ALMA Observations of CO Emission from Luminous Lyman-break Galaxies at z = 6.0293–6.2037

For J0235–0532, the CO(6–5) emission line is marginally detected at the expected frequency from the systemic redshift, while for the other two targets, the CO(6–5) emission is not significantly detected. Figure 1 shows the ALMA spectra of our z = 6 luminous LBGs around their CO(6–5) emission line as expected from their systemic redshift measured by Harikane et al. (2020b) with the far-infrared (FIR) emission lines of [O iii] and [C ii]. The spectrum of J0235–0532 is extracted by placing a single beam aperture around the peak position of the CO emission in the CO(6–5) moment-zero map (velocity-integrated map; Figure 2), because the CO emission is not spatially resolved in the ALMA data. We fit Gaussian functions to the observed spectrum of J0235–0532 from 97.3 to 97.8 GHz and obtain the best-fit Gaussian function as presented in Figure 1. The integrated flux of this line calculated from the best-fit function is 0.0652 ± 0.0175 Jy km s⁻¹, indicating that the CO(6–5) emission line of J0235–0532 shows a marginal detection at the ≃4σ significance level. Reassuringly the velocity width of the CO line is comparable to those of the previously detected [C ii] and [O iii] lines (Harikane et al. 2020b). Because the CO(6–5) is not significantly detected for J1211–0118 and J0217–0208 (Figures 1 and 2), we extract their spectra by placing a beam aperture based on the coordinates of their rest UV continuum emission. The upper limits of their CO(6–5) line fluxes are calculated from the square root of the sum of the squared flux density errors in the range of ±250 km s⁻¹ around the expected CO(6–5) frequency from their systemic redshift. The range of ±250 km s⁻¹ is comparable to twice the FWHM of their [O iii] and [C ii] emission lines (their FWHMs are about 170–370 km s⁻¹; Table 1 of Harikane et al. 2020b). The integrated emission line flux or the upper limit for each target and the observed FWHM of the detected line are presented in Table 3.

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Table 3. Summary of Our Observational Results

	J1211–0118	J0235–0532	J0217–0208
CO(6–5) integrated flux (Jy km s−1)	<0.0713	0.0652 ± 0.0175	<0.0609
CO(6–5) FWHM (km s−1)	⋯	237 ± 51	⋯
LCO(6−5) (107 L⊙)	<3.41	2.88 ± 0.773	<3.95
$L{{\prime} }_{\mathrm{CO}(6-5)}$ (109 K km s−1 pc2)	<3.22	2.72 ± 0.73	<3.74
fν,430 μm (μJy)	<52.3	<29.1	<22.1
LIR (1011 L⊙)	${3.6}_{-1.9}^{+34.4}$	${5.8}_{-5.8}^{+19.4}$ a	${2.0}_{-0.3}^{+5.9}$
Tdust (K)	${40}_{-14}^{+44}$	50–80 (fixed)	${31}_{-9}^{+26}$
${f}_{\mathrm{CMB}}^{\mathrm{CO}(6-5)}$	0.72	0.79–0.89	0.55
SFRIR (M⊙ yr−1)	${39}_{-21}^{+375}$	${63}_{-63}^{+211}$ a	${22}_{-3}^{+64}$
SFRtot (M⊙ yr−1)	${88}_{-21}^{+375}$	${112}_{-64}^{+211}$	${98}_{-5}^{+65}$
Mgas (1010 M⊙)	<8.99	7.59 ± 4.74	<10.4
ΣSFR (M⊙ yr−1 kpc−2)	${9.6}_{-2.3}^{+41.5}$	${18.9}_{-10.7}^{+35.6}$	${48.2}_{-2.5}^{+31.6}$
Σgas (103 M⊙ pc−2)	<9.9	12.8 ± 8.0	<5.1
fgas b	<0.59	${0.55}_{-0.23}^{+0.12}$	<0.46
tdep (Gyr)	<1.02	${0.68}_{-0.47}^{+0.90}$	<1.06

Notes. The upper limits are 3σ.

^a These values are the 3σ upper limits when T_dust = 50 K, and the upper error takes into account the case when T_dust = 80 K. For details, see the text in Section 4.2. ^b The quoted uncertainties in the gas fraction do not include the systematic uncertainty associated with the stellar mass estimates.

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Figure 3 compares the ALMA spectra of our z = 6 luminous LBGs around the CO(6–5) emission line with those around the [O iii] and [C ii] emission lines. Although the S/N of the CO(6–5) line of J0235–0532 is not high, the redshifts based on the CO, [O iii], and [C ii] lines are broadly consistent with each other. Some previous studies have shown significant velocity shifts between [O iii] and [C ii] (e.g., BDF-3299 in Carniani et al. 2017; see Hashimoto et al. 2019; Bakx et al. 2020 as counterexamples); our results suggest that the velocity shift in J0235–0532, if any, is smaller than the previous results.

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Figure 4 presents the CO(6–5) emission contours of J0235–0532 with the Subaru HSC z-band image probing the rest-frame UV continuum emission. Although the positions of CO(6–5) and UV continuum appear to be slightly offset, this may be caused by the relatively low S/N of the CO emission. We estimate the uncertainties of the CO peak position by running a suite of Monte Carlo simulations in the same way as Harikane et al. (2020b). We add artificial noises to the actual data according to a Gaussian random distribution with a standard deviation equal to the 1σ noise of the data, and remeasure the peak positions one thousand times to estimate the uncertainties of the CO peak position. We find that the CO peak position is consistent with that of the UV continuum within the 2σ uncertainties. In Figure 4, we also present the [C ii] and [O iii] positions obtained in Harikane et al. (2020b), confirming that the CO peak position is also consistent with those of [C ii] and [O iii].

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For a sanity check of the position of J0235–0532 in the HSC astrometry, which has been calibrated against the Pan-STARRS first data release (DR1) catalog (Chambers et al. 2016), we use nearby ($\lt 1^{\prime}$) bright stars whose positions are accurately measured in the Gaia early data release 3 (EDR3) catalog (Gaia Collaboration et al. 2016, 2021). ²³ We confirm that the positional differences of the nearby bright stars between the HSC data and the Gaia EDR3 catalog are only within <001 with no systematic offsets, which is consistent with similar comparison results in previous work on a much larger scale (Section 6.3 of Aihara et al. 2019).

