THE SIZES OF z ∼ 6–8 LENSED GALAXIES FROM THE HUBBLE FRONTIER FIELDS ABELL 2744 DATA

We measure sizes for a bright subset of i₈₁₄-dropout and Y₁₀₅-dropout galaxies in Ishigaki et al. (2015), which are constructed from the version 1.0 release data of Epoch 1 and Epoch 2 of the Abell 2744 main cluster and parallel fields. We use the 30 mas pixel⁻¹ image mosaics with the standard correction. For NIR images of the parallel field, the mosaics corrected for time-variable background sky emission are used. See Ishigaki et al. (2015) for details of the sample construction. Briefly, $z\sim 6$–7 galaxies, or i₈₁₄ dropouts, are selected by

$$\begin{eqnarray*}\begin{array}{ccccccccccccccc} {{i}_{814}}-{{Y}_{105}} & \gt & 0.8, \\ {{Y}_{105}}-{{J}_{125}} & \lt & 0.8, \\ {{i}_{814}}-{{Y}_{105}} & \gt & 2({{Y}_{105}}-{{J}_{125}})+0.6, \\ {\rm S}/{\rm N}({{Y}_{105}}) & \gt & 5.0,{\rm S}/{\rm N}({{J}_{125}})\gt 5.0, \\ {\rm S}/{\rm N}({{B}_{435}}) & \lt & 2.0,{\rm S}/{\rm N}({{V}_{606}})\lt 2.0, \\ \end{array}\end{eqnarray*}$$

and $z\sim 8$ galaxies, or Y₁₀₅ dropouts, are selected by

$$\begin{eqnarray*}\begin{array}{ccccccccccccccc} {{Y}_{105}}-{{J}_{125}} & \gt & 0.5, \\ {{J}_{125}}-{{H}_{160}} & \lt & 0.4, \\ {\rm S}/{\rm N}({{J}_{125}}) & \gt & 3.5,{\rm S}/{\rm N}(J{{H}_{140}})\gt 3.5, \\ {\rm S}/{\rm N}({{B}_{435}}) & \lt & 2.0,{\rm S}/{\rm N}({{V}_{606}})\lt 2.0, \\ {\rm S}/{\rm N}({{i}_{814}}) & \lt & 2.0. \\ \end{array}\end{eqnarray*}$$

As a result, 18 i₈₁₄ dropouts and 11 Y₁₀₅ dropouts are selected in the main cluster field and 17 i₈₁₄ dropouts and 4 Y₁₀₅ dropouts in the parallel field. It is worth noting that most of the Y₁₀₅ dropouts in the cluster field are highly clustered in a small region. If this overdense region is a forming cluster, they could have some different properties from galaxies in the average field. This overdensity has been reported by some previous studies (e.g., Zheng et al. 2014; Ishigaki et al. 2015).

Galaxy size measurements require high signal-to-noise ratios (S/N). To maximize the number of galaxies for size measurements, we coadd the Y₁₀₅, J₁₂₅, and JH₁₄₀ images to make a deep UV-continuum image for the size analysis of i₈₁₄ dropouts, and coadd the J₁₂₅, JH₁₄₀, and H₁₆₀ images for Y₁₀₅ dropouts. In this process, we use ${\tt sWarp}$ (Bertin et al. 2002) and exploit the inverse variance weight images in the release data set, considering differences in exposure time and zero point properly. The $5\sigma$ limiting magnitudes of the coadded images are 29.27 (Y₁₀₅ + J₁₂₅ + JH₁₄₀) and 29.10 (J₁₂₅ + JH₁₄₀ + H₁₆₀) for the main cluster field and 29.36 (Y₁₀₅ + J₁₂₅ + JH₁₄₀) and 29.07 (J₁₂₅ + JH₁₄₀ + H₁₆₀) for the parallel field.

We then run ${\tt SExtractor}$ (Bertin & Arnouts 1996) on the coadded images, and select galaxies brighter than the $10\sigma$ limiting magnitudes for size measurements. After removing a small number of objects falling in the region of bright intracluster light unsuitable for reliable size measurement, we are left with 31 $z\sim 6$–7 galaxies and 8 $z\sim 8$ galaxies as given in Tables 1 and 2. One of the $z\sim 6$–7 galaxies is as faint as ${{M}_{{\rm UV}}}\simeq -16.6$ with a high magnification of 11.4, where ${{M}_{{\rm UV}}}$ is the rest-frame UV (${{\lambda }_{{\rm rest}}}\simeq 1500\ \overset{\circ}{\rm A}$) absolute magnitude. This is the faintest $z\gtrsim 6$ galaxy whose size has ever been measured. Ishigaki et al. (2015) have also selected six $z\sim 9$ galaxies, but none of them is included in our analysis because they are either too faint or too close to a bright star.

