On the Faint End of the Galaxy Luminosity Function in the Epoch of Reionization: Updated Constraints from the HST Frontier Fields

The photometric catalogs of high-z galaxies used in the present paper are provided by the ASTRODEEP team (Castellano et al. 2016c; Merlin et al. 2016b; Di Criscienzo et al. 2017), and all the lensing models are provided by the FFs team on the project website.¹¹

The high-z sample comprises all sources with ${H}_{160,\mathrm{int}}\geqslant 27.5$ from the ASTRODEEP catalogs of FFs clusters A2744, M0416 (Castellano et al. 2016a; Merlin et al. 2016b), M0717 and M1149 (Di Criscienzo et al. 2017)¹² , where ${H}_{160,\mathrm{int}}$ is the demagnified apparent magnitude at the HST F160W band (H band). The model described in Section 2.2 will use sample galaxies with 5.0 < z < 7.0, while the model described in Section 2.3 will use sample galaxies with 5.0 < z < 9.5; see details in the relevant sections. The original samples presented by ASTRODEEP team have redshifts up to ∼10. However, for objects z ≳ 9.5, their redshifts may be not correctly measured. Moreover, samples with z ≳ 9.5 are only detectable in one band. Considering these reasons we do not select samples with z > 9.5.

All catalogs include photometry from the available HST ACS and WFC3 bands (B435, V606, I814, Y105, J125, JH140, H160; see, e.g., Lotz et al. 2017) and from deep K-band (Brammer et al. 2016) and IRAC 3.6 and 4.5 μm data (PI Capak). Sources are detected on the H160 band after removing foreground light both from bright cluster galaxies and the diffuse intracluster light (ICL) as described in detail in Merlin et al. (2016b). Foreground light is also removed from the HST bands before estimating photometry with SExtractor (Bertin & Arnouts 1996) in dual-image mode. Photometry from the lower-resolution Ks and IRAC images has been obtained with T-PHOT v2.0 (Merlin et al. 2015, 2016a). Photometric redshifts for all the sources have been measured with six different techniques based on different codes and assumptions. The FFs sources are then assigned the median of the six available photometric redshift estimates in order to minimize systematics and improve the accuracy. The final typical error on the photo-z is ∼0.04 × (1 + z) (C16a; Di Criscienzo et al. 2017). For the four clusters A2744, M0416, M0717, and M1149, more than 10% of the point-like objects brighter than 28.8, 28.8, 28.5, and 28.7, respectively, could be successfully resolved.

In the top panel of Figure 1 we plot the observed H-band apparent magnitude, H₁₆₀, versus redshift for our selected sample galaxies (galaxies with photometric redshifts between 5.0 and 9.5 and with demagnified H-band magnitudes larger than 27.5 in either of lensing models) in the four FFs clusters. There are 73 (87), 51 (62), 73 (76), and 34 (47) galaxies with 5.0 < z < 7.0 (5.0 < z < 9.5) in clusters A2744, M0416, M0717, and M1149, respectively. In Appendix A we list the unique ID in the ASTRODEEP catalog of all these objects, so that properties like the released SEDs can be found directly.

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The magnification for each observed source is estimated on the basis of the relevant photometric redshift from shear and mass surface density values at its barycenter of the light distribution. All models made available on the STSCI website¹³ are used.

Compared to C16a in this paper we update the A2744 and M0416 high-z samples by exploiting the improved v3 lensing models now available, and we include in the analysis number counts from other two additional clusters, M0717 and M1149. In Table 1 we list the clusters and the corresponding lensing models used in this paper. In Figure 2 we plot the distributions of the magnification factors of our selected galaxy samples in each cluster, for our adopted lensing models. In different lensing models an identified galaxy could have different magnifications, and hence different demagnified magnitudes. Therefore, for a given cluster we can reconstruct different number counts (galaxy number per magnitude bin) when using different lensing models. The median of these number counts are our fiducial number counts. In the middle and bottom panels of Figure 1 we show the number counts of the faint (${H}_{160,\mathrm{int}}\gt 27.5$) galaxies with 5.0 < z < 7.0 and 5.0 < z <9.5 in the fields of four FFs clusters (we do not use the data of the parallel blank fields), as a function of ${H}_{160,\mathrm{int}}$.

