The z ∼ 6 Luminosity Function Fainter than −15 mag from the Hubble Frontier Fields: The Impact of Magnification Uncertainties

In making use of various gravitational lensing models to derive constraints on the prevalence of extremely faint galaxies at high redshift, it is important to obtain an estimate of how predictive the lensing models are for the true magnification factors.

One way of addressing this issue is the fully end-to-end approach pursued by Meneghetti et al. (2016) and involves constructing highly realistic mock data sets, analyzing the mock data sets using exactly the same approach as are used on the real observations, and then quantifying the performance of the different methods by comparing with the actual magnification maps. While each of the methods did fairly well in reproducing the magnification maps to magnification factors of ∼10, the best performing methods for reconstructing the magnification maps of clusters were the parametric models, with perhaps the best reconstructions achieved by the GLAFIC models, the Sharon/Johnson models, and the CATS models.

An alternate way of addressing this issue is by comparing the public lensing models against each other. Here we pursue such an approach. We treat one of the models as the truth and then to quantify the effectiveness of the other magnification models taken as a set for predicting that model's magnifiction map. We consider both cases in which the true mass profile of the HFF clusters is considered (1) to lie among the parametric class of models built on NFW-type mass profiles and (2) to lie among the non-parametric class of models, which allows for more freedom in the modeling process. We take the former models to include the GLAFIC, CATS, Sharon/Johnson, and Zitrin-NFW models, and the latter to include the Bradac (2009), Grale (Liesenborgs et al. 2006; Sebesta et al. 2016), and Zitrin-LTM (Zitrin et al. 2012, 2015).¹⁰ A brief description of the general properties of the public lensing models can be found in Table 1. In performing this test, we assume that the median of the magnification models provides our best means for predicting magnifications in the model we are treating as the truth. The truth model is always excluded when constructing the median magnification map for this test.

Alternatively treating each of the magnification models as the truth, we then quantify what the median magnification factor is in the truth model as a function of the median magnification factors from the other models. For perfectly predictive models, the magnification factors in the truth model would be precisely centered around the median magnification factors from the other models. In practice, this is not true, given the difficulty in predicting the precise locations of the rare regions around the cluster with the highest magnification factors. While one can control for these uncertainties through the use of quantities like the median, even the median will overpredict the true magnification, due to the impact of model "noise" on the medians and the possibility for chance overlap in the high-magnification regions across the models.

For the most general results, we take a geometric mean of the median magnification factors considering each model as the truth and then plot the results in the upper panel of Figure 3. Results on the predictive power of the parametric (GLAFIC, CATS, Sharon/Johnson, Zitrin-NFW) and non-parametric (Grale, Bradac, Zitrin-LTM) models are presented separately with magenta and blue colored lines. The dashed and dotted lines give the "true" magnifications recovered versus median magnification factors for best and worst performing clusters. Meanwhile, the solid line between the dashed and dotted lines gives the geometric mean of the "true" magnifications across all four clusters considered here. The lower panel of Figure 3 shows the position-to-position scatter around the median magnification in the model treated as the truth. From this exercise, it is clear that the magnification maps have excellent predictive power to magnification factors of ∼10 in all cases and perhaps to even higher magnification factors assuming that the magnification profiles of HFF clusters are as well behaved as in the parametric models. The scatter, however, is already very large at magnification factors of 10. We will extend this exercise in a future work (R. J. Bouwens et al. 2017, in preparation).

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The exercise we perform in this section shows similarities in philosophy to the analyses that Priewe et al. (2017) pursue, in comparing magnification models over the HFF clusters with each other to determine the probable errors in the individual magnification maps. One prominent conclusion from that study was that differences between the magnification maps were almost always larger than the estimated errors in the magnification for a given map, pointing to large systematics in the construction of some subset of the individual maps. This provides some motivation for the tests we perform here and in future sections in this paper and confirmation of the importance of this study. Other powerful tests of the predictive power of the magnification maps, and the challenges, were provided by observations of SNe Ia (Rodney et al. 2015).

The purpose of this subsection is to illustrate the impact of magnification errors on the derived LFs from the HFF clusters. Two different example LFs are considered for this exercise: (1) one with a faint-end slope of −2 and a turn-over at −15 and (2) another with a fixed faint-end slope α of −2 and no turn-over.

How well can we recover these LFs given uncertainties in the magnification maps? We can evaluate this by generating a mock catalog of sources for each of the first four clusters from the HFF program using one set of magnification models ("input" models) and then attempting to recover the LF using another set of magnification models ("recovery" models). These catalogs include positions and magnitudes for all the individual sources in each cluster. In computing the impact of lensing, the redshifts are fixed to z = 6 for all sources. The input magnification models are taken to be the GLAFIC models for this exercise. Following previous work (e.g., Ishigaki et al. 2015; Oesch et al. 2015), each galaxy in the image plane is treated as coming from an independent volume of the universe, allowing us to construct the input catalogs from the model magnification maps alone (and therefore not requiring use of the deflection maps). This choice does not bias the LF results in our analysis relative to analyses that account for multiple imaging of the same galaxies (from the source plane), since both the cosmological volume and total number of background galaxies is increased in proportion to the overcounting. The selection efficiencies of sources are accounted for when creating the mock catalogs, as estimated in Appendix B. In performing this exercise, we ignore errors in our estimates of the selection efficiencies and small number statistics at the faint end of the LF.

One can try to recover the input LFs from these mock catalogs, using various magnification models. Sources are binned according to luminosity using the "recovery" magnification model. The selection volumes available in each luminosity bin are also estimated as described in Appendix B using this "recovery" magnification model. To demonstrate the overall self-consistency in our approach, we show the recovered LFs using the same magnification model as we used to construct the input catalogs in the top two panels in Figure 4.

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What is the impact if different lensing models are used to recover the LF than those used to construct the mock catalogs? The lowest two rows of panels in Figure 4 show the results using the latest magnification models by Grale, CATS, Zitrin-LTM, and the median of the CATS, Sharon/Johnson, and Zitrin-NFW models where available.

These simulation results demonstrate that the recovery process appears to work very well for input LFs with faint-end slopes of −2 (left panels in Figure 4) independent of the magnification model, with all recovered LFs showing a form that is very similar to that of the input LFs.

Very different results are, however, obtained in our attempts to recover input LFs with a turn-over at −15 mag (right panels in Figure 4) using magnification models that are different from the input model. For all four magnification models we consider, the recovered LFs look very similar to the LF example we just considered. All recovered LFs show a steep faint-end slope to −11 mag. What is striking is that they do not reproduce the turn-over present in the input model at −15 mag. There are some differences in the recovered LFs depending on how similar the input magnification model is to the recovery model, with effective faint-end slopes of −2, −1.8, and −1.7 achieved with the Grale and Zitrin-LTM models, the CATS models, and the median parametric models, respectively. The GLAFIC magnification model is not used when constructing the median parametric magnification model.

Both examples demonstrate that the faint-end slope for the recovered LFs tends to gravitate toward a value of −2. It is useful to provide a brief explanation as to why. For a power-law LF, i.e., ${L}^{\alpha }$, and ignoring any dependence of the selection efficiency on magnification factor, one expects the surface density of sources on the sky to depend on magnification factor μ as ${L}^{\alpha }/\mu \,{dL}{| }_{L={L}_{\mathrm{obs}}/\mu }\propto {\mu }^{-\alpha -2}\propto {\mu }^{-(2+\alpha )}$, where L and L_obs represent the intrinsic and observed luminosities, respectively. For faint-end slopes shallower than −2, one therefore expects a systematic decrease in the surface density of sources on the sky as the magnification increases; for faint-end slopes of −2, one expects no dependence on source magnification; and for faint-end slopes steeper than −2, one expects a systematic increase in the surface density of sources as the magnification increases. All of the above statements are for the intrinsic surface densities. The observed surface densities will be impacted by the magnification-dependent selection efficiencies $S(\mu )$.

We illustrate this expected dependence on the magnification factor in Figure 5 for an LF with a faint-end slope of −1.35 by laying down sources behind the HFF clusters using the glafic magnification model. The surface density of the sources versus magnification factor can then be recovered using a variety of other models. At sufficiently high magnification factors, the uncertainties in the magnification factors become large, washing out any dependence on the magnification factor. This results in a relatively constant surface density of sources and a faint-end slope of −2.

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Two other examples of the impact of large magnification errors on LF results are presented in Figures 16 and 17 in Appendices C and D, utilizing an input LF with a faint-end slope of −1.3. For each of these examples, the recovered LF closely matches the input LF; dramatically, however, faintward of −15 mag (and even −16 mag for some models), the recovered LFs steepen and asymptote again toward a faint-end slope of approximately −2 (or steeper if sources are resolved), even if the actual slope of the LF is much shallower (or the LF turns over!).

Each of these examples demonstrate that, regardless of the input LF, a faint-end slope α of approximately −2 will be recovered whenever the magnification uncertainties are large. One cannot, therefore, use the consistent recovery of a steep faint-end slope based on a large suite of lensing models to argue that the actual LF maintains a steep form to extremely low luminosities (as was done by L17 using their Figure 12). The presented examples show that this is not a valid argument.

How then can one interpret LF results from lensing clusters when a steep LF $\alpha \sim -2$ is found? As we have demonstrated, such a result could be indicative of the LFs truly being steep or simply an artifact of large magnification uncertainties. To determine which is the case, the safest course of action is to simulate all steps in the LF recovery process, to determine the impact of magnification uncertainties on the shape of the LF, and finally to interpret the recovered LFs from the observations. While we showed a few examples here, we formalize the process in the next section.

