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

Shoichiro Mizukoshi  Institute of Astronomy, Graduate School of Science, The
University of Tokyo, 2-21-1 Osawa, Mitaka, Tokyo 181-0015, Japan  Kotaro Kohno
Institute of Astronomy, Graduate School of Science, The University of Tokyo,
2-21-1 Osawa, Mitaka, Tokyo 181-0015, Japan  Research Center for the Early
Universe, Graduate School of Science, The University of Tokyo, 7-3-1 Hongo,
Bunkyo-ku, Tokyo 113-0033, Japan  Fumi Egusa Institute of Astronomy, Graduate
School of Science, The University of Tokyo, 2-21-1 Osawa, Mitaka, Tokyo
181-0015, Japan  Bunyo Hatsukade Institute of Astronomy, Graduate School of
Science, The University of Tokyo, 2-21-1 Osawa, Mitaka, Tokyo 181-0015, Japan
Takeo Minezaki Institute of Astronomy, Graduate School of Science, The
University of Tokyo, 2-21-1 Osawa, Mitaka, Tokyo 181-0015, Japan  Toshiki
Saito Max Planck Institute for Astronomy, K$\ddot{o}$nigstuhl 17, 69117
Heidelberg, Germany  Yoichi Tamura Division of Particle and Astrophysical
Science, Graduate School of Science, Nagoya University, Nagoya 464-8602,
Japan.  Daisuke Iono National Astronomical Observatory of Japan, 2-21-1 Osawa,
Mitaka, Tokyo 181-8588, Japan  The Graduate University for Advanced Studies
(SOKENDAI), 2-21-1 Osawa, Mitaka, Tokyo 181-0015, Japan  Junko Ueda National
Astronomical Observatory of Japan, 2-21-1 Osawa, Mitaka, Tokyo 181-8588, Japan
Yuichi Matsuda National Astronomical Observatory of Japan, 2-21-1 Osawa,
Mitaka, Tokyo 181-8588, Japan  The Graduate University for Advanced Studies
(SOKENDAI), 2-21-1 Osawa, Mitaka, Tokyo 181-0015, Japan  Ryohei Kawabe
National Astronomical Observatory of Japan, 2-21-1 Osawa, Mitaka, Tokyo
181-8588, Japan  The Graduate University for Advanced Studies (SOKENDAI),
2-21-1 Osawa, Mitaka, Tokyo 181-0015, Japan  Minju M. Lee Max-Planck-Institut
f$\ddot{u}$r Extraterrestrische Physik (MPE), Giessenbachstr., D-85748
Garching, Germany  Min S. Yun Department of Astronomy, University of
Massachusetts, Amherst, MA 01003, USA  Daniel Espada Departamento de Física
Teórica y del Cosmos, Campus de Fuentenueva, Universidad de Granada,
E18071–Granada, Spain

###### Abstract

We present a detailed investigation of millimeter-wave line emitters ALMA
J010748.3-173028 (ALMA-J0107a) and ALMA J010747.0-173010 (ALMA-J0107b), which
were serendipitously uncovered in the background of the nearby galaxy VV 114
with spectral scan observations at $\lambda$ = 2 – 3 mm. Via Atacama Large
Millimeter/submillimeter Array (ALMA) detection of CO(4–3), CO(3–2), and [C
i](1–0) lines for both sources, their spectroscopic redshifts are
unambiguously determined to be $z=2.4666\pm 0.0002$ and $z=2.3100\pm 0.0002$,
respectively. We obtain the apparent molecular gas masses $M_{\rm gas}$ of
these two line emitters from [C i] line fluxes as $(11.2\pm 3.1)\times
10^{10}M_{\odot}$ and $(4.2\pm 1.2)\times 10^{10}M_{\odot}$, respectively. The
observed CO(4–3) velocity field of ALMA-J0107a exhibits a clear velocity
gradient across the CO disk, and we find that ALMA-J0107a is characterized by
an inclined rotating disk with a significant turbulence, that is, a
deprojected maximum rotation velocity to velocity dispersion ratio $v_{\rm
max}/\sigma_{v}$ of $1.3\pm 0.3$. We find that the dynamical mass of
ALMA-J0107a within the CO-emitting disk computed from the derived kinetic
parameters, $(1.1\pm 0.2)\times 10^{10}\ M_{\odot}$, is an order of magnitude
smaller than the molecular gas mass derived from dust continuum emission,
$(3.2\pm 1.6)\times 10^{11}\ M_{\odot}$. We suggest this source is magnified
by a gravitational lens with a magnification of $\mu\gtrsim 10$, which is
consistent with the measured offset from the empirical correlation between CO-
line luminosity and width.

submillimeter: galaxies – galaxies: ISM – galaxies: high-redshift – galaxies:
kinematics and dynamics – galaxies: starburst

††journal: ApJ

##  1 Introduction

In our universe, galaxies form stars most actively at $z=1-3$ (Hopkins &
Beacom, 2006; Madau & Dickinson, 2014), and their molecular gas content is a
key parameter because stars are formed in molecular gas. Therefore, extensive
observations of rotational CO lines, which have been established as a useful
measure of cold molecular gas mass $M_{\rm gas}$ (e.g., Bolatto et al., 2013),
have been made for various samples of galaxies that are pre-selected based on
their physical properties, such as stellar mass ($M_{\star}$) and star
formation rate (SFR), (e.g., Tacconi et al., 2020). This approach has
successfully revealed the evolution of molecular gas components by measuring
the molecular gas fraction $f_{\rm gas}\equiv M_{\rm gas}/(M_{\rm
gas}+M_{\star})$ in galaxies across cosmic time (e.g., Tacconi et al., 2018,
and references therein). Despite its success, it is also necessary to conduct
a blind search of CO-line-emitting galaxies without any priors. This can be
accomplished by unbiased spectral scan observations of a region of the sky,
which often target known deep fields such as the Hubble Ultra Deep Field
(HUDF), where rich multi-wavelength datasets are available. This “deep-field
scanning approach” is capable of uncovering galaxies that were not present in
standard optical/near-infrared deep surveys, and thus is considered less
biased than the pointed approach (Carilli & Walter, 2013; Tacconi et al.,
2020) . Following the pioneering spectral scan observations of the Hubble Deep
Field North (HDF-N) using the IRAM Plateau de Bure Interferometer (Decarli et
al., 2014), the Atacama Large Millimeter/submillimeter Array (ALMA) has been
exploited to conduct spectral scan observations of the HUDF (e.g., Walter et
al., 2016; Aravena et al., 2019), SSA22 (Hayatsu et al., 2017, 2019), and
lensing clusters (e.g., Yamaguchi et al., 2017; González-López et al., 2017)
to uncover millimeter-wave line-emitting galaxies and constrain CO-line
luminosity functions as a function of redshift and, therefore, the cosmic
molecular gas mass density evolution (e.g., Decarli et al., 2020).

