Size–Stellar Mass Relation and Morphology of Quiescent Galaxies at z ≥ 3 in Public JWST Fields

We use all available images of JWST/NIRCam from the F115W filter to the F444W filter (see Table 3 of V23 for the full list of available filters on each field). We use each filter's cutouts of 40 × 40 as input object images. The variance images generated by the Grizli pipeline (Brammer & Matharu 2021; Brammer et al. 2022) are used to make sigma images. The sigma image can sometimes be overestimated or underestimated; thus, it is renormalized by a correction factor, which makes the sigma image of the regions without any sources equal to its fluctuation. Object masks are generated using the photutils module. Objects more than 1'' away from the target objects are masked.

The point-spread function (PSF) is an essential factor in conducting the morphological fitting precisely. Previous studies use the theoretical PSFs (e.g., Suess et al. 2022) based on the WebbPSF software (Perrin et al. 2012, 2014), those based on the software with the real images (e.g., Zhuang & Shen 2024), such as PSFEx (Bertin 2011), or natural stars (e.g., Ding et al. 2022; Yang et al. 2022). This study utilizes the theoretical PSFs from WebbPSF smoothed to match their radial flux profile to the median profile of natural stars of each field. This approach copes with the possible difference in the profile between the theoretical PSFs and the natural stars reported recently (Ding et al. 2022; Ono et al. 2023) and is free from the noises that exist in the stacked star images.

The theoretical PSFs are first generated for each filter with WebbPSF. Here, in-flight sensing measurements are used by retrieving the Optical Path Difference files at the nearest dates to the observation date. The obtained PSFs are rotated to match the position angle (P.A.) Next, we derive the radial flux profile of natural stars. Stars are selected from the source catalog of each field by focusing on the unresolved objects based on the FLUX_RADIUS and with the small ellipticity from SEP, which is a similar approach to that employed in PSFEx. The detailed selection is summarized in Appendix A. We then smooth the PSFs from WebbPSF with the Moffat profile to match its one-dimensional radial profile with that of the median-stacked natural stars. The half-width at half-maximum (HWHM) of the used PSF is 003–008, which is larger at a longer wavelength (see Appendix Table 3). Appendix Figure 12 shows the one-dimensional radial profiles of smoothed PSFs used in this study in CEERS as an example.

In fitting the Sérsic profile to images, we use GALFIT (Peng et al. 2002, 2010). It uses real object images, sigma images, mask images, and PSFs as inputs and returns the best-fit parameters. We first fit the F277W image of each galaxy, which shows the light profile at around the rest frame 0.5 μm, similar to van der Wel et al. (2014) for our galaxies (λ_rest = 0.49–0.72 μm depending on their redshift). In the fitting, the magnitude m, effective radius r_eff, Sérsic index n, axis ratio q, P.A., and the central pixel coordinates are set as the free parameters. We allow these parameters in the following range to avoid the unrealistic cases: −3 mag < m–m_SEP < 3 mag, 0.1 pixel < r_eff < 500 pixel, 0.2 < n < 8, and 0.0001 < q < 1, where m_SEP is the Kron magnitude from SEP. Next, we fit the images in the other filters. In this case, we fix the axis ratio and P.A. as the best-fit value in the F277W images as in Nedkova et al. (2021) since they are not likely to depend on the wavelength. This enables us to derive the effective radius and the Sérsic index consistently in the different wavelengths. We get the Sérsic index in the edge of the parameter range (n = 8) for some galaxies. In that case, we refit the images of all filters, assuming n = 4. The sky background of the images is also set to be a free parameter, but the result does not change significantly even if we fix it to the value derived independently. Also, surrounding objects within 1'' from the target galaxies are simultaneously fit by the Sérsic profile.