As mentioned above, the significance of the CO(6–5) line from J0235–0532 is only about 4σ. However, the observed CO peak position on the sky is consistent with that of the UV continuum, and the observed CO frequency is also in good agreement with those of the previously detected FIR emission lines; the probability of these events occurring simultaneously by chance is lower than the estimate above. We calculate the combined probability that these three events occur simultaneously by chance based on Fisher's method (Fisher 1970; see also Finke et al. 2015; Mulders et al. 2018; Kikuchihara et al. 2022; cf. Heard & Rubin-Delanchy 2018). First, the probability that the CO detection is a false positive can be calculated from the significance of the CO line from J0235–0532. We obtain a false-positive probability (p-value) of p₁ ≃ 9.7 × 10⁻⁵, assuming that the flux measurement errors follow a Gaussian distribution. Second, the probability that the CO peak position is consistent with the previously detected source position can be calculated from the ratio of the area corresponding to the 2σ range to that of the obtained ALMA data. The p-value for this event is estimated to be p₂ ≃ 7.3 × 10⁻⁴. Third, the probability that the detected line frequency coincides with those of the previously detected FIR lines can be calculated from the ratio of twice the FWHM frequency range to that of the four SPWs. The p-value for this event is estimated to be p₃ ≃ 2.1 × 10⁻². From these individual p-values, we calculate a test statistic (TS) for the combined probability,

$$\begin{eqnarray}&&\mathrm{TS}=-2\sum _{i=1}^{k}\mathrm{ln}{p}_{i},\end{eqnarray} \tag{ 1 }$$

where k = 3 in this case. By comparing this TS with the χ² distribution with 2k degrees of freedom, ${\chi }_{2k}^{2}$, we obtain a combined p-value p_com ≃ 3.4 × 10⁻⁷ from

$$\begin{eqnarray}&&{p}_{\mathrm{com}}={\int }_{\mathrm{TS}}^{\infty }{\chi }_{2k}^{2}(x){dx}.\end{eqnarray} \tag{ 2 }$$

We then solve the equation

$$\begin{eqnarray}&&{p}_{\mathrm{com}}={\int }_{{\rm{S}}/{{\rm{N}}}_{\mathrm{com}}}^{\infty }\displaystyle \frac{1}{\sqrt{2\pi }}\exp \left(-\displaystyle \frac{{x}^{2}}{2}\right){dx},\end{eqnarray} \tag{ 3 }$$

to obtain an equivalent Gaussian standard deviation of S/N_com ≃ 5.0 as the combined significance. In this paper, this signal for J0235–0532 is regarded as the CO(6–5) line. However, because the combined significance is still not very high, it is necessary to secure a firm detection of this CO line with follow-up observations.

Although the CO peak position is consistent with that of the UV continuum, the apparent offset (with large uncertainties) might indicate a hint of photoevaporation of photodissociation regions (PDRs; e.g., Carniani et al. 2017; Decataldo et al. 2017; Vallini et al. 2017). A partial displacement between the UV continuum tracing H ii regions and CO(6–5) emission tracing dense clumps within giant molecular clouds (GMCs; e.g., McKee & Ostriker 2007) is expected when PDRs are photoevaporated. (For details, see discussion in Vallini et al. 2017.) This scenario can be verified by observing the CO emission from J0235–0532 with a higher S/N and better resolution.

From the integrated CO(6–5) emission line flux, we obtain the CO(6–5) luminosity in units of L_⊙ by using Equation (18) of Casey et al. (2014). We also calculate the CO(6–5) luminosity in units of K km s⁻¹ pc² defined by Equation (19) of Casey et al. (2014). These equations are presented in Appendix A. In these calculations, the effect of the cosmic microwave background (CMB) is taken into account by dividing the observed integrated flux by a factor of f_CMB (da Cunha et al. 2013),

$$\begin{eqnarray}&&{f}_{\mathrm{CMB}}=1-\displaystyle \frac{{B}_{\nu }[{T}_{\mathrm{CMB}}(z)]}{{B}_{\nu }[{T}_{\mathrm{exc}}]},\end{eqnarray} \tag{ 4 }$$

where B_ν is the Planck function, T_CMB(z) = 2.73(1 + z) K is the temperature of the CMB, and T_exc is the excitation temperature of the CO(6–5) transition. Assuming local thermal equilibrium (LTE), T_exc is equal to the kinetic temperature of the gas, T_kin, and then to the dust temperature, T_dust, i.e., T_exc = T_kin = T_dust. ²⁴ Here we use T_dust estimated in Section 4.2. The obtained CO luminosities or upper limits, as well as the f_CMB values, are presented in Table 3. ²⁵ Note, as a caveat, that this prescription assumes a uniform kinetic temperature for CO- and dust continuum-emitting regions. However, in reality, PDRs have a kinetic temperature profile that depends on the radiation field and the gas density. If the kinetic temperature of the CO-emitting regions is higher than adopted here, T_exc would be higher and thus f_CMB would be higher (Section 2.4 of Vallini et al. 2015; see also Section 4.3 of Vallini et al. 2018). In this sense, the da Cunha et al. (2013) prescription may provide a pessimistic estimate of the fraction of the intrinsic flux observed.

The dust continuum emission from our z = 6 luminous LBGs at λ_obs ≃ 3 mm (λ_rest ≃ 430 μm) is not significantly detected. The 3σ upper limits of their dust continuum flux densities are 52.3 μJy for J1211–0118, 29.1 μJy for J0235–0532, and 22.1 μJy for J0217–0208. Their dust continuum emission maps are presented in Appendix C.

In order to characterize the properties of their dust continuum emission, we combine our ALMA results at λ_rest ≃ 430 μm with the results of Harikane et al. (2020b) at shorter wavelengths of λ_rest ≃ 90–160 μm, and fit modified blackbody spectral energy distributions (SEDs) to the observed flux densities by varying T_dust and L_IR. We calculate the intrinsic dust continuum flux densities of a modified blackbody SED by using Equation (A5), and then obtain the expected dust continuum flux densities of the modified blackbody, ${f}_{\nu }^{(\exp )}$, from ${f}_{\nu }^{(\mathrm{int})}$ by considering the CMB heating and attenuation effects based on the prescription of da Cunha et al. (2013) in the same way as described in Section 4.1. In the dust continuum SED fitting, we require that T_dust be higher than the CMB temperature at the redshift of the galaxy (∼20 K at z ∼ 6).

