Dust evolution processes in normal galaxies at z > 6 detected by ALMA

Abstract

1 INTRODUCTION

Dust grains in the interstellar medium (ISM) play an important role in radiative and chemical processes. Dust absorbs stellar light and re-emits it in the far-infrared (FIR), thus affecting the spectral energy distribution (SED) of a galaxy (e.g. Schaerer et al. 2015; Yajima et al. 2015, for recent modelling). Dust also activates H₂ formation since the dust surface serves as a reaction site of H₂ formation (Gould & Salpeter 1963). In addition, since dust is composed of metals (silicon, carbon, oxygen, iron, etc.), it traces the metal enrichment in galaxies (Lisenfeld & Ferrara 1998; Dwek 1998). Therefore, the understanding of dust enrichment in galaxies is of fundamental importance in clarifying the evolutions of galaxies and the ISM.

Galaxies are enriched with metals as a result of their star formation activities. When supernovae (SNe) and asymptotic giant branch (AGB) star winds inject metals in the ISM, a fraction of the metals are condensed into dust (Kozasa, Hasegawa & Nomoto 1989; Todini & Ferrara 2001; Nozawa et al. 2003; Bianchi & Schneider 2007; Gall, Andersen & Hjorth 2011). Above a certain metallicity level (typically ≈0.1 Z_⊙), grain growth by the accretion of gas phase metals in the dense ISM becomes the dominant source of dust (Dwek 1998; Hirashita 1999; Inoue 2003; Zhukovska, Gail & Trieloff 2008; Inoue 2011; Valiante et al. 2011; Mattsson & Andersen 2012; Asano et al. 2013; Ferrara, Viti & Ceccarelli 2016; Popping, Somerville & Galametz 2016). This process is referred to as ‘accretion’ in this paper. Furthermore, SN shocks destroy the dust by sputtering (Dwek & Scalo 1980; Nozawa, Kozasa & Habe 2006). In this paper, we consider those processes since they dominate the increase or decrease of the dust mass in a galaxy. Indeed, the models including these processes are successful in reproducing the evolution of dust content as a function of metallicity (Kuo, Hirashita & Zafar 2013; de Bennassuti et al. 2014; Rémy-Ruyer et al. 2014). However, applications of dust enrichment models have been limited to bright objects at high redshift, especially at z > 5, since it was extremely difficult to detect dust emission from such distant galaxies.

Recent observations by the Atacama Large Millimetre/submillimetre Array (ALMA) have been providing a new window to the studies of origin and evolution of dust in the Universe. Because of its high sensitivity, ALMA has the potential for detection of dust emission from high-redshift ‘normal’ galaxies represented by Lyman break galaxies (LBGs) and Lyman α emitters (LAEs) at high redshift (z ≳ 5; e.g. Capak et al. 2015). ALMA has expanded the possibility of high-z dust studies that had been previously limited to very bright objects such as quasars and submillimetre (submm) galaxies. The current frontier of such dust hunting in high-z LBGs and LAEs is z > 6, where most of the observations ended up with non-detection of dust continuum (Ouchi et al. 2013; Ota et al. 2014; Maiolino et al. 2015; Schaerer et al. 2015; Aravena et al. 2016; Bouwens et al. 2016). Watson et al. (2015) recently detected a lensed LBG at z = 7.5, A1689-zD1 (see also Knudsen et al. 2016). Willott et al. (2015) also reported a firm and a tentative detection of two LBGs at z ∼ 6. These cases provide important clues to the origin and evolution of dust in the Universe.

