EVOLUTION OF THE SIZES OF GALAXIES OVER 7 < z < 12 REVEALED BY THE 2012 HUBBLE ULTRA DEEP FIELD CAMPAIGN

Low surface brightness in the outskirts of a galaxy may not be correctly measured by GALFIT, leading to systematically low measured half-light radii and/or total magnitudes. The differences in PSF sizes of the filters used in this study¹⁵ may also cause systematic effects on size measurements. In order to quantify and correct for any such systematic effects, we use the following simulations.

First, we use GALFIT to produce galaxy images whose Sérsic index n is fixed at 1.0, half-light radius r_e is randomly chosen between 0.5 and 10.5 pixels, and total magnitude is randomly chosen between 26 and 30 mag. Then we convolve them with a PSF image which is a composite of bright and unsaturated stellar objects in the HUDF (Pirzkal et al. 2005). Figure 1 shows the measured PSFs for the J₁₂₅ + J₁₄₀, J₁₄₀ + H₁₆₀, and H₁₆₀ images. The PSF-convolved galaxy images are inserted into empty regions of the original images before being analyzed in an identical manner to the true galaxy sample.

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Figure 2 displays the results of size measurements of our simulated galaxies. The panels show $r_e^{\rm (in)}$ versus $r_e^{\rm (out)}$ for each image at two different magnitude ranges (26 < m^(out) < 27 and 27 < m^(out) < 28). We see that measurements for all images give low systematic offsets for objects with sizes smaller than ∼4 pixels, although at larger sizes the profiles are progressively underestimated as the surface brightness of the objects decreases. The systematics are also seen to be larger for the fainter objects. We also use these simulation results for estimating statistical errors in the measurements.

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Figure 3 shows the results for measured total magnitudes compared to input magnitude. This time the results are displayed in two size bins ($0.5 < r_e^{\rm (out)} < 2.5$ pixels, $2.5 < r_e^{\rm (out)} < 4.5$ pixels) for each image. The results for the smaller size bin show that the total measured magnitude is robust down to ∼28 mag. For objects fainter than this the measured magnitude is systematically fainter than the intrinsic value, and the statistical errors increase. The trend is similar for both size bins but the results for larger objects show greater systematic offsets and statistical uncertainties.

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Note that the input axis ratio is fixed at b/a = 1 in the simulations. To investigate the effect of changing this input value on the systematic and statistical uncertainties, we created images with n = 1, r_e = 2.5 pixels, which is comparable to the size of the stacked objects and the z ∼ 12 object, total magnitude m⁽ⁱⁿ⁾ = 26.5–28.5 mag, and axis ratios ranging from b/a = 0.2 to 1. We convolve these images with the PSF, insert them into blank sky regions, and run GALFIT to measure their best-fit parameters in a similar way to that noted above. The results for the H₁₆₀ image are shown in Figure 4. We find that the output r_e is systematically smaller than its input value and that this difference increases for fainter galaxies (by up to ∼30% on average at m⁽ⁱⁿ⁾ = 28.5). However, this bias does not depend significantly on the input axis ratio. Thus, a simple correction for this bias can be made. However, if the axis ratio is smaller than 1, the statistical uncertainties do increase. At the faintest magnitude, the standard deviation increases by ∼40% if the input axis ratio is 0.2. Although this means the statistical uncertainties in the measured, circularized radii will then be underestimated, the effect is small in most cases.

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In summary, our simulations show that GALFIT measurements of half-light radii and total magnitudes are systematically underestimated for faint objects. We correct for systematic effects in the half-light radii and total magnitudes using the measured offsets in Figures 2 and 3, respectively. Note that the errors on r_e and total magnitude reported in this paper are also taken from these simulations.

We perform surface brightness profile fitting for our samples at z ∼ 7–12, using GALFIT and making use of our simulation results to correct for any systematic effects. We analyze each of the objects with >15σ detections individually (9 z₈₅₀-dropouts, 6 Y₁₀₅-dropouts), as well as the z = 11.9 object, which is formally detected at ∼8σ. We extend the analysis to fainter magnitudes using stacked observations. The fainter z₈₅₀- and Y₁₀₅-dropouts are split into two luminosity bins before stacking ($0.12 < L/L^\ast _{z=3} < 0.3$ and $0.048 < L/L^\ast _{z=3} < 0.12$), whereas we group all z ∼ 9 candidates into a single stack.

