HUBBLE FRONTIER FIELDS FIRST COMPLETE CLUSTER DATA: FAINT GALAXIES AT z ∼ 5–10 FOR UV LUMINOSITY FUNCTIONS AND COSMIC REIONIZATION

Using SWarp (Bertin et al. 2002), we make two detection images that are the coadded data of $(J_{125} + {JH}_{140} + H_{160})$ and $({JH}_{140} + H_{160})$ for our i- and Y-dropout candidates and YJ-dropout candidates, respectively. We match the PSFs of these band images in the same manner as the WFC3-IR images (Section 2) and produce the detection images. To apply the criteria of no blue-continuum detections for our dropout selections, we do not match the PSFs of the blue bands whose wavelengths are shorter than the redshifted Lyα-break feature of our dropout candidates. Because the PSF-unmatched images in the blue bands provide upper limits on the flux densities that are stronger than PSF-homogenized images for point-like sources of high-z galaxies, we use the PSF-unmatched data of the individual ACS images to obtain the upper limits.

We construct our source catalogs from the HFF images using SExtractor (Bertin & Arnouts 1996) in a total area of 9.5 arcmin² where all of the WFC3 and ACS images are available. We run SExtractor in dual-image mode for each set of images. In the cluster field, we set DEBLEND_NTHRESH to 16 and DEBLEND_MINCONT to a small value of 0.0005 in order to detect objects even in highly crowded regions. In the parallel field, we use more conservative values, ${\tt DEBLEND\_NTHRESH} = 32$ and ${\tt DEBLEND\_MINCONT} = 0.005$, because the parallel field is not crowded.⁷ The number of objects identified in the detection images is ∼4300 in total. The colors of the objects are measured with magnitudes of MAG_APER (m_AP), which are estimated from the flux density within a fixed circular aperture. The aperture diameters used for m_AP are two times the FWHMs of the PSFs. We adopt the diameters of 036 (038) and ∼02 for the PSF-matched images and for the PSF-unmatched blue-band images in the cluster (parallel) field, respectively. The detection limits are also defined with 036 (038) diameter apertures for the PSF-matched images and ∼02 diameter apertures for the PSF-unmatched images in the cluster (parallel) field.

We apply an aperture correction that is defined by the following procedure. We create a median stacked J₁₂₅-band image of our dropout candidates selected in Section 3.2 and measure the aperture flux of the stacked dropout candidate as a function of aperture size. Because the flux almost levels off at around a 12 diameter, we regard the flux within a 12-diameter aperture as the total flux corresponding to the total magnitude m_tot. In the stacked image, m_AP is fainter than m_tot by 0.82 mag. We thus estimate the total magnitudes with m_tot = m_AP − c_AP, where c_AP is the aperture correction factor of 0.82 mag. We also make median stacked images for bright and faint subsamples of our dropout candidates and obtain c_AP values. We confirm that the values of c_AP do not depend on luminosity beyond the statistical uncertainties in the magnitude range of our dropout candidates. Thus we apply one aperture correction factor of c_AP = 0.82 for all of our dropout candidates.

To check the accuracy of our aperture correction, we compare m_tot with the magnitude of MAG_AUTO (m_AUTO), which is calculated with the Kron elliptical aperture (Kron 1980). Figure 2 presents m_AUTO − m_tot as a function of m_AP and indicates that m_tot is comparable to m_AUTO for bright dropout candidates with m_AP < 27 mag. The values of m_AUTO − m_tot have significant scatters at the faint magnitudes, which is mainly due to uncertainties in determining the Kron elliptical apertures of faint sources. We adopt m_tot for our estimates of total magnitudes because we expect that the m_tot values are more reliable than m_AUTO for faint sources.

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For the selection of i dropouts at z ∼ 6–7, we use the color criteria defined by Atek et al. (2014b):

$$\begin{eqnarray} &&i_{814} - Y_{105} > 0.8, \end{eqnarray} \tag{ 1 }$$

$$\begin{eqnarray} &&i_{814} - Y_{105} > 0.6 + 2(Y_{105} - J_{125}), \end{eqnarray} \tag{ 2 }$$

$$\begin{eqnarray} &&Y_{105} - J_{125} < 0.8. \end{eqnarray} \tag{ 3 }$$

For objects fainter than the 3σ limiting magnitude in i₈₁₄, the i₈₁₄ 3σ upper limiting magnitude is replaced with the i₈₁₄ magnitude (see Atek et al. 2014b). For secure source detection, we apply the source identification thresholds of the >5σ significance levels both in the Y₁₀₅ and J₁₂₅ bands. From our i-dropout candidate catalog, we remove sources detected at the >2σ level in either the B₄₃₅ or V₆₀₆ band.

For Y dropouts at z  ∼  8, we adopt the following criteria (Schenker et al. 2013):

$$\begin{eqnarray} &&Y_{105} - J_{125} > 0.5, \end{eqnarray} \tag{ 4 }$$

$$\begin{eqnarray} &&J_{125} - H_{160} < 0.4. \end{eqnarray} \tag{ 5 }$$

As described in Schenker et al. (2013), we use the 1σ upper limiting magnitude in the Y₁₀₅ band. In this selection, sources with >3.5σ levels in both the J₁₂₅ and JH₁₄₀ bands are regarded as real objects. We reject sources detected at 2σ in the optical bands. Additionally, we apply a criterion that no more than one of the optical bands shows a detection above the 1.5σ level. We use a collective $\chi _{{\rm opt}}^2$ value to eliminate contamination, the details of which are described in Section 3.3 of Bouwens et al. (2011) and Section 3.2 of Schenker et al. (2013). The corrective $\chi _{{\rm opt}}^2$ is defined by $\chi _{{\rm opt}}^2\equiv \sum _j {\rm SGN}(f_j)({\rm SNR}_j)^2$, where f_j is the flux density in the jth band, SNR_j is the signal-to-noise ratio of the source in the jth band, and SGN(f_j) is a sign function: SGN(f_j) = 1 if f_j > 0 and −1 if f_j < 0. The j index runs across B₄₃₅, V₆₀₆, and i₈₁₄. We remove objects with $\chi _{{\rm opt}}^2 > 5.0$ from our dropout candidates if they are brighter than the 10σ limit in the JH₁₄₀ band and remove ones with $\chi _{{\rm opt}}^2 > 2.5$ if they are fainter than the 5σ limit. A linear interpolation is used for objects with JH₁₄₀ between the 5σ and 10σ limits.

For YJ dropouts at z  ∼  9, we use the following criteria:

$$\begin{eqnarray} &&(Y_{105} + J_{125})/2 - {JH}_{140} > 0.75, \end{eqnarray} \tag{ 6 }$$

$$\begin{eqnarray} &&(Y_{105} + J_{125})/2 - {JH}_{140} > 0.75 + 0.8 \nonumber\\ && \quad \times ({JH}_{140} - H_{160}), \end{eqnarray} \tag{ 7 }$$

$$\begin{eqnarray} &&J_{125} - H_{160} < 1.15, \end{eqnarray} \tag{ 8 }$$

$$\begin{eqnarray} &&{JH}_{140} - H_{160} < 0.6. \end{eqnarray} \tag{ 9 }$$

We replace the Y₁₀₅ or J₁₂₅ magnitude with the 1σ upper limiting magnitude if an object is fainter than the 1σ magnitude in Y₁₀₅ or J₁₂₅, following Oesch et al. (2013). For the YJ dropouts, we require detection significance levels beyond 3σ in the JH₁₄₀ and H₁₆₀ bands and 3.5σ in at least one of the JH₁₄₀ and H₁₆₀ bands. From our YJ-dropout sample, we remove sources detected at the 2σ level in at least one of the optical bands and sources with $\chi ^2_{{\rm opt}} > 2.8$. These criteria are similar to those defined by Oesch et al. (2013), but we slightly relax the criteria to include dropout candidates at z  ∼  9.5.

We select i, Y, and YJ dropouts with the selection criteria shown above. Figure 3 shows the two-color diagrams for our dropout candidates, together with the expected tracks of high-redshift star-forming galaxies with UV slopes of β = −2 and −3 (see Meurer et al. 1999 for the definition of β). Our dropout samples consist of 35 i-dropout, 15 Y-dropout, and six YJ-dropout candidates. These dropout candidates are listed in Tables 2–4. Figure 4 shows cutout images of our dropout candidates. Note that two out of six YJ-dropout candidates are the objects that are also selected as Y-dropout candidates. The number of dropout candidates in the cluster field is comparable to that in the parallel field, although the cluster field is subject to strong lensing effects. The number of dropout candidates is affected by two effects of the lensing magnification. One is the magnification of surface brightness that enhances the observed brightness of the dropout candidates. The other is the magnification of the observed area, which reduces the effective survey area on the source plane. In the number of dropout candidates, these two effects compensate for one another (Coe et al. 2014). Albeit the small statistics of a single HFF pointing, this would be one of the reasons why the number of dropout candidates is similar in the cluster and the parallel fields. More quantitative arguments of the lensing effects are presented in Section 5.