Figure 3 shows an example of the result of the fitting for all available images of a galaxy, which is the typical one of our sample with the redshift and the stellar mass close to the median values. Appendix B also summarizes the fitting results in the F200W filter and F277W filter and the best-fit parameters in the F277W filter of all galaxies used in Sections 4 and 5 (see Section 3.5 for the selection). Some galaxies have compact residuals with both positive and negative pixels. These residuals are often seen in HST images of compact galaxies with high Sérsic index, too (e.g., Marsan et al. 2019; Lustig et al. 2021; Forrest et al. 2022). Most residuals are just 10% of the flux of the original images at most. The median reduced chi-square of the fitting for targets is ${\chi }_{\nu }^{2}=1.0-1.2$ in each filter, suggesting that the assumption of the single Sérsic profile is overall good.

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To estimate realistic uncertainties of the obtained parameters, we conduct the following procedure. The PSF-convolved Sérsic profile of the best-fit parameters for each galaxy is first inserted into a noise image for each filter and each field. The noise images are created by randomly cutting out the real images without any sources. We apply GALFIT to this mock image in the same manner as for the real image and obtain the best fit. This procedure is repeated for 100 different noise images, and the uncertainty of each parameter is taken as the range between the 16th and 84th percentile values of this distribution. We find that the original uncertainty reported by GALFIT is two to five times more underestimated depending on wavelength than that derived by this procedure.

The quality of the GALFIT best-fit model depends on the S/N of the images (e.g., see Figure 6 in van der Wel et al. 2012). Thus, the different brightness and different profiles of galaxies will have a different bias and quality in the fitting. Here, we evaluate the bias and quality of the fitting using mock galaxy images following the Sérsic profile.

First, 3000 different single Sérsic profiles with various ranges of parameters are prepared. Parameters for each profile are randomly selected in the following range: 21 mag < m < 29 mag, 0002 < r_eff < 2'', and 0.1 < q < 1. The P.A. is set to be random. Since we are focusing on quiescent galaxies, which are like to have bulge-like structures as supported by the literature (e.g., Lustig et al. 2021) and Section 5.1 of this paper, this section assumes the Sérsic index as n = 3–5. For reference, Appendix C shows the case of n = 1–3. Its trend is broadly consistent with the case with n = 3–5, except for the smallest input size (0002 < r_eff < 002). Next, after convolving with the corresponding PSF, we insert them into 30 different noise images for each. In this section, the CEERS F277W image is used as an example. We note that the results do not significantly change even if we use images in different filters. In total, we generate 90,000 mock Sérsic profile images. We fit the suite of mock galaxies using GALFIT in the same manner as for the real galaxies and compare the obtained parameters with the input parameters.

Figure 4 shows the relative uncertainty of the effective radius, Sérsic index, and axis ratio. As an example, we here focus on the trend at ∼26.3 mag, which corresponds to the S/N = 50 detection in the CEERS F277W image (see V23 for the depth of images). First, the effective radii are slightly overestimated in smaller input sizes (20% in 0002 < r_eff < 002) and slightly underestimated in larger input sizes (20% in 02 < r_eff < 2''). The Sérsic index is underestimated (10%–40%, equal to δ n ∼ 0.4 − 1.6 for n = 3–5) regardless of the input size and axis ratio. The axis ratio is well reproduced for 002 < r_eff < 02, but underestimated in larger or smaller sizes, in particular larger axis ratio (q > 0.4). Fainter (brighter) objects are expected to have more (less) significant scatter and bias in all parameters.

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As described in the following sections, our sample typically has r_eff ∼ 0.6 kpc (∼01) and q ∼ 0.6–0.8. Therefore, this test implies that our sample with our S/N cut of S/N > 50 is typically expected to have only a negligible offset in the effective radii and axis ratio (<10%) and a slight offset in the Sérsic index (<30%).

It is known that the size of galaxies has a dependency on the observed wavelength at z < 3 (e.g., van der Wel et al. 2014; Kawinwanichakij et al. 2021; Suess et al. 2022). Our sample spans a wide redshift range; thus, obtaining the effective radius in the same rest-frame wavelength for all galaxies is important for a fair comparison with the literature at lower redshift.