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Is there evidence of a "turnover" in the faint end of the high-z galaxy LFs from the available FFs data? To investigate this problem we adopt the following reference LF model: a standard Schechter formula modulated by a term that rapidly drops when the absolute UV magnitude M_UV is much fainter than the turnover magnitude ${M}_{\mathrm{UV}}^{{\rm{T}}}$, and rapidly approaches unity when ${M}_{\mathrm{UV}}\ll {M}_{\mathrm{UV}}^{{\rm{T}}}$:

$$\begin{eqnarray}&&\begin{array}{l}{\rm{\Phi }}({M}_{\mathrm{UV}},z)=0.4\mathrm{ln}(10){{\rm{\Phi }}}^{* }\exp [{10}^{-0.4({M}_{\mathrm{UV}}-{M}_{\mathrm{UV}}^{* })}]\\ \,\times \,{10}^{-0.4(1+\alpha )({M}_{\mathrm{UV}}-{M}_{\mathrm{UV}}^{* })}0.5[(1-\mathrm{erf}({M}_{\mathrm{UV}}-{M}_{\mathrm{UV}}^{{\rm{T}}})],\end{array}\end{eqnarray} \tag{ 1 }$$

where erf is the error function. At the ${M}_{\mathrm{UV}}^{{\rm{T}}}$ the LF drops to half the value of a standard Schechter LF. In addition to the three redshift-dependent Schechter parameters Φ*, ${M}_{\mathrm{UV}}^{* }$, α, we introduce here a new parameter ${M}_{\mathrm{UV}}^{{\rm{T}}}$. The ${M}_{\mathrm{UV}}^{* }$ is mainly determined by observations of bright galaxies and by large volume galaxy surveys, while it is unconstrained in our lensed samples of ultra-faint galaxies. For this reason, and to focus on the LF turnover-relevant parameters only, we directly adopt the parameterization of ${M}_{\mathrm{UV}}^{* }$ from Section 5.1 of (B15).

$$\begin{eqnarray}&&{M}_{\mathrm{UV}}^{* }=-20.95+0.01\times (z-6),\end{eqnarray} \tag{ 2 }$$

discarding its uncertainties. We keep the Φ_* and α as free parameters that will be constrained together with ${M}_{\mathrm{UV}}^{{\rm{T}}}$ from the FFs data.

In Y16 we have developed a physically motivated analytical model that describes the faint end of the high-z galaxy LFs during the EoR. The model calibrates the "star formation efficiency" (defined as the star formation rate to halo dark matter mass ratio) - halo mass relation using the Schechter formula of observed LF at redshift ∼5, then computes the luminosity of a halo according to its mass and formation time at any redshifts (Mason et al. 2015; see also Trenti et al. 2010 and Tacchella et al. 2013). Considering the probability distribution of a halo's formation time (Giocoli et al. 2007), and the possibility of its star formation being quenched (if the circular velocity of this halo is smaller than a presumed circular velocity criterion ${v}_{c}^{* }$ and it is located in ionized regions), the LF is then derived from halo mass function. In this model, the LF does not necessarily decrease monotonically at its faint end but has complex shapes; see Figure 6 and Figure 7 in Y16.

The Y16 model has two free parameters, the escape fraction of ionizing photons, f_esc, and the critical circular velocity, ${v}_{c}^{* }$. The galaxy number counts are sensitive to ${v}_{c}^{* }$ but less sensitive to f_esc; therefore we combine the number counts with the measured Thomson scattering optical depth to CMB photons, τ, to obtain the joint constraints.

Here, we summarize the procedure adopted to derive constraints on theoretical parameters from the observed galaxy number counts. A more detailed discussion can be found in C16a.