Table 1.  Dropout Candidates at z ∼ 6–7 in the Abell 2744 Field

ID	R.A.a	Decl.a	${{i}_{814}}-{{Y}_{105}}$	${{Y}_{105}}-{{J}_{125}}$	J125b	${{J}_{125}}-{{H}_{160}}$	Magnificationc, d	Photo-z	Referencee
Cluster Field
HFF1C-i1	3.593804	−30.415447	$\gt 2.37$	0.05 ± 0.07	26.10 ± 0.05	$-0.17_{-0.05}^{+0.05}$	$3.73_{-0.21}^{+0.24}$	6.6 ± 0.8	1, 3
HFF1C-i2	3.570654	−30.414659	1.33 ± 0.12	0.11 ± 0.05	26.21 ± 0.03	$-0.08_{-0.06}^{+0.06}$	$1.62_{-0.06}^{+0.06}$	6.0 ± 0.7	1, 3
HFF1C-i3	3.606222	−30.386644	1.07 ± 0.09	0.09 ± 0.04	26.25 ± 0.04	$0.07_{-0.06}^{+0.06}$	$1.69_{-0.05}^{+0.05}$	5.8 ± 0.7	1, 3
HFF1C-i4	3.606385	−30.407282	1.69 ± 0.21	0.00 ± 0.05	26.37 ± 0.04	$-0.33_{-0.07}^{+0.07}$	$2.25_{-0.10}^{+0.12}$	6.3 ± 0.7	1, 3
HFF1C-i5	3.580452	−30.405043	$\gt 2.10$	0.17 ± 0.07	26.60 ± 0.05	$-0.22_{-0.09}^{+0.09}$	$5.64_{-0.39}^{+0.40}$	6.8 ± 0.8	1, 3
HFF1C-i6	3.597834	−30.395961	$\gt 1.71$	0.27 ± 0.09	26.79 ± 0.07	$-0.27_{-0.10}^{+0.10}$	$2.87_{-0.19}^{+0.19}$	7.0 ± 0.8	1, 3
HFF1C-i7	3.590761	−30.379408	0.95 ± 0.13	−0.16 ± 0.09	27.06 ± 0.08	$-0.13_{-0.12}^{+0.12}$	$1.87_{-0.05}^{+0.06}$	5.9 ± 0.7	⋯
HFF1C-i9	3.601072	−30.403991	1.11 ± 0.27	0.00 ± 0.11	27.26 ± 0.09	$-0.19_{-0.15}^{+0.15}$	$3.56_{-0.23}^{+0.26}$	5.9 ± 0.7	1, 3
HFF1C-i10	3.600619	−30.410296	$\gt 1.43$	−0.06 ± 0.11	27.29 ± 0.08	$-0.45_{-0.18}^{+0.18}$	$11.43_{-1.20}^{+1.60}$	6.4 ± 0.7	3
HFF1C-i11	3.603426	−30.383219	0.87 ± 0.16	−0.13 ± 0.11	27.29 ± 0.09	$-0.15_{-0.15}^{+0.15}$	$1.71_{-0.05}^{+0.04}$	5.8 ± 0.7	3
HFF1C-i12	3.603214	−30.410350	$\gt 1.36$	−0.03 ± 0.12	27.32 ± 0.09	$0.10_{-0.14}^{+0.14}$	$3.88_{-0.21}^{+0.29}$	6.3 ± 0.7	1, 2, 3
HFF1C-i13	3.592944	−30.413328	$\gt 1.25$	−0.09 ± 0.20	27.35 ± 0.15	$-0.03_{-0.16}^{+0.16}$	$6.85_{-0.54}^{+0.60}$	6.1 ± 0.7	3
HFF1C-i14	3.585016	−30.413084	0.85 ± 0.19	−0.23 ± 0.14	27.45 ± 0.12	$0.14_{-0.16}^{+0.16}$	$2.94_{-0.17}^{+0.18}$	$5.7_{-1.1}^{+0.7}$	⋯
HFF1C-i15	3.576889	−30.386329	$\gt 0.96$	0.13 ± 0.18	27.45 ± 0.16	$-0.27_{-0.19}^{+0.19}$	$2.77_{-0.13}^{+0.15}$	$6.1_{-0.7}^{+0.8}$	3
HFF1C-i16	3.609003	−30.385283	1.35 ± 0.33	−0.07 ± 0.14	27.56 ± 0.12	$-0.12_{-0.19}^{+0.19}$	$1.59_{-0.04}^{+0.04}$	6.1 ± 0.7	3
HFF1C-i17	3.604563	−30.409364	$\gt 0.91$	0.12 ± 0.16	27.62 ± 0.11	$-0.25_{-0.22}^{+0.22}$	$2.94_{-0.16}^{+0.19}$	6.1 ± 0.8	3
Parallel Field
HFF1P-i1	3.474802	−30.362578	$\gt 1.80$	0.46 ± 0.07	26.52 ± 0.05	$-0.20_{-0.07}^{+0.07}$	1.04	7.3 ± 0.8	⋯
HFF1P-i2	3.480642	−30.371175	1.76 ± 0.34	−0.02 ± 0.09	26.95 ± 0.07	$-0.45_{-0.11}^{+0.11}$	1.05	6.3 ± 0.7	⋯
HFF1P-i3	3.487575	−30.364380	1.27 ± 0.33	0.32 ± 0.11	27.06 ± 0.08	$0.08_{-0.10}^{+0.10}$	1.05	5.8 ± 0.7	⋯
HFF1P-i4	3.488924	−30.394630	$\gt 1.39$	0.25 ± 0.11	27.14 ± 0.08	$0.03_{-0.11}^{+0.11}$	1.05	6.7 ± 0.8	⋯
HFF1P-i5	3.482550	−30.371559	1.19 ± 0.29	0.17 ± 0.11	27.15 ± 0.09	$0.23_{-0.10}^{+0.10}$	1.05	$5.8_{-1.4}^{+0.7}$	⋯
HFF1P-i6	3.483960	−30.397152	$\gt 1.57$	0.00 ± 0.11	27.20 ± 0.09	$-0.07_{-0.12}^{+0.12}$	1.05	6.3 ± 0.7	⋯
HFF1P-i7	3.467582	−30.396908	$\gt 1.39$	0.15 ± 0.12	27.23 ± 0.09	$-0.19_{-0.13}^{+0.13}$	1.04	6.8 ± 0.8	⋯
HFF1P-i8	3.467097	−30.387686	1.28 ± 0.25	−0.24 ± 0.11	27.30 ± 0.10	$-0.10_{-0.13}^{+0.13}$	1.04	6.0 ± 0.7	⋯
HFF1P-i9	3.489520	−30.399528	$\gt 1.41$	0.05 ± 0.13	27.32 ± 0.10	$-0.26_{-0.14}^{+0.14}$	1.05	6.6 ± 0.8	⋯
HFF1P-i10	3.466056	−30.394409	1.10 ± 0.30	0.00 ± 0.14	27.43 ± 0.11	$-0.10_{-0.15}^{+0.15}$	1.04	6.0 ± 0.7	⋯
HFF1P-i11	3.460587	−30.366320	0.92 ± 0.34	0.05 ± 0.18	27.70 ± 0.14	$-0.09_{-0.18}^{+0.18}$	1.04	$5.8_{-1.1}^{+0.7}$	⋯
HFF1P-i12	3.455844	−30.366359	1.03 ± 0.36	0.00 ± 0.18	27.70 ± 0.14	$0.41_{-0.16}^{+0.16}$	1.03	$4.5_{-3.9}^{+1.6}$	⋯
HFF1P-i13	3.488139	−30.367864	$\gt 0.91$	0.12 ± 0.19	27.73 ± 0.15	$0.01_{-0.18}^{+0.18}$	1.06	$5.8_{-5.2}^{+1.0}$	⋯
HFF1P-i14	3.486988	−30.399579	0.91 ± 0.33	−0.07 ± 0.18	27.75 ± 0.15	$-0.02_{-0.19}^{+0.19}$	1.05	$5.9_{-1.1}^{+0.7}$	⋯
HFF1P-i16	3.477238	−30.385998	$\gt 1.02$	−0.22 ± 0.21	27.98 ± 0.18	$0.03_{-0.22}^{+0.22}$	1.05	6.5 ± 0.7	⋯

Notes.

^aCoordinates are in J2000. ^bTotal magnitude. ^cThe magnification errors in the parallel field are less than 1%. ^dMedian value of the magnification distribution. ^eReferences: (1) Atek et al. (2014b), (2) Zheng et al. (2014), (3) Atek et al. (2014a).

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Table 2.  Dropout Candidates at $z\sim 8$ in the Abell 2744 Field

ID	R.A.a	Decl.a	${{Y}_{105}}-{{J}_{125}}$	${{J}_{125}}-{{H}_{160}}$	JH140b	$J{{H}_{140}}-{{H}_{160}}$	Magnificationc, d	Photo-z	Referencee
Cluster Field
HFF1C-Y1	3.604518	−30.380467	1.17 ± 0.06	0.04 ± 0.04	25.91 ± 0.02	$0.01_{-0.04}^{+0.04}$	$1.49_{-0.04}^{+0.04}$	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	$-0.06_{-0.08}^{+0.08}$	$1.61_{-0.05}^{+0.05}$	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	$-0.07_{-0.08}^{+0.08}$	$2.25_{-0.09}^{+0.09}$	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	$0.08_{-0.10}^{+0.10}$	$1.49_{-0.04}^{+0.04}$	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	$0.19_{-0.10}^{+0.10}$	$1.60_{-0.05}^{+0.05}$	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	$0.17_{-0.12}^{+0.12}$	$1.48_{-0.04}^{+0.04}$	$7.9_{-6.1}^{+0.9}$	2, 6
Parallel Field
HFF1P-Y1	3.474918	−30.362542	0.61 ± 0.11	0.04 ± 0.08	26.93 ± 0.07	$0.07_{-0.08}^{+0.08}$	1.05	$7.5_{-1.5}^{+0.8}$	⋯
HFF1P-Y2	3.459245	−30.367360	0.73 ± 0.18	0.02 ± 0.13	27.44 ± 0.11	$0.13_{-0.13}^{+0.13}$	1.04	$7.6_{-1.7}^{+0.8}$	⋯

Notes.