The LF recovery results presented in Section 3 (Figure 4) illustrate the impact that errors in the magnification maps can have on the recovered LFs. Input LFs, of very different forms, can be driven toward a faint-end slope α of −2 at the faint end, after accounting for the impact of magnification errors. The results from Section 3.2 demonstrate the importance of forward modeling the entire LF recovery process to ensure that both the results and uncertainties are reliable.

We then utilize our forward-modeling approach to derive constraints on the $z\sim 6$ LF. The basic idea behind our approach follows closely from the simulations we ran in the previous section and is illustrated in Figure 6. We begin by treating one of the public magnification models as providing an exact representation of reality. In conjunction with an input LF, those models are used to create a mock data set for the four HFF clusters considered. The mock data set is then interpreted using other magnification models for the clusters to determine the distribution of sources versus UV luminosity M_UV and also to recover the UV LF. As illustrated by the LF recovery experiments presented in Section 3.2, the recovery could be done with the models individually or by using some combination of models like the median.

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There are many advantages to using the present procedure to derive accurate errors on the overall shape of the UV LF. Probably the most significant of these is inherent in the end-to-end nature of the present procedure and our relying significantly on forward modeling to arrive at accurate errors on the observational results. Through the construction of many mock data sets using plausible magnification models and recovery using other similarly plausible models, the proposed procedure allows us to determine the full range of allowed LFs.

In addition, the advocated procedure provides us with a natural means to account for "noise" in the lensing model magnification maps. The presence of lensing model "noise" is obvious looking at the range in magnification factors across the various models (e.g., compare the GLAFIC and GRALE critical lines in Figure 6 or see the lower panel in Figure 3). Such noise can even be present in a median magnification model created from the combination of many individual models because there will be regions where the high-magnification regions of the lensing maps simply overlap due to chance coincidence. Analogous to considerations of low-significance sources in imaging observations, the robustness of specific magnification factors in the median map can be assessed, by considering comparisons with independent determinations of the same map. Through the treatment of one of the public magnification models as the truth, the present forward-modeling approach effectively formalizes such a technique to determine the robustness of specific features in the magnification maps. The advantage of the current procedure is that this robustness can be determined using the full magnification maps available for each cluster (and not just at a limited number of positions where candidate high-magnification $\mu \gt 10$ sources happen to be found in the real observations), while also allowing us to derive more reliable likelihood distributions (with realistic errors).

Another advantage of our forward modeling procedure is that it explicitly incorporates source selection. This is important, since the selection efficiency S could depend on the magnification factor μ in the sense that the most magnified sources would also be the most incomplete, as is likely the case for the brightest and most extended objects, due to the impact of lensing shear on source detection (Oesch et al. 2015). If the same situation applied to the faintest sources in the HFFs (and one did not utilize a procedure that included forward modeling or an explicit correction), the recovered LFs would be biased. This is due to the fact that the actual surface density of the sources on the sky is proportional to $S({\mu }_{\mathrm{true}})$, but it is assumed to be proportional to $S({\mu }_{\mathrm{model}})$ and ${\mu }_{\mathrm{true}}\lt {\mu }_{\mathrm{model}}$ at high magnifications $\mu \gt 10$ (Figure 3). The recovered LF would therefore be higher by the factor $S({\mu }_{\mathrm{true}})/S({\mu }_{\mathrm{model}})$.

We would expect such an issue to affect recovered LFs, in all cases where sources have non-zero size (since the selection efficiency would then depend on the magnification factor). For example, if the completeness of sources shows an inverse correlation with the magnification factor as Oesch et al. (2015) find, i.e., $S(\mu )\propto {\mu }^{-0.3}$ (e.g., in their Figure 3), a direct approach would recover a faint-end slope that is ${\rm{\Delta }}\alpha \sim 0.3$ steeper than in reality (Figure 16 from Appendix C), at very low luminosities, where $\mu \gt 10$ (where 〈${\mu }_{\mathrm{true}}$〉 is typically less than 〈${\mu }_{\mathrm{model}}$〉).

Remarkably, there is no evidence that this issue is even considered in many recently derived LFs, which is worrisome given the size assumptions that were made. This is most problematic for analyses pushing to very low luminosities, i.e., $\gt -15\,\mathrm{mag}$, while quoting tiny statistical uncertainties (e.g., L17 who quote statistical uncertainties on α of ±0.04 versus this bias, which is ∼8× larger).

We perform our forward modeling simulations at the catalog level, to ensure that the time requirements on these simulations are manageable. This involves the construction of catalogs of sources with precise positions and apparent magnitudes. Both in the construction of the mock catalogs and in recovering the LF from these catalogs, the selection efficiency S must be accounted for, which is, in general, a function of the apparent magnitude m and magnification factor μ, i.e., $S(m,\mu )$. For the lowest luminosity z ∼ 5–8 galaxies, there is little evidence to suggest that these sources show significant spatial extension (Bouwens et al. 2017), which implies that we can credibly treat their selection efficiencies as just a function of the apparent magnitudes, i.e., S(m). We describe our procedure for estimating S(m) in this case in Appendix B.¹¹

In putting together the mock observed catalogs for each LF parameter set we are considering, i.e., ${\phi }^{* }$, α, and a third parameter δ to be introduced in the next section, ∼2 × 10⁵ sources are inserted at random positions (yet uniformly in the source plane) across the four HFF cluster fields we are considering (Figure 6). Sources are included in the catalogs in proportion to their estimated selection efficiencies $S(m,\mu )$ (Appendix B), their implied volume densities (according to the model LF), and cosmological volume element (proportional to the area divided by the magnification factor). It is the inclusion of sources in the catalogs in inverse proportion to the magnification factor that ensures that galaxies are distributed uniformly within the source plane (since our catalog construction process does not consider the deflection maps from the lensing models or multiple imaging).¹² All sources in the input catalogs are assumed to have the same input redshift z = 6.

During the recovery process, sources are placed into individual bins in UV luminosity using a "recovery" magnification model, which we take to be the median of the parametric magnification models.¹³ During the recovery process, the redshift is taken to have the same mean value as assumed in constructing the mock catalogs, i.e., z = 6.0, but with a $1\sigma$ uncertainty of 0.3. This is to account for the impact of uncertainties in the estimated redshifts of individual sources on the recovered LFs.

We use the results of the simulations we run for each parameter set (each with ∼2 × 10⁵ sources) to establish the expectation values for the number of sources per luminosity bin (in the same 0.5 mag intervals used in the previous section). We then compute the likelihood of a given parameterization of the LF by comparing the observed number of galaxies per bin in luminosity (considering all four clusters at the same time) with the expected numbers assuming a Poissonian distribution, as

$$\begin{eqnarray*}&&{{\rm{\Pi }}}_{i}{e}^{-{N}_{\exp ,i}}\displaystyle \frac{{({N}_{\exp ,i})}^{{N}_{\mathrm{obs},i}}}{({N}_{\mathrm{obs},i})!}\end{eqnarray*}$$

where ${N}_{\exp ,i}$ is the expected number of sources in a given bin in intrinsic UV luminosity. The observed number of sources in a given bin in UV luminosity ${N}_{\mathrm{obs},i}$ is computed from the median parametric magnification maps, as done in Section 2 (Figure 2). A flow chart showing one example of our forward-modeling approach is provided in Figure 6.

Use of a parametric form to the LF is particularly useful for examining the overall LF constraints on the shape of the UV LF in lensed fields, due to uncertainties that exist on the magnification factors, and hence luminosities, of individual sources used in the construction of the LF. This makes it difficult to place sources in specific bins of the UV LF (resulting in each bin showing a larger error).

For the parametric modeling we do of the LF, we start with a general Schechter form for the LF:

$$\begin{eqnarray*}&&{\phi }^{* }(\mathrm{ln}(10)/2.5){10}^{-0.4(M-{M}^{* })(\alpha +1)}{e}^{-{10}^{-0.4(M-{M}^{* })}}.\end{eqnarray*}$$

However, we modify the basic form of the Schechter function by multiplying the general Schechter form by the following expression faintward of −16 mag:

$$\begin{eqnarray*}&&{10}^{-0.4\delta {(M+16)}^{2}}.\end{eqnarray*}$$

Positive values of δ result in the LF turning over faintward of −16 mag, while negative values of δ result in the LF becoming steeper faintward of −16 mag. An illustration of this parameterization is provided in Figure 7.

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Our use of −16 mag allows us to test for possible curvature in the shape of the LF at $\gt -16\,\mathrm{mag}$, as predicted by some models (e.g., Jaacks et al. 2013; Kuhlen et al. 2013; Ocvirk et al. 2016). −16 mag is also just faintward of luminosities probed in field studies (i.e., −16.77 mag: Bouwens et al. 2015a). Finally, since $\gt -16$ mag corresponds to the luminosity of $\mu \gt 10$ faint sources in the HFFs, our fitting for a curvature parameter δ allows us to investigate how well the shape of the LF can be recovered in the regime where the magnification factors are large.

With this parameterization, the turn-over luminosity M_T (i.e., where ${(d\phi /{dM})}_{M={M}_{T}}=0$) can be easily shown to be

$$\begin{eqnarray}&&{M}_{T}=-16-\displaystyle \frac{\alpha +1}{2\delta }\end{eqnarray} \tag{ 2 }$$

assuming that $\delta \gt 0$. For sufficiently small values for δ, i.e., $\delta \sim 0.05$, a turn-over in the LF would be so faint as to be impractical to confirm, and below what would be expected theoretically (Section 6.3). For typical models (Section 6.3), δ is expected to be $\delta \gtrsim 0.08$, resulting in turnover magnitudes $\lt -$ 10.

