UV LUMINOSITY FUNCTIONS AT REDSHIFTS z ∼ 4 TO z ∼ 10: 10,000 GALAXIES FROM HST LEGACY FIELDS^{* †}

3.1.1.  HST Photometry

3.1.2. Photometry on Ground-based Imaging Data

3.1.3. IRAC Photometry

3.2.1. Lyman Break Selection Criteria

As in previous work, we construct the bulk of our high-redshift samples using two color Lyman-break-like criteria. Substantial spectroscopic follow-up work has shown that this approach is quite effective at identifying large samples of star-forming galaxies at $z\gtrsim 3$ (Steidel et al. 1999; Bunker et al. 2003; Dow-Hygelund et al. 2007; Popesso et al. 2009; Vanzella et al. 2009; Stark et al. 2010).

Lyman break samples typically take advantage of three pieces of information in identifying probable sources at high redshift: (1) color information from two adjacent passbands necessary to locate the position and measure the amplitude of the Lyman break, (2) color information redward of the break needed to define the intrinsic color of the source (thereby distinguishing the selected high-redshift sources from intrinsically red galaxies), and (3) evidence that sources show essentially no flux blueward of the spectral break.

Our selection is constructed to take advantage of all three pieces of information and to do so in a suitably optimal manner, within the context of simple color criteria. The most noteworthy gains can be achieved by taking advantage of the additional wavelength leverage provided by the deep near-IR and mid-IR observations for constraining the intrinsic colors of candidate sources. This allows us to go beyond what is possible from the Lyman-break-like selection utilized in Giavalisco et al. (2004b) and Bouwens et al. (2007). Obviously, the colors that provide us with the most significant leverage in probing the intrinsic colors of the sources are those we would use to provide optimal measurements of the spectral slope β (e.g., we use the same ${{i}_{775}}-{{J}_{125}}$ color below in constructing our color criterion for the $z\sim 4$ selection as would be optimal for deriving β for $z\sim 4$ galaxies; Bouwens et al. 2012b, 2014b).

While one could consider selecting $z\sim 4$–10 samples based on the best-fit photometric redshift or redshift likelihood contours (e.g., McLure et al. 2010; Finkelstein et al. 2012; Bradley et al. 2014; see Figure 1, right panel), Lyman break selection procedures can be simpler to apply and offer a slight advantage in terms of operational transparency. This makes such a selection procedure easier to reproduce by both theorists and observers, as follow-up studies by Shimizu et al. (2014), Lorenzoni et al. (2013), and Schenker et al. (2013) utilizing our color criteria all illustrate.

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Despite the present procedural choice, photometric redshift techniques also work quite well, particularly when used with a well-calibrated prior or as refinements to the redshift estimate, as direct comparisons between LF determinations conducted using Lyman-break-like selection criteria (e.g., Schenker et al. 2013) and photometric redshift selection criteria (e.g., McLure et al. 2013) illustrate. Indeed, we will be utilizing photometric redshift techniques in Section 3.2 to redistribute sources across our CANDELS-UDS/COSMOS/EGS $z\sim 5$, $z\sim 6$, $z\sim 7$, and $z\sim 8$ samples based on our best-estimate redshifts from the HST+ground-based+Spitzer/IRAC observations.

3.2.2. XDF, HUDF09-1, HUDF09-2, CANDELS-GS, CANDELS-GN, ERS, BoRG/HIPPIES

In this section we describe the selection criteria we employ for data sets with deep observations in the Y band with HST, i.e., the XDF, HUDF09-1, HUDF09-2, CANDELS-GS, CANDELS-GN, ERS, and BoRG/HIPPIES fields.

We have constructed two-color selection criteria so that the lower-redshift boundary is approximately the same for sources independent of their spectral slope. For those areas where the Y-band observations are available in the Y₁₀₅-band filter, we use one set of criteria, while for those areas where the Y₀₉₈ is available, we employ an alternate set of selection criteria. The specific color criteria we have developed are presented in Table 2.

Table 2.  Criteria Utilized in Selecting Our $z\sim 4$–10 Samples^a

	Data Set
Sample	XDF, HUDF09-Ps		CANDELS-UDS
< z >	CANDELS-GS+GN	ERS, BoRG/HIPPIESb	COSMOS,EGS
4	$({{B}_{435}}$ – V606 > 1) ∧ (i775 – J125 < 1) ∧	$({{B}_{435}}$ – V606 > 1) ∧ (i775 – J125 < 1) ∧
	$({{B}_{435}}$ – V606 > 1.6(i775 – J125) +1) ∧	$({{B}_{435}}$ – V606 > 1.6(i775 – J125) +1) ∧
	(not in $z\sim 5$ selection)	(not in $z\sim 5$ selection)
5	$({{V}_{606}}$ – i775 > 1.2) ∧ $({{z}_{850}}$ – H160 < 1.3) ∧	$({{V}_{606}}$ – i775 > 1.2) ∧ $({{z}_{850}}$ – H160 < 1.3) ∧	$(({{V}_{606}}$ – I814 > 1.3) ∧ $({{I}_{814}}$ – H160 < 1.25) ∧
	$({{V}_{606}}$ – i775 > 0.8(z850 – ${{H}_{160}})$ +1.2) ∧	$({{V}_{606}}$ – i775 > 0.8(z850 – ${{H}_{160}})$ +1.2) ∧	$({{V}_{606}}$ – I814 > $0.72({{I}_{814}}$ – ${{H}_{160}})$ + $1.3)$ ∧
	($z\sim 5$ non-detection criterion)c ∧	($z\sim 5$ non-detection criterion)c ∧	$({{f}_{u}}$/efu < $2.5)$ ∧
	(not in $z\sim 6$ selection)	(not in $z\sim 6$ selection)	(4.2 < zphot < 5.5) ∧ (J125 < 26.7))∨e
			(other LBGs with 4.2 < zphot < 5.5)d
6	$({{i}_{775}}$ – z850 > 1.0) ∧ (Y105 – H160 < 1.0) ∧	$({{i}_{775}}$ – z850 > 1.0) ∧ (Y098 – H160 < 1.0) ∧	(I814 – J125 > 0.8) ∧ $({{J}_{125}}$ – H160 < 0.4) ∧
	$({{i}_{775}}$ – z850 > 0.78(Y105 – H160) +1.0) ∧	$({{i}_{775}}$ – z850 > 0.6(Y098 – H160) +1.0) ∧	(I814 – J125 > 2(J125 – H160) +0.8) ∧
	($z\sim 6$ non-detection criterion)c ∧	($z\sim 6$ non-detection criterion)c ∧	$({{f}_{ubg}}$/efubg < 2.5) ∧
	(not in $z\sim 7$ selection)	(not in $z\sim 7$ selection)	(5.5 < zphot < 6.3) ∧ (J125 < 26.7))∨e
			(other LBGs with 5.5 < zphot < 6.3)d
7	$({{z}_{850}}$ – Y105 > 0.7) ∧ $({{J}_{125}}$ – H160 < 0.45) ∧	$({{z}_{850}}$ – Y098 > 1.3) ∧ $({{J}_{125}}$ – H160 < 0.5) ∧	$({{I}_{814}}$ – J125 > 2.2) ∧ $({{J}_{125}}$ – H160 < 0.4) ∧
	$({{z}_{850}}$ – Y105 > 0.8(J125 – H160) +0.7) ∧	$({{z}_{850}}$ – J125 > 0.8(J125 – H160) +0.7) ∧	$({{I}_{814}}$ – J125 > 2(J125 – H160) +2.2) ∧
	$(({{I}_{814}}$ – J125 > 1.0)∨ $(SN({{I}_{814}})$ < 1.5)) ∧	$(({{I}_{814}}$ – J125 > 1.0)∨ $(SN({{I}_{814}})$ < 1.5)) ∧	$({{f}_{ubgvri}}$/efubgvri < 2.5) ∧
	($z\sim 7$ non-detection criterion)c ∧	($z\sim 7$ non-detection criterion)c ∧	$(6.3\lt {{z}_{phot}}\lt 7.3)\;\wedge \;$
	(not in $z\sim 8$ selection)	(not in $z\sim 8$ selection)	(${{J}_{125,AB}}$ < 26.7)∨e
			(other LBGs with 6.3 < zphot < 7.3)d
8	(Y105 – J125 > 0.45) ∧ $({{J}_{125}}$ – H160 < 0.5) ∧	$({{Y}_{098}}$ – J125 > 1.3) ∧ $({{J}_{125}}$ – H160 < 0.5) ∧	$({{I}_{814}}$ – J125 > 2.2) ∧ $({{J}_{125}}$ – H160 < 0.4) ∧
	$({{Y}_{105}}$ – J125 > 0.75(J125 – H160) +0.525) ∧	$({{Y}_{098}}$ – J125 > 0.75(J125 – H160) +1.3) ∧	$({{I}_{814}}$ – J125 > 2(J125 – H160) +2.2) ∧
	($z\sim 8$ non-detection criterion)c	($z\sim 8$ non-detection criterion)c,f	$({{f}_{ubgvri}}$ / efubgvri < 2.5) ∧
			(7.3< zphot < 9.0) ∧ $({{H}_{160,{\rm AB}}}$ < 26.7)∨e
			(other LBGs with 7.3 < zphot < 9.0)d
10	$({{J}_{125}}$ – H160 > 1.2) ∧	(J125 – H160 > 1.2) ∧	(J125 – H160 > 1.2) ∧
	$(({{H}_{160}}$ – [3.6] < 1.4)∨	((H160 – $[3.6]$ < 1.4)∨	((H160 – [3.6] < 1.4) ∨
	$({\rm S/N}([3.6])$ < 2)) ∧	$({\rm S/N}([3.6])$ < 2)) ∧	$({\rm S/N}([3.6])$ < 2)) ∧
	($z\sim 10$ non-detection criterion)c	($z\sim 10$ non-detection criterion)c	$({{f}_{ubgvriz}}$/efubgvriz < 2.5) ∧
			($\chi _{V,I}^{2}$ < 2)
All	(Stellarity Criterion)g	(Stellarity Criterion)g	(Stellarity Criterion)g
	$(\chi _{Y+J+JH+H}^{2}\gt 25)$ h	$(\chi _{Y+J+JH+H}^{2}\gt 25)$ h	$(\chi _{Y+J+JH+H}^{2}\gt 25)$ h

^aThroughout this table, ∧ and ∨ represent the logical AND and OR symbols, respectively, and S/N represents the signal-to-noise ratio. The ${{\chi }^{2}}$ statistic is as defined in Section 3.2 (see also Bouwens et al. 2011b). In the application of these criteria, flux in the dropout band is set equal to the $1\sigma$ upper limit in cases of a non-detection. ^bThe BoRG/HIPPIES data set is only used in searches for $z\sim 8$ galaxies. ^cThe optical non-detection criteria are as follows: $({\rm SN}(B)\lt 2)$ [$z\sim 5$], $({\rm SN}(B)\lt 2)\;\wedge$ $(({{V}_{606}}-{{z}_{850}}\gt 2.7)$ $\vee ({\rm SN}(V)\lt 2))$ [$z\sim 6$], $({\rm SN}(B)\lt 2)$ $\wedge \;({\rm SN}(V)\lt 2)$ $\wedge \;({\rm SN}(i)\lt 2)$ $\wedge \;(\chi _{bvi}^{2}\lt 3)$ [$z\sim 7$], $({\rm SN}(B)\lt 2)$ $\wedge \;({\rm SN}(V)\lt 2)$ $\wedge \;({\rm SN}(i)\lt 2)$ $\wedge \;({\rm SN}(I)\lt 2)$ $\wedge \;(\chi _{b,v,i,I}^{2}\lt 3)$ [$z\sim 8$], and $(\chi _{b,v,i,I,z,Y}^{2}\lt 3)$ $\;\wedge \;({\rm SN}(B)\lt 2)$ $\wedge \;({\rm SN}(V)\lt 2)$ $\wedge \;({\rm SN}(i)\lt 2)$ $\wedge \;({\rm SN}(I)\lt 2)$ $\wedge \;({\rm SN}(z)\lt 2)$ $\wedge \;({\rm SN}(Y)\lt 2)$ [$z\sim 10$]. For our $z\sim$ 7–10 selections, we also require that the optical ${{\chi }^{2}}$ be less than 4 and 3 in fixed $0\buildrel{\prime\prime}\over{.} 35$-diameter and 0 2-diameter apertures, respectively. We also impose a stricter optical non-detection criterion for the faintest sources in each of our selections (i.e., where the total detection significance is defined by $\chi _{Y,J,JH,H}^{2}\lt 64$). These criteria are ${\rm SN}(B)\lt 1$ ($z\sim 5$), $({\rm SN}(B)\lt 1)$ $\wedge \;(({{V}_{606}}-{{z}_{850}}\gt 2.3)$ $\vee (\chi _{B,V}^{2}\lt 2))$ ($z\sim 6$), $\chi _{B,V,i}^{2}\lt 2$ ($z\sim 7$), and $\chi _{B,V,i,I}^{2}\lt 2$ ($z\sim 8$). ^dWe also include sources in our $z\sim 5$, $z\sim 6$, $z\sim 7$, and $z\sim 8$ selections, respectively, if they satisfy any of our $z\sim 5$, $z\sim 6$, and $z\sim$ 7–8 LBG criteria, and the photometric redshifts we estimate for the sources are $4.2\lt z\lt 5.5$, $5.5\lt z\lt 6.3$, $6.3\lt z\lt 7.3$, and $7.3\lt z\lt 9.0$, respectively, with a total measured magnitude of ${{J}_{125,AB}}\lt 26.7$, ${{J}_{125,AB}}\lt 26.7$, ${{J}_{125,AB}}\lt 26.7$, and ${{H}_{160,{\rm AB}}}\lt 26.7$. See Section 3.2.3. ^eWhile we select sources to ∼26.7 mag, we only include sources brightward of 26.5 mag in our LF determinations. ^fWe required sources identified within the BoRG/HIPPIES data set to satisfy an even more stringent optical non-detection criterion (${\rm SN}(V)\lt 1.5$) to effectively exclude all low-redshift interlopers from our selection. ^gWe require that the measured stellarity of sources (as measured from the detection image) be less than 0.9 to exclude stars from our samples (0 = extended source and 1 = point source). We also exclude particularly compact sources, with detection-image stellarities less than 0.9 if its HST+ground-based+Spitzer photometry is significantly better fit with a stellar SED than a $z\geqslant 3$ galaxy (${\Delta}{{\chi }^{2}}\gt 2$) and the measured stellarity in either the J₁₂₅ or H₁₆₀ band is at least 0.8. The stellarity requirement is only imposed within 1 mag of the detection limit of the sample, i.e., 26.5 mag for the CANDELS/WIDE data sets, 27.0 mag for the CANDELS/DEEP data sets, 28.0 mag for the HUDF09-1+HUDF09-2 data sets, and 28.5 mag for the XDF data set. ^hEven more stringent requirements are made on the detection significance of sources in data sets shallower than the XDF. Candidates are required to have a total S/N in the Y₁₀₅, J₁₂₅, JH₁₄₀, and H₁₆₀ bands of 5.5 in the HUDF09-Ps and CANDELS data set and 6.0 in the BoRG/HIPPIES data set. For $z\sim 8$ and $z\sim 10$ selections, only the ${{J}_{125}}J{{H}_{140}}{{H}_{160}}$ and $J{{H}_{140}}{{H}_{160}}$ fluxes, respectively, are used in assessing the detection significance of candidate sources. The $z\sim 10$ candidates over the CANDELS-UDS/COSMOS/EGS fields are required to have an rms S/N of 2.0 in the Spitzer/IRAC 3.6 and 4.5 μm imaging to ensure they are real.

