Physical Characterization of Serendipitously Uncovered Millimeter-wave Line-emitting Galaxies at $z\sim 2.5$ behind the Local Luminous Infrared Galaxy VV 114

We analyzed the ALMA data listed in Table 1, which targeted VV 114 and include J0107a and J0107b within the FoV. The data sets of bands 3 and 4 were selected based on the frequency range, in which some CO and [C i] emission lines should be included if the redshift estimate of $z=2.467$ (Tamura et al., 2014) was correct. The data sets of bands 6 and 7 were used to measure the dust continuum emissions. We selected these data sets based on the relative position of VV114 in the FoV in order to detect our targets, which is often in the edge of the FoV. We also selected the data sets with relatively long integration time, more than 1000 s, to achieve high signal-to-noise ratio (S/N). We used channels without emission lines to analyze continuum emission. We conducted standard calibration and imaging using the Common Astronomy Software Applications package (CASA versions 5.1.0 and 5.4.0; McMullin et al., 2007).

For imaging, we used the CASA task tclean with a parameter threshold of 1–$3\ \sigma$. Briggs weighting with a robust parameter of robust = 0.5 was adopted for the band 3 and 4 data, while robust = 2.0 was adopted for band 6 and 7 data, for which the beam size was significantly smaller than for bands 3 and 4. The data of the [C i] line (see Section 2.2) and dust continuum (see Section 2.3) are strongly affected by the emission from VV114. Therefore, we set a mask as a box around VV 114 for these data to effectively remove the side lobes.

Figure 2 shows the primary-beam-corrected spectra of J0107a and J0107b over the entire range of band 3 and 4 data listed in Table 1. These spectra were obtained from a single pixel ($0\farcs 35\times 0\farcs 35$) at the line peak (the same position for all lines within uncertainties) to make these line peaks clear and easy to identify. Note that the peak spectra were used only for line identification and redshift determination, but not for measuring line width and flux.

With the use of Gaussian fittings for the peak line spectra, we identify three redshifted emission lines both in the spectra of J0107a and J0107b, as CO(3–2), CO(4–3), and [C i](³P₁–³P₀) ([C i](1–0) hereafter) at $z(\rm{J0107a})=2.4666\pm 0.0002$ and $z(\rm{J0107b})=2.3100\pm 0.0002$, respectively. The detection of multiple emission lines yields unambiguous redshifts of the two sources, and this confirms the line identification of Tamura et al. (2014), which is based on a photometric redshift analysis using infrared-to-radio data. The angular diameters corresponding to $1^{\prime\prime}$ at these redshifts are 8.3 kpc and 8.4 kpc, respectively.

The zoom-in spectrum of each detected line with the best-fit Gaussian profiles is shown in Figure 3. Each spectrum in this figure was made by taking an aperture of $\sim 3^{\prime\prime}\times 3^{\prime\prime}$ square centered at the peak position. The line width of each line is measured as the full-width-at-half-maximum (FWHM) of these Gaussian profiles.

We made channel maps for each line to measure line properties and to create moment maps. We set the velocity resolution of the channel map which includes CO(4–3) of J0107a to be 20 km s^-1, and 50 km s^-1 for other maps. The derived physical properties, achieved peak S/N in channel maps, and typical noise level for each line are listed in Table 2.

To create all moment maps (0th, 1st, and 2nd), we included approximately $\pm 150$ km s^-1 around the line center. In addition, to create the 1st and 2nd moment maps, we set a masking threshold with a range of 2 – 4 $\sigma$ of these line data. We arbitrarily set these clipping thresholds for each data cube in order to make these maps clear. We do not adopt this clipping procedure for 0th moment maps, hence the total fluxes of all lines are correctly measured. In fact, the total fluxes derived from moment maps are well consistent with those from integration of the Gaussian fitting for line spectra (Figure 3) for all emission lines. We measure the integrated line fluxes $S_{\rm{obs}}$ and beam deconvolved source sizes of each line using the CASA task imfit for 0th moment maps with the default setting.

