MORPHOLOGIES OF ∼190,000 GALAXIES AT z = 0–10 REVEALED WITH HST LEGACY DATA. I. SIZE EVOLUTION

The first sample is made of $176,152$ HST/WFC3-IR-detected galaxies with photometric redshifts (hereafter photo-z galaxies) at z = 0–6 taken from Skelton et al. (2014). These galaxies are identified in five Cosmic Assembly Near-infrared Deep Extragalactic Legacy Survey (CANDELS) fields (Grogin et al. 2011; Koekemoer et al. 2011) and detected in stacked images of the ${J}_{125},{{JH}}_{140}$, and H₁₆₀ bands of WFC3/IR, which yields roughly a stellar-mass-limited sample. The photometric properties and the results of spectral energy distribution (SED) fitting for all of the sources are summarized in Skelton et al. (2014). The HST images and catalogs are publicly released at the 3D-HST website.⁵ The catalogs include the spectroscopic redshifts on the basis of the HST/WFC3 G141 grism observation (Brammer et al. 2012). We use galaxies whose physical quantities and photometric redshifts are well derived from SED fitting (specifically, sources with use_phot $=1$ in the public catalogs). Table 2 summarizes the number of galaxies at each redshift in the photo-z galaxy sample that we use. In this paper, we assume the Salpeter (1955) initial mass function (IMF). To obtain the Salpeter IMF values of stellar masses (M_*) and star-formation rates (SFRs), we multiply the Chabrier (2003) IMF values from the Skelton et al. (2014) catalog by a factor of 1.8. We divide the sample of photo-z galaxies at z = 0–4 into two subsamples of star-forming galaxies (SFGs) and QGs by the rest-frame UVJ color criteria of Muzzin et al. (2013). Because the UVJ color criteria are not tested for $z\gt \;4$ sources, we do not apply these color criteria to the photo-z galaxies at $z\gt 4$. Muzzin et al. (2013) find that the QG fraction is small, 10%, at $z\sim 3.5$, and it is likely that a QG fraction at the early cosmic epoch of $z\gt 4$ is negligibly small, perhaps $\lt 10$%. We thus regard all of the $z\gt 4$ photo-z galaxies as SFGs. The total numbers of SFGs and QGs are 165,517 and 10,631, respectively. The H₁₆₀ magnitude at the 50% completeness is ∼26.5 mag for the photo-z galaxies in the deep CANDELS fields. The details of the completeness estimates and values are presented in Skelton et al. (2014).

Table 2.  Number of Photo-z Galaxies for Our Size Measurements

	${N}_{{\mathtt{GALFIT}}}/{N}_{\mathrm{SFG},\ \mathrm{QGs}}$
Field	$z=0-1$	$z=1-2$ b	$z=2-3$ b	$z=3-4$	$z=4-5$	$z=5-6$
(1)	(2)	(3)	(4)	(5)	(6)	(7)
SFGs, ${r}_{{\rm{e}}}^{\mathrm{Opt}}(4500-8000$Å)
HUDF 09+12	294 (397)	368 (611)	8 (157)	⋯	⋯	⋯
HUDF 09-Pa	2168 (3467)	2024 (4707)	66 (1451)	⋯	⋯	⋯
GOODS-S Deep	1753 (2793)	1402 (3847)	37 (1296)	⋯	⋯	⋯
GOODS-S Wide	2790 (4724)	2267 (6313)	66 (2518)	⋯	⋯	⋯
GOODS-N Deep	3270 (5106)	1778 (5168)	71 (2094)	⋯	⋯	⋯
GOODS-N Wide	3903 (5939)	2731 (6611)	139 (2477)	⋯	⋯	⋯
UDS	5157 (9175)	5433 (15771)	209 (6094)	⋯	⋯	⋯
AEGIS	6441 (11074)	5833 (13943)	278 (6418)	⋯	⋯	⋯
COSMOS	6856 (11385)	3594 (9754)	179 (3915)	⋯	⋯	⋯
${N}_{\mathrm{total}}(z)$	32632 (54060)	25430 (66725)	1053 (26420)	⋯	⋯	⋯
${N}_{\mathrm{total}}$	59115 (147205)
SFGs, ${r}_{{\rm{e}}}^{\mathrm{UV}}(1500-3000$Å)
HUDF 09+12	⋯	145 (611)	79 (157)	34 (69)	19 (33)	12 (26)
HUDF 09-Pa	⋯	777 (4707)	624 (1451)	432 (936)	160 (453)	102 (177)
GOODS-S Deep	⋯	776 (3847)	633 (1296)	348 (696)	101 (347)	40 (702)
GOODS-S Wide	⋯	1297 (6313)	1154 (2518)	535 (1147)	138 (487)	44 (213)
GOODS-N Deep	⋯	1235 (5168)	784 (2094)	389 (987)	154 (446)	66 (174)
GOODS-N Wide	⋯	1711 (6611)	1114 (2477)	412 (962)	165 (516)	47 (167)
UDS	⋯	2730 (15771)	1747 (6094)	678 (2266)	180 (716)	52 (176)
AEGIS	⋯	3158 (13943)	2182 (6418)	952 (2768)	228 (873)	84 (281)
COSMOS	⋯	2413 (9754)	1642 (3915)	939 (2048)	192 (765)	54 (296)
${N}_{\mathrm{total}}(z)$	⋯	14242 (66725)	9959 (26420)	4719 (11879)	1337 (4636)	501 (1796)
${N}_{\mathrm{total}}$	30765 (165517)
QGs, ${r}_{e}^{\mathrm{Opt}}(4500-8000\,{\rm \AA})$
HUDF09+12	55 (85)	17 (34)	0 (3)	⋯	⋯	⋯
HUDF09-P	262 (513)	107 (320)	1 (50)	⋯	⋯	⋯
GOODS-S Deep	190 (367)	85 (242)	1 (43)	⋯	⋯	⋯
GOODS-S Wide	324 (621)	107 (363)	4 (67)	⋯	⋯	⋯
GOODS-N Deep	246 (458)	84 (290)	2 (90)	⋯	⋯	⋯
GOODS-N Wide	242 (480)	94 (289)	2 (123)	⋯	⋯	⋯
UDS	302 (876)	214 (778)	5 (165)	⋯	⋯	⋯
AEGIS	369 (849)	260 (762)	6 (243)	⋯	⋯	⋯
COSMOS	721 (1206)	148 (490)	5 (115)	⋯	⋯	⋯
${N}_{\mathrm{total}}(z)$	2711 (5455)	1116 (3568)	26 (899)
${N}_{\mathrm{total}}$	3853 (9922)

Notes. Columns: (1) Field. (2)–(7) Number of photo-z galaxies that have S/N $\geqslant 15$ and reliable GALFIT outputs in each redshift range. The value in parentheses is the number of photo-z galaxies in the parent sample.

^aTotal number of objects in the HUDF 09-P1 and HUDF 09-P2 fields. ^bThe actual redshift range is $2\leqslant z\leqslant 2.1$ ($1.2\leqslant z\leqslant 2$) for the ${r}_{{\rm{e}}}^{\mathrm{Opt}}$ (${r}_{{\rm{e}}}^{\mathrm{UV}}$) measurement. ^cThe numbers of QGs with ${r}_{{\rm{e}}}^{\mathrm{UV}}$ are not shown here because of the rarity at $z\gtrsim 2-3$ and the UV faintness.

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The second sample consists of 10,454 LBGs at z = 4–10 made by Y. Harikane et al. (2015, in preparation) in the CANDELS, the Hubble Ultra Deep Field 09+12 (HUDF 09+12; Beckwith et al. 2006; Bouwens et al. 2011; Illingworth et al. 2013; Ellis et al. 2013) fields,⁶ and the parallel fields of Abell 2744 and MACS 0416 in the Hubble Frontier Fields (e.g., Coe et al. 2014; Ishigaki et al. 2014; Oesch et al. 2014; Atek et al. 2015). The numbers of our LBGs are summarized in Table 3. These LBGs are selected with the color criteria similar to those of Bouwens et al. (2014b). We perform source detections by SExtractor (Bertin & Arnouts 1996) in coadded images constructed from bands of ${Y}_{098}{Y}_{105}{J}_{125}{H}_{160}$, ${J}_{125}{H}_{160}$, and H₁₆₀ for the $z\sim 4-7$, 8, and 10 LBGs, respectively. The JH₁₄₀ band is included in the coadded image for the $z\sim 7-8$ LBGs in the HUDF 09+12 field. The flux measurements are carried out in Kron (1980)-type apertures with a Kron parameter of 1.6 whose diameter is determined in the H₁₆₀ band. In two-color diagrams, we select objects with a Lyman break, no extreme-red stellar continuum, and no detection in passbands bluer than the spectral drop. See Y. Harikane et al. (2015, in preparation) for more details of the source detections and LBG selections.