Figure 5 presents the results of Sérsic profile fitting for the 9 bright z₈₅₀-dropouts. Shown, from left to right, are the 3'' × 3'' cut-outs of the original image, the best-fit model produced by GALFIT, the residual images (original image − best-fit profile), and the segmentation maps used for masking all the neighboring objects during the profile fitting. Figure 6 similarly shows the results for the 6 bright Y₁₀₅-dropouts. All the objects are cleanly subtracted in the residual images. Note, however that three of the objects (two of the brightest z₈₅₀-dropouts, UDF12-4258-6567 and UDF12-3746-6327, as well as one of the Y₁₀₅-dropouts, UDF12-3952-7173) are significantly blended with neighboring objects in the original images.¹⁶ In addition, one of the Y₁₀₅-dropouts, UDF12-4470-6442, shows two cores. The uncertainties in the derived profile parameters for these objects will therefore be larger than for other isolated objects.

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Although some of the objects in our samples have been analyzed in a previous paper by Oesch et al. (2010a), as those authors do not provide a list of sizes for each candidate explicitly, a direct comparison is not possible. The only exception is UDFz-42566566. In Section 2 of their paper, they mention that they have treated this object as a two-component source and measured their sizes separately, and reported that the fainter component has a half-light radius of 0.5 kpc and the brighter one has a half-light radius of 0.8 kpc. We have selected the fainter component as UDF12-4258-6567, and obtained its half-light radius, r_e = 0.48 ± 0.03 kpc, which is in good agreement with the result by Oesch et al. (2010a). Note that the brighter component is not included in our sample, due to our rigorous non-detection criterion in the optical.

Figure 7 shows the profile fitting result for the z ∼ 12 object, UDF12-3954-6284. Since the magnitude of the z ∼ 12 object in H₁₆₀ measured with 050 diameter aperture is 29.2 mag, corresponding to S/N ∼8, the profile fitting for this object is quite challenging. Actually, the best-fit model galaxy profile seems more elongated than that in the original image shown in Figure 7, which would overestimate of its total magnitude. At least, the residual image in Figure 7 does not clearly show any noticeable residuals around the central position, although the uncertainties of the fitting parameters are relatively large as inferred from the moderate S/N ratio. If we measure the curve of growth for this object, using progressively larger circular apertures, we find that the magnitude saturates at 28.8 mag within an aperture diameter ∼045. We also find by this robust method that the half-light of the source is covered by about 035 diameter aperture, and after considering the PSF broadening effect, we obtain its half-light radius, r_hl = 0.45 kpc, which is consistent with the GALFIT measurement within ∼1σ, and is also nearly equal to the value reported by Bouwens et al. (2013), ∼0.5 kpc.

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Additionally, we note that this object has an unusual morphology. It is visually confirmed that the z ∼ 12 object has a diffuse filamentary structure stretching from northeast to southwest, although the significance is very low. This has been already mentioned very recently by Bouwens et al. (2013). Figure 8 shows the cutout H₁₆₀ images of this object from various subsets and the full data. The diffuse structure is seen in the full (2009+2012) data and in the 2009 data. The 2009b cutout also shows a low-S/N filament, and the 2009a cutout has a similar pattern along the same direction. If this diffuse filament is indeed associated with the source at z ∼ 12, it corresponds to its bright UV continuum and/or Lyα, which would suggest that this object is experiencing a major merger event, leading to their high star formation activity. This star formation enhancement may explain the visibility of such a high-redshift galaxy.

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Figures 9 and 10 show profile fitting results for the stacked z₈₅₀-dropouts and Y₁₀₅-dropouts, respectively, whose UV luminosities are $L=(0.12\hbox{--}0.3)L^\ast _{z=3}$ (top) and $L=(0.048\hbox{--}0.12)L^\ast _{z=3}$ (bottom). Also shown in Figure 11 is the profile fitting result for the stacked z ≃ 9 candidates. Note that we also make averaged (not median-stacked) images and perform profile fitting using GALFIT, which yields similar fitting results, although for some of the average stacks, GALFIT does not provide a reasonable fit due to severe confusion with neighboring objects.