Mancini et al. (2015) modelled dust evolution in z ≳ 6 galaxies. They adopted a cosmological hydrodynamics simulation and apply a dust evolution model that includes dust formation in stellar ejecta, dust growth by accretion and dust destruction by shocks as post-processing. By assuming a typical accretion time-scale of nearby galaxies (2 Myr at solar metallicity), they compared the calculated redshift evolution dust mass with the observation data of A1689-zD1, finding that the predicted dust mass is too small in comparison with the observation. Thus, they adopted a shorter accretion time-scale as 0.2 Myr to reproduce the observation, concluding that extremely efficient accretion is required. Michałowski (2015) also concluded a necessity of grain growth by accretion for the same object using a much simpler model. Although Mancini et al. (2015)'s model is based on a realistic semi-analytic model, they basically fixed the parameters concerning the dust production and destruction. Since those parameters may be uncertain, it is still worth making an effort of constraining the time-scales and efficiencies of dust production and destruction using A1689-zD1, in order to quantify how fast each process driving dust evolution works. By modelling this object, we can also address the reason for the non-detection of most sources at z > 6. In other words, we will be able to clarify what kind of dust evolution process determines the detection and non-detection by ALMA. Thus, the predictions in this paper could be useful for the planning and interpretations of future ALMA observations of high-z galaxies.

This paper is organized as follows. In Section 2, we present the dust enrichment model and explain how we calculate the corresponding dust emission flux. In Section 3, we apply the model to LBGs at z > 6 detected by ALMA. We discuss our results and parameter dependence in Section 4. In Section 5, we give conclusions on ALMA-detected LBGs. We adopt (h, Ω_m, |$\Omega _\Lambda$|⁠) = (0.7, 0.3, 0.7) for the cosmological parameters.

2 METHOD

The purpose of this paper is to constrain the dust enrichment and destruction processes using stellar and dust luminosities of high-redshift (z > 6) normal galaxies detected by ALMA. To this aim, we use a simple one-zone chemical evolution model, from which we obtain the evolution of dust mass and stellar mass. Further, we derive the dust temperature by assuming the radiative equilibrium of dust. Based on the stellar mass, dust mass and dust temperature estimated, we further derive the fluxes at rest-frame ultraviolet (UV) and FIR wavelengths. These fluxes are compared with observational data that provide constraints to our model. As a consequence, we obtain an insight into the relevant dust enrichment and destruction time-scales (or efficiencies).

2.1 Dust enrichment model

For the convenience of analytic treatment, we adopt the instantaneous recycling approximation (Tinsley 1980). The mass ejection rate of stellar ejecta E and that of metals E_Z are both proportional to the SFR as |$E=\mathcal {R} \psi$| and |$E_Z=\mathcal {Y}_Z \psi$|⁠, where |$\mathcal {R}$| and |$\mathcal {Y}_Z$| are the returned fraction and metal yield of stars, respectively. We evaluate ∫R and |$\mathcal {Y}_Z$| based on equations (8) and (10) in Inoue (2011) and the fitting formulae of remnant mass w(m, Z) and metal yield m_Z(m) from the same reference, using 5 M_⊙ as the turn-off mass. Consequently, we obtain |$\mathcal {R}=0.1644$| and |$\mathcal {Y}_Z=0.0139$|⁠. These values are not sensitive to the turn-off mass chosen (Hirashita & Kuo 2011) and the results below are far more sensitive to the parameters directly related to the dust evolution (i.e. f_in, τ_SN and τ_acc).

2.2 FIR and UV luminosity

Although the stellar mass is estimated more precisely using data at longer wavelengths, we use UV magnitude for the following reasons. Since LBGs are UV-selected galaxies, they always have data in the rest UV, while the availability of longer wavelength data depends on how extensively those objects are followed up. More importantly, the above formula is already tested for a sample of LBGs by Schaerer et al. (2015). Their formula is derived by SED fitting including nebular emission to a sample of LBGs at z ∼ 7, which means that their formula is applicable to the galaxies of interest in this paper. Therefore, we take advantage of their convenient expression to make our formulation analytically manageable. In addition, we compare the stellar masses estimated by the spectral fitting in Watson et al. (2015) and by the formula in Schaerer et al. (2015) and find that they differ by a factor of 1.4. This is small enough and does not affect our conclusions.