Table 3.  Number of LBGs for Our Size Measurements

	${N}_{{\mathtt{GALFIT}}}/{N}_{\mathrm{LBG}}$
Field	$z\sim 4$	$z\sim 5$	$z\sim 6$	$z\sim 7$	$z\sim 8$	$z\sim 10$
(1)	(2)	(3)	(4)	(5)	(6)	(7)
HUDF 09+12	160 (348)	43 (130)	26 (86)	13 (50)	9 (24)	0 (2)
HUDF 09-P1	⋯	41 (95)	12 (30)	2 (9)	2 (7)	0 (0)
HUDF 09-P2	⋯	30 (90)	8 (37)	4 (23)	0 (16)	0 (0)
GOODS-S Deep	1046 (1872)	292 (696)	122 (311)	55 (203)	11 (57)	1 (1)
GOODS-S Wide	294 (510)	73 (142)	20 (51)	9 (31)	3 (21)	0 (0)
GOODS-N Deep	868 (1655)	279 (630)	48 (135)	35 (111)	12 (28)	1 (2)
GOODS-N Wide	522 (800)	106 (222)	25 (68)	12 (231)	3 (28)	1 (1)
UDS	⋯	152 (310)	39 (65)	12 (25)	⋯	⋯
AEGIS	⋯	189 (381)	47 (101)	11 (28)	⋯	⋯
COSMOS	⋯	209 (348)	40 (80)	11 (27)	⋯	⋯
HFF-Abell2744P	⋯	15 (37)	12 (26)	4 (7)	2 (7)	⋯
HFF-MACS0416P	⋯	30 (134)	23 (106)	5 (18)	4 (10)	⋯
${N}_{\mathrm{total}}(z)$	2890 (5185)	1459 (3215)	422 (1096)	173 (763)	46 (195)	3 (6)
${N}_{\mathrm{total}}$	4993 (10454)

Note. Columns: (1) Field. (2)–(7) Number of LBGs that have S/N $\geqslant 15$ and reliable GALFIT outputs in each redshift range. The value in parentheses is the number of LBGs in the parent sample.

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The H₁₆₀ magnitude at 50% completeness is ∼28 mag for the LBGs in the deep CANDELS fields (Bouwens et al. 2014b). The details of the completeness estimates and values are presented in Y. Harikane et al. (2015, in preparation).

Several previous studies on galaxy size have included a galaxy at $z\sim 12$ selected in the photo-z technique (Ellis et al. 2013). In this study, we do not use the galaxy at $z\sim 12$ because the redshift of the source is under debate (e.g., Bouwens et al. 2013; Brammer et al. 2013; Capak et al. 2013; Ellis et al. 2013; Pirzkal et al. 2013).

Some analyses and discussions in this work require the M_* of the photo-z galaxies and the LBGs. For the photo-z galaxies, we take M_* values from Skelton et al. (2014). For the LBGs, we derive stellar masses, adopting an empirical relation between UV magnitude ${M}_{\mathrm{UV}}$ and M_*. First, we calculate ${M}_{\mathrm{UV}}$ from the total magnitudes in the LBG detection images (Section 3), assuming that the typical redshifts are $\langle z\rangle \sim 3.8,4.9,5.9,6.8,7.9$, and 10.4. The stellar masses are obtained by converting their ${M}_{\mathrm{UV}}$ through the empirical relation from González et al. (see also the updated result of González et al. 2014):

$$\begin{eqnarray} \mathrm{log}{M}_{*} & = & -39.6+1.7\times \mathrm{log}{L}_{1500},\\ & = & -4.50-0.68\times {M}_{\mathrm{UV}}, \end{eqnarray} \tag{ 1 }$$

where ${L}_{1500}$ is the luminosity at the rest-frame 1500 Å. This empirical relation is derived under assumptions similar to ours (the Salpeter IMF and no nebular emission lines included in the SED).

To test whether this empirical relation (Equation (1)) of ${L}_{\mathrm{UV}}$–M_* is reliable and consistent with the M_* estimates of the photo-z galaxy sample, we compare this empirical relation with the ${M}_{\mathrm{UV}}$–M_* relations derived from the photo-z galaxies.

We estimate ${M}_{\mathrm{UV}}$ from the absolute UV magnitudes at a wavelength of 2800 Å from the photo-z catalog, assuming that the majority of star-forming galaxies have a flat UV spectrum of ${f}_{\nu }=$ constant. We present the ${M}_{\mathrm{UV}}$–M_* relations of the photo-z galaxies in Figure 1. The UV magnitude correlates well with M_*, suggesting the existence of the "star-formation main sequence" (e.g., Daddi et al. 2007; Lee et al. 2012; Whitaker et al. 2012; Steinhardt et al. 2014).

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Figure 1 shows that the slopes of the relations appear to be flatter at a bright range of ${M}_{\mathrm{UV}}\lesssim -22$ than at a faint ${M}_{\mathrm{UV}}$ range. Similar flat slopes are reported by a large survey area of the CANDELS fields (Stark et al. 2009; Lee et al. 2011; Salmon et al. 2015). Because our LBGs used in this analysis have magnitudes of ${M}_{\mathrm{UV}}\geqslant -22$, we fit $\mathrm{log}{M}_{*}=a+{{bM}}_{\mathrm{UV}}$ to the ${M}_{\mathrm{UV}}$–M_* relation at ${M}_{\mathrm{UV}}\geqslant -22$, where a and b are free parameters. The best-fit functions for the photo-z galaxies are

$$\begin{eqnarray} \mathrm{log}{M}_{*}[{M}_{\odot }] & = & 1.46-0.43\times {M}_{\mathrm{UV}}(z=0-1),\\ & = & 0.82-0.45\times {M}_{\mathrm{UV}}(z=1-2),\\ & = & 1.12-0.43\times {M}_{\mathrm{UV}}(z=2-3),\\ & = & -0.22-0.49\times {M}_{\mathrm{UV}}(z=3-4),\\ & = & -2.10-0.58\times {M}_{\mathrm{UV}}(z=4-5),\\ & = & -2.45-0.59\times {M}_{\mathrm{UV}}(z=5-6). \end{eqnarray} \tag{ 2 }$$

If we assume that the magnitudes of 1500–1700 Å are the same for typical LBGs with ${f}_{\nu }=$ constant, the slopes b of −0.58 ± 0.02 and −0.59 ± 0.03 at $z\sim 4-6$ roughly agree with that of Equation (1) (i.e., $b=-0.68\pm 0.08$). We thus conclude that the empirical relation (Equation (1)) is reliable and consistent with the M_* estimates of the photo-z galaxy sample. Moreover, no strong evolution in the ${M}_{\mathrm{UV}}$–M_* relation is found at $z\gtrsim 4$ in Equation (2) and Figure 1. We use Equation (1) to estimate the M_* of our $z\gtrsim 4$ LBGs.

We investigate the effects of morphological K correction in our size measurements, comparing our r_e at different wavelengths. Because the HST imaging data cover up to the H₁₆₀ band, we can study the rest-frame UV morphology for galaxies at $z\gtrsim 3$. Understanding the effects of morphological K correction is considerably important in evaluating the size evolution of star-forming galaxies over a wide redshift range of $0\lesssim z\lesssim 10$. The sizes in the rest-frame UV and optical stellar continuum emission, ${r}_{{\rm{e}}}^{\mathrm{UV}}$ and ${r}_{{\rm{e}}}^{\mathrm{Opt}}$, tracing different stellar populations, would yield a large difference in r_e. Here we make a comparison between ${r}_{{\rm{e}}}^{\mathrm{UV}}$ and ${r}_{{\rm{e}}}^{\mathrm{Opt}}$ of the SFGs at $1.2\lesssim z\lesssim 2.1$ where both radii can be measured with the HST data.

Figure 4 shows the differences between ${r}_{{\rm{e}}}^{\mathrm{UV}}$ and ${r}_{{\rm{e}}}^{\mathrm{Opt}}$ of the SFGs as a function of stellar mass. Although we find a large scatter, the median values of $({r}_{{\rm{e}}}^{\mathrm{UV}}-{r}_{{\rm{e}}}^{\mathrm{Opt}})/{r}_{{\rm{e}}}^{\mathrm{Opt}}$ are less than 20% in all stellar mass bins. This indicates that the differences in statistical r_e measurements are small for star-forming galaxies with $\mathrm{log}{M}_{*}=9-11\;{M}_{\odot }$ at $z\sim 1-2$.