The sample galaxies of each cluster in the specified redshift range are divided into n_b bins according to their demagnified magnitudes. Suppose that in the ith bin there are ${N}_{\mathrm{obs}}^{i}$ galaxies. For a given LF model with parameter set ${\boldsymbol{a}}$, we perform Monte Carlo simulations to calculate the probability of observing such a number of galaxies in this bin, ${p}_{1}({N}_{\mathrm{obs}}^{i}| {\boldsymbol{a}})$. In the Monte Carlo simulations, we include the completeness as a function size and magnitude of the image. The image size of each input galaxy is derived from its luminosity using an intrinsic galaxy radius–luminosity relation given in Huang et al. (2013). This relation is comparable with the relation in Bouwens et al. (2017a, 2017b). We note that Kawamata et al. (2018) found a steeper relation slope for galaxies down to ${M}_{\mathrm{UV}}\sim -12.3$ in FFs, although at this moment we do not check the influence on our results. ${p}_{1}({N}_{\mathrm{obs}}^{i}| {\boldsymbol{a}})$ depends on both LF models and lensing models. We use the mean probability of different lensing models (see their Equation (3)) except when comparing different lensing models.

We then build the following combined likelihood:

$$\begin{eqnarray}\begin{array}{rcl}L & = & {L}_{1}\times {L}_{2}\\ & = & \left[\displaystyle \prod _{i}^{{n}_{b}}{p}_{1}({N}_{\mathrm{obs}}^{i}| {\boldsymbol{a}})\right]\times {L}_{2},\end{array}\end{eqnarray} \tag{ 3 }$$

where L₁ is the likelihood from our FFs observations, and L₂ is the likelihood of additional observations that can help to improve the constraints.

For the empirical model, we build L₂ from the constructed LF data points of wide blank fields at z ∼ 6,

$$\begin{eqnarray}&&{L}_{2}=\displaystyle \prod _{j}\displaystyle \frac{1}{\sqrt{2\pi {\sigma }_{{\rm{\Phi }},j}^{2}}}\exp \left[-\displaystyle \frac{{({{\rm{\Phi }}}_{j}-{\rm{\Phi }}({\boldsymbol{a}}))}^{2}}{2{\sigma }_{{\rm{\Phi }},j}^{2}}\right].\end{eqnarray} \tag{ 4 }$$

Introducing this L₂ is necessary, because although the gravitational lensing surveys are deeper, usually they have smaller effective volume, while the blank field surveys have large volume, and thereby are helpful for reducing the uncertainties.

For the physically motivated model, we build the L₂ from the measured CMB scattering optical depth,

$$\begin{eqnarray}&&{L}_{2}({\boldsymbol{a}})=\displaystyle \frac{1}{\sqrt{2\pi {\sigma }_{\tau }^{2}}}\exp \left[-\displaystyle \frac{{({\tau }_{\mathrm{obs}}-\tau ({\boldsymbol{a}}))}^{2}}{2{\sigma }_{\tau }^{2}}\right].\end{eqnarray} \tag{ 5 }$$

The constraints on parameter set ${\boldsymbol{a}}$ are obtained by looking for the minimum of ${\chi }^{2}=-2\mathrm{log}(L)$ and its variations corresponding to different C.L. given by chi-squared distribution.

In this subsection we investigate the constraints on parameters in the empirical model described in Section 2.2 at $z\sim 6$ by analyzing galaxy samples with 5 < z < 7 in ASTRODEEP catalogs.

Using a collection of wide and deep blank field HST surveys data, including the CANDELS, HUDF09, HUDF12, ERS, and BoRG/HIPPIES fields, B15 have constructed the LFs from z ∼ 4 to z ∼ 10. We use their stepwise maximum likelihood determination of the z ∼ 6 data points to build the L₂ (see Table 5 of B15). We vary Φ_* in the range ${{\rm{\Phi }}}_{* }\leqslant 1.5\times {10}^{-3}$ Mpc⁻³, α in the range $-2.5\leqslant \alpha \leqslant -1.6$ and ${M}_{\mathrm{UV}}^{{\rm{T}}}$ in the range $-18\leqslant {M}_{\mathrm{UV}}^{{\rm{T}}}\leqslant -10$.

