Search for Optically Dark Infrared Galaxies without Counterparts of Subaru Hyper Suprime-Cam in the AKARI North Ecliptic Pole Wide Survey Field

To select AKARI sources without HSC counterparts securely and investigate their physical properties, we compile multiwavelength data. AKARI NEP-W was observed by many facilities, and thus we have abundant data sets from ultraviolet (UV) to radio (see, e.g., Kim et al. 2012; Oi et al. 2020, and references therein for a full description). In addition to the AKARI NEP-W catalog (Kim et al. 2012) and HSC catalog (Oi et al. 2020) we particularly utilized the data sets that are summarized in Table 1. The area coverage of each data set is summarized in Figure 1.

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We used NIR data provided by Jeon et al. (2014), who conducted deep imaging with J and H bands taken by the Florida Multi-object Imaging Near-IR Grism Observational Spectrometer (FLAMINGOS: Elston et al. 2006) on the Kitt Peak National Observatory (KPNO) 2.1 m telescope. This photometric catalog contains 295,383 sources with 5σ detection limits of 21.6 and 21.3 mag in the J and H bands, respectively.

For MIR data, we used a catalog of Nayyeri et al. (2018), who provided 3.6 μm (ch1) and 4.5 μm (ch2) data taken with the Infrared Array Camera (IRAC: Fazio et al. 2004) on board the Spitzer Space Telescope (Werner et al. 2004). This photometric catalog contains 380,858 sources with 5σ detection limits of 6.45 and 3.95 μJy in the 3.6 μm and 4.5 μm bands, respectively.

Regarding the FIR data, we utilized a recent catalog of 250, 350, and 500 μm data (Pearson et al. 2017, C. Pearson 2020, in preparation). The data were taken with the Spectral and Photometric Imaging Receiver instrument (SPIRE: Griffin et al. 2010) on board the Herschel Space Observatory (Pilbratt et al. 2010). This SPIRE catalog contains 4820 sources with 5σ detection limits of 9.0, 7.5, and 10.8 mJy at 250, 350, and 500 μm, respectively (Barrufet et al. 2020). Note that AKARI NEP-D was observed by the Photoconductor Array Camera and Spectrometer (PACS: Poglitsch et al. 2010) at 100 and 160 μm, and the catalog (including 1384 and 630 sources detected at 100 and 160 μm, respectively) is available (Pearson et al. 2019). But since the survey footprint is not so large (∼0.7 deg², see Figure 1), we did not use the catalog in this work.³⁰

Eventually, we used 21 photometric data in total; g, r, i, z, y, J, and H bands and 2.4 (N2), 3.2 (N3), 3.6 (ch1), 4.1 (N4), 4.5 (ch2), 7.0 (S7), 9.0 (S9W), 11.0 (S11), 15.0 (L15), 18.0 (L18W), 24.0 (L24), 250, 350, and 500 μm, even though some of them are often upper limits (e.g., HSC five-bands).

The procedure for sample selection is summarized in Figure 2. The candidates for AKARI sources without HSC counterparts were drawn from the AKARI NEP-W sample in Kim et al. (2012), who provided 114,794 sources detected by the IRC. The 5σ detection limits of N2, N3, N4, S7, S9W, S11, L15, L18W, and L24 are 15.42, 13.30, 13.55, 58.61, 67.30, 93.76, 133.1, 120.2, and 274.4 μJy, respectively. Note that we selected candidates and obtained their multiwavelength information by cross-matching with several catalogs in which coordinates in the AKARI NEP-W catalog were always quoted. In this work, we attempt a cross-matching with a catalog by using a search radius that is much larger than the typical size of a point-spread function for objects in the catalog. We then determine a search radius for cross-matching with a catalog as a 3σ deviation from mean separation (ΔR.A. and Δdecl.) between AKARI NEP-W and that catalog, in the same manner as S. J. Kim et al. (2020, in preparation).