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Table 2. Dropout Candidates at z ∼ 6–7 in the HFF A2744 Fields

ID	R.A.	Decl.	i814 − Y105	Y105 − J125	J125a	Magnificationb	Photo-z	Referencec
	(J2000)	(J2000)
Cluster Field
HFF1C-i1	3.593804	−30.415447	>2.37	0.05 ± 0.07	26.10 ± 0.05	$3.73^{+0.24}_{-0.21}$	6.6 ± 0.8	1, 3
HFF1C-i2	3.570654	−30.414659	1.33 ± 0.12	0.11 ± 0.05	26.21 ± 0.03	1.62 ± 0.06	6.0 ± 0.7	1, 3
HFF1C-i3	3.606222	−30.386644	1.07 ± 0.09	0.09 ± 0.04	26.25 ± 0.04	1.69 ± 0.05	5.8 ± 0.7	1, 3
HFF1C-i4	3.606385	−30.407282	1.69 ± 0.21	0.00 ± 0.05	26.37 ± 0.04	$2.25^{+0.12}_{-0.10}$	6.3 ± 0.7	1, 3
HFF1C-i5	3.580452	−30.405043	>2.10	0.17 ± 0.07	26.60 ± 0.05	5.64 ± 0.39	6.8 ± 0.8	1, 3
HFF1C-i6	3.597834	−30.395961	>1.71	0.27 ± 0.09	26.79 ± 0.07	2.87 ± 0.19	7.0 ± 0.8	1, 3
HFF1C-i7	3.590761	−30.379408	0.95 ± 0.13	−0.16 ± 0.09	27.06 ± 0.08	$1.87^{+0.06}_{-0.05}$	5.9 ± 0.7	⋅⋅⋅
HFF1C-i8	3.585321	−30.397958	>1.10	0.13 ± 0.15	27.19 ± 0.10	$4.60^{+0.43}_{-0.48}$	6.8 ± 0.8	1, 3
HFF1C-i9	3.601072	−30.403991	1.11 ± 0.27	0.00 ± 0.11	27.26 ± 0.09	$3.56^{+0.26}_{-0.23}$	5.9 ± 0.7	1, 3
HFF1C-i10	3.600619	−30.410296	>1.43	−0.06 ± 0.11	27.29 ± 0.08	$11.43^{+1.60}_{-1.20}$	6.4 ± 0.7	3
HFF1C-i11	3.603426	−30.383219	0.87 ± 0.16	−0.13 ± 0.11	27.29 ± 0.09	$1.71^{+0.04}_{-0.05}$	5.8 ± 0.7	3
HFF1C-i12	3.603214	−30.410350	>1.36	−0.03 ± 0.12	27.32 ± 0.09	$3.88^{+0.29}_{-0.21}$	6.3 ± 0.7	1, 2, 3
HFF1C-i13	3.592944	−30.413328	>1.25	−0.09 ± 0.20	27.35 ± 0.15	$6.85^{+0.60}_{-0.54}$	6.1 ± 0.7	3
HFF1C-i14	3.585016	−30.413084	0.85 ± 0.19	−0.23 ± 0.14	27.45 ± 0.12	$2.94^{+0.18}_{-0.17}$	$5.7^{+0.7}_{-1.1}$	⋅⋅⋅
HFF1C-i15	3.576889	−30.386329	>0.96	0.13 ± 0.18	27.45 ± 0.16	$2.77^{+0.15}_{-0.13}$	$6.1^{+0.8}_{-0.7}$	3
HFF1C-i16	3.609003	−30.385283	1.35 ± 0.33	−0.07 ± 0.14	27.56 ± 0.12	1.59 ± 0.04	6.1 ± 0.7	3
HFF1C-i17	3.604563	−30.409364	>0.91	0.12 ± 0.16	27.62 ± 0.11	$2.94^{+0.19}_{-0.16}$	6.1 ± 0.8	3
HFF1C-i18	3.590518	−30.379763	>0.95	−0.02 ± 0.25	28.09 ± 0.19	$1.94^{+0.06}_{-0.05}$	$6.1^{+0.9}_{-1.4}$	3
Parallel Field
HFF1P-i1	3.474802	−30.362578	>1.80	0.46 ± 0.07	26.52 ± 0.05	1.04	7.3 ± 0.8	⋅⋅⋅
HFF1P-i2	3.480642	−30.371175	1.76 ± 0.34	−0.02 ± 0.09	26.95 ± 0.07	1.05	6.3 ± 0.7	⋅⋅⋅
HFF1P-i3	3.487575	−30.364380	1.27 ± 0.33	0.32 ± 0.11	27.06 ± 0.08	1.05	5.8 ± 0.7	⋅⋅⋅
HFF1P-i4	3.488924	−30.394630	>1.39	0.25 ± 0.11	27.14 ± 0.08	1.05	6.7 ± 0.8	⋅⋅⋅
HFF1P-i5	3.482550	−30.371559	1.19 ± 0.29	0.17 ± 0.11	27.15 ± 0.09	1.05	$5.8^{+0.7}_{-1.4}$	⋅⋅⋅
HFF1P-i6	3.483960	−30.397152	>1.57	0.00 ± 0.11	27.20 ± 0.09	1.05	6.3 ± 0.7	⋅⋅⋅
HFF1P-i7	3.467582	−30.396908	>1.39	0.15 ± 0.12	27.23 ± 0.09	1.04	6.8 ± 0.8	⋅⋅⋅
HFF1P-i8	3.467097	−30.387686	1.28 ± 0.25	−0.24 ± 0.11	27.30 ± 0.10	1.04	6.0 ± 0.7	⋅⋅⋅
HFF1P-i9	3.489520	−30.399528	>1.41	0.05 ± 0.13	27.32 ± 0.10	1.05	6.6 ± 0.8	⋅⋅⋅
HFF1P-i10	3.466056	−30.394409	1.10 ± 0.30	0.00 ± 0.14	27.43 ± 0.11	1.04	6.0 ± 0.7	⋅⋅⋅
HFF1P-i11	3.460587	−30.366320	0.92 ± 0.34	0.05 ± 0.18	27.70 ± 0.14	1.04	$5.8^{+0.7}_{-1.1}$	⋅⋅⋅
HFF1P-i12	3.455844	−30.366359	1.03 ± 0.36	0.00 ± 0.18	27.70 ± 0.14	1.03	$4.5^{+1.6}_{-3.9}$	⋅⋅⋅
HFF1P-i13	3.488139	−30.367864	>0.91	0.12 ± 0.19	27.73 ± 0.15	1.06	$5.8^{+1.0}_{-5.2}$	⋅⋅⋅
HFF1P-i14	3.486988	−30.399579	0.91 ± 0.33	−0.07 ± 0.18	27.75 ± 0.15	1.05	$5.9^{+0.7}_{-1.1}$	⋅⋅⋅
HFF1P-i15	3.484920	−30.376917	>0.85	0.11 ± 0.20	27.82 ± 0.16	1.05	$5.9^{+0.8}_{-0.7}$	⋅⋅⋅
HFF1P-i16	3.477238	−30.385998	>1.02	−0.22 ± 0.21	27.98 ± 0.18	1.05	6.5 ± 0.7	⋅⋅⋅
HFF1P-i17	3.481928	−30.389557	>0.99	−0.21 ± 0.22	27.99 ± 0.19	1.05	$6.2^{+0.7}_{-0.8}$	⋅⋅⋅

Notes. ^aTotal magnitudes estimated with the aperture correction. ^bThe magnification errors in the parallel field are less than 1% based on our model extrapolation estimates. Note that the errors in the parallel field would be underestimated because A2744 is a complex merging cluster. ^cReferences. (1) Atek et al. 2014b; (2) Zheng et al. 2014; (3) Atek et al. 2014a.