We begin by looking at the constraints we can set on the shape of the $z\sim 6$ LF by restricting our analysis to $z\sim 6$ samples found behind the first four HFF clusters. Such an exercise is useful since it allows us to examine the LF constraints we obtain from our HFF search results, entirely independent of results in the field.

In deriving best-fit LF results, we use the following approach. We fix M* to the same value Bouwens (2016) found at $z\sim 6$, i.e., −20.94 mag, set the curvature δ to be zero, and then fit for ${\phi }^{* }$ and α. We use a Markov Chain Monte Carlo (MCMC) algorithm where we start with the field LF results from Bouwens et al. (2015a), i.e., ${\phi }^{* }=0.0005$ Mpc⁻³ and $\alpha =-1.87$, to determine how the likelihood of various parameter combinations varies as a function of ${\phi }^{* }$ and α. For each set of parameters α and ${\phi }^{* }$, we repeat the simulations described in Section 4.2 to calculate the likelihood of those parameters. For those simulation, we consistently use magnification maps from one team to create the mock observations and then recover the results using the median magnification maps formed from the parametric models. On the basis of the grid of likelihoods we derive, we determine the most likely values for ${\phi }^{* }$ and α, while also determining the covariance matrix that best fits the same likelihood grid. From the covariance matrix, we estimate errors on ${\phi }^{* }$ and α.

To determine the impact that errors in the magnification maps can have on the derived values for ${\phi }^{* }$ and α, we repeat the exercise from the above paragraph seven times. In each case, we treat the magnification maps from a different team as the truth and proceed to derive constraints on ${\phi }^{* }$ and α using the results from the first four HFF clusters. Because two of the HFF clusters we examine do not have Zitrin-NFW magnification maps available, i.e., MACS 0717 and MACS 1149, we make use of the Zitrin-LTM-Gauss models instead. The results are presented in Table 2.

Table 2.  Best-fit Constraints on the z ∼ 6 UV LF

Input		ϕ* (10−3
Model	${M}_{\mathrm{UV}}^{* }$ a	Mpc−3)	αb	δc	MTd
	HFF Observations Alone + M* from field LF results (Section 5.1)
GLAFIC	−20.94	0.69 ± 0.04	−1.90 ± 0.03	0.00	⋯
CATS	−20.94	0.66 ± 0.04	−1.91 ± 0.02	0.00	⋯
Grale	−20.94	0.68 ± 0.06	−1.98 ± 0.03	0.00	⋯
Bradac	−20.94	0.70 ± 0.05	−1.89 ± 0.03	0.00	⋯
Sharon/Johnson	−20.94	0.68 ± 0.04	−1.91 ± 0.02	0.00	⋯
Zitrin-NFW	−20.94	0.58 ± 0.01	−1.92 ± 0.03	0.00	⋯
Zitrin-LTM	−20.94	0.67 ± 0.05	−1.95 ± 0.02	0.00	⋯
Mean	−20.94	0.66 ± 0.06	−1.92 ± 0.04	0.00	⋯
Mean Parametric	−20.94	0.65 ± 0.06	−1.91 ± 0.03	0.00	⋯
Fiducial (Section 5.2): HFF Observations + CANDELS/HUDF/HUDF-Parallel (Bouwens et al. 2015)
GLAFIC	−20.94	0.57 ± 0.05	−1.92 ± 0.04	0.07 ± 0.16	>−14.2
CATS	−20.94	0.58 ± 0.05	−1.91 ± 0.04	0.17 ± 0.20	>−14.9
Grale	−20.94	0.63 ± 0.07	−1.95 ± 0.03	0.16 ± 0.30	>−15.2
Bradac	−20.94	0.57 ± 0.05	−1.92 ± 0.04	0.21 ± 0.32	>−15.3
Sharon/Johnson	−20.94	0.57 ± 0.05	−1.92 ± 0.03	0.12 ± 0.21	>−14.9
Zitrin-NFW	−20.94	0.56 ± 0.06	−1.91 ± 0.03	0.07 ± 0.20	>−14.6
Zitrin-LTM	−20.94	0.58 ± 0.05	−1.93 ± 0.03	0.14 ± 0.25	>−15.1
Mean	−20.94	0.58 ± 0.06	−1.92 ± 0.04	0.14 ± 0.24	⋯
Mean Parametric	−20.94	0.57 ± 0.05	−1.92 ± 0.04	0.11 ± 0.20	⋯
Idem (but Estimating Completeness from Conventional Size–Luminosity Relations: Section 5.4)
GLAFIC	−20.94	0.54 ± 0.06	−1.93 ± 0.04	−0.08 ± 0.18	⋯
CATS	−20.94	0.54 ± 0.06	−1.93 ± 0.04	−0.08 ± 0.22	⋯
Grale	−20.94	0.58 ± 0.06	−1.94 ± 0.03	−0.27 ± 0.27	⋯
Bradac	−20.94	0.52 ± 0.06	−1.91 ± 0.04	−0.25 ± 0.28	⋯
Sharon/Johnson	−20.94	0.54 ± 0.06	−1.93 ± 0.04	−0.03 ± 0.22	⋯
Zitrin-NFW	−20.94	0.53 ± 0.06	−1.91 ± 0.04	−0.21 ± 0.23	⋯
Zitrin-LTM	−20.94	0.55 ± 0.05	−1.93 ± 0.04	−0.25 ± 0.27	⋯
Mean	−20.94	0.54 ± 0.06	−1.93 ± 0.04	−0.17 ± 0.26	⋯
Mean Parametric	−20.94	0.54 ± 0.06	−1.92 ± 0.04	−0.10 ± 0.22	⋯
Literature Including HFF Observations
Atek et al. (2015b)	−20.9 ± 0.7	${0.28}_{-0.18}^{+0.59}$	$-{2.04}_{-0.13}^{+0.17}$ f
Livermore et al. (2017)	$-{20.82}_{-0.05}^{+0.04}$	${0.23}_{-0.02}^{+0.02}$	$-{2.10}_{-0.03}^{+0.08}$	⋯	$\gt -{11.1}_{-0.8}^{+0.4}$ e
Literature Before HFF Observations
Bouwens et al. (2015a)	−20.94 ± 0.20	${0.50}_{-0.16}^{+0.22}$	−1.87 ± 0.10
Finkelstein et al. (2015)	$-{21.13}_{-0.31}^{+0.25}$	${0.19}_{-0.08}^{+0.09}$	−2.02 ± 0.10

Notes.

^aFixed ^bThe faint-end slopes we derive are moderately dependent on the assumptions we make about the intrinsic size distribution of very low luminosity galaxies. Nevertheless, motivated by the results from a companion paper (Bouwens et al. 2017), where extremely faint z ∼ 5–8 galaxies were found to have a size distribution consistent with point sources, we used this assumption in deriving results for the faint end of the UV LF. However, since obtaining direct constraints on the size distribution and hence completeness of extremely faint galaxies over the HFF clusters is difficult, we could underestimate the volume density of faint sources. This could result in faint end slopes that are steeper by Δα ∼ 0.08 (if we adopt 20% larger sizes than the Shibuya et al. (2015) size–luminosity relation instead of assuming galaxies to be point sources). ^cBest-fit curvature in the shape of the UV LF faintward of −16 mag (see Figure 7). For HFF-only determinations, the curvature is fixed to 0 for simplicity. ^dBrightest luminosity at which the current constraints from the HFF permit a turn-over in the z ∼ 6 LF (within the 68% confidence intervals). ^eThis is the luminosity where according to Figure 12 of L17, L17 find ${\rm{\Delta }}({\rm{BIC}})=2$, where BIC denotes the Bayesian Information Criteria (analogous to ${\rm{\Delta }}{\chi }^{2}$ for their usage). Strictly speaking, it is closer to an 84% confidence limit than a 68% confidence limit. ^fLF constraints obtained for a combined z ∼ 6–7 sample.

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The faint-end slope α results we obtain from the HFF clusters alone inhabit the range from −1.89 to −1.98 depending on which magnification model we treat as reality. The faint-end slope we estimate averaging the results from all of the models is −1.92 ± 0.04, while the faint-end slope α we find using the parametric models is −1.91 ± 0.03. The quoted uncertainty includes median statistical error added in quadrature with the standard deviation among the faint-end slope determinations for the different magnification models.

These results are interesting in that they are consistent with our own results over the field, i.e., Bouwens et al. (2015a), where $\alpha =-1.87\pm 0.10$, as well as other estimates in the literature (Yan & Windhorst 2004; Bouwens et al. 2007; Calvi et al. 2013; Bowler et al. 2015; Finkelstein et al. 2015) which generally lie in the range from approximately −1.8 to approximately −2.0.

It is worthwhile to emphasize the value of the test we perform in the previous paragraph. Because we consider the use of searches behind lensing clusters for constraining the faint end of the UV LFs, it is essential that we derive constraints from the lensing clusters in isolation of those obtained from field searches to verify that no major systematics appear to be present in the LF results from the lensing clusters. This is relevant, since recent determinations of the faint-end slope α to the LFs from field and cluster search results seem to show a substantial discrepancy (Figure 1).