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The new color criteria we have developed are not directly comparable to those previously developed to work with optical/ACS observations (Giavalisco et al. 2004b; Bouwens et al. 2007), though we remark that the ${{B}_{435}}-{{V}_{606}}$, ${{V}_{606}}-{{i}_{775}}$, ${{i}_{775}}-{{z}_{850}}$ color criteria that we utilize (to identify the existence of a Lyman break) are almost identical to previous criteria. Our color criteria are most similar in spirit to the z = 4 criteria previously developed by Castellano et al. (2012), though Castellano et al. (2012) use a ${{V}_{606}}-{{H}_{160}}$ color to quantify the color of galaxies redward of the break rather than an ${{i}_{775}}-{{J}_{125}}$ color. The advantage of using the ${{i}_{775}}-{{J}_{125}}$ colors over the ${{V}_{606}}-{{H}_{160}}$ colors is the cleaner measurement it provides of the slope of the UV continuum for candidate $z\sim 4$ galaxies (though the wavelength leverage it provides is less).

The $z\sim 7$ and $z\sim 8$ color criteria we utilize here are very similar to the criteria we had previously applied in Bouwens et al. (2011b) to the HUDF and ERS data sets. See Figures 2 and 3 and Figures 6 and 7 from Bouwens et al. (2011b). The $z\sim 8$ selection criteria we employ over the ERS+BoRG+HIPPIES fields utilize a less stringent ${{Y}_{098}}-{{J}_{125}}\gt 1.3$ cut than the ${{Y}_{098}}-{{J}_{125}}\gt 1.75$ cut utilized in the standard BoRG search (e.g., Bradley et al. 2012), making our selection slightly more susceptible to contamination by low-mass stars. However, such sources should be largely excluded by the stellarity criterion we discuss below (see also Section 3.5.5).

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Finally, our $z\sim 10$ selection criteria are identical to those previously presented by Bouwens et al. (2011a) and Oesch et al. (2012a, 2014).

In applying these criteria, we set the flux in the dropout band to be equal to the $1\sigma$ upper limit in cases of a non-detection.

In isolation, the color criteria we present in Table 2 would allow for the selection of sources at least one unit higher in redshift than our desired high-redshift boundaries for these selections (e.g., our $z\sim 4$ selection criteria could allow us to select sources from $z\sim 3.5$ to $z\sim 5.5$). Fortunately, we can impose a high-redshift boundary for each of our selections by explicitly requiring that sources not satisfy the selection criteria for the sample just above it in redshift. This ensures that our selections are both essentially complete and disjoint from one another.

To keep contamination from lower-redshift sources to a minimum, we require that sources in our $z\sim 5$ and $z\sim 6$ selections be undetected ($\lt 2\sigma$) in B₄₃₅-band imaging data for our fields, if they are available. For our $z\sim 6$ selections, we require the ${{V}_{606}}-{{z}_{850}}$ color to be redder than 2.6 or for sources to be undetected ($\lt 2\sigma$) in the V₆₀₆-band imaging data (similar to Bouwens et al. 2006). For our $z\sim 7$–10 selections, we calculate an optical "${{\chi }^{2}}$" for each candidate source (Bouwens et al. 2011a), as $\chi _{{\rm opt}}^{2}={{{\Sigma}}_{i}}{\rm SGN}({{f}_{i}}){{({{f}_{i}}/{{\sigma }_{i}})}^{2}},$ where f_i is the flux in band i in a consistent aperture, ${{\sigma }_{i}}$ is the uncertainty in this flux, and SGN(f_i) is equal to 1 if ${{f}_{i}}\gt 0$ and −1 if ${{f}_{i}}\lt 0$. The ${{B}_{435}}{{V}_{606}}{{i}_{775}}$ flux measurements (where available) were used in calculating $\chi _{{\rm opt}}^{2}$ for our $z\sim 7$ selections, while the ${{B}_{435}}{{V}_{606}}{{i}_{775}}{{I}_{814}}$ and ${{B}_{435}}{{V}_{606}}{{i}_{775}}{{I}_{814}}{{z}_{850}}{{Y}_{105}}$ observations were used in computing $\chi _{{\rm opt}}^{2}$ for our $z\sim 8$ and $z\sim 10$ selections, respectively. $\chi _{{\rm opt}}^{2}$ is computed on the basis of the flux measurements in small-scalable apertures; any candidate with a measured ${{\chi }_{{\rm opt}}}$ in excess of 3 is excluded from our selections.

For our highest-redshift selections, i.e., $z\sim 7$–10, we also computed a $\chi _{{\rm opt}}^{2}$ for sources in 0 35-diameter apertures and especially small $0\buildrel{\prime\prime}\over{.} 2$-diameter apertures (before PSF smoothing to maximize the S/N) and required sources to be less than 3 and 4, respectively. An even lower threshold of 2 for $\chi _{{\rm opt}}^{2}$ was used in selecting $z\sim 7$–8 sources over the HUDF09-1 field, owing to the lack of B₄₃₅-band observations over that field. Finally, for the faintest $z\sim 5$–8 candidates in each of our selections with a co-added significance of the detections in the Y₀₉₈, Y₁₀₅, J₁₂₅, JH₁₄₀, and H₁₆₀ bands less than 8 (i.e., $\chi _{Y,J,JH,H}^{2}\lt 64$), we used the even stricter requirements on the flux in the optical bands listed in footnote a of Table 2.

For our deepest field, the XDF, sources are required to be detected at $5\sigma$ in a ${{\chi }^{2}}$ stack of all the HST observations redward of the break (in a fixed 0 36-diameter aperture). This is to ensure source reality. For sources over the deep HUDF09-1 and HUDF09-2 fields and the wider-area CANDELS and ERS fields, we require sources to be detected at $5.5\sigma$. For sources over the BoRG/HIPPIES fields, we require sources to be detected at $6\sigma$. Our use of more stringent criteria for our shallower fields is quite reasonable, given the much smaller number of exposures in these data and therefore noise that is less Gaussian in its characteristics (e.g., see Schmidt et al. 2014).¹³

For sources that are at least 1 mag brightward of the nominal detection limit for our samples (i.e., 26.5 mag for the CANDELS/WIDE data sets, 27.0 mag for the CANDELS/DEEP data sets, 28.0 mag for the HUDF09-1+HUDF09-2 data sets, and 28.5 mag for the XDF data set), the SExtractor stellarity parameter for sources (from the SExtractor detection image) is required to be less than 0.9 (where 0 corresponds to very extended sources and 1 corresponds to point sources). We also exclude particularly compact sources, with measured stellarities (from the detection image) greater than 0.5 if its HST photometry was significantly better fit to a stellar SED than a $z\geqslant 3$ galaxy (${\Delta}{{\chi }^{2}}\gt 2$) and if the measured stellarity in either the J₁₂₅ or H₁₆₀ image is greater than 0.8. The templates we use for our stellar SED fits are from the SpeX prism library of low-mass stars (Burgasser et al. 2004) extended to 5 μm using the derived spectral types and the known J – [3.6] or J – [4.5] colors of these spectral types (Patten et al. 2006; Kirkpatrick et al. 2011).

A careful visual inspection was performed on all of the candidate $z\sim 4$–10 galaxies that otherwise satisfy our selection criteria to exclude obvious artifacts (e.g., diffraction spikes, spurious "sources" on the wings of ellipticals) or any sources that seemed likely to be associated with bright foreground sources.¹⁴ We also verified that none of the sources in our selection were previously included in the catalogs of candidate low-mass stars from Holwerda et al. (2014b) or were associated with SNe identified during the CANDELS observations (Rodney et al. 2014).

Finally, we made minor corrections to our final catalogs to account for multi-component galaxies (e.g., Ouchi et al. 2009a) being split into multiple sources by SExtractor. To minimize the impact of this on our final results, we combine selected sources in a given sample whose centers are closer than 0 5 to each other, merging their total photometry (occuring <∼2% of sources).

3.2.3. CANDELS-UDS, CANDELS-COSMOS, CANDELS-EGS Fields

Because of the lack of deep HST imaging in B₄₃₅, z₈₅₀, or Y₀₉₈/Y₁₀₅ band over the CANDELS-UDS, CANDELS-COSMOS, and CANDELS-EGS fields (Table 1 and Figure 2), it is not possible to select $z\sim 5$–8 galaxies over those fields using the same color criteria as we utilized over our primary search fields (i.e., the XDF, CANDELS-GN, and CANDELS-GS).

Our procedure for selecting our samples of $z\sim 5$, $z\sim 6$, $z\sim 7$ and $z\sim 8$ galaxies over these is therefore more involved and makes significant use of the ground-based observations. We describe our procedure in the paragraphs that follow. The first step was to identify all those sources that plausibly corresponded to star-forming galaxies at $z\sim 5$–8 through the systematic selection of Lyman-break-like galaxies at $z\sim 5$, $z\sim 6$, and $z\sim 7$–8. The criteria we used to do this preselection are presented in Appendix B.

In the second step, we obtained photometry on each of these sources in deep ground-based Subaru+CFHT+VLT+VISTA+ Spitzer/IRAC observations that are available over our search fields. We then used the EAZY photometric redshift code (Brammer et al. 2008) to estimate redshifts for all the sources. The photometry utilized in deriving the photometric redshifts included flux measurements from the HST ${{V}_{606}}{{I}_{814}}{{J}_{125}}J{{H}_{140}}{{H}_{160}}$+Subaru-SuprimeCam BgVriz+CFHT/Megacam ugriyz+UltraVISTA YJHK_s data sets for the CANDELS COSMOS field, HST ${{V}_{606}}{{I}_{814}}{{J}_{125}}J{{H}_{140}}{{H}_{160}}$+Subaru-SuprimeCam BVriz+CFHT/Megacam u+UKIRT/WFCAM K_s+VLT/HAWKI/HUGS YK_s data sets for the CANDELS UDS field, and the HST ${{V}_{606}}{{I}_{814}}{{J}_{125}}J{{H}_{140}}{{H}_{160}}$+CFHT/Megacam ugriyz+CFHT/WIRCam K_s+Spitzer/IRAC 3.6 μm + 4.5 μm data sets for the CANDELS EGS field. No consideration of the Spitzer/IRAC photometry is made for sources over the CANDELS-UDS and CANDELS-COSMOS fields owing to the availability of deep Y-band observations to distinguish $z\sim 7$ sources from $z\sim 8$ sources.¹⁵

Sources with photometric redshifts in the range z = 4.2–5.5, z = 5.5–6.3, z = 6.3–7.3, and z = 7.3–9.0 were tentatively assigned to our $z\sim 5$, $z\sim 6$, $z\sim 7$, and $z\sim 8$ selections, respectively. These redshift ranges were chosen to ensure a good match with the mean redshifts for the color selections defined in Section 3.2.2. Our photometric redshift fitting is conducted using the EAZY_v1.0 template set supplemented by SED templates from the Galaxy Evolutionary Synthesis Models (GALEV; Kotulla et al. 2009). Nebular continuum and emission lines were added to the later templates using the Anders & Fritze-v. Alvensleben (2003) prescription, a $0.2\;{{Z}_{\odot }}$ metallicity, and a rest-frame EW for Hα of 1300 Å.¹⁶

We only included galaxies in our $z\sim 6$, $z\sim 7$, and $z\sim 8$ samples brightward of ${{J}_{125}}=26.7$ mag (z = 6–7) and ${{H}_{160}}=26.7$ mag (z = 8). However, only sources brightward of 26.5 mag are used in our LF determinations (Section 4). This was to ensure good redshift separation, given the limited depth of the I₈₁₄-band observations and ground-based z- and Y-band observations (Figure 2). As we demonstrate with the simulations in Section 4.1 (illustrated in Figure 4), we can effectively split sources into different redshift subsamples to 26.5 mag.