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Table 3. Dropout Candidates at z  ∼  8 in the HFF A2744 Fields

ID	R.A.	Decl.	Y105 − J125	J125 − H160	JH140a	Magnificationb	Photo-z	Referencec
	(J2000)	(J2000)
Cluster Field
HFF1C-Y1	3.604518	−30.380467	1.17 ± 0.06	0.04 ± 0.04	25.91 ± 0.02	1.49 ± 0.04	8.0 ± 0.9	1, 2, 4, 5, 6
HFF1C-Y2	3.603378	−30.382254	1.26 ± 0.14	0.09 ± 0.07	26.62 ± 0.05	1.61 ± 0.05	8.2 ± 0.9	2, 4, 6
HFF1C-Y3	3.596091	−30.385833	1.30 ± 0.16	−0.00 ± 0.08	26.67 ± 0.05	2.25 ± 0.09	8.2 ± 0.9	2, 4, 6
HFF1C-Y4	3.606461	−30.380996	1.12 ± 0.13	−0.09 ± 0.08	26.94 ± 0.06	1.49 ± 0.04	8.0 ± 0.9	2, 4, 6
HFF1C-Y5	3.603859	−30.382263	1.92 ± 0.34	0.35 ± 0.10	26.98 ± 0.07	1.60 ± 0.05	8.4 ± 0.9	2, 4, 6
HFF1C-Y6	3.606577	−30.380924	1.09 ± 0.22	0.40 ± 0.12	27.15 ± 0.08	1.48 ± 0.04	$7.9^{+0.9}_{-6.1}$	2, 6
HFF1C-Y7	3.588980	−30.378668	1.12 ± 0.19	−0.08 ± 0.11	27.28 ± 0.09	$1.82^{+0.06}_{-0.05}$	7.9 ± 0.9	2, 4, 6
HFF1C-Y8	3.603997	−30.382304	1.05 ± 0.29	0.17 ± 0.17	27.59 ± 0.12	1.60 ± 0.05	$7.9^{+0.9}_{-6.1}$	2, 6
HFF1C-Y9	3.592349	−30.409892	0.60 ± 0.39	−0.24 ± 0.32	28.00 ± 0.24	$9.82^{+0.65}_{-0.57}$	$7.3^{+0.8}_{-1.9}$	2, 6
HFF1C-Y10	3.605263	−30.380604	0.76 ± 0.39	0.28 ± 0.27	28.03 ± 0.17	1.48 ± 0.04	$7.6^{+0.8}_{-6.4}$	3
HFF1C-Y11	3.605062	−30.381463	0.84 ± 0.30	0.02 ± 0.22	28.06 ± 0.18	$1.53^{+0.04}_{-0.05}$	$7.7^{+0.8}_{-6.4}$	2
Parallel Field
HFF1P-Y1	3.474918	−30.362542	0.61 ± 0.11	0.04 ± 0.08	26.93 ± 0.07	1.05	$7.5^{+0.8}_{-1.5}$	⋅⋅⋅
HFF1P-Y2	3.459245	−30.367360	0.73 ± 0.18	0.02 ± 0.13	27.44 ± 0.11	1.04	$7.6^{+0.8}_{-1.7}$	⋅⋅⋅
HFF1P-Y3	3.479684	−30.366359	0.85 ± 0.33	0.15 ± 0.21	27.71 ± 0.14	1.05	$7.8^{+0.9}_{-6.4}$	⋅⋅⋅
HFF1P-Y4	3.457192	−30.379281	0.84 ± 0.45	0.17 ± 0.29	28.02 ± 0.19	1.03	$7.8^{+0.9}_{-6.6}$	⋅⋅⋅

Notes. ^aTotal magnitudes estimated with the aperture correction. ^bThe magnification errors in the parallel field are less than 1%. ^cReferences. (1) Atek et al. 2014b; (2) Zheng et al. 2014; (3) Zheng et al. 2014 possible candidates; (4) Coe et al. 2014; (5) Laporte et al. 2014; (6) Atek et al. 2014a.

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Table 4. Dropout Candidates at z  ∼  9 in the HFF A2744 Fields

ID	R.A.	Decl.	$(Y_{105}+J_{125})/2-{JH}_{140}$	${JH}_{140}-H_{160}$	H160a	Magnificationb	Photo-z	Referencec
	(J2000)	(J2000)
Cluster Field
HFF1C-YJ1	3.592512	−30.401486	>1.22	0.55 ± 0.30	27.37 ± 0.16	$14.40^{+1.20}_{-1.06}$	$9.6^{+1.0}_{-7.1}$	1, 2
HFF1C-Y2d	3.603380	−30.382255	0.78 ± 0.07	−0.05 ± 0.07	26.67 ± 0.05	1.61 ± 0.05	8.2 ± 0.9	3, 4, 5
HFF1C-Y5d	3.603859	−30.382262	1.13 ± 0.10	0.19 ± 0.08	26.78 ± 0.05	1.60 ± 0.05	8.4 ± 0.9	3, 4, 5
Parallel Field
HFF1P-YJ1	3.488893	−30.396183	1.61 ± 0.25	−0.26 ± 0.16	27.67 ± 0.11	1.05	8.7 ± 1.0	⋅⋅⋅
HFF1P-YJ2	3.473522	−30.384024	>1.28	0.26 ± 0.22	27.70 ± 0.11	1.04	$8.8^{+1.0}_{-1.7}$	⋅⋅⋅
HFF1P-YJ3	3.474445	−30.368728	>1.79	−0.34 ± 0.23	28.14 ± 0.17	1.04	$8.9^{+1.0}_{-6.9}$	⋅⋅⋅

Notes. ^aTotal magnitudes estimated with the aperture correction. ^bThe magnification errors in the parallel field are less than 1%. ^cReferences. (1) Zitrin et al. 2014; (2) Oesch et al. 2014; (3) Zheng et al. 2014; (4) Coe et al. 2014; (5) Atek et al. 2014a. ^dIdentified by our two selections for Y dropouts and YJ dropouts.

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We estimate photometric redshifts of our dropout candidates using the Bayesian photometric redshift code bpz (Benítez 2000). Tables 2–4 include the photometric redshifts. The redshift ranges are 5.7–7.3, 7.2–8.4, and 8.1–9.6 for the i-, Y-, and YJ-dropout candidates, except a candidate with the ID of HFF1P-i12 at z = 4.5. The three samples of dropout candidates cover z  ∼  5–10. We confirm that the ranges of the photometric redshifts are consistent with the redshifts defined by the dropout selections, z  ∼  6–7, 8, and 9. In the remainder of this paper, we refer to i-, Y-, and YJ-dropout candidates as z  ∼  6–7, z  ∼  8, and z  ∼  9 dropouts, respectively.

Recently, Atek et al. (2014b), Zheng et al. (2014), Coe et al. (2014), and Atek et al. (2014a) have identified a total of 16, 18, 7, and 58 dropouts at z  >  6 in the A2744 cluster field, respectively. We recover 10, 12, 6, and 25 dropouts of their samples. Laporte et al. (2014) analyze the spectral energy distribution of a z  ∼  8 dropout, which is identified in our selection with the ID of HFF1C-Y1 and in all of the other three studies. Our dropout with the ID of HFF1C-Y9 is also found in Zheng et al. (2014) as one of the three multiple images. Although we find HFF1C-Y9 by the Y-dropout selection, Zheng et al. (2014) identify the multiply imaged object as an i dropout. The different selections of Y and i dropouts are explained by the fact that the photometric redshift of HFF1C-Y9 is 7.3, which is the border value of the i-dropout and Y-dropout redshift ranges. Zitrin et al. (2014) have reported a triply imaged z  ∼  10 dropout in the cluster field (see also Oesch et al. 2014). We recover one of the multiple images as a YJ dropout with the ID of HFF1C-YJ1. In Section 4.4, we discuss the multiple images, including HFF1C-Y9 and HFF1C-YJ1, with our mass model.

We do not recover 7, 6, 1, and 33 dropouts found in Atek et al. (2014b), Zheng et al. (2014), Coe et al. (2014), and Atek et al. (2014a), respectively. The difference between our and their samples can be explained with the following three reasons. First, we use the full-depth images of HFF A2744 observations, while Atek et al. (2014b), Zheng et al. (2014), and Coe et al. (2014) only use the relatively shallow ACS images observed in HST Cycle 17 (GO 11689, PI: Dupke). Second, the limiting magnitude definitions are different. We take into account the space-dependent limiting magnitudes in the cluster field as described in Section 2. Third, there are differences in galaxy selection techniques. Zheng et al. (2014) and Coe et al. (2014) do not use the well-tested color selections but instead sophisticated photometric redshifts for the selections. Although we concur with all nine confident z  >  8 dropouts from Zheng et al. (2014), we recover only one z  ∼  7–8 dropout from Zheng et al. (2014). Similarly, we recover only 19 out of 50 dropouts at z  ∼  7 from Atek et al. (2014a). This is possibly because our selection criteria for our z < 8 dropouts are more conservative than those of Zheng et al. (2014) and Atek et al. (2014a).