In addition to these dedicated spectral scan observations of deep fields,
there are mounting examples of serendipitous detection of millimeter-wave line
emitters. They were detected using ALMA and the NOrthern Extended Millimeter
Array (NOEMA), and in the data from the second Plateau de Bure High-z Blue-
Sequence Survey (PHIBSS2), within a field of view (FoV) of, for example, a
nearby galaxy (e.g., Tamura et al., 2014) and high-$z$ source (e.g., Swinbank
et al., 2012; Gowardhan et al., 2017; Wardlow et al., 2018; Lenkić et al.,
2020). Currently, the nature of such serendipitously uncovered millimeter-wave
line emitters remains unexplored given the very limited number of such
sources, but it is important to characterize the known line emitters because
we can learn the types of galaxies that can be selected from monotonically
increasing numbers of spectral cubes in the ALMA science archive over time.

Here, we report a detailed investigation of millimeter-wave line emitters ALMA
J010748.3-173028 (ALMA-J0107a) and ALMA J010747.0-173010 (ALMA-J0107b), which
were serendipitously uncovered around the nearby galaxy VV 114 with spectral
scan observations at $\lambda$ = 2 – 3 mm. We display the positions of
ALMA-J0107a and ALMA-J0107b in Figure 1 (top), in which an HST $I$-band image
of VV 114 is shown. As VV 114 is one of the best-studied archetypical luminous
infrared galaxies (LIRGs) in the local region (e.g., Iono et al., 2013; Saito
et al., 2015, 2017), a number of spectral scan observations have been
conducted using ALMA. The discovery of ALMA-J0107a was first reported by
Tamura et al. (2014), based on a single line detection in ALMA band 3.
Although the multi-wavelength counterpart identification at the position of
ALMA-J0107a favors a CO(3–2) line at $z=2.467$, the proximity of ALMA-J0107a
to the local LIRG VV 114 ($\sim 10\arcsec$ from the eastern nucleus of VV 114)
hampers reliable photometric constraints at near-to-far-infrared bands and
therefore requires other transitions of CO to obtain an unambiguous
spectroscopic redshift. Tamura et al. (2014) also found the hard X-ray source
at the position of ALMA-J0107a in Chandra/ACIS-I data (see Figure 1 (a)),
which suggested the presence of an active galactic nucleus (AGN). ALMA-J0107b
is also serendipitously detected in the line scan of the VV 114 field at
$(\alpha,\delta)_{\rm{J2000}}=(01^{\rm{h}}07^{\rm{m}}46^{\rm{s}}.99,-17\tcdegree
30^{\prime}10\farcs 09)$, and we report it in this paper. Figure 1(a and b)
shows multiwavelength images of both ALMA-J0107a and ALMA-J0107b. It contains
three Spitzer/IRAC band images and the Chandra/ACIS-I image.

The remainder of this paper is organized as follows: The reduced ALMA data and
reduction procedures are described in Section 2 with the derived line spectra.
The physical quantities derived from the observed lines and continuum
emissions are summarized in Section 3. Section 4 is devoted to kinematic
modeling of the observed CO velocity field of ALMA-J0107a. After discussing
the nature of the millimeter-wave line emitters in Section 5, we summarize our
findings in Section 6. We assume a $\Lambda$CDM cosmology with
$\Omega_{\mathrm{m}}=0.3,\Omega_{\mathrm{\Lambda}}=0.7$, and $H_{0}=68$ km s-1
Mpc-1.

##  2 Data analysis

###  2.1 Selection, reduction, and imaging

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.

Table 1: The details of ALMA archive data we used in this study.

Project ID | Band | Max baseline length | Frequency (GHz) | Integration time (s) | Target a | Main use b
---|---|---|---|---|---|---
2013.1.01057.S | 3 |  $650.3\,$m | 84.08-87.79 / 97.91-99.79 | 2268.0 | a | CO(3–2), continuum
2013.1.01057.S | 3 |  $650.3\,$m | 87.81-91.56 / 99.81-103.55 | 1360.8 | a | CO(3–2), continuum
2013.1.01057.S | 3 |  $783.5\,$m | 91.56-95.31 / 103.56-107.31 | 2721.600 | b | CO(3–2), continuum
2013.1.01057.S | 4 |  $538.9\,$m | 130.74-134.49 / 142.74-146.49 | 1149.120 | a | CO(4–3), continuum
2013.1.01057.S | 4 |  $538.9\,$m | 138.24-141.99 / 150.24-153.99 | 1149.120 | b | CO(4–3), continuum
2013.1.01057.S | 4 |  $538.9\,$m | 126.99-130.74 / 138.99-142.74 | 1149.120 | a | [C i](1–0), continuum
2013.1.01057.S | 4 |  $538.9\,$m | 134.49-138.24 / 146.49-150.24 | 1149.120 | b | [C i](1–0), continuum
2015.1.00973.S | 6 |  $641.5\,$m | 245.09-248.93 / 259.44-263.14 | 1814.400 | a / b | continuum
2015.1.00902.S | 6 |  $867.2\,$m | 211.87-215.07 / 226.05-228.23 | 2721.600 | a / b | continuum
2013.1.00740.S | 7 |  $1.6\,$km | 325.67-329.50 / 337.67-341.49 | 1332.197 | a / b | continuum
2013.1.00740.S | 7 |  $1.6\,$km | 334.89-338.70 / 346.87-350.49 | 2124.645 | a / b | continuum

Note. —

a target a represents J0107a and target b represents J0107b.

b Suggested emission lines are detected only with the target suggested in this
table.