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To ensure that each of these candidate $z\sim 5$–8 galaxies was robust, we required that each of these sources show a <2.5σ detection blueward of the break. To this end, inverse-variance-weighted fluxes were derived for each source blueward of the Lyman break. Included in the inverse-variance-weighted measurements for the samples in brackets below were the CFHT Megacam u and Subaru Suprime-Cam B (CANDELS-UDS $z\sim 5$), CFHT u and Subaru B (CANDELS-UDS $z\sim 6$), CFHT u and Subaru BVr (CANDELS-UDS $z\sim 7$), CFHT u and Subaru BVri (CANDELS-UDS $z\sim 8$), CFHT MegaCam u (COSMOS $z\sim 5$), Subaru Bg and CFHT ug $({\rm COSMOS}\;z\sim 6),$ Subaru BgVr and CFHT ugr (COSMOS $z\sim 7$), Subaru BgVri and CFHT ugriy (COSMOS $z\sim 8$), CFHT u (EGS $z\sim 5$), CFHT ug (EGS $z\sim 6$), CFHT ugr (EGS $z\sim 7$), and CFHT ugriy (EGS $z\sim 8$) flux measurements, respectively. We also excluded candidate $z\sim 6$, $z\sim 7$, and $z\sim 8$ galaxies from our selection where flux in the HST V₆₀₆, V₆₀₆, and ${{V}_{606}}+{{I}_{814}}$ bands, respectively, was greater than 1.5σ. Exclusion of sources with detections blueward of the break only had a modest effect on the size of the $z\sim 5$, $z\sim 6$, $z\sim 7$, and $z\sim 8$ samples we derived from the wide-area CANDELS fields (removing 2%, 7%, 8%, and 21% of the sources from the $z\sim 5$, $z\sim 6$, $z\sim 7$, and $z\sim 8$ samples, respectively).¹⁷

We used a similar strategy for excluding stars from our CANDELS-UDS, COSMOS, and EGS fields to what we utilized for selections over the XDF, HUDF09-Ps, ERS, CANDELS-GN+GS, and BoRG/HIPPIES fields (Section 3.2.2). The only procedural difference with the present fields is that we also make use of the ground-based+Spitzer/IRAC photometry we obtain for sources in ascertaining whether their SEDs are more consistent with that of a star or a $z\sim 5$–8 galaxy. Using simulations where we added point-like sources to the real data with input fluxes taken from random stars in the SpeX prism library of late-type stars (Burgasser et al. 2004), we found that our schema was successful at excluding 97%, 97%, and 94% of ${{H}_{160,{\rm A}B}}=26.0$–26.5 stars from our selection over the CANDELS-UDS, CANDELS-COSMOS, and CANDELS-EGS fields, respectively (with late L and early T type stars being the most challenging to exclude).

As a check on the fidelity of our $z\sim 5$–8 samples, we also derived fluxes for sources in our samples in larger $1\buildrel{\prime\prime}\over{.} 8$-diameter apertures than we used for our fiducial selection. Co-adding the fluxes of sources blueward of the break while weighting by the inverse variance, we found that 94% of the sources in our samples remain undetected at $\lt 2.5\sigma$ even in larger $1\buildrel{\prime\prime}\over{.} 8$-diameter apertures. To interpret these findings, we repeated this experiment on the mock images we created in Section 4.1 and found similar incompleteness levels, strongly arguing that the slight detection rate we found for our $z\sim 5$–8 samples in the larger apertures could be explained as resulting from noise and imperfectly subtracted nearby neighbors.¹⁸

We further stacked the optical V₆₀₆-band observations (blueward of the break for $z\geqslant 7$ galaxies) for all 107 $z\sim 7$ and $z\sim 8$ candidates from the wide-area fields and found no detection ($\lt 1\sigma$). Similar stack results in the I₈₁₄ band for our CANDELS-UDS/COSMOS/EGS $z\sim 8$ samples yielded no detection.

Our selection of $z\sim 10$ galaxies over these fields is very similar to our selection of $z\sim 10$ galaxies from HST fields with Y-band imaging (Section 3.2.2). Again, we require that sources satisfy a ${{J}_{125}}-{{H}_{160}}\gt 1.2$ color cut, show a $6\sigma$ detection in the H₁₆₀ band, be undetected in a stack of the optical/ACS data ($\chi _{{\rm opt}}^{2}\lt 3$), and also be detected at $\geqslant 6\sigma$ in the H₁₆₀ band. However, we also require that sources remain undetected ($\lt 2\sigma$) in whatever Y-band observations were available over our search fields (i.e., from HAWK-I and VISTA over the CANDELS-UDS and CANDELS-COSMOS fields, respectively), that sources also remain undetected ($\lt 2.5\sigma$) in a stack of the optical ground-based Subaru+CFHT observations available over each field, and that sources be detected at $\gt 2\sigma$ in the available 3.6 μm + 4.5 μm IRAC imaging over the CANDELS fields from the SEDS program (Ashby et al. 2013) to ensure source reality.

Applying the selection criteria from Section 3.2 to XDF, HUDF09-Ps, ERS, BoRG/HIPPIES, and CANDELS data sets results in 5859, 3001, 857, 481, 217, and 6 sources in our $z\sim 4$, $z\sim 5$, $z\sim 6$, $z\sim 7$, $z\sim 8$, and $z\sim 10$ samples. Our total $z\sim 4$–10 sample includes ∼10,400 sources. The individual number of high-redshift candidates in each field is provided in Table 4.

The surface density of galaxies we find in our different redshift samples is presented in Figure 5 as a function of magnitude. While it is clear that some field-to-field variations exist in the surface density of galaxies in our different samples (e.g., $z\sim 4$ galaxies in the HUDF seem to be underdense relative to our other search fields), overall the surface density of galaxies as a function of magnitude is fairly similar for each of our search fields, over magnitude ranges where our search is largely complete. We discuss field-to-field variations in detail in Section 4.5. In Table A1 in Appendix C, we tabulate the average surface density of galaxies in our different samples as a function of magnitude.

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Our best estimate of the approximate redshift distribution for our different high-redshift samples is shown in the left panel of Figure 1 and is based on the simulations we describe in Section 4.1 for the XDF, HUDF09-1, and HUDF09-2 fields. The mean redshift for galaxies in our $z\sim 4$, $z\sim 5$, $z\sim 6$, $z\sim 7$, and $z\sim 8$ samples is 3.8, 4.9, 5.9, 6.8, and 7.9, respectively. From these simulations, it is clear that our selection criteria are quite effective in dividing high-redshift galaxies into discrete redshift slices. In the right panel of Figure 1, we also present the redshift distributions we derive for our XDF, HUDF09-1, and HUDF09-2 samples using the photometric redshift code EAZY (Brammer et al. 2008). Photometric redshifts are estimated based on the HST photometry (for our $z\sim 4$–8 samples) and HST+Spitzer photometry (for our $z\sim 10$ sample). As is clear from the figure, our simple color–color selections result in essentially the same subdivision of sources by redshift as one would find if one relied on a photometric redshift code to do the selection.

We include our complete $z\sim 4$, $z\sim 5$, $z\sim 6$, $z\sim 7$, $z\sim 8$, and $z\sim 10$ catalogs in Table 3, with coordinates and rest-frame UV luminosities. We have also provided our best-estimate redshifts for each of the $z\sim 5$–8 candidates we identified over the CANDELS-UDS/COSMOS/EGS fields. Photometric redshift estimates are also provided for $z\sim 4$–10 candidates over XDF, HUDF09-Ps, ERS, CANDELS-GS/GN, and BoRG/HIPPIES by applying the photometric redshift software EAZY (Brammer et al. 2008) and the template set from Section 3.2.3 to the HST photometry we have available for these candidates. To improve the accuracy of the photometric redshift estimates for our $z\sim 4$ CANDELS-GN+GS+ERS samples (where the lack of photometric constraints blueward of the B₄₃₅ band can impact the results), we have also incorporated the U-band photometry of these candidates from KPNO (Capak et al. 2004) and VLT/VIMOS (Nonino et al. 2009) using mophongo in fixed 1 2-diameter apertures.

The present compilation of $z\sim 4$–10 galaxy candidates from the XDF, HUDF09-1, HUDF09-2, ERS, and the five CANDELS fields contains ∼10,400 $z\sim 4$–10 candidates and is the largest such compilation obtained to date based on HST observations. Previously, the largest such samples of galaxies found in HST observations were reported in Bouwens et al. (2007; 6714 sources over the range z = 4–6) and Bouwens et al. (2014b; 4004 sources over the range z = 4–8).

A substantial fraction (∼30%–70%) of the sources from the current catalogs appeared in previous wide-area selections. A total of 2331, 586, and 206 of the $z\sim 4$, $z\sim 5$, and $z\sim 6$ candidates (44%, 34%, and 37% of this sample, respectively) from our wide-area CANDELS+ERS selections were previously reported by Bouwens et al. (2007). For $z\sim 7$–8 selections over the CANDELS-GS, 59 and 28 of the $z\sim 7$ and $z\sim 8$ candidates (19% and 27% of this sample, respectively), were previously reported by Bouwens et al. (2011b), Oesch et al. (2012b), Grazian et al. (2012), Yan et al. (2012), Lorenzoni et al. (2013), Schenker et al. (2013), and McLure et al. (2013). A total of 22 of the present $z\sim 7$ candidates over the CANDELS-UDS and CANDELS-EGS fields (35% of our sample) previously appeared in Grazian et al. (2012) or McLure et al. (2013). The brightest $z\sim 7$ candidate we find in the CANDELS-UDS field is the well-known "Himiko" z = 6.595 Lyα-emitting galaxy previously reported by Ouchi et al. (2009a). The brightest three $z\sim 6$ and brightest two $z\sim 7$ galaxies from our CANDELS-COSMOS catalog were previously identified by Willott et al. (2013) and Bowler et al. (2014), respectively.

A total of 11 of the 23 $z\sim 8$ candidates we identified over the BoRG/HIPPIES fields and similar data sets (i.e., 48%) were previously identified as $z\sim 8$ candidates by Bradley et al. (2012), McLure et al. (2013), and Schmidt et al. (2014). The reason our catalogs include many $z\sim 8$ candidates not included in the Bradley et al. (2012) and Schmidt et al. (2014) compilation is our use of one additional data set not previously considered (i.e., a parallel field outside of Abell 1689) and our selecting sources with slightly weaker ${{Y}_{098}}-{{J}_{125}}$ breaks and slightly redder ${{J}_{125}}-{{H}_{160}}$ colors (consistent with our $z\sim 8$ selection from the ERS data set). While excluding these sources may allow Bradley et al. (2012) and Schmidt et al. (2014) to identify a marginally cleaner selection of $z\sim 8$ galaxies, Bradley et al. (2012) and Schmidt et al. (2014) potentially miss a modest fraction of the luminous $z\sim 8$ galaxies over the BoRG/HIPPIES search fields (i.e., those having significantly redder ${{J}_{125}}-{{H}_{160}}$ colors than would be selected by their criteria).¹⁹

For fainter $z\sim 4$–8 samples from the XDF, HUDF09-1, and HUDF09-2 data sets, our samples again show very good overlap. A total of 209, 139, 92, 75, and 45 of the present sample of $z\sim 4$, $z\sim 5$, $z\sim 6$, $z\sim 7$, and $z\sim 8$ candidates (41%, 43%, 55%, 72%, and 71% of this sample, respectively) were previously reported by Bouwens et al. (2007), Wilkins et al. (2010), Bouwens et al. (2011b), Schenker et al. (2013), and McLure et al. (2013). The reason the current selection contains many sources that were not previously found by Bouwens et al. (2007) is due to our ability to probe to greater depths with WFC3/IR than was possible using the HST/ACS optical camera alone and deeper optical observations now available over the XDF/HUDF and HUDF09-2 fields.

The present $z\gt 6$ sample is thus far the most comprehensive in the literature, including some 698 $z\sim 7$–8 high-quality candidates based on all search fields.