We place three cluster-scale halos at the positions of the three brightest galaxies in the core of the cluster. We adopt the Navarro–Frenk–White (NFW) profiles (Navarro et al. 1997) for the mass distributions of the cluster-scale halos. The radial profiles of NFW are described as

$$\begin{eqnarray} &&\rho (r) = \frac{\rho _s}{(r/r_s)(1+r/r_s)^2}, \end{eqnarray} \tag{ 10 }$$

where ρ_s is the characteristic density and r_s is the scale radius. The scale radius is defined by

$$\begin{eqnarray} &&r_s = \frac{r_{\rm vir}}{c_{\rm vir}}, \end{eqnarray} \tag{ 11 }$$

where r_vir is the virial radius of the cluster-scale halo and c_vir is the concentration parameter. The scale radius and the characteristic density are related to the virial mass M_vir and the concentration parameter with the equations

$$\begin{eqnarray} r_s &= \frac{1}{c_{\rm vir}} \left(\frac{3 M_{\rm vir}}{4 \pi \Delta (z) \bar{\rho }(z)} \right)^{1/3}, \end{eqnarray} \tag{ 12 }$$

$$\begin{eqnarray} \rho _s &= \frac{\Delta (z) \bar{\rho }(z) c_{\rm vir}^3}{3 m_{\rm nfw}(c_{\rm vir})}, \end{eqnarray} \tag{ 13 }$$

$$\begin{eqnarray} m_{\rm nfw} (c_{\rm vir}) &= \displaystyle \int ^{c_{\rm vir}}_0 \frac{r}{(1+r)^2} dr, \end{eqnarray} \tag{ 14 }$$

where Δ(z) is the nonlinear overdensity (e.g., Nakamura & Suto 1997) and $\bar{\rho }(z)$ is the mean matter density of the universe at redshift z. The surface mass density Σ(r') is obtained as a function of radius $r^{\prime } \equiv \sqrt{x^2 + y^2}$ by integrating ρ(r) along the line of sight:

$$\begin{eqnarray} &&\Sigma (r^{\prime }) = \int ^{+\infty }_{-\infty } \rho (r^{\prime },z)dz. \end{eqnarray} \tag{ 15 }$$

In the above discussion, we assumed spherical halos. We then introduce an ellipticity e in the isodensity contour by replacing r' in Σ(r') (Oguri 2010):

$$\begin{eqnarray} &&\Sigma (r^{\prime }): r^{\prime } \rightarrow \sqrt{\frac{\tilde{x}^2}{(1-e)}+(1-e)\tilde{y}^2}, \end{eqnarray} \tag{ 16 }$$

where $\tilde{x}$ and $\tilde{y}$ are defined by the following equations with the position angle θ_e (measured east of north) of the isodensity contours:

$$\begin{eqnarray} &&\tilde{x} = x\cos \theta _e + y \sin \theta _e \end{eqnarray} \tag{ 17 }$$

$$\begin{eqnarray} &&\tilde{y} = -x\sin \theta _e + y \cos \theta _e. \end{eqnarray} \tag{ 18 }$$

We use the ellipsoidal halos in this work. Each cluster-scale halo has four free parameters: M_vir, c_vir, e, and θ_e.

To estimate contributions from cluster member galaxy halos, we identify cluster member galaxies with spectroscopic redshifts z_spec, photometric redshifts z_photo, and B₄₃₅ − V₆₀₆ colors. First, we use z_spec as presented in Table 5 of Owers et al. (2011). Galaxies at 0.28 < z_spec < 0.34 are regarded as cluster member galaxies. For objects with no z_spec, we apply color criteria

$$\begin{eqnarray} -\!\frac{1}{18}(B_{435} - V_{606}) + 2 &< V_{606}, \end{eqnarray} \tag{ 19 }$$

$$\begin{eqnarray} -\!\frac{1}{18}(B_{435} - V_{606}) + 2.4 & \;{>}\; V_{606}, \end{eqnarray} \tag{ 20 }$$

$$\begin{eqnarray} V_{606} &< 24, \end{eqnarray} \tag{ 21 }$$

and select galaxies on the red sequence of the cluster redshift. Figure 5 presents the B₄₃₅ − V₆₀₆ versus V₆₀₆ color–magnitude diagram of our objects, together with the color criteria of the red-sequence galaxies shown with the solid-line box. Finally, for objects that are selected neither by z_spec nor the red-sequence criteria, we refer z_photo estimated with bpz (Section 3). We select member galaxies with the z_photo criterion and the relaxed color–magnitude criteria

$$\begin{eqnarray} 0.09 < z_{\rm photo} &< 0.4, \end{eqnarray} \tag{ 22 }$$

$$\begin{eqnarray} -\frac{1}{22}(B_{435} - V_{606}) + 1.2 &< V_{606}, \end{eqnarray} \tag{ 23 }$$

$$\begin{eqnarray} -\frac{1}{22}(B_{435} - V_{606}) + 3 &\; {>}\; V_{606}, \end{eqnarray} \tag{ 24 }$$

$$\begin{eqnarray} V_{606} &\;{<}\; 26. \end{eqnarray} \tag{ 25 }$$

The dashed-line box in Figure 5 indicates the boundary of the relaxed color–magnitude criteria.

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We describe halo mass distributions of these member galaxies by the sum of pseudo-Jaffe ellipsoids (PJE; Keeton 2001, see also Jaffe 1983). In this model, the mass profile is characterized by the velocity dispersion σ and the truncation radius r_trun. We assume that the parameters of σ and r_trun are scaled with the galaxy luminosity L in the i₈₁₄ band:

$$\begin{eqnarray} \frac{\sigma }{\sigma _*} &= \left(\frac{L}{L_*} \right)^{1/4}, \end{eqnarray} \tag{ 26 }$$

$$\begin{eqnarray} \frac{r_{\rm trun}}{r_{\rm trun,*}} &= \left(\frac{L}{L_*} \right)^\eta, \end{eqnarray} \tag{ 27 }$$

where L_* is the normalization luminosity at the cluster redshift, and σ_*, r_{trun, *}, and η are free parameters. The mass-to-light ratio is constant for η = 0.5. The ellipticities and position angles of the member galaxy halos are determined from the shapes of the member galaxies in the JH₁₄₀ band.

Although the mass distribution of A2744 is mostly explained by the contributions from the three cluster-scale halos (Section 4.1) and the member galaxy halos (Section 4.2), we include perturbation induced by external sources to improve our mass model. If the perturbation is weak, its potential can be described by (e.g., Kochanek 1991)

$$\begin{eqnarray} &&\phi = \frac{1}{2} r^2 \kappa + \frac{1}{2} r^2 \gamma \cos 2(\theta - \theta _\gamma), \end{eqnarray} \tag{ 28 }$$

where κ is the constant convergence and γ is the constant tidal shear. The amplitude of the potential ϕ is defined for a given fiducial source redshift z_{s, fid}. We refer to this potential as PRT. In this paper, γ and θ_γ are free parameters. We fix z_{s, fid} and κ; z_{s, fid} ≡ 2.0 and κ ≡ 0.

To constrain the mass-model parameters of the cluster, we use the positions of multiply imaged systems. We identify multiply imaged galaxies based on their colors and morphologies while iteratively refining the mass-model parameters. In total, we use the positions of 67 multiple images of 24 systems, as summarized in Table 6. Seventeen out of the 24 systems, IDs 1.1–17.2 (Table 6), are the same as those listed in Table 1 of the document provided by the HFF map-making team (PI: K. Sharon; see also Johnson et al. 2014).⁹ Similarly, four high-redshift systems from the 24 systems, IDs 19.1–22.2, are listed in Table 3 of Atek et al. (2014b), among which IDs 19.1, 19.2, and 19.3 are identified as z  ∼  6–7 dropouts in Section 3.2. The images of IDs 19.1, 19.2, and 19.3 are referred to as HFF1C-i5, HFF1C-i6, and HFF1C-i8 in Table 2. We also find three new sets of multiple images, IDs 18, 23, and 24.

We search for the best-fit mass model that reproduces the positions of the multiple images. We optimize the 23 free parameters described in Sections 4.1–4.3 based on a χ² minimization with the downhill-simplex algorithm (see Oguri 2010 for more details). We assign a positional error of 04 in the image plane for each multiple image, following Oguri (2010), and obtain the best-fit parameters as summarized in Table 5. The best-fit model has χ² = 52.8 for 41 degrees of freedom (dof), suggesting that our model is reasonable.