We now proceed to derive constraints on the overall form of the UV LF at $z\sim 6$ combining constraints from the field with those available from the HFF clusters. For simplicity, we keep the characteristic luminosity M* fixed to the value −20.94 mag that we found in our earlier wide-area field study (Bouwens et al. 2015a), given the lack of substantial information in the HFF cluster program for constraining this parameter due to the small volume probed.

In combining constraints from the field and from the HFF clusters, we need to allow for some error in the normalization of both the field and HFF cluster results, as a result of large-scale structure variations ("cosmic variance": Robertson et al. 2014; see also Somerville et al. 2004 and Trenti & Stiavelli 2008) and also small systematic errors in the estimates of the volume densities of galaxies in each of our probes. We assume an uncertainty of ∼20% in the volume density in both the field and HFF LF results.

The 20% uncertainty we assume to be present in the normalization of the LF results at both the bright and faint ends includes an ∼10% uncertainty in our estimates of the selection volume and ∼10% systematic error due to uncertainties in the total magnitude measurements (reflecting a ∼0.1 mag systematic error in the magnitude measurements). The inclusion of such an error is relevant given the existence of real errors in the estimated volume densities of galaxies using photometric criteria. Small differences appear to be guaranteed, given that the filters available for the selection of galaxies from the field are different (in particular, including a deep "z"-band filter) from those available over the HFF clusters (which do not include a deep "z" band filter). Important factors contributing to these uncertainties are (1) likely differences between the assumed sizes and SEDs of galaxies versus redshift in the observations versus those in the simulations and (2) uncertainties in the contamination rate of observed samples.¹⁴

Similar to our LF results using the HFF observations alone, we derive confidence regions on the $z\sim 6$ LF results using an MCMC-type procedure where we explore a limited region in the ${\phi }^{* }$-α-δ parameter space and calculate the likelihood of each point in parameter space using the forward modeling simulations we describe in Section 4.1. Our calculated likelihoods explicitly include a marginalization across the 20% volume density uncertainties we assume. These likelihoods are then multiplied by the likelihoods on the same Schechter parameters derived by Bouwens et al. (2015a) at $z\sim 6$ using results from the full CANDELS program, the HUDF, and the HUDF parallels (again after marginalizing the Bouwens et al. (2015a) over ${\phi }^{* }$ to account for a 20% uncertainty in the volume density of sources). Finally, this 3D likelihood grid is fit to derive the most likely values for ${\phi }^{* }$, α, and δ and also the covariance matrix.

As in our determinations of the LF parameters from the HFF programs alone, we repeat this exercise seven different times, treating each of the magnification models from different teams as reality and recovering the LF results using the median magnification map. We present our constraints on each of the LF parameters in Table 2. The faint-end slope α we estimate averaging all of our models and just the parametric models are −1.92 ± 0.04 and −1.92 ± 0.04, respectively.

As in our determinations using only the HFF observations themselves, the faint-end slopes α we derive at $z\sim 6$ are fairly consistent with LF results in the field. Our obtaining consistent results for all seven of the magnification model families we consider is not especially surprising, given the fact that individual sources would be expected to scatter in almost the same direction as the dominant slope of the LF, i.e., approximately −2 (the expected slope of the LF from scatter) versus $\sim -1.9$ (the actual slope of the LF).

Despite general agreement on the most likely value for α at $z\sim 6$, we find a broad range of values for the curvature parameter δ, from 0.07 to 0.21, with $1\sigma$ uncertainties ranging from 0.16 to 0.32. None of the magnification models we considered point toward our having even modest evidence, i.e., $\delta \gt 0$ at $\gt 2\sigma$, for the $z\sim 6$ LF showing a turn-over at the faint end.

We find the smallest uncertainties on δ assuming that the GLAFIC magnification models represent reality. Slightly larger uncertainties on δ are found assuming that the CATS, Sharon/Johnson, and Zitrin-NFW models represent reality, while the largest uncertainties on δ are found, assuming that the Zitrin-LTM, Bradac, and Grale models represent reality. The size of the uncertainty on δ is a function of how similar the various magnification models are to the median magnification model formed from the parametric models. Given the similarity of assumptions utilized in the GLAFIC, CATS, Sharon/Johnson, and Zitrin-NFW models, it is not surprising that their magnification maps agree best with median magnification maps constructed using similar assumptions.

The results in Table 2 for the different magnification models indicate the general range of constraints one could obtain on the form of the $z\sim 6$ LF: the results for the non-parametric models indicate the errorbars on the LFs if the mass distribution in clusters do not strictly follow the assumptions made in the parametric models, while results for the parametric models indicate the likely error bars, if the mass profiles in the HFF clusters generally do adhere to those assumptions.

Uncertainties in the redshifts of the lensed $z\sim 6$ galaxies also contribute to the error in δ. To estimate the impact, we kept the redshifts of lensed background souces fixed while rerunning the forward-modeling simulations from our MCMC chain. Comparing the uncertainties we derive in δ to the uncertainties we derive including errors in the redshift, we find a typical increase of 0.01 in the uncertainty on δ, i.e., from 0.15 to 0.16 in the case of the GLAFIC simulations. If we assume that uncertainties in the deflection maps and redshifts both add in quadrature, this suggests that errors in the photometric redshift errors contribute ∼12% of the fractional error in δ, i.e., (0.16² − 0.15²)/0.16²$\,\sim 0.12$.

To help visualize what our present LF results mean, we have made use of our parametrized constraints to derive 68% and 95% confidence intervals on the volume density of galaxies as a function of the UV luminosity M_UV. These results are presented both in Figure 8 and also in Table 3.

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Table 3.  68% and 95% Confidence Intervals on the UV LF at z ∼ 6 (Section 5.2) Adopting the Functional Form in Section 4.3

	ϕ(M) $[{\mathrm{log}}_{10}$(#/Mpc3/mag])
	Lower Bound		Upper Bound
${M}_{\mathrm{UV},{AB}}$	95%	68%	68%c	95%c
Case 1 (GLAFIC)a
−16.75	−1.88	−1.82	−1.69	−1.63
−16.25	−1.71	−1.64	−1.50	−1.42
−15.75	−1.54	−1.47	−1.30	−1.23
−15.25	−1.38	−1.30	−1.13	−1.05
−14.75	−1.29	−1.18	−0.95	−0.84
−14.25	−1.31	−1.12	−0.73	−0.54
−13.75	−1.43	−1.13	−0.49	−0.18
−13.25	−1.65	−1.18	−0.22	0.25
−12.75	−1.95	−1.29	0.07	0.73
−12.25	−2.34	−1.45	0.37	1.26
−11.75	−2.82	−1.66	0.70	1.84
−11.25	−3.37	−1.93	1.04	2.48
−10.75	−4.01	−2.24	1.40	3.17
−10.25	−4.74	−2.60	1.79	3.91
Case 2 (Bradac)b
−16.75	−1.85	−1.79	−1.66	−1.60
−16.25	−1.67	−1.60	−1.46	−1.39
−15.75	−1.50	−1.42	−1.27	−1.20
−15.25	−1.34	−1.27	−1.11	−1.04
−14.75	−1.36	−1.21	−0.90	−0.75
−14.25	−1.57	−1.27	−0.65	−0.35
−13.75	−1.93	−1.42	−0.36	0.16
−13.25	−2.43	−1.66	−0.04	0.75
−12.75	−3.08	−1.99	0.31	1.42
−12.25	−3.88	−2.40	0.68	2.18
−11.75	−4.81	−2.90	1.09	3.02
−11.25	−5.88	−3.48	1.52	3.95
−10.75	−7.10	−4.17	1.98	4.95
−10.25	−8.46	−4.93	2.47	6.04

Notes.

^aFor case 1, we assume that the differences between the magnifications in the median parametric model and the GLAFIC model are a good representation of the typical errors in the magnification models we utilize. ^bFor case 2, we assume that differences between the magnifications in the median parametric model and the Bradac model are a good representation of the typical errors in the magnification models we utilize. Similar confidence regions are obtained if one uses the Grale model instead of the Bradac model. ^cIf 50% of faint sources at z ∼ 6–8 are significantly spatially extended (intrinsic half-light radii >30 mas), the 68%-likelihood upper bounds on the implied LF constraints could increase by ∼0.3 dex (Bouwens 2017). The actual upper bound on the volume density could be much higher if the completeness is substantially less than 50%.

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One question that has recently been of significant interest in the literature regards whether there is a flattening or turn-over in the UV LF at the faint end. This question is important because the answer could indicate to us whether there is a sufficient number of extremely faint galaxies to produce the photons necessary to drive the reionization of the universe.

Fortunately, using the likelihood contours for δ-α-${\phi }^{* }$ and Equation (2), we can directly determine the brightest point in the LF where a turn-over is permitted (with the 68% confidence intervals). The results do depend somewhat on which magnification model we assume to be representative of reality (and therefore which of the panels we consider from Figure 8). Nevertheless, we find that the HFF observations allow for a turn-over in the LF as bright as −14.2 mag to −15.3 mag (within the 68% confidence intervals). The allowed turn-over luminosities we estimate assuming different magnification models are presented in Table 2 and also in Figure 9.

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In a companion paper (Bouwens et al. 2017), we showed that the slope of the LF at low luminosities is strongly dependent on the size of very faint galaxies (see Figure 2 from that paper). We constrained the sizes of faint galaxies ($\gt -16\,\mathrm{mag}$) by taking advantage of the large samples of such galaxies available behind the first four HFF clusters at z ∼ 2–3 and z ∼ 5–8. We found no evidence to indicate that these galaxies were significantly resolved, looking at (1) the prevalence of high magnification sources as a function of the predicted shear and (2) their stacked profile along the expected shear axis. The slope varied dramatically from $\alpha \sim -1.9$ for the small sizes we found (taking 7.5 mas to be representative of the half-light radius) to $\alpha \sim -2.65$ (taking 120 mas as representative). The resulting luminosity density from integrating the LF to −14 changed by a factor of 40. Clearly the size assumed for faint galaxies is a key parameter that is central to a reliable determination of the LF.