We obtain the effective radius in the rest frame 0.5 μm and the wavelength dependence of the effective radius by fitting the measured effective radii in the different wavelengths to the following form:

$$\begin{eqnarray}&&\mathrm{log}{r}_{\mathrm{eff}}({\lambda }_{\mathrm{rest}})=\gamma \mathrm{log}\left(\displaystyle \frac{{\lambda }_{\mathrm{rest}}}{0.5\,\mu {\rm{m}}}\right)+\mathrm{log}{r}_{\mathrm{eff},0.5\mu {\rm{m}}},\end{eqnarray} \tag{ 1 }$$

where γ and r_eff,0.5μm correspond to the strength of the wavelength dependence of the effective radius ($\gamma ={\rm{\Delta }}\mathrm{log}{r}_{\mathrm{eff}}/{\rm{\Delta }}\mathrm{log}{\lambda }_{\mathrm{rest}}$) and the effective radius at the rest frame 0.5 μm. λ_rest is the wavelength in the rest frame according to the photometric redshift. We note that the uncertainty in photometric redshift (see Section 2) induces only negligible impact on the rest-frame wavelength λ_rest (∼3%). Following the literature, we use the effective radius along the semimajor axis as the proxy of the galaxy size. In the fitting, we use the effective radii of the images in all bands where the target source is detected with an S/N > 50. This detection threshold is to assure the measurement quality in each band and is motivated by the mock simulation described in Section 3.4. In addition, the best effective radii at the edge of the parameter range (i.e., r_eff = 0.1 pixel or 500 pixels) are not used here. After these selections, we require that at least three bands be used. The number of data points in this fitting depends on the field and the target's brightness, which spans from three to seven.

We successfully determine γ and r_eff,0.5μm of 26 quiescent galaxies, which we use in the following analysis. Two galaxies (#7912 and #362) are removed from the discussion because they are too compact and have the best effective radii at the edge of the parameter range in all wavelengths at least longer than 2.7 μm and thus do not have enough size measurements. Among 26 galaxies, all of them are selected by the UVJ diagram, and 13 of them are selected by the NUVUVJ diagram. We remind the reader that Appendix B summarizes the fitting result and the best-fit parameters of all of them.

Figure 5 shows the strength of the wavelength dependence of the effective radius γ as a function of the stellar mass. Most quiescent galaxies have a negative correlation between the wavelength and the size, i.e., smaller sizes in longer wavelengths. The median values of UVJ-selected and NUVUVJ-selected samples is $\gamma =-{0.38}_{-0.08}^{+0.08}$ and $\gamma =-{0.33}_{-0.11}^{+0.10}$, respectively. This negative correlation between wavelength and size agrees with the lower redshift results (e.g., van der Wel et al. 2014; Kawinwanichakij et al. 2021; Suess et al. 2022). van der Wel et al. (2014) report the average strength of the wavelength dependence of the effective radius of quiescent galaxies at 0 < z < 2 as γ = −0.25, less significant than our value. This could be due to the redshift evolution or the different stellar mass distribution between our sample and van der Wel et al. (2014). Given the small size of our sample, we do not attempt to model the correlation between the stellar mass and the strength of the wavelength dependence of the effective radius as reported in Kawinwanichakij et al. (2021), but we note that most of our sample is in good agreement with the relationship at the highest redshift bin (0.8 < z < 1.0) in Kawinwanichakij et al. (2021) with some outliers with the stronger negative gradient at lower redshift and lower mass.

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We show quiescent galaxies at z ≥ 3 in the size–mass plane in Figure 6. The effective radii at the rest frame 0.5 μm obtained in Section 3.5 are used. First of all, we can see that many quiescent galaxies of our sample have effective radii of less than 1 kpc, which is significantly smaller than those of star-forming galaxies at z < 3 (e.g., van der Wel et al. 2014). Their effective radii appear to correlate with their stellar mass, i.e., more massive galaxies have larger radii. This is consistent with the size–mass relation of quiescent galaxies in lower redshift studies (e.g., van der Wel et al. 2014; Mowla et al. 2019b), which will be discussed in the following section.