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In the brightest luminosity bin, $L=(0.3\hbox{--}1)L^\ast _{z=3}$, we do not perform a stacking analysis, since the numbers of the dropouts are small (3 for z₈₅₀-dropouts and 3 for Y₁₀₅-dropouts) and the stacked images are significantly confused by neighboring objects. Instead, we calculate their average sizes and magnitudes; r_e = 0.64 ± 0.32 kpc and M_UV = −20.17 ± 0.32 mag (z ∼ 7) and r_e = 0.65 ± 0.28 kpc and M_UV = −20.14 ± 0.14 mag (z ∼ 8).

The best-fit parameters are summarized in Table 4 for the z₈₅₀-dropouts, Table 5 for the Y₁₀₅-dropouts, and Table 6 for the z > 8.5 candidates. The weighted means of half-light radii for the z₈₅₀-dropouts and Y₁₀₅-dropouts with $L=(0.05\hbox{--}1)L^\ast _{z=3}$ are 0.32 ± 0.07 kpc and 0.34 ± 0.07 kpc, respectively. In the next section, we present the size–luminosity relation, and investigate the redshift evolution of galaxy sizes and SFR surface densities based on these results.

Table 4. Surface Brightness Profile Fitting Results for Bright z₈₅₀-dropouts

Object ID	$m_{\rm UV}^{\rm (ap)}$ a	nb	mUVc	MUVd	ree
	(mag)		(mag)	(mag)	(kpc)
S/N >15, $L/L^\ast _{z=3} = 0.3\hbox{--}1$
UDF12-3746-6328	27.38	1.0	26.25 ± 0.06	−20.54 ± 0.06	1.09 ± 0.09
UDF12-4258-6567	27.41	1.0	26.74 ± 0.05	−20.22 ± 0.05	0.48 ± 0.03
UDF12-4256-7314	27.73	1.0	27.23 ± 0.05	−19.75 ± 0.05	0.36 ± 0.05
S/N >15, $L/L^\ast _{z=3} = 0.12\hbox{--}0.3$
UDF12-4219-6278	28.09	1.0	27.74 ± 0.10	−19.13 ± 0.10	0.25 ± 0.07
UDF12-3677-7536	28.22	1.0	27.82 ± 0.11	−19.00 ± 0.11	0.37 ± 0.09
UDF12-4105-7156	28.30	1.0	27.98 ± 0.13	−18.98 ± 0.13	0.20 ± 0.08
UDF12-3958-6565	28.33	1.0	27.87 ± 0.11	−19.02 ± 0.11	0.40 ± 0.10
UDF12-3744-6513	28.38	1.0	28.02 ± 0.13	−18.85 ± 0.13	0.15 ± 0.09
UDF12-3638-7162	28.41	1.0	28.05 ± 0.14	−18.79 ± 0.14	0.10 ± 0.08
Stack
UDF12z-stack1 (L/L* = 0.12–0.3)	28.33	1.0	27.94 ± 0.12	−18.93 ± 0.12	0.28 ± 0.09
UDF12z-stack2 (L/L* = 0.048–0.12)	29.20	1.0	28.73 ± 0.26	−18.19 ± 0.26	0.35 ± 0.14

Notes. ^aMeasured in 046 diameter aperture with the stack of the J₁₂₅ and J₁₄₀ images. ^bSérsic index. This is fixed, not measured. ^cTotal magnitude measured by GALFIT. The systematic effect is considered. ^dTotal absolute magnitude calculated using z_photo if available, otherwise z = 6.7, which corresponds to the peak of the selection function for z₈₅₀-dropouts. ^eCircularized half-light radius $r_e = a \sqrt{b/a}$, where a is the radius along the major axis, and b/a is the axis ratio.