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Similarly, van der Wel et al. (2014) have found that ${r}_{{\rm{e}},\mathrm{major}}$ is typically smaller in redder bands for SFGs at $z\sim 0-2$ (see also, e.g., Szomoru et al. 2011; Wuyts et al. 2012). This trend is more significant in more massive SFGs. The smaller size in redder bands could be interpreted as heavier dust attenuation in the galactic central regions in bluer bands (e.g., Kelvin et al. 2012) and or inside-out disk formation (e.g., Bezanson et al. 2009; Brooks et al. 2009; Naab et al. 2009; Nelson et al. 2012; Patel et al. 2013). We confirm the wavelength dependence even in our r_e in the most massive M_* bin, as shown in Figure 4. van der Wel et al. (2014) have parameterized the wavelength dependence of ${r}_{{\rm{e}},\mathrm{major}}$ as a function of redshift and stellar mass. Following the formula, the size difference fraction $({r}_{{\rm{e}}}^{\mathrm{UV}}-{r}_{{\rm{e}}}^{\mathrm{Opt}})/{r}_{{\rm{e}}}^{\mathrm{Opt}}$ is calculated to be ∼30% for $z\sim 2$ galaxies with $\mathrm{log}{M}_{*}=11\;{M}_{\odot }$.

Note that the difference of stellar population becomes smaller at $z\gt 2$ than at $z\sim 1-2$, which is because the short cosmic age of $z\gt 2$ provides a smaller stellar-age difference and less metal enrichment than that of $z\sim 1-2$. This agreement of ${r}_{{\rm{e}}}^{\mathrm{UV}}$ and ${r}_{{\rm{e}}}^{\mathrm{Opt}}$ suggests that the statistical ${r}_{{\rm{e}}}^{\mathrm{UV}}$ values represent the typical sizes of stellar-component distribution for star-forming galaxies of SFGs and LBGs at $z\gtrsim 3$ with a small systematic uncertainty of $\lesssim 30$%.⁸

We examine the effect of morphological K correction in more detail by investigating the evolutionary trends of r_e and size-relevant quantities in the rest-frame optical and UV emission for the photo-z galaxies at $z\sim 0-6$. Figure 5 presents the redshift evolution of r_e, n, and the star-formation rate surface density (SFR SD), ${{\rm{\Sigma }}}_{\mathrm{SFR}}$. The SFR SD is derived in the effective radius and calculated by

$$\begin{eqnarray}&&{{\rm{\Sigma }}}_{\mathrm{SFR}}\;[{M}_{\odot }\;{\mathrm{yr}}^{-1}\;{\mathrm{kpc}}^{-2}]=\displaystyle \frac{\mathrm{SFR}/2}{\pi {r}_{{\rm{e}}}^{2}},\end{eqnarray} \tag{ 3 }$$

where a factor of $1/2$ corrects for the SFR value, which is derived from the total magnitudes. For the photo-z galaxies, we use SFRs taken from the catalog of Skelton et al. (2014). For LBGs, we compute SFRs from ${L}_{\mathrm{UV}}$ using the relation of Kennicutt (1998a):

$$\begin{eqnarray}&&\mathrm{SFR}\;[{M}_{\odot }\;{\mathrm{yr}}^{-1}]=1.4\times {10}^{-28}{L}_{\nu }\;[\mathrm{erg}\ {{\rm{s}}}^{-1}\;{\mathrm{Hz}}^{-1}].\end{eqnarray} \tag{ 4 }$$

van der Wel et al. (2014) have already examined the r_e evolution at 5000 Å in the rest frame for galaxies at $0\lesssim z\lesssim 3$ in the 3D-HST+CANDELS sample. In our study, we extend this analysis of $0\lesssim z\lesssim 3$ to $z\gtrsim 4$, using the photo-z galaxies and the LBGs.

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In Figure 5, the median values of these quantities are in good agreement between the measurements in the rest-frame optical and UV emission of the SFGs at $2\lesssim z\lesssim 3$. Additionally, the evolutionary tracks at $z\lesssim 3$ smoothly connect with those at $z\gtrsim 3$. We also find no strong dependence of these evolutionary trends on stellar mass. These agreements confirm a small effect of morphological K correction on the median r_e values.

We examine the redshift evolution of median, average, and modal r_e of our galaxies to evaluate statistical differences and selection biases. We define four ${L}_{\mathrm{UV}}$ bins for these analyses. The ${L}_{\mathrm{UV}}$ bins are 1–10, 0.3–1, 0.12–0.3, and 0.048–0.12 ${L}_{\mathrm{UV}}/{L}_{z=3}^{*}$, where ${L}_{z=3}^{*}$ is the characteristic UV luminosity of LBGs at $z\sim 3$ (${M}_{\mathrm{UV}}=-21$, Steidel et al. 1999).⁹ To investigate the r_e distribution shape, in Figure 6 we plot the r_e distribution of SFGs and LBGs at $z\sim 1-6$ in the bin of ${L}_{\mathrm{UV}}=0.3-1\;{L}_{z=3}^{*}$ that has good r_e measurement accuracies whose typical reduced ${\chi }^{2}$ values are the smallest among the ${L}_{\mathrm{UV}}$ bins of the r_e measurements. We fit the r_e with the log-normal distribution

$$\begin{eqnarray}&&p({r}_{{\rm{e}}})=\displaystyle \frac{1}{{r}_{{\rm{e}}}{\sigma }_{\mathrm{ln}{r}_{{\rm{e}}}}\sqrt{2\pi }}\mathrm{exp}\left[-\displaystyle \frac{{\mathrm{ln}}^{2}({r}_{{\rm{e}}}/\bar{{r}_{{\rm{e}}}})}{2{\sigma }_{\mathrm{ln}{r}_{{\rm{e}}}}^{2}}\right],\end{eqnarray} \tag{ 5 }$$

where $\bar{{r}_{{\rm{e}}}}$ and ${\sigma }_{\mathrm{ln}{r}_{{\rm{e}}}}$ are the peak of r_e and the standard deviation of $\mathrm{ln}{r}_{{\rm{e}}}$, respectively. We fit the log-normal functions to the r_e distribution data with the two free parameters of $\bar{{r}_{{\rm{e}}}}$ and ${\sigma }_{\mathrm{ln}{r}_{{\rm{e}}}}$, and we present the best-fit log-normal functions in Figure 6 for the data of good statistics, the SFGs at $z\sim 1-3$, and the LBGs at $z\sim 4-6$ in the ${L}_{\mathrm{UV}}=0.3-1.0{L}_{\mathrm{UV}}^{*}$ bin. The r_e distributions of the high-z star-forming galaxies are well represented by the log-normal distribution. The reduced ${\chi }^{2}$ values are 0.006, 0.003, 0.004, 0.005, and 0.011 for the SFGs at $z=1-2$ and 2–3 and the LBGs at $z\sim 4$, 5, and 6, respectively. Figure 7 is the same as Figure 6, but for all of our galaxies. Figure 7 indicates that the r_e distributions are well fitted by the log-normal functions in the wide range of redshift, $z\sim 0-6$, and the UV luminosity, $\sim 0.12-10{L}_{\mathrm{UV}}^{*}$. Note that log-normal functions cannot be fitted to the data of the $z\gtrsim 7$ galaxies and some low-z galaxies in Figure 7 because of the small statistics. Moreover, the fitting result of $z\sim 0-1$ is only obtained for the ${r}_{{\rm{e}}}^{\mathrm{Opt}}$ distribution in the luminosity bin of $0.12-0.1{L}_{\mathrm{UV}}^{*}$ because of the poor statistics of the other luminosity bin data.

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Because the r_e distributions follow the log-normal functions, the average, median, and modal values of $\mathrm{ln}{r}_{{\rm{e}}}$ should be the same. However, in previous studies, the size evolution is discussed with the average, median, and modal values of r_e in the linear space (e.g., Bouwens et al. 2004; Oesch et al. 2010; Grazian et al. 2012; Ono et al. 2013; Holwerda et al. 2014). Here we obtain r_e measurements with different statistics choices in the linear space, following the previous studies, and evaluate the differences in the size evolution results. We derive size growth rates based on average, median, and modal r_e in a bin of $0.3-1\;{L}_{\mathrm{UV}}/{L}_{z=3}^{*}$, estimating the modal r_e by fitting the size distributions with a log-normal function. In Figure 8, we compare our r_e measurements with those of the previous studies that apply the different statistics.¹⁰ We confirm that our results are consistent with those of the previous studies. Moreover, Figure 8 indicates that galaxy sizes decrease from $z\sim 0$ to ∼6 in any statistical choices of average, median, and mode.

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We fit ${r}_{{\rm{e}}}={B}_{z}{(1+z)}^{{\beta }_{z}}$ for the average, median, and modal r_e values given by our and previous studies, where B_z and ${\beta }_{z}$ are free parameters. The fitting is performed for the combination of ${r}_{{\rm{e}}}^{\mathrm{UV}}$ and ${r}_{{\rm{e}}}^{\mathrm{Opt}}$ as well as for ${r}_{{\rm{e}}}^{\mathrm{UV}}$ only. Table 6 summarizes the best-fit B_z and ${\beta }_{z}$ values. Table 7 is a summary of the samples and ${\beta }_{z}$ values from our and previous studies for LBGs with $z\gtrsim 4$. Our average, median, and modal r_e values scale as $\propto {(1+z)}^{-1\sim -1.3}$, indicating that, again, the choices of statistics in r_e measurements has no significant effect on size growth rates. This conclusion is consistent with the result that ${\sigma }_{\mathrm{ln}{r}_{{\rm{e}}}}$ shows no significant evolution, as discussed in Section 6.1.1.