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We see that several galaxies deviate from the trend. Some of the less massive galaxies ($\mathrm{log}({M}_{\star }/{M}_{\odot })\lt 10.5$) have r_eff > 2 kpc in the UVJ-selected case (#788, #7399, #8967). This may mean that these galaxies are in a period of morphological transformation from star-forming galaxies and larger than quiescent galaxies or the contaminants of star-forming galaxies. On the other hand, we can not rule out that this is due to the difficulty in fitting the Sérsic profile to these galaxies. For example, #788 has an elongated structure, which can be a sign of a merger (Appendix Figure 13). Also, #7399 can be affected by nearby sources. #8967 has an arclet shape, but it is not likely strongly lensed since the redshift of the companion galaxy (z = 3.49) is consistent with #8967. We note that removing these galaxies from the sample does not affect our conclusions. In contrast, there are galaxies with significantly small sizes (r_eff < 0.1 kpc, #185 and #1427 in PRIMER). AGNs can cause the apparent compact morphology, but we do not have clear evidence that this is due to their contribution because they are not detected in X-ray (X-UDS, Kocevski et al. 2018). The limiting flux of X-UDS survey is 1.4 × 10⁻¹⁶ erg cm⁻² s⁻¹ in the 0.5–2 keV. This corresponds to the limiting luminosity in the rest frame 2–10 keV of L_X = (1.5 − 1.6) × 10⁴³ erg s⁻¹ at their redshift, which implies that they are at least not likely powerful unobscured X-ray AGNs. We find that removing these galaxies from the sample does not affect our conclusions, either.

We note that our stellar mass is robust against the codes we use. The stellar mass of our sample measured with eazy-py is compared to that measured with FAST++. ¹⁸ They are in good agreement with the offset of $\mathrm{log}({M}_{\star ,\mathrm{EAZY}}/{M}_{\odot })-\mathrm{log}({M}_{\star ,\mathrm{FAST}++}/{M}_{\odot })$ = ${0.069}_{-0.066}^{+0.075}\ ({0.067}_{-0.027}^{+0.033})$ for our UVJ (NUVUVJ)-selected quiescent galaxies, respectively. van der Wel et al. (2014), Mowla et al. (2019b), and Nedkova et al. (2021), whose size–mass relation will be compared with ours, use the stellar mass measured with FAST. Thus, this supports that there is no significant bias in comparing our size–mass relation with them.

Motivated by the appearance of the existence of size–mass relation, we attempt to analytically fit our sample following the literature (van der Wel et al. 2014; Mowla et al. 2019b; Kawinwanichakij et al. 2021; Nedkova et al. 2021). First, following van der Wel et al. (2014), the effective radii distribution is assumed to be a log-normal distribution $N(\mathrm{log}{R}_{\mathrm{eff}},{\sigma }_{\mathrm{log}{R}_{\mathrm{eff}}})$, where $\mathrm{log}{R}_{\mathrm{eff}}$ is the mean of the effective radius and ${\sigma }_{\mathrm{log}{R}_{\mathrm{eff}}}$ is the intrinsic scatter of the distribution. We assume that the mean of the effective radius R_eff depends on the stellar mass as

$$\begin{eqnarray}&&{R}_{\mathrm{eff}}/\mathrm{kpc}\,=\,A{\left(\displaystyle \frac{{M}_{\star }}{5\times {10}^{10}{M}_{\odot }}\right)}^{\alpha },\end{eqnarray} \tag{ 2 }$$

where A is the effective radius at M_⋆ = 5 × 10¹⁰ M_⊙ and α is the slope of the size–mass relation. We consider the uncertainty δ r_eff of the observed effective radius r_eff by assuming it follows a Gaussian distribution. The δ r_eff includes not only the observed uncertainty of r_eff but also the uncertainty of the stellar mass, assuming 0.21 dex, the scatter of stellar mass from eazy-py (Gould et al. 2023, and also see the discussion in Section 4.1). The latter is included by converting it to the uncertainty of the effective radius by assuming the typical slope (α = 0.6) of the size–mass relation at lower redshift (e.g., Mowla et al. 2019b).