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Table 5. Surface Brightness Profile Fitting Results for Bright Y₁₀₅-dropouts

Object ID	$m_{\rm UV}^{\rm (ap)}$ a	nb	mUVc	MUVd	ree
	(mag)		(mag)	(mag)	(kpc)
S/N >15, $L/L^\ast _{z=3} = 0.3\hbox{--}1$
UDF12-3879-7072	27.21	1.0	26.80 ± 0.03	−20.29 ± 0.03	0.34 ± 0.02
UDF12-4470-6443	27.69	1.0	27.13 ± 0.08	−19.95 ± 0.08	0.59 ± 0.08
UDF12-3952-7174	28.10	1.0	26.95 ± 0.10	−20.18 ± 0.10	1.03 ± 0.19
S/N >15, $L/L^\ast _{z=3} = 0.12\hbox{--}0.3$
UDF12-4314-6285	28.10	1.0	27.73 ± 0.08	−19.27 ± 0.08	0.33 ± 0.06
UDF12-3722-8061	28.28	1.0	27.89 ± 0.10	−19.20 ± 0.10	0.19 ± 0.06
UDF12-3813-5540	28.33	1.0	27.99 ± 0.11	−19.21 ± 0.11	0.10 ± 0.05
Stack
UDF12y-stack1 (L/L* = 0.12–0.3)	28.37	1.0	27.98 ± 0.11	−19.15 ± 0.11	0.31 ± 0.08
UDF12y-stack2 (L/L* = 0.048–0.12)	29.47	1.0	28.90 ± 0.29	−18.23 ± 0.29	0.36 ± 0.13

Notes. ^aMeasured in 050 diameter aperture with the stack of the J₁₄₀ and H₁₆₀ images. ^bSérsic index. This is fixed, not measured. ^cTotal magnitude measured by GALFIT. The systematic effect is considered. ^dTotal absolute magnitude calculated using z_photo if available, otherwise z = 8.0, which corresponds to the peak of the selection function for Y₁₀₅-dropouts. ^eCircularized half-light radius $r_e = a \sqrt{b/a}$, where a is the radius along the major axis, and b/a is the axis ratio.

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Table 6. Surface Brightness Profile Fitting Results for z > 8.5 Candidates

Object ID	R.A.a	Decl.a	$m_{\rm UV}^{\rm (ap)}$ b	nc	mUVd	MUVe	ref
	(h:m:s)	(d:m:s)	(mag)		(mag)	(mag)	(kpc)
UDF12-3954-6284	3:32:39.54	−27:46:28.4	29.24	1.0	28.47 ± 0.25	−20.41 ± 0.25	0.32 ± 0.14
Stack
UDF12hz-stack			29.69	1.0	29.11 ± 0.49	−18.21 ± 0.49	0.35 ± 0.16

Notes. ^aCoordinates are in J2000. ^bMeasured in 050 diameter aperture with the H₁₆₀ image. ^cSérsic index. This is fixed, not measured. ^dTotal magnitude measured by GALFIT. The systematic effect is considered. For UDF12-3954-6284, this might be overestimated, since the best-fit model galaxy profile seems more elongated than that in the original image shown in Figure 7. If we measure the curve of growth for this source, the magnitude saturates at 28.8 mag (see Section 4.2). ^eTotal absolute magnitude. We calculate it with z_photo = 11.9 for UDF12-3954-6284, considering IGM absorption shortward of its Lyα wavelength. We use the average photometric redshift, z_photo = 9.0 for the stacked object. ^fCircularized half-light radius $r_e = a \sqrt{b/a}$, where a is the radius along the major axis, and b/a is the axis ratio.