On the basis of our findings in a companion study, can we assume that $\gt -16\,\mathrm{mag}$ galaxies are all entirely unresolved? Unfortunately, we cannot due to the impact of surface brightness selection effects on the recovered samples. The impact is sufficient that we cannot rule out (87% confidence) sources having half-light radii of 30 mas, which is approximately the size we would predict for faint sources extrapolating conventional size–luminosity relations.

Given this fact, it is therefore quite logical to repeat the present determination of the LF, but this time assuming a conventional size–luminosity relation. We will then treat the results as providing an upper bound on the $z\sim 6$ LF results, given uncertainties in the size distribution. To this end, we suppose that the median half-light radii of galaxies follow the following correlation with luminosity ${r}_{{hl}}={(0\buildrel{\prime\prime}\over{.} 14)(L/{L}_{z=3}^{* })}^{0.27}$, which is in good agreement with the size–luminosity relation found by Shibuya et al. (2015). In addition, we adopt the sizes of galaxies that exhibit a log-normal distribution with 0.3 dex $1\sigma$ scatter. We assume galaxies to have a Sérsic profile and for the Sérsic indices to have a log-normal distribution with a median of 1.5 and scatter of 0.3 dex. The axial ratio is also assumed to be log-normal with a median value of 1.8 and a scatter of 0.3 dex. A random position angle is adopted for sources. Finally, the pixel-by-pixel profiles for all sources in the Monte-Carlo catalogs are computed. The impact of gravitational lensing is included using the latest deflection maps from the CATS team.

The simulated galaxies are then added to the real observations, and we utilize the same procedure for catalog creation and source selection as we use on the real data. We then rederive the selection volumes in the same way as before (i.e., see Appendix B) and repeat the determination of the LF results using the outlined forward-modeling procedure. Compared to the situation where point-source sizes are assumed, the selection volumes we derive are lower, increasing the overall volume density of sources inferred at lower luminosities $\gt -18\,\mathrm{mag}$.

The approximate impact of this use of larger sizes for faint sources is to increase the volume density of sources at approximately −17, −15, and −14 by factors of ∼1.6, ∼2, and ∼3.3, respectively. The amplitude of the correction increases toward lower luminosities due to the correlation between surface brightness and luminosity implied by conventional size–luminosity relations (where $R\propto {L}^{0.27}$: e.g., Shibuya et al. 2015), i.e., $L/{R}^{2}\propto L/{({L}^{0.27})}^{2}\propto {L}^{0.46}$.¹⁵

Based on this scaling, one would expect 0.01 L* and 0.001 L* galaxies to have ∼8× and ∼24× lower surface brightnesses, respectively, than more luminous L* galaxies. Since gravitational lensing preserves surface brightness, it should not be easy to select extremely low luminosity galaxies behind lensing clusters, if conventional relations held. We should emphasize, however, that it is not clear, however, that conventional relations hold down to such low luminosities, i.e., ${M}_{\mathrm{UV}}\gt -16$. The light profile in galaxies may be dominated by just a single super star cluster or two in this regime, as suggested by our results in Bouwens et al. (2017).

The derived LF results we derive for the stated size assumptions are presented in Table 2. The results are similar to our fiducial results. However, they do nevertheless give faint-end slopes α that are ${\rm{\Delta }}\alpha \sim 0.01$ steeper and curvature parameters δ that are approximately 0.2 lower. With the present size assumptions, δ inferred for the $z\sim 6$ LF is formally negative for all of the magnification models we consider. As in Section 5.3, we emphasize that a possible upturn in the LF (i.e., $\delta \lt 0$) is not a statistically robust result. If we force δ to be 0, the faint-end slope we derive is ${\rm{\Delta }}\alpha \sim 0.03$ steeper for the typical lensing model. If we allow for such a change in α, the present tension in the faint-end slope α versus L17 would decrease to $3\sigma$.

The exercise in this section demonstrates the sensitivity of the curvature parameter in the LF δ—and in fact the whole question as to where or if the UV LF turns over—to the form of the size–luminosity relation. We caution that the conclusions here are based on an extrapolation of sizes seen at significantly higher luminosities and that the indications from our recent work on sizes in the HFF clusters (Bouwens et al. 2017) suggest that galaxies at luminosities $\gt -16$ may be very small. In such a case, the completeness corrections will be much smaller.

To determine if galaxies produce enough ionizing photons to drive the reionization of the universe, we require constraints on the luminosity density in the rest-frame UV that include the contribution from all galaxies.

We can compute confidence intervals on the luminosity densities in the rest-frame UV by using the constraints from the analyses performed in the previous section and then marginalizing over ${\phi }^{* }$, α, and δ. We compute the results to a number of different limiting luminosities M_UV, i.e., −17, −15, −13, −10, and −3 mag,¹⁶ the first four of which commonly appear in the literature, in considering whether galaxies might drive cosmic reionization, particularly including the contribution at very low luminosities. We have presented the results we obtain in Figure 10 and also in Table 4 to these faint-end limits. Our results imply a luminosity density of ${10}^{26.38\pm 0.05}$ erg s⁻¹ Hz⁻¹ Mpc⁻³ to −15 mag.

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Table 4.  68% and 95% Confidence Intervals on the UV Luminosity Density at z ∼ 6 to Various Limiting Luminosities (Section 5.5)

	${\mathrm{log}}_{10}{\rho }_{\mathrm{UV}}$ (UV Luminosity Density)
	(erg s−1 Hz−1 Mpc−3)
	Lower Bound		Upper Bound
Faint-end Limit	95%	68%	68%	95%
Case 1 (GLAFIC)a
${M}_{\mathrm{UV}}\lt -17$	26.13	26.17	26.25	26.28
${M}_{\mathrm{UV}}\lt -15$	26.28	26.33	26.42	26.47
${M}_{\mathrm{UV}}\lt -13$	26.35	26.40	26.54	26.65
${M}_{\mathrm{UV}}\lt -10$	26.36	26.42	26.93	28.65
${M}_{\mathrm{UV}}\lt -3$ b	26.36	26.42	31.10	41.23
Case 2 (Bradac)a
${M}_{\mathrm{UV}}\lt -17$	26.13	26.18	26.26	26.29
${M}_{\mathrm{UV}}\lt -15$	26.29	26.33	26.43	26.47
${M}_{\mathrm{UV}}\lt -13$	26.33	26.39	26.57	27.00
${M}_{\mathrm{UV}}\lt -10$	26.33	26.39	27.39	31.62
${M}_{\mathrm{UV}}\lt -3$ b	26.33	26.39	34.35	55.95

Notes.

^aSame assumptions as in Table 3. ^bThe −3 mag limit is included here for illustrative value and takes as its inspiration results from O'Shea et al. (2015) and Ocvirk et al. (2016), which predict sources to such faint magnitudes.

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Not surprisingly, our LF results allow for essentially an arbitrarily high contribution from very faint galaxies to the UV luminosity density, particularly including a contribution from galaxies to −3 mag. These results also provide fairly firm $1\sigma$ and $2\sigma$ lower bounds on the luminosity density. We find 1 σ and 2 σ lower limits on luminosity density in the rest-frame UV of $\sim {10}^{26.40}$ erg s⁻¹ Hz⁻¹ Mpc⁻³ and $\sim {10}^{26.35}$ erg s⁻¹ Hz⁻¹ Mpc⁻³, respectively.

The $1\sigma$ and $2\sigma$ lower bounds we find on the luminosity density integrating to arbitrarily faint luminosities is not especially higher than what we find integrating to −15 mag. These results indicate that it is not yet possible to argue for the discovery of a significant additional reservoir of photons from galaxies faintward of −15 mag (Atek et al. 2015a, 2015b), as has been the claim in one recent study (L17).

Finally, to conclude this section, we derive a binned representation of the results from the first four HFF clusters. Binned representations of the LF results from the HFFs should be very reliable at high luminosities, where errors in the magnification maps are smaller. Binned representations retain the advantage that they are much more model independent probes of the LF shape as a function of luminosity.

In our binned representation of the LF, we adopt bins of width 0.5 mag and determine the binned LF results ${\phi }_{m}$ to be as follows.

$$\begin{eqnarray}&&{\phi }_{m}=\displaystyle \frac{{N}_{m}}{{V}_{m}},\end{eqnarray} \tag{ 3 }$$

where N_m is the number of sources in absolute magnitude bin m after correcting for the estimated magnification.

We derive the selection volumes V_m in a given magnitude bin using the following equation.

$$\begin{eqnarray}&&{V}_{m}={\int }_{A}{\int }_{{dz}}C(z,m,\mu )\displaystyle \frac{1}{\mu (A)}\displaystyle \frac{{dV}(z)}{{dA}}{dzdA},\end{eqnarray} \tag{ 4 }$$

where A denotes the area, V denotes the volume, C denotes the estimated completeness, and μ denotes the magnification factor. Our estimates of the completeness are provided in Appendix B; the completeness C does not appear to be a strong function of the magnification factor μ in data sets as deep as the HFFs, if we take the results of Bouwens et al. (2017) to be indicative.