^aCoordinates are in J2000. ^bTotal magnitude. ^cThe magnification errors in the parallel field are less than 1%. ^dMedian value of the magnification distribution. ^eReferences: (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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To correct for the GL effect on our galaxies, we use the cluster mass map obtained from Ishigaki et al. (2015), in which the mass model is constructed with the pubic software ${\tt glafic}$ (Oguri 2010) using all available multiply lensed objects in the literature plus three newly identified objects. This map is found to be in good agreement with those provided by other research teams, which are posted on the public HFF website.⁶

While ${\tt GALFIT}$ is widely used for analyzing luminosity profiles of faint galaxies, in this paper we employ ${\tt glafic}$ for size measurements as well, because it enables detailed light profile fitting similar to ${\tt GALFIT}$ for lensed, hence distorted, galaxies. This package finds the best-fit Sérsic profile parameters for a given, lensed galaxy image by simulating many lensed images with different profile parameters taking account of the GL effect at the position of the galaxy and fitting with them on the observed image. That is, distorted Sérsic profiles are directly compared on the image plane. In this sense, this method is more direct than one in which observed (distorted) images are fit with undistorted profiles created with ${\tt GALFIT}$ and then the best-fit radius and magnitude are corrected for the GL effect simply by dividing by the magnification factor (e.g., Laporte et al. 2014).

In the course of profile fitting, each simulated image is convolved with the point-spread function (PSF) when compared with the observed light profile. The PSF profile for each coadded image is determined by stacking four stellar objects in the same image. In case there are nearby foreground galaxies near the target galaxy, we carefully mask these nearby galaxies during the profile fitting.

In this paper, we fix the Sérsic index to n = 1, the value widely used in previous studies (e.g., Ono et al. 2013). We use the half-light radius to express galaxy size following previous studies. We have thus six free parameters to determine: positions, half-light radius, flux, ellipticity, and position angle. The upper limit of the ellipticity is set to 0.9. For each galaxy, Sérsic profile fitting is repeated with different initial parameter sets until the best-fit parameters converge. The output fluxes (half-light radii) are converted into absolute magnitudes (radii in physical length) assuming that all the $z\sim 6$–7 and $z\sim 8$ galaxies are located at z = 6.1 and z = 8.0, respectively.

Some galaxies, e.g., CY-1, appear to consist of multiple components or contain sub-clumps. Although we treat these complex galaxies as single objects, we first fit each system with multiple Sérsic profiles simultaneously in order to obtain reliable sky background estimates for the field. This is because the sky background value obtained by this multi-component fitting is more stable and robust than single profile fitting of these complex systems. We then perform single profile fitting with the fixed sky background value derived above. Results of light profile fitting for all dropout galaxies are shown in Figures A1–A6 in the Appendix.

There are four very small (${{r}_{{\rm e}}}\lt 0.08$ kpc) galaxies at z ∼ 6–7: C-i10, C-i13, C-i15, and C-i17. However, we find only one (C-i10) to be actually resolved and calculate for the remaining three an upper limit to the half-light radius, from the examination below: for each object, we replace the real image with a stellar image and conduct a profile fitting in the same manner as for the real object considering the GL effect. This stellar image is taken from the same field without any luminosity correction. We repeat this five times using five different stellar images. Then, if the average of the five output half-light radii is significantly smaller than the half-light radius of the real image, we consider this galaxy to be resolved. If not, we consider it to be unresolved and adopt the average stellar half-light radius for the upper limit. We also confirm that C-i16 is resolved.

We note that C-i5 and C-i6 are lensed images of the same galaxy. We adopt the average values of these two images for the parameters of this galaxy.

There are three main sources of errors in our size and magnitude measurements: statistical and systematic errors in light profile fitting, internal statistical errors in lensing magnification and distortion of our mass map, and external systematic errors in lensing effects coming from different assumptions made in creating mass maps. In the HFF project, the last source of errors can be estimated by comparing lensing properties of eight public mass maps that are constructed independently (see Figure 11 of Ishigaki et al. 2015). These mass maps enable us to discuss the validity of our mass map and to estimate systematic errors caused by differences in the method of mass map construction.

The largest among the three is the error in light profile fitting. It is well known that the size and brightness of very faint galaxies such as those in our sample tend to be biased depending on their size and magnitude even when they are measured from careful profile fitting. For each galaxy, we correct for these systematic errors with the following Monte Carlo approach. (1) We generate a model galaxy image by randomly changing Sérsic parameter values around the best-fit values, and put it randomly near the observed position of the galaxy if it is in the main cluster field, where the GL effect varies from position to position, or randomly in the entire image if it is in the parallel field, where the GL effect is small and nearly constant. In this process, we do not change the GL effect, which depends on its position, because the purpose here is to estimate only photometric errors. (2) We measure its size and magnitude in exactly the same manner as for the observed galaxy. (3) We repeat (1) and (2) to make a table of 100–300 input and measured sizes and magnitudes. (4) From (3), we extract input half-light radii whose measured radii and magnitudes are virtually equal to that of the observed galaxy to create their histogram. The median and the standard deviation of the histogram are adopted as the true radius and its error of the galaxy, respectively. Its true magnitude and error are determined in a similar manner. As an example, the results of the simulations for C-Y1 (the most precise case) and P-i16 (the least precise case) are shown in Figure 1.

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Errors in the mass map affect size and magnitude measurements primarily through changes in magnification. Ishigaki et al. (2015) have found that the change in magnification due to the internal errors in our mass map is negligibly small, ∼5%, for the main cluster field and less than 1% for the parallel field, as given in Tables 3 and 4.