2.3 Observational data

We compare the calculated νf_{d, ν}/f_UV with observations. As mentioned in the Introduction, we chose normal galaxies at z > 6 (i.e. the current frontier for dust hunting of normal galaxies) identified in the optical with the dust continuum detected. This sample provides us with opportunity to access ‘normal’ galaxies with moderate SFRs ∼10 M_⊙ yr⁻¹ and enables us to derive a more general picture of dust enrichment in the earliest Universe accessible by ALMA, not biased to submm galaxies or quasars (Mancini et al. 2015). So far, A1689-zD1 (Watson et al. 2015; Knudsen et al. 2016) and CLM1 (Willott et al. 2015) satisfy this criterion. A1698-zD1 is strongly lensed by a factor of 9.5 and the redshift (7.5) is determined by the spectroscopic break feature at the rest-frame UV wavelengths. The dust continuum of CLM1 at z = 6.2 was tentatively detected at the 2σ level by ALMA. Because of the rareness of the sample, we use this flux as detected, keeping in mind that this might overestimate the dust emission (or the efficiency of dust enrichment) in our model. Willott et al. (2015) also reported a more significant detection of dust continuum emission for WMH 5; however, the emitting regions of dust continuum and stars are clearly different in this object, indicating that this object is a merging system between a dusty galaxy and a dust-free galaxy. Therefore, the UV radiation observed is not likely to be the heating source of the dust in this object, which means that our model cannot be applied to it. In Table 1, we list the observational data adopted in this paper.

3 RESULTS

3.1 Overall behaviour

In Fig. 1, we plot the normalized flux νf_{d, ν}/f_UV versus the normalized time t/τ_SF. To separate the effect of each process, we fix two parameters in the set (f_in, ζ_{acc, 0}, ζ_SN) and change the other in each panel. In Fig. 1(a), we observe that the effect of f_in appears at the early stage and all the cases converge to the same evolution at later times. Thus, we can conclude that dust condensation in stellar ejecta has an influence on the early evolution. This is consistent with the conclusion by Hirashita (1999) that the stellar dust condensation dominates the dust abundance in the early (low-metallicity) stage of galaxy evolution. Besides, from equation (3), we find that dust growth by accretion and SN destruction are dependent on M_d. Thus, as the dust abundance becomes large, the evolution is determined by these two processes; this is why all the cases with different f_in converge to the same evolution.

Figure 1.

Normalized dust emission flux νf_ν/f_UV as a function of the normalized time t/τ_SF. The horizontal lines at νf_ν/f_UV = 1.35 and 0.34 are the observed values of A1689-zD1 and CLM1, respectively. The shaded area shows the error of each object. (a) The dotted, dashed and dot–dashed lines show the results for f_in = 0.5, 0.1, and 0.01, respectively, with ζ_SN = 0.1 and ζ_{acc, 0} = 0.01. (b) The dotted, dashed and dot–dashed lines are for ζ_SN = 1, 0.1 and 0.01, respectively, with ζ_{acc, 0} = 0.01 and f_in = 0.1. (c) The long-dashed, triple-dot–dashed, dot–dashed, dashed and dotted lines are for ζ_{acc, 0} = 10⁻⁴, 10⁻³, 0.01, 0.1 and 1, respectively, with ζ_SN = 0.1 and f_in = 0.1. (d) Metallicity evolution (dotted line). The horizontal solid line shows Z = 0.1 Z_⊙: LGBs at z > 6 typically have metallicities below this level (Mancini et al. 2015).

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In Figs 1(b) and (c), we show the effect of changing ζ_SN and ζ_{acc, 0}, respectively. In contrast to the cases with different f_in, the effects of the variations of these two parameters are prominent at a late stage of evolution as expected from the discussion in the previous paragraph. SN destruction decreases and accretion increase the dust emission relative to the stellar emission, as expected. Thus, we conclude that SN destruction and dust accretion are important when the system is enriched with dust and metals. We will characterize the metallicity level at which these processes, especially accretion, become dominant in Section 4.3.