Table 2: Summary of the analysis of detected emission lines in this study.

| J0107a |  |  | J0107b |  |
---|---|---|---|---|---|---
| CO(4–3) | CO(3–2) | [C i](1–0) | CO(4–3) | CO(3-2) | [C i](1–0)
$\nu_{\rm{obs}}$ (GHz) | 132.99 | 99.749 | 141.97 | 139.29 | 104.47 | 148.69
Line peak (mJy beam-1) | $2.29\pm 0.11$ | $1.88\pm 12$ | $0.93\pm 0.13$ | $1.77\pm 0.10$ | $1.06\pm 0.12$ | $0.38\pm 0.05$
rms (mJy beam-1) | $0.54$ | $0.42$ | $0.46$ | $0.33$ | $0.37$ | $0.32$
beam size | $1\farcs 23\times 1\farcs 02$ | $1\farcs 24\times 1\farcs 03$ | $1\farcs 37\times 1\farcs 00$ | $1\farcs 22\times 0\farcs 955$ | $1\farcs 16\times 0\farcs 916$ | $1\farcs 02\times 0\farcs 870$
deconvolved source size | $1\farcs 3\times 0\farcs 7$ | $1\farcs 8\times 1\farcs 0$ | $1\farcs 1\times 0\farcs 3$ | $0\farcs 7\times 0\farcs 5$ | $0\farcs 7\times 0\farcs 4$ | $1\farcs 0\times 0\farcs 3$
$S_{\rm{obs}}$ (Jy km s-1) | $3.0\pm 0.2$ | $2.2\pm 0.2$ | $1.2\pm 0.2$ | $1.3\pm 0.1$ | $0.95\pm 0.1$ | $0.51\pm 0.1$
Luminosity ($10^{10}$ K km s-1 pc2) | $5.7\pm 0.4$ | $7.2\pm 0.5$ | $2.0\pm 0.3$ | $2.1\pm 0.2$ | $2.8\pm 0.3$ | $0.75\pm 0.1$
Line FWHM (km s-1) | $193\pm 11$ | $165\pm 13$ | $217\pm 34$ | $190\pm 13$ | $164\pm 21$ | $176\pm 27$
Peak S/N in channel maps | $\sim\,20$ | $\sim\,15$ | $\sim\,10$ | $\sim 11$ | $\sim 9$ | $\sim 3$

Note. — The line peaks are those of line spectra in Figure 3. The source size
and velocity-integrated flux density ($S_{\rm{obs}}$) are measured with the
CASA task imfit and we applied primary beam correction for $S_{\rm{obs}}$. The
velocity resolution is 20 km s-1 for the data of CO(4–3) of J0107a and 50 km
s-1 for other data. Source sizes have 5–10 % errors for each axis. The peak
S/N is the approximate ratio of line peak in Figure 2 and rms noise for each
line.

###  2.2 Line identification

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.

###  2.3 Continuum measurement

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.

Figure 1: Top: HST image of VV114 and the periphery, which is taken in the
$I$-band (filter [F814W]). The two white boxes indicate the regions for which
ALMA images for two sources are presented in Figures 4 and 5; the east one is
for J0107a, and the north one is for J0107b. The white crosses in these boxes
indicate the positions of J0107a and J0107b, respectively. Bottom:
$10^{\prime\prime}\times 10^{\prime\prime}$ multiwavelength (infrared to
X-ray) images of J0107a (a) and J0107b (b). Contours show the CO(4–3)
integrated intensities and are drawn at 10, 30, and 50 $\sigma$ for J0107a,
and 5, 10, and 20 $\sigma$ for J0107b. The $\sigma$ for each source are
$\sigma=0.018\ \mathrm{Jy\ beam^{-1}\ km\ s^{-1}}$ for J0107a, and
$\sigma=0.035\ \mathrm{Jy\ beam^{-1}\ km\ s^{-1}}$ for J0107b (see Section 2.2
and Figures 4, 5).  Figure 2:  The band 3 and 4 spectra for J0107a (a) and
J0107b (b). The gaps seen in the band 3 spectra are due to ones between two
spectral windows.  Figure 3: The three spectral lines, CO(4–3), CO(3–2), [C
i](1–0), in each of J0107a (a) and J0107b (b). The black solid curve drawn in
each plot is the result of Gaussian fitting to each line. The velocity
resolution of data is 20 km s-1 for CO(4–3) of J0107a and 50 km s-1 for the
others.  Figure 4:  CO, [C i], and continuum images of J0107a. Each panel
depicts a $10\arcsec\times 10\arcsec$ region centered on the CO(4–3) peak
position. (a1)(a2)(a3): Velocity-integrated intensity images of CO(4–3),
CO(3–2), and [C i](1–0) lines. Contour levels are 10, 20, and 30 $\sigma$,
where 1 $\sigma$ = 0.018, 0.031, and 0.031 Jy beam-1 km s-1 for CO(4–3),
CO(3–2), and [C i](1–0) lines, respectively. (a4)(a5)(a6): Intensity-weighted
mean radial velocity images of CO(4–3), CO(3–2), and [C i](1–0) lines. Contour
levels are -60, -40, -20, 0, 20, and 40 km s-1 for CO(4–3), -40, -20, 0, 20,
and 40 km s-1 for CO(3–2), and -20, 0, 20, and 40 km s-1 for [C i](1–0).
Negative contours are dashed. (a7)(a8)(a9): Intensity-weighted velocity
dispersion images of CO(4–3), CO(3–2), and [C i](1–0) lines. Contour levels
are 20, 40, and 60 km s-1 for CO(4–3) and CO(3–2), and 20 and 40 km s-1 for [C
i](1–0). (a10)(a11)(a12)(a13): Continuum images of 3.0 mm (band 3), 2.2 mm
(band 4), 1.3 mm (band 6), and 0.89 mm (band 7). Contour levels are 1 $\sigma$
(band 3), 2 and 3 $\sigma$ (band 4), and 2, 3, and 5 $\sigma$ (bands 6 and 7).
Each noise level is given in Table 3 (rms in mJy beam-1). The band 7 continuum
image (a13) has relatively large noise in the left side because J0107a is near
the east edge of the field.  Figure 5: The same as Figure 4 but for J0107b.
Contour levels in (b1), (b2), and (b3) are 5, 10, 20 $\sigma$, where 1
$\sigma$ = 0.035, 0.046, and 0.028 Jy beam-1 km s-1 for CO(4–3), CO(3–2), and
[C i](1–0) lines, respectively. Contour levels in other maps are the same as
those in Figure 4. The band 6 (b12) and 7 (b13) continuum images are near the
edges of the observed fields.