Moment maps for the three emission lines are presented in Figures 4 and 5.

We analyzed the entire ALMA band 6, 7 and 3, 4 data using the line-free channels to obtain the dust continuum properties (Table 3). We obtain flux values using the CASA task imfit by setting the same area as we set to obtain line spectra in Figure 3. The rms noise levels are measured using the CASA task imstat with algorithm biweight over the entire FoV. Here, we did not detect the continuum of J0107a in band 3 and of J0107b in bands 3 and 4; hence, we present the 3–$\sigma$ upper limits for flux values in these bands. The intensity maps for the continuum in these bands are presented in Figures 4 and 5. Because both J0107a and J0107b are located close to the edge of the band 7 FoV, these maps appear noisy. However, the noise levels in the vicinity of the sources are consistent with those measured over the entire field, suggesting no significant effect of the position on the map.

The molecular gas mass can be calculated from both the CO and [C i] line luminosities. The molecular gas mass derived from the CO line luminosity has an uncertainty caused by the excitation and the choice of a CO-to-H₂ conversion factor $\alpha_{\mathrm{CO}}$. Although the molecular gas mass derived from [C i](1–0) has a similar uncertainty due to the [C i] abundance, this can be used for reasonable estimation of H₂ and molecular gas mass because of its simple partition function and chemistry (Alaghband-Zadeh et al., 2013; Saito et al., 2020). In this section, we first calculate $M_{\rm{gas}}$ with the [C i] line luminosity, and compare these results with $M_{\rm{gas}}$ which are derived with CO line luminosity and typical $\alpha_{\mathrm{CO}}$ for high-redshift galaxies. Hereafter, we adopt some parameters and equations for submillimeter galaxies (SMGs) from previous studies, such as CO line luminosity ratio, dust temperature $T_{\rm{dust}}$, Equations (3.2), and (8), because the properties of J0107a and J0107b based on the analysis in this study, such as submillimeter fluxes and star formation rate, are SMG-like (see Table 3 and Section 3.3 about J0107a).

We calculate the $\rm H_{2}$ mass using the formula given by Alaghband-Zadeh et al. (2013) (and references therein):

where $D_{L}$ is the luminosity distance in Mpc (20656 Mpc for J0107a and 19073 Mpc for J0107b), $S_{\rm{C_{I}}}$ is the flux density of the [C i](1–0) line, $A_{10}=7.93\times 10^{-8}\ \mathrm{s^{-1}}$ is the Einstein A coefficient, $X_{\mathrm{C_{I}}}=[\mathrm{C_{I}}]/[\mathrm{H_{2}}]=(5.0-7.9)\times 10^{-5}$ is the [C i] abundance relative to $\rm H_{2}$ for SMGs at $z>2.5$ (Valentino et al., 2018, and the references therein), and $Q_{10}=Q_{10}(n,T_{\rm{ex}})=0.49\pm 0.02$ (Alaghband-Zadeh et al., 2013) is the partition function, which depends on the gas excitation conditions (Papadopoulos & Greve, 2004) and here the excitation temperature is assumed to be about 30 K (Alaghband-Zadeh et al., 2013). While $A_{10}$ is naturally a physical constant, $X_{\rm{C_{I}}}$ and $Q_{10}$ are empirical parameters.

By multiplying the $\rm H_{2}$ mass, calculated from Eq. (1), by 1.36 to account for the $\rm He$ contribution, we derive the total molecular gas mass $M_{\rm gas}$:

The dominant factor in the error is the uncertainty in $X_{\mathrm{C_{I}}}$.