Figure 5 shows the results of fitting the modified blackbody SED to the observed SEDs. For J1211–0118 and J0217–0208, modified blackbody SEDs fit well with the observed SEDs. The best-fit IR luminosities and dust temperatures are (L_IR, T_dust) = (${3.6}_{-1.9}^{+34.4}\times {10}^{11}\,{L}_{\odot }$, ${40}_{-14}^{+44}$ K) for J1211–0118 and (${2.0}_{-0.3}^{+5.9}\times {10}^{11}\,{L}_{\odot }$, ${31}_{-9}^{+26}$ K) for J0217–0208, which are consistent with the results of Harikane et al. (2020b). Considering the large T_dust uncertainties, the reason why the CO emission lines are not detected for these targets may be that the CMB attenuation effect for these targets is relatively large (f_CMB is small) due to low T_dust. Because our observations only add an upper limit to the observed SEDs on the longer wavelength side of the SED peak, the parameter constraints do not become stronger than the previous work. The two parameters of L_IR and T_dust are still degenerate, which will be greatly improved if deep observations of the dust continuum emission at shorter wavelengths than the SED peak are conducted. Note that another method has been proposed recently to determine T_dust and the dust mass assuming dust to be in radiative equilibrium if the source size of dust continuum emission is obtained (Inoue et al. 2020). Alternatively, it would be possible to have independent estimates of L_IR and T_dust, as well as the dust mass, based on the dust continuum and [C ii] line luminosities by adopting the method recently presented in Sommovigo et al. (2021).

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For J0235–0532, although the allowed parameter ranges are determined based on the upper limits of the flux densities, the constraints obtained on L_IR and T_dust are not stringent. Following Harikane et al. (2020b), we adopt T_dust = 50 K for this galaxy without continuum detection as a fiducial value (see also Hashimoto et al. 2019) for comparisons with previous studies, which yields a 3σ upper limit of L_IR < 5.8 × 10¹¹ L_⊙. We also consider the case of a higher dust temperature of T_dust = 80 K as a systematic uncertainty. This is because J0235–0532 has the highest [O iii]/[C ii] luminosity ratio (Table 1), possibly suggesting a relatively high dust temperature. In fact, previous observations of nearby galaxies have shown that SFGs with higher [O iii]/[C ii] ratios tend to have higher T_dust values (Walter et al. 2018), although the [O iii]/[C ii] ratios of their SFGs are not as high as those of J0235–0532. More recently, dust continuum observations of a z = 8.31 galaxy with a similarly high [O iii]/[C ii] ratio, MACS0416-Y1, have suggested a possibility that its dust temperature may be extremely high, exceeding 80 K, although its physical origin is still under discussion (Bakx et al. 2020). One possible physical explanation for very high dust temperatures is that part of their dust is locked in molecular clouds and/or young star clusters that host active star formation. Based on hydrodynamic simulations, Behrens et al. (2018) have shown that, in such a situation, dust is heated by the strong interstellar radiation fields and can show a very high temperature, efficiently emitting FIR continuum, which can explain the high FIR luminosity without invoking mechanisms for massive dust production at high redshifts (see also, e.g., Arata et al. 2019; Sommovigo et al. 2020). The case of higher T_dust for J0235–0532 yields a more conservative upper limit of L_IR < 2.5 × 10¹² L_⊙ (3σ).

In Figure 6, we compare the CO(6–5) and IR luminosities of our luminous LBGs at z = 6 with those of nearby sources at z < 0.1 (Liu et al. 2015) as well as dusty star-forming galaxies (DSFGs; Riechers et al. 2010; Strandet et al. 2017; D'Odorico et al. 2018; Apostolovski et al. 2019; Casey et al. 2019) and quasars (Wang et al. 2010, 2011, 2016; Venemans et al. 2017a; Carniani et al. 2019) at comparable redshifts of z ∼ 5–7 to our targets. For nearby sources, the correlation between CO and IR luminosities has been found over a wide luminosity range (Liu et al. 2015), which can be interpreted as an integrated KS relation because the CO and IR luminosities are correlated with gas mass and SFR, respectively (e.g., Magdis et al. 2017). We find that our result for J0235–0532, which is the only one of our targets showing CO(6–5) detection at the ≃4σ significance level, is broadly consistent with previous results owing to the relatively large uncertainty on the IR luminosity. For J1211–0118 and J0217–0208, whose CO emission is not detected, our results are also consistent with previous results. In other words, the obtained upper limits on CO luminosity for these two sources are not deep enough to know whether they deviate from the correlation between $L{{\prime} }_{\mathrm{CO}}$ and L_IR seen in low-z sources or not, which can be distinguished by much deeper CO observations.

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Note that the excitation of the CO spectral line energy distribution (SLED) varies as a function of gas density, radiation field, Mach number within GMCs, and presence of shocks (e.g., Vallini et al. 2018; Pensabene et al. 2021), and thus the CO(6–5) emission line, which traces dense gas with critical density of n_crit = 2.9 × 10⁵ cm⁻³, would trace a fraction of the total molecular gas, i.e., dense clumps within GMCs. Thus, the relation between $L{{\prime} }_{\mathrm{CO}(6-5)}$ and L_IR would not be entirely related to the M_gas–SFR relation, and the interpretation as an integrated version of the KS relation could be partially hampered. In Sections 5.2 and 5.3, we convert $L{{\prime} }_{\mathrm{CO}(6-5)}$ to $L{{\prime} }_{\mathrm{CO}(1-0)}$ by adopting the average CO SLED for SFGs at lower redshifts to obtain the estimates of gas mass and gas surface density from $L{{\prime} }_{\mathrm{CO}(1-0)}$, and compare them with the KS relation found in the local universe, although the systematic uncertainties in such conversions are not small (Section 5.4).

In this study, we have observed CO(6–5) emission for the three luminous LBGs at z = 6 with comparable total SFRs of ∼100 M_⊙ yr⁻¹. As a result, CO(6–5) is marginally detected in J0235–0532 at the ≃4σ significance level, but not in the other two LBGs. In this section, we discuss physical reasons for this difference.

Because the [C ii] emission has also been detected for these LBGs in Harikane et al. (2020b), we calculate the line ratio of CO(6–5) to [C ii] as well as the ratio of the [C ii] to IR luminosity, which are useful for obtaining constraints on the physical properties of PDRs in galaxies such as the density of hydrogen nuclei, n_H, and the incident far-ultraviolet (FUV) radiation field, U_UV, with 6–13.6 eV based on comparisons with theoretical models for PDRs. For the PDR modeling, we use the Photodissociation Region Toolbox (PDRT; Kaufman et al. 1999, 2006; Pound & Wolfire 2008), ²⁶ which calculates various line and continuum intensity ratios for combinations of n_H and U_UV by solving for the equilibrium chemistry, thermal balance, and radiation transfer through a PDR layer in a self-consistent way. Specifically, we use the wk2006 model of the PDRT with solar metallicity for comparisons with previous results.