The constraints on empirical model parameters are shown in Figure 3. In the 2D ${M}_{\mathrm{UV}}^{{\rm{T}}}$-α contour map, the ${{\rm{\Phi }}}_{* }$ has been marginalized, and in the ${M}_{\mathrm{UV}}^{{\rm{T}}}$-${{\rm{\Phi }}}_{* }$ contour map the α has been marginalized. We can see that the upper boundary of ${M}_{\mathrm{UV}}^{{\rm{T}}}$ is always open. To obtain the final constraints we marginalize both ${{\rm{\Phi }}}_{* }$ and α, and we have ${M}_{\mathrm{UV}}^{{\rm{T}}}\gtrsim -14.6$ at 1σ C.L. and ${M}_{\mathrm{UV}}^{{\rm{T}}}\gtrsim -15.2$ at 2σ C.L.. Still, we only find the lower boundary of ${M}_{\mathrm{UV}}^{{\rm{T}}}$, the upper boundary is open. This implies that no evidence is found in the existing data for the four FFs clusters of a LF turnover at z ≈ 6. We summarize the constraints in Table 2. We have tested that in Equation (3) if we use the L₂ derived from Finkelstein et al. (2015) LF at z ∼ 6 we obtained quite similar results on ${M}_{\mathrm{UV}}^{{\rm{T}}}$. If we remove L₂, i.e., using only the FFs data, while we restrict ${{\rm{\Phi }}}_{* }$, α and ${M}_{\mathrm{UV}}^{{\rm{T}}}$ vary in range as specified in last paragraph, we would obtain ${M}_{\mathrm{UV}}^{{\rm{T}}}\gtrsim -15.2$ and ${M}_{\mathrm{UV}}^{{\rm{T}}}\gtrsim -15.9$ at 1σ and 2σ C.L., respectively.

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Table 2.  Constraints on ${M}_{\mathrm{UV}}^{{\rm{T}}}$ and ${v}_{c}^{* }$, and the Halo Mass and Absolute UV Magnitude Corresponding to ${v}_{c}^{* }$ Constraints

		ALL	GLAFIC	CATS	Sharon	Williams	Zitrin-LTM-Gauss
${M}_{\mathrm{UV}}^{{\rm{T}}}$	1σ	$\gtrsim -14.6$	≳−14.6	≳−12.9	≳−13.7	≳−14.3	≳−11.8
	2σ	≳−15.2	≳−15.2	≳−14.3	≳−14.7	≳−14.9	≳−13.2
${v}_{c}^{* }/\mathrm{km}\ {{\rm{s}}}^{-1}$	1σ	$\lesssim 50$	≲48	≲40	≲45	≲45	≲34
	2σ	≲59	≲56	≲49	≲56	≲54	≲45
${M}_{h}/{M}_{\odot }(z=5)$	1σ	≲5.6 × 109	≲4.9 × 109	≲2.9 × 109	≲4.1 × 109	≲4.1 × 109	≲1.8 × 109
	2σ	≲9.2 × 109	≲7.9 × 109	≲5.3 × 109	≲7.9 × 109	≲7.0 × 109	≲4.1 × 109
${M}_{\mathrm{UV}}(z=5)$	1σ	≳−14.2	≳−14.0	≳−13.2	≳−13.7	≳−13.7	≳−12.4
	2σ	≳−15.0	≳−14.8	≳−14.1	≳−14.8	≳−14.6	≳−13.7
Mh/M⊙(z = 9.5)	1σ	≲2.4 × 109	≲2.1 × 109	≲1.2 × 109	≲1.8 × 109	≲1.8 × 109	≲7.6 × 108
	2σ	≲4.0 × 109	≲3.4 × 109	≲2.3 × 109	≲3.4 × 109	≲3.0 × 109	≲1.8 × 109
MUV(z = 9.5)	1σ	≳−14.0	≳−13.8	≳−13.0	≳−13.6	≳−13.6	≳−12.3
	2σ	≳−14.8	≳−14.6	≳−14.0	≳−14.6	≳−14.4	≳−13.6

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We plot the LF corresponding to the constraints at z ∼ 6 in Figure 4 by curves and filled regions. As a reference and consistency check, we also plot the B15 LF data, and the LF data constructed from A2744, M0416, and M0717 and their corresponding parallel blank field in A15 at z ∼ 7, which is one of the deepest LFs, and is consistent with other LFs in the overlap magnitude range. Moreover, the L17, B17, and I18 LFs are also plotted in Figure 4.