We derived simple stepwise constraints on the UV LF brightward of −16 mag, by dividing the number of sources in each absolute magnitude bin by the computed selection volume. The results are presented in Table 5 and Figure 11. The upper limits we quote on the volume densities of sources include a possible $1.6\times$ underestimate, if the sizes of sources follow the size–luminosity relation presented in Section 5.4.

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Table 5.  Binned Determination of the Rest-frame UV LF at $z\sim 6$ (Section 5.6)^a

${M}_{\mathrm{UV},{AB}}$ b	${\phi }_{k}$ (10−3 Mpc−3 mag−1)
−20.75	${0.0002}_{-0.0002}^{+0.0002}$
−20.25	${0.0009}_{-0.0004}^{+0.0004}$
−19.75	${0.0007}_{-0.0004}^{+0.0004}$
−19.25	${0.0018}_{-0.0006}^{+0.0006}$
−18.75	${0.0036}_{-0.0009}^{+0.0009}$
−18.25	${0.0060}_{-0.0012}^{+0.0012}$
−17.75	${0.0071}_{-0.0014}^{+0.0066}$ c
−17.25	${0.0111}_{-0.0022}^{+0.0102}$ c
−16.75	${0.0170}_{-0.0039}^{+0.0165}$ c
−16.25	${0.0142}_{-0.0054}^{+0.0171}$ c
−15.75	${0.0415}_{-0.0069}^{+0.0354}$ c
−15.25	${0.0599}_{-0.0106}^{+0.0757}$ d
−14.75	${0.0817}_{-0.0210}^{+0.1902}$ d
−14.25	${0.1052}_{-0.0434}^{+0.5414}$ d
−13.75	${0.1275}_{-0.0747}^{+1.6479}$ d
−13.25	${0.1464}_{-0.1077}^{+5.4369}$ d
−12.75	${0.1584}_{-0.1343}^{+19.8047}$ d

Notes.

^aThese LF results are simple estimates, representing the number of sources at a given luminosity (using the median magnification maps) after division by the selection volumes. No account is made for scatter resulting from errors in the magnification maps. Errors in the magnification maps can be best handled using the forward modeling simulations and methodology we consider in Section 4.2, leading to the results presented in Table 3. ^bUpper limits are $1\sigma$. ^cThe 1σ upper limits indicate the upper limits if one adopts the larger size for sources assumed in Section 5.4. ^d68% confidence intervals on the z ∼ 6 UV LF at >−16 mag we achieve using forward modeling and observations of the first four HFF clusters in Section 4.3. The quoted constraints give the geometric mean of our results using the GLAFIC, CATS, and Sharon/Johnson parametric models as inputs. The 1σ upper limits indicate the upper limits if one adopts the larger size for sources assumed in Section 5.4 (resulting in ∼0.01 and ∼0.2–0.3 more negative values for α and δ). The error bars are not independent.

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Faintward of −16 mag, we include results in Table 5 and Figure 11 by taking the geometric mean of the best-fit LF results using the GLAFIC, CATS, and Sharon/Johnson models. We present those results to luminosities plausibly probed by the present study. $1\sigma$ errors on the results at the faint end of the $z\sim 6$ LF are similarly taken to be the geometric mean of the confidence intervals on the LF fits for the same parametric magnification models. The upper limits we quote on the LF results include the impact of the larger sizes quoted in Section 5.4, which we find to result in a 0.01–0.03 steeper value of the faint-end slope α and to lower the inferred δ by ∼0.2–0.3.

Before looking into comparisons of our new observational constraints with theory, it is important first to compare the present results with previous observational results where available to evaluate and understand any differences. Doing so helps clarify the gains that are made with our newer analysis and enables others to also understand what new factors led to the changes from prior work.

6.1.1. Atek et al. (2015b)

6.1.2. L17

As can be seen from Figure 11, the differences between our results and those from L17 are more substantial than those with Atek et al. (2015b) discussed above. These differences with L17 are particularly large for the LFs at the fainter magnitudes $\gt -17$, where much of the current interest lies since this is a region that is uniquely accessible using the HFF clusters. As a result, the discussion of the reasons for these differences is necessarily much more extensive than that for Atek et al. (2015b).

To ensure that comparisons with the LF results from L17 were made using a consistent luminosity scheme, we carefully cross-matched sources from our catalogs with those from L17. We also computed apparent magnitudes for individual sources using the tabulated absolute magnitudes, redshifts, and magnification factors that L17 provide in their Tables A1–A3. It is these derived apparent magnitudes that we compare to our own photometry and that of other groups.

Comparing the total magnitudes, we derive for sources using our scaled aperture scheme to the L17 apparent magnitudes, we find a 0.43 mag median difference, with the L17 apparent magnitudes being fainter than ours, both for relatively bright ${H}_{160,{AB}}\lt 28$ sources and also for the fainter ${H}_{160,{AB}}\gt 28$ sources. If we instead estimate total magnitudes for sources by taking the flux in fixed apertures that would enclose 70% of the flux for point sources, as performed by the HUDF12 team (McLure et al. 2013; Schenker et al. 2013) and derive an inverse variance-weighted total magnitude from the Y₁₀₅, J₁₂₅, JH₁₄₀, and H₁₆₀ bands, we find differences of 0.2 mag in the median, with the L17-inferred magnitudes being fainter, comparing magnitudes for the faintest sources (i.e., >28 mag). The L17 magnitudes show a similar offset relative to the published photometry of Atek et al. (2015b).

Given that the HUDF12 apparent magnitude measurement scheme should give a fairly conservative lower limit on the total fluxes for individual candidates, these comparisons suggest that L17 systematically underestimate the luminosity of individual sources in their catalog by at least ∼0.2 mag, if not more (taking our scaled-aperture magnitudes as the baseline).

Given this range in values, we adopt a shift of the binned $z\sim 6$ LF of L17 brightward by 0.3 mag to compare volume density measurements at luminosities closer to what we measure. After doing so, we find that the L17 stepwise results appear to be a factor of ∼3–4× higher than our own results in the luminosity range −17 to −14.5 mag (see Figure 11). After considering various explanations for these differences, it seems likely that the sizeable discrepancies arise due to the excess of sources L17 at the completeness limit, i.e., there are some ∼22 sources in the 29.25 mag bin (${\rm{\Delta }}m=0.12\,\mathrm{mag}$) versus ∼7 sources per bin brightward of 29.2 mag (this can be clearly seen in Figure 12, which is adapted from Figure 13 of Bouwens et al. 2017). When one combines such an excess with the large intrinsic half-light radii that L17 assume (median of $0\buildrel{\prime\prime}\over{.} 09$) results, it seems clear that L17 may significantly overestimate the volume density implied by galaxies at the faint end of their probe. In Bouwens et al. (2017), we demonstrated through extensive simulations that the assumptions of such large sizes for faint galaxies would result in much higher inferred volume densities for sources (by factors of >5) than if smaller sizes (i.e., <10 mas) were used. The effect of using large sizes for faint galaxies can be seen in Figure 2 from Bouwens et al. (2017).

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Based on recent results from a number of papers (Bouwens et al. 2017; see also Kawamata al. 2015; Laporte et al. 2016), it is now clear that the use of such large sizes for very faint galaxies is not realistic. Several lines of evidence indicate that the intrinsic half-light radii of very low luminosity sources (i.e., $\gt -16\,\mathrm{mag}$) are very small, i.e., ≲003, with many sources appearing to have sizes ≲001 (Bouwens et al. 2017). Small sizes imply small completeness corrections and therefore much lower volume densities for faint galaxies. It is important to verify that the sizes assumed in selection volume simulations are realistic, given how sensitive the selection volume estimates (and hence LFs) for faint galaxies are to the sizes assumed for faint galaxies (see Figure 13 of Bouwens et al. 2017).

The very high volume densities L17 find for $\gt -17.5\,\mathrm{mag}$ galaxies in their $z\sim 6$ LF seem likely to have impacted the analyses they performed regarding a possible turn-over at the faint end, as we explain in Section 6.2. For similar reasons, the present constraints on the faint-end slope α that we determine, i.e., $\alpha =-1.92\pm 0.04$ are shallower than those obtained by L17. Given that the constraints can also depend on the gravitational lensing model assumed, it is best to compare faint-end slopes assuming the same gravitational lensing model. If we take their formal constraints as measured assuming the GLAFIC gravitational lensing model, our estimated faint-end slope α is −1.92 ± 0.04 versus the L17 faint-end slope at $z\sim 6$ of −2.10 ± 0.03. The difference in the derived slope is 3.5σ, combining the errors from both measurements of the slope.

The actual differences between our faint-end slope α estimates and L17's estimates are likely even larger than 3.5σ, if we compare like measurements with like. L17 combine their HFF measurements with the Finkelstein et al. (2015) field constraints, which prefer −2.02 ± 0.10. Re-estimating the faint-end slope α from the L17 HFF results alone, we derive their faint-end slopes to be −2.15 to −2.3 (see Appendix E here or Section 6.2 of Bouwens et al. 2017). This is significantly (≥4.5σ) steeper than our mean estimate using the HFF data alone of −1.92 ± 0.04 (Section 5.2; Table 2).

One issue we examined in this study concerned the existence of a possible turn-over in the $z\sim 6$ UV LF at the faint end. This question is of great interest for the theoretical models, as we shall see in the next section, and so any observational claims regarding where a turnover occurs—or does not occur—require a very high degree of careful analysis and credibility if they are to be of real value to the theoretical modelers.