##  3 Results

###  3.1 Molecular gas mass by emission lines

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):

$\displaystyle M_{\mathrm{H_{2}}}=1375.8\
\frac{D_{L}^{2}}{1+z}\left(\frac{X_{\mathrm{C_{I}}}}{10^{-5}}\right)^{-1}\left(\frac{A_{10}}{10^{-7}\
\mathrm{s^{-1}}}\right)^{-1}$ $\displaystyle\times
Q_{10}^{-1}\frac{S_{\mathrm{C_{I}}}}{\mathrm{Jy\ km\ s^{-1}}}\ \ [M_{\odot}],$
(1)

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
$M_{\rm gas}$:

$\displaystyle M_{\rm gas}({\rm J0107a})$ $\displaystyle=$
$\displaystyle(11.2\pm 3.1)\times 10^{10}\quad M_{\odot},$ $\displaystyle
M_{\rm gas}({\rm J0107b})$ $\displaystyle=$ $\displaystyle(4.2\pm 1.2)\times
10^{10}\quad M_{\odot}.$

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:

$\frac{M_{\rm{gas}}}{M_{\odot}}=\alpha_{\mathrm{CO}}\
\frac{L^{\prime}(\mathrm{CO})_{1-0}}{\mathrm{K~{}km\ s^{-1}~{}pc^{2}}}.$ (2)

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.

Table 3: Continuum fluxes for the targets.

Facility | Center | beam size |  $S_{\rm{J0107a}}$ [mJy] |  $S_{\rm{J0107b}}$ [mJy] | rms [mJy beam-1]
---|---|---|---|---|---
ALMA/Band 3 | 3.0 mm | $1\farcs 14\times 1\farcs 07$ | $<0.27$ | $<0.27$ | 0.09
ALMA/Band 4 | 2.2 mm | $1\farcs 16\times 0\farcs 95$ | $0.61\pm 0.12$ | $<0.42$ | 0.14
ALMA/Band 6 | 1.3 mm | $0\farcs 57\times 0\farcs 50$ | $3.4\pm 0.3$ | $0.96\pm 0.2$ | 0.07
ALMA/Band 7 | 887 $\mu$m | $0\farcs 18\times 0\farcs 15$ | $7.9\pm 1$ | $5.1\pm 1.2$ | 0.15
SMA | 1.3 mm |  | $5.2\pm 1.3$ |  | 1.21

Note. — Upper limits are 3 $\sigma$. The SMA data in the last row are from
Tamura et al. (2014).

###  3.2 Molecular gas mass by dust continuum

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:

$\displaystyle\frac{M_{\rm{gas}}}{10^{10}M_{\odot}}=1.78\,(1+z)^{-4.8}\frac{S_{\nu_{\rm{obs}}}}{\mathrm{mJy}}\left(\frac{\nu_{\rm{obs}}}{353\
\mathrm{GHz}}\right)^{-3.8}$ $\displaystyle\times\left(\frac{6.7\times
10^{19}}{\alpha_{\mathrm{353\,GHz}}}\right)\frac{\Gamma_{0}}{\Gamma_{\rm{RJ}}}\left(\frac{D_{L}}{\mathrm{Gpc}}\right)^{2},$
(3)

where

$\frac{\alpha_{\mathrm{353\,GHz}}}{\mathrm{erg~{}s^{-1}~{}Hz^{-1}}~{}M_{\odot}^{-1}}=L_{\mathrm{353\,GHz}}/M_{\mathrm{gas}}\
,$ (4)
$\Gamma_{\rm{RJ}}(T_{\rm{dust}},\nu_{\rm{obs}},z)=\frac{h\nu_{\rm{obs}}(1+z)/kT_{\rm{dust}}}{e^{h\nu_{\rm{obs}}(1+z)/kT_{\rm{dust}}}-1}\
,$ (5)

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:

$M_{\rm{gas}}\,(\mathrm{J0107a})=(3.2\pm 1.6)\times 10^{11}\ M_{\odot}\ .$

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:

$M_{\rm{gas}}\,(\mathrm{J0107b})=(9.5\pm 4.8)\times 10^{10}\ M_{\odot}\ .$

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.

###  3.3 Far-IR luminosity and SFR

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):

$\displaystyle
L_{\rm{FIR}}=4\pi\Gamma[\beta+4]\zeta[\beta+4]D_{L}^{2}\left(\frac{h\nu_{\rm{rest}}}{kT_{\rm{dust}}}\right)^{-(\beta+4)}$
$\displaystyle\times\left(e^{h\nu_{\rm{rest}}/kT_{\rm{dust}}}-1\right)S_{\rm{obs}}\nu_{\rm{obs}}\
,$ (6)

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

$L_{\rm{FIR}}(\mathrm{J0107a})=(10.3\pm 0.8)\times 10^{12}\ \ L_{\odot}\ ,$
$L_{\rm{FIR}}(\mathrm{J0107b})=(3.0\pm 0.6)\times 10^{12}\ \ L_{\odot}\ .$

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:

$\log_{10}\left(\frac{SFR}{M_{\odot}\,\mathrm{yr^{-1}}}\right)=\log_{10}\left(\frac{L_{\rm{FIR}}}{L_{\odot}}\right)+\log_{10}(1.3)-10,$
(7)

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.