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We investigate the relation between size and luminosity, i.e., the size–luminosity relation, at each redshift. Figure 12 presents the size–luminosity relation for our z₈₅₀-dropout and Y₁₀₅-dropout galaxies at z ∼ 7–8. Our z > 8.5 galaxy candidates are not shown, because we cannot constrain the relation with only two measurements (one from an individual object and one from the stack). In Figure 12, fainter galaxies have a smaller half-light radius than brighter galaxies. This trend is the same as those of local galaxies (de Jong & Lacey 2000) as well as high-z dropout galaxies studied by Grazian et al. (2012). Because the luminosity of a galaxy depends on two physical quantities (surface brightness and size), one needs to clarify which quantity is dominant in shaping the size–luminosity relation. We define SFR surface density, Σ_SFR, as the average SFR in a circle whose radius is r_e,

$$\begin{equation} \Sigma _{\rm SFR} \,[M_\odot \, {\rm yr}^{-1} \, {\rm kpc}^{-2}] \equiv \frac{{\rm SFR} / 2}{\pi r_e^2}. \end{equation} \tag{ 2 }$$

A multiplicative factor of 1/2 is applied since the SFR is estimated from the total magnitude in the rest-frame UV. In the case that dust extinction is negligible, a rest-frame UV luminosity density approximately correlates with an SFR (Kennicutt 1998a),

$$\begin{equation} {\rm SFR} \,(M_\odot \, {\rm yr}^{-1}) = 1.4 \times 10^{-28} L_\nu \, ({\rm erg}\,{\rm s}^{-1} \,{\rm Hz}^{-1}). \end{equation} \tag{ 3 }$$

From Equations (2) and (3), we obtain

$$\begin{equation} M_{\rm UV} = - 2.5 \log \left(\frac{\Sigma _{\rm SFR} \, r_e^2}{1.4 \times 10^{-28} \cdot 2 \cdot \left({\rm 10\ pc} \, ({\rm cm})\right)^2} \right) -48.6. \end{equation} \tag{ 4 }$$

In Figure 12, we show constant SFR surface densities of Σ_SFR = 0.1, 0.3, 1, 3, and 10 (M_☉ yr⁻¹ kpc⁻²) with dashed lines. We find that most of the individual galaxies and stacked galaxies fall in the range of Σ_SFR ≃ 1–10 within their uncertainties. These results indicate that both bright and faint z ∼ 7–8 galaxies have similar SFR surface densities of Σ_SFR ≃ 1–10, and that the size–luminosity relation at each redshift is mainly determined by the size of galaxies that have a similar SFR surface density.

We then investigate the size evolution. Since the half-light radius depends on luminosity, as displayed by the size–luminosity relation, we carefully compare the half-light radii of our dropout galaxies within a fixed magnitude range. Figure 13 presents the average half-light radius as a function of redshift for our dropout galaxies at z ∼ 7–12 with $(0.3\hbox{--}1) L^\ast _{z=3}$ and $(0.12\hbox{--}0.3) L^\ast _{z=3}$, together with dropout galaxies at z ∼ 4–8 taken from the literature. Our measurements of average half-light radii are consistent with those from the previous studies at z ∼ 7–8, where we see overlap with previous measurements. Note that the sizes of our z ∼ 7 galaxies in the fainter luminosity bin are marginally discrepant with that estimated by Oesch et al. (2010a). As discussed earlier, unfortunately a galaxy-by-galaxy comparison is not possible. There are two possible reasons for this difference. First, the current paper deals with UDF12 data of much greater depth than that analyzed by Oesch et al. (2010a) who used only the first epoch of the HUDF09 survey. Second, our sample selection criteria are somewhat different as discussed by Schenker et al. (2013). Figure 13 indicates that the average half-light radius decreases with redshift from z ∼ 4 to 8, which is consistent with the claims of previous studies (e.g., Ferguson et al. 2004; Bouwens et al. 2004; Hathi et al. 2008; Oesch et al. 2010a).

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UDF12 provides us with the deepest ever near-infrared images of the HUDF, allowing our study to extend the dynamic range of redshift in the size evolution analysis from z ∼ 8 to z ∼ 12, and identifies that the decreasing trend holds up to z ∼ 12 as shown in Figure 13, if the putative z ∼ 12 source is real. Although the statistical uncertainty of the measurements is large, the half-light radius of z ∼ 12 is r_e = 0.32 ± 0.14 kpc in the luminosity bin of $(0.3\hbox{--}1) L^\ast _{z=3}$, which is significantly smaller than that of z ∼ 6 by a factor of three. Note that we can only plot the half-light radius at z ∼ 12 on the panel of $(0.3\hbox{--}1) L^\ast _{z=3}$ in Figure 13, because there are no z ∼ 12 galaxies with $(0.12\hbox{--}0.3) L^\ast _{z=3}$ in the UDF12 data. Similarly, our stack of z > 8.5 galaxies have a luminosity fainter than $(0.12\hbox{--}0.3) L^\ast _{z=3}$, which is too faint to be compared with the baseline of the average half-light radii at z ∼ 4–6. However, our results of z > 7 galaxies at these faint magnitudes, which are shown as gray filled circles in the bottom panel of Figure 13, are consistent with the decreasing trend of the half-light radius, albeit with large errors.