Because the [C ii] emission comes from not only PDRs but also H ii regions, we need to subtract the contribution of [C ii] emission from H ii regions for comparisons with the results of the PDRT calculation. For this purpose, we refer to Figure 9 of Cormier et al. (2019), which presents the dependence of the fraction of [C ii] emission from H ii regions, ${f}_{[{\rm{C}}\,{\rm\small{II}}]}^{(\mathrm{ion})}$, on gas-phase metallicity (see also Figure 4 of Croxall et al. 2017; Sutter et al. 2019; Rybak et al. 2021; see also theoretical results such as Katz et al. 2017; Olsen et al. 2017; Ferrara et al. 2019; Pallottini et al. 2019). Based on analyses of interstellar medium absorption lines detected in the stacked spectrum of z ∼ 6 luminous LBGs with M_UV ≃ −23 mag including one of our targets, J1211–0118, Harikane et al. (2020a) have found that their gas-phase metallicity is close to solar. ²⁷ By combining these two previous results, ${f}_{[{\rm{C}}\,{\rm\small{II}}]}^{(\mathrm{ion})}$ of our z = 6 luminous LBGs would be about ${f}_{[{\rm{C}}\,{\rm\small{II}}]}^{(\mathrm{ion})}\simeq 0.3$. Because the observed ${f}_{[{\rm{C}}\,{\rm\small{II}}]}^{(\mathrm{ion})}$ values have a scatter of ≃0.1–0.2, here we consider it as a systematic uncertainty. We also apply a factor-of-two correction for the observed CO flux considering line luminosity from both sides of each optically thick cloud for comparisons with the results of the PDRT calculation, as suggested by Kaufman et al. (1999) (see also Wang et al. 2016; Rybak et al. 2019; Shao et al. 2019).

Because the ${f}_{[{\rm{C}}\,{\rm\small{II}}]}^{(\mathrm{ion})}$ values are correlated with the [C ii]/[N ii] 122 μm luminosity ratio as presented in Figure 10 of Cormier et al. (2019), we can also evaluate ${f}_{[{\rm{C}}\,{\rm\small{II}}]}^{(\mathrm{ion})}$ from [C ii]/[N ii]. However, the [N ii] emission has not been detected in any of our targets (Harikane et al. 2020b), and the lower limits on the [C ii]/[N ii] ratios are not so stringent. Calculating the 3σ lower limits on the [C ii]/[N ii] ratios based on Table 1 of Harikane et al. (2020b), we obtain L_{[C II]}/L_{[N II]} > 0.36–2.3. We confirm that the expected ranges of the ${f}_{[{\rm{C}}\,{\rm\small{II}}]}^{(\mathrm{ion})}$ values from the lower limits of the [C ii]/[N ii] ratios are consistent with those expected from the gas metallicity.

By taking account of these points, in the top left panel of Figure 7, we compare our ALMA results for L_CO(6−5)/L_{[C II]} versus L_{[C II]}/L_IR with the results of the PDRT calculation. Because the PDRT does not include the CMB temperature, the observed CO and IR luminosities are corrected for the CMB effect (Section 4). ²⁸ In the same way as in Figure 6, we adopt T_dust = 50 K for J0235–0532 as a fiducial value and consider up to T_dust = 80 K as a systematic uncertainty, yielding a conservative lower limit of L_{[C II]}/L_IR. We find that the n_H value of J0235–0532 is higher than those of J1211–0118 and J0217–0208. Because we only obtain the lower limit for the L_{[C II]}/L_IR ratio of J0235–0532 due to the nondetection of the dust continuum emission, it is unclear whether the incident FUV radiation is stronger in J0235–0532 than in the others or not.

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For comparisons of the n_H and U_UV values of our z = 6 luminous LBGs with those of other sources at lower redshifts, in the top right panel of Figure 7, we show previous observation results for L_CO(1−0)/L_{[C II]} versus L_{[C II]}/L_IR of luminous infrared galaxies (LIRGs), ultraluminous infrared galaxies (ULIRGs; Rosenberg et al. 2015), quasars (Benford et al. 1999; Maiolino et al. 2005; Iono et al. 2006; Wagg et al. 2012, 2014; Leipski et al. 2013; Stefan et al. 2015; Wang et al. 2016; Venemans et al. 2017b; compiled by Shao et al. 2019), local SFGs such as spiral galaxies, and Galactic star-forming regions (Stacey et al. 1991), as well as the results of the PDRT model calculation. For easier comparison, the bottom panel of Figure 7 is the same as the top left panel but with the colored shaded regions that roughly correspond to the locations of low-z sources in the previous work with CO(1–0) observations shown in the top right panel. We find that the relatively high n_H value of J0235–0532, n_H ≳ 10⁵ cm⁻³ depending on U_UV, is consistent with those of LIRGs and ULIRGs with relatively low n_H values in that population, as well as with those of quasars and Galactic star-forming regions with high n_H and U_UV values considering the case of high T_dust. We also find that J1211–0118 and J0217–0208, likely showing moderate U_UV values with n_H upper limits around 10⁵ cm⁻³, are consistent with nuclear regions of local SFGs and Galactic star-forming regions with relatively low n_H values.

It should be noted that there are two systematic uncertainties in this comparison. One is related to ${f}_{[{\rm{C}}\,{\rm\small{II}}]}^{(\mathrm{ion})}$. As mentioned above, the relation between ${f}_{[{\rm{C}}\,{\rm\small{II}}]}^{(\mathrm{ion})}$ and metallicity has a scatter of about 0.1–0.2 (e.g., Figure 9 of Cormier et al. 2019). In our L_CO(6−5)/L_{[C II]} versus L_{[C II]}/L_IR figures, we show the blue arrow in the upper right corner that corresponds to the amount of shift when ${f}_{[{\rm{C}}\,{\rm\small{II}}]}^{(\mathrm{ion})}$ is increased by 0.1. We confirm that this systematic uncertainty does not significantly affect the results. The other systematic uncertainty is the effect of the CMB on the [C ii] emission. As discussed in Section 6.1 of Harikane et al. (2020b), the [C ii] emission may also be affected by the CMB attenuation due to the high CMB temperature at z ∼ 6 (see also González-López et al. 2014; Lagache et al. 2018; Laporte et al. 2019). Figure 1 of Kohandel et al. (2019) shows the CMB suppression effect of the [C ii] emission with different gas temperatures as a function of gas number density. Although only upper limits are derived for n_H of J1211–0118 and J0217–0208, with a conservative gas density value of 10⁴ cm⁻³, we obtain the effect of the CMB on the [C ii] emission line flux of ${f}_{\mathrm{CMB}}^{[\mathrm{CII}]}=0.71$–0.86 at 30–40 K (Kohandel et al. 2019; see also Pallottini et al. 2015; Vallini et al. 2015), which is comparable to the dust temperature. If the gas density and/or the gas temperature is higher, then the CMB effect is smaller, suggesting that the impact of this systematic uncertainty is comparable to or smaller than that of the ${f}_{[{\rm{C}}\,{\rm\small{II}}]}^{(\mathrm{ion})}$ scatter.