The present $z\sim 10$ sample contains six candidates in total and is almost identical to the Oesch et al. (2014) $z\sim 10$ sample, with one $z\sim 10$ candidate over the CANDELS-GS field, three $z\sim 10$ candidates over the CANDELS-GN field, and two $z\sim 10$ candidates over the XDF data set. One of the six $z\sim 10$ candidates from the present $z\sim 10$ sample (XDFyj-40248004) was classified as a $z\sim 9$ candidate in Oesch et al. (2013a). The earlier analyses of Ellis et al. (2013) and Oesch et al. (2013a) had only identified one plausible $z\sim 10$ candidate each,²⁰ while McLure et al. (2013) did not identify any $z\sim 10$ candidates over our search fields.²¹

We carefully considered many possible sources of contamination for our $z\sim 4$–10 samples. Potential contaminants include stellar sources, time-variable events like SNe, spurious sources, EELGs, and photometric scatter. We discuss possible contamination by each of these sources in the subsections that follow.

3.5.1. Stars

3.5.2. Transient Sources or Supernovae

3.5.3. Lower-redshift Galaxies

3.5.4. Extreme Emission Line Galaxies

3.5.5. Establishing the Contamination from Low-redshift Galaxies by Adding Noise to Real Data

3.5.6. Spurious Sources

3.5.7. Summary

We first consider a simple stepwise (binned) determination of the UV LFs at $z\sim 4$–8. The baseline approach in the literature for these type of determinations is to use the stepwise maximum likelihood (SWML) approach of Efstathiou et al. (1988). With this approach, the goal is to find the maximum likelihood LF shape that best reproduces the available constraints. Since the focus with this approach is on reproducing the shape of the LF, this approach is largely robust against field-to-field variations in the normalization of the LF and hence large-scale structure effects.

As in Bouwens et al. (2007) and Bouwens et al. (2011b), we can write the stepwise LF ${{\phi }_{k}}$ as ${\Sigma}{{\phi }_{k}}W(M-{{M}_{k}}),$ where k is an index running over the magnitude bins, where M_k corresponds to the absolute magnitude at the center of each bin, where

$$\begin{eqnarray}\qquad \quad W(x)=\begin{array}{ccccccccccccccc} 0, & x\lt -0.25 \\ 1, & -0.25\lt x\lt 0.25 \\ 0, & x\gt 0.25. \\ \end{array}\end{eqnarray} \tag{ 1 }$$

and where x gives the position within a magnitude bin (for a 0.5 mag binning scheme). The goal then is to find the LF that maximizes the likelihood of reproducing the observed source counts over our various search fields. The likelihood $\mathcal{L}$ can be expressed analytically as

$$\begin{eqnarray}&&\mathcal{L}={{{\Pi}}_{field}}{{{\Pi}}_{i}}p({{m}_{i}})\end{eqnarray} \tag{ 2 }$$

where

$$\begin{eqnarray}&&p({{m}_{i}})={{\left( \frac{{{n}_{expected,i}}}{{{{\Sigma}}_{j}}{{n}_{expected,j}}} \right)}^{{{n}_{observed,i}}}}\end{eqnarray} \tag{ 3 }$$

where the above products run over the different search fields and magnitude interval i used in the LF determinations, ${{n}_{expected,i}}$ is the expected number of sources in magnitude interval i for a given LF, and ${{n}_{observed,i}}$ is the observed number of sources in magnitude interval i. The quantity ${{n}_{observed,i}}$ is derived using the apparent magnitude of the source closest to 1600 Å, which occurs in the i₇₇₅ band for sources in our $z\sim 4$ samples, in the Y₁₀₅ band for our $z\sim 5$ and $z\sim 6$ samples,²² in the J₁₂₅ band for our $z\sim 7$ samples, and in the H₁₆₀ band for our $z\sim 8$ samples. For the ERS and CANDELS UDS/COSMOS/EGS wide-area samples, where no Y₁₀₅ coverage is available, we make use instead of the Y₀₉₈- and J₁₂₅-band magnitudes, respectively, for our $z\sim 5$ and $z\sim 6$ samples. We apply a small correction to the apparent magnitude of individual sources (typically ≲0.1 mag) so that it corresponds to an effective rest-frame wavelength 1600 Å. The correction we apply is based on the biweight mean β Bouwens et al. (2014b) derive for galaxies with a given absolute magnitude and redshift. The quantity ${{n}_{observed,i}}$ is also corrected for contamination using the simulations we describe in Section 3.5.5.

Similar to our previous work, we compute the number of sources expected in a given magnitude interval i assuming a model LF as

$$\begin{eqnarray}&&{{n}_{expected,i}}={{{\Sigma}}_{j}}{{\phi }_{j}}{{V}_{i,j}}\end{eqnarray} \tag{ 4 }$$

where ${{V}_{i,j}}$ is the effective volume over which one could expect to find a source of absolute magnitude j in the observed magnitude interval i. We estimate ${{V}_{i,j}}$ for a given search field using an extensive suite of Monte Carlo simulations where we add sources with an absolute magnitude j to the different search fields and then see whether we select a source with apparent magnitude i. The ${{V}_{i,j}}$ factors implicitly correct for flux-boosting type effects that are important near the detection limits of our samples, whereby faint sources scatter to brighter apparent fluxes and thus into our samples.

Computing the relevant ${{V}_{i,j}}$ values for all of our samples and search fields required our running an extensive suite of Monte Carlo simulations. In these simulations, large numbers of artificial sources were inserted into the input data (typically ∼50 arcmin⁻² in each simulation). Catalogs were then constructed from the data and sources selected. To ensure that the candidate galaxies in these simulations had realistic sizes and morphologies, we randomly selected similar-luminosity $z\sim 4$ galaxies from the Hubble Ultra Deep Field to use as a template to model the two-dimensional spatial profile for individual sources. We assigned each galaxy in our simulations a UV color using the β versus M_UV determinations of Bouwens et al. (2014b), with an intrinsic scatter in β varying from 0.35 brightward of −20 mag to 0.20 faintward of −20 mag. This matches the intrinsic scatter in β measured for brightest $z\sim 4-5$ sources by Bouwens et al. (2009, 2012b) and Castellano et al. (2012), as well as the decreased scatter in β for the faintest sources (Rogers et al. 2014). Finally, the templates were artificially redshifted to the redshift in the catalog using our well-tested "cloning" software (Bouwens et al. 1998, 2003a) and inserted these sources into the real observations. In projecting galaxies to higher redshift, we scaled source size approximately as ${{(1+z)}^{-1.2}}$ to match that seen in the observations (Oesch et al. 2010a; Grazian et al. 2012; Ono et al. 2013; Holwerda et al. 2014a; Kawamata et al. 2014). We verified through a series of careful comparisons that the source sizes we utilized were similar to those in the real observations, as a function of both redshift and luminosity (Appendix D).

In calculating the effective selection volumes over the CANDELS-UDS, COSMOS, and EGS search areas, we also simulated realistic images of our mock sources in the ground-based and Spitzer/IRAC observations, by covolving the H₁₆₀-band images of these sources by the appropriate kernels to match the broader-PSF and adding these sources to the real observations and extracting their fluxes using the same photometric procedure as we applied to the real observations. Finally, we made use of the full set of flux information we were able to derive for the mock sources (HST+ground-based+Spitzer/IRAC) to estimate photometric redshifts for these sources and hence determine whether sources fell within our redshift selection windows. As with the real observations, mock sources were excluded from the selection, if they were detected at >2.5σ significance in passbands blueward of the break. We note that in producing simulated IRAC images for the mock sources, we assume a rest-frame EW of 300 Å for Hα+[N II] emission and 500 Å for [O III]+Hβ emission over the entire range z = 4–9, a flat rest-frame optical color, and an H₁₆₀-optical continuum color of 0.2–0.3 mag, to match the observational results of Shim et al. (2011), Stark et al. (2013), González et al. (2012, 2014), Labbé et al. (2013), Smit et al. (2014a, 2014b), and Oesch et al. (2013b).

After deriving the shape of the LF at each redshift using this procedure, we set the normalization by requiring that the total number of sources predicted on the basis of our LFs be equal to the total number of sources observed over our search fields. Applying the above SWML procedure to the observed surface densities of sources in our different search fields, we determined the maximum likelihood LFs.

We elected to use a 0.5 mag binning scheme for the LFs at $z\sim 4$–8, consistent with past practice. To cope with the noise in our SWML LF determinations that result from deconvolving the transfer function (implicit in the ${{V}_{i,j}}$ term in Equation (4)) from the number counts ${{n}_{observed,i}}$, we have adopted a wider binning scheme at the faint end of the LF. This issue also causes the uncertainties we derive on the bright end of the LF to remain somewhat large at all redshifts (as uncertainties in the measured flux for individual sources allow for the possibility that the observed source counts could arise from "picket fence"-type LFs with the bulk of sources concentrated in just the odd or even stepwise LF intervals).

In deriving the LF from such a diverse data set, it is essential to ensure that our LF determinations across this data set are generally self-consistent. We therefore derived the UV LFs at $z\sim 5$, $z\sim 6$, $z\sim 7$, and $z\sim 8$ separately from the wide-area UDS+COSMOS+EGS CANDELS observations, from the CANDELS-DEEP region within the CANDELS-GN and GS, from the CANDELS-WIDE region within the CANDELS-GN and GS, and from the BoRG/HIPPIES observations. As we demonstrate in Figure A3 in Appendix E, we find broad agreement between our LF determinations from all four data sets, suggesting that the impact of systematics on our LF results is quite limited in general.

Table 4.  Total Number of Sources in Our $z\sim 4$, $z\sim 5$, $z\sim 6$, $z\sim 7$, $z\sim 8$, and $z\sim 10$ Samples Used in Deriving the Present High-redshift LFs

	Area	$z\sim 4$	$z\sim 5$	$z\sim 6$	$z\sim 7$	$z\sim 8$	$z\sim 10$
Field	(arcmin2)	#	#	#	#	#	#
HUDF/XDF	4.7	357	153	97	57	30	2
HUDF09-1	4.7	⋯	91	38	22	18	0
HUDF09-2	4.7	147	77	32	23	17	0
CANDELS-GS-DEEP	64.5	1590	471	198	77	27	1
CANDELS-GS-WIDE	34.2	451	117	43	5	3	0
ERS	40.5	815	205	61	47	6	0
CANDELS-GN-DEEP	68.3	1628	634	188	134	51	2
CANDELS-GN-WIDE	65.4	871	282	69	39	18	1
CANDELS-UDS	151.2	⋯	270	33	18	6	0
CANDELS-COSMOS	151.9	⋯	320	48	15	9	0
CANDELS-EGS	150.7	⋯	381	50	44	9	0
BORG/HIPPIES	218.3	⋯	⋯	⋯	⋯	23	0
Total	959.1	5859	3001	857	481	217	6

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After considering the LF results from each of our fields separately, we combine our search results from all fields under consideration to arrive at stepwise LFs at $z\sim 4$–8 for our overall sample. The results are presented in Figure 6 and in Table 5. Broadly speaking, the LF determinations over the range $z\sim 4$–8 show clear evidence for a steady buildup in the volume density and luminosity of galaxies with cosmic time.

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Table 5.  Stepwise Determination of the Rest-frame UV LF at $z\sim 4$, $z\sim 5$, $z\sim 6$, $z\sim 7$, $z\sim 8$, and $z\sim 10$ Using the SWML Method (Section 4.1)

${{M}_{1600,AB}}$ a	${{\phi }_{k}}$ (Mpc−3 mag−1)	${{M}_{1600,AB}}$ a	${{\phi }_{k}}$ (Mpc−3 mag−1)	${{M}_{1600,AB}}$ a	${{\phi }_{k}}$ (Mpc−3 mag−1)
$z\sim 4$ galaxies		$z\sim 6$ galaxies		$z\sim 8$ galaxies
−22.69	0.000003 ± 0.000004	−22.52	0.000002 ± 0.000002	−22.87	<0.000002b
−22.19	0.000015 ± 0.000009	−22.02	0.000015 ± 0.000006	−22.37	<0.000002b
−21.69	0.000134 ± 0.000023	−21.52	0.000053 ± 0.000012	−21.87	0.000005 ± 0.000003
−21.19	0.000393 ± 0.000040	−21.02	0.000176 ± 0.000025	−21.37	0.000013 ± 0.000005
−20.69	0.000678 ± 0.000063	−20.52	0.000320 ± 0.000041	−20.87	0.000058 ± 0.000015
−20.19	0.001696 ± 0.000113	−20.02	0.000698 ± 0.000083	−20.37	0.000060 ± 0.000026
−19.69	0.002475 ± 0.000185	−19.52	0.001246 ± 0.000137	−19.87	0.000331 ± 0.000104
−19.19	0.002984 ± 0.000255	−18.77	0.001900 ± 0.000320	−19.37	0.000533 ± 0.000226
−18.69	0.005352 ± 0.000446	−17.77	0.006680 ± 0.001380	−18.62	0.001060 ± 0.000340
−18.19	0.006865 ± 0.001043	−16.77	0.013640 ± 0.004200	−17.62	0.002740 ± 0.001040
−17.69	0.010473 ± 0.002229	$z\sim 7$ galaxiesc		$z\sim 10$ galaxies
−16.94	0.024580 ± 0.003500	−22.66	<0.000002b	−22.23	<0.000001b
−15.94	0.025080 ± 0.007860	−22.16	0.000001 ± 0.000002	−21.23	0.000001 ± 0.000001
$z\sim 5$ galaxies		−21.66	0.000033 ± 0.000009	−20.23	0.000010 ± 0.000005
−23.11	0.000002 ± 0.000002	−21.16	0.000048 ± 0.000015	−19.23	<0.000049b
−22.61	0.000006 ± 0.000003	−20.66	0.000193 ± 0.000034	−18.23	0.000266 ± 0.000171
−22.11	0.000034 ± 0.000008	−20.16	0.000309 ± 0.000061	⋯	⋯
−21.61	0.000101 ± 0.000014	−19.66	0.000654 ± 0.000100	⋯	⋯
−21.11	0.000265 ± 0.000025	−19.16	0.000907 ± 0.000177	⋯	⋯
−20.61	0.000676 ± 0.000046	−18.66	0.001717 ± 0.000478	⋯	⋯
−20.11	0.001029 ± 0.000067	−17.91	0.005840 ± 0.001460	⋯	⋯
−19.61	0.001329 ± 0.000094	−16.91	0.008500 ± 0.002940	⋯	⋯
−19.11	0.002085 ± 0.000171	⋯	⋯	⋯	⋯
−18.36	0.004460 ± 0.000540	⋯	⋯	⋯	⋯
−17.36	0.008600 ± 0.001760	⋯	⋯	⋯	⋯
−16.36	0.024400 ± 0.007160	⋯	⋯	⋯	⋯