Table 3.  Fitting Results for Dropouts at $z\sim 6{\rm --}7$

IDa	$m_{{\rm UV}}^{{\rm AUTO}}$ b	n	mUVc	MUV	red	e	Magnificatione, f
	(mag)		(mag)	(mag)	(kpc)
Cluster Field
HFF1C-i1	$26.20\pm 0.01$	1.0	$27.68_{-0.05}^{+0.04}$	$-19.05_{-0.05}^{+0.04}$	$0.16_{-0.02}^{+0.02}$	0.31	$3.73_{-0.21-0.55}^{+0.24+0.65}$
HFF1C-i2	$26.23\pm 0.01$	1.0	$26.79_{-0.05}^{+0.04}$	$-19.93_{-0.05}^{+0.04}$	$0.15_{-0.03}^{+0.03}$	0.90	$1.62_{-0.06-0.14}^{+0.06+0.15}$
HFF1C-i3	$26.34\pm 0.01$	1.0	$26.76_{-0.05}^{+0.04}$	$-19.96_{-0.05}^{+0.04}$	$0.41_{-0.02}^{+0.02}$	0.19	$1.69_{-0.05-0.18}^{+0.05+0.33}$
HFF1C-i4	$26.75\pm 0.02$	1.0	$27.45_{-0.05}^{+0.04}$	$-19.27_{-0.05}^{+0.04}$	$0.14_{-0.02}^{+0.02}$	0.37	$2.25_{-0.10-0.18}^{+0.12+1.01}$
HFF1C-i5	$26.71\pm 0.02$	1.0	$28.71_{-0.06}^{+0.06}$	$-18.01_{-0.06}^{+0.06}$	$0.14_{-0.02}^{+0.02}$	0.42	$5.64_{-0.39-0.69}^{+0.40+0.11}$
HFF1C-i6	$27.06\pm 0.02$	1.0	$28.25_{-0.07}^{+0.08}$	$-18.47_{-0.07}^{+0.08}$	$0.09_{-0.03}^{+0.02}$	0.65	$2.87_{-0.19-0.09}^{+0.19+0.73}$
$^{*}$HFF1C-i7	$26.40\pm 0.01$	1.0	$27.13_{-0.15}^{+0.09}$	$-19.59_{-0.15}^{+0.09}$	$0.73_{-0.07}^{+0.10}$	0.65	$1.87_{-0.05-0.07}^{+0.06+0.80}$
$^{*}$HFF1C- i9	$27.14\pm 0.02$	1.0	$28.00_{-0.17}^{+0.10}$	$-18.72_{-0.17}^{+0.10}$	$0.55_{-0.06}^{+0.07}$	0.54	$3.56_{-0.23-0.43}^{+0.26+0.58}$
HFF1C-i10	$27.83\pm 0.04$	1.0	$30.07_{-0.09}^{+0.09}$	$-16.65_{-0.09}^{+0.09}$	$0.03_{-0.01}^{+0.01}$	0.85	$11.4_{-1.20-2.55}^{+1.60+0.06}$
HFF1C-i11	$27.71\pm 0.04$	1.0	$28.00_{-0.07}^{+0.08}$	$-18.72_{-0.07}^{+0.08}$	$0.15_{-0.04}^{+0.03}$	0.40	$1.71_{-0.05-0.24}^{+0.04+0.46}$
HFF1C-i12	$26.44\pm 0.01$	1.0	$27.91_{-0.11}^{+0.11}$	$-18.81_{-0.11}^{+0.11}$	$0.57_{-0.06}^{+0.05}$	0.47	$3.88_{-0.21-0.26}^{+0.29+5.12}$
HFF1C-i13	$27.69\pm 0.04$	1.0	$29.58_{-0.07}^{+0.06}$	$-17.14_{-0.07}^{+0.06}$	$\lt 0.04$	0.78	$6.85_{-0.54-1.31}^{+0.60+1.65}$
HFF1C-i14	$26.65\pm 0.01$	1.0	$28.53_{-0.14}^{+0.12}$	$-18.19_{-0.14}^{+0.12}$	$0.23_{-0.07}^{+0.07}$	0.51	$2.94_{-0.17-0.16}^{+0.18+0.56}$
HFF1C-i15	$27.61\pm 0.03$	1.0	$28.96_{-0.11}^{+0.09}$	$-17.76_{-0.11}^{+0.09}$	$\lt 0.06$	0.85	$2.77_{-0.13-0.00}^{+0.15+4.65}$
HFF1C-i16	$27.82\pm 0.04$	1.0	$28.37_{-0.19}^{+0.09}$	$-18.35_{-0.19}^{+0.09}$	$0.08_{-0.04}^{+0.04}$	0.90	$1.59_{-0.04-0.16}^{+0.04+0.61}$
HFF1C-i17	$27.89\pm 0.04$	1.0	$29.19_{-0.11}^{+0.09}$	$-17.53_{-0.11}^{+0.09}$	$\lt 0.08$	0.89	$2.94_{-0.16-0.23}^{+0.19+3.70}$
Parallel Field
$^{*}$HFF1P-i1	$26.36\pm 0.01$	1.0	$26.75_{-0.07}^{+0.05}$	$-19.97_{-0.07}^{+0.05}$	$0.52_{-0.06}^{+0.05}$	0.68	1.04
HFF1P-i2	$27.28\pm 0.03$	1.0	$27.20_{-0.07}^{+0.08}$	$-19.52_{-0.07}^{+0.08}$	$0.40_{-0.05}^{+0.04}$	0.42	1.05
HFF1P-i3	$26.87\pm 0.02$	1.0	$26.92_{-0.10}^{+0.10}$	$-19.80_{-0.10}^{+0.10}$	$0.83_{-0.07}^{+0.09}$	0.27	1.05
HFF1P-i4	$26.91\pm 0.02$	1.0	$27.35_{-0.11}^{+0.12}$	$-19.37_{-0.11}^{+0.12}$	$0.60_{-0.10}^{+0.07}$	0.55	1.05
HFF1P-i5	$27.43\pm 0.03$	1.0	$27.19_{-0.09}^{+0.09}$	$-19.53_{-0.09}^{+0.09}$	$0.64_{-0.06}^{+0.07}$	0.21	1.05
HFF1P-i6	$27.04\pm 0.02$	1.0	$27.21_{-0.11}^{+0.11}$	$-19.51_{-0.11}^{+0.11}$	$0.60_{-0.11}^{+0.09}$	0.70	1.05
HFF1P-i7	$27.23\pm 0.02$	1.0	$27.06_{-0.12}^{+0.12}$	$-19.66_{-0.12}^{+0.12}$	$0.81_{-0.10}^{+0.08}$	0.50	1.04
HFF1P-i8	$27.34\pm 0.03$	1.0	$27.26_{-0.10}^{+0.10}$	$-19.46_{-0.10}^{+0.10}$	$0.56_{-0.08}^{+0.07}$	0.44	1.04
HFF1P-i9	$27.37\pm 0.03$	1.0	$27.54_{-0.12}^{+0.11}$	$-19.18_{-0.12}^{+0.11}$	$0.41_{-0.09}^{+0.08}$	0.57	1.05
HFF1P-i10	$27.70\pm 0.04$	1.0	$27.45_{-0.14}^{+0.12}$	$-19.27_{-0.14}^{+0.12}$	$0.45_{-0.11}^{+0.10}$	0.74	1.04
HFF1P-i11	$26.80\pm 0.02$	1.0	$28.35_{-0.10}^{+0.12}$	$-18.37_{-0.10}^{+0.12}$	$0.12_{-0.05}^{+0.07}$	0.90	1.04
HFF1P-i12	$27.62\pm 0.03$	1.0	$27.58_{-0.18}^{+0.23}$	$-19.14_{-0.18}^{+0.23}$	$0.80_{-0.18}^{+0.13}$	0.45	1.03
HFF1P-i13	$27.45\pm 0.03$	1.0	$27.39_{-0.18}^{+0.20}$	$-19.33_{-0.18}^{+0.20}$	$1.04_{-0.17}^{+0.18}$	0.31	1.06
HFF1P-i14	$27.67\pm 0.04$	1.0	$27.22_{-0.16}^{+0.15}$	$-19.50_{-0.16}^{+0.15}$	$1.00_{-0.14}^{+0.16}$	0.26	1.05
HFF1P-i16	$27.83\pm 0.04$	1.0	$28.07_{-0.19}^{+0.17}$	$-18.65_{-0.19}^{+0.17}$	$0.34_{-0.13}^{+0.14}$	0.90	1.05

Notes.