There is another noticeable difference between J0235–0532 and the other two LBGs. J0235–0532 has the highest [O iii]/[C ii] luminosity ratio ([O iii]/[C ii] = 8.9 ± 1.7) among our targets, as shown in Figure 5 of Harikane et al. (2020b). In the first place, these three z = 6 LBGs have significantly higher [O iii]/[C ii] ratios than z ∼ 0 galaxies with comparable total SFRs. ²⁹ Harikane et al. (2020b) have discussed the physical reason for this based on comparisons with the results of model calculations for both H ii regions and PDRs with CLOUDY (Ferland et al. 1998, 2017) and concluded that high ionization parameters and/or low PDR covering fractions can explain high-z galaxy results including the high [O iii]/[C ii] ratios and low L_{[C II]}/SFR ratios. Harikane et al. (2020b) have also found that high n_H, low C/O ratios, and the CMB attenuation effect can reproduce a part of the high-z galaxy results.

Because the CO emission originates from different regions from [O iii]-/[C ii]-emitting regions (e.g., Figure 31.2 of Draine 2011), it is difficult to make a direct comparison between our results and the results of Harikane et al. (2020b). The least we can say is that our results suggest a relatively high n_H in PDRs of J0235–0532 compared to the other two LBGs, which would be consistent with the results of Harikane et al. (2020b), although in their study it is not enough to explain the high-z galaxy results. In addition, U_UV of J0235–0532 may be higher than those of the other two LBGs, which would be consistent with the high [O iii]/[C ii] ratio and thus the relatively high ionization parameter, although deeper dust continuum observations are required for constraining L_{[C II]}/L_IR to reach a conclusion on this point.

Interestingly, this is in line with what is expected from theoretical models. The high-J CO lines trace regions of relatively high density more directly connected to star formation. At such a high density, the self-shielding effect prevents the dissociation of molecules and at the same time the high temperature produced by the strong UV radiation suggested from the high [O iii]/[C ii] ratio is expected to boost the high-J CO emission (Vallini et al. 2018). In this case, the dust temperature is also likely to be high (Behrens et al. 2018). In fact, the dust continuum is not detected only for J0235–0532, which is consistent with the possibility that the dust temperature of J0235–0532 may be very high. To confirm this picture, it would be interesting to carry out deep observations to detect high-J CO emission from SFGs with high [O iii]/[C ii] ratios and/or high T_dust, such as MACS1149-JD1 at z = 9.1096 ([O iii]/[C ii] ≳ 19; Hashimoto et al. 2018; Laporte et al. 2019), MACS0416-Y1 at z = 8.3118 ([O iii]/[C ii] = 8.6 ± 2.5 and T_dust > 80 K; Tamura et al. 2019; Bakx et al. 2020), and SXDF-NB1006-2 at z = 7.2120 ([O iii]/[C ii] ≳ 10; Inoue et al. 2016). Note that careful estimates of their [O iii]/[C ii] luminosity ratios have been provided recently by considering the surface brightness dimming effect (Carniani et al. 2020); they still show relatively high [O iii]/[C ii] values of 4.2 ± 1.4 for MACS1149-JD1, 8 ± 2 for MACS0416-Y1, and 4.3 ± 1.4 for SXDF-NB1006-2. We confirm that these sources also have high [O iii]/[C ii] surface brightness ratios (Vallini et al. 2021).

We constrain molecular gas masses in our z = 6 luminous LBGs based on our CO(6–5) results, although the systematic uncertainties are not small, particularly in the CO SLED, which has not been investigated well for SFGs at high redshifts. Here we present conservative constraints on molecular gas masses in our targets by taking account of such uncertainties and compare with previous results for lower-z sources.

The total gas mass for molecular clouds, M_gas, can be estimated from the CO(1–0) luminosity in units of K km s⁻¹ pc², $L{{\prime} }_{\mathrm{CO}(1-0)}$, by using Equation (4) of Solomon & Vanden Bout (2005),

$$\begin{eqnarray}&&{M}_{\mathrm{gas}}={\alpha }_{\mathrm{CO}}L{{\prime} }_{\mathrm{CO}(1-0)},\end{eqnarray} \tag{ 5 }$$

where α_CO is the conversion factor from $L{{\prime} }_{\mathrm{CO}(1-0)}$ to M_gas. We assume a fixed value of α_CO = 4.5 M_⊙ (K km s⁻¹ pc²)⁻¹, ³⁰ which is consistent with previous results for the Milky Way (Bolatto et al. 2013), z ∼ 1–2 SFGs (Daddi et al. 2010; Carilli & Walter 2013), and even an LBG at z = 5.7 (HZ10; Pavesi et al. 2019). ³¹ The $L{{\prime} }_{\mathrm{CO}(1-0)}$ values of our z = 6 luminous LBGs can be estimated from $L{{\prime} }_{\mathrm{CO}(6-5)}$ by using the average CO SLED for SFGs. Specifically, we assume that the integrated flux of CO(6–5) is comparable to that of CO(5–4), i.e., I_CO(6−5) ≃ I_CO(5−4), and adopt the average integral CO line flux ratio of I_CO(5−4)/I_CO(1−0) ≃ 5.8 ± 3.3, which is measured for z ∼ 1–2 SFGs (Daddi et al. 2015). The large uncertainty of I_CO(5−4)/I_CO(1−0) is estimated from the standard deviation of the integrated CO flux ratios of the z ∼ 1–2 SFGs reported in Daddi et al. (2015). ³² We then calculate $L{{\prime} }_{\mathrm{CO}(1-0)}$ ³³ and obtain M_gas constraints as summarized in Table 3.