In the following sections, we use this best-fit mass model to estimate the lensing effects in both the cluster and parallel fields. Because the parallel field has no multiple images to constrain the parameters of the mass model, it should be noted that the lensing effects in the parallel field are estimated from the extrapolation of the mass model determined at the cluster field. However, the extrapolation gives negligibly small uncertainties in our final results because of the very small magnification factors (see Section 4.5).

Figure 6 displays the critical lines for z = 8 sources and the positions of our z  ∼  5–10 dropouts in the cluster field. Because most of the dropouts are located far from the critical lines, the magnification factors are generally small, μ  ∼  1.5–2, as presented in Tables 2 and 4. However, some of the dropouts are placed near the critical line and highly magnified. In particular, the magnification factors of three dropouts, HFF1C-i10, HFF1C-Y9, and HFF1C-YJ1, are estimated to be μ  ∼  10–14. The intrinsic absolute magnitudes of these three dropouts are −17.00, −16.66, and −17.06 mag, respectively. Figure 7 shows the positions of our z  ∼  5–10 dropouts in the parallel field. The magnification factors in the parallel field are almost the same and near unity, typically ∼1.05. Thus, the lensing effects in the parallel field are negligibly small, and the parallel field may be regarded as a blank field. Although the magnification of the parallel field is very small, we adopt the lensing magnifications to our dropouts both in the cluster and in the parallel fields in our analysis. We estimate the errors of the magnification factors with a Markov chain Monte Carlo method. These errors are shown in Tables 2 and 4.

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Our mass model predicts that three systems in our dropout samples have counterimages. We discuss the positions and redshifts of these multiple images using our mass model. Because the predicted positions of multiple images depend on the source redshifts, we estimate the redshifts of multiple images from the positions in the image plane.

HFF1C-YJ1 at z  ∼  9. HFF1C-YJ1 is a highly magnified dropout at z  ∼  9 that is also reported in Zitrin et al. (2014). Zitrin et al. (2014) claim two counterimages of this dropout, which we refer to as HFF1C-YJ1-2 and HFF1C-YJ1-3. In Figure 8, we show the predicted positions of the counterimages of HFF1C-YJ1 whose redshifts are assumed to be z = 4–12. The positions of the counterimages are consistent with our estimates if the redshift of HFF1C-YJ1 is z > 6. In fact, the predicted positions of the counterimages are largely separated, >2'', if HFF1C-YJ1 resides at z = 4. Thus, our mass model predicts that the redshift of HFF1C-YJ1 is z > 6, which is consistent with the findings of Zitrin et al. (2014). The multiple image positions rule out the possibility that HFF1C-YJ1 is a low redshift galaxy at z < 4 and imply that the HFF1C-YJ1 system is a strong candidate for a high redshift galaxy. Combining the photometric redshift result, we find that the system of HFF1C-YJ1, HFF1C-YJ1-2, and HFF1C-YJ1-3 is located at z ≃ 9.6.

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HFF1C-Y9 at z  ∼  8. HFF1C-Y9 is a highly magnified dropout at z  ∼  8. Zheng et al. (2014) find two counterimages of this dropout, which are referred to as HFF1C-Y9-2 and HFF1C-Y9-3 in this paper. Figure 9 presents the predicted positions of the counterimages of HFF1C-Y9. Our mass model predicts that HFF1C-Y9 has more than two counterimages if HFF1C-Y9 resides at z = 6–8. However, we cannot investigate the positions of these additional counterimages because of the bright galaxies along the lines of sight. We compare the observed images of HFF1C-Y9-2 and HFF1C-Y9-3 with the predicted positions. The observed images fall into the predicted positions for the system at z = 4–6, implying that HFF1C-Y9 would be a source at a redshift slightly lower than that estimated in the dropout selection.

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HFF1C-i5, -i6, and -i8 at z  ∼  6–7. We find three multiple images of an i dropout, which are HFF1C-i5, HFF1C-i6, and HFF1C-i8. We use the positions of these three multiple images for the construction of our mass model. Their IDs in Table 6 are 19.1, 19.2, and 19.3, respectively. In addition to these three multiple images, Atek et al. (2014b) report another counterimage, which is named Image 5.4 in Table 3 of Atek et al. (2014b). In our paper, we refer to Image 5.4 as HFF1C-i5-2. In Figure 10, we plot the predicted positions of these four multiple images. The predicted position of HFF1C-i5-2 is about 8'' away from the images reported by Atek et al. (2014b). Instead, it is close to the position predicted by Jauzac et al. (2014), shown with the green circle in Figure 10. The observed images of the other three multiple images lie near the predicted positions at z = 6 and 8. Our mass model predicts that the best-fit value of their redshift is z = 7.94, as shown in Table 6.

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Mass models of A2744 are also made by other groups (e.g., Lam et al. 2014). Eight public mass models are accessible through the Space Telescope Science Institute (STScI) Web site¹⁰ that are made by the five independent groups M. Bradač (PI), The Clusters As TelescopeS (CATS) team (co-PIs J. P. Kneib, P. Natarajan; see Richard et al. 2014), J. Merten & A. Zitrin (co-PIs), K. Sharon (PI; see Johnson et al. 2014), and L. Williams (PI). Figure 11 compares the magnification factors of our mass model and these public mass models at the positions of our dropout candidates in the cluster field. The vertical axes show Δμ/μ, where μ is the magnification factor from our mass model and Δμ ≡ μ_other − μ is the difference between magnification factors of our model and a public mass model (μ_other). In the cluster field, the magnification factors from our mass model are broadly consistent with those from the public mass models. We especially find excellent agreement with the CATS and Zitrin-NFW models. The Merten group extends their mass model to the parallel field using weak lensing data covering both the cluster and parallel fields. The magnification factors in the parallel field from the Merten model are ∼1.08–1.22. Our mass model estimates the magnifications of the dropouts in the parallel field to be ∼1.05, which is consistent with those from the Merten model.

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One of the major sources of contamination in high-redshift dropout galaxy samples is galaxies at z  ∼  2 whose Balmer break mimics a Lyα break in the spectra of high-z star-forming galaxies. Although bright z  ∼  2 interlopers are removed by detection of a blue continuum in deep optical images, faint interlopers are selected because of the photometric uncertainties. Here we estimate the expected number of such contaminants that meet our dropout selection criteria, basically following the method described in Section 3.3 of Schenker et al. (2013, see also Section 3.1 of Ouchi et al. 2004).

To obtain the expected number of contaminants, we make use of our catalog of bright objects detected in the HFF fields. We assume that bright objects with 22 < H₁₆₀ < 25 that do not satisfy the dropout selection criteria are bright interlopers and that faint interlopers have the same color distribution as that of bright ones. We create artificial objects with faint magnitudes of H₁₆₀ = 25.0–29.5 using the mkobjects package in the iraf (Tody 1986, 1993) software. Their number counts are matched to the observed number counts extrapolated from the bright magnitudes. These artificial objects are placed in random positions in our HFF images. We conduct the source extraction and the dropout selection in the same manner as our dropout galaxy identification in the real HFF data. The artificial objects selected as dropouts are regarded as contaminants. We derive the number of contamination objects as a function of magnitude, which is used in our UV luminosity function estimates in Section 5.3. We find that the fraction of total number of contaminants to our dropout candidates down to 29.5 magnitude is ∼27%. This fraction is relatively larger than the contamination rate estimated by some previous studies, which are, for example, ∼7% at z ∼ 7–8 in Bouwens et al. (2011) and ∼23% at z  ∼  9–10 in Bouwens et al. (2014a). In Section 6, we discuss the discrepancy between the UV luminosity densities and the Thomson scattering optical depth. Because the differences in the number counts given by the contamination estimates are at most 30%, which is smaller than the discrepancy discussed in Section 6, our conclusion does not change.

The gravitational lensing effects are important in interpreting the observational results of our dropouts in the HFF fields. The brightness of dropouts is magnified, and multiple images appear for some of the dropouts. Thus, the number counts of our HFF dropout candidates are changed from those of the blank field by the gravitational lensing effects, especially in the cluster field. Moreover, to derive UV luminosity functions, one needs to correct for the selection incompleteness of dropouts that is a function of both magnitude and source redshift. In our study, we carry out Monte Carlo simulations in the image plane to evaluate the gravitational lensing effects as well as the selection completeness. The simulations contain all lensing effects: magnification, distortion, and multiplication of images. This is called the image plane technique.

There is another method for the luminosity function estimates that is referred to as the source plane technique (Atek et al. 2014b). The source plane technique determines an absolute magnitude of each dropout candidate with a magnification factor to derive the number of dropout candidates per unit source plane volume. However, distortion and multiplication effects are not included in the source plane technique. Note that our method for the image plane technique is self consistent and more complete than the source plane technique.