Table 6.  Summary of the Best-fit Size Growth Rates

Data points	Sample	${L}_{\mathrm{UV}}/{L}_{z=3}^{*}$	Bz	${\beta }_{z}$	BH	${\beta }_{H}$
			(kpc)		(kpc)
(1)	(2)	(3)	(4)	(5)	(6)	(7)
Median	All	1–10	4.78 ± 0.68	−0.84 ± 0.11	3.80 ± 0.40	−0.62 ± 0.08
		0.3–1	5.45 ± 0.31	−1.10 ± 0.06	4.33 ± 0.17	−0.86 ± 0.04
		0.12–0.3	4.44 ± 0.19	−1.22 ± 0.05	3.46 ± 0.13	−0.97 ± 0.05
	w/o ${r}_{{\rm{e}}}^{\mathrm{Opt}}$	1–10	4.05 ± 0.59	−0.78 ± 0.08	3.09 ± 0.36	−0.56 ± 0.06
		0.3–1	5.21 ± 0.28	−1.15 ± 0.07	3.54 ± 0.29	−0.80 ± 0.05
		0.12–0.3	3.54 ± 0.58	−1.11 ± 0.11	2.45 ± 0.32	−0.78 ± 0.08
Average	All	1–10	5.80 ± 0.65	−0.79 ± 0.10	4.91 ± 0.42	−0.61 ± 0.07
		0.3–1	5.85 ± 0.33	−0.95 ± 0.07	4.83 ± 0.20	−0.74 ± 0.04
		0.12–0.3	5.52 ± 0.43	−1.17 ± 0.07	4.29 ± 0.27	−0.87 ± 0.05
	w/o ${r}_{{\rm{e}}}^{\mathrm{Opt}}$	1–10	11.3 ± 4.44	−1.22 ± 0.25	7.48 ± 2.37	−0.85 ± 0.18
		0.3–1	10.9 ± 2.94	−1.36 ± 0.18	6.90 ± 1.50	−0.95 ± 0.13
		0.12–0.3	6.82 ± 2.25	−1.31 ± 0.20	4.37 ± 1.18	−0.91 ± 0.14
Mode	All	1–10	4.00 ± 0.49	−0.78 ± 0.08	3.07 ± 0.30	−0.55 ± 0.57
		0.3–1	4.45 ± 0.89	−1.26 ± 0.17	2.97 ± 0.45	−0.89 ± 0.12
		0.12–0.3	3.28 ± 0.18	−1.23 ± 0.07	2.56 ± 0.11	−1.00 ± 0.05
	w/o ${r}_{{\rm{e}}}^{\mathrm{Opt}}$	1–10	10.9 ± 3.94	−1.14 ± 0.25	7.45 ± 2.14	−0.80 ± 0.17
		0.3–1	3.00 ± 0.19	−1.01 ± 0.05	2.15 ± 0.10	−0.71 ± 0.03
		0.12–0.3a	⋯	⋯	⋯	⋯

Notes. Columns: (1) Statistics of r_e. (2) Sample used in the fits for the size evolution. "All" denotes the use of all samples in an ${L}_{\mathrm{UV}}$ bin, and "w/o ${r}_{{\rm{e}}}^{\mathrm{Opt}}$" represents the exclusion of the data points of ${r}_{{\rm{e}}}^{\mathrm{Opt}}$. (3) Bins of ${L}_{\mathrm{UV}}$ in units of ${L}_{z=3}^{*}$. (4) B_z of ${B}_{z}{(1+z)}^{{\beta }_{z}}$. (5) ${\beta }_{z}$ of ${B}_{z}{(1+z)}^{{\beta }_{z}}$. (6) B_H of ${B}_{H}h{(z)}^{{\beta }_{H}}$, where $h(z)\equiv H(z)/{H}_{0}=\sqrt{{{\rm{\Omega }}}_{m}{(1+z)}^{3}+{{\rm{\Omega }}}_{{\rm{\Lambda }}}}$. (7) ${\beta }_{H}$ of ${B}_{H}h{(z)}^{{\beta }_{H}}$.

^aThe ${\chi }^{2}$ minimization is not converged.

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Table 7.  Summary of the LBG Size Growth Rates from Previous Studies

References	Number	Redshift Range	${\beta }_{z}$ of ${(1+z)}^{{\beta }_{z}}$	Statistics	Size Measurements
(1)	(2)	(3)	(4)	(5)	(6)
Bouwens et al. (2004)	⋯ $(2929)$	$2-6$	−1.05 ± 0.21	Average	SExtractor
Ferguson et al. (2004)	⋯ $(773)$	2–5	$\sim -1.5$	Average	SExtractor
Ravindranath et al. (2006)	1333 $(4694)$	3–5	⋯	⋯	GALFIT
Hathi et al. (2008a)	61 $(61)$	3–6	$\sim -1.5$	Average	SExtractor
Conselice & Arnold (2009)	583 $(583)$	$4-6$	⋯	⋯	SExtractor
Oesch et al. (2010)	21 $(21)$	$7-8$	−1.12 ± 0.17	Average	SExtractor, GALFIT
Grazian et al. (2012)	⋯ $(153)$	7	⋯	⋯	SExtractor
Mosleh et al. (2012)	⋯ $(218)$	$4-7$	−1.20 ± 0.11	Median	GALFIT
Huang et al. (2013)	1012 $(1356)$	4–5	$\sim -1$	Mode	SExtractor, GALFIT
Ono et al. (2013)	15 $(81)$	7–10	$-{1.30}_{-0.14}^{+0.12}$	Average	GALFIT
Curtis-Lake et al. (2014)	1318 $(3738)$	4–9	−0.31 ± 0.26	Mode	SExtractor
Holwerda et al. (2014)	8 $(8)$	9–10	−1.0 ± 0.1	Average	GALFIT
Kawamata et al. (2014) a	39 $(39)$	6–8	−1.24 ± 0.1	Average	glafic
This work	4993 $(10454)$	4–10	−1.10 ± 0.06	Median	GALFIT
			−0.95 ± 0.07	Average	GALFIT
			−1.26 ± 0.17	Mode	GALFIT
incl. Photo-z SFGs	89880 $(312722)$ b	$0-6$	⋯	⋯	⋯

Notes. Columns: (1) Reference. (2) Number of galaxies whose size is measured in the reference. The values in parentheses are the number of galaxies in the parent sample. (3) Redshift range for size measurements of LBGs. (4) Best-fit ${\beta }_{z}$ of ${(1+z)}_{z}^{\beta }$ for a bright (${L}_{\mathrm{UV}}\sim 0.3-1\;{L}_{z=3}^{*}$) galaxy sample. (5) Statistics for deriving a representative r_e at a redshift. "Mode" corresponds to the peak of the size distribution derived by fitting with a log-normal function (Equation (5)). (6) Method or software used to measure galaxy sizes.

^aSample galaxies are selected in a field of a galaxy cluster. This study corrects for the gravitational lensing effects of magnification and shear with their mass model. ^bThe value is the total number of SFGs whose sizes are well measured in ${r}_{{\rm{e}}}^{\mathrm{Opt}}$ and ${r}_{{\rm{e}}}^{\mathrm{UV}}$. See Table 2.

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Most previous studies have employed average values for representative r_e. However, Figure 7 indicates that the median measurements trace the typical galaxy sizes parameterized by $\bar{{r}_{{\rm{e}}}}$ better than the average values do. Because the small samples of $z\gtrsim 7$ galaxies do not allow us to estimate modal r_e values, we use median values for our main analyses, unless otherwise specified.

Figure 8 compares the r_e values of SFGs and LBGs at $z\sim 4-6$ with a bright UV luminosity. In any statistics choice, we find that the r_e values of SFGs and LBGs are comparable within a scatter of $\lesssim 30$%. These results indicate that star-forming galaxies selected by photo-z and dropout techniques statistically give similar r_e values and that the bias from the different selection techniques is as small as $\lesssim 30$% in the r_e determination.

The Sérsic index represents the SB profiles of galaxies. A high n means a cuspier SB distribution, indicating the existence of a central bulge. On the other hand, a lower n suggests a disk-like light profile with a flatter SB distribution at the central galactic region. The Sérsic index depends on observed wave bands and stellar populations (e.g., color), which have been revealed by detailed structural analyses with multiple passbands for local galaxies (e.g., Häußler et al. 2013; Vika et al. 2013, 2014). Vulcani et al. (2014) have reported that n tends to be larger in redder bands for blue galaxies because of a bulge component with old stellar ages and or dust attenuation at the central region.