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Before making comparisons between B17 and our results, it is necessary to clarify a dissimilarity between the definition of the "turnover magnitude" between B17 and our work. In B17, the turnover magnitude is the absolute magnitude at which the LF's derivative is zero, while in our work it is defined as the absolute magnitude where the LF decreases to half of the Schechter LF. Moreover, their LF end is modulated by a term ${10}^{-0.4\delta {({M}_{\mathrm{UV}}+16)}^{2}}$, which could decrease gently even when M_UV is higher than the turnover magnitude, depending on δ. However, we assume the modulation term $0.5[1-\mathrm{erf}({M}_{\mathrm{UV}}-{M}_{\mathrm{UV}}^{{\rm{T}}})]$, and at ${M}_{\mathrm{UV}}\gt {M}_{\mathrm{UV}}^{{\rm{T}}}$ the LF drops rapidly. For the above reason, we do not directly compare our ${M}_{\mathrm{UV}}^{{\rm{T}}}$ with B17. Instead, we plot our constrained LF together with the LF constructed in B17; see Figure 4. From this figure, the B17 LF constraint could be approximately translated into ${M}_{\mathrm{UV}}^{{\rm{T}}}\gtrsim -14$ of our model, a bit deeper than what we found in our work, ${M}_{\mathrm{UV}}^{{\rm{T}}}\gtrsim -14.6$. This difference might be due to the different methodologies adopted to take into account systematic effects. We use the median of the number counts from different lensing models as the true number count, while B17 incorporates systematics estimated by the difference between the various models. As a check, in our work if we drop the galaxies with magnification factors >100 in observations we obtain constraints ${M}_{\mathrm{UV}}^{{\rm{T}}}\gtrsim -14.8$ (−15.4) at 1σ (2σ) C.L., quite similar to the model without dropping galaxies.

The L17 constraint on LF turnover is deeper than ours, i.e., no turnover is seen until ${M}_{\mathrm{UV}}\sim -12.5$ at $z\sim 6$ in their work. The difference between our results and those of L17 could be due to the different methodologies adopted: (a) they built the source catalog and subtracted the ICL in a very different way than in our case. (b) They assumed different galaxy size distributions. (c) In our case, we first construct the number count for each lensing model independently, then take the median of the number counts reconstructed from different lensing models; while in L17, for the image of each galaxy they take the flux-weighted magnification of different lensing models, then construct the LF.

We now investigate the systematic differences between the various lensing models. In Figure 5 we show the α-${M}_{\mathrm{UV}}^{{\rm{T}}}$ constraints by using only one lensing model each time, ignoring version discrepancies. Each of "GLAFIC," "CATS," "Sharon," "Williams" and "Zitrin-LTM-Gauss" is shown by one column in Figure 5. We choose these five lensing models because they are available for all the four clusters; and at least for A2744 and M0416 the versions are equal or later than v3.0. For each lensing model, from top to bottom, the panels are α-${M}_{\mathrm{UV}}^{{\rm{T}}}$ constraints, the sum of the number counts for the four FFs clusters, and the LFs corresponding to the constraints (see Table 2).

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Indeed, the discrepancies between lensing models are rather evident, especially for the ${M}_{\mathrm{UV}}^{{\rm{T}}}$ boundaries. This is because these lensing models use different mass distribution and observations as constraint inputs; as a result, although the number counts (the middle panels of Figure 5) are basically consistent with each other at ${H}_{160,\mathrm{int}}\lesssim 32$, at the faintest end they are rather different from. Detailed investigations about the systematics among lensing models could be found in Acebron et al. (2017), Meneghetti et al. (2017), Priewe et al. (2017), and the references of each lensing model listed below in Table 1. In all the cases, the upper boundaries are open, implying that no turnover is apparent.