Table 4: Physical quantities derived by data analysis.

Target | $z$ |  $M_{\rm{gas}}$([C i]) [$M_{\odot}$] |  $M_{\rm{gas}}$(CO) [$M_{\odot}$] |  $M_{\rm{gas}}(\mathrm{dust})$ [$M_{\odot}$] |  $L_{\rm{FIR}}$ [$L_{\odot}$] | SFR [$M_{\odot}$ yr-1]
---|---|---|---|---|---|---
J0107a | $2.4666\pm 0.0002$ | $(11.2\pm 3.1)\times 10^{10}$ | $(13.4\pm 1.7)\times 10^{10}$ | $(32\pm 16)\times 10^{10}$ | $10.3\times 10^{12}$ | $1.3\times 10^{3}$
J0107b | $2.3100\pm 0.0002$ | $(4.2\pm 1.2)\times 10^{10}$ | $(5.0\pm 0.7)\times 10^{10}$ | $(9.5\pm 4.8)\times 10^{10}$ | $3.0\times 10^{12}$ | $3.9\times 10^{2}$

Note. — $z$ : spectroscopic redshift; $M_{\rm{gas}}$([C i]) : molecular gas
mass derived by [C i](1–0) line intensity; $M_{\rm{gas}}$(CO) : molecular gas
mass derived by CO(4–3) line luminosity with $\alpha_{\mathrm{CO}}=0.8$;
$M_{\rm{gas}}(\mathrm{dust})$ : molecular gas mass derived by dust continuum
emission; $L_{\rm{FIR}}$ : far-infrared luminosity derived by dust continuum
emission; SFR : star formation rate derived by $L_{\rm{FIR}}$. SFR have
typical uncertainties of about a factor of 4.5.

##  4 Kinematic modeling

In this section, we perform kinematic modeling of a rotating disk to derive a
rotation curve and estimate some dynamical properties. Here, we discuss only
J0107a. Although the CO(4–3) line of J0107b indicates the rotational motion
(see Figure 5), the S/N is not enough for the kinematic modeling.

###  4.1 Method and results

We model the disk rotation of J0107a using the Markov chain Monte Carlo (MCMC)
method with GalPaK3D (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
GalPaK3D. 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 GalPaK3D 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 GalPaK3D, 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 GalPaK3D model
in Table 5. Figure 6 shows the output rotation curve, and Figure 7 shows a
comparison of the GalPaK3D 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 GalPaK3D. 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.

Table 5:  The results of the GalPaK3D kinematic modeling of J0107a.  J0107a | value | 95% CI
---|---|---
flux [Jy beam-1] | $1.83\pm 0.02$ | $[1.78,1.88]$
half-light radius [kpc] | $4.91\pm 0.1$ | $[4.71,5.13]$
inclination [deg] | $63.2\pm 1.7$ | $[59.8,66.8]$
pa [deg] | $290.4\pm 1.5$ | $[287.5,293.2]$
turnover radius [kpc] | $0.697\pm 0.772$ | $[0.05,2.71]$
maximum velocity [km s-1] | $69.31\pm 4.79$ | $[64.45,82.53]$
velocity dispersion [km s-1] | $54.57\pm 1.78$ | $[51.48,58.35]$
Figure 6: The model rotation curve of J0107a calculated from GalPaK3D (black
solid curve). The horizontal blue solid line indicates the calculated maximum
velocity of rotation and the vertical blue dashed line indicates the twice of
half-light radius.  Figure 7: Comparison of the observed (left panels) and
modeled (middle panels) CO(4-3) data of J0107a, along with their residuals
(right panels). (a): The observed CO(4–3) integrated intensity map. The
contours show 5, 10, 20, and 30 $\sigma$ of the data, and contours with the
same levels are also shown in (b). (b): The modeled CO(4–3) integrated
intensity map from the best-fit GalPaK3D results. (c): The residual of the
CO(4-3) integrated intensity map, (a)-(b). The contours show $\pm 5\sigma$ of
the data. (d): The observed CO(4-3) velocity map. The contour interval in (d),
(e), and (f) is 20 km s-1, and negative contours are dashed. (e): The modeled
CO(4-3) velocity map from the GalPaK3D model. The velocities in the position-
velocity diagram in Figure 8 are extracted along the dotted line shown in (d)
and (e). (f): The residual of the CO(4-3) velocity map, (d)-(e). Figure 8:
The position-velocity diagram of J0107a along the major axis with PA =
$110.4\tcdegree$ (the dashed lines in Figure 7 (d, e)). The background color
is from the ALMA CO(4-–3) data while the contours are from GalPaK3D results.
Contour levels are 2, 3.5, 5, and 6 mJy beam-1.

###  4.2 Dynamical mass

In the calculation of a dynamical mass, $M_{\rm{dyn}}$, we use the results of
GalPaK3D. 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:

$M_{\rm{dyn}}\,(\mathrm{J0107a})=\frac{rv_{\rm{max}}^{2}}{G}=(1.1\pm
0.2)\times 10^{10}\ \ M_{\odot}\ .$

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.

##  5 Discussion

Here, we mainly discuss the physical properties of J0107a. Specifically, we
focus on the ratio of maximum rotation velocity $v_{\rm{max}}$ to velocity
dispersion $\sigma_{v}$, $v_{\rm{max}}/\sigma_{v}$, to assess the dynamic
hotness of the gas disk, and the molecular gas fraction
$f_{\rm{gas}}=M_{\rm{gas}}/(M_{\rm{gas}}+M_{\star})$ , where $M_{\star}$ is
the stellar mass of the galaxy, to characterize the evolutionary state of the
system. The possible cause of the discrepancy between $M_{\mathrm{gas}}$ and
$M_{\mathrm{dyn}}$, which is found to be of an order of magnitude, is also
discussed in this section.