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We first narrowed the sample to sources within the footprint observed by the HSC. This is because the HSC data do not cover quite the whole region of the AKARI NEP-W (see Figure 1), and hence some objects are just unobserved by the HSC. As a result, 109,734 objects were left out. We then removed 84,076 objects with nm³¹ > 0 that were reported as having optical counterparts in Kim et al. (2012). They used optical catalogs provided by Hwang et al. (2007) and Jeon et al. (2010) in which the optical data were taken with MegaCam (Boulade et al. 2003) on the Canada–France–Hawaii Telescope (CFHT) and SNUCAM (Im et al. 2010) at the Maidanak observatory, respectively. Following that, we cross-identified the sample with the HSC catalog (Oi et al. 2020), which has a sensitivity roughly five times deeper than optical catalogs used in Kim et al. (2012), providing optically faint AKARI sources. By adopting a search radius of 18, 20,446 objects were cross-identified.

For 25,658 − 20,446 = 5212 AKARI sources, we removed contaminants. Because objects with magnitude brighter than ∼16 may be saturated in the MegaCam/SNUCAM/HSC images, they were removed from those catalogs before the cross-matching with the AKARI NEP-W catalog. Therefore, some AKARI sources in our sample are expected to be bright stars/galaxies. In order to remove bright objects, we cross-identified the sample with Gaia DR2 (Gaia Collaboration et al. 2016, 2018) using a search radius of 16. The Gaia DR2 catalog contains point sources with a g-band magnitude of 3–16. As a result, 534 bright objects were removed. We then extracted AKARI objects with 5σ detections in at least one AKARI band and with clean photometry flag³² (i.e., flag_bandname_mag = 0). We also applied nfl = 0 to select objects unaffected by cosmic rays and/or multiplexer bleed trails in the NIR bands (see Section 2.2 in Kim et al. 2012), which yielded 3734 objects.

Finally, we conducted a visual inspection to select reliable candidates in which we supplementarily used multiwavelength images in J and H bands (FLAMINGOS) and ch1 and ch2 (IRAC) in addition to HSC and AKARI images. Consequently, 583 objects were selected as AKARI sources without HSC counterparts. Figure 3 shows examples of postage stamp images for our sample. Why were 3734 − 583 = 3151 objects removed? One reason is that the edges of the fields of view of HSC may be degraded by artifacts. The AKARI NEP-W consists of 4–7 exposures by the HSC with a field of view of 15 in diameter through dithering observations. Therefore, objects around the edges of each exposure frame would be missed from the HSC catalog created by the HSC pipeline (Bosch et al. 2018) although they exist on the HSC image (see Oi et al. 2020 for details). Nevertheless, we should note that the visual inspection might be highly dependent on the classifier. The number of objects selected in this work may have an uncertainty. Hence, the population census results such as number counts and volume density of AKARI sources without HSC counterparts should be addressed in future work, and we focus on an overview of their physical properties in this work.

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For those AKARI sources without HSC counterparts, we compiled a multiwavelength data set up to 500 μm. We cross-identified the sample with FLAMINGOS, IRAC/Spitzer, and SPIRE/Herschel. In our sample, AKARI coordinates of each source were always used for the cross-matching. We employed 19, 23, and 66 as the search radius for cross-matching with FLAMINGOS, IRAC, and SPIRE, respectively. As stated, these search radii were determined to be 3σ deviations from the mean separation between AKARI objects without HSC counterparts and FLAMINGOS/IRAC/SPIRE catalogs. Accordingly, 70, 338, and 35 objects were cross-identified with FLAMINGOS, IRAC, and SPIRE, respectively. We found that 2/70, 4/338, and 2/35 AKARI objects have two candidate counterparts for FLAMINGOS, IRAC, and SPIRE, respectively. In this study, we chose the closest object as the counterpart for such cases. Among AKARI bands, 425/583 (73%), 109/583 (19%), and 59/583 (10%) objects are detected in the NIR, MIR-S, and MIR-L, respectively, We note that 23 objects with L24 detection naturally satisfy the DOG criterion, ${(r-[24\mu {\rm{m}}])}_{\mathrm{AB}}\gt 7.5$ (Dey et al. 2008), owing to the limiting magnitude in the HSC r band (26.7) and AKARI 24 μm (17.8), suggesting that our sample selection may preferentially select dusty galaxies (see also Section 4.2).