^aAsterisks indicate galaxies with multiple cores. ^b MAG_AUTO magnitude from SExtractor. ^cTotal apparent magnitude from light profile fitting with glafic. ^dCircularized effective radius, $r_{{\rm e}}^{{\rm maj}}\sqrt{1-e}$, where $r_{{\rm e}}^{{\rm maj}}$ is the radius along the major axis and e the ellipticity. ^eBest-fit value of magnification. ^fThe first error shows the error in our mass map, and the second error shows the variation in magnification among the other eight public mass maps.

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On the other hand, there is a significant variation in magnification among the eight public mass maps (see Figure 11 of Ishigaki et al. 2015). We estimate this systematic error at the position of each dropout by calculating the standard deviation of the eight magnification values excluding the highest and lowest ones. The errors obtained are shown in Tables 3 and 4. The error bars plotted in Figure 2 are a quadratic sum of this systematic error and the error in profile fitting.

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Table 4.  Fitting Results for Dropouts at $z\sim 8$

IDa	$m_{{\rm UV}}^{{\rm AUTO}}$ b	n	mUVc	MUV	red	e	Magnificatione, f
	(mag)		(mag)	(mag)	(kpc)
Cluster Field
$^{*}$HFF1C-Y1	$25.97\pm 0.02$	1.0	$26.20_{-0.03}^{+0.03}$	$-20.95_{-0.03}^{+0.03}$	$0.24_{-0.02}^{+0.02}$	0.70	$1.49_{-0.04-0.11}^{+0.04+0.56}$
$^{*}$HFF1C-Y2	$25.99\pm 0.02$	1.0	$26.48_{-0.13}^{+0.05}$	$-20.67_{-0.13}^{+0.05}$	$0.62_{-0.05}^{+0.05}$	0.60	$1.61_{-0.05-0.17}^{+0.05+0.55}$
HFF1C-Y3	$26.58\pm 0.03$	1.0	$27.13_{-0.10}^{+0.06}$	$-20.01_{-0.10}^{+0.06}$	$0.38_{-0.03}^{+0.03}$	0.20	$2.25_{-0.09-0.26}^{+0.09+3.06}$
HFF1C-Y4	$26.66\pm 0.03$	1.0	$27.27_{-0.12}^{+0.09}$	$-19.88_{-0.12}^{+0.09}$	$0.27_{-0.05}^{+0.06}$	0.65	$1.49_{-0.04-0.11}^{+0.04+0.62}$
$^{*}$HFF1C-Y5	$26.16\pm 0.02$	1.0	$26.61_{-0.13}^{+0.10}$	$-20.54_{-0.13}^{+0.10}$	$0.84_{-0.09}^{+0.08}$	0.46	$1.60_{-0.05-0.17}^{+0.05+0.54}$
HFF1C-Y6	$26.46\pm 0.02$	1.0	$26.59_{-0.20}^{+0.15}$	$-20.55_{-0.20}^{+0.15}$	$0.92_{-0.10}^{+0.14}$	0.44	$1.48_{-0.04-0.11}^{+0.04+0.63}$
Parallel Field
$^{*}$HFF1P-Y1	$27.15\pm 0.03$	1.0	$27.14_{-0.09}^{+0.09}$	$-20.00_{-0.09}^{+0.09}$	$0.24_{-0.07}^{+0.05}$	0.65	1.05
HFF1P-Y2	$27.07\pm 0.03$	1.0	$27.48_{-0.14}^{+0.11}$	$-19.66_{-0.14}^{+0.11}$	$0.21_{-0.07}^{+0.07}$	0.90	1.04

Notes.

^aAsterisks indicate galaxies with multiple cores. ^b MAG_AUTO magnitude from SExtractor. ^cTotal apparent magnitude from light profile fitting with glafic. ^dCircularized effective radius, $r_{{\rm e}}^{{\rm maj}}\sqrt{1-e}$, where $r_{{\rm e}}^{{\rm maj}}$ is the radius along the major axis and e the ellipticity. ^eBest-fit value of magnification. ^fThe first error shows the error in our mass map, and the second error shows the variation in magnification among the other eight public mass maps.

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The best-fit values and errors for all the dropout galaxies are summarized in Tables 3 and 4.

Figure 2 shows the distribution of our galaxies in the ${{r}_{{\rm e}}}$-${{M}_{{\rm UV}}}$ plane for the two redshift ranges. Also plotted are the galaxies selected in Ono et al. (2013) who investigated the size distribution of HUDF 12 galaxies over similar redshift ranges using GALFIT. Because of the difference in the dropout band (i₈₁₄ versus z₈₅₀), the redshift distribution of our $z\sim 6$–7 galaxies is somewhat different from that of the HUDF 12 galaxies that are distributed around $z\approx 7$ (Figure 3). However, since there seems to be little size evolution between $z\sim 6$ and $z\sim 7$ (see Figure 6), we merge these two samples into one to improve the statistics.

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The merged sample is less affected by cosmic variance than the HUDF 12 sample. Calculation based on Robertson et al. (2014) finds that at $z\sim 7$ the cosmic variance in the number density of ${{M}_{{\rm UV}}}=-20$ galaxies reduces from 50% to 30% by adding our sample to the HUDF 12 one. Similarly, a reduction from 60 to 40% is expected at $z\sim 8$.

Similar to Ono et al. (2013), we find a positive correlation between half-light radius and luminosity in our sample for both redshift ranges. The average size of our galaxies as a function of luminosity is also in a rough agreement with the result of Ono et al. (2013), once the statistical uncertainties are taken into account. However, at both redshift ranges, the correlation seen in our sample is much weaker than that of Ono et al. (2013). For example, around ${{M}_{{\rm UV}}}\sim -19$ to −20, the sizes of our $z\sim 6$–7 galaxies are distributed over 0.1–1.0 kpc. This is perhaps a consequence of the much larger sample size and less cosmic variance of our sample as compared with the sample of Ono et al. (2013).

Huang et al. (2013) have also found a large scatter in size for dropout galaxies at $z\sim 4$ and 5. A relatively weak correlation between size and luminosity may be common at high redshifts.

Our sample contains two very bright and compact galaxies, C-Y1 at $z\sim 8$ and C-i2 at $z\sim 6$–7. Both galaxies have also been identified in previous papers analyzing the HFF data. C-Y1 has been reported by Laporte et al. (2014) to have ${{r}_{{\rm e}}}=0.35\pm 0.15\ {\rm kpc}$ and ${{M}_{{\rm UV}}}=-20.88\pm 0.04$, consistent with our measurements. C-i2 has been identified by Atek et al. (2014b) with ${{M}_{{\rm UV}}}=-20.45\pm 0.03$, also consistent with our value.