Following van der Wel et al. (2014), we calculate the total likelihood for three parameters (A, α, and ${\sigma }_{\mathrm{log}{R}_{\mathrm{eff}}}$) as

$$\begin{eqnarray}&&{ \mathcal L }\,=\,{\rm{\Sigma }}\mathrm{ln}\left[W\cdot \langle N(\mathrm{log}{R}_{\mathrm{eff}},{\sigma }_{\mathrm{log}{R}_{\mathrm{eff}}}),N(\mathrm{log}{r}_{\mathrm{eff}},\delta \mathrm{log}{r}_{\mathrm{eff}})\rangle \right],\end{eqnarray} \tag{ 3 }$$

where W is the weight to avoid the dominance of lower-mass galaxies. We use the inverse of the stellar mass function of quiescent galaxies at 3 < z < 3.5 in Weaver et al. (2023) as the weight. The stellar mass function itself has uncertainty due to effects such as cosmic variance (e.g., Moster et al. 2011; Steinhardt et al. 2021), SED modeling uncertainties (e.g., Marchesini et al. 2009; Pacifici et al. 2023), photometric redshift, and Poisson noise. The weight fluctuates according to the uncertainty of the stellar mass function to include these effects.

Several studies report that the size–mass relation bends at $\mathrm{log}({M}_{\star }/{M}_{\odot })=10-10.5$ at z < 2 (e.g., Mowla et al. 2019a; Kawinwanichakij et al. 2021; Nedkova et al. 2021; Cutler et al. 2022). To avoid the deviation from our assumption in Equation (2), we only fit the galaxies with $\mathrm{log}({M}_{\star }/{M}_{\odot })\gt 10.3$. This is the same stellar mass limit imposed in van der Wel et al. (2014). In addition, dusty star-forming galaxies could be contaminating our sample (e.g., Pérez-González et al. 2023). They could have larger rest-frame optical sizes than typical quiescent galaxies (e.g., ∼2 kpc at z ∼ 3 in Gillman et al. 2023), driving a larger scatter of the relation. Thus, we randomly remove 20% of the galaxies from the sample in the fitting, which is the contamination fraction of UVJ quiescent galaxies in Schreiber et al. (2018). Following the literature (e.g., van der Wel et al. 2014), we randomly select those to be removed by weighting galaxies by the ratio of the stellar mass function of star-forming galaxies and that of quiescent galaxies at 3 < z < 3.5 in Weaver et al. (2023). Even if we do not account for these potential contaminants, we find that the best-fit parameters are consistent within their 1σ uncertainty. In the fitting, the Markov Chain Monte Carlo (MCMC) method is employed using emcee (Foreman-Mackey et al. 2013). The fitting procedure, including fluctuating the weight W and randomly removing the possible contaminants, is repeated 500 times. The best-fit values of A (the effective radius at M_⋆ = 5 × 10¹⁰ M_⊙), α (the slope of the size–mass relation), and ${\sigma }_{\mathrm{log}{R}_{\mathrm{eff}}}$ (the intrinsic scatter of the size–mass relation) are obtained from the total of their chain.

Figure 6 shows the best fit of the size–mass relation, and Table 1 summarizes the best-fit values of the parameters. The best-fit α confirms that the positive correlation between the stellar mass and the effective radius exists in both UVJ-selected and NUVUVJ-selected galaxies, even considering the 1σ uncertainty. The intrinsic scatter is ∼0.2 dex for both the UVJ-selected and NUVUVJ-selected sample, similar to those reported at z < 3 (e.g., van der Wel et al. 2014; Kawinwanichakij et al. 2021).

Table 1. Summary of the Obtained Best-fit Parameters of the Size–Mass Relation

Sample	Na	$\mathrm{log}(A/\mathrm{kpc})$ b	αb	${\sigma }_{\mathrm{log}{R}_{\mathrm{eff}}}$ b
UVJ	22	$-{0.18}_{-0.06}^{+0.06}$	${0.32}_{-0.21}^{+0.20}$	${0.18}_{-0.05}^{+0.07}$
UVJ (z < 3.3)	12	$-{0.14}_{-0.08}^{+0.08}$	${0.43}_{-0.35}^{+0.29}$	${0.16}_{-0.06}^{+0.09}$
UVJ (z > 3.3)	10	$-{0.26}_{-0.17}^{+0.15}$	${0.39}_{-0.49}^{+0.43}$	${0.28}_{-0.11}^{+0.13}$
NUVUVJ	12	$-{0.24}_{-0.09}^{+0.08}$	${0.54}_{-0.28}^{+0.24}$	${0.20}_{-0.08}^{+0.13}$
vDW14 (z = 2.75) c	⋯	−0.06 ± 0.03	0.79 ± 0.07	0.14 ± 0.03

Notes.