We employed CIGALE to conduct detailed SED modeling in a self-consistent framework by considering the energy balance between the UV/optical and IR. This code enables us to handle many parameters, such as star formation history (SFH), single stellar population (SSP), attenuation law, AGN emission, dust emission, and radio synchrotron emission (see, e.g., Boquien et al. 2014, 2016; Buat et al. 2014, 2015; Ciesla et al. 2017; Lo Faro et al. 2017; Toba et al. 2019a, 2020b; Burgarella et al. 2020). Note that one of the purposes of this work is to compare the physical properties between AKARI sources with and without HSC counterparts. Wang et al. (2020) constructed an AKARI L18W-selected sample with HSC counterparts, and also conducted the SED fitting with CIGALE to derive their physical properties. Therefore, to compare the resultant quantities under the same conditions, we chose the same models with the same parameter ranges as those that Wang et al. (2020) adopted, except for AGN fraction in the AGN module and γ in the dust module (see below). Parameter ranges used in the SED fitting are tabulated in Table 2.

Table 2.  Parameter Ranges Used in the SED Fitting with CIGALE

Parameter	Value
Delayed SFH
τmain (Myr)	5000.0
τburst (Myr)	20000
fburst	0.00, 0.01
Age (Myr)	1000, 5000, 10000
SSP (Bruzual & Charlot 2003)
IMF	Salpeter (1955)
Metallicity	0.02
Dust Attenuation (Charlot & Fall 2000)
log (${A}_{{\rm{V}}}^{\mathrm{ISM}}$)	−2, −1.7, −1.4, −1.1, −0.8, −0.5,
	−0.2, 0.1, 0.4, 0.7, 1.0
slope_ISM	−0.9, −0.7, −0.5
slope_BC	−1.3, −1.0, −0.7
AGN Emission (Fritz et al. 2006)
${R}_{\max }/{R}_{\min }$	60
τ9.7	0.3, 6.0
β	−0.5
γ	4.0
θ	100.0
ψ	0.001, 60.100, 89.990
fAGN	0.1, 0.3, 0.5, 0.7, 0.9
Dust Emission (Draine et al. 2014)
qPAH	0.47, 1.77, 2.50, 5.26
Umin	0.1, 1.0, 10, 50
α	1.0, 1.5, 2.0, 2.5, 3.0
γ	0.01, 0.1, 1.0
Photometric Redshift
zphoto	0.8, 1.0, 1.5, 2.0, 2.2, 2.4, 2.6, 2.8, 3.0,
	3.2, 3.4, 3.6, 3.8, 4.0, 4.2

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We used a delayed SFH model, assuming a single starburst with an exponential decay (Ciesla et al. 2015, 2016), where we fixed e-folding times of the main stellar population (τ_main) and the late starburst population (τ_burst), while we parameterized the age of the main stellar population in the galaxy.

We utilized the stellar templates with solar metallicity provided by Bruzual & Charlot (2003) assuming the IMF of Salpeter (1955), and the standard default nebular emission model included in CIGALE (see Inoue 2011).

Dust attenuation is modeled by using the model of Charlot & Fall (2000) with two different power-law attenuation curves that are parameterized by the power-law slope of the attenuation in the interstellar medium (ISM) and birth clouds (BC). We also separately parameterized the V-band attenuation in the ISM (${A}_{{\rm{V}}}^{\mathrm{ISM}}$).

For AGN emission, we used models provided by Fritz et al. (2006). In order to avoid a degeneracy of AGN templates in the same manner as in Ciesla et al. (2015) and Toba et al. (2019b), we fixed certain parameters that determine the number density distribution of the dust within the dust torus, i.e., ratio of the maximum to minimum radii of the torus (${R}_{\max }/{R}_{\min }$), density profile along the radial and the polar distance coordinates parameterized by β and γ (see Equation (3) in Fritz et al. 2006), and opening angle (θ). We parameterized the optical depth at 9.7 μm (τ_9.7) and ψ parameter (the angle between the equatorial axis and the line of sight) that corresponds to our viewing angle of the torus. We further parameterized AGN fraction (f_AGN), which is the contribution of AGNs to the total IR luminosity (Ciesla et al. 2015). Note that we adopted a discrete interval for f_AGN, unlike Wang et al. (2020), where f_AGN is a key parameter to investigate the dependence of the fractional AGN contribution on IR luminosity and redshift. Our sample is basically detected in only a few AKARI bands (and the remaining bands give upper limits). This may often not be enough to constrain ${f}_{\mathrm{AGN}}$ precisely. Hence we reduced the number of possible values of f_AGN considered.