On the other hand, $M_{\rm{gas}}$ can be derived from the CO(1–0) luminosity and $\alpha_{\rm{CO}}$ as:

We here estimate $M_{\rm{gas}}$ by assuming $\alpha_{\rm{CO}}=0.8$ (Downes & Solomon, 1998), which is commonly adopted for high-redshift galaxies, and converting CO(4–3) and CO(3–2) luminosities to that of CO(1–0) based on SMGs’ line ratios ($L^{\prime}(\mathrm{CO})_{4-3}/L^{\prime}(\mathrm{CO})_{1-0}=0.32\pm 0.05$, and $L^{\prime}(\mathrm{CO})_{3-2}/L^{\prime}(\mathrm{CO})_{1-0}=0.60\pm 0.11$; Birkin et al., 2021). The results are $M_{\mathrm{gas}}\,(\mathrm{J0107a})=(13.4\pm 1.7)\times 10^{10}\ M_{\odot}$ and $M_{\mathrm{gas}}\,(\mathrm{J0107b})=(5.0\pm 0.7)\times 10^{10}\ M_{\odot}$ for CO(4–3) luminosities, and $M_{\mathrm{gas}}(\mathrm{J0107a})=(9.2\pm 1.9)\times 10^{10}\ M_{\odot}$ and $M_{\rm{gas}}(\mathrm{J0107b})=(3.6\pm 0.8)\times 10^{10}\ M_{\odot}$ for CO(3–2) luminosities. These results are consistent with the $M_{\rm{gas}}$ based on [C i] within uncertainties.

For local galaxies, ULIRGs, and redshifted SMGs observed at wavelengths longer than 250 $\mu$m, in which we may assume that the emission is in the Rayleigh–Jeans tail, Scoville et al. (2016) presented that $M_{\rm{gas}}$ can be derived from the dust continuum emission as follows:

where

and $\Gamma_{0}=\Gamma_{\rm{RJ}}(T_{\rm{dust}},353\ \mathrm{GHz},0)$.

We use our ALMA/band 6 data for calculation of $M_{\rm{gas}}$ of J0107a. Because our current data set cannot constrain $T_{\rm{dust}}$, we assume $T_{\rm{dust}}$ = 40 K which is given by the surveys of SMGs and galaxies in $0<z<4$ (da Cunha et al., 2015; Schreiber et al., 2018). Here, we should note that this $T_{\rm{dust}}$ is luminosity-weighted and is likely higher than mass-weighted $T_{\rm{dust}}$, such as that suggested in Scoville et al. (2014). Consequently, $M_{\rm{gas}}$ of J0107a is calculated as below:

Here, we adopt $\alpha_{\mathrm{353\,GHz}}=8.4\times 10^{19}\ \mathrm{erg~{}s^{-1}~{}Hz^{-1}}~{}M_{\odot}^{-1}$, which is derived as the mean value for $z\sim 2$ SMGs in Scoville et al. (2016). We also set a typical uncertainty of the derived $M_{\mathrm{gas}}$ as $\pm 50\%$ (Scoville et al., 2014). If $T_{\rm{dust}}=30$ K and 50 K are adopted, $M_{\rm{gas}}$ becomes $M_{\rm{gas}}(\mathrm{J0107a})=(3.6\pm 1.8)\times 10^{11}\ M_{\odot}$ and $M_{\rm{gas}}\,(\mathrm{J0107a})=(3.0\pm 1.5)\times 10^{11}\ M_{\odot}$, respectively. Hence, the result may change from approximately 6% to 12% with a $T_{\rm{dust}}$ error of 10 K. We calculate the $M_{\rm{gas}}$ of J0107b in the same manner as follows:

The result of J0107b also changes approximately 6% to 12% with a $T_{\rm{dust}}$ error of 10 K.

The similarity between the cold gas mass derived from the lines and from the dust continuum supports the derived cold gas mass.