This decreasing trend may be explained by the evolution of the host dark halo radius. According to the analytic model in the hierarchical structure formation framework of ΛCDM (see, e.g., Mo et al. 1998; Mo & White 2002; Ferguson et al. 2004), the virial radius of a dark matter halo is given by

$$\begin{equation} r_{\rm vir} = \left(\frac{G M_{\rm vir}}{100 H(z)^2} \right)^{1/3}, \end{equation} \tag{ 5 }$$

where H(z) is the Hubble parameter and M_vir is the virial mass of the halo. The virial radius is also expressed as a function of the circular velocity of the dark halo by

$$\begin{equation} r_{\rm vir} = \frac{v_{\rm vir}}{10 H(z)}. \end{equation} \tag{ 6 }$$

Since H(z) is approximated by ∼(1 + z)^3/2 in a flat universe at high redshifts, the redshift evolution of the virial radius is r_vir∝H(z)^−2/3 ∼ (1 + z)⁻¹ for constant halo mass and r_vir∝H(z)⁻¹ ∼ (1 + z)^−1.5 for constant velocity.

Figure 13 shows the radius–redshift relation of dark matter halos for the case of constant halo mass and constant velocity. Previous studies investigating the radius–redshift relation in the redshift range 4 < z < 8 reach two different conclusions. Bouwens et al. (2004, 2006) claim that the relation is roughly (1 + z)⁻¹, suggestive that the sizes of disks scale with constant halo mass, while Ferguson et al. (2004) and Hathi et al. (2008) argue that (1  +  z)^−1.5, i.e., the constant velocity case, is preferable. We fit the radius–redshift relation over a wider range of redshift (extending to z ∼ 12) with a function of (1 + z)^s, where s is a free parameter. We take into account our size measurements: for the brighter bin, the average sizes of z₈₅₀- and Y₁₀₅-dropouts and the size of the z ∼ 12 source, and for the fainter bin, the measured sizes of the stacks of z₈₅₀- and Y₁₀₅-dropouts with $L=(0.12\hbox{--}0.3)L^\ast _{z=3}$. In addition, we use the results reported in the literature: the average sizes at z = 2.5 (Bouwens et al. 2004) and the average sizes at z = 4–6 (Oesch et al. 2010a).¹⁷ We fit the following two functions to the data, log r_e = slog (1 + z) + a₁ for $L=(0.3\hbox{--}1)L^\ast _{z=3}$ and log r_e = slog (1 + z) + a₂ for $L=(0.12\hbox{--}0.3)L^\ast _{z=3}$, where s, a₁, and a₂ are free parameters. Varying the three parameters, we search for the best-fitting set of (s, a₁, a₂) that minimizes χ². The best-fit parameters are $s = -1.30^{+0.12}_{-0.14}$, $a_1 = 1.00^{+0.09}_{-0.07}$, and $a_2 = 0.88^{+0.08}_{-0.09}$. Note that the best-fit parameters do not change significantly even if the putative z ≃ 12 object is in the fainter UV luminosity bin, which might be the case when its H₁₆₀ magnitude is boosted by a strong line emission as Ellis et al. (2013) proposed (see also Brammer et al. 2013). We have checked that exclusion of the putative z ≃ 12 object produces no significant change in our results, but it is nevertheless interesting that its size conforms with the trend established at slightly lower redshifts. We note that these results are clearly consistent with the redshift trend derived by Oesch et al. (2010a) from the early UDF09 data, as they reported s = −1.12 ± 0.17 for galaxies with luminosities in the range $L = (0.3\hbox{--}1)L^\ast _{z=3}$, and s = −1.32  ±  0.52 for the fainter galaxies with $L = (0.12\hbox{--}0.3) L^\ast _{z=3}$.¹⁸ However, our derived value for s is more accurate, both because of the improved data and galaxy samples provided by UDF12, and because we have chosen to fit a single value of s to both the bright and more modest luminosity galaxies.