Dust grain emission is modeled by Draine et al. (2014). The model is parameterized by the mass fraction of PAHs (${q}_{\mathrm{PAH}}$), the minimum radiation field (U_min), and the power-law slope of the radiation field distribution (α) (see Equation (4) in Draine et al. 2014). We also parameterized the fraction illuminated with a variable radiation field ranging from U_min to U_max (γ), although Wang et al. (2020) fixed this at γ = 1. We confirmed that parameterizing γ gives a better fit to the FIR parts of the spectra.

We also parameterized redshift to estimate photometric redshift (z_photo), because by definition our sample is optically too faint to obtain spectroscopic redshifts. Since the optically dark galaxies tend to be located at $z\gt 2$ (e.g., Franco et al. 2018; Wang et al. 2019; Williams et al. 2019; Yamaguchi et al. 2019), we optimized the range of z_photo, which reduces the computing time. Aside from being an excellent SED-fitting tool, CIGALE is known to be a good estimator of z_photo, since it utilizes a large number of models covering the whole SED including the MIR–FIR regime. For example, Małek et al. (2014) calculated z_photo for AKARI sources in the AKARI Deep Field South. They demonstrated the accuracy of z_photo by using the normalized median absolute deviation defined as ${\sigma }_{{\rm{\Delta }}z/(1+{z}_{\mathrm{spec}})}=1.48\times$ median($| {\rm{\Delta }}z|$/(1 + z_spec)) in the same manner as Ilbert et al. (2006). The resultant ${\sigma }_{{\rm{\Delta }}z/(1+{z}_{\mathrm{spec}})}$ is 0.056, lower than what was obtained by software using mainly optical to NIR data (Ilbert et al. 2006; see also Barrufet et al. 2020). On the other hand, the above AKARI sources have optical counterparts, i.e., they are moderately bright in the optical, and the sample is limited to the local universe (z < 0.25). Since we constrain the optical SEDs based only on upper limits, we need to investigate the influence of our lack of optical data points on the accuracy of z_photo (see Section 4.6.1).

In order to ensure the reliability of the derived physical quantities including z_photo, we extracted AKARI sources with 5σ detections in at least five bands between the NIR and FIR, which yields 142/583 objects for SED fitting. If the signal-to-noise ratio (S/N) at a certain band is greater than 5.0, we used the photometry at that band. Otherwise, we placed 10σ upper limits³³ that are drawn from each catalog (see Section 2.1).

Before the SED fitting, we check the IRAC ch1 and ch2 color ([3.6]–[4.5] in Vega magnitudes) of our sample objects. Figure 4 shows the IRAC color of 338 AKARI sources without HSC counterparts. We also plotted IRAC colors of AKARI sources with HSC counterparts where we used a multiwavelength merged AKARI catalog provided by S. J. Kim et al. (2020, in preparation). We removed stars with stellarity parameter (which was measured from CFHT r-band or Maidanak R-band images) greater than 0.8 (see Kim et al. 2012). This left 42,264 objects for the comparison.

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We find that [3.6]–[4.5] colors of AKARI sources without HSC counterparts (i.e., our sample) are systematically redder than those of AKARI sources with HSC counterparts. Blecha et al. (2018) reported that W1 (3.4 μm) and W2 (4.6 μm) color (almost the same as IRAC ch1 and ch2 color) taken with the the Wide-field Infrared Survey Explorer (Wright et al. 2010) is a good indicator of AGN activity and nuclear obscuration. The AGN luminosity and hydrogen column density peak during the galaxies' coalescence, where W1 – W2 color ranges from 0.8 to 1.6 (see Figure 1 in Blecha et al. 2018). This suggests that a fraction of objects in our sample could correspond to a luminous, obscured AGN phase.

Figure 5 shows examples of the SED fitting with CIGALE. We confirm that 91/142 (∼64%) objects have reduced χ² < 3.0 while 112/142 (∼79%) objects have reduced χ² <5.0. This means that the data are moderately well fitted with the combination of the stellar, AGN, and SF components by CIGALE. We also confirm that the probability distribution function (PDF) of redshift does not have prominent secondary peaks for ∼90% of our sample, where a peak that is 30% of the primary peak is considered to be a secondary peak (see top panels for objects that show no secondary peak and bottom panels for objects that do show a secondary peak in the PDF in Figure 5). The typical uncertainty of z_photo is about 20% (see also Section 4.6.1). Hereafter, we will focus on a subsample of 112 AKARI galaxies with reduced χ² of their SED fitting smaller than 5.0, in the same manner as Toba et al. (2019b).