Metallicity is often used as a measure for the evolutionary stage of a galaxy. In Fig. 1, we show the evolution of metallicity as a function of t/τ_SF. We observe that the metallicity reaches 0.1 Z_⊙ at t/τ_SF = 0.15. According to the theoretical calculations by Mancini et al. (2015), the metallicity of the LBGs is less than 0.1 Z_⊙. Therefore, we first put a constraint on the metallicity as Z < 0.1 Z_⊙. Since the galaxies detected by ALMA may be biased to evolved ones as we discuss later, we also examine a case where we do not put any constraint on the metallicity.

3.2 Constraint on the dust evolution parameters

Before analysing the result, we first discuss the probable ranges of the three parameters (f_in, ζ_{acc, 0}, ζ_SN). As compiled in Kuo et al. (2013), the value of the condensation efficiency f_in, expected from theoretical calculations, has large uncertainty. Thus, we investigate the value from 0.01 to 0.5 with f_in = 0.1 being the fiducial value. Also, McKee (1989) and Lisenfeld & Ferrara (1998) show that β_SN = M_g/(τ_SNψ) = τ_SF/τ_SN (which is equivalent with 1/ζ_SN in our paper) is ≈10. For a wide parameter survey, we investigate the range 0.01 ≲ ζ_SN ≲ 1. We are interested in as efficient accretion as suggested by Mancini et al. (2015). To consider such efficient accretion, we focus on ζ_{acc, 0} down to 10⁻⁴ (see also the discussion in Section 4.1).

In the upper panel of Fig. 2, we show the area on the (ζ_{acc, 0}, ζ_SN) plane in which the set of (f_in, ζ_{acc, 0}, ζ_SN) of A1689-zD1 satisfies the above condition (equation 14) at Z < 0.1 Z_⊙ (expected metallicity for LBGs by Mancini et al. 2015). We find that for f_in ≲ 0.1, ζ_{acc, 0} much smaller than ζ_SN is necessary to explain the observational data. This is because only when the accretion time-scale is shorter than the SN time-scale, the dust could have chance to grow before being destroyed by SNe. If we assume a standard value of ζ_SN ∼ 0.1 as suggested for nearby galaxies (McKee 1989), small values of ζ_{acc, 0} (<0.004) are required. For large f_in > 0.5, however, the constraint on ζ_{acc, 0} is not stringent since the dust produced by stellar sources makes it easier to increase νf_{d, ν}/f_UV up to the observed value regardless of accretion. Nevertheless, even for f_in = 0.5, we need ζ_{acc, 0} < 0.01 if SN destruction is efficient (ζ_SN ≲ 0.05). As we discuss later, theoretical dust condensation calculations indicate that f_in is likely to be less than 0.5, which means that dust growth by accretion is important to explain the dust emissions detected by ALMA for A1689-zD1.

Figure 2.

Ranges of the time-scale parameters for accretion and SN destruction (ζ_{acc, 0}, ζ_SN) constrained by A1689-zD1 for f_in = 0.5, 0.1 and 0.01. The region on the right and lower side (shaded region) of each line satisfies the criterion. The solid, dashed and dotted lines are the boundary lines for f_in = 0.5, 0.1 and 0.01, respectively. The upper and lower panel are with and without the criterion of metallicity Z > 0.1 Z_⊙, respectively. The asterisk refers to the standard accretion and destruction values of nearby galaxies, (ζ_{acc, 0}, ζ_SN) = (0.012, 0.1)

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Next, we remove the condition for the metallicity in the lower panel of Fig. 2 by allowing a possibility that LBGs detected by ALMA are more metal-rich than expected (i.e. Z > 0.1 Z_⊙). We can see that the trend of f_in = 0.5 barely changes, meaning that the large efficiency of stellar dust production produces a sufficient amount of dust in the early phase of evolution as shown in Fig. 1(a). In contrast, the condition for ζ_{acc, 0} is much more relaxed for f_in = 0.01 and 0.1. This means that allowing for a metal-rich stage is essential for accretion, since the efficiency of accretion is proportional to the metallicity. As shown later, ζ_{acc, 0} ∼ 0.1 is similar to the value inferred for nearby galaxies. Thus, if the LBGs detected by ALMA at z > 6 are solar-metallicity objects, their rich dust content can be explained by assuming an accretion efficiency expected for nearby galaxies.