We first estimate the completeness of dropouts identified by our selections in the HFF images, where the completeness depends on redshift and magnitude. We create a mock catalog of ∼1, 000, 000 galaxies uniformly distributed at z = 5.0–10.4 in the magnitude range of 25.0–30.5 mag. To define the UV continuum colors of the galaxies, we assume a spectral UV slope of β = −2.0, which is the same as that used in Schenker et al. (2013). IGM attenuation is given with the prescription given by Madau et al. (1996). For the galaxies' intrinsic surface brightness profiles, we adopt a Sérsic index of 1.0 and half-light radii of ≃ 0.6 kpc and ≃ 0.3 kpc for bright (M_UV ≲ −19.5 mag) and faint (M_UV ≳ −19.5 mag) dropout candidates, respectively, which are motivated by recent size measurements for z  ∼  7–8 dropout candidates (Ono et al. 2013, see also Oesch et al. 2010). We assume a uniform distribution of the intrinsic ellipticity in the range of 0.0–0.9 because the observed ellipticities of z  ∼  3–5 dropouts have roughly uniform distributions (Ravindranath et al. 2006).

Then, we produce simulated images of the galaxies that include the HST PSFs and the A2744's gravitational lensing effects with writeimage command of glafic. We randomly select about 3000 simulated images of galaxies in a magnitude bin of Δm = 0.5 and place these simulated galaxy images at random positions on the real HFF images to make simulated HFF images. In the same manner as the procedure for the identification of our real dropouts (Section 3.1), we perform source extractions for the simulated HFF images with SExtractor and construct photometric catalogs. Applying the color selection criteria used in Section 3.2 and the magnitude-dependent contamination rates estimated in Section 5.1, we obtain simulated dropout galaxies. We make a simulated dropout galaxy sample for the magnitude bin. We conduct the same simulations over the given magnitude range of 25.0–30.5 mag and derive the completeness for our i-, Y-, and YJ-dropout selections that depend on redshift and magnitude. Figure 12 shows the completeness derived from our simulations. The left (right) panels indicate the selection windows in the cluster (parallel) field. At bright magnitudes of ∼25–27 mag, our i-, Y-, and YJ-dropout selection criteria provide a high completeness sample of star-forming galaxies at 6.0 < z < 7.4, 7.4 < z < 8.6, and 8.2 < z < 9.4, respectively.

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These completeness estimates are then used to predict the observed galaxy number counts. We calculate the predicted number counts from the completeness estimates and a UV luminosity function expressed as a Schechter function with a set of parameters, (M_*, ϕ_*, α) and repeat it for various sets of Schechter parameters covering a wide parameter space. In this way, we obtain the predicted number counts for various sets of Schechter parameters. Because the completeness values are estimated with all of the observational effects in the image plane, these predicted number counts include the lensing magnifications, magnification, distortion, and multiplication as well as the corresponding detection incompleteness in the redshift and magnitude space.

Using the predicted number counts, we search for the best-fit Schechter parameters that reproduce the observed number counts of our dropout candidates. We adopt a maximum likelihood method, assuming a Poisson distribution of the number counts. The likelihood is written as

$$\begin{equation} {\cal L} \propto \prod _{\rm field} \prod _i n_{{\rm exp},i}^{n_{{\rm obs},i}} e^{-n_{{\rm exp},i}}, \end{equation} \tag{ 29 }$$

where n_{exp, i} is the expected number count from a given Schechter function in a magnitude interval i, and n_{obs, i} is the observed number count in the magnitude interval. Constraining the Schechter parameters, we simultaneously fit both the observed HFF number counts and the UV luminosity function data points obtained in the previous studies. For our dropouts at z  ∼  6–7, we compare our number counts of z  ∼  6–7 with the luminosity function data points of z  ∼  7 in previous studies, assuming that the UV luminosity function does not rapidly change in z  ∼  6–7. We take the previous blank-field survey results from the studies of CANDELS, HUDF09, HUDF12, ERS, and BORG/HIPPIES (Bouwens et al. 2014b), UltraVISTA+UKIDSS UDS (Bowler et al. 2014), BoRG (Bradley et al. 2012), SDF+GOODS-N (Ouchi et al. 2009), and HUDF12/XDF+CANDELS (Oesch et al. 2013). We regard M_*, ϕ_*, and α as free parameters for the fitting of number counts at z  ∼  6–7 and z  ∼  8. Because the statistics of the z  ∼  9 luminosity function is poor, we choose ϕ_* for a free parameter and M_* and α to be fixed to the best-fit values of z  ∼  8. Maximizing the Poisson likelihood, we obtain the best-fit parameters of $(M_\ast, \log \phi _\ast {\rm (Mpc^{-3})}, \alpha) =(-20.45^{+0.1}_{-0.2}, -3.30^{+0.10}_{-0.20}, -1.94^{+0.09}_{-0.10})$ for the z  ∼  6–7 dropout candidates, $(-20.45^{+0.3}_{-0.2}, -3.65^{+0.15}_{-0.25}, -2.08^{+0.21}_{-0.12})$ for the z  ∼  8 dropout candidates, and (− 20.45[fixed], −4.00 ± 0.15, −2.08[fixed]) for the z  ∼  9 dropout candidates. Table 7 summarizes these parameters, together with those obtained by the previous studies. We find that our results are consistent with the previous results within the 1σ uncertainties. Figure 13 shows the 1σ confidence intervals on the α versus M_* plane for the UV luminosity functions at z  ∼  6–7 and z  ∼  8, respectively. To test our results, we also perform Schechter function fittings without the results from the HUDF09+ERS data at z  ∼  6–7 and z  ∼  8. We confirm that the fitting results without the HUDF09+ERS data are consistent with the previous results, although the uncertainties are substantially larger because of the small statistics of the HFF samples.

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Table 7. Best-fit Schechter Parameters of Luminosity Functions

Reference	M*	log ϕ*	α
		(Mpc−3)
z  ∼  6–7
This work	$-20.45^{+0.1}_{-0.2}$	$-3.30^{+0.10}_{-0.20}$	$-1.94^{+0.09}_{-0.10}$
Atek et al. (2014a)	$-20.63^{+0.69}_{-0.56}$	−3.34 ± 0.36	$-1.88^{+0.17}_{-0.20}$
Bouwens et al. (2014b)	−21.04 ± 0.26	$-3.65^{+0.27}_{-0.17}$	−2.06 ± 0.12
Bouwens et al. (2011)	−20.14 ± 0.26	−3.07 ± 0.26	−2.01 ± 0.21
Ouchi et al. (2009)	−20.10 ± 0.76	−3.16 ± 0.68	−1.72 ± 0.65
Schenker et al. (2013)	$-20.14^{+0.36}_{-0.48}$	$-3.19^{+0.27}_{-0.24}$	$-1.87^{+0.18}_{-0.17}$
z  ∼  8
This work	$-20.45^{+0.3}_{-0.2}$	$-3.65^{+0.15}_{-0.25}$	$-2.08^{+0.21}_{-0.12}$
Bouwens et al. (2014b)	−19.97 ± 0.34	−3.19 ± 0.30	−1.86 ± 0.27
Bouwens et al. (2011)	−20.10 ± 0.52	−3.22 ± 0.43	−1.91 ± 0.32
Bradley et al. (2012)	$-20.26^{+0.29}_{-0.34}$	$-3.37^{+0.26}_{-0.29}$	$-1.98^{+0.23}_{-0.22}$
Schenker et al. (2013)	$-20.44^{+0.47}_{-0.35}$	$-3.50^{+0.35}_{-0.32}$	$-1.94^{+0.21}_{-0.24}$
z  ∼  9
This work	−20.45 (fixed)	−4.00 ± 0.15	−2.08 (fixed)
Oesch et al. (2013)	−18.8 ± 0.3	−2.94 (fixed)	−1.73 (fixed)
Bouwens et al. (2014a)	−20.04 (fixed)	$-3.95^{+0.39}_{-0.56}$	−2.06 (fixed)

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The top and bottom panels of Figures 14–16 present the best-fitting number counts and Schechter functions, respectively. In the bottom panels, we present recent studies of z  ∼  7 and 8 luminosity functions, Schenker et al. (2013), Finkelstein et al. (2014), and Atek et al. (2014a), for comparison. The best-fitting results broadly agree with the observed number counts. The observed number counts of z  ∼  8 at the bright end are larger than the best-fit function. It is probably caused by the field-to-field variance because our effective survey area is only ≃ 6 arcmin² in the source plane. In fact, eight of the z  ∼  8 dropouts are found within a small region with a radius of 6'' (corresponding to a physical length of ∼30 kpc at z = 8). This overdensity of z  ∼  8 dropouts is originally claimed by Zheng et al. (2014) with the early optical images shallower than our full-depth data by ∼1 magnitudes. Because one cannot remove a number of foreground interlopers with the shallow early optical data, the existence of overdensity is an open question (see the discussion in Coe et al. 2014). In our study with the full-depth HFF data deep enough to remove such foreground interlopers reliably, there is the overdensity of z  ∼  8 dropouts, indicating that the overdensity is real. If this is true, the existence of the overdensity would significantly enhance the source number counts of dropouts at z ∼ 8 in the HFF fields.