^aDerived at a rest-frame wavelength of 1600 Å. ^bUpper limits are $1\sigma$. ^cThe CANDELS-EGS field contains a much larger number of luminous (${{M}_{{\rm UV},AB}}\lt -21.41$) $z\sim 7$ galaxy candidates than the other CANDELS fields (7 vs. 1, 2, 3, and 4) and may represent an extreme overdensity. Therefore, as an alternative to the present determination, we also provide a stepwise determination of the $z\sim 7$ LF in Table A3 in Appendix E, which excludes the CANDELS-EGS data set.

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We next attempt to represent the UV LFs at $z\sim 4$, $z\sim 5$, $z\sim 6$, $z\sim 7$, $z\sim 8$, and $z\sim 10$ using a Schechter-like parameterization (${{\phi }^{*}}({\rm ln} (10)/2.5)$ ${{10}^{-0.4(M-{{M}^{*}})(\alpha +1)}}$ ${{e}^{-{{10}^{-0.4(M-{{M}^{*}})}}}}$). Schechter functions exhibit a power-law-like slope α at the faint end, with an exponential cutoff brightward of some characteristic magnitude M^*. The Schechter parameterization has proven to be remarkably effective in fitting the LF of galaxies at both low and high redshifts (e.g., Blanton et al. 2003; Reddy & Steidel 2009).

The procedure we use to determine the best-fit Schechter parameters is that of Sandage et al. (1979: STY79) and has long been the method of choice in the literature. Like the SWML procedure of Efstathiou et al. (1988), this approach determines the LF shape that would most likely reproduce the observed surface density of galaxies in our many search fields. The approach is therefore highly robust against large-scale structure variations across the survey fields. As with the SWML approach, one must normalize the LF derived using this method in some way, and for this we require that the total number of sources observed across our search fields match the expected numbers.

We can make use of essentially the same procedure to derive the maximum likelihood Schechter parameters as we used for the stepwise LF in the previous section, after we convert model Schechter parameters to the equivalent stepwise LF. For this calculation, we adopt a 0.1 mag binning scheme in comparing the stepwise LF to the surface density of sources in our search fields. A 0.1 mag binning scheme is sufficiently high resolution that it will yield essentially the same results as estimates made without binning the observations at all (e.g., Su et al. 2011).

Our maximum likelihood results for the Schechter fits at $z\sim 4$, $z\sim 5$, $z\sim 6$, $z\sim 7$, and $z\sim 8$ are presented in Figure 7. Meanwhile, our best-fit Schechter parameters are presented in Table 6 using the XDF+HUDF09-Ps+ERS+CANDELS-GN+CANDELS-GS data set alone and using the full data set considered here. These Schechter parameters are also provided for the $z\sim 7$ and $z\sim 8$ LFs based on the full data set but excluding the CANDELS-EGS field, to indicate what the results would be excluding the large number of bright ($\sim -21.7$ mag) galaxies found in that CANDELS field. Finally, in Table A2 in Appendix E, we also present determinations of the Schechter parameters from the CANDELS-GN+XDF+HUDF09-Ps fields and CANDELS-GS+ERS+XDF+HUDF09-Ps fields separately.

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Table 6.  STY79 Determinations of the Schechter Parameters for the Rest-frame UV LFs at $z\sim 4$, $z\sim 5$, $z\sim 6$, $z\sim 7$, $z\sim 8$, and $z\sim 10$ (Section 4.2)

Dropout			${{\phi }^{*}}$
Sample	$\lt z\gt$	$M_{{\rm UV}}^{*}$ a	$({{10}^{-3}}$ Mpc−3)	α
Reddy & Steidel (2009)
U	3.0	−20.97 ± 0.14	1.71 ± 0.53	−1.73 ± 0.13
XDF+HUDF09-Ps+CANDELS-GN+GS+ERS
B	3.8	−20.88 ± 0.08	$1.97_{-0.29}^{+0.34}$	−1.64 ± 0.04
V	4.9	−21.10 ± 0.15	$0.79_{-0.18}^{+0.23}$	−1.76 ± 0.06
i	5.9	−21.10 ± 0.24	$0.39_{-0.14}^{+0.21}$	−1.90 ± 0.10
z	6.8	−20.61 ± 0.31	$0.46_{-0.21}^{+0.38}$	−1.98 ± 0.15
Y	7.9	−20.19 ± 0.42	$0.44_{-0.24}^{+0.52}$	−1.81 ± 0.27
J	10.4	−20.92 (fixed)	$0.013_{-0.005}^{+0.007}$	−2.27 (fixed)
All Fields (Excluding CANDELS-EGS)b
z	6.8	−20.77 ± 0.28	$0.34_{-0.14}^{+0.24}$	−2.03 ± 0.13
Y	7.9	−20.21 ± 0.33	$0.45_{-0.21}^{+0.42}$	−1.83 ± 0.25
All Fields
B	3.8	−20.88 ± 0.08	$1.97_{-0.29}^{+0.34}$	−1.64 ± 0.04
V	4.9	−21.17 ± 0.12	$0.74_{-0.14}^{+0.18}$	−1.76 ± 0.05
i	5.9	−20.94 ± 0.20	$0.50_{-0.16}^{+0.22}$	−1.87 ± 0.10
z	6.8	−20.87 ± 0.26	$0.29_{-0.12}^{+0.21}$	−2.06 ± 0.13
Y	7.9	−20.63 ± 0.36	$0.21_{-0.11}^{+0.23}$	−2.02 ± 0.23
J	10.4	−20.92 (fixed)	$0.008_{-0.003}^{+0.004}$	−2.27 (fixed)

^aDerived at a rest-frame wavelength of 1600 Å. ^bWhile our simulation results (Figure 4) suggest that it is possible to identify $z\sim 7$ and $z\sim 8$ galaxies using the available observations over the CANDELS EGS field (albeit with some intercontamination between $z\sim 7$ and $z\sim 8$ samples), the lack of deep Y-band observations over this search field makes the results slightly less robust than over the other CANDELS fields. Our quantification of the stepwise LFs at $z\sim 7$ and $z\sim 8$ from all fields (but excluding the CANDELS-EGS data set) is presented in Table A3 in Appendix E.

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These results suggest that a good fraction of the evolution in the UV LF at $z\gt 4$ may involve an evolution in both the normalization of the LF ${{\phi }^{*}}$ and the faint-end slope α. Evolution in ${{\phi }^{*}}$ would be expected, if galaxies in arbitrarily massive halos in the early universe were capable of reaching the same maximum luminosity at essentially all epochs, independent of redshift. Evolution in the faint-end slope α is also expected owing to the steepening of the halo mass function toward early times (e.g., Trenti et al. 2010; see Section 5.5).

These general conclusions are not significantly impacted by possible systematic errors in our analysis technique. Even if we make factor-of-2 changes in the contamination rate across all of our search fields, we only find ${\Delta}M\lesssim 0.1$ changes in the characteristic magnitude M^* at $z\sim 4$–7 and ${\Delta}{{{\rm log} }_{10}}\;{{\phi }^{*}}\lesssim 0.1$ changes in the normalization ${{\phi }^{*}}$. While the impact on our faint-end slope α estimates is larger, i.e., ${\Delta}\alpha$ changes of 0.01, 0.03, 0.02, and 0.10 at $z\sim 4$, $z\sim 5$, $z\sim 6$, and $z\sim 7$, respectively, these uncertainties are small relative to the overall evolution apparent from $z\sim 7$ to $z\sim 4$. Larger changes in the characteristic magnitude M^* are potentially possible (i.e., ${\Delta}{{M}^{*}}\gtrsim 0.1$), but for this to occur, contaminating sources must be systematically undercorrected so as to leave a 20% excess in the number of bright galaxies. Small (∼10%) systematic errors in the selection volumes (per 0.5 mag interval) also likely have a small impact on the best-fit Schechter parameters.

Some earlier studies have argued that a simple ${{\phi }^{*}}$ evolutionary model may allow for a better representation of the evolution of the LF than an evolution in M^* (Beckwith et al. 2006; van der Burg et al. 2010). At slightly higher redshifts ($z\gtrsim 6$), McLure et al. (2010), Bouwens et al. (2011b), McLure et al. (2013), and Oesch et al. (2014) all indicated that ${{\phi }^{*}}$ evolution may provide a slightly better description of the evolution of the UV LF. Of course, even distinguishing evolution in ${{\phi }^{*}}$ from M^* over the range $z\sim 6$–8 can be challenging (as McLure et al. 2013 note explicitly).

While a pure ${{\phi }^{*}}$ evolutionary model seems quite effective at fitting the evolution at the bright end of the LF to high redshift, such a model does not capture the considerable steepening the UV LF experiences over a wide-luminosity baseline. Fitting this steepening requires either evolution in α or evolution in M^* as had been preferred by Bouwens et al. (2007). Yan & Windhorst (2004) effectively captured both aspects of the approximate evolution with their best-fit LF at $z\sim 6$ (though they offer no clear justification in their analysis for their decision to fix M^* to the $z\sim 3$ value and to exclusively use the faint-end slope α to model possible shape changes in the UV LF).

The present evolutionary scenario in ${{\phi }^{*}}$ and α would appear to be quite different in form from the evolutionary scenario proposed by Bouwens et al. (2007) and McLure et al. (2009, 2011), which preferred evolution in the characteristic luminosity (particularly over the redshift range $z\sim 4$–6), with some evolution in ${{\phi }^{*}}$ and α at $z\gt 6$ (Bouwens et al. 2011b; McLure et al. 2013). However, in detail, an ${{\phi }^{*}}$ + α evolutionary scenario is not as different from M^* evolution as one might think given their different parameterizations. Changes in the characteristic magnitude M^* produce a similar steepening of the UV LF, as one can accomplish through changes in the faint-end slope α.

Moreover, as we show in Section 5.3, the evolution in the UV luminosity we find for a galaxy (at a fixed cumulative number density) under the present ${{\phi }^{*}}$ + α evolutionary scenario is essentially identical to what Bouwens et al. (2008, 2011b) found previously invoking an evolution in the characteristic magnitude M^* (Figure 17). Unless one has very wide-area data to obtain tight constraints on the bright end of the LF at high redshift (such as one has with the wide-area CANDELS data set), one can trade off changes in the characteristic magnitude M^* for changes in both α and ${{\phi }^{*}}$ (without appreciably affecting the goodness of fit). We discuss these issues in more detail in Appendixes F.2, F.3, F.6, and Figure A5.

An alternate way of looking at the evolution in the UV LF is by rescaling the volume densities of our derived LFs so that they have the same normalization at −21.1 mag. We chose to rescale the LFs so they have the same normalization at this luminosity, which approximately corresponds to the value of M^* at $z\sim 4$–7. This allows us to look for systematic changes in the shape of the UV LF without relying on a specific parameterization of the LF. The results are presented in Figure 8, and it is clear that the LF adopts an increasingly steep form at higher redshift. It is also clear that the volume density of galaxies at $z\sim 4$–7 does not fall off precipitously until brightward of −22.5 mag.

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Before moving on to a discussion of possible non-Schechter-like features in the LF field-to-field variations, or our LF constraints at $z\sim 10$, it is useful to compare the present LF results with previous results from our own team (e.g., Bouwens et al. 2007, 2011b; Oesch et al. 2012b, 2014), as well as those from other groups. We include a comprehensive set of comparisons to previous results in Figure 9.

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Overall, we find broad agreement with previous LF results over the full redshift range $z\sim 4$–10. However, there are also some noteworthy differences, particularly with regard to the volume densities of the most luminous $z\sim 6$–8 galaxies. In our current results, we find a higher volume density for luminous (${{M}_{{\rm UV},{\rm AB}}}\lt -20.5$) $z\sim 6$–8 galaxies than reported earlier.