The upper and lower panel of Fig. 3 show the results for CLM1. Comparing Figs 2 and 3, we observe that the condition for dust formation and SN destruction is similar for both objects. That is, for small dust condensation efficiency in stellar ejecta (⁠|$f_\mathrm{in}\lesssim 0.1$|⁠), both objects need very efficient dust accretion (⁠|$\zeta _\mathrm{acc,0}\lesssim 0.004$|⁠) under the metallicity constraint Z < 0.1 Z_⊙. This constraint on the accretion time-scale is much relaxed if we allow the metallicity to be higher than 0.1 Z_⊙. The fact that we obtain very similar constraints for both objects indicates that the detection of LGBs by ALMA requires a similar range for (f_in, ζ_{acc, 0}, ζ_SN).

Figure 3.

Same as Fig. 2 but for the other object, CLM1.

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4 DISCUSSION

4.1 Likely values for the parameters

Knudsen et al. (2016) derived the SFR of A1689-zD1 as 12 M_⊙ yr⁻¹ from the submm continuum. Although the gas has not been detected, Watson et al. (2015) derived a gas mass as 2 × 10⁹ M_⊙ based on the Schmidt—Kennicutt law, which is equivalent to assuming a star formation time-scale of 1.7 × 10⁸ yr. Mancini et al. (2015) showed, using their semi-analytic models of galaxy formation and evolution, that A1689-zD1 requires very short accretion time-scale, τ_{acc, 0} = 0.2 Myr. Thus, the literature value indicates that ζ_{acc, 0} ∼ 1.2 × 10⁻³. Our constraint obtained in Fig. 2 shows |$\zeta _\mathrm{acc,0}\lesssim 4\times 10^{-3}$| if we impose the metallicity constraint Z < 0.1 Z_⊙ as indicated by Mancini et al. (2015). Thus, we confirm their conclusion about the necessity of strong accretion quantitatively (see also Michałowski 2015). Such a high accretion efficiency is realized if the molecular clouds are very dense (Kuo, Hirashita & Zafar 2013; Schneider, Hunt & Valiante 2016). However, we also show that if we allow larger metallicities, we obtain ζ_{acc, 0} ≲ 5 × 10^{− 2} with f_in = 0.1 and ζ_SN = 0.1. If we adopt the above star formation time-scale, this corresponds to τ_{acc, 0} ∼ 8.5 × 10⁶ yr, which is consistent with the accretion time-scale derived for nearby galaxies (Hirashita & Kuo 2011). Note that the dust emission is only tentatively detected for CLM1. In other words, the detection of dust continuum by ALMA at z ≳ 6 indicates that the object should satisfy either of the following conditions: (i) very efficient condensation in stellar ejecta (⁠|$f_\mathrm{in}\gtrsim 0.5$|⁠), (ii) very efficient dust growth by accretion with |$\zeta _\mathrm{acc,0}\lesssim 4\times 10^{-3}$| or (iii) high metallicity ( ≳ solar metallicity).

Recently, Ferrara et al. (2016) proposed another reason for the ‘deficit’ of dust emission for high-z LBGs. Since the ISM in high-z galaxies has a high pressure, high-z LBGs host dense molecular clouds (Pallottini et al. 2016). Thus, a large fraction of dust is ‘hidden’ in the shielded environment of the dense molecular clouds. Since such a hidden component of dust has dust temperatures as low as the CMB temperature at the redshift of the LBGs, its FIR emission is weak. If this deficit of dust emission is common for high-z LBGs, more extreme conditions (i.e. higher efficiencies) for dust enrichment than shown in this paper would be necessary for ALMA detection of dust emission.