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Figure 6 shows the stellar mass for AKARI sources with and without HSC counterparts. We find that the stellar mass of our sample galaxies is systematically larger than that of AKARI sources with HSC counterparts (Wang et al. 2020). The average stellar mass of AKARI sources with and without HSC counterparts is $\mathrm{log}({M}_{* }/{M}_{\odot })=10.8$ and 11.3, respectively. A two-sided Kolmogorov–Smirnov (K-S) test rules out the hypothesis that the two samples are drawn from the same distribution at >99.9% significance.

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One caution here is that the procedure of sample selection in this work differs from that in Wang et al. (2020); AKARI–HSC objects in Wang et al. (2020) require both L18W and Herschel detections, while our sample is not necessarily detected by them. Therefore, we extracted 22 objects that satisfy the sample selection criteria in Wang et al. (2020). Their average $\mathrm{log}({M}_{* }/{M}_{\odot })$ is 11.7, also significantly larger than for the sample in Wang et al. (2020), which is also confirmed by the K-S test with >99.9% significance.

Figure 6 also shows objects with the Bayesian information criterion (BIC; Schwarz 1978) greater than 2, which indicates that those objects need the AGN component to give a better SED fitting (see Section 4.6.2 for more detail). We confirmed that there is no systematic difference between those objects and others.

We then compare the V-band attenuation in the ISM (${A}_{{\rm{V}}}^{\mathrm{ISM}}$) for AKARI sources with and without HSC counterparts. Before the comparison, we should note that the dust attenuation may depend on the stellar mass (e.g., Buat et al. 2012, 2015), and the stellar mass distributions in the two samples are different as discussed in Section 4.1. Therefore, we extracted objects in an overlapped stellar mass range, i.e., $10\lt \mathrm{log}({M}_{* }/{M}_{\odot })\lt 12$, in the AKARI sample with/without HSC counterparts (see Figure 6).

Figure 7 shows ${A}_{{\rm{V}}}^{\mathrm{ISM}}$ for AKARI sources with and without HSC counterparts. These objects are samples matched by stellar mass. The average ${A}_{{\rm{V}}}^{\mathrm{ISM}}$ of AKARI sources with and without HSC counterparts is 1.26 and 5.16 mag, respectively. We find that the attenuation of our sample is systematically much larger than that of AKARI sources with HSC counterparts (Wang et al. 2020), which is supported by the K-S test with >99.9% significance. This result seems to be reasonable, given the fact that our sample is undetected even by the HSC, and its optical light is expected to be suppressed heavily by enshrouding dust.

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One caution is that the redshift distributions of AKARI objects in Wang et al. (2020) and in this study are different (see Section 4.3). Given an overlapped redshift range (0.8 <z < 2.0) between two samples (see Figure 8), the average ${A}_{{\rm{V}}}^{\mathrm{ISM}}$ of the sample in Wang et al. (2020) is 1.77 mag and that in this work is 5.23 mag. This suggests that our sample is intrinsically affected by dust extinction compared with AKARI with HSC counterparts.

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Figure 8 shows the IR luminosity contributed by AGNs as a function of redshift. We find that AKARI objects without HSC counterparts tend to be located at higher redshifts than those with HSC counterparts: the mean redshift of AKARI objects with and without HSC counterparts is z ∼ 0.46 and 1.34, respectively. We also find that objects with L18W and SPIRE detections tend to have larger AGN luminosity than others in our sample.

Note that the parent sample for these objects is the same, i.e., a flux-limited sample drawn from the AKARI NEP-W catalog (Kim et al. 2012). Thus AKARI sources are expected to lie in the same sequence on the redshift–luminosity plane, regardless of HSC detection. We confirm that samples in Wang et al. (2020) and this work are continuously distributed in that plane, indicating that CIGALE securely estimated redshift and luminosity. Indeed, given the overlapped redshift range of 0.8 < z < 2.0, the mean AGN luminosity of our sample is $\mathrm{log}({L}_{\mathrm{IR}}(\mathrm{AGN})/{L}_{\odot })\sim 11.2$, comparable to the sample in Wang et al. (2020).