This significant update to our measurement of the volume density of high-luminosity $z\geqslant 6$ galaxies is the direct result of our lacking sufficiently deep ($H\gtrsim 25.5$) near-IR observations over very wide areas prior to the CANDELS program. With the new searches, we can now probe ∼15 times more volume than was possible in our earlier $z\sim 7$–8 study (Bouwens et al. 2011b) and ∼3 times more volume than in our earlier Bouwens et al. (2007) $z\sim 4$–6 LF analysis. Our new LF results agree quite well with our earlier results, if we only consider the LF constraints over a range that was well constrained by previous observations (Figure 10).

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Differences with our previous $z\sim 6$ constraints (Bouwens et al. 2007) can be attributed to the large increases in search volume, the availability of near-IR coverage to do a proper two-color selection of $z\sim 6$ galaxies, and consistent coverage with HST to minimize the impact of systematics on our results, as one can verify by using the new WFC3/IR information to k-correct previous results (Appendix F.2). The explanation for the observed differences with previous ground-based results (McLure et al. 2009; Willott et al. 2013) is less clear but can at least partially be explained by uncertainties in deriving total luminosities from the z-band fluxes (both from the intergalactic medium (IGM) correction and k-correcting the results to 1600 Å: another ∼0.13 mag correction) and also possibly large field-to-field variance (Bowler et al. 2015). In any case, it is encouraging that our $z\sim 6$ catalog and the Willott et al. (2013) catalogs agree quite well over the search fields where there is overlap (the brightest $z\sim 6$ galaxies we find over the CANDELS COSMOS and EGS fields are exactly the same $z\sim 6$ candidates as found by Willott et al. [2013] and we only miss one of the Willott et al. [2013] candidates over that field). It is also encouraging that new wide-area search results from Bowler et al. (2015) utilizing both the UltraVISTA and UDS fields are consistent with our determinations.

At $z\sim 7$, our LF results also indicate a much higher volume density of bright sources than indicated previously in Bouwens et al. (2011b). However, this was largely due to our reliance on LF constraints available from the ground (e.g., from Ouchi et al. 2009b; Castellano et al. 2010). If there was an overcorrection for contamination in those studies or the total magnitudes derived for sources were systematically fainter (∼0.1–0.2 mag) than those found here, it could explain the observed differences. We remark that our present constraints on M^* and α are in excellent agreement with the Bouwens et al. (2011b) constraints on those quantities if only the HST search results from that study are considered (Figure 8 from Bouwens et al. 2011b and Figure A6 in Appendix F.3).

Our new LF constraints also show a higher volume density of sources at $\sim -21.5$ mag than was previously found in the McLure et al. (2013), Schenker et al. (2013), or Bowler et al. (2014) studies. Differences with the McLure et al. (2013) and Schenker et al. (2013) results appear to occur as a result of ∼0.2 mag bias in the total magnitudes measured by McLure et al. (2013) and Schenker et al. (2013) for the brightest sources (see Figure A8 in Appendix H). Differences relative to Bowler et al. (2014) determinations over the UltraVISTA and UDS fields can potentially be explained, if Bowler et al. (2014) overestimated their completeness at the faint end of their probe, but we note that our constraint at −22.2 mag is consistent with the Bowler et al. (2014) determinations. Moreover, it is encouraging that we identify exactly the same two $z\sim 7$ galaxies over the CANDELS COSMOS area we probe as Bowler et al. (2014) find as part of their wide-area search ($z\sim 7$ candidates 268576 and 271028 from Bowler et al. [2014] lie over that region of the CANDELS COSMOS field that lacks deep optical ACS data and hence are not included in our search).

At $z\sim 8$, our new LF results are generally in excellent agreement with all previous HST studies. However, we do note a slight excess at the bright end of the $z\sim 8$ LF relative to previous studies. This excess derives from three particularly bright (${{H}_{160,{\rm AB}}}\sim 25$ mag) $z\gt 7$ candidate galaxies found over the CANDELS EGS program. Each of these candidates appears very likely to be at $z\gt 7$, as they each have $[3.6]-[4.5]$ colors of ∼0.8 mag, very similar to that found by Ono et al. (2012), Finkelstein et al. (2013), and Laporte et al. (2014). One of these bright candidates has been spectroscopically confirmed to lie at z = 7.73 (Oesch et al. 2015).

For a more extensive set of comparisons with previous work, we refer the reader to Appendix F.

Several previous studies (Bowler et al. 2012, 2014) have presented evidence that the UV LF at $z\sim 7$ is not well represented by a Schechter function, but is rather better represented by a double power law:

$$\begin{eqnarray*}&&\phi (M)=\frac{{{\phi }^{*}}}{{{10}^{0.4(\alpha +1)(M-{{M}^{*}})}}+{{10}^{0.4(\beta +1)(M-{{M}^{*}})}}}\end{eqnarray*}$$

Bowler et al. (2014) derive their constraints on the bright end of the $z\sim 7$ LF from the UltraVISTA+UDS fields. If true, the Bowler et al. (2014) claim would be interesting, as it would imply that the UV LF at $z\sim 7$ does not cut off abruptly at a specific luminosity (as it would if the LF were exponential), perhaps indicating that mass quenching and dust extinction were not as important early in the history of the universe as they were at later times.

The depth, area, and redshift range provided by our present samples put us in an unprecedented position to examine the general shape of the $z\sim 4$–8 UV LF and to see whether the UV LF is better represented by a Schechter-like function, a power law, a double power law, or some other functional form.²³ More precise constraints will eventually be possible, of course, integrating current HST constraints with even wider-area probes of the LF.²⁴

There are at least two different facets to this endeavor. The first regards the shape of the UV LF at the bright end. Does the UV LF show an exponential-like cutoff at the bright end, or is the bright end of the LF better represented by a steep power law? This was the question Bowler et al. (2014) attempted to answer. The second regards the shape of the UV LF at the faint end. Does the effective slope of the UV LF asymptote to a constant power-law slope (after modulation by an exponential), or does the effective slope of the UV LF show some dependence on luminosity even at very faint magnitudes? This second question was considered by Muñoz & Loeb (2011), as Figure 3 from their paper illustrates quite well.

Perhaps the easiest way to look for deviations from a Schechter-like form of the UV LF is by comparing the stepwise maximum likelihood LF to the Schechter LF determined using the STY technique and computing the residuals as a function of luminosity. The result is shown in Figure 11. The lack of a significant trend relative to the best-fit Schechter functions suggests that the UV LFs at $z\sim 4$–8 can be described reasonably well with a Schechter function.

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We can look at the overall functional form of the UV LF more directly by computing the effective slope of the LFs as a function of luminosity. This will allow us to assess whether other functional forms, i.e., a double power law, a rolling power law, or a simple power law, also provide a reasonable representation of the UV LF. As with determinations of the UV LF itself, the effective slope of the LF can be derived over a limited range in luminosity using the same maximum likelihood technique as we used on the UV LF itself (i.e., by STY79). For simplicity, we only attempt to derive these slopes at six distinct luminosities along the LF, i.e., −22.5, −21.5, −20.5, −19.5, −18.5, and −17.5.

In deriving the effective slope of the UV LF at these luminosities, we only consider sources 0.75 mag brighter and fainter than these luminosities, providing us with a total luminosity baseline for these slope measurements of 1.5 mag. Since this luminosity baseline is slightly longer than the separation between our slope measurements, we caution that the slope measurements we derive will not be entirely independent of each other. The longer luminosity baseline for each slope determination is quite useful, though, given the reductions in uncertainty on each slope measurement.

The result is shown in the panel of Figure 12 for the $z\sim 4$, $z\sim 5$, $z\sim 6$, $z\sim 7$, and $z\sim 8$ LFs. Also shown in this figure (open black circle) is the effective slope of the $z\sim 7$ LF at –22 mag by comparing our −21.6 mag z = 7 LF constraint with the volume density of −22.5 mag $z\sim 7$ galaxies obtained by Bowler et al. (2014). It is clear from these results that the effective slope of the LF at $z\sim 6$, $z\sim 7$, and $z\sim 8$ is generally steeper than it is at $z\sim 4$–5. We already saw this in the fit results from the previous section.

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To obtain a more precise constraint on the general shape of the LFs at $z\sim 4$–8, we can try to combine the constraints from the LFs at $z\sim 4$, $z\sim 5$, $z\sim 6$, $z\sim 7$, and $z\sim 8$ considered individually. Motivated by the results from Section 4.2, perhaps the best way of accomplishing this is to assume that the effective slope results at z = 4–8 LFs are all the same, modulo a change in the zero point (e.g., ${\Delta}{{\alpha }_{z=5}}$, ${\Delta}{{\alpha }_{z=6}}$, ${\Delta}{{\alpha }_{z=7}}$, ${\Delta}{{\alpha }_{z=8}}$). If we do so and find the offsets that minimize the overall differences in the inverse-variance-weighted mean corrected offset at each redshift (specifically minimizing ${{{\Sigma}}_{M}}{{{\Sigma}}_{z}}{{({{\alpha }_{M,z}}+{\Delta}{{\alpha }_{z}}-\overline{{{\alpha }_{M}}+{\Delta}\alpha })}^{2}}/\sigma {{({{\alpha }_{M,z}})}^{2}},$ where ${{\alpha }_{M,z}}$ indicates the effective slope measurements at a given absolute magnitude M and redshift z), we find the following offsets in slope ${\Delta}{{\alpha }_{z}}$ for the $z\sim 5$, $z\sim 6$, $z\sim 7$, and $z\sim 8$ LFs relative to the $z\sim 4$ LF: 0.01, 0.27, 0.37, and 0.64, respectively.

We then apply these offsets to the slopes of the $z\sim 4$, $z\sim 5$, $z\sim 6$, $z\sim 7$, and $z\sim 8$ LFs shown in the top panel and show the result in the lower left panel of Figure 12. For context, we also show in Figure 12 the luminosity dependence we would expect adopting the typical Schechter function results derived in the previous section (shaded curve), with ${{M}^{*}}=-21.07$ and $\alpha =-1.73$. Overall, the constraints we have on the slope of the UV LF as a function of luminosity all appear to be remarkably similar to each other (after one removes the general offset in slope).

It is interesting to try to combine the constraints we have available on the $z\sim 4$–8 LFs to examine the overall form of the UV LF at $z\gtrsim 4$. We examine the z = 4–6 case and the z = 7–8 case separately, given possible evolution in both the shape and functional form of the LF. In the two cases, we compute the inverse-variance-weighted mean effective slope and variance as a function of luminosity (after removing the zero-point offset in effective slope).

The estimated 68% confidence intervals on the effective slope of the z = 4–6 and z = 7–8 LFs are indicated in the upper right and lower right panels of Figure 12 with the dark gray and light gray regions, respectively. In general, we find that our luminosity-dependent slope results are in broad agreement with the expectations of a Schechter function. At the low-luminosity end, we see no evidence for the effective slope of the LF being especially steeper at −19.5 mag than at −17.5 mag. This argues against the effective slope of the UV LF being strongly luminosity dependent, as one might expect if there is curvature in the halo mass function or if galaxy formation were less efficient at lower masses (e.g., Muñoz & Loeb 2011).

At high luminosities, the z = 4–6 UV LFs show evidence for a similar exponential-like cutoff at bright magnitudes to that present in a Schechter function (compare the dark gray region in the upper left panel of Figure 12 with the light gray region). Not surprisingly, at z = 7–8, our overall constraints on the shape of the UV LF at high luminosities are much weaker and clearly not sufficient to constrain the functional form of the LF. However, our results do seem consistent with that observed at z = 4–6 and also adopting a Schechter function (compare the light gray region in the lower left panel with the dark gray region). For context, we also show the effective slope results implied from the Bowler et al. (2014) double power-law fit (shown in the lower right panel of Figure 12 as the solid black line), i.e., with $\alpha =-2.1$, $\beta =-4.2$, ${{M}^{*}}=-20.3$, and ${{\phi }^{*}}=3.9\times {{10}^{-4}}$ Mpc⁻³ mag⁻¹. While it is reasonable to imagine that the UV LF may exhibit a slightly non-Schechter shape at early enough times, we find no strong evidence for such a behavior here.

It is interesting to ask why our conclusions appear to differ from those of Bowler et al. (2014). For the purpose of this discussion, we compare our LF constraints with the double power-law fit they find for their $z\sim 7$ LF in Figure 13. While we find good agreement between the Bowler et al. (2014) power-law fit and our results at both the bright and faint ends, our LF is in excess of their double power-law fit at moderately high luminosities (−21.7 mag), suggesting that this is the origin of our different conclusions.