In star-forming galaxies (SFGs), most infrared emissions are believed to be emitted from warm dust around young stars, and, thus, their SFR is calculated from the infrared luminosity $L_{\rm IR}$ (Kennicutt, 1998; Carilli & Walter, 2013). Although we can also assume AGN to be a heat source, Brown et al. (2019) reported that the AGN contribution to $L_{\rm{IR}}$ is typically smaller than $10$% when $L_{\rm{IR}}\sim 10^{13}\ L_{\odot}$. Here, we estimate the SFR of J0107a and J0107b using the far-infrared luminosity $L_{\rm{FIR}}$ because of the lack of $L_{\rm{IR}}$ data. By assuming the gray body model $F_{\nu}\propto\nu^{\beta}B(\nu,T)$, we derive $L_{\rm{FIR}}$ with the formula in De Breuck et al. (2003):

where $\beta$ is the beta index, $\Gamma[x]$ is the Gamma function, and $\zeta[x]$ is the zeta function. Assuming $\beta=1.5$ and $T_{\rm{dust}}=40$ K, we calculate $L_{\rm{FIR}}$ of J0107a and J0107b using this formula as

Here, we used band 6 data, in which both targets are clearly identified. For a sanity check, we also calculate $L_{\mathrm{FIR}}$ with other continuum data. For J0107a, we derived $L_{\mathrm{FIR}}<12.3\times 10^{12}\ L_{\odot}$, $L_{\mathrm{FIR}}=(10.2\pm 2.0)\times 10^{12}\ L_{\odot}$, and $L_{\mathrm{FIR}}=(9.0\pm 1.1)\times 10^{12}\ L_{\odot}$ with band 3, 4, and 7 data, respectively. These results are all consistent with each other. We calculate for J0107b with the same manner as $L_{\mathrm{FIR}}<12.3\times 10^{12}\ L_{\odot}$, $L_{\mathrm{FIR}}<7.2\times 10^{12}\ L_{\odot}$, and $L_{\mathrm{FIR}}=(5.8\pm 1.4)\times 10^{12}\ L_{\odot}$ with band 3, 4, and 7 data, respectively. They are also comparable to each other. $L_{\mathrm{FIR}}$ of J0107b derived from band 7 data is a little larger than that from band 6 data. This may be because of poorer fitting in band 7 data due to apparent multiple peaks.

Regarding $L_{\rm{FIR}}$, the results change by a factor of 2 when $T_{\rm{dust}}$ changes by 10 K. The results also depend on $\beta$. Chapin et al. (2009) derived $\beta=1.75$ for 29 SMGs with a median redshift $z=2.7$ with a 1.1 mm survey. The Planck Collaboration et al. (2011) similarly suggested $\beta=1.8\pm 0.1$ based on the all-sky observation results, and Scoville et al. (2014) adopted $\beta=1.8$ after this result. If we adopt $\beta=1.8$ for our calculation of $L_{\rm{FIR}}$, the results with $T_{\rm{dust}}=40$ K increase by about a factor of 1.5 ($L_{\mathrm{FIR}}(\mathrm{J0107a})=1.7\times 10^{13}\ L_{\odot}$ and $L_{\mathrm{FIR}}(\mathrm{J0107b})=5.0\times 10^{12}\ L_{\odot}$, respectively). Consequently, the total systematic uncertainty of $L_{\mathrm{FIR}}$, due to the selection of $T_{\mathrm{dust}}$ and $\beta$, is about a factor of 3.

The relationship between SFR and $L_{\rm{FIR}}$ is shown in Genzel et al. (2010) as follows:

where $\log(1.3)$ is a correction factor between $L_{\rm IR}$ and $L_{\rm FIR}$ (Graciá-Carpio et al., 2008), and the typical uncertainty of this equation is about $\pm 50\%$ (Genzel et al., 2010). The derived SFRs are $1.3\times 10^{3}$ $M_{\odot}$ yr^-1 and $3.9\times 10^{2}$ $M_{\odot}$ yr^-1 for J0107a and J0107b, respectively, and the typical systematic uncertainties of these SFRs are about a factor of 4.5, which is mainly attributed to the uncertainties of $L_{\mathrm{FIR}}$ and the Equation (7). The physical quantities derived from our data analysis are summarized in Table 4 and discussed in Section 5.