The above conditions for ALMA detection may be modified if we consider gas inflow and outflow from galaxies. We adopted a closed-box model in this paper for simplicity. As mentioned at the beginning of Section 2.1, if we include the effect of inflow and outflow, we need to consider the dilution of metals and dust (Feldmann 2015). However, if we define the total baryonic mass available in the end as M₀ and define the star formation time-scale by the total gas mass including what the galaxy obtain in its entire lifetime, we could use the same framework. Moreover, we have confirmed that we obtain a similar constraint on the accretion time-scale to that obtained by Mancini et al. (2015), who took into account the build-up of galaxy mass. Thus, we expect that our conclusion does not change significantly even if we take into account the realistic gas inflow and outflow that occurred in the course of galaxy evolution.

4.2 Dependence on grain properties

Our model assumes dust properties such as the grain radius a, the dust material density s, the emissivity index β and the mass absorption coefficient at 158 μm κ₁₅₈. Here we investigate how much the constraints obtained for the parameters (such as shown in Fig. 2) are affected by the assumed dust properties. Since similar results are obtained for both galaxies, we concentrate on A1689-zD1 in this subsection. We also fix the value of f_in to 0.1.

A larger grain radius leads to a more efficient emission, thus a lower temperature (see equation 10; the emission scales with a³ while the absorption scales with a²). Thus, we may obtain a severer constraint on the parameters for a larger grain. We here investigate the case of a = 1 μm in Fig. 4(a). We find that the constraints on ζ_{acc, 0} and ζ_SN differ only by a factor of ∼2 between a = 0.1 and 1 μm.

Figure 4.

Same as Fig. 2 but for various grain radii and material properties. Only the boundary for f_in = 0.1 is shown. Upper panel: grain size dependence. The solid and dashed lines refer to a = 0.1 and 1 μm, respectively. Lower panel: dust material dependence. The lines from top to bottom refer to AC, graphite, silicate, SN_dest and SN_con, respectively.

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Variation of the dust material affects the values of s, β and κ₁₅₈. For the dust material, we adopted graphite above, but we here investigate other grain species listed as possible dust species by Hirashita et al. (2014); silicate, amorphous carbon (AC), SN_con (theoretically expected dust material properties for condensation in SN ejecta; Nozawa et al. 2003) and SN_des (theoretically expected values with reverse shock destruction in an SN remnant; Nozawa et al. 2007). The dust properties are summarized in Table 2. The constraints obtained for each grain species are compared in Fig. 4(b). The longest accretion time-scale is allowed for AC among all the species, since AC has the highest emissivity, thus achieving the observed ALMA flux most easily. For the opposite reason, SN_con requires the shortest accretion time-scale.

4.3 Importance of metallicity measurements

In the above, two solutions were broadly permitted to explain the observed dust abundance in the two high-z galaxies. One is an extremely high condensation efficiency f_in typically ≥0.5, although this solution is not supported by theoretical condensation calculations and the other is an efficient accretion. As shown in Asano et al. (2013), the rise of dust abundance occurs after the system is enriched with metals to a certain ‘critical’ metallicity, since efficient accretion occurs only when a sufficient amount of accreting materials (i.e. metals) is available.