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The middle panels of Figures 14–16 show the histograms of the number of dropout galaxies used for the UV luminosity function determinations. We also show the number of dropout galaxies newly identified in the observations of the HDF12 at z  ∼  7–8 (Schenker et al. 2013) and those of CLASH at z  ∼  9.2 (Bouwens et al. 2014a). These histograms indicate that our HFF samples enable us to probe the UV luminosity functions down to a faint UV magnitude of ≃ − 17, which is comparable to the survey limits of the deepest blank-field observations of the HUDF (Ellis et al. 2013; Oesch et al. 2013), thanks to the gravitational lensing effects. In addition to the lensing effects, the HFF A2744 observations provide two new ultradeep imaging regions of the cluster and parallel fields, allowing us to significantly increase the number of z  ∼  9 dropouts (see the z  ∼  9 dropout samples in UDF12 and CANDELS Oesch et al. 2013).

To investigate the ionizing sources for the cosmic reionization, we first estimate the UV luminosity densities ρ_UV from our UV luminosity functions. The value ρ_UV is calculated by

$$\begin{eqnarray} &&\rho _{{\rm UV}}(z) = \int ^{M_{\rm trunc}}_{-\infty } \Phi (M_{\rm UV}) L(M_{\rm UV}) dM_{\rm UV}, \end{eqnarray} \tag{ 30 }$$

where M_trunc is the truncation magnitude of the UV luminosity function where no galaxies exist beyond this magnitude. Because the M_trunc parameter is not constrained by observations, in this study we assume two M_trunc values that bracket the plausible range of the parameter: M_trunc = −17 mag corresponding to the current observational limit and M_trunc = −10 mag being the predicted magnitude of minimum-mass halos that can host star-forming galaxies (Faucher-Giguère et al. 2011).

We use the best-fit Schechter functions shown in Section 5 and those in the literature for z  ∼  4–6 (Bouwens et al. 2007), z  ∼  9.2 (Bouwens et al. 2014a), and z  ∼  10.4 (Bouwens et al. 2014b). For comparison purposes, we plot the data for z  ∼  7–8 taken from Schenker et al. (2013) and McLure et al. (2013). Because the data of Schenker et al. (2013) and McLure et al. (2013) are included in our luminosity function estimates via Bouwens et al. (2014b) data points, we do not use the data points of Schenker et al. (2013) and McLure et al. (2013) for the fitting analyses carried out in Section 6.2. The top and bottom left panels of Figure 17 present ρ_UV as a function of redshift under the assumptions of M_trunc = −17 and −10, respectively. The solid and dashed lines show the best-fit functions of ρ_UV with the two fitting methods detailed in Section 6.2. We confirm that our ρ_UV values at z  ∼  6–9 are broadly consistent with the previous results and that there is a rapid decrease of ρ_UV from z  ∼  8 toward high redshifts, which is claimed by Oesch et al. (2013) and Bouwens et al. (2014b). With the improved measurements of ρ_UV in our study, this trend of rapid decrease is strengthened. To test whether the rapid decrease is confirmed with the HFF data alone, we derive the luminosity function at z  ∼  9 with the HFF data alone and estimate ρ_UV at z  ∼  9. The gray circles in the left panels of Figure 17 indicate the ρ_UV obtained with our HFF data alone. Although the uncertainty is large, the HFF data independently support the rapid decrease of ρ_UV from z  ∼  8. A similar analysis is found in Oesch et al. (2014). They derive the luminosity function at z  ∼  10 from the HFF cluster data alone. We plot ρ_UV at z  ∼  10 calculated from the luminosity function derived by Oesch et al. (2014) with the gray squares. These plots are also consistent with the rapid decrease from z  ∼  8.

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Robertson et al. (2014) estimate the cosmic variance uncertainties of the high-redshift galaxies in the A2744 cluster field. The uncertainties are ∼35% at z  ∼  7 and ≳ 65% at z  ∼  10. Our errors of ρ_UV slightly increase by the cosmic variance uncertainties. However, our conclusion does not change because our ρ_UV at z  ∼  9 is smaller than ρ_UV at z  ∼  8 by a factor of two, which is significantly larger than the uncertainties of the cosmic variance.

The evolution of the ionized hydrogen fraction in the IGM, $Q_{{\rm H{\, \scriptsize{II}}}}$, is described by the following ionization equation (e.g., Robertson et al. 2013):

$$\begin{eqnarray} &&\dot{Q}_{{\rm H{\, \scriptsize{II}}}} = \frac{\dot{n}_{{\rm ion}}}{\langle n_{\rm H}\rangle } - \frac{Q_{{\rm H{\, \scriptsize{II}}}}}{t_{{\rm rec}}}, \end{eqnarray} \tag{ 31 }$$

where the dots denote time derivatives.

The first term in the right-hand side of Equation (31) is a source term proportional to the ionizing photon emissivity. The variables $\dot{n}_{{\rm ion}}$ and 〈n_H〉 are the production rate of ionizing photons and the mean hydrogen number density, respectively. They are defined by

$$\begin{eqnarray} \dot{n}_{{\rm ion}} &=& \int ^{M_{\rm trunc}}_{-\infty } f_{\rm esc}(M_{\rm UV}) \xi _{\rm ion}(M_{\rm UV}) \Phi (M_{\rm UV}) L(M_{\rm UV}) dM_{\rm UV}\nonumber \\ &\equiv & \left< f_{\rm esc} \xi _{\rm ion} \right> \rho _{\rm UV}, \end{eqnarray} \tag{ 32 }$$

$$\begin{eqnarray} \langle n_{\rm H}\rangle &=& \frac{X_{\rm p} \Omega _{\rm b} \rho _{\rm c}}{m_{\rm H}}. \end{eqnarray} \tag{ 33 }$$

Here, X_p is the primordial mass fraction of hydrogen, ρ_c is the critical density, and m_H is the mass of the hydrogen atom. Note that f_esc and ξ_ion are parameters that appear in the product form for our analysis. If one assumes that f_esc and ξ_ion depend on M_UV, <f_escξ_ion > is a magnitude-averaged value defined in Equation (32).

The second term in the right-hand side of Equation (31) is a sink term due to recombinations; t_rec is the averaged gas recombination time,

$$\begin{eqnarray} &&t_{{\rm rec}} = \frac{1}{C_{\rm H{\, \scriptsize{II}}} \alpha _{\rm B}(T) (1 + Y_{\rm p} / 4X_{\rm p}) \langle n_{\rm H}\rangle (1+z)^3}, \end{eqnarray} \tag{ 34 }$$

where α_B is the case-B hydrogen recombination coefficient, T is the IGM temperature at a mean density, and Y_p is the primordial helium mass fraction. Substituting Equations (32)–(34) into Equation (31), we obtain

$$\begin{eqnarray} \dot{Q}_{{\rm H{\, \scriptsize{II}}}} &=& A \left(\frac{\rho _{\rm UV}}{10^{26}\ {\rm ergs\ s^{-1}\ Hz^{-1}\ Mpc^{-3}}} \right) - \frac{Q_{{\rm H{\, \scriptsize{II}}}}}{t_{{\rm rec}}}, \end{eqnarray} \tag{ 35 }$$

$$\begin{eqnarray} A &=& 2.06\ {\rm Gyr^{-1}} \left(\frac{ \left< f_{{\rm esc}}\xi _{{\rm ion}} \right>}{0.2 \times 10^{25.2}\ {\rm erg\ Hz^{-1}}} \right), \end{eqnarray} \tag{ 36 }$$

$$\begin{eqnarray} t_{\rm rec} &=& 3.19 \times 10^2 \ {\rm Gyr}\ (1+z)^{-3} \left(\frac{C_{\rm H{\, \scriptsize{II}}}}{3} \right)^{-1}. \end{eqnarray} \tag{ 37 }$$

Once the evolution of $Q_{{\rm H{\, \scriptsize{II}}}}$ is determined by these equations, τ_e at a redshift z is estimated (e.g., Kuhlen & Faucher-Giguère 2012) from

$$\begin{eqnarray} \tau _e(z) &=& \int ^z_0 \frac{c(1+z^{\prime })^2}{H(z^{\prime })}Q_{{\rm H{\, \scriptsize{II}}}} \sigma _{\rm T} \langle n_{\rm H}\rangle \nonumber\\ && \times (1+\eta Y_{\rm p}/4X_{\rm p}) dz^{\prime }, \end{eqnarray} \tag{ 38 }$$

where c is the speed of light, H(z) is the Hubble parameter, and σ_T is the Thomson scattering cross section. We assume that helium is singly ionized (η = 1) at z > 4 and doubly ionized (η = 2) at z < 4 (Kuhlen & Faucher-Giguère 2012). The value of τ_e is measured to be $\tau _e = 0.091^{+0.013}_{-0.014}$ (Planck Collaboration et al. 2014) from the combination of the Planck temperature power spectrum, the WMAP polarization low-multipole (l ⩽ 23) likelihood (Bennett et al. 2013), and the high-resolution ground-based CMB data (e.g., Reichardt et al. 2012; Story et al. 2013).