There is no faint and large galaxy found in our sample. At $z\sim 8$, it is unlikely that we are missing a significant fraction of such galaxies due to selection bias because our galaxies are distributed in the ${{r}_{{\rm e}}}$–${{M}_{{\rm UV}}}$ plane well separate from the boundary of 50% detection completeness shown by the dashed and dashed-dotted lines. However, since some of our galaxies at $z\sim 6$–7 are close to the completeness lines, their size distribution can be affected by the completeness effect. These completeness lines are determined by the following process. (1) We generate model galaxies over wide magnitude and size ranges and put them randomly on the detection image. (2) We run SExtractor on this image in exactly the same manner as for the original image. (3) The detection rate is found to decrease with increasing size. The size at which the detection rate is 50% is obtained as a function of absolute magnitude.

The blue points with error bars in each panel of Figure 2 indicate the average size and the scatter in the given luminosity bin for the merged sample of this study and Ono et al. (2013). The average correlations at two redshifts agree well with each other, indicating that the size–luminosity relation does not significantly evolve from $z\sim 6$–7 to ∼8. Therefore, we combine the two redshift ranges together to examine the dependence of the size–luminosity relation on two other galaxy properties, UV color and multiplicity.

UV Color. Figure 4 plots half-light radii against UV luminosities for all the galaxies in the combined sample, colored according to the UV slope β, which is defined as ${{f}_{\lambda }}\propto {{\lambda }^{\beta }}$ with ${{f}_{\lambda }}$ being the UV-continuum flux density with respect to wavelength λ. We calculate β using the equations given in Bouwens et al. (2013): $\beta =-2.0+4.39({{J}_{125}}-{{H}_{160}})$ for $z\sim 6$–7 and $\beta =-2.0+8.98(J{{H}_{140}}-{{H}_{160}})$ for $z\sim 8$. For each redshift, the range of β in our sample is consistent with those of Bouwens et al. (2013) and Dunlop et al. (2013).

We find that largest ($\gt 0.8$ kpc) galaxies tend to be bright (${{M}_{{\rm UV}}}\lesssim -19.5$) while the remaining galaxies have a wide range of luminosity with a weak correlation with size, as already seen in each panel of Figure 2 with smaller statistics. We also find that the largest ($\gt 0.8$ kpc) galaxies are mostly red and the smallest ($\lt 0.08$ kpc) galaxies are mostly blue, while the remaining do not show a very strong trend. There are mainly three factors that make galaxies red: high dust extinction, old age, and high metallicity. Since these three characteristics are often seen in evolved galaxies, selecting largest galaxies may lead to effectively picking out evolved galaxies at the redshifts studied here.

We note that the faintest galaxy in the sample, C-i10, with ${{M}_{{\rm UV}}}=-16.6$, is also the bluest ($\beta =-4.00$) and smallest (${{r}_{{\rm e}}}=0.03\ {\rm kpc}$) object. This interesting galaxy, detected thanks to the high lensing magnification of $\mu \simeq 11$, may be a very young galaxy with an extremely low metallicity. This galaxy has also been reported by Atek et al. (2014a) to have ${{M}_{{\rm UV}}}=-15.80\pm 0.16$.

Multiplicity. We also examine whether galaxies with multiple cores have any preference in size or luminosity. Multiple cores can be regarded as a sign of a recent merging event. Many papers have estimated the fraction of galaxies with multiple cores at high redshift (e.g., Ravindranath et al. 2006; Lotz et al. 2008; Oesch et al. 2010; Guo et al. 2012; Law et al. 2012; Jiang et al. 2013). For example, Ravindranath et al. (2006) have reported that 30% of $z\sim 3$ LBGs have multiple cores, and Jiang et al. (2013) have found that 40%–50% of bright (${{M}_{{\rm UV}}}\leqslant -20.5$) galaxies at $5.7\leqslant z\leqslant 7.0$ have multiple cores.

We identify galaxies with multiple cores in our sample by visual inspection, considering the claim by Jiang et al. (2013) that while galaxies at $z\gtrsim 6$ are too small and faint for quantitative morphological analysis, visual inspection is still valid for examining whether or not a galaxy has multiple cores. Galaxies with multiple cores are marked with a large square in Figure 4, and marked with a star in Figures A1–A6. Ten galaxies, or 19% of the sample, are found to have multiple cores. This fraction is similar to that derived by Oesch et al. (2010) at similar redshifts. As seen in the galaxy images summarized in the Appendix, for most of the galaxies with multiple cores, the primary cores are distinct compared to the secondary or later cores, which perhaps implies relatively minor mergers.

As can be seen from Figure 4, most of the galaxies with multiple cores are bright (${{M}_{{\rm UV}}}\lesssim -20$), qualitatively consistent with the trend seen in the sample of Oesch et al. (2010) that brighter galaxies tend to have multiple cores. More specifically, in the sample of Oesch et al. (2010), the brightest and fourth-brightest galaxies have multiple cores among the 16 $z\sim 7$ galaxies. In our sample, three of the four brightest galaxies (${{M}_{{\rm UV}}}\leqslant -20.5$) at $z\sim 8$ have multiple cores. On the other hand, we find that the sizes of galaxies with multiple cores are distributed widely from 0.2 to 1 kpc.

At $z\sim 6$–7, the bright galaxies ($-20\leqslant {{M}_{{\rm UV}}}\leqslant -19.0$) from the cluster field are on average smaller than those from the parallel field. We find about a factor of two difference in the average size of those galaxies ($0.32\pm 0.23$ kpc for the cluster field and $0.67\pm 0.20$ kpc for the parallel field). This discrepancy might be caused by sample variance and/or cosmic variance.

Figure 5 shows the average half-light radius as a function of UV luminosity for $2.5\leqslant z\lesssim 9$–10 LBGs and for $z\sim 0$ spirals and $z\sim 0.5$ irregulars for comparison. Galaxies from our merged sample are plotted as orange ($z\sim 6$–7) and red ($z\sim 8$) filled circles. Huang et al. (2013), Jiang et al. (2013), Ono et al. (2013), and Holwerda et al. (2014) have used GALFIT to measure sizes and luminosities, while Bouwens et al. (2004) have used half-light radii based on Kron-style magnitudes, and Oesch et al. (2010) and Grazian et al. (2012) based on SExtractor.

From this figure, we find that the average size around ${{M}_{{\rm UV}}}=-20.4$ gradually becomes smaller with redshift from $z\sim 2.5$ to $z\sim 7$ but the evolution from $z\sim 7$ to $z\sim 9$–10 is not significant. The slopes of the size–luminosity relation for $z\sim 6$–8 galaxies seem to be steeper than those for $z\sim 4$–5 galaxies, although the statistical uncertainty is still large. This may indicate that fainter, or less massive, galaxies grow in size more rapidly over $z\sim 4$–8. It is worth noting that among the two local ($z\lesssim 0.5$) galaxy populations, irregular galaxies have a steep slope similarly to those of $z\sim 6$–8 galaxies.