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How reliable are our $z\sim 7$ LF constraints at $\sim -21.7$ mag? In the luminosity interval −21.91 mag to −21.41 mag, we find 16 galaxy candidates in total (3, 1, 1, 4, and 7 from the CANDELS-GS, GN, UDS, COSMOS, and EGS fields, respectively), so the uncertainties from shot noise (0.12 dex) are relatively limited. In addition, all 16 appear to be relatively robust $z\sim 6.3$–7.3 galaxy candidates, as inferred from the tests we run in Appendix G and Sections 3.2.2–3.2.3 (for distinguishing stars and galaxies). Nevertheless, there are other issues that could have an impact. If the large number of bright sources in the CANDELS-EGS field represents a rare overdensity and we exclude that field, if our total magnitude estimates are too bright by 0.1 mag, or if the completeness is underestimated (and it is instead 100%), then our $\sim -21.7$ mag point in the $z\sim 7$ LF would be lower by 0.15 dex, 0.09 dex, and 0.14 dex, respectively. Even if we assume all three issues to be the case (as a worst-case scenario), our LF estimates (open dotted circle in Figure 13) would only be lower by 0.38 dex and still be in tension with the Bowler et al. (2014) $\sim -21.7$ mag point by ∼0.4 dex. Such a constraint, however, would be in excellent agreement with a Schechter fit to the LF at $z\sim 7$ (but we note that fits to other functional forms would also be possible given the uncertainties).

One generic concern for the determination of any LF is the presence of large-scale structure. As a result of such structure, the volume density of sources seen in one's survey fields can lie significantly above or below that of the cosmic average—resulting in sizable field-to-field variations. While normally these field-to-field variations introduce considerable uncertainties in our LF determinations, the availability of deep HST+ground-based observations over five independent survey fields allows us to largely overcome this issue. We estimate the overall uncertainty on our LF results to be just 10%, by using the Trenti & Stiavelli (2008) cosmic variance calculator and accounting for the fact that we have observational constraints over five independent ∼150 arcmin² search fields.

Owing to the large number of independent search fields, we can perform a different test. Instead of our results on the UV LF being significantly limited by the impact of cosmic variance, we can use the current samples to set interesting constraints on the amplitude of the field-to-field variations themselves. For simplicity, we assume that we can capture all variations in the LF through a change in its normalization ${{\phi }^{*}}$, keeping the characteristic magnitude M^* and faint-end slope α for galaxies at a given redshift fixed. The best-fit values for ${{\phi }^{*}}$ we derive for sources in each field relative to that found for all fields are shown in Figure 14 for sources in all five samples considered here. Bouwens et al. (2007) previously attempted to quantify the differences in surface densities of $z\sim 4$, $z\sim 5$, and $z\sim 6$ sources over GN and GS (see also Bouwens et al. 2006; Oesch et al. 2007). Uncertainties on the value of ${{\phi }^{*}}$ in a field relative to the average of all search fields are calculated based on the number of sources in each field assuming Poissonian uncertainties, allowing for small (∼10%) systematic errors in the calculated selection volumes field-to-field.

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While the volume density of high-redshift candidates in most wide-area fields does not differ greatly (typically varying ≲20% field-to-field), there are still sizable differences present for select samples field-to-field. One of the largest deviations from the cosmic average occurs for $z\sim 7$ galaxies over the EGS field where the volume density appears to be almost double what it is over the CANDELS-GS, COSMOS, or UDS fields, for example. The CANDELS-GN also shows a similar excess at $z\sim 7$ relative to these other fields (see also Finkelstein et al. 2013). The relative surface density of $z\sim 4$, $z\sim 5$, and $z\sim 6$ candidates over the CANDELS-GN and GS fields is similar to what Bouwens et al. (2007) found previously (see Table B1 from that work), with the GS field showing a slight excess in $z\sim 4$ and $z\sim 6$ candidates relative to GN and the GN field showing an excess of $z\sim 5$ candidates.

Generally, however, the observed field-to-field variations are well within the expected ∼20% variations in volume densities for the large volumes probed in the present high-redshift samples.

We also took advantage of our large search areas to set constraints on the UV LF at $z\sim 10$. Only a small number of $z\sim 10$ candidates were found, but they still provide, along with the upper limits, a valuable addition to the $z\sim 4$–8. In doing so, we slightly update the recent LF results of Oesch et al. (2014) to consider the additional search area provided by the CANDELS-UDS, CANDELS-COSMOS, and CANDELS-EGS fields.

Owing to the fact that the majority of our search fields contain zero $z\sim 10$ candidates, we cannot use the bulk of the present fields to constrain the shape of the LF, making the SWML and STY fitting techniques less appropriate. In such cases, it can be useful to simply derive the UV LF assuming that the source counts are Poissonian distributed (given that field-to-field variations will be smaller than the very large Poissonian uncertainties). One then maximizes the likelihood of both the stepwise and model LFs by comparing the observed surface density of $z\sim 10$ candidates with the expected surface density of $z\sim 10$ galaxies in the same way as we have done before (e.g., Bouwens et al. 2008).

Figure 15 shows the constraints we derive on the stepwise LF at $z\sim 10$ based on the present searches (the $z\sim 10$ results are also provided in Table 5). A 1 mag binning scheme is used, given the very small number of $z\sim 10$ candidates in the present search. Also included in Figure 15 is our best-fit Schechter function at $z\sim 10$. For the latter fit, we fix the characteristic magnitude M^* equal to −20.92 and the faint-end slope α to −2.27, consistent with the approximate characteristic magnitude M^* and faint-end slope α we estimate based on the LF fitting formula we present in Section 5.1.

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The best-fit ${{\phi }^{*}}$ we estimate using our $z\sim 10$ search over all of our search fields is $0.000008_{-0.000003}^{+0.000004}$ Mpc⁻³. We tabulate this value of ${{\phi }^{*}}$ in Table 6. As we will discuss in Appendix F.5, the best-fit parameters we derive here are consistent with what Oesch et al. (2014) derived previously from a search over the CANDELS-GN+GS+XDF+HUDF09-Ps fields. These parameters are also consistent with the $10\times$ evolution in volume density that Oesch et al. (2013a, 2014) find from $z\sim 10$ to $z\sim 8$.

As in previous work (e.g., Bouwens et al. 2008), it is useful to take the present constraints on the UV LF and condense them into a fitting formula for describing the evolution of the UV LF with cosmic time. This enterprise has utility not only for extrapolating the present results to $z\gt 8$ but also for interpolating between the present LF determinations at $z\sim 4$, $z\sim 5$, $z\sim 6$, $z\sim 7$, and $z\sim 8$ when making use of a semi-empirical model. We will assume that each of the three Schechter parameters (M^*, α, ${{{\rm log} }_{10}}{{\phi }^{*}}$) depends linearly on redshift when deriving this formula. The resultant fitting formula is as follows:

$$\begin{eqnarray*}\begin{array}{rcl} M_{{\rm UV}}^{*} & = & (-20.95\pm 0.10)+(0.01\pm 0.06)(z-6) \\ {{\phi }^{*}} & = & (0.47_{-0.10}^{+0.11}){{10}^{(-0.27\pm 0.05)(z-6)}}{{10}^{-3}}{\rm Mp}{{{\rm c}}^{-3}} \\ \alpha & = & (-1.87\pm 0.05)+(-0.10\pm 0.03)(z-6). \\ \end{array}\end{eqnarray*}$$

Constraints from Reddy & Steidel (2009) on the faint-end slope of the LF at $z\sim 3$ were included in deriving the above best-fit relations. As is evident from these relations, the evolution in the faint-end slope α is significant at $3.4\sigma$. The evolution in the normalization ${{\phi }^{*}}$ of the LF is significant at $5.4\sigma$. We find no significant evolution in the value of M^*.

Given the considerable degeneracies that exist between the Schechter parameters, it is also useful to derive the best-fit model if we fix the characteristic magnitude M^* to some constant value and assume that all of the evolution in the effective shape of the UV LF is due to evolution in the faint-end slope α. For these assumptions, the resultant fitting formula is as follows:

$$\begin{eqnarray*}\begin{array}{rcl} M_{{\rm UV}}^{*} & = & (-20.97\pm 0.06)\quad {\rm (fixed)} \\ {{\phi }^{*}} & = & (0.44\pm 0.06){{10}^{(-0.28\pm 0.02)(z-6)}}{{10}^{-3}}{\rm Mp}{{{\rm c}}^{-3}} \\ \alpha & = & (-1.87\pm 0.04)+(-0.100\pm 0.018)(z-6). \\ \end{array}\end{eqnarray*}$$

From this fitting formula, we can see that the steepening in the effective shape of the UV LF (as seen in Figure 8) appears to be significant at 5.7σ.

The apparent evolution in the faint-end slope α is quite significant. Even if we allow for large factor-of-2 errors in the contamination rate or sizable (∼10%) uncertainties in the selection volume (as we consider in Section 4.2), the formal evolution is still significant at $2.9\sigma$, while the apparent steepening of the UV LF presented in Figure 8 remains significant at $5\sigma$ (instead of $5.7\sigma$).

The best-fit faint-end slopes α we find in the present analysis are presented in Figure 16. The faint-end slope α we determine is equal to −1.87 ± 0.10, −2.06 ± 0.13, and −2.02 ± 0.23 at $z\sim 6$, $z\sim 7$, and $z\sim 8$, respectively. Faint-end slopes α of $\sim -2$ are very steep, and the integral flux from low-luminosity sources can be very large since the luminosity density in this case is formally divergent. While clearly the UV LF must cut off at some luminosity, the UV light from galaxies fainter than −16 should dominate the overall luminosity density (Bouwens et al. 2012a).

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In combination with the results at somewhat lower redshifts, the present results strongly argue for increasingly steep faint-end slopes α at higher redshifts. Results from Section 5.1 suggest that this evolution is significant at $3.1\sigma$ if we consider just the formal evolution in the faint-end slope α itself. The evolution is significant at $5.7\sigma$ if we consider the evolution in the shape of the UV LF (Figure 8).

While consistent with previous results, the present results suggest slightly steeper faint-end slopes α than reported in Bouwens et al. (2011b), McLure et al. (2013), and Schenker et al. (2013) at $z\sim 7$. These steeper faint-end slopes are a direct consequence of the somewhat brighter values for M^* that we find in the current study and the trade-off between fainter values for M^* and steeper faint-end slopes α. These results only serve to strengthen earlier findings suggesting that the faint-end slope α is steeper at $z\sim 7$ (and likely $z\sim 8$) than it is at $z\sim 3.$ Similar conclusions have been drawn from follow-up work on gamma-ray hosts (Robertson & Ellis 2012; Tanvir et al. 2012; Trenti et al. 2012b, 2013).

Given the apparent evolution of the UV LF, one might ask how rapidly the UV luminosity or SFR of an individual galaxy likely increases with cosmic time. Fortunately, we can make progress on this question using a number density matching procedure,²⁵ by ordering galaxies in terms of their observed UV luminosities and following the evolution of those sources with a fixed cumulative number density.

For convenience, we adopt the same integrated number density 2 × 10⁻⁴ Mpc⁻³ (the approximate cumulative number density for L^* galaxies) for this question as Papovich et al. (2011; see also Lundgren et al. 2014) had previously considered in quantifying the growth in the SFR of an individual galaxy with cosmic time. Dust corrections are performed using the measured β values for galaxies at $z\sim 4$–8 (Bouwens et al. 2014b) and the well-known IRX-β relationship from Meurer et al. (1999).

The results are presented in Figure 17. The UV luminosity at a fixed cumulative number density evolves as ${{M}_{{\rm UV}}}(z)=-20.40+0.37(z-6)$. Interestingly enough, the evolution in the UV luminosity we infer for galaxies at some fixed cumulative number density is almost identical to what Bouwens et al. (2011b) had previously inferred for the evolution in the characteristic magnitude M^* with redshift (i.e., $-20.29+0.33(z-6)$; dotted black line).

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Upon reflection, it is clear why this must be so. For pure luminosity evolution, one would expect both the characteristic magnitude ${{M}^{*}}$ of the UV LF and the UV luminosity of individual galaxies to evolve in exactly the same manner. Even though we now see that such a scenario does not work for the brightest, rarest galaxies, one can nevertheless roughly parameterize the evolution of fainter galaxies assuming pure luminosity evolution. For these galaxies, the Bouwens et al. (2008, 2011b) fitting formula for M^* evolution works remarkably well in describing their steadily increasing UV luminosities. In this way, the modeling of the evolution of the LF using M^* evolution by Bouwens et al. (2008, 2011b)—a treatment built on by Stark et al. (2009)—effectively foreshadowed later work using a sophisticated cumulative number density matching formalism to trace the star formation history of individual systems at $z\gt 2$ (Papovich et al. 2011; Lundgren et al. 2014).

The SFR for a galaxy in this number density matched scenario evolves as ${\rm SFR}=(15.8{\mkern 1mu} {{M}_{\odot }}/{\rm yr}){{10}^{-0.24(z-6)}}$. The evolution in the SFR is remarkably similar to the relations found by Papovich et al. (2011) and Smit et al. (2012). Not surprisingly, the best-fit trends for galaxies with L^*-like volume densities (i.e., at ∼2 × 10⁻⁴ Mpc⁻³) show little dependence on the parameterization of the Schechter function and whether one fits the evolution through a change in M^* or a change in ${{\phi }^{*}}$ and α.