We assume that ρ_UV is approximated by a logarithmic double power law,

$$\begin{eqnarray} &&\rho _{\rm UV}(z) = \frac{2 \rho _{{\rm UV}, z=8}}{10^{a(z-8)} + 10^{b(z-8)}}, \end{eqnarray} \tag{ 39 }$$

where ρ_{UV, z = 8} is a normalization factor, and a and b determine the slopes of ρ_UV(z). This double power-law function recovers the rapid decrease of ρ_UV from z  ∼  8 toward high redshifts.

With the analytic reionization models described with Equations (35)–(39), we carry out a χ² fitting to the observational data of τ_e and ρ_UV to search for reionization models allowed by these observational constraints. There are six free parameters in the fit, ρ_{UV, z = 8}, a, b, <f_escξ_ion >, and $C_{\rm H{\, \scriptsize{II}}}$. Because there is no observational data point of ρ_UV at z  >  11, we extrapolate the best-fit ρ_UV function of z < 11 to z = 30. At z  >  30, we assume ρ_UV = 0. In conjunction with this assumption, we regard that τ_e(z = 30) should agree with the τ_e value from the CMB measurements.

For the data of ρ_UV in the fitting, we use all of the ρ_UV data points presented in Figure 17 (Section 6.1), except for those given by Schenker et al. (2013) and McLure et al. (2013) at z = 7 and 8. Note that the data from these two studies are already included in our ρ_UV estimates via our best-fit UV luminosity functions in Section 5.3. We, thus, use a total of eight ρ_UV data points for the fitting.

The fitting ranges of $C_{\rm H{\, \scriptsize{II}}}$ and <f_escξ_ion > are 1.0–9.9 and 0–10^25.2 erg⁻¹ Hz, respectively. The range of <f_escξ_ion > is motivated by the estimate of spectral properties of high-redshift galaxies in Robertson et al. (2013). We calculate the χ² value by simply summing up the χ² value of each data point of ρ_UV and τ_e and obtain the best-fitting parameters. We refer to this fitting method as a simple χ² (SC) method.

From the χ² minimization of the SC method, we find the best-fit parameters for M_trunc = −17 and −10. The best-fit parameters and the χ² values are shown in Table 8. Figure 18 presents Δχ² values on the <f_escξ_ion > versus $C_{\rm H{\, \scriptsize{II}}}$ plane calculated by the SC method. The Δχ² is determined by $\Delta \chi ^2 \equiv \chi ^2 - \chi ^2_{\rm min}$, where $\chi ^2_{\rm min}$ is the minimum χ² value. In Figure 17, we show the best-fit functions of ρ_UV(z) and τ_e(z) for M_trunc = −17 and −10. Figure 17 indicates that the best-fit ρ_UV(z) agrees with the data points but that the best-fit τ_e is significantly lower than the one of the CMB measurement. The χ² values and the dof shown in Table 8 suggest that the probabilities of these χ² values occurring by chance are 0.4% and 1.8% with M_trunc = −17 and −10, respectively. It indicates that our analytic reionization models may not be good enough to explain the reionization history and sources of reionization.

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The best-fit functions by the SC method are weighted by the ρ_UV data more strongly than the τ_e(z) data because the number of data points of ρ_UV is eight, while that of τ_e(z) is one. Here, we calculate χ² values by another method that gives equal weight to the ρ_UV and τ_e data sets. In this method, we divide the χ² values of the ρ_UV data by eight, that is, the number of the data points. We refer to this method as an equally weighted χ² (EWC) method. The best-fit parameters and functions by the EWC method are shown in Table 8 and Figure 17, respectively. In the case of EWC, the best-fit τ_e falls in the error range of the CMB measurement, as we expect. However, there is a discrepancy between the best-fit ρ_UV(z) and the observational ρ_UV data points, especially at z ≳ 9, where the observational ρ_UV data exhibit the rapid decrease toward high z.

Even if we change the weights of the fitting and allow the large parameter space of <f_escξ_ion > and $C_{\rm H{\, \scriptsize{II}}}$, we have found that no single model can reproduce both the τ_e and ρ_UV data points from the observations. This is because the data points of ρ_UV decrease too rapidly at z  >  8 to contribute to adding to τ_e. This conclusion is in contrast with the claims of the pioneering study of Robertson et al. (2013), who find a parameter space of the similar analytic models explaining the observational τ_e and ρ_UV data available in 2013. While the best measurement value of τ_e is almost unchanged since then, the rapid decrease of ρ_UV at z  >  8 is clearly identified by the subsequent observational studies, including our HFF work. The strong constraints on the evolution of ρ_UV at z  > 8 probably allow us to find the discrepancy between the analytic models and the observational data.

There are three possible explanations for the discrepancy between the models and the observational data of ρ_UV and τ_e. First, the decrease of ρ_UV at z > 11 may not be as rapid as that found at z = 8–11. Because there is no ρ_UV data at z > 11, in our model we extrapolate the best-fit power-law ρ_UV(z) of z = 8–11 toward z = 30. If the real ρ_UV values at z > 11 are larger than this extrapolation, the τ_e value becomes larger, which eases the tension between the model prediction and the observations. The slow decrease of ρ_UV at z  >  11 may be made by a M_trunc fainter than −10 mag or a luminosity function slope (α) steeper than the values found at z  ∼  6–8 (Table 7). In other words, the discrepancy that we find may suggest that faint galaxies dominate at z  >  11 even more than at z  ∼  6–8. It is also possible that the rapid decrease would be weakened by the luminosity function slope steepening or the M_trunc becoming fainter at z  ∼  9–10. Second, the evolution of <f_escξ_ion > or $C_{\rm H{\, \scriptsize{II}}}$ can increase $Q_{\rm H{\, \scriptsize{II}}}$, as suggested by Kuhlen & Faucher-Giguère (2012). If the <f_escξ_ion > value becomes large toward high z, $Q_{\rm H{\, \scriptsize{II}}}$ (accordingly τ_e) could be boosted. Similarly, the small $C_{\rm H{\, \scriptsize{II}}}$ would enhance τ_e, although $C_{\rm H{\, \scriptsize{II}}}$ can be as low as unity by definition. Third, another source of ionizing photons besides massive stars of galaxies may exist, which contributes to the cosmic reionization significantly. X-ray sources such as X-ray binaries and faint active galactic nuclei would not leave a clear signature in the ρ_UV measurements but provide a fraction of ionizing photons via X-ray that are necessary for the cosmic reionization (Fragos et al. 2013; Madau et al. 2004; Mesinger et al. 2013).

The bottom right panel of Figure 17 shows $Q_{\rm H{\, \scriptsize{II}}}$ as a function of redshift, reproduced by our best-fit models. In Figure 17, we also plot $Q_{\rm H{\, \scriptsize{II}}}$ estimated from the observational results of the Lyα forest transmission (Mesinger 2010; McGreer et al. 2011), Lyα near-zone sizes around high-redshift quasars (Carilli et al. 2010; Bolton et al. 2011), Lyα damping wing absorption in a GRB spectrum (Totani et al. 2006; McQuinn et al. 2008), evolution of the Lyα luminosity function and Lyα emitter clustering (Ota et al. 2008; Ouchi et al. 2010; Konno et al. 2014), and the Lyα emitting galaxy fraction evolution (Pentericci et al. 2011; Schenker et al. 2012; Ono et al. 2012). Because these measurements have uncertainties too large to constrain our model parameters, we do not use these measurements for our model fitting. However, Figure 17 illustrates that our best-fit models are in good agreement with most of the $Q_{\rm H{\, \scriptsize{II}}}$ measurements.