^a The number of galaxies used in deriving the best-fit parameters of the size–mass relation. ^b Their uncertainties are taken from the 16th to 84th percentiles of the total chain in the MCMC fitting. ^c The best-fit parameters of early-type galaxies at 2.5 < z < 3.0 reported in van der Wel et al. (2014).

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We split UVJ-selected quiescent galaxies into two subsamples based on their photometric redshift, i.e., z < 3.3 and z > 3.3. The threshold corresponds to the median of the redshift of the sample. Figure 7 shows the size–mass relation of these subsamples and their best fit. Table 1 summarizes the best-fit values of the parameters. Though the uncertainty is more considerable than the total UVJ-selected sample, we see a positive correlation between the stellar mass and the effective radius even in these subsamples. The effective radius at the fixed stellar mass is smaller in the high-redshift sample than in the low-redshift sample. This suggests a potential redshift evolution of the sizes of quiescent galaxies at z ≥ 3.

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Figure 8 shows the evolution of the effective radius at M_⋆ = 5 × 10¹⁰ M_⊙ (i.e., A) and the slope α. In this figure, we compare our values with those of quiescent galaxies at lower redshift (z < 3) (van der Wel et al. 2014; Mowla et al. 2019b; Nedkova et al. 2021). We note that all of the above studies select quiescent galaxies based on the UVJ diagram. We confirm that the effective radius at M_⋆ = 5 × 10¹⁰ M_⊙ of this study is smaller than those at z < 3, implying that the size of the quiescent galaxies monotonically increases since 3 < z < 4. Our measured relations are ∼0.2 dex below those at z ∼ 2 from the literature and have the smallest amplitude in the studies. Moreover, our measurement is in good agreement with the extrapolation of the redshift evolution of the effective radii suggested in van der Wel et al. (2014), which is R_e = 5.6 × (1 + z)^−1.48 kpc or R_e = 4.3 × h(z)^−1.29 kpc. The UVJ-selected sample subdivided by redshift is also consistent with these extrapolations.

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The slope of the size–mass relation of quiescent galaxies does not seem to evolve with redshift. Those drawn from NUVUVJ-selected and UVJ-selected quiescent galaxies—also divided in redshift bins—are consistent. Our values are also consistent with those at z < 3 reported in Mowla et al. (2019b) and Nedkova et al. (2021). Also, this slope is consistent with the high-mass end slope of the relation in Cutler et al. (2022). However, our values are slightly flatter than those in van der Wel et al. (2014), particularly for the UVJ-selected sample. The possible origin of this difference will be discussed in Section 6.

Lastly, Figure 9 compares our size–mass relation with those at lower redshift (van der Wel et al. 2014; Kawinwanichakij et al. 2021; Nedkova et al. 2021) and the size measurements of individual quiescent galaxies at z ≥ 2.5 in the literature (Gobat et al. 2012; Kubo et al. 2018; Saracco et al. 2020; Esdaile et al. 2021; Lustig et al. 2021; Forrest et al. 2022; Carnall et al. 2023b). Using the K_s -band image of the AO-assisted Very Large Telescope/High Acuity Wide-field K-band Imager (HAWK-I), we also obtain the structural parameters of a quiescent galaxy at z = 4.01 (SXDS-27434) reported in Tanaka et al. (2019) and Valentino et al. (2020). Its effective radius is estimated as ${r}_{{\rm{e}}}\,=\,{0.74}_{-0.09}^{+0.20}\,\mathrm{kpc}$ (see Appendix D for further details). Though the observed wavelength differs among the sample, and many of these values are not corrected for the wavelength dependence of the size (except Lustig et al. 2021; Forrest et al. 2022), they are overall consistent with our measurements. Some of the quiescent galaxies in Lustig et al. (2021) are larger than our size–mass relation, and this might be due to the redshift difference because of lower redshift for most of them than ours. It should be noted that all z ≥ 4 measurements in Kubo et al. (2018) and Carnall et al. (2023b) are below our size–mass relations, as is SXDS-27434 at z = 4.01. This is in agreement with our z ≥ 4 quiescent galaxies (#2876 at z = 4.146, #185 at z = 4.508, and #10084 at z = 4.633) with two-thirds smaller than our relations, which are 0.42 kpc at $\mathrm{log}({M}_{\star }/{M}_{\odot })=11.0$ on average. Such a systematically small size for the z ≥ 4 sample may imply that the size evolution of quiescent galaxies starts at z ≥ 4.