###  5.1 Dynamical properties of the gas disk

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: Ratio of rotational velocity to velocity dispersion
($v_{\rm{max}}/\sigma_{v}$) versus redshift. The sky-blue pluses indicate SFGs
(data from Swinbank et al., 2017; Lee et al., 2019), the green squares
indicate millimeter-wave line emitter samples in Kaasinen et al. (2020), the
brown up triangles indicate 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), and the red circle indicates J0107a. In Kaasinen et al.
(2020), they treated their $\sigma_{v}$ as upper limits, but we here do not
show $v_{\mathrm{max}}/\sigma_{v}$ of these samples as lower limits (see
details in Section 5.1). We also show the fitting results for the distribution
of SFGs with low stellar masses ($10^{9}<M_{\star}/M_{\odot}<10^{10}$) and
with high stellar masses ($10^{10}<M_{\star}/M_{\odot}<10^{11}$), which are
presented by Simons et al. (2017). They are indicated with a gray line and a
light-green line, respectively, and the shaded areas indicate the $1\sigma$ of
these fitting lines.

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.

###  5.2 Molecular gas fraction

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:

$\frac{M_{\star}}{M_{\odot}}=\frac{1}{5.8}\left(\frac{L_{H}}{L_{\odot}}\right)\
.$ (8)

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:

$L_{H}\,\sim\,4\pi D_{L}^{2}\times S_{\rm{obs}}\ d\nu<5.8\times 10^{11}\ \
L_{\odot}\ ,$

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:

$M_{\star}\,(\mathrm{J0107a})<1.0\times 10^{11}\ \ M_{\odot}\ .$

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

$f_{\rm{gas}}\,(\mathrm{J0107a})>0.5\ .$

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).

###  5.3 Why does the molecular gas mass exceed the dynamical mass in J0107a?

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:

$8<\frac{M_{\rm{gas}}+M_{\star}}{M_{\rm{dyn}}}<46.$

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

$\mu=3.5\times\frac{L^{\prime}(\mathrm{CO})_{\mathrm{apparent}}}{\mathrm{10^{11}\
K\ km\ s^{-1}\ pc^{2}}}\left(\frac{400\ \mathrm{km\ s^{-1}}}{\Delta
v_{\mathrm{FWHM}}}\right)^{1.7}\ ,$ (9)

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

$\mu(\mathrm{J0107a})\,\sim\,21\ \ .$

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 $(M_{\rm{gas}}+M_{\star})/M_{\rm{dyn}}$.

Figure 10: Plot of $L^{\prime}_{\mathrm{CO(1-0)}}$ versus FWHM of CO line. Red
and blue circles indicate J0107a and J0107b, respectively. The data of local
SFGs, which are indicated with sky-blue pluses, are taken from Bothwell et al.
(2014) and Saintonge et al. (2017). The data of $z\sim 1-2$ SFGs, which are
indicated with pink down triangles, are taken from Daddi et al. (2010),
Magnelli et al. (2012), Magdis et al. (2012), and Tacconi et al. (2013). The
data of $z\sim 2-4$ SMGs and $z\sim 2.1-3.5$ SMGs, which are indicated with
green and purple up triangles, respectively, are taken from Harris et al.
(2012) and citation therein. The data of millimeter-wave line emitters, which
are indicated with green squares, are taken from Aravena et al. (2019) and
Kaasinen et al. (2020), while the sources in Kaasinen et al. (2020) are all
contained in the samples of Aravena et al. (2019), and we use the updated
values from Kaasinen et al. (2020) for some of these objects. The $z\sim
2.1-3.5$ SMGs are thought to be magnified with gravitational lens effect
(Harris et al., 2012). The linewidths of all SMGs in Harris et al. (2012) is
that of CO(1–0), whereas most of all linewidths of local SFGs and $z\sim 1-2$
SFGs in this figure are that of CO(2–1) or CO(3–2). We also show a power-law
fit for intrinsic line luminosity versus linewidth with light-green dashed
line (Harris et al., 2012) and gray dot dashed line (Bothwell et al., 2013),
and for magnified line luminosity with purple dashed line ($\mu=21$ from
Harris et al., 2012) and black dot dashed line ($\mu=5.9$ from Bothwell et
al., 2013).

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.

##  6 Conclusion

We conducted analysis of ALMA band 3, 4, 6, and 7 data of J0107a and J0107b,
which are serendipitously discovered millimeter-wave line-emitting galaxies in
the same field of the nearby galaxy VV114. In addition, we performed kinematic
modeling of J0107a and investigated its physical properties. Our findings and
conclusions are as follows:

1\. We identify three emission lines, CO(4–3), CO(3–2), and [C i](1–0), for
each of them. In addition, we detect dust continuum emission of J0107a in
bands 4, 6, and 7, and that of J0107b in band 6 and 7.

2\. By fitting the Gaussian to CO spectra, we derive the redshifts of J0107a
and J0107b as $z=2.4666\pm 0.0002$ and $z=2.3100\pm 0.0002$, respectively.

3\. We obtain $M_{\rm{gas}}$ of our targets with the use of [C i] line fluxes
as $M_{\rm{gas}}(\mathrm{J0107a})=(11.2\pm 3.1)\times 10^{10}\ M_{\odot}$ and
$M_{\rm{gas}}(\mathrm{J0107b})=(4.2\pm 1.2)\times 10^{10}\ M_{\odot}$,
respectively. Moreover, using $\alpha_{\rm{CO}}=0.8$ (Downes & Solomon, 1998)
and CO(1–0) luminosity which is converted from CO(4–3), we derive
$M_{\rm{gas}}$ independently as $M_{\rm{gas}}(\mathrm{J0107a})=(13.4\pm
1.7)\times 10^{10}\ \ M_{\odot}$ and $M_{\rm{gas}}(\mathrm{J0107b})=(5.0\pm
0.7)\times 10^{10}\ \ M_{\odot}$, respectively. These results are consistent
with $M_{\rm{gas}}$ derived with [C i] line fluxes within uncertainties.