We model the disk rotation of J0107a using the Markov chain Monte Carlo (MCMC) method with GalPaK^3D (Bouché et al., 2015). We use the CO(4–3) data cube (2D image and frequency dimension) for the modeling, which has a better S/N than those of CO(3–2) or [C i](1–0). The algorithm directly compares the data cube with a disk parametric model with ten free parameters: coordinates of the galaxy center ($x_{c},y_{c},z_{c}$), flux, half-light radius, inclination angle ($i$), position angle (PA), turnover radius of the rotation curve, deprojected maximum rotation velocity, and intrinsic velocity dispersion. We first perform modeling with ten free parameters, and then repeat it by setting the initial values for $x_{c}$, $y_{c}$, $z_{c}$, and $i$. We assume a rotational velocity with a hyperbolic tanh profile in this model. The uncertainty is the 95% confidence interval (CI) calculated from the last 60% of the MCMC chain for 20,000 iterations. We here set the random scale of MCMC chain as 0.4 to obtain its acceptance rate of 30–50%, which is suggested by GalPaK^3D. By doing this, we achieved the acceptance rate of $\sim 38\%$ in this modeling. Furthermore, by setting a beam size, the effect of beam smearing is considered in the modeling.

The initial value for $i$ is set to be $60\tcdegree$ based on the major to minor axis ratio of the beam-deconvolved source size of the 1.3 mm continuum. We confirm that setting initial values to be $20\tcdegree$ or $40\tcdegree$ does not largely change the results, and hereafter present results with $i=60\tcdegree$ only. We then obtain a GalPaK^3D model, and the inclination angle converges to approximately $63\tcdegree$. This result indicates that J0107a is unlikely to be face-on, and the major/minor ratio of the model ($=2.2\pm 0.1$) is consistent with the observed source sizes of 1.3 mm, 0.89 mm, and CO(4-3) emission. In addition, the half-light radius of the model is consistent with the deconvolved half width at half maximum of J0107a ($=5.4\pm 0.5$ kpc), which is derived by the CASA task imfit for the CO(4–3) source in the band 4 data. We also obtain the maximum rotation velocity as $v_{\rm{max}}=69.3\pm 4.8$ km s^-1, and the velocity dispersion as $\sigma_{v}=54.6\pm 1.8$ km s^-1. In GalPaK^3D, velocity dispersion is estimated by assuming three components, one of which is an intrinsic velocity dispersion (Bouché et al., 2015), and $\sigma_{v}$ of J0107a here is the result for the intrinsic velocity dispersion. Therefore, we hereafter treat $\sigma_{v}$ of J0107a as an intrinsic value. We summarize the results of the GalPaK^3D model in Table 5. Figure 6 shows the output rotation curve, and Figure 7 shows a comparison of the GalPaK^3D model with the observational data of J0107a on CO(4–3) intensity and velocity maps. Figure 8 shows the position-velocity diagram of J0107a and the contours of its model by GalPaK^3D. Both of the velocities in this diagram are extracted along the lines shown in the black dotted lines in Figure 7(d) and (e), whose PA is $110.4\tcdegree$. As we can see in Figure 8, the velocity of J0107a changes continuously from $v\sim-120\ \mathrm{km\ s^{-1}}$ to $\sim+120\ \mathrm{km\ s^{-1}}$, and there are apparently two peaks at $v\sim-20\ \mathrm{km\ s^{-1}}$ and $v\sim+40\ \mathrm{km\ s^{-1}}$ in this diagram. This result suggests that J0107a is less likely to be a galaxy merger. All of these results suggest that J0107a has a rotating disk. Additionally, we show the plot of cross-correlations in the Markov chain in Appendix Figure 11.