Note that Zanella et al. (2018) have reported a linear correlation between the [C ii] luminosity and the gas mass for z ∼ 2 SFGs and obtained a conversion factor from the [C ii] luminosity and the gas mass, ${\alpha }_{[{\rm{C}}\,{\rm\small{II}}]}=31\,{M}_{\odot }{L}_{\odot }^{-1}$. By adopting this conversion factor, we estimate the gas mass of J0235–0532 from the [C ii] luminosity to be only about 1.3 × 10¹⁰ M_⊙, which is significantly smaller than that obtained from the CO luminosity. This may suggest that the conversion factor α_{[C II]} or α_CO for high-z luminous LBGs is different from those at low redshifts, or that the CO SLED is different from those for z ∼ 1–2 SFGs, although it is difficult to clarify these possibilities with the currently available data. In this study, we adopt the estimates based on the CO luminosity, because it is more commonly used in previous studies and would thus be more appropriate for comparisons.

In Figure 8, we compare total SFRs and M_gas estimates of our z = 6 luminous LBGs with dusty starbursts and other SFGs over a wide range of redshifts from z ∼ 0 to z ∼ 6 (Béthermin et al. 2015; Scoville et al. 2016; Saintonge et al. 2017) including HZ10, LBG-1, AzTEC-3, and CRLE (Riechers et al. 2010, 2014; Pavesi et al. 2018; see also Pavesi et al. 2019). This figure should be interpreted as an integrated KS relation in a more direct sense than the L_IR versus $L{{\prime} }_{\mathrm{CO}}$ plot presented as Figure 6 (Section 4.2). Following Kennicutt (1998a), we adopt the molecular gas mass as a proxy for the total gas mass for high-SFR sources including our z = 6 luminous LBGs, because such sources in the local universe show that the disks are molecular-dominated (Sanders & Mirabel 1996; see also Kennicutt & De Los Reyes 2021). Because we only obtain the upper limit of SFR_IR for J0235–0532, we present the sum of SFR_UV and the 2σ upper limit of SFR_IR in the case of T_dust = 50 K as its total SFR, and consider the higher dust temperature up to T_dust = 80 K as well as the minimum SFR case of the SFR_UV alone with no SFR_IR in the relatively large error bars as systematic uncertainties. We find that our CO-based results for J0235–0532 are in broad agreement with SFGs at various redshifts with similar M_gas including HZ10 and LBG-1. J1211–0118 and J0217–0208 are consistent with the previous results with similar SFRs, although their M_gas values are upper limits.

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Although the currently available data for our z = 6 luminous LBGs do not resolve their internal structures in detail, we estimate their sizes to calculate their global SFR surface densities and gas surface densities for comparisons with the KS relation for the average surface densities of SFGs in the local universe.

The sizes of star-forming regions of our z = 6 luminous LBGs for calculating global SFR surface densities are measured with the HSC z-band images, which trace the rest UV continuum emission. We fit Sérsic profiles (Sersic 1968) to the observed surface brightness distributions by using GALFIT ver. 3.0.5 (Peng et al. 2002, 2010), ³⁴ which convolves a galaxy model profile with a point-spread function (PSF) profile and optimizes the fitting parameters based on the Levenberg–Marquardt algorithm for χ² minimization (e.g., Press et al. 1992). We use a PSF image for the position of each of our z = 6 luminous LBGs downloaded from the PSF picker website of the HSC survey. ³⁵ The output parameters include the centroid coordinates of the objects, their total magnitude, the half-light radius along the semimajor axis, the axis ratio, and the position angle. The Sérsic index n is fixed at 1.0. ³⁶ We calculate the circularized half-light radius, ${r}_{{\rm{e}}}=\sqrt{q}\,{r}_{{\rm{e}},\mathrm{maj}}$, where q is the axis ratio and r_e,maj is the half-light radius along the semimajor axis, because it is widely used in size measurements in previous high-z galaxy studies (e.g., Mosleh et al. 2012; Newman et al. 2012; Ono et al. 2013; Shibuya et al. 2015; Kawamata et al. 2018). The obtained r_e values are presented in Table 1.

Note that, for J0235–0532, the output axis ratio obtained with GALFIT is enclosed between star symbols, indicating that a numerical convergence issue may have occurred in the fitting for this particular source (for details, see Section 10 of the GALFIT user's manual). As an alternative method, for J0235–0532, we use SExtractor ver. 2.8.6 (Bertin & Arnouts 1996) ³⁷ to calculate the observed half-light radius, ${r}_{{\rm{e}}}^{(\mathrm{obs})}$, by using circular apertures that contain half of the light from a galaxy, and correct it for the PSF broadening according to

$$\begin{eqnarray}&&{r}_{{\rm{e}}}=\sqrt{{r}_{{\rm{e}}}^{(\mathrm{obs})\,2}-{r}_{\mathrm{PSF}}^{2}},\end{eqnarray} \tag{ 7 }$$

where r_PSF is the half-light radius of the PSF image (e.g., Oesch et al. 2010; Holwerda et al. 2020; Bowler et al. 2021; Roberts-Borsani et al. 2022). For this PSF broadening correction, we use the PSF images downloaded from the PSF picker website. The obtained r_e value for J0235–0532 is also presented in Table 1. We also measure the half-light radii of J1211–0118 and J0217–0208 with SExtractor and confirm that the results are consistent with those obtained with GALFIT.

With the obtained r_e values, we define SFR surface density, Σ_SFR, as the average SFR in a circular region whose half-light radius is r_e (Equation (A6)). The obtained Σ_SFR values are listed in Table 3.

Because the resolution of our ALMA data is not high enough to estimate the sizes of CO-emitting regions, we calculate their gas surface densities by assuming that the sizes of CO-emitting regions are comparable to those of star-forming regions. In fact, Tacconi et al. (2013) have reported that molecular gas and UV/optical light distributions of z ∼ 1–2 SFGs show comparable sizes, in agreement with similar findings in z ∼ 0 SFGs (e.g., Regan et al. 2001; Leroy et al. 2008). With the r_e values obtained above, we define the gas surface density, Σ_gas, in a similar way to Σ_SFR (Equation (A7)). ³⁸ The obtained Σ_gas values are also presented in Table 3.