Our results confirm that n values of QGs are significantly higher than those of SFGs at $z\lesssim 2$ in the Redshift-Sersic index right panel of Figure 5. For massive SFGs with $\mathrm{log}{M}_{*}=10-11\;{M}_{\odot }$, n values monotonically increase from $n\sim 1-1.5$ at $z\sim 1$ to $n\sim 2-3$ at $z\sim 0$. The evolutionary trend of n for the massive SFGs is similar to that of the QGs at $z\sim 0-2$, which is consistent with previous results (see the discussions in van Dokkum et al. 2010; Naab et al. 2009; Pastrav et al. 2013).

At $2\lesssim z\lesssim 3$, the n values of the SFGs at the rest-frame optical wavelengths are slightly smaller than those at the rest-frame UV wavelengths by ${\rm{\Delta }}n\lesssim 0.5$, which is similar to the results of Vulcani et al. (2014) for local objects.

Interestingly, in Figure 5, we find that typical SFGs have a value of $n\sim 1-1.5$ at the wide redshift range of $z\sim 1-6$, albeit with the large scatter of individual galaxies. A similar claim is made by, for example, Morishita et al. (2014), but only for $z\sim 1-3$ star-forming galaxies (Figure 5). Our results newly suggest that the typical Sérsic indices of star-forming galaxies are $n\sim 1-1.5$ at $z\sim 3-6$.

This constant n guarantees that we use a fixed n value of 1.5 in the size measurements for LBGs (Section 3).

We investigate the size–luminosity r_e–${L}_{\mathrm{UV}}$ relation and its dependence on redshift. Figure 9 and Table 8 represent the size–luminosity relation at z = 0–8 for the SFGs and LBGs, where ${L}_{\mathrm{UV}}$ is presented with ${M}_{\mathrm{UV}}$. We cannot examine the size–luminosity relation at $z\sim 10$ because the number of $z\sim 10$ LBGs is only three. A large area of ∼910 arcmin² in the HST fields allows us to derive the r_e–${L}_{\mathrm{UV}}$ relation in a wide range of magnitude, $-23\lesssim {M}_{\mathrm{UV}}\lesssim -17$ mag, even for $z\sim 4$ LBGs. Figure 9 shows that r_e has a negative correlation with ${{\rm{M}}}_{\mathrm{UV}}$ at $0\lesssim z\lesssim 8$.

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Table 8.  Size–Luminosity Relation at z = 0–8

${M}_{\mathrm{UV}}$	${r}_{{\rm{e}}}^{\mathrm{Opt}}$	${M}_{\mathrm{UV}}$	${r}_{{\rm{e}}}^{\mathrm{UV}}$	${M}_{\mathrm{UV}}$	${r}_{{\rm{e}}}^{\mathrm{UV}}$
(mag)	(kpc)	(mag)	(kpc)	(mag)	(kpc)
(1)	(2)	(3)	(4)	(5)	(6)
$z=0-1$ SFGs		$z=1-2$ SFGs		$z\sim 4$ LBGs
−21.0	${3.896}_{-2.273}^{+4.513}$	−21.0	${2.090}_{-1.139}^{+2.847}$	−23.0	${1.536}_{-0.878}^{+1.041}$
−20.0	${3.380}_{-1.497}^{+2.036}$	−20.0	${1.794}_{-0.962}^{+2.418}$	−22.0	${1.223}_{-0.600}^{+1.100}$
−19.0	${2.504}_{-1.045}^{+1.566}$	−19.0	${1.232}_{-0.634}^{+1.464}$	−21.0	${1.058}_{-0.529}^{+0.841}$
−18.0	${1.837}_{-0.810}^{+1.234}$	−18.0	${0.940}_{-0.477}^{+1.056}$	−20.0	${0.733}_{-0.325}^{+0.638}$
−17.0	${1.340}_{-0.570}^{+0.983}$	−17.0	${0.723}_{-0.352}^{+1.016}$	−19.0	${0.589}_{-0.265}^{+0.510}$
−16.0	${1.074}_{-0.467}^{+0.791}$	−16.0	${0.606}_{-0.404}^{+1.108}$	−18.0	${0.509}_{-0.204}^{+0.468}$
−15.0	${0.854}_{-0.359}^{+0.680}$	−15.0	${0.461}_{-0.309}^{+0.398}$	−17.0	${0.438}_{-0.247}^{+0.325}$
$z=1-2$ SFGs		$z=2-3$ SFGs		$z\sim 5$ LBGs
−22.0	${3.035}_{-0.448}^{+0.237}$	−22.0	${1.428}_{-0.632}^{+1.602}$	−22.0	${1.025}_{-0.352}^{+1.044}$
−21.0	${1.958}_{-1.026}^{+2.304}$	−21.0	${1.443}_{-0.727}^{+1.664}$	−21.0	${0.788}_{-0.359}^{+0.686}$
−20.0	${1.982}_{-0.734}^{+1.414}$	−20.0	${1.076}_{-0.545}^{+1.177}$	−20.0	${0.595}_{-0.287}^{+0.393}$
−19.0	${1.360}_{-0.589}^{+1.071}$	−19.0	${0.813}_{-0.382}^{+0.787}$	−19.0	${0.519}_{-0.232}^{+0.496}$
−18.0	${1.113}_{-0.399}^{+0.817}$	−18.0	${0.685}_{-0.333}^{+0.749}$	−18.0	${0.506}_{-0.284}^{+0.426}$
−17.0	${1.028}_{-0.451}^{+0.893}$	−17.0	${0.509}_{-0.263}^{+0.573}$	−17.0	${0.356}_{-0.076}^{+0.095}$
−16.0	${0.937}_{-0.378}^{+1.017}$	$z=3-4$ SFGs		$z\sim 6$ LBGs
$z=2-3$ SFGs		−22.0	${1.473}_{-0.705}^{+1.687}$	−22.0	${1.053}_{-0.696}^{+0.841}$
−21.0	${2.878}_{-0.908}^{+3.607}$	−21.0	${1.054}_{-0.495}^{+1.154}$	−21.0	${0.635}_{-0.274}^{+0.717}$
−20.0	${1.253}_{-0.098}^{+0.488}$	−20.0	${0.778}_{-0.374}^{+0.765}$	−20.0	${0.565}_{-0.287}^{+0.400}$
−19.0	${1.066}_{-0.710}^{+0.543}$	−19.0	${0.620}_{-0.278}^{+0.538}$	−19.0	${0.584}_{-0.327}^{+0.424}$
−18.0	${1.625}_{-0.997}^{+9.075}$	−18.0	${0.572}_{-0.269}^{+0.577}$	−18.0	${0.371}_{-0.198}^{+0.222}$
⋯	⋯	$z=4-5$ SFGs		$z\sim 7$ LBGs
⋯	⋯	−22.0	${1.081}_{-0.436}^{+1.082}$	−21.0	${0.737}_{-0.421}^{+0.320}$
⋯	⋯	−21.0	${0.892}_{-0.427}^{+0.769}$	−20.0	${0.489}_{-0.268}^{+0.956}$
⋯	⋯	−20.0	${0.708}_{-0.354}^{+0.703}$	−19.0	${0.467}_{-0.207}^{+0.572}$
⋯	⋯	−19.0	${0.444}_{-0.302}^{+0.362}$	$z\sim 8$ LBGs
⋯	⋯	$z=5-6$ SFGs		−21.0	${0.419}_{-0.262}^{+1.981}$
⋯	⋯	−22.0	${0.975}_{-0.425}^{+3.757}$	−20.0	${0.425}_{-0.173}^{+1.331}$
⋯	⋯	−21.0	${0.716}_{-0.286}^{+0.674}$	−19.0	${0.243}_{-0.068}^{+0.225}$
⋯	⋯	−20.0	${0.678}_{-0.328}^{+0.756}$	−18.0	${0.356}_{-0.218}^{+1.194}$

Note. Columns: (1), (3), (5) UV magnitude. (2), (4), (6) Median effective radius at the rest-frame optical or UV wavelength. The lower and upper limits indicate the 16th and 84th percentiles of the r_e distribution, respectively.

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The r_e–${L}_{\mathrm{UV}}$ relation is fitted by

$$\begin{eqnarray}&&{r}_{{\rm{e}}}={r}_{0}{\left(\displaystyle \frac{{L}_{\mathrm{UV}}}{{L}_{0}}\right)}^{\alpha },\end{eqnarray} \tag{ 6 }$$

where r₀ and α are free parameters. The r₀ value represents the effective radius at a luminosity of L₀, which is similar to the parameter γ used in, for example, Newman et al. (2012). The α value is the slope of the r_e–${L}_{\mathrm{UV}}$ relation. We select L₀ to the best-fit Schechter parameter M^* at $z\sim 3$ that corresponds to ${M}_{\mathrm{UV}}=-21.0$, following the arguments of Huang et al. (2013).