4\. We also calculate $M_{\mathrm{gas}}$ of our targets with the dust
continuum emission in 1.3 mm as $M_{\rm{gas}}(\mathrm{J0107a})=(3.2\pm
1.6)\times 10^{11}\ M_{\odot}$ and $M_{\rm{gas}}(\mathrm{J0107b})=(9.6\pm
4.8)\times 10^{10}\ M_{\odot}$, respectively. These
$M_{\rm{gas}}(\mathrm{dust})$ is consistent with $M_{\rm{gas}}$, which is
derived with [C i] line intensity, within uncertainties.

5\. We make the rotating disk model of J0107a with GalPaK3D. This model
reproduces not only the moment maps of J0107a, but also the position-velocity
diagram of it well. This result suggests that J0107a is likely to have a
rotating disk. Using the results of this kinematic modeling, we obtain the
dynamical mass of J0107a as $M_{\rm{dyn}}=(1.1\pm 0.2)\times 10^{10}\
M_{\odot}$.

6\. We utilize the results of kinematic modeling to calculate the ratio of
maximum rotation velocity $v_{\rm{max}}$ to velocity dispersion $\sigma_{v}$,
namely $v_{\rm{max}}/\sigma_{v}$, and derive $v_{\rm{max}}/\sigma_{v}=1.3\pm
0.3$. The $v_{\rm{max}}/\sigma_{v}$ of J0107a is quantitatively comparable to
that of SMGs at similar redshifts.

7\. By comparing $M_{\rm{dyn}}$ and the sum of $M_{\rm{gas}}$ and $M_{\star}$,
we propose that J0107a is magnified by the gravitational lens effect, and
detected fluxes might be magnified by a factor of more than 10. This is
consistent with the excess CO luminosity of J0107a compared to the expectation
from the CO linewidth.

We thank the anonymous referee for a number of insightful comments and
suggestions that greatly improved the quality of this paper. We also thank
Hideki Umehata and Ken Tadaki for helpful discussions on stellar properties of
dust-obscured starburst galaxies. Data analysis was partly carried out on the
common-use data analysis computer system at the Astronomy Data Center (ADC) of
the National Astronomical Observatory of Japan (NAOJ). This paper makes use of
the following ALMA data: ADS/JAO.ALMA#2013.1.01057.S, #2015.1.00973.S,
#2015.1.00902.S, and #2013.1.00740.S. ALMA is a partnership of ESO
(representing its member states), NSF (USA) and NINS (Japan), together with
NRC (Canada), MOST and ASIAA (Taiwan), and KASI (Republic of Korea), in
cooperation with the Republic of Chile. The Joint ALMA Observatory is operated
by ESO, AUI/NRAO and NAOJ. This work was supported by JSPS KAKENHI Grant
Number JP17H06130 and by the NAOJ ALMA Scientific Research Grant Number
2017-06B. FE is supported by JSPS KAKENHI Grant Number 17K14259. DE
acknowledges support from a Beatriz Galindo senior fellowship (BG20/00224)
from the Ministry of Science and Innovation.

##  Appendix A cross-correlations between the GaLPak3D results

Figure 11: The cross-correlation plot of Markov chain in GalPaK3D modeling
with initial inclination angle of $i=60\tcdegree$. We use the last 10,000
iteration data of total 20,000 iterations to avoid using the data before MCMC
burn-in. The gray dots indicate individual iteration data and the density of
these data is presented with contours. The solid and dashed lines indicate the
average (50th percentile) and 2–$\sigma$ width (2.5th and 97.5th percentiles)
of each parameter, respectively.

