. No smoothing of the ACS optical images is performed prior to making these formal flux measurements. This ensures that we obtain the highest possible S/N for these flux measurements—effectively taking advantage of the sharper PSF in the ACS images.

The χ²_opt distributions observed for our samples also provide a convenient way of ascertaining the actual number of bona fide z ⩾ 7 galaxies in our search fields and therefore the contamination rate (by inference). Looking at Figure 19, we see that an exceedingly small number of contaminants are expected to have χ²_opt <0 from the simulations, and so essentially all of sources satisfying our LBG criteria with χ²_opt <0 are bona fide z ∼ 7–8 galaxies. However, we also know the χ²_opt <0 requirement introduces an incompleteness of ∼50% (simply because ∼50% of the sources will have positive values of χ²_opt and ∼50% negative values—given our definition of χ²_opt). Therefore, the actual number of bona fide z ⩾ 7 candidates is approximately double the number of candidates with χ²_opt <0. Moreover, since we quantify the χ²_opt distribution for each search field in three different apertures (035 diameter circular, 018 diameter circular, scalable Kron), this provides us with three different estimates of the actual number of z ⩾ 7 candidates.

Reassuringly, the number of high-redshift candidates in our samples is in good agreement with the numbers derived from the above argumentation. Multiplying the number of z ∼ 7 candidates with χ²_opt < 0 by 2, we expect 28.7, 15.3, 15.3, and 8.7 z ∼ 7 candidates in the HUDF09, HUDF09-1, HUDF09-2, and ERS fields, respectively, versus the 29, 17, 14, and 13 candidates in our observed samples. Similarly, we expect 28, 15.3, 10.7, and 6 z ∼ 8 candidates, respectively, in the above fields versus the 24, 14, 15, and 6 candidates in our observed samples.

It is instructive to contrast the present criteria for rejecting low-redshift contaminants with other frequently used methods. Perhaps the most common of these is one relying just upon the 2σ detections of the sources in individual optical bands for removing low-redshift contaminants from high-redshift selections (e.g., Bouwens et al. 2007; Wilkins et al. 2011). The difficulty with this approach is that it relies upon the optical data being very deep in each individual band (which is rarely the case).

As an example of the limitations of this approach we found that had we only used a 2σ-optical rejection criterion in selecting z ∼ 7 and z ∼ 8 galaxies, a large fraction (∼30%–40%) of our z ∼ 7–8 samples over the HUDF09-1, HUDF09-2, and ERS fields would have consisted of contaminants. Such large contamination rates significantly reduce the value of the samples for quantitative work.

The red lines in Figure 19 show the χ²_opt distribution expected for low-redshift contaminants not detected in any optical band at >2σ. The area under the red curves in Figure 19 is comparable to that under the blue curves for data sets like HUDF09-1 or the ERS fields. Clearly, contamination from low-redshift sources can be quite substantial and have a significant impact in fields with shallower optical data (e.g., recall the discussion of contamination in the Wilkins et al. (2010, 2011) sample in Sections 4.1.2 and 4.1.3). This is perhaps an even more important concern than uncertainties in the size distribution at faint magnitudes (see, e.g., Bouwens et al. 2004; Grazian et al. 2011). Careful use of measures like χ²_opt is required to limit contamination (or correct for it statistically in a robust way).