Under the metallicity constraint Z < 0.1 Z_⊙, values of ζ_{acc, 0} smaller than 4 × 10⁻³ appropriate for the sample of z > 6 galaxies indicate that the critical metallicity is ∼0.06 Z_⊙. Therefore, if we find that the metallicity is 0.06 Z_⊙ < Z < 0.1 Z_⊙, it would strongly support the scenario of efficient dust growth by accretion in those galaxies. Otherwise, for Z < 0.06 Z_⊙, efficient stellar dust production would be the way to explain the dust emission detected by ALMA. Taking away the metallicity constraint, we obtained a constraint |$\zeta _\mathrm{acc,0}\lesssim 0.1$| (i.e. a similar efficiency of accretion to the one in nearby galaxies) for ζ_SN ∼ 0.1, leading to a critical metallicity of ∼0.3 Z_⊙. Thus, if the metallicity proves to be larger than 0.3 Z_⊙, accretion naturally explains the high dust luminosity, but it is not necessary to assume an extremely high accretion efficiency compared with the one in nearby galaxies.

It is generally difficult to detect metal emission lines for galaxies at z ≳ 6. Nagao et al. (2012) proposed to use FIR fine-structure lines to assess the metallicity. Future sensitive submm observations may constrain the metallicities in high-z LBGs. Alternatively, the grain size distribution would serve to constrain the dominant dust production process, since the dust grains produced by stars have large sizes (≈0.1 μm), while there should be a large abundance of small grains produced by shattering if dust growth by accretion is actively occurring (Asano et al. 2013). Probably, extinction curves, although they are not easy to be derived, could provide the information on the grain size distribution (Asano et al. 2014). Observations of dense molecular gas would directly probe the existence of clouds hosting dust growth by accretion, since extremely dense molecular clouds may enhance the efficiency of dust growth (Kuo et al. 2013). Indeed, Pallottini et al. (2016) showed in their simulation a dense and metal-enriched region in the central part of an LBG. Such a dense region may host dust growth. Observations of highly excited molecular line whose critical density is high may serve to identify such a dense region (Vallini et al. 2016). Thus, if observations of dense molecular clouds prove that molecular clouds are dense (by high-excitation molecular lines, etc.), they would support the possibility of efficient accretion.

5 CONCLUSION

Recent ALMA observations have started to detect normal galaxies even at z > 6. We investigated what constraint we can get for the dust enrichment in normal z > 6 galaxies detected by ALMA. To this aim, we used a theoretical model that includes major processes driving dust evolution in a galaxy, including dust condensation in stellar ejecta, dust growth by the accretion of gas-phase metals and SN destruction. To cancel out unknown quantities such as the total baryonic content and the star formation time-scale, we consider the flux at the ALMA band normalized to the UV flux as a function of time normalized to the star formation time-scale. Using the observational data for A1689-zD1 and CLM1, we obtained the range of the time-scales (or efficiencies) of the above mentioned processes by examining if the observed level of the ALMA flux normalized to the UV flux is achieved. We find that if we assume extremely high condensation efficiency in stellar ejecta (f_in ≳ 0.5), stardust may be enough to explain the observed ALMA flux in their early evolutionary stage, unless dust destruction by SNe in those galaxies is stronger than that in nearby galaxies. If we assume a condensation efficiency expected from the theoretical calculations (f_in ≲ 0.1), dust growth by accretion is required. If the metallicities of those objects are lower than 0.1 Z_⊙ as suggested by a previous theoretical model (Mancini et al. 2015), strong dust growth (ten times stronger than assumed for nearby galaxies) is derived. Alternatively, if we allow for solar metallicity, only moderate accretion whose efficiency is comparable to that in nearby galaxies is required to explain the ALMA detections. These results indicate that the normal galaxies detected by ALMA at z > 6 are biased to objects (i) with high dust condensation efficiency in stellar ejecta, (ii) with strong dust growth or (iii) with efficient dust growth because of fast metal enrichment up to solar metallicity. We finally argue that measurement of metallicity will provide a key to distinguish among these possibilities.

Acknowledgments

We are grateful to the anonymous referee for useful comments. HH thanks the Ministry of Science and Technology for support through grant MOST 105-2112-M-001-027-MY3.

REFERENCES

APPENDIX: CRITICAL METALLICITY FOR DUST GROWTH BY ACCRETION