The left panel of Figure 10 shows the redshift evolution of r₀ and α. We parameterize the size growth rate by fitting r₀ with a function of ${B}_{z}{(1+z)}^{{\beta }_{z}}$. The best-fit function is $6.9{(1+z)}^{-1.20\pm -0.04}$ kpc, which does not significantly change even with and without the ${r}_{{\rm{e}}}^{\mathrm{Opt}}$ results. We also carry out fitting with a function of ${B}_{H}h{(z)}^{{\beta }_{H}}$, where B_H and ${\beta }_{H}$ are free parameters and $h(z)\equiv H/{H}_{0}=\sqrt{{{\rm{\Omega }}}_{m}{(1+z)}^{3}+{{\rm{\Omega }}}_{{\rm{\Lambda }}}}$. Here the fitting of the h(z)-form functions are conducted because these h(z)-form functions could be a more realistic physical treatment, as claimed by, for example, van der Wel et al. (2014). The fitting results yield the best-fit function of $5.3\;h{(z)}^{-0.97\pm 0.04}$ kpc that is plotted in the left panel of Figure 10. Although we do not use the r_e estimate of $z\sim 10$ for the fitting, the $z\sim 10$ data point is placed on the extrapolation of the best-fit function.

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The evolution of r₀ is similar to those of the median r_e values that are presented in Figure 8. Here we plot r_e as a function of redshift in Figure 11, which is the same as Figure 8, but for the median r_e values of three different UV luminosity samples. We fit the functions and find that the best-fit ${\beta }_{z}$ are −1.22 ± 0.05, −1.10 ± 0.06, and −0.84 ± 0.11 in ${L}_{\mathrm{UV}}/{L}_{z=3}^{*}=0.12-0.3,0.3-1$, and 1–10, respectively (Table 6). The best-fit ${\beta }_{z}$ values are comparable to the one of r₀.

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In contrast to the r_e evolution, there is no significant evolution of α (Equation (6)) at z = 0–8 found in the right panel of Figure 10. We calculate the weighted-average value of α with our data points over z = 0–8 and obtain $\alpha =0.27\pm 0.01$. Figure 10 compares the α estimates of z = 0–8 obtained in the previous studies. The α measurements of local spiral and or disk galaxies are comparable to $\alpha \sim 0.27$ (de Jong & Lacey 2000; Shen et al. 2003; Courteau et al. 2007). At $z=0-3$, van der Wel et al. (2014) have revealed that the slopes of the size–stellar mass relations do not evolve. Adopting Equation (2) to calculate ${L}_{\mathrm{UV}}$ from the stellar masses, we obtain the r_e–${L}_{\mathrm{UV}}$ relation and evolution similar to our results. At $z\gt 4$, there are several α measurements reported by Curtis-Lake et al. (2014), Huang et al. (2013), Jiang et al. (2013), and Grazian et al. (2012). However, these data points of α are largely scattered (the right panel of Figure 10). Nevertheless, our α values fall within the scatter of the previous measurements.

Our results of the r_e (or r₀) evolution and the constant α suggest that the r_e–${L}_{\mathrm{UV}}$ relation of star-forming galaxies is unchanged but with a decreasing offset of r_e from z = 0 to 8. Because the morphological evolution trend of star-forming galaxies is simple, our results are a benefit to studies using Monte Carlo simulations for luminosity function determinations that require an assumption of high-z galaxy sizes (e.g., Ishigaki et al. 2014; Oesch et al. 2014). Moreover, these morphological evolution trends are important constraints on the parameters of galaxy-formation models.

Note that there is a possible source of systematics given by the cosmological SB dimming effect by which we would underestimate r_e (Section 3). To estimate the effect of the cosmological SB dimming, we measure the r_e of $z\sim 4-8$ LBGs with stacked images that accomplish the detection limit deeper than the individual images by a factor of $\sim 20-30$. The r_e values measured in the stacked images roughly reproduce the size–luminosity relation of Figure 9, suggesting that there are no signatures of systematics in the r_e values measured by our GALFIT profile fitting technique. There is another possibility of the cosmological SB dimming effect. If a large population of diffuse high-z galaxies exists that are not identified in our HST images, we would underestimate the r_e values. However, it is unlikely that such a diffuse high-z population exists. This is because the luminosity functions of $z\sim 4-6$ LBGs derived with HST data agree with those obtained by ground-based observations (Beckwith et al. 2006), whose PSF FWHM is $\sim 1\prime\prime$, corresponding to 3–4 kpc in radius at $z\sim 4-6$. In other words, at these redshifts, there is no diffuse population with a radius up to $\sim 3-4$ kpc that is significantly larger than our size measurements of ${r}_{{\rm{e}}}\lesssim 1$ kpc (see, e.g., Figure 6). We therefore conclude that our results of size measurements are not significantly changed by the cosmological SB dimming effect.

We examine the redshift evolution of SFR SD ${{\rm{\Sigma }}}_{\mathrm{SFR}}$. Figure 12 shows ${{\rm{\Sigma }}}_{\mathrm{SFR}}$ as a function of redshift. Figure 12 is the same as Figure 5, but for all of our galaxies up to z = 8 with the binning of ${L}_{\mathrm{UV}}$ values. Figure 12 shows that ${{\rm{\Sigma }}}_{\mathrm{SFR}}$ gradually increases by redshift from $z\sim 0$ to 8. This evolutional trend and the ${{\rm{\Sigma }}}_{\mathrm{SFR}}$ values are consistent with those of $z\sim 4-8$ previously reported by, for example, Oesch et al. (2010) and Ono et al. (2013). Our results of the ${{\rm{\Sigma }}}_{\mathrm{SFR}}$ evolution suggest that the ${{\rm{\Sigma }}}_{\mathrm{SFR}}$ of typical high-z galaxies continuously increases from $z\sim 0$ to 8.

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In Figure 12, we also find that the increase rate per redshift becomes small at $z\gtrsim 4$ in the regime of $\mathrm{log}{{\rm{\Sigma }}}_{\mathrm{SFR}}\sim 0.5-1\;{M}_{\odot }$ yr⁻¹ kpc⁻². We obtain the ${{\rm{\Sigma }}}_{\mathrm{SFR}}$ evolution curve using  Equation (3) with the inputs of the best-fit function ${r}_{0}=6.9{(1+z)}^{-1.20\pm -0.04}$ (Section 5.2) and the SFR estimated from the ${L}_{\mathrm{UV}}$ value via Equation (4). Figure 12 presents the ${{\rm{\Sigma }}}_{\mathrm{SFR}}$ evolution curve. As expected, the ${{\rm{\Sigma }}}_{\mathrm{SFR}}$ evolution curve follows the ${{\rm{\Sigma }}}_{\mathrm{SFR}}$ data points. In other words, the slow ${{\rm{\Sigma }}}_{\mathrm{SFR}}$ evolution at $z\gtrsim 4$ is explained by the simple power-law galaxy size evolution of ${r}_{0}=6.9{(1+z)}^{-1.20\pm -0.04}$.

In Figure 13, we examine the dependence of ${{\rm{\Sigma }}}_{\mathrm{SFR}}$ on SFR and M_*. The left and right panels of Figure 13 show ${{\rm{\Sigma }}}_{\mathrm{SFR}}$ as functions of SFR and M_*, respectively. For comparison, we also plot SDSS galaxies with an exponential SB profile in Lackner & Gunn (2012) and the Milky Way (Kennicutt & Evans 2012). These local galaxies are placed in the regime of low ${{\rm{\Sigma }}}_{\mathrm{SFR}}$ values. Obviously, Figure 13 reproduces the result of Figure 12 that ${{\rm{\Sigma }}}_{\mathrm{SFR}}$ is typically higher for high-z galaxies than for low-z galaxies. In the ${{\rm{\Sigma }}}_{\mathrm{SFR}}$–SFR diagram of Figure 13, ${{\rm{\Sigma }}}_{\mathrm{SFR}}$ positively correlates with SFR. This is because the ${{\rm{\Sigma }}}_{\mathrm{SFR}}$ and SFR values are related by Equation (3). The slopes of the ${{\rm{\Sigma }}}_{\mathrm{SFR}}$–SFR relation appear similar at z ∼ 2–8. On the other hand, we find that the ${{\rm{\Sigma }}}_{\mathrm{SFR}}$–M_* diagram of Figure 13 shows no strong dependence of ${{\rm{\Sigma }}}_{\mathrm{SFR}}$ on M_* (see also, e.g., Wuyts et al. 2011). These two diagrams suggest that ${{\rm{\Sigma }}}_{\mathrm{SFR}}$ increases toward high z, keeping the similar ${{\rm{\Sigma }}}_{\mathrm{SFR}}$–SFR and ${{\rm{\Sigma }}}_{\mathrm{SFR}}$–M_* relations over z ∼ 2–8.