In the calculation of a dynamical mass, $M_{\rm{dyn}}$, we use the results of GalPaK^3D. As for the radius, we regard the twice of half-light radius, which corresponds to $r=9.8\pm 0.2$ kpc ($=1.2\pm 0.02$ arcsec.), as the radius of J0107a. We use this $r$ and the best-fit value of $v_{\mathrm{max}}$ from the model to estimate $M_{\rm{dyn}}$, and derive it as follows:

Here, $G$ is the gravitational constant, and $1.1\times 10^{10}\ M_{\odot}$ is the 50th percentile of the results of $M_{\rm{dyn}}$ for each 12,000 iterations after MCMC burn-in. This is an order of magnitude smaller than $M_{\mathrm{gas}}(\mathrm{dust})$ derived in Section 3.2. The possible origins of this discrepancy are discussed in the next section.

We obtain $v_{\rm{max}}/\sigma_{v}=1.3\pm 0.1$ using the best-fit kinetic parameters in Section 4. Here, 1.3 is the 50th percentile of the results for each 12,000 iterations after MCMC burn-in, similar to $M_{\rm{dyn}}$. The error of $v_{\rm{max}}/\sigma_{v}$ is also calculated with the use of each result of 12,000 iterations. As a more conservative estimate on the error in $\sigma_{v}$, we adopt the error of the spectral line width derived by the Gaussian fitting (Table 2). The largest error is 34 km s^-1 (in FWHM) for the [C i] line, which corresponds to 13 km s^-1 in standard deviation $\sigma$ when corrected for the inclination angle. Consequently the error of $v_{\rm{max}}/\sigma_{v}$ can be estimated as 0.3, which is still very small. We hereafter adopt this error for conservative discussion. This result suggests that $\sigma_{v}$ of J0107a is compareble to $v_{\rm{max}}$, and the disk is rather turbulent.

Figure 9 shows the relation between $v_{\rm{max}}/\sigma_{v}$ of SFGs (data from Swinbank et al., 2017; Lee et al., 2019), SMGs (data from Alaghband-Zadeh et al., 2012; Hodge et al., 2012; De Breuck et al., 2014; Tadaki et al., 2019; Jiménez-Andrade et al., 2020), millimeter-wave line emitter samples in Kaasinen et al. (2020), and J0107a.

Burkert et al. (2010) mentioned that $z\sim 1.5$–$3.5$ SFGs have large gas velocity dispersions of $30$–$120$ km s^-1 and ratios of $v_{\rm{max}}/\sigma_{v}\sim 1$–$6$. This is supported by the observations of $z=0.3$–$1.7$ SFGs (Swinbank et al., 2017), and SFGs in a protocluster at $z\sim 2.5$ (Lee et al., 2019). Alaghband-Zadeh et al. (2012) investigated the H$\alpha$ velocity fields of SMGs in $z\sim 2$–3 and derived $v_{\rm{max}}/\sigma\sim 1$–3 (average is $1.9\pm 0.2$). Kaasinen et al. (2020) also reported $v_{\rm{max}}/\sigma_{v}\sim 5$ for their millimeter-wave line emitter samples in $z=1.4$–$2.7$, and noted that this value was higher than those of typical high-redshift galaxies due to a selection bias. They also noted that the velocity dispersions of their samples were global estimates that include dispersions due to motion along the line-of-sight (i.e. due to motion inside a thick disk, or, motions due to warps), and they treated their velocity dispersions as upper limits. However, this discussion can be applied to the $\sigma_{v}$ of other samples, which were derived with, for example, line width, and does not significantly affect our results. We therefore do not strictly consider it here. Because of this reason, we here do not treat $\sigma_{v}$ of Kaasinen’s samples as upper limits, as with other samples.

Consequently, from a quantitative point of view, the $v_{\rm{max}}/\sigma_{v}$ ratio of J0107a seems smaller than those of other line emitters, and at the same level as those of SMGs at similar redshifts.