We noted in Section 5.3 that we were not able to find clear and compelling evidence for a turn-over at the faint end of the LF. Using our likelihood contours for δ-α-${\phi }^{* }$, we concluded that a turn-over in the LF is allowed faintward of −14.2 mag or −15.3 mag (at 68% confidence), depending on whether we assumed smaller or larger uncertainties in the magnification maps.

Since we could not find compelling evidence for a turn-over at the faint end of the LF (Section 5.3), we proceeded to examine what constraints we can place on the presence of a turn-over as well as the luminosity at which a turn-over could occur. Using our search results, we concluded that any possible turn-over in the LF would need to occur at $\gtrsim -14.2\,\mathrm{mag}$ or $\gtrsim -15.3\,\mathrm{mag}$ (within the 68% confidence intervals), depending on the assumptions we made about errors in the magnification maps.

The present conclusions parallel to those of Castellano et al. (2016b), who also concluded that the UV LF at $z\sim 6$ appeared unlikely to show a turn-over at $\lt -15\,\mathrm{mag}$ (68% confidence), after factoring in the uncertainties in the magnification maps. We now proceed to examine in more detail what constraints we can place on the presence of a turn-over as well as the luminosity at which a turn-over could occur. We also look in more detail at the L17 result since they claim strong evidence against a turn-over to very low luminosities.

L17 concluded based on their analysis of $z\sim 6$ galaxies behind the first two HFF clusters that they have "positive" and "strong" evidence that any turn-over in the LF must be fainter than −11.1 mag and −12 mag using the criteria ${\rm{\Delta }}({\rm{BIC}})=2$ and ${\rm{\Delta }}({\rm{BIC}})=6$, respectively (where we draw these numbers from their Figure 12). BIC denotes the Bayesian Information Criterion (Schwarz 1978). In the way L17 apply BIC, BIC is effectively just equal to ${\rm{\Delta }}{\chi }^{2}$, and the above limits on the turn-over luminosity translate to nominal confidence levels of 84% and 98.5%, respectively. On the basis of the L17 fit results assuming different magnification models, L17 reported an uncertainty in their allowed turn-over luminosity of ${}_{-0.8}^{+0.4}$ mag and ${}_{-0.6}^{+0.3}$ mag, respectively. While our conclusions regarding the lack of a bright turnover (at $\sim -15$ or $\sim -14\,\mathrm{mag}$) are consistent with the much lower limit claimed by L17, there are a number of reasons for being concerned about the validity of their constraints on the luminosity of a possible turn-over and their strong statements that it cannot occur at brighter magnitudes.

To be more specific, L17 use their LF results to claim evidence against a turn-over in the $z\sim 6$ LF 3.1 mag fainter than what we do, despite their examination of only half the HFF data set and with a resulting smaller $z\sim 6$ sample (we use four clusters versus their two). How could they claim stronger constraints? Part of this could be because we have recognized, and applied, the very large uncertainties that are inherent in using very highly magnified sources, as indicated in Figure 8, whereas their analysis does not include this very large source of systematic error. Remarkably, however, it appears that the primary reason for the difference lies elsewhere, and not with the faintest sources they report. It appears to arise from the very high volume densities they estimate for the UV LF in the range −17 to −15 (which lie in significant excess of our own determinations and those of Atek et al. 2015b, by factors of ∼3–4: see Figure 11).

When compared to brighter points at $\lt -18$ in the LF, the volume density of sources L17 report over the luminosity range from −17.5 to −15 is sufficiently high (with small error bars) as to suggest an LF form, which steepens further at lower luminosities, i.e., $\delta \lt 0$, rather than one that retains a fixed faint-end slope and then flattens toward lower luminosities, i.e., $\delta \gt 0$ (using our formalism). Such a shape, with a "concave up" feature around approximately −17.5 mag is quite unusual (this region is indicated in Figure 11 with the dashed green line). Given this, we suspect that it would be very difficult indeed for L17 to find evidence for a turn-over at intermediate luminosities (since their LF is suggesting the opposite curvature). Given the statistical weight of this upward change of slope at approximately −17.5 to −15 mag, L17 would find that they needed to probe very faint indeed to find a luminosity where a turn-over was allowed.

A probable explanation for this derives from what we found in the L17 apparent magnitude distribution in Figure 12. The faintest sources are subject to very large completeness corrections. Normally, such sources would only contribute to the faintest bin in the LF, but by virtue of a diversity of magnification factors relevant to sources in this bin, they impact all the fainter bins in the LF. The large number of sources in the 29.13–29.25 mag bin exacerbates this effect, and results in large contributions to the LF over a broad range. We will discuss this further in a future paper (R. J. Bouwens et al. 2017, in preparation), but the current result suggests the need for particular conservatism in selections near the detection limit, when taking advantage of gravitational lensing.

There is a second piece of evidence L17 present that could argue against a possible turn-over in the LF at $\sim -15$ to $\sim -14\,\mathrm{mag}$. This involves their $z\sim 6$ LF at approximately −12.4 mag (A2744_z6_2830). Based on this candidate, L17 estimate a volume density of ∼6 galaxies per Mpc³ at $\sim -12.5\,\mathrm{mag}$, which would disfavor a turn-over in the LF at any luminosity down to this limit. This candidate galaxy thus assumes a critically important role in their conclusions, and it is therefore very important both to consider and examine, as to its robustness. Interestingly, this $z\sim 6$ candidate from L17 is also detected in our catalogs, though we do not include it in our $z\sim 6$ sample, since the integrated likelihood of the candidate lying at $z\lt 4$ is ∼50% and hence it does not meet our selection criteria. Nonetheless, given its importance, we need to give consideration to this object.

On the basis of our own photometry, we estimate the source to have a total apparent magnitude of 28.9 mag. Using the median magnification factor ${61}_{-15}^{+44}$ we derive for the source based on the latest publicly available magnification models (weighting each type equally) and a redshift of $z={6.11}_{-1.22}^{+1.05}$ (L17's estimate), we calculate an absolute magnitude of −13.4 mag for the source. This is ∼1 mag brighter than what L17 estimate for the same source. L17's reported luminosity is based on a magnification factor of ${110.0}_{-22.2}^{+129.0}$. This high magnification factor appears to be at the high end of the publicly available models, as we discuss in Appendix F, even though L17's estimate is purportedly a median of those same model results. By contrast, our magnification estimates agree very well (<2% difference) with L17's initial estimate of ${60.4}_{-22.2}^{+129.0}$. It is unclear why L17 changed their median magnification estimate from 60.4 (their version 1 value) to 110 (their published value) since the median magnification is clearly much closer to 60 than it is to 110 (as is obvious from Figure 19 and the entire discussion in Appendix F). We have verified that this is the case both from our own calculations and using the public magnification calculator.¹⁷ Despite this described tension and this faint candidate not being in our $z\sim 6$ selection, we can examine the implications this source would have for our UV LF results if it is indeed at $z\sim 6$.

Taking this single source and dividing by the selection volume in the range $-13.5\lt {M}_{\mathrm{UV},{AB}}\lt -13$, we estimate a volume density of ${0.28}_{-0.22}^{+0.64}$ Mpc⁻³ mag⁻¹. Interestingly, this is completely consistent, as illustrated in Figure 13 (left panel), with the LF constraints we find at $\sim -13.4$ mag treating the GLAFIC magnification model as reality and recovering the LF results using the median magnification model. It is also consistent (right panel to Figure 13) with the LF results we derive using the other magnification models, e.g., Zitrin-LTM, which suggest a magnification factor of 19, with the associated volume density of ∼0.06 Mpc⁻³ mag⁻¹, instead of the magnification factor of ∼110 adopted by L17. It is not surprising that we recover lower volume densities than those recovered by L17, given our use of smaller sizes for the faint $\gt -16$ mag population as now appears to be appropriate (Bouwens et al. 2017). Smaller sizes translate into a higher completeness and selection volume.

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Since this source is not included in our LF results, we also checked the impact it would have on our results if it had been included. We found that it only had a moderate impact on the allowed luminosity of a turn-over. We find our constraints on a possible turn-over in the UV LF change by 0.8 mag (becoming fainter) assuming that GLAFIC-versus-median model are typical of the true magnification errors and by ∼0.4 mag (becoming fainter) assuming that the Bradac-versus-median or Grale-versus-median models are more typical of the errors. If we ask which luminosities we can exclude for a turn-over at 95% confidence, the excluded luminosities are $\lt -14.7$ mag for the GLAFIC-versus-median case (2.7 mag brighter than what L17 report for this limit).

The considerations discussed in this section suggest that it is not possible using current observational data to definitively rule out the presence of a turn-over in the LF as bright as −15 mag and especially at −14 mag. L17 (despite a smaller $z\sim 6$ sample) had previously claimed strong evidence against a turn-over brightward of $\sim -12\,\mathrm{mag}$. From the present discussion, it appears their conclusions were impacted by the large numbers of faint sources incorporated into their LF results near the completion limit (Figure 12) and their size assumptions. This resulted in an apparent upturn in their $z\sim 6$ LF at $\gtrsim -17.5\,\mathrm{mag}$, strongly disfavoring a turn-over in their LF results until very low luminosities. A reassessment of L17's faintest source using the public magnification models shows a wide range of estimated magnifications, mostly lower than their value and many consistent with a turn-over at higher luminosities.

The observational constraints we have obtained here are obviously of great value for comparing against the predictions for the form of the LF at the faint end, as provided by many different teams using simulations, theoretical models, and on the basis of observations of the nearby universe.