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We derive the r_e–UV slope β relation to investigate the dependence of galaxy sizes on stellar population. The β parameter is defined by ${f}_{\lambda }\propto {\lambda }^{\beta }$, where ${f}_{\lambda }$ is a galaxy spectrum at ∼1500–3000 Å, which is a coarse indicator of the stellar population and extinction of galaxies. A small β means a blue spectral shape, suggesting young stellar ages, low metallicity, and or dust extinction.

For the SFGs, we calculate β via

$$\begin{eqnarray}&&\beta =-\displaystyle \frac{{m}_{1700}-{m}_{2800}}{2.5\mathrm{log}1700/2800}-2,\end{eqnarray} \tag{ 7 }$$

where m₁₇₀₀ and m₂₈₀₀ are the total magnitudes at wavelengths of 1700 and 2800 Å in the rest frame, respectively. These magnitudes are taken from the catalog of Skelton et al. (2014). For the $z\sim 4$, 5, and 6 LBGs, we derive β, fitting the function of ${f}_{\lambda }\propto {\lambda }^{\beta }$ to the magnitude sets of ${i}_{775}{I}_{814}{z}_{850}{Y}_{105}{J}_{125}$, ${z}_{850}{Y}_{105}{J}_{125}{H}_{160}$, and ${Y}_{105}{J}_{125}{H}_{160}$, respectively, in the same manner as Bouwens et al. (2014a). For the $z\sim 7$ and 8 LBGs, we estimate β using

$$\begin{eqnarray}&&\beta =-2.0+4.59({J}_{125}-{H}_{160})(\mathrm{for}\;z\sim 7),\end{eqnarray} \tag{ 8 }$$

$$\begin{eqnarray}&&\beta =-2.0+8.68({{JH}}_{140}-{H}_{160})(\mathrm{for}\;z\sim 8).\end{eqnarray} \tag{ 9 }$$

Figure 14 represents the r_e–β relation in the bin of ${L}_{\mathrm{UV}}/{L}_{z=3}^{*}=0.3-1$. We find that the ${L}_{\mathrm{UV}}$–beta relation is poorly determined for the $z\sim 8$ LBGs, due to the small statistics, and the $z\sim 8$ result is not presented. In Figure 14, we identify clear trends that smaller galaxies have a bluer UV spectral shape at $0\lesssim z\lesssim 7$. This is consistent with the results of z ∼ 6–8 LBGs reported by Kawamata et al. (2014). This r_e–β correlation indicates that young and forming galaxies typically have a small size. We find a negative correlation between r_e and β for the $z=5-6$ SFGs. The negative correlation trend appears to be simply due to the small sample, which is not statistically significant.

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Here we investigate the properties of the r_e distributions in Section 6.1.1 and estimate SHSRs in Section 6.1.2. Combining these results and theoretical models, we discuss the host DM halos and the stellar dynamics in Section 6.1.3.

6.1.1. Log-normal Distribution of r_e

In Section 4.2, we find that the r_e distributions of our galaxies are well fitted by the log-normal functions in the wide range of redshift, $z\sim 0-6$, and luminosity.

Figure 15 shows the best-fit ${\sigma }_{\mathrm{ln}{r}_{{\rm{e}}}}$ values as a function of redshift. Size measurement uncertainties ${\sigma }_{\mathrm{ln}{r}_{{\rm{e}}},\mathrm{err}}$ would broaden the width of the r_e distribution. We estimate typical ${\sigma }_{\mathrm{ln}{r}_{{\rm{e}}},\mathrm{err}}$ in each z and ${L}_{\mathrm{UV}}$ bin. We correct ${\sigma }_{\mathrm{ln}{r}_{{\rm{e}}}}$ for the size measurement uncertainties through ${\sigma }_{\mathrm{ln}{r}_{{\rm{e}}}}={({\sigma }_{\mathrm{ln}{r}_{{\rm{e}}},\mathrm{obs}}{\;}^{2}-{\sigma }_{\mathrm{ln}{r}_{{\rm{e}}},\mathrm{err}}{\;}^{2})}^{0.5}$, where ${\sigma }_{\mathrm{ln}{r}_{{\rm{e}}},\mathrm{obs}}$ is the observed width of the r_e distribution. We find that ${\sigma }_{\mathrm{ln}{r}_{{\rm{e}}}}$ values fall in the range of $\sim 0.45-0.75$ with no clear evolutional trend at $z\sim 0-6$. Our ${\sigma }_{\mathrm{ln}{r}_{{\rm{e}}}}$ values are slightly larger than the estimates for local disks in de Jong & Lacey (2000), Shen et al. (2003), and Courteau et al. (2007) and for late-type galaxies at $z\sim 0-3$ in van der Wel et al. (2014). These differences would be explained by the choices of the wavelengths for the galaxy size measurements because these previous studies measure galaxy sizes in the rest-frame optical wavelength. In fact, if we change from the rest-frame UV luminosity to optical wavelength sizes for the size distribution, we obtain moderately small ${\sigma }_{\mathrm{ln}{r}_{{\rm{e}}}}$ values. However, differences of ∼20%–30% still remain beyond the error bars in Figure 15. These ∼20%–30% differences are probably explained by the sample and measurement technique differences. We also compare the ${\sigma }_{\mathrm{ln}{r}_{{\rm{e}}}}$ estimates of $z\sim 4-5$ LBGs given by Huang et al. (2013) and find a moderately large difference by a factor of 1.5. However, the scatters of our measurements and the statistical uncertainties in the estimates of Huang et al. (2013)  are too large to conclude on the differences.

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6.1.2. SHSR

We estimate the SHSRs that are defined with the ratio of ${r}_{{\rm{e}}}/{r}_{\mathrm{vir}}$, where ${r}_{\mathrm{vir}}$ is the virial radius of a host DM halo.

The ${r}_{\mathrm{vir}}$ value is calculated by

$$\begin{eqnarray}&&{r}_{\mathrm{vir}}={\left(\displaystyle \frac{2{{GM}}_{\mathrm{vir}}}{{{\rm{\Delta }}}_{\mathrm{vir}}{{\rm{\Omega }}}_{{\rm{m}}}(z)H{(z)}^{2}}\right)}^{1/3},\end{eqnarray} \tag{ 10 }$$

where ${{\rm{\Delta }}}_{\mathrm{vir}}=18{\pi }^{2}+82x-39{x}^{2}$ and $x={{\rm{\Omega }}}_{m}(z)-1$ (Bryan & Norman 1998). We obtain the virial mass of a DM halo, ${M}_{\mathrm{vir}}$, from the stellar mass, M_*, of individual galaxies by using the relation determined by the abundance-matching analyses (Behroozi et al. 2010, 2013). Figure 16 shows ${r}_{{\rm{e}}}/{r}_{\mathrm{vir}}$ as a function of redshift and its dependence on ${L}_{\mathrm{UV}}$ at $z\sim 0-8$. The $z\sim 10$ data point is omitted because of the small statistics. In Figure 16, we find that ${r}_{{\rm{e}}}/{r}_{\mathrm{vir}}$ is ∼2% for the star-forming galaxies and ∼0.5% for the QGs. Interestingly, the ${r}_{{\rm{e}}}/{r}_{\mathrm{vir}}$ of the star-forming galaxies is almost constant with redshift, albeit with large uncertainties at $z\gtrsim 5$. The no significant evolution of ${r}_{{\rm{e}}}/{r}_{\mathrm{vir}}$ is reported by Kawamata et al. (2014) based on a compilation of data from the literature for star-forming galaxies at $z\gtrsim 2$. Our systematic structural analyses seamlessly confirm the report of no large evolution from $z\sim 0$ with the homogenous data sets and the same analysis technique over the wide redshift range. Figure 16 also indicates that there is no strong dependence of ${r}_{{\rm{e}}}/{r}_{\mathrm{vir}}$ in the wide luminosity range of ${L}_{\mathrm{UV}}\sim 0.12-10{L}_{z=3}^{*}$.

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We compare our ${r}_{{\rm{e}}}/{r}_{\mathrm{vir}}$ estimates with those of previous studies. Because the previous studies chose different statistics for ${r}_{{\rm{e}}}/{r}_{\mathrm{vir}}$ estimates, we present average, median, and modal ${r}_{{\rm{e}}}/{r}_{\mathrm{vir}}$ for our galaxies with $0.3-1\;{L}_{\mathrm{UV}}/{L}_{z=3}^{*}$ in Figure 17, with the previous results.