It should be noted again that the above simple constant mass or constant velocity models assume that the stellar to halo size ratio does not change over this redshift range (Mo et al. 1998). To properly interpret our result, more realistic models are needed which carefully treat the stellar to halo size ratio, as well as consider effects on galaxy sizes from galaxy mergers, torquing, and feedback, based on a hierarchical galaxy formation scenario over the full redshift range (e.g., Somerville et al. 2008). These size measurements of high-redshift galaxies provide a launching point for a theoretical understanding of the structure of such galaxies, which has only recently been attempted but is of critical importance in understanding their properties (e.g., Muñoz & Furlanetto 2012).

Figure 14 presents the average star formation surface density, Σ_SFR, as a function of redshift. Note that the measurements are shown up to z ∼ 8 in Figure 14, because the uncertainty of the z ∼ 12 measurements are too large to place a meaningful constraint. Our galaxies at z > 7 have Σ_SFR ∼ 2. Σ_SFR appears to increase toward high redshifts. In fact, this increase of Σ_SFR is expected from the decreasing trend of galaxy size at a given luminosity (Figure 13). Since Σ_SFR is proportional to the UV luminosity density in the case of no dust extinction, Figure 14 indicates that UV luminosity surface brightness is higher for z ∼ 7–8 galaxies than for z ∼ 4–5 galaxies by a factor of 2–3. Figure 14 has data points of dust-corrected Σ_SFR. The dust-extinction corrected Σ_SFR is significantly larger than uncorrected Σ_SFR at z ∼ 4–5, while almost no difference is found at z > 6. Oesch et al. (2010a) claim that the dust-extinction corrected Σ_SFR is roughly constant over the redshift range of z ∼ 3–7. Extending their study, we find this constant Σ_SFR up to z ∼ 8.¹⁹ Dotted and dashed lines in Figure 14 show the Σ_SFR values for $L^\ast _{z=3}$ and $0.3L^\ast _{z=3}$ expected from the best-fit size evolution of (1 + z)^s. These lines imply that one would find galaxies with extremely high Σ_SFR at z ≳ 10 and beyond. A simple increase of the average Σ_SFR is not expected, however. This is because the typical UV luminosity, L*, is fainter at higher redshifts and galaxies with $(0.3\hbox{--}1.0) L^\ast _{z=3}$ are quite rare at z ≳ 10. In this sense, even higher-redshift galaxies cannot take an extremely high average Σ_SFR beyond ∼3–10 M_☉ yr⁻¹ kpc⁻². In the local universe, the SFR surface densities of normal disk galaxies are about 0.01 M_☉ yr⁻¹ kpc⁻², smaller than what we have found for z ∼ 4–8 dropouts. The centers of normal disk galaxies, on the other hand, reach about 1 M_☉ yr⁻¹ kpc⁻² (Figure 6 of Kennicutt 1998b; see also Momose et al. 2010), which is comparable to Σ_SFR of z ∼ 4–8 dropouts. Note that some local starbursts show comparable Σ_SFR to z ∼ 4–8 dropouts. However, because local starbursts, especially nuclear starbursts, have high Σ_SFR up to 100–1000 M_☉ yr⁻¹ kpc⁻² (Kennicutt 1998b), the star formation surface density of dropout galaxies at z ∼ 4–8 is significantly smaller than those of the extreme population found in the local universe, which indicates that star formation in dropout galaxies is not as rapid as that of local extreme starbursts. Speculatively, because high-z dropouts are metal- and dust-poor galaxies (e.g., Bouwens et al. 2012), gas cooling in a given amount of molecular clouds of high-z dropouts would be less efficient than that of local starbursts.

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