Plotted in Figure 6 is the average half-light radius of bright ($(0.3{\rm --}1)L_{z=3}^{*}$) galaxies as a function of redshift. In the calculation of the average radii for the merged samples of this work and Ono et al. (2013), we reduce the weight of the samples from the cluster field according to the uncertainty in the mass model. For the cluster-field sample at each redshift, the uncertainty in magnification is calculated from the average of the variances in magnification among the eight public mass maps at the positions of the sample galaxies. This uncertainty is then converted into the uncertainty in radius and is quadratically added to the statistical error in the average radius for this sample, thus resulting in a reduction of the weight compared with the parallel-field and Ono et al.'s (2013) samples for which only the statistical error is considered. We include the $z\sim 12$ object given in Ono et al. (2013). The data of Ferguson et al. (2004) are not included because their sample includes brighter ($\lt 5L_{z=3}^{*}$) galaxies. We find that the average size of bright galaxies decreases from $z\sim 2.5$ to $6$–7, in agreement with previous results (e.g., Bouwens et al. 2004; Hathi et al. 2008; Oesch et al. 2010; Grazian et al. 2012; Huang et al. 2013; Ono et al. 2013), while the evolution between $z\sim 6$–7 and ∼8 is insignificant. We note that the average radius of our z ∼ 6–7 combined sample may be underestimated because all but one are in the range of (0.3–0.5) L^*_z = 3.

The small open circles in Figure 6 indicate the mode of the log-normal distribution of ${{r}_{{\rm e}}}$ for $z\sim 4$–8 LBGs over the same luminosity range obtained by Curtis-Lake et al. (2014), who have adopted a non-parametric, curve-of-growth method to measure sizes. Their values are slightly but systematically higher than the other measurements except for $z\sim 4$, leading them to conclude that the typical galaxy size does not significantly evolve over $z\sim 4$–8. The reason for this systematic difference is not clear, although we find in our $z\sim 6$–7 sample that adopting the mode instead of the average results in a 0.15 kpc decrease. Our $z\sim 8$ sample is too small to calculate a modal value.

Fitting the size evolution of ${{r}_{{\rm e}}}\propto {{\left( 1+z \right)}^{-m}}$ to the data except those of Curtis-Lake et al. (2014) who adopted the modal values gives $m=1.24\pm 0.1$, which is consistent with previous results based on average r_e measurements (Oesch et al. 2010; Ono et al. 2013). Analytic models of dark-halo evolution predict that the virial radius scales with redshift as ${{\left( 1+z \right)}^{-1}}$ for halos with a fixed mass and as ${{\left( 1+z \right)}^{-1.5}}$ for halos with a fixed circular velocity (e.g., Ferguson et al. 2004). The value we find, $m=1.24$, is in the middle of these two cases. However, any previous attempts to link galaxies to dark matter halos using an observed redshift scaling of half-light radius have implicitly made a non-trivial assumption that half-light radius linearly scales with virial radius.

In order to obtain further insights into disk evolution in dark matter halos, we take a different approach. We combine the so-called abundance matching analysis that connects the stellar mass and halo mass with the observed relation between stellar mass and luminosity. Specifically, we adopt the abundance matching result of Behroozi et al. (2013), and the observed stellar mass–luminosity relations of Reddy & Steidel (2009) for $z\sim 2.5$ galaxies and González et al. (2011) for $z\sim 4$–7 galaxies⁷ to calculate the halo mass of galaxies in Figure 6 from their UV luminosity. The estimated halo mass of ${{M}_{{\rm UV}}}=-20.2$ galaxies at z = 6.0 is ${\rm log} ({{M}_{h}}/{{M}_{\odot }})=11.2$, in good accordance with the recent clustering result, ${\rm log} ({{M}_{h}}/{{M}_{\odot }})=11.0_{-0.6}^{+0.4}$, by Barone-Nugent et al. (2014). Then, the virial radius is calculated by

$$\begin{eqnarray}&&{{r}_{{\rm vir}}}={{\left( \frac{2{\rm G}{{{\rm M}}_{{\rm vir}}}}{{{{\Delta }}_{{\rm vir}}}{{{\Omega }}_{m}}(z)H{{(z)}^{2}}} \right)}^{1/3}},\end{eqnarray} \tag{ 1 }$$

where ${{{\Delta }}_{{\rm vir}}}\approx (18{{\pi }^{2}}+82x-39{{x}^{2}})/{{{\Omega }}_{m}}(z)$ and $x={{{\Omega }}_{m}}(z)-1$ (Bryan & Norman 1998). The uncertainties in the estimation of virial radii are not considered here. We exclude the $z\sim 12$ data because the analysis result of Behroozi et al. (2013) does not extend to that redshift. We note that Kravtsov (2013) have conducted a similar analysis for local galaxies, and have found a linear relation between half-mass radius and virial radius over eight orders of magnitude in stellar mass. Our analysis represents the first analysis of the evolution of the relation of the galaxy and halo sizes over a wide redshift range based on the abundance matching technique.

Figure 7 shows the ratio of half-light radius to virial radius for galaxies over $z\sim 2.5$–9.5. When limited to the data of average r_e measurements, we find the ratio to be virtually constant at 3.3 ± 0.1% over the entire redshift range. The modal data of Curtis-Lake et al. (2014) give systematically higher ratios over z ∼ 5–8, perhaps showing a slight decrease toward $z\sim 4$, but the differences from $3.3\%$ are mostly within the $1{\rm --}2\sigma$ errors. Thus, the assumption of a constant ${{r}_{{\rm e}}}/{{r}_{{\rm vir}}}$ ratio appears to be broadly consistent with the data. Our analysis shows that the halo mass of $(0.3{\rm --}5)L_{z=3}^{*}$ galaxies mildly decreases with redshift. This, combined with the constant ${{r}_{{\rm e}}}/{{r}_{{\rm vir}}}$ ratio found here, results in the redshift evolution of $1\lt m\lt 1.5$.

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According to the disk formation model by Mo et al. (1998), ${{r}_{{\rm e}}}/{{r}_{{\rm vir}}}$ is described as

$$\begin{eqnarray}&&\frac{{{r}_{{\rm e}}}}{{{r}_{{\rm vir}}}}=\frac{1.678}{\sqrt{2}}\left( \frac{{{j}_{{\rm d}}}}{{{m}_{{\rm d}}}}\lambda \right)f_{{\rm c}}^{-1/2}{{f}_{{\rm R}}}\end{eqnarray} \tag{ 2 }$$

$$\begin{eqnarray}&&\equiv \;\frac{1.678}{\sqrt{2}}\lambda ^{\prime} ,\end{eqnarray} \tag{ 3 }$$

for a self-gravitating disk embedded in an NFW dark matter halo, where ${{r}_{{\rm e}}}=1.678{{R}_{{\rm d}}}$ (R_d is the scale length of the exponential disk), m_d and j_d are, respectively, the fractions of the mass and angular momentum within the halo belonging to the disk, λ is the spin parameter of the halo, f_c is a function of the halo concentration ${{c}_{{\rm vir}}}$, and f_R is a function concerning baryonic contraction. In the context of this model, our ratio (for average r_e) gives $\lambda ^{\prime} =3.3\%\times \sqrt{2}/1.678\simeq 0.028$.