We compare our LF constraints with the following cosmological simulation or theoretical model results.

Renaissance (O'Shea et al. 2015): O'Shea et al. (2015) report some of the first results from the "Renaissance" simulations. The "Renaissance" simulations are zoom-in simulations of a ${(28.4\mathrm{Mpc}/{\rm{h}})}^{3}$ volume of the universe, powered by the Enzo code (Bryan et al. 2014), and self-consistently following the evolution of gas and dark matter, including H₂ formation and destruction from photodissociation. Star-formation and supernovae physics are included and ionizing and UV radiation are produced as predicted by Starburst99 (Leitherer et al. 1999). Individual dark-matter particles in the simulations have masses of $2.9\times {10}^{4}$ M_⊙, meaning that the smallest halos that are resolved in the simulation are 2 × 10⁶ M_⊙ (∼70 particles/halo). Many details of the physical implementation of the implementation of the physics and also sub-grid recipes are provided in Xu et al. (2013, 2014) and Chen et al. (2014). In the "Renaissance" simulations, flattening in the UV LF is a direct result of the decreasing fraction of baryons converted to stars in the lowest mass halos, due to the impact of radiative feedback and less efficient cooling processes. While it is not yet possible to follow the results of these simulations to $z\sim 6$, results are available at $z\sim 12$ and this is the redshift we use for comparisons.

CoDa (Ocvirk et al. 2016): the Cosmic Dawn (CoDa) simulations are full gravity + hydrodynamic simulations of a large $\sim {(100\mathrm{Mpc})}^{3}$ volume of the universe using the RAMSES code (Teyssier 2002). The simulations include standard prescriptions for star formation and supernovae explosions following standard recipes (Ocvirk et al. 2008; Governato et al. 2009, 2010). One new feature of the CoDa simulations is the inclusion of radiative transfer into the simulations, in the sense that hydrodynamics and radiative transfer are now fully coupled. As a result, the effects of photoionization heating on low-mass galaxies are fully included in the CoDa simulations. Ocvirk et al. (2016) report that radiative feedback plays a big role in suppressing star formation in low-mass galaxies and modulating the faint end of the LF.

Cosmic Reionization On Computers (CROC; Gnedin 2014, 2016): the LF results for the CROC are based on gravity + hydrodynamical simulations using the Adaptive Refinement Treement code (Kravtsov 1999; Kravtsov et al. 2002; Rudd et al. 2008). The CROC simulations include a wide variety of physical processes, including gas cooling and heating processes, molecular hydrogen chemistry, star formation, stellar feedback, radiative transfer of ionizing and UV light from stars. These simulations are conducted in 20 h⁻¹ Mpc boxes at a variety of resolutions. The effective slope of CROC LFs continue to flatten toward fainter magnitudes and reach a peak at approximately −12 mag. However, the peak at approximately −12 mag is reported not to be a robust prediction of the simulation and to depend on the minimum particle size in the simulations. The impact of radiative feedback is less important in the CROC simulations than in CoDa.

Finlator et al. (2015, 2016, 2017): the Finlator et al. (2015, 2016, 2017) LF results are based on a cosmological simulation of galaxy formation in a ${(7.5{h}^{-1})}^{3}$ Mpc³ volume of the universe including both gravity and hydrodynamics as implemented in the GADGET-3 code (Springel 2005). To this code, gas cooling is added through collisional excitation of hydrogen and helium as in Katz et al. (1996), and metal line cooling is implemented using the collisional ionization equilibrium tables of Sutherland & Dopita (1993). Star formation is added using the Kennicutt–Schmidt law, with supernovae feedback included following the "ezw" prescription from Davé et al. (2013) and metal enrichment from supernovae as implemented as in Oppenheimer & Davé (2008). Flattening in the Finlator et al. (2015, 2016, 2017) LFs occurs mostly due to less efficient gas cooling at lower halo masses.

DRAGONS (Liu et al. 2016): the LF results from Liu et al. (2016) are based on the Dark-ages Reionization And Galaxy-formation Observables from Numerical Simulations (DRAGONS)¹⁸ project, which builds semi-numerical models of galaxy formation on top of halo trees derived from N-body simulations done over different box sizes to probe a large dynamical range. The semi-numerical models include gas cooling physics, star-formation prescriptions, feedback, and merging prescriptions, among other components of the model. The turn-over in the LF results of Liu et al. (2016) at approximately −12 mag correspond to the approximate halo masses ∼10⁸ M_⊙ where the gas temperature is 10⁴ K. Above this temperature, atomic cooling processes become efficient. In earlier data sets, Muñoz & Loeb (2011) had looked at what constraints could be placed on this mass using earlier LFs of Bouwens et al. (2007).

Jaacks et al. (2013): the simulation results in Jaacks et al. (2013) are powered by the GADGET-3 (Springel 2005) gravity+hydrodynamics code run in three box sizes (10, 33.75, and 100 ${h}^{-1}$ Mpc) and three different particle sizes (9 × 10⁵, 2 × 10⁷, and 3 × 10⁸ h⁻¹ M_⊙). Radiation cooling is included in the simulations by H, He, and metals (Choi & Nagamine 2009), a UV background self-shielding effect, and heating by a uniform UV background (Faucher-Giguère et al. 2009). Supernovae feedback is implemeneted by a momentum-driven wind model (Choi & Nagamine 2011). Star formation in the simulations is governed by the SFR-versus-H₂ model of Krumholz et al. (2009) rather than in terms of the total density in cold gas. The implementation of this in the GADGET-3 code is as described in Thompson et al. (2014) and is similar to the implementation of the same recipe by Kuhlen et al. (2013) in the Enzo code. As a result of the lower gas density in molecular hydrogen H₂ in fainter, lower-mass galaxies, the LFs predicted by Jaacks et al. (2013) show a turn-over at approximately −15.4 ± 0.6 mag.

Dayal et al. (2014): the LF results from Dayal et al. (2014) are based on a semi-analytic model, which follows the evolution of galaxies in the merger tree constructed from the extended Press–Schechter theory (Lacey & Cole 1993). The star-formation rate in individual galaxies proceeds at such a rate as to balance the impact of supernovae feedback in expelling all the gas from a galaxy. Flattening in the UV LF is partially the result of a similar flattening in the halo mass function, as well as lower efficiency for star formation in the lower-mass halos that contribute to the low luminosity end of the LF.

Yue et al. (2016): Yue et al. (2016) derive their LF results assuming a non-evolving stellar mass–halo mass relation. Yue et al. (2016) adopt a very similar approach to what Mason et al. (2015) employ in predicting the galaxy LF (see also Trenti et al. 2010 and Tacchella et al. 2013). Yue et al. (2016) start with the halo mass function, break up the star-formation history of each halo into segments according to which the halo grows in mass by a factor of two, and then assume that the SFR must be such to maintain a constant stellar mass-halo mass relation which they calibrate to the $z\sim 5$ LF of Bouwens (2015a). Yue et al. (2016) then look into the impact that radiative feedback could have during the epoch of reionization. Yue et al. (2016) derive their LF results assuming that halos below some fixed circular velocity would have their star formation quenched.

Finally, we also include a comparison with the empirical results on the faint end of the LF at high redshift:

Boylan-Kolchin et al. (2015): Boylan-Kolchin et al. (2015) arrive at constraints on the faint-end of the LF at $z\sim 7$ by leveraging deep probes of the color–magnitude relationship of nearby dwarf galaxies, which allow one to estimate the luminosity of these sources at $z\sim 7$. By comparing the distribution of inferred luminosities of these dwarfs with the expected numbers extrapolating $z\sim 7$ LFs to −10 mag, Boylan-Kolchin et al. (2015) infer a break in the LF at $\sim -13\,\mathrm{mag}$ from a faint-end slope of approximately −2 to approximately −1.2.

We present comparisons of the predicted LFs from both sophisticated hydrodynamical simulations, various semi-analytic models, and the empirical results of Boylan-Kolchin et al. (2015) in Figures 14 and 15.

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Overall, we find reasonable agreement between our observational results and the predicted LFs from both hydrodynamic simulations and various semi-empirical theoretical models. The predicted LFs from the CoDa simulation (Ocvirk et al. 2016) and from two semi-analytical models Yue et al. (2016) and Dayal et al. (2014), against which we compare, fall slightly below our observational constraints at −15.5 mag by ∼0.2–0.3 dex, but otherwise are in reasonable agreement with our results.

In particular, we find that our observational constraints allow for the existence of a flattening or turn-over in the $z\sim 6$ LF at $\gt -15\,\mathrm{mag}$ as predicted in the theoretical models, due to a variety of physical processes, including a greater role for radiative feedback (O'Shea et al. 2015; Ocvirk et al. 2016; Yue et al. 2016) and less efficient cooling in lower mass halos where atomic cooling processes would be less important (Wise et al. 2014; Gnedin 2016; Liu et al. 2016). The present results suggest that these physical processes can impact the shape of the LF at $\gt -15\,\mathrm{mag}$, as is predicted, and there is no fundamental disagreement with observational results to $\gt -14\,\mathrm{mag}$ (contrary to reports from L17).

Our observational results are also fully consistent with the abundance matching constraints obtained by Boylan-Kolchin et al. (2015), which suggest a break in the faint end slope of the LF at −13 mag. The toy LF from Boylan-Kolchin et al. (2015) is almost entirely contained with our 68% confidence intervals, suggesting that the flattening they infer from analyses of nearby dwarf galaxies is fully consistent with the HFF observations of faint $z\sim 6$ galaxies.