In the bottom panels of Figure 5 we also plot the absolute magnitudes of the faintest galaxies in each lensing model with vertical lines. Usually, the ${M}_{\mathrm{UV}}^{{\rm{T}}}$ constraints are shallower than these faintest magnitudes. We check the influence of the faintest galaxies on the ${M}_{\mathrm{UV}}^{{\rm{T}}}$ constraints. We find that in Figure 5, the fainest galaxy (referring to the demagnified magnitude) for the Williams lensing model is in the M0717 field, while for other lensing models it is in the M0416 field.

For the GLAFIC and CATS lensing models, the faintest galaxy is the same one whose observed apparent magnitude ${H}_{160}=28.1$, and demagnified magnitudes ${H}_{160,\mathrm{int}}=32.2$ and 32.9 in these two lensing models, respectively. For the Sharon, Williams, and Zitrin-LTM-Gauss lensing models, the faintest galaxies have H₁₆₀ = 28.0, 28.6, and 26.7, and ${H}_{160,\mathrm{int}}=33.9$, 32.5 and 34.6 respectively. An investigation of the influence of the photometric errors is featured in Appendix B.

Although we have checked all the galaxies one-by-one visually and do not find any reason to consider the above faint galaxies past our checkup spurious objects, we still check their influences. When including (removing) them in (from) samples, we obtain the 2σ C.L. constraints: ${M}_{\mathrm{UV}}^{T}\gtrsim$−15.2 (−15.3), −14.3 (−14.5), −14.7 (−15.5), −14.9 (−15.2), and −13.2 (−14.1) for lensing models GLAFIC, CATS, Sharon, Williams, and Zitrin-LTM-Gauss, respectively. The changes to the Sharon and Zitrin-LTM-Gauss models are the most obvious, almost up to 1 magnitude.

We then investigate the constraints we can put on the physically motivated model. Since in this model both f_esc and ${v}_{c}^{* }$ are redshift-independent parameters, we use all the data in $z=5.0-9.5$. In the top panel of Figure 6 we show the constraints on f_esc and ${v}_{c}^{* }$, using the combination of galaxy number counts in the FFs fields and the latest Planck2016 Thomson scattering optical depth to CMB photons: τ = 0.058 ± 0.012 (Planck Collaboration et al. 2016b). Compared with C16a, the smaller τ helps us to obtain tighter constraints on f_esc, say f_esc ≲ 57% (2σ C.L.) after marginalizing ${v}_{c}^{* }$. When marginalizing f_esc we find ${v}_{c}^{* }\lesssim 59$ km s⁻¹ (2σ C.L.), which corresponds to a halo mass ${M}_{h}\lesssim 4.0\times {10}^{9}\,{M}_{\odot }$ and $9.2\times {10}^{9}\,{M}_{\odot }$ at z = 9.5 and 5, respectively. Given the halo mass, using the star formation efficiency–halo mass relation constructed in Y16, and the halo assembly history, we can derive its mean luminosity. We therefore translate the halo mass constraints into absolute UV magnitude constraints, ${M}_{\mathrm{UV}}\gtrsim -14.8$ and −15.0. They are slightly tighter than those reported in C16a (see Figure 3 there). Additional constraints are listed in Table 2, where we present both 1σ and 2σ constraints. The LFs at z ∼ 6 and 8 in the Y16 model corresponding to no feedback, ${v}_{c}^{* }=50$ and 59 km s⁻¹, for f_esc = 0.15, are shown in the bottom panels of Figure 6.

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We also find that different clusters do not contribute equally to the final constraint. If we respectively remove A2744, M0416, M0717, and M1149 each time, we obtain ${v}_{c}^{* }\,\lesssim$ 65, 61, 62, and 58 km s⁻¹ (all at the 2σ C.L.). As seen in Figure 1, compared with other clusters the M1149 has less contribution to the number counts in the faintest magnitude bin, therefore it has less influence on the final constraint.

We also investigate the discrepancies between different lensing models in this case. When using one lensing model at a time, as mentioned in the last subsection, we obtain the 2σ C.L. constraints: ${v}_{c}^{* }\lesssim 56$ (GLAFIC), 49 (CATS), 56 (Sharon), 54 (Williams), and 45 (Zitrin-LTM-Gauss) km s⁻¹ respectively; see Table 2.