Figure 9 plots Σ_SFR of our z = 6 luminous LBGs as a function of Σ_gas. For comparison, we also present normal spiral (disk) galaxies and starbursts in the local universe with the best-fit relation between their Σ_SFR and Σ_gas (the KS relation; de los Reyes & Kennicutt 2019; Kennicutt & De Los Reyes 2021; see also Kennicutt 1998a), ³⁹ as well as the z = 5.3–5.7 sources, HZ10, LBG-1, AzTEC-3, and CRLE (Riechers et al. 2010, 2014; Pavesi et al. 2018; see also Pavesi et al. 2019). We find that J0235–0532 and HZ10 are almost at the same position in this plane, located below the local KS relation, suggesting that J0235–0532 and HZ10 have very high gas surface densities with relatively low star formation efficiencies. We also find that J1211–0118 and J0217–0208 are consistent with the local KS relation, although their obtained Σ_gas values are upper limits. Note that in Figure 8 their data points are consistent with the integrated KS relation, while in Figure 9 they are below the local KS relation. The reason for this is that our z = 6 luminous LBGs have smaller gas-emitting regions and/or larger star-forming regions than those of local starbursts. In Figure 9, we confirm that the dusty starbursts at comparable redshifts, AzTEC-3 and CRLE, are located above the local KS relation. These results may indicate that the scatter of the KS relation is larger with increasing redshift, at least at large Σ_gas of ∼10⁴ M_⊙ pc⁻², possibly suggesting that star formation in high-z galaxies with high Σ_gas is diverse, ranging from bursty to slow. However, the number of high-z data points is still limited; this needs to be examined by investigating more objects at high redshifts in the future. Averaging the four data points for the z = 5–6 galaxies of J0235–0532, HZ10, AzTEC-3, and CRLE, we find that the z = 5–6 KS relation at Σ_gas ∼ 10⁴ M_⊙ pc⁻² on average is consistent with the KS relation in the local universe. Again, the number of high-z sources whose Σ_SFR and Σ_gas are estimated is limited yet. It would be interesting to compare the observational results for a larger sample of high-z galaxies with those of theoretical studies in the future (e.g., Ferrara et al. 2019; Dubois et al. 2021).

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Note that the r_e values measured in the rest UV continuum images are used in the calculations of both the SFR and gas surface densities for our z = 6 luminous LBGs. If their CO sizes are significantly larger than the rest UV sizes, the currently presented Σ_gas values correspond to the upper limits (e.g., Kaasinen et al. 2020). More quantitative discussion about this point is presented in Section 5.4. In order to obtain more accurate Σ_gas values with no such systematic uncertainties, high-resolution deep CO observations are necessary.

In Sections 5.2 and 5.3, we obtain the total gas mass estimates for our z = 6 luminous LBGs based on our CO(6–5) observation results by carefully considering the uncertainties suggested from the previous observation results, and discuss the KS relation. However, we caution that the gas mass estimates have substantial systematic uncertainties.

One is the CO SLED uncertainty. In our discussion above, we adopt the previously observed CO SLED results for z ∼ 1–2 SFGs and consider the significant amount of scatter seen in observations of individual objects (Daddi et al. 2015). However, our z = 6 luminous LBGs may be experiencing more bursty star formation with higher gas density and thus the CO SLED could be more excited. For example, previous CO observations of nearby starbursts have revealed that their integrated CO flux ratios are about I_CO(6−5)/I_CO(1−0) ≃ 8–20 (Figure 1 of Mashian et al. 2015). ⁴⁰ Based on the ALMA Spectroscopic Survey in the Hubble Ultra Deep Field (ASPECS), Boogaard et al. (2020) have shown that SFGs at $\left\langle z\right\rangle =2.5$ have higher CO excitation than those at $\left\langle z\right\rangle =1.2$ as well as the results of Daddi et al. (2015), suggesting the increased CO excitation at higher redshifts (e.g., Figure 7 of Boogaard et al. 2020). Theoretically, Vallini et al. (2018) have developed a semianalytical model for GMCs where the CO lines are excited, and postprocessed a state-of-the-art zoom-in cosmological simulation of a main-sequence galaxy at z = 6, Althaea, with M_star ≈ 10¹⁰ M_⊙ and SFR ≈ 100 M_⊙ yr⁻¹ (Pallottini et al. 2017), which is in line with the nature of the galaxies discussed in this paper, showing that the CO SLED of Althaea has a peak at around the upper-state rotational quantum number of J_up ≃ 6. Specifically, Althaea has CO luminosities of about $L{{\prime} }_{\mathrm{CO}(1-0)}\simeq {10}^{9.2}$ K km s⁻¹ pc² and $L{{\prime} }_{\mathrm{CO}(6-5)}\simeq {10}^{8.9}$ K km s⁻¹ pc², yielding the CO luminosity ratio of $L{{\prime} }_{\mathrm{CO}(1-0)}/L{{\prime} }_{\mathrm{CO}(6-5)}\simeq 2.0$. In contrast, the CO luminosity ratio that we have adopted in our discussion above is $L{{\prime} }_{\mathrm{CO}(1-0)}/L{{\prime} }_{\mathrm{CO}(6-5)}=6.2\pm 3.5$ (Equation (6) in Section 5.2). Although their theoretical result is just for one z = 6 galaxy, their upcoming results with the SERRA simulation show that the physical mechanisms exciting the CO SLED (i.e., high density and high turbulence) are common in more than 100 high-z galaxies (A. Pallottini 2022, in preparation; see also Pallottini et al. 2019). If the CO SLEDs of our z = 6 luminous LBGs are similar to that of Althaea, the gas mass estimates become lower by a factor of about 1/3. This should be examined by observing several CO emission lines with different excited states from high-z SFGs.

Another systematic uncertainty comes from the CO-to-H₂ conversion factor, α_CO. In our discussion above, we adopt the fixed value of α_CO = 4.5 M_⊙ (K km s⁻¹ pc²)⁻¹, which is consistent with the previous observational results such as for the Milky Way, z ∼ 1–2 SFGs, and HZ10 at z = 5.7 (Daddi et al. 2010; Bolatto et al. 2013; Carilli & Walter 2013; Pavesi et al. 2019). However, it is known that the CO-to-H₂ conversion factor becomes smaller in galaxies with more active star formation. In fact, LIRGs and ULIRGs show low CO-to-H₂ conversion factors of α_CO ≃ 0.8 M_⊙ (K km s⁻¹ pc²)⁻¹ (Downes & Solomon 1998), which is often adopted in previous studies of high-z dusty starbursts (e.g., Greve et al. 2005; Riechers et al. 2010; Aravena et al. 2016; see also Wagg et al. 2009). Our z = 6 luminous LBGs may also have small α_CO values compared to the adopted one. From a theoretical point of view, Vallini et al. (2018) have shown that the simulated z = 6 galaxy Althaea has a small α_CO value of α_CO = 1.5 M_⊙ (K km s⁻¹ pc²)⁻¹ (see their Figure 12; see also Narayanan et al. 2012). If our z = 6 luminous LBGs have small α_CO values comparable to Althaea, the gas mass estimates based on the CO(6–5) results are further reduced by a factor of 1/3. Interestingly, if we adopt the CO SLED and α_CO for Althaea, the obtained gas mass estimate for J0235–0532 from CO(6–5) is consistent with that obtained from the [C ii] luminosity (Section 5.2).