The total IR luminosity of AKARI sources without HSC counterparts is also larger than that of those with HSC counterparts, with mean $\mathrm{log}({L}_{\mathrm{IR}}/{L}_{\odot })$ of 11.4 and 12.2, respectively. Actually, about 65% and 15% of our sample are ultraluminous IR galaxies (ULIRGs: Sanders & Mirabel 1996) and hyperluminous IR galaxies (HyLIRGs: Rowan-Robinson 2000) with L_IR greater than 10¹² and 10¹³ L_☉, respectively. This is in good agreement with the fact that dusty galaxies with extreme optical–IR color tend to be luminous in the IR (e.g., Tsai et al. 2015; Toba & Nagao 2016; Toba et al. 2018, 2020a; Fan et al. 2020).

Figure 9 shows the AGN fraction (f_AGN), L_IR(AGN)/L_IR. The mean AGN fraction of AKARI sources with and without HSC counterparts is 0.06 and 0.22, respectively, indicating that our sample tends to harbor AGNs. We also find that f_AGN of objects with L18W and SPIRE detection is 0.16, smaller than the average of all objects in our sample. On the other hand, given the overlapped redshift range (0.8 < z < 2.0), the mean f_AGN of AKARI sources with and without HSC counterparts is 0.12 and 0.23, respectively. This could indicate that one of the reasons for the difference in f_AGN between the two samples may be the difference in sample selection and redshift. Other possible uncertainties caused by a limited number of MIR data are discussed in Section 4.6.2.

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Finally, we compare the SFR in two samples as shown in Figure 10. The SFR is converted from dust luminosity using a relation provided by Kennicutt (1998; see also Hirashita et al. 2003).

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We find that the SFR of our sample is systematically higher than that of AKARI with HSC objects. The average SFR of AKARI objects with and without HSC counterparts is about 91 and 758 M_☉ yr⁻¹, respectively. If we focus on our sample with L18W and SPIRE detections, the mean SFR is 2240 M_☉ yr⁻¹ because of the requirement of FIR detection. These results are consistent with previous works reporting that SFRs of dusty galaxies tend to be high (e.g., Ikarashi et al. 2017; Toba et al. 2017d; Yamaguchi et al. 2019; Fan et al. 2020). On the other hand, if we compare SFRs of the sample in this work and Wang et al. (2020) in an overlapped redshift range (0.8 < z < 2.0), the mean SFR of AKARI with and without HSC counterparts is 319 and 207 M_☉ yr⁻¹, respectively. This indicates that the observed difference in SFR may be due to the difference in redshift.

4.6.1. Influence of Data Set without Optical Photometry on the SED-based Photometric Redshift

We showed that our estimate of z_photo is expected to be secure in some ways (see Sections 2.3 and 4.3), but these arguments are indirect. Although Małek et al. (2014) demonstrated the accuracy of z_photo based on CIGALE for an AKARI sample at z_spec < 0.25, they utilized optical data points for z_photo estimation, and the redshift range for their sample differs from that for our sample. We need to investigate how the lack of optical data affects the accuracy of z_photo for AKARI objects at z > 0.8. Hence, we estimate z_photo of AKARI objects with z_spec in the band-merged catalog (S. J. Kim et al. 2020, in preparation). In order to calculate ${z}_{\mathrm{photo}}$ under the same condition as we performed so far, we extracted objects with z_spec > 0.8 and with five-band detections in NIR–FIR, yielding 100 objects. Note that although the sample is detected by the HSC, we "artificially" multiplied the HSC flux densities by 10 and treated them as 10σ upper limits. We then executed the SED fitting with CIGALE in exactly the same manner as what we described in Section 2.3.