We will take advantage of our new LF determinations at $z\sim 4$–10 to provide updated measurements of the UV luminosity density at $z\sim 4$–10. As in previous work (Bouwens et al. 2007, 2008, 2011b; Oesch et al. 2012a), we only derive the UV luminosity density to the limiting luminosity probed by the current study at $z\sim 8$, i.e., −17 mag (0.03 $L_{z=3}^{*}$), to keep these determinations as empirical as possible. Since this is slightly fainter than what one can probe in searches for galaxies at $z\sim 10$, we make a slight correction to our $z\sim 9$ and $z\sim 10$ results. The best-fit faint-end slope $\alpha =-2$ we find at $z\sim 8$ is assumed in this correction. The use of even steeper faint-end slopes (i.e., −2.3) as implied by our LF fitting formula in Section 5.1 would yield similar results, only increasing the luminosity density by ∼0.015 dex.

In combination with our estimates of the luminosity density, we also take this opportunity to provide updated measurements of the SFR density at $z\sim 4$–10. In making these estimates of the SFR density at $z\sim 4$–10, we correct for dust extinction using the well-known IRX-β relationship (Meurer et al. 1999) combined with the latest measurements of β from Bouwens et al. (2014b). As before, we assume that the extinction A_UV at rest-frame UV wavelengths is $4.43+1.99\beta$, with an intrinsic scatter of 0.35 in the β distribution. This is consistent with what has been found for bright galaxies at $z\sim 4$–5 (Bouwens et al. 2012b; Castellano et al. 2012). The new β determinations from Bouwens et al. (2014b) utilize large (>4000-source) samples constructed from the XDF, HUDF09-1, HUDF09-2, ERS, CANDELS-GN, and CANDELS-GS data sets and were constructed to provide much more accurate and robust measurements of the β distribution than has been provided in the past. The mean dust extinction we estimate based on the Meurer et al. (1999) law for the observed β distribution is 2.4, 2.2, 1.8, 1.66, 1.4, and 1.4 (in units of ${{L}_{{\rm IR}}}/{{L}_{{\rm UV}}}+1,$ where L_IR and L_UV are the bolometric and UV luminosities of a galaxy, respectively) for the observed galaxies at $z\sim 4$, $z\sim 5$, $z\sim 6$, $z\sim 7$, $z\sim 8$, and $z\sim 10$, respectively.

The dust-corrected UV luminosity densities are then converted into SFR densities using the canonical Kennicutt (1998) and Madau et al. (1998) relation:

$$\begin{eqnarray}&&{{L}_{{\rm UV}}}=\left( \frac{{\rm SFR}}{{{M}_{\odot }}{\rm y}{{{\rm r}}^{-1}}} \right)8.0\times {{10}^{27}}{\rm erg}\;{{{\rm s}}^{-1}}\;{\rm H}{{{\rm z}}^{-1}}\end{eqnarray} \tag{ 5 }$$

where a 0.1–$125\;{{M}_{\odot }}$ Salpeter initial mass function (IMF) and a constant SFR for ages of $\gtrsim 100$ Myr are assumed. In light of the very high EWs of the ${\rm H}\alpha$ and [O III] emission lines in $z\sim 4$–8 galaxies (Schaerer & de Barros 2009; Shim et al. 2011; Stark et al. 2013; Schenker et al. 2013; González et al. 2012, 2014; Labbé et al. 2013; Smit et al. 2014a), it is probable that the adopted conversion factors underestimate the actual SFRs (perhaps by as much as a factor of 2; Castellano et al. 2014).

Our updated results on both the luminosity density and SFR density are presented in Table 7 and Figure 18. As before, we have included select results from the literature (Schiminovich et al. 2005; Reddy & Steidel 2009) to show the trends at $z\lt 4$, as well as presenting recent determinations of the SFR density at $z\sim 0.5$–2.0 from IR-bright sources (Daddi et al. 2009; Magnelli et al. 2009, 2011). We also include select $z\geqslant 6$ results from the literature for comparison with previous results (Zheng et al. 2012; Coe et al. 2013; Ellis et al. 2013; McLure et al. 2013; Bouwens et al. 2014a).

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Table 7.  UV Luminosity Densities and Star Formation Rate Densities to −17.0 AB mag (0.03 $L_{z=3}^{*};$ see Section 5.4)^b

		${{{\rm log} }_{10}}\mathcal{L}$	log10 SFR density
Dropout		(ergs s−1	(M⊙ Mpc−3 yr−1)
Sample	<z >	Hz−1 Mpc−3)	Dust Uncorrected	Dust Corrected
B	3.8	26.52 ± 0.06	$-1.38\pm$ 0.06	−1.00 ± 0.06
V	4.9	26.30 ± 0.06	$-1.60\pm$ 0.06	−1.26 ± 0.06
i	5.9	26.10 ± 0.06	$-1.80\pm$ 0.06	−1.55 ± 0.06
z	6.8	25.98 ± 0.06	$-1.92\pm$ 0.06	−1.69 ± 0.06
Y	7.9	25.67 ± 0.06	$-2.23\pm$ 0.07	−2.08 ± 0.07
J	10.4	24.62$_{-0.45}^{+0.36}$	−3.28 $_{-0.45}^{+0.36}$	−3.13 $_{-0.45}^{+0.36}$

^aIntegrated down to 0.05 $L_{z=3}^{*}$. Based on LF parameters in Table 2 of Bouwens et al. (2011b; see also Bouwens et al. 2007; see Section 5.4). The SFR density estimates assume $\gtrsim 100$ Myr constant SFR and a Salpeter IMF (e.g., Madau et al. 1998). Conversion to a Chabrier (2003) IMF would result in a factor of ∼1.8 (0.25 dex) decrease in the SFR density estimates given here.

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We observe very good agreement with previous results over the full range in redshift $z\sim 4$–10. The most noteworthy changes occur at $z\sim 5,$ where the volume density we find is higher than estimated previously (Bouwens et al. 2007) and better matches the evolutionary trend connecting the $z\sim 4$ and $z\sim 6$ results. The improved robustness of the present $z\sim 5$ results is likely a direct consequence of the significantly broader wavelength baseline available to select $z\sim 5$ galaxies over the $z\sim 4.5$–5.5 volume than was available in the earlier purely optical/ACS data set (e.g., see discussion in Duncan et al. 2014).

It is interesting to compare the current observational results with what is found from large hydrodynamical simulations and also from simple theoretical models. Such comparisons are useful for interpreting the present results and also for ascertaining whether any of our observational results are unexpected or challenge the current paradigm in any way. We first describe the models, and then in the following subsections we discuss comparisons with our new LF results.

The first set of cosmological hydrodynamical simulations we consider are those from Jaacks et al. (2012). These results provide a very detailed investigation as to how the shape of the UV LF might evolve with cosmic time. Jaacks et al. (2012) make use of some large simulations done on a modified version of the GADGET-3 code (Springel 2005) that includes cooling by H+He+metal line cooling, heating by a modified Haardt & Madau (1996) spectrum (Katz et al. 1996), an Eisenstein & Hu (1999) initial power spectrum, "Pressure model" star formation (Schaye & Dalla Vecchia 2008), SN feedback, and a multiple-component variable velocity wind model (Choi & Nagamine 2011). Simulations are done with a range of box sizes from 10 h⁻¹ to 100 h⁻¹ Mpc ($2\times {{600}^{3}}$ or $3\times {{400}^{3}}$ particles).

As an alternative to the results from large hydrodynamical simulations, we make use of a much more simple-minded theoretical model using a conditional LF (CLF; Yang et al. 2003; Cooray & Milosavljević 2005) formalism where one derives the LF from the halo mass function using some mass-to-light kernel. We adopt the same CLF model as Bouwens et al. (2008) had previously used in their analysis of the UV LF, but we have modified the model to include a faster evolution in the mass-to-light (M/L) ratio of halos, i.e., $\propto {{(1+z)}^{-1.5}}.$ This evolution better reproduces changes in the observed UV LF from $z\sim 8$ to $z\sim 4$. The ${{(1+z)}^{-1.5}}$ factor also matches the expected evolution of the dynamical timescale. A detailed description of this model is provided in Appendix I. The advantage of this approach is that it can give us insight into the extent to which the evolution in the UV LF is driven by the growth of dark matter halos themselves and to what extent the evolution arises from changes in the M/L ratio of those halos and hence gas-dynamical processes (e.g., gas cooling or SFR timescales).

Finally, we consider the predictions by Tacchella et al. (2013), which are based on a minimal model that also links the evolution of the UV galaxy LF to that of the dark matter halo mass function. The model is constructed by assuming that a halo of mass M_h at redshift z has a stellar mass ${{M}_{*}}=\epsilon ({{M}_{h}})*{{M}_{h}}$, of which a small fraction (10%) is formed at the halo assembly time z_a, while the remaining is formed at a constant rate from z_a to z. Since halos have shorter assembly times as redshift increases, the ratio of UV light to halo mass increases with redshift. $\epsilon ({{M}_{h}})$ describes the efficiency of the accretion in forming stars; Tacchella et al. (2013) calibrate it at z = 4 via abundance matching.

Before conducting detailed comparisons of the observational results with the above theoretical models, we first present a comparison of the binned LF results with the first two theoretical models to illustrate the broad overall agreeement between the two sets of results (Figure 19).

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5.5.1. Expected Evolution of the Faint-end Slope

5.5.2. Expected Evolution in the Characteristic Luminosity?

Our discovery of modest numbers of highly luminous galaxies in each of our high-redshift samples, even at $z\sim 10$, provides strong evidence against a rapid evolution in the luminosity where the UV LF cuts off. Over the redshift range $z\sim 4-7$, we find no significant evolution in M^* (see Table 6). Over the slightly wider redshift range $z\sim 4$–8, our best-fit estimate for the evolution in the characteristic magnitude M^* is just $d{{M}^{*}}/dz\sim 0.01\pm 0.06$ (see the fitting formula in Section 5.1) or just $d{{M}^{*}}\sim 0.25\pm 0.37$ from $z\sim 8$ to $z\sim 4$. Given the observed luminosity of the brightest $z\sim 10$ candidates found over the CANDELS fields (Oesch et al. 2014), i.e., −21.4 mag, it seems unlikely that the bright-end cutoff M^* is especially fainter than ${{M}^{*}}\sim -20$ (limiting the evolution in M^* to $\lesssim$ 1 mag over the redshift range $z\sim 4$–10).

This implies that whatever physical mechanism imposes a cutoff at the bright end of $z\gtrsim 4$ UV LFs, this cutoff luminosity does not vary dramatically with redshift, at least out to $z\sim 7$. Indeed, for the three mechanisms discussed by Bouwens et al. (2008) to impose a cutoff at the bright end of the UV LF, i.e., heating from an AGN (Croton et al. 2006), the inefficiency of gas cooling for high-mass halos (e.g., Binney 1977; Rees & Ostriker 1977; Silk 1977), and the increasing importance of dust attenuation for the most luminous and likely most massive galaxies (Bouwens et al. 2009, 2012b, 2014b; Pannella et al. 2009; Reddy et al. 2010; Finkelstein et al. 2012), there is no obvious reason any of these mechanisms should depend significantly on redshift or cosmic time.

Indeed, the results of the simulations or theoretical models bear out these expectations. The best-fit characteristic magnitudes M^* derived from the Jaacks et al. (2012) simulations show very little evolution with cosmic time. Jaacks et al. (2012) derive −21.15, −20.85, and −21.0 for their $z\sim 6$, $z\sim 7,$ and $z\sim 8$ LFs, respectively.

Simple fits to our CLF results also show only limited evolution in the characteristic magnitude M^* with redshift, even out to $z\sim 10$. The characteristic magnitudes we derive from fitting the model LFs at $z\sim 4$–10 (minimizing the square of the logarithmic residuals) are presented in Figure 20 for comparison with our observational determinations of this same quantity (Table 6). Both a model assuming fixed M/L ratio for the halos (black line) and a model with M/L ratio evolving as the dynamical time (${{(1+z)}^{-3/2}};$ blue line) are considered.

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It is useful to contrast these results with a CLF model where no cutoff is imposed at the bright end of the UV LF and where there is no evolution in the M/L ratio of halos. For the model described in Appendix I, this could be achieved by replacing the $(1+(M/{{m}_{c}}))$ term in Equation (I2) by unity and renormalizing the M/L ratio so that M^* for the model LF is equal to −21 at $z\sim 4$. The evolution in the characteristic magnitude M^* we would predict for this model is shown with the dashed gray line in Figure 20.

At sufficiently high redshift, it seems clear from the gray line that we would expect the characteristic magnitude M^* to be fainter owing to evolution in the halo mass function. In practice, however, the evolution in the characteristic magnitude M^* may be more limited if the bright-end cutoff to the UV LF is instead set by a physical process that becomes dominant at some mass threshold (e.g., dust obscuration or quenching), as the dotted black line in Figure 20 illustrates. Even less evolution would be expected in the characteristic magnitude M^* with cosmic time if halos at higher redshifts had systematically lower M/L ratio, as illustrated by the blue line in this same figure.

In reality, of course, we should emphasize that almost all LFs predicted by simulations or CLF models can only be approximately modeled using a Schechter-function-like parameterization, and therefore there can be considerable ambiguity in actually extracting the Schechter parameters from the model results and hence representing their evolution with cosmic time.