Furthermore, in the calculations of the gas surface densities of our z = 6 luminous LBGs, we assume that the sizes of CO-emitting regions are comparable to those of star-forming regions, which is also a source of systematic uncertainties. As mentioned in Section 5.3, some previous observational studies for z ∼ 0–2 SFGs have shown that this is the case (Regan et al. 2001; Leroy et al. 2008; Tacconi et al. 2013), while some other studies have shown that the gas sizes are larger. For our z = 6 luminous LBGs, Carniani et al. (2020) have reported their [C ii] sizes (FWHMs along the major axis) in their Table A1. If the [C ii] sizes are comparable to those of CO-emitting regions and the CO luminosities are comparable to the current measurements, the gas sizes become larger by about a factor of 3, and thus Σ_gas becomes smaller by about a factor of 1/10. This can be tested by deep observations of low-J CO emission, which better traces the molecular gas distribution and the total gas mass.

If we adopt the CO SLED and α_CO for Althaea for total gas mass estimates, and use the previously obtained [C ii] sizes as their gas sizes, then the estimated gas surface densities of our z = 6 luminous LBGs become smaller by about two orders of magnitude than presented in Figure 9. Figure 9 adds this possibility on the Σ_SFR–Σ_gas plot. In this case, our z = 6 luminous LBGs are located above the KS relation, which means that they are experiencing bursty star formation. Because their [O iii]/[C ii] ratios are relatively high compared to local galaxies with similar total SFRs (Section 5.1), this interpretation may be physically more reasonable (Vallini et al. 2021; see also Ferrara et al. 2019). This issue is expected to be clarified by future follow-up observations.

We constrain gas fractions for our z = 6 luminous LBGs. The gas fraction is defined as

$$\begin{eqnarray}&&{f}_{\mathrm{gas}}=\displaystyle \frac{{M}_{\mathrm{gas}}}{{M}_{\mathrm{gas}}+{M}_{\mathrm{star}}},\end{eqnarray} \tag{ D1 }$$

where M_star is the stellar mass. For our z = 6 luminous LBGs, M_star can be roughly estimated from M_UV by using the relation between M_star and M_UV for SFGs at similar redshifts, e.g., Equation (2) of Shibuya et al. (2015),

$$\begin{eqnarray}&&\mathrm{log}{M}_{\mathrm{star}}=-2.45-0.59{M}_{\mathrm{UV}}+\mathrm{log}{\beta }_{\mathrm{SC}},\end{eqnarray} \tag{ D2 }$$

where β_SC = 1/1.64 ≃ 0.61 is the factor to convert from M_star with the Salpeter (1955) IMF to that with the Chabrier (2003) IMF (Madau & Dickinson 2014; see also Table E1). We present the obtained f_gas constraints in Table 3. Note that our f_gas constraints do not include the systematic uncertainty in the stellar mass estimates from the UV luminosity, which is about ±0.5 dex due to differences in stellar population properties such as star formation history (Shibuya et al. 2015). For more robust discussion, deep rest-frame optical data that can probe the stellar continuum emission are required.

In Figure 11, we present f_gas of our z = 6 luminous LBGs with those of lower-z SFGs (Béthermin et al. 2015; Scoville et al. 2016; Saintonge et al. 2017) as well as the z = 5.3–5.7 sources, HZ10, LBG-1, AzTEC-3, and CRLE (Riechers et al. 2010, 2014; Pavesi et al. 2018; see also Pavesi et al. 2019) as a function of total SFR. We find that J0235–0532 has a comparable gas fraction to lower-z SFGs with similar SFRs. We also find that the obtained upper limits on the gas fractions of J1211–0118 and J0217–0208 are consistent with lower-z SFGs with similar SFRs. Compared to HZ10, the gas fraction of J0235–0532 is consistent owing to the large uncertainties, while those of J1211–0118 and J0217–0208 are significantly lower, although their total SFRs are comparable.

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Figure 12 shows the gas depletion timescale, t_dep = M_gas/SFR, as a function of redshift. For comparison, we also present the results for the four z = 5.3–5.7 sources of HZ10, LBG-1, AzTEC-3, and CRLE (Riechers et al. 2010, 2014; Pavesi et al. 2018; see also Pavesi et al. 2019), the average results for z = 4.4–5.9 LBGs with M_star = 10^8.4−11 M_⊙ obtained in the ALMA Large Program to Investigate [C ii] at Early Times (ALPINE; Dessauges-Zavadsky et al. 2020), and other SFGs including dusty starbursts over a wide range of redshifts (Béthermin et al. 2015; Aravena et al. 2016; Schinnerer et al. 2016; Scoville et al. 2016; Magdis et al. 2017; Saintonge et al. 2017).

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As expected from Figure 8, t_dep of J0235–0532 is comparable to those of lower-z SFGs at z ∼ 2–3. Based on previous results for lower-z SFGs, Tacconi et al. (2013) have suggested a redshift dependence of the gas depletion timescale in the form of t_dep ∝ (1 + z)^−1.0, which is shallower than what is expected if t_dep is proportional to the dynamical timescale, t_dep ∝ (1 + z)^−1.5 (Davé et al. 2011, 2012; see also Saintonge et al. 2013). Our results for J0235–0532 show that the t_dep value is likely to be larger than expected from the previously reported redshift dependences. For the other two z = 6 luminous LBGs, J1211–0118 and J0217–0208, we have obtained upper limits on their t_dep, indicating that their t_dep values can be significantly shorter than that for J0235–0532. In other words, there is a possibility that t_dep values of high-z SFGs are not necessarily as large as those of lower-z SFGs, suggesting that J0235–0532 may be an outlier with large t_dep. Similar arguments can be made at slightly lower redshifts based on the results of HZ10 and LBG-1 as well as the ALPINE results. Because the previous results for lower-z SFGs show a large scatter of t_dep, it would be interesting to investigate a typical t_dep value by observing more high-z SFGs with better sensitivities in future studies to characterize the typical star formation properties in high-z SFGs.