Simons et al. (2017) presented that the $v_{\rm{max}}/\sigma_{v}$ ratio of an SFG is affected by stellar mass; hence, we show the fitting results for the distribution of $v_{\rm{max}}/\sigma_{v}$ ratios of typical SFGs with low stellar mass ($M_{\star}\sim 10^{9}-10^{10}\ M_{\odot}$), and high stellar mass ($M_{\star}\sim 10^{10}-10^{11}\ M_{\odot}$), respectively, in Figure 9. In this figure, the $v_{\rm{max}}/\sigma_{v}$ ratios of almost all line emitters and SMGs, which have stellar masses of approximately $10^{11}\ M_{\odot}$, are consistent with the fitting result for high-$M_{\star}$ SFGs within uncertainties. The SFG samples in this figure have a very wide range of stellar mass ($M_{\star}\sim 10^{6}-10^{11}\ M_{\odot}$), but their $v_{\rm{max}}/\sigma_{v}$ are also consistent with the fitting lines within uncertainties.

The $v_{\rm{max}}/\sigma_{v}$ ratio of J0107a is consistent with the fitting for both low and high stellar masses within the error. The stellar mass of J0107a is derived as $M_{\star}<1.0\times 10^{11}\ M_{\odot}$ (see Section 5.2), but considering the gravitational lensing (see Section 5.3), it may be more appropriate to conclude that the $v_{\rm{max}}/\sigma_{v}$ ratio of J0107a is consistent with the fitting for low stellar mass, whereas it is difficult to constrain the range of $M_{\star}$ by the $v_{\rm{max}}/\sigma_{v}$ ratio alone.

To obtain $f_{\rm{gas}}$, we at first calculate $M_{\star}$. Hainline et al. (2011) used a sample of $\sim 70$ SMGs to derive the ratio between $M_{\star}$ and the rest-frame near-infrared ($H$-band) luminosity $L_{H}$ for two star formation histories, instantaneous starburst (IB) and constant star formation (CSF), by taking the average of best-fit model ages over the sample. Consequently, they obtained the relation between $M_{\star}$ and $L_{H}$ of SMGs for the population synthesis model of Bruzual & Charlot (2003) and CSF history as below within a factor of 2–3:

Considering $\lambda_{H}=1.65\,\mu$m, the observed wavelength is $5.72\,\mu$m with $z=z(\mathrm{J0107a})$. Therefore, we first use the intensity data in the nearest band, IRAC/$5.8\,\mu$m data (Tamura et al., 2014), which is equal to $S_{\rm{obs}}<0.1$ mJy (upper limit), and obtain the $L_{H}$ of J0107a as follows:

where $d\nu=4.4\times 10^{13}\,$Hz is the band width of the IRAC/5.8 $\mu$m band, which is calculated by the band width in the wavelength scale $d\lambda=0.41~{}\mu$m in $z=z(\mathrm{J0107a})$.

Consequently, $M_{\star}$ is calculated as follows:

We can also estimate $M_{\star}$ with the use of IRAC/4.5 $\mu$m and IRAC/3.6 $\mu$m data (Tamura et al., 2014), as $M_{\star}=7.1\times 10^{10}\ M_{\odot}$ and $5.5\times 10^{10}\ M_{\odot}$, respectively, but we adopt the result from IRAC/5.8 $\mu$m in order to avoid possible extinction effect. We note again that the photometry of these data is uncertain due to the foreground emission from VV114, and we should regard the $M_{\star}$ as an upper limit.

The $f_{\rm{gas}}$ of J0107a is then calculated using the results of previous calculations as

This fraction may contain additional uncertainty caused by differential magnification (Hezaveh et al., 2012; Serjeant, 2012) between stellar emission and gas emission, considering the possibility of strong magnification of J0107a (see Section 5.3).