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For local galaxies, Kravtsov (2013) obtain ${r}_{{\rm{e}}}/{r}_{\mathrm{vir}}=1.50\pm 0.07$% by the fitting of size–luminosity relations. This result of z = 0 is consistent with our results at a similar redshift of $z\sim 0.5$ within the $1\sigma$ uncertainty (Figure 17). For high-z galaxies, Kawamata et al. (2014) calculate ${r}_{{\rm{e}}}/{r}_{\mathrm{vir}}$ values with the average statistics. In Figure 17, the gray symbols of Kawamata et al. (2014)ʼs estimates agree with the blue crosses of our results. We find that the results of our and previous studies fall in the ${r}_{{\rm{e}}}/{r}_{\mathrm{vir}}$ range of ${r}_{{\rm{e}}}/{r}_{\mathrm{vir}}=1.0\%-3.5\%,$regardless of statistics choices.

Motivated by the no large evolution of ${r}_{{\rm{e}}}/{r}_{\mathrm{vir}}$, we calculate a $\lt{r}_{{\rm{e}}}/{r}_{\mathrm{vir}}\gt$ that is a ${r}_{{\rm{e}}}/{r}_{\mathrm{vir}}$ value weighted averaged over $z\sim 0-8$. We obtain $\lt{r}_{{\rm{e}}}/{r}_{\mathrm{vir}}\gt=2.76\pm 0.47$%, 1.92 ± 0.09%, and 1.13 ± 0.06% for our ${r}_{{\rm{e}}}/{r}_{\mathrm{vir}}$ estimates of average, median, and modal statistics, respectively.

The $\lt{r}_{{\rm{e}}}/{r}_{\mathrm{vir}}\gt$ value from our average statistics results is in good agreement with that of Kawamata et al. (2014): 3.3 ± 0.1%.

6.1.3. DM Halo and Stellar Disk

As we discuss in Section 6, a number of observational results suggest that typical high-z star-forming galaxies have disk-like stellar components in dynamics and morphology at $z\sim 0-6$. Thus we compare our results with the disk-formation model of Mo et al. (1998):

$$\begin{eqnarray}&&\displaystyle \frac{{r}_{{\rm{e}}}}{{r}_{\mathrm{vir}}}=\displaystyle \frac{1.678}{\sqrt{2}}\left(\displaystyle \frac{{j}_{{\rm{d}}}}{{m}_{{\rm{d}}}}\lambda \right)\displaystyle \frac{{f}_{{\rm{R}}}(\lambda ,{c}_{\mathrm{vir}},{m}_{{\rm{d}}},{j}_{{\rm{d}}})}{\sqrt{{f}_{c}({c}_{\mathrm{vir}})}},\end{eqnarray} \tag{ 11 }$$

where 1.678 is a coefficient for converting the scale length of the exponential disk ${R}_{{\rm{d}}}$ to r_e. The ${j}_{{\rm{d}}}$ (${m}_{{\rm{d}}}$) value is an angular momentum (mass) ratio of a central disk to a host DM halo. The ${f}_{c}({c}_{\mathrm{vir}})$ and ${f}_{{\rm{R}}}(\lambda ,{c}_{\mathrm{vir}},{m}_{{\rm{d}}},{j}_{{\rm{d}}})$ are functions related to halo and baryon concentrations, respectively. The ${c}_{\mathrm{vir}}$ is the halo concentration factor. The full functional forms of ${f}_{c}({c}_{\mathrm{vir}})$ and ${f}_{{\rm{R}}}(\lambda ,{c}_{\mathrm{vir}},{m}_{{\rm{d}}},{j}_{{\rm{d}}})$ are found in Mo et al. (1998). The SHSR ${r}_{{\rm{e}}}/{r}_{\mathrm{vir}}$ with a fixed ${j}_{{\rm{d}}}/{m}_{{\rm{d}}}$ shows little or no dependence on ${m}_{{\rm{d}}}$ and ${j}_{{\rm{d}}}$. If we use λ and ${c}_{\mathrm{vir}}$ values well constrained by numerical simulations (e.g., Vitvitska et al. 2002; Davis & Natarajan 2009; Prada et al. 2012), we can constrain ${j}_{{\rm{d}}}/{m}_{{\rm{d}}}$.

Figure 17 presents ${r}_{{\rm{e}}}/{r}_{\mathrm{vir}}$ regions corresponding to ${j}_{{\rm{d}}}/{m}_{{\rm{d}}}=0.5$, 1.0, and 1.5. To determine these regions, we randomly change the λ and ${c}_{\mathrm{vir}}$ values within the $\lambda =0.038-0.045$ (Vitvitska et al. 2002; Davis & Natarajan 2009) and ${c}_{\mathrm{vir}}$ ranges at $\mathrm{log}{M}_{\mathrm{vir}}=11-13\;{M}_{\odot }$ in Figure 12 of Prada et al. (2012), respectively. We also assume the conservative range of $0.05\leqslant {m}_{{\rm{d}}}\leqslant 0.1$ (e.g., Mo et al. 1998). Substituting these numbers and our results of ${r}_{{\rm{e}}}/{r}_{\mathrm{vir}}$ (Section 6.1.2) into Equation (11), we obtain ${j}_{{\rm{d}}}/{m}_{{\rm{d}}}=0.7-0.8$. Note that our estimates of ${r}_{{\rm{e}}}/{r}_{\mathrm{vir}}$ fall in ${j}_{{\rm{d}}}/{m}_{{\rm{d}}}\sim 0.5-1$ at $z\sim 0-8$, regardless of the statistical choices (Figure 17).

This result of ${j}_{{\rm{d}}}/{m}_{{\rm{d}}}\sim 0.5-1$ indicates that a central galaxy acquires more than half of the specific angular momentum from a host DM halo. Our ${j}_{{\rm{d}}}/{m}_{{\rm{d}}}$ values are comparable to the estimates with kinematical data for nearby disks (${j}_{{\rm{d}}}/{m}_{{\rm{d}}}\sim 0.8$; Romanowsky & Fall 2012; Fall & Romanowsky 2013). Moreover, Genel et al. (2015) predict ${j}_{{\rm{d}}}/{m}_{{\rm{d}}}\sim 1$ for $z\sim 0$ late-type galaxies with the Illustris simulations (Genel et al. 2014; Vogelsberger et al. 2014a, 2014b). These independent studies for $z\sim 0$ galaxies confirm that our estimate of ${j}_{{\rm{d}}}/{m}_{{\rm{d}}}\sim 0.5-1$ is correct at $z\sim 0$ and suggest that the conclusion of no significant evolution of ${j}_{{\rm{d}}}/{m}_{{\rm{d}}}$ over $z\sim 0-8$ would be reliable. Genel et al. (2015) have revealed that galactic winds with high mass-loading factors (active galactic nucleus feedback) enhance (suppress) ${j}_{{\rm{d}}}/{m}_{{\rm{d}}}$. This suggests that the no significant evolution of ${j}_{{\rm{d}}}/{m}_{{\rm{d}}}$ at $0\lesssim z\lesssim 8$ would place important constraints on parameters of galaxy feedback models.

In Section 6.1.2, we obtain that the SHSR of QGs is ∼0.5%, which is about four times smaller than that of star-forming galaxies. If we naively assume that QGs follow Equation (11) with one-fourth of the specific angular momentum of the star-forming galaxies, we obtain ${j}_{{\rm{d}}}/{m}_{{\rm{d}}}\sim 0.1-0.25$. This value is comparable to ${j}_{{\rm{d}}}/{m}_{{\rm{d}}}\sim 0.1$ for nearby ellipticals in Fall & Romanowsky (2013) and ${j}_{{\rm{d}}}/{m}_{{\rm{d}}}\sim 0.3$ for $z\sim 0$ early-type galaxies predicted in Genel et al. (2015). This small specific angular momentum of QGs would be explained by the loss of angular momentum via dynamical friction during merger events and or weak feedback (e.g., Scannapieco et al. 2008; Zavala et al. 2008).

Our study has shown a wide variety of morphological measurement results, supplemented by the theoretical models. It should be noted that these results are based on the structural analyses for major stellar components of the galaxies because we mask substructures such as star-forming clumps (e.g., Guo et al. 2014; Murata et al. 2014; Tadaki et al. 2014) in our analyses. The signatures of the morphological variety could emerge in dispersions of internal colors and SB profiles in recent structural analyses at $z\sim 2-3$ (e.g., Boada et al. 2015; Morishita et al. 2015). Moreover, we find that the SFR SD, ${{\rm{\Sigma }}}_{\mathrm{SFR}}$, increases toward high z in Figures 12 and 13. This fact suggests that star-forming galaxies at high z would tend to have a high gas mass density if we assume the Kennicutt–Schmidt law (Kennicutt 1998b). The gas-rich disks may enhance the formation of star-forming clumps through the process of disk instabilities (e.g., Genzel et al. 2011). The detailed analyses and results for the clumpy stellar subcomponents are presented in paper II (T. Shibuya et al. 2015, in preparation).