Although λ and ${{c}_{{\rm vir}}}$ are well determined by N-body simulations (Vitvitska et al. 2002; Davis & Natarajan 2009; Prada et al. 2012), ${{j}_{{\rm d}}}$ and ${{m}_{{\rm d}}}$ are not reliably predicted because complicated baryonic precesses have to be considered. Since the size ratio, that is, $\lambda ^{\prime}$, is proportional to ${{j}_{{\rm d}}}/{{m}_{{\rm d}}}$ while, with a fixed ${{j}_{{\rm d}}}/{{m}_{{\rm d}}}$, being dependent more weakly on m_d through f_R and essentially insensitive to j_d, we here aim to find out what value of ${{j}_{{\rm d}}}/{{m}_{{\rm d}}}$ is reasonable. The shaded bands in Figure 7 are the ratios calculated from Equation (2) using the simulation results of λ by Vitvitska et al. (2002) and Davis & Natarajan (2009) and ${{c}_{{\rm vir}}}$ by Prada et al. (2012) for three ${{j}_{{\rm d}}}/{{m}_{{\rm d}}}$ values over a conservative m_d range of $0.05\leqslant {{m}_{{\rm d}}}\leqslant 0.1$. We find that the observed size ratio is consistent with the model when ${{j}_{{\rm d}}}\sim {{m}_{{\rm d}}}$ is assumed.

The ratio of $3.3\%$ is larger than that for local galaxies reported by Kravtsov (2013), $1.5\%$, implying that the ratio decreases from $z\sim 2.5$ to the present day. Indeed, Kravtsov (2013) have compared the local value with a theoretically plausible value for galaxies at the era of disk formation, $\simeq 3.2\%,$ and attributed the lower local ratio to pseudo growth of dark matter halos since disk formation. In any case, our finding of a constant ${{r}_{{\rm e}}}/{{r}_{{\rm vir}}}$ ratio of $3.3\%$ over a wide redshift range of $2.5\lesssim z\lesssim 9.5$ strongly constrains disk formation models.

The physical state of star formation of a galaxy is effectively described by the total SFR and the ${{{\Sigma }}_{{\rm SFR}}}$. While the former is just the scale of star formation, the latter corresponds to the intensity of star formation and is useful for discussing the mode of star formation. We calculate the SFR and ${{{\Sigma }}_{{\rm SFR}}}$ for our galaxies with Equation (3) from Kennicutt (1998) and Equation (4) from Ono et al. (2013), respectively:

$$\begin{eqnarray}&&\frac{SFR}{{{M}_{\odot }}\;{\rm y}{{{\rm r}}^{-1}}}=1.4\times {{10}^{-28}}\frac{{{L}_{\nu }}}{{\rm erg}\ {{{\rm s}}^{-1}}\ {\rm H}{{{\rm z}}^{-1}}}\end{eqnarray} \tag{ 4 }$$

$$\begin{eqnarray}&&{{{\Sigma }}_{{\rm SFR}}}=\frac{SFR/2}{\pi r_{{\rm e}}^{2}}.\end{eqnarray} \tag{ 5 }$$

The weighted-log-average ${{{\Sigma }}_{{\rm SFR}}}$s of $(0.3{\rm --}1)L_{z=3}^{*}$ galaxies considering the uncertainty in the mass model are $4.1\;{{M}_{\odot }}\;{\rm y}{{{\rm r}}^{-1}}\;{\rm kp}{{{\rm c}}^{-2}}$ at $z\sim 6$–7 and $5.6\;{{M}_{\odot }}\;{\rm y}{{{\rm r}}^{-1}}\;{\rm kp}{{{\rm c}}^{-2}}$ at $z\sim 8$, slightly higher than $3.5\ {{M}_{\odot }}\;{\rm y}{{{\rm r}}^{-1}}\;{\rm kp}{{{\rm c}}^{-2}}$ at $z\sim 7$ and $3.2\ {{M}_{\odot }}\;{\rm y}{{{\rm r}}^{-1}}\;{\rm kp}{{{\rm c}}^{-2}}$ at $z\sim 8$ by Ono et al. (2013).

As found from Figure 4, our galaxies are forming stars with a rate of $SFR\sim 1{\rm --}10\ {{M}_{\odot }}{\rm y}{{{\rm r}}^{-1}}$ and with a surface intensity of ${{{\Sigma }}_{{\rm SFR}}}\sim 1{\rm --}50\ {{M}_{\odot }}\;{\rm y}{{{\rm r}}^{-1}}\;{\rm kp}{{{\rm c}}^{-2}}$. This ${{{\Sigma }}_{{\rm SFR}}}$ range is slightly wider toward higher values than reported by Ono et al. (2013) based on HUDF 12, reflecting the fact that our galaxies are distributed over a wider area in the size–luminosity plane than those of Ono et al. (2013). Our sample extends especially toward smaller half-light radii.

We show in Figure 8 the distribution in the SFR–${{{\Sigma }}_{{\rm SFR}}}$ plane of our galaxies and various types of local galaxies, in order to examine in what sense the state of star formation of z ∼ 6–8 galaxies are similar or dissimilar to local ones. A comparison to normal galaxies finds that our galaxies have much higher (typically three orders of magnitude higher) ${{{\Sigma }}_{{\rm SFR}}}$s than normal galaxies in spite of having modest SFRs similar to that of the Milky Way. In other words, z ∼ 6–8 galaxies are forming stars at similar rates to local normal galaxies but in 10³ times smaller areas.

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Our galaxies are roughly comparable in ${{{\Sigma }}_{{\rm SFR}}}$ to average infrared-selected galaxies and to blue compact galaxies, while falling between these two galaxy populations in SFR. Circumnuclear regions resemble our galaxies the most. Circumnuclear regions are not the whole bodies of galaxies but starbursting rings at the center of a certain type of galaxies in which gas is effectively fed along bars. This resemblance may suggest that z ∼ 6–8 galaxies have a similar amount of cold gas of a similarly high density to that of circumnuclear regions.

Ono et al. (2013) have found that the ${{{\Sigma }}_{{\rm SFR}}}$s of z ∼ 7–8 galaxies are comparable to those of infrared-selected galaxies and circumnuclear regions. While we confirm this finding above, we also find that infrared-selected galaxies are scaled-up systems in terms of SFR.

Finally, we find that our galaxies have similar ${{{\Sigma }}_{{\rm SFR}}}$s, but slightly lower SFRs than star-forming clumps of $z\sim 2$ galaxies taken from Genzel et al. (2011; black dots in Figure 8). In this sense, z ∼ 6–8 galaxies may be considered to be scaled-down systems of clumps.

In this subsection we have neglected dust extinction. If we use UV slope to calculate the extinction at 1600 Å, A₁₆₀₀, according to Meurer et al.'s (1999) formula, we find a median ${{A}_{1600}}=1.2$ in our sample, corresponding to a factor 2.9 increase in SFR (and ${{{\Sigma }}_{{\rm SFR}}}$). Therefore, if Meurer et al.'s (1999) formula is still applicable to z ∼ 6–8 galaxies (although they could have very different stellar populations and dust properties from local starbursts), then a significant fraction of our galaxies enter the region in the SFR–${{{\Sigma }}_{{\rm SFR}}}$ plane occupied by local infrared-selected galaxies.