Considering the dark matter contribution, $M_{\rm{dyn}}$ should generally be consistent with, or larger than, $M_{\rm{gas}}+M_{\star}$ within uncertainties. However, we find that, for J0107a, $M_{\rm{dyn}}$ is an order of magnitude smaller than $M_{\rm{gas}}$. This discrepancy cannot be attributed to the $M_{\rm{gas}}$ over-estimation because all $M_{\rm{gas}}$, yielded with the data of [C i], CO lines and dust continuum, are significantly larger than $M_{\rm{dyn}}$. Including $M_{\star}$, the discrepancy becomes even larger:

The most plausible reason for this result is brightening due to the gravitational lens effect. Harris et al. (2012) reported that 11 SMGs identified by H-ATLAS at $z\sim 2.1$–$3.5$ are amplified by the gravitational lens effect. Harris et al. (2012) claimed that the lens magnification rate $\mu$ can be estimated from the empirical relation between the CO linewidth and luminosity for unlensed systems as

where $L^{\prime}(\mathrm{CO})_{\mathrm{apparent}}$ is the apparent luminosity of CO(1–0) , and $\Delta v_{\mathrm{FWHM}}$ is the FWHM of the observed CO(1–0) spectra. Subsequently, $\mu$ of J0107a is calculated as

Here we assume the linewidth ratio, $\Delta v_{\rm{CO(1-0)}}/\Delta v_{\rm{CO(3-2)}}=1.15\pm 0.06$ (Ivison et al., 2011), and derive $\Delta v_{\rm{FWHM}}(\rm{J0107a})\sim 189$ km s^-1, while $L^{\prime}(\rm{CO})_{\rm{apparent}}$ is calculated with CO(4–3) which has the best S/N in our dataset, and the SMG line ratio ($L^{\prime}(\mathrm{CO})_{4-3}/L^{\prime}(\mathrm{CO})_{1-0}=0.32\pm 0.05$; Birkin et al., 2021). In the same manner, $\mu(\rm{J0107b})$ is calculated to be $\mu(\rm{J0107b})\sim 7.9$. Figure 10 shows the relationship between $L^{\prime}(\rm{CO})$ and $\Delta v_{\rm{FWHM}}$ for local SFGs (Bothwell et al., 2014; Saintonge et al., 2017), high-$z$ SFGs (Daddi et al., 2010; Magnelli et al., 2012; Magdis et al., 2012; Tacconi et al., 2013), high-$z$ SMGs (Harris et al., 2012), millimeter-wave line emitters (Aravena et al., 2019; Kaasinen et al., 2020), and our targets. The positions of J0107a and J0107b are higher than the average for SFGs, unlensed SMGs, and line emitters at similar redshifts. We also draw two types of relations for intrinsic CO line luminosity versus CO linewidth: one from Harris et al. (2012) with $\mu=1$ and $\mu=21$, and the other from Bothwell et al. (2013) with $\mu=1$ and $\mu=5.9$. The calculated magnification rates of J0107a from these relations, 21 and 5.9, respectively, differ by a factor of more than 3. One reason for this large difference is that both of these relations are empirical, and affected by the difference of the sample. The other reason is the uncertainty in the calculation from Bothwell et al. (2013). In this calculation, the parameter $C$, which parameterize the kinematics of the galaxy (Bothwell et al., 2013), depends on the galaxy’s mass distribution and velocity field, and it takes a wide range from $C\leq 1$ to $C\geq 5$ (Erb et al., 2006). We here adopt $C=2.1$, as suggested in Bothwell et al. (2013), but it may include large uncertainty. In our case, the magnification rate from Harris et al. (2012) is reasonable considering the consistency with the ratio of $(M_{\rm{gas}}+M_{\star})/M_{\rm{dyn}}$.

Such a high magnification must result in a highly perturbed morphology of the magnified image (e.g., Tamura et al., 2015). In fact, we can find a hint of an elongated arc-like structure seen in the 0$\farcs$2 resolution 887 $\mu$m continuum image (see Figure 4 (a13)), although the presence of a lens source is unclear because of contamination by the nearby bright source VV 114. This is also true in the images of Spitzer/IRAC and Chandra/ACIS-I, which are shown in Figure 1 (a and b). From this point of view, the intrinsic radius of J0107a might be smaller than that we adopt here. In this case, $M_{\rm{dyn}}$ of J0107a would become even smaller than the result in Section 4.2. Further, deeper, and higher-angular-resolution ALMA observations of this source will clarify the presence of a strong lens and the true size of J0107a.