Figure 11 shows the relative difference between z_photo and z_spec, i.e., (z_spec − z_photo)/(1 + z_spec) as a function of z_spec. The mean and standard deviation of (z_spec − z_photo)/(1 + z_spec) is 0.09 ± 0.45, and the accuracy of z_photo, ${\sigma }_{{\rm{\Delta }}z/(1+{z}_{\mathrm{spec}})}$ is 0.23, which are worse than those reported by Małek et al. (2014). We find that ∼56% of objects have $| ({z}_{\mathrm{spec}}-{z}_{\mathrm{photo}})| /(1+{z}_{\mathrm{spec}})\lt 0.2$. This may be partly because the AKARI–HSC objects at z > 0.8 are biased toward optically bright type 1 AGNs, for which z_photo is harder to estimate precisely than for other galaxy types. Actually, ∼90% of objects plotted in Figure 11 are spectroscopically confirmed type 1 AGNs, which may induce a large deviation of $({z}_{\mathrm{spec}}-{z}_{\mathrm{photo}})/(1+{z}_{\mathrm{spec}})$.

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4.6.2. Bayesian Information Criterion

Another potential issue raised by the SED fitting with a limited number of data points in the MIR may be an uncertainty in the AGN contribution to the total SED, although we used upper limits at an MIR band if an object is not detected at that band. To test the requirement to add an AGN component to the SED fitting, we compute the BIC for two fits that are derived with and without an AGN component. The BIC is defined as BIC = χ² + k × ln(n), where χ² is nonreduced chi-square, k is the number of degrees of freedom (DOF), and n is the number of photometric data points used for the fitting. We then compare the results of two SED fittings without/with an AGN model by using ΔBIC = BIC_woAGN – BIC_wAGN. The resultant ΔBIC tells whether the AGN model is required to give a better fit when taking into account the difference in DOF (e.g., Ciesla et al. 2018; Buat et al. 2019, see also Aufort et al. 2020).

Figure 12 shows the histogram of ΔBIC for AKARI sources without HSC counterparts. If ΔBIC is larger than two, this indicates that adding the AGN component provides a better fit than not adding it (Liddle 2004; Stanley et al. 2015; see also Ciesla et al. 2018; Buat et al. 2019, who set a higher threshold for ΔBIC). Otherwise, there is no significant difference between two fits with/without an AGN model. We find that only 31% of object fits satisfy ΔBIC > 2. This suggests that it may be difficult to constrain the AGN activity for many cases, and thus AGN fraction may have a large uncertainty given a limited number of photometric detections in the MIR regime. On the other hand, Figures 6–10 also show objects with ΔBIC > 2. We find that there is no systematic difference between objects with ΔBIC > 2 and others, especially for AGN fraction (see Figure 9), which suggests that overall trends discussed in Section 4.4 may not be changed even if taking the BIC into account.

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4.6.3. Mock Analysis

Finally, we check whether or not the derived physical properties in this work can actually be estimated reliably, given the uncertainty of the photometry. We conduct a mock analysis that is a procedure provided by CIGALE (see, e.g., Buat et al. 2012, 2014; Ciesla et al. 2015; Lo Faro et al. 2017; Boquien et al. 2019; Toba et al. 2019b, for more detail).

Figure 13 shows the differences in z_photo, V-band attenuation in the A_V ISM, stellar mass, SFR, and L_IR(AGN) derived from CIGALE in this work and those derived from the mock catalog as a function of redshift. The mean and standard deviations of Δz_photo, ${\rm{\Delta }}{A}_{{\rm{V}}}^{\mathrm{ISM}}$, ${\rm{\Delta }}\mathrm{log}\,{M}_{* }$, ${\rm{\Delta }}\mathrm{log}\,\mathrm{SFR}$, and ${\rm{\Delta }}\mathrm{log}\,{L}_{\mathrm{IR}}$(AGN) are ${\rm{\Delta }}{z}_{\mathrm{photo}}=0.03\pm 0.47$, ${\rm{\Delta }}{A}_{{\rm{V}}}^{\mathrm{ISM}}\,=-0.23\pm 1.17$, ${\rm{\Delta }}\mathrm{log}\,{M}_{* }=0.08\pm 0.37$, ${\rm{\Delta }}\mathrm{log}\,\mathrm{SFR}\,=0.03\pm 0.43$, and ${\rm{\Delta }}\mathrm{log}\,{L}_{\mathrm{IR}}$(AGN) =0.11 ± 0.30. In particular, we can see a large deviation for objects at z < 1.5. This trend can also be seen in Figure 11. These results indicate that any physical quantities derived by the SED fitting especially for objects at z < 1.5 may be significantly affected by the limited number of photometric data points. On the other hand, we could ensure reliable physical quantities for our sample at z > 1.5.

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