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The Sérsic index measured in F277W, the filter close to the rest frame 0.5 μm at z ∼ 3–4, is shown in the top panel of Figure 10. Here, we only focus on the objects with successful measurements of the Sérsic index (i.e., the Sérsic index is not fixed to n = 4 in the fitting). The measured Sérsic indices range from n = 0.5–6, with median values of $n={3.42}_{-0.57}^{+0.49}$ and $n={3.61}_{-0.47}^{+0.51}$ for the UVJ-selected and NUVUVJ-selected quiescent galaxies, respectively. These values and uncertainties are computed by bootstrapping 5000 times, considering the uncertainty of the Sérsic index of each galaxy. The top panel of Figure 10 shows the median of two subsamples divided according to their redshift, as done in Section 4.2. They do not show significant redshift evolution in their median values and are in good agreement with lower redshift results (e.g., van der Wel et al. 2012; Marsan et al. 2019; Esdaile et al. 2021; Lustig et al. 2021). This suggests that most quiescent galaxies have bulge-dominant structures as early as z ∼ 4.

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One point that we should mention from our sample is that some quiescent galaxies have a low Sérsic index of n ∼ 0.5–2. To investigate this further, we next derive the disk fraction, defined as the number of galaxies with n < 2 divided by the total number of galaxies. These values and their uncertainties are also computed by bootstrapping. The bottom panel of Figure 10 shows the disk fraction as a function of redshift. Though both bins at lower redshifts are consistent with zero, the disk fraction increases toward higher redshifts. In particular, the high redshift bin (z > 3.3) for UVJ-selected quiescent galaxies has a nonzero value (${0.36}_{-0.18}^{+0.09}$) even considering the 1σ uncertainty. Our sample suggests that some fraction of quiescent galaxies at high redshift can be disk-dominant morphology, in line with the literature (e.g., Toft et al. 2017; Newman et al. 2018; Fudamoto et al. 2022).

The observed axis ratio of each galaxy projected to the images depends on its inclination angle, but we can trace the intrinsic shape of the sample by taking the median and removing the effect from the inclination angle. We find that the median axis ratio in F277W is $q={0.61}_{-0.09}^{+0.12}$ and $q={0.78}_{-0.20}^{+0.04}$ for the UVJ-selected and NUVUVJ-selected samples, respectively. These values and uncertainties are again obtained from bootstrapping, as for the median value of the Sérsic index (Section 5.1). The NUVUVJ-selected sample has a higher median value than the UVJ-selected sample, though with a notably larger uncertainty. Figure 11 shows the individual values of our sample as a function of their redshift and the median value for the 2.8 < z ≤ 3.3 and 3.3 < z < 4.6 samples. We do not see any significant dependence of the axis ratio on either the redshift or the stellar mass.

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Our median values are in good agreement with the measurement of quiescent galaxies at z < 3 (e.g., van der Wel et al. 2012; Straatman et al. 2015; Lustig et al. 2021) and at z ∼ 3–4 (Straatman et al. 2015; Esdaile et al. 2021). Coupled with the median Sérsic index, these values provide further evidence that the majority of quiescent galaxies have bulge-like structures as early as z ∼ 4.