## References

  * Alaghband-Zadeh et al. (2012) Alaghband-Zadeh, S., Chapman, S. C., Swinbank, A. M., et al. 2012, MNRAS, 424, 2232, doi: 10.1111/j.1365-2966.2012.21386.x
  * Alaghband-Zadeh et al. (2013) —. 2013, MNRAS, 435, 1493, doi: 10.1093/mnras/stt1390
  * Aravena et al. (2019) Aravena, M., Decarli, R., Gónzalez-López, J., et al. 2019, ApJ, 882, 136, doi: 10.3847/1538-4357/ab30df
  * Birkin et al. (2021) Birkin, J. E., Weiss, A., Wardlow, J. L., et al. 2021, MNRAS, 501, 3926, doi: 10.1093/mnras/staa3862
  * Bolatto et al. (2013) Bolatto, A. D., Wolfire, M., & Leroy, A. K. 2013, ARA&A, 51, 207, doi: 10.1146/annurev-astro-082812-140944
  * Bothwell et al. (2013) Bothwell, M. S., Smail, I., Chapman, S. C., et al. 2013, MNRAS, 429, 3047, doi: 10.1093/mnras/sts562
  * Bothwell et al. (2014) Bothwell, M. S., Wagg, J., Cicone, C., et al. 2014, MNRAS, 445, 2599, doi: 10.1093/mnras/stu1936
  * Bouché et al. (2015) Bouché, N., Carfantan, H., Schroetter, I., Michel-Dansac, L., & Contini, T. 2015, AJ, 150, 92, doi: 10.1088/0004-6256/150/3/92
  * Brown et al. (2019) Brown, A., Nayyeri, H., Cooray, A., et al. 2019, ApJ, 871, 87, doi: 10.3847/1538-4357/aaf73b
  * Bruzual & Charlot (2003) Bruzual, G., & Charlot, S. 2003, MNRAS, 344, 1000, doi: 10.1046/j.1365-8711.2003.06897.x
  * Burkert et al. (2010) Burkert, A., Genzel, R., Bouché, N., et al. 2010, ApJ, 725, 2324, doi: 10.1088/0004-637X/725/2/2324
  * Carilli & Walter (2013) Carilli, C. L., & Walter, F. 2013, ARA&A, 51, 105, doi: 10.1146/annurev-astro-082812-140953
  * Chapin et al. (2009) Chapin, E. L., Pope, A., Scott, D., et al. 2009, MNRAS, 398, 1793, doi: 10.1111/j.1365-2966.2009.15267.x
  * da Cunha et al. (2015) da Cunha, E., Walter, F., Smail, I. R., et al. 2015, ApJ, 806, 110, doi: 10.1088/0004-637X/806/1/110
  * Daddi et al. (2010) Daddi, E., Bournaud, F., Walter, F., et al. 2010, ApJ, 713, 686, doi: 10.1088/0004-637X/713/1/686
  * De Breuck et al. (2003) De Breuck, C., Neri, R., Morganti, R., et al. 2003, A&A, 401, 911, doi: 10.1051/0004-6361:20030171
  * De Breuck et al. (2014) De Breuck, C., Williams, R. J., Swinbank, M., et al. 2014, A&A, 565, A59, doi: 10.1051/0004-6361/201323331
  * Decarli et al. (2014) Decarli, R., Walter, F., Carilli, C., et al. 2014, ApJ, 782, 78, doi: 10.1088/0004-637X/782/2/78
  * Decarli et al. (2020) Decarli, R., Aravena, M., Boogaard, L., et al. 2020, ApJ, 902, 110, doi: 10.3847/1538-4357/abaa3b
  * Downes & Solomon (1998) Downes, D., & Solomon, P. M. 1998, ApJ, 507, 615, doi: 10.1086/306339
  * Erb et al. (2006) Erb, D. K., Steidel, C. C., Shapley, A. E., et al. 2006, ApJ, 646, 107, doi: 10.1086/504891
  * Genzel et al. (2010) Genzel, R., Tacconi, L. J., Gracia-Carpio, J., et al. 2010, MNRAS, 407, 2091, doi: 10.1111/j.1365-2966.2010.16969.x
  * González-López et al. (2017) González-López, J., Bauer, F. E., Aravena, M., et al. 2017, A&A, 608, A138, doi: 10.1051/0004-6361/201730961
  * Gowardhan et al. (2017) Gowardhan, A., Riechers, D. A., Daddi, E., et al. 2017, ApJ, 838, 136, doi: 10.3847/1538-4357/aa65d2
  * Graciá-Carpio et al. (2008) Graciá-Carpio, J., García-Burillo, S., Planesas, P., Fuente, A., & Usero, A. 2008, A&A, 479, 703, doi: 10.1051/0004-6361:20078223
  * Hainline et al. (2011) Hainline, L. J., Blain, A. W., Smail, I., et al. 2011, ApJ, 740, 96, doi: 10.1088/0004-637X/740/2/96
  * Harris et al. (2012) Harris, A. I., Baker, A. J., Frayer, D. T., et al. 2012, ApJ, 752, 152, doi: 10.1088/0004-637X/752/2/152
  * Hayatsu et al. (2017) Hayatsu, N. H., Matsuda, Y., Umehata, H., et al. 2017, PASJ, 69, 45, doi: 10.1093/pasj/psx018
  * Hayatsu et al. (2019) Hayatsu, N. H., Ivison, R. J., Andreani, P., et al. 2019, Research Notes of the American Astronomical Society, 3, 97, doi: 10.3847/2515-5172/ab3228
  * Hezaveh et al. (2012) Hezaveh, Y. D., Marrone, D. P., & Holder, G. P. 2012, ApJ, 761, 20, doi: 10.1088/0004-637X/761/1/20
  * Hodge et al. (2012) Hodge, J. A., Carilli, C. L., Walter, F., et al. 2012, ApJ, 760, 11, doi: 10.1088/0004-637X/760/1/11
  * Hopkins & Beacom (2006) Hopkins, A. M., & Beacom, J. F. 2006, ApJ, 651, 142, doi: 10.1086/506610
  * Iono et al. (2013) Iono, D., Saito, T., Yun, M. S., et al. 2013, PASJ, 65, L7, doi: 10.1093/pasj/65.3.L7
  * Ivison et al. (2011) Ivison, R. J., Papadopoulos, P. P., Smail, I., et al. 2011, MNRAS, 412, 1913, doi: 10.1111/j.1365-2966.2010.18028.x
  * Jiménez-Andrade et al. (2020) Jiménez-Andrade, E. F., Zavala, J. A., Magnelli, B., et al. 2020, ApJ, 890, 171, doi: 10.3847/1538-4357/ab6dec
  * Kaasinen et al. (2020) Kaasinen, M., Walter, F., Novak, M., et al. 2020, ApJ, 899, 37, doi: 10.3847/1538-4357/aba438
  * Kennicutt (1998) Kennicutt, Robert C., J. 1998, ARA&A, 36, 189, doi: 10.1146/annurev.astro.36.1.189
  * Lee et al. (2019) Lee, M. M., Tanaka, I., Kawabe, R., et al. 2019, ApJ, 883, 92, doi: 10.3847/1538-4357/ab3b5b
  * Lenkić et al. (2020) Lenkić, L., Bolatto, A. D., Förster Schreiber, N. M., et al. 2020, AJ, 159, 190, doi: 10.3847/1538-3881/ab7458
  * Madau & Dickinson (2014) Madau, P., & Dickinson, M. 2014, ARA&A, 52, 415, doi: 10.1146/annurev-astro-081811-125615
  * Magdis et al. (2012) Magdis, G. E., Daddi, E., Béthermin, M., et al. 2012, ApJ, 760, 6, doi: 10.1088/0004-637X/760/1/6
  * Magnelli et al. (2012) Magnelli, B., Saintonge, A., Lutz, D., et al. 2012, A&A, 548, A22, doi: 10.1051/0004-6361/201220074
  * McMullin et al. (2007) McMullin, J. P., Waters, B., Schiebel, D., Young, W., & Golap, K. 2007, in Astronomical Society of the Pacific Conference Series, Vol. 376, Astronomical Data Analysis Software and Systems XVI, ed. R. A. Shaw, F. Hill, & D. J. Bell, 127
  * Papadopoulos & Greve (2004) Papadopoulos, P. P., & Greve, T. R. 2004, ApJ, 615, L29, doi: 10.1086/426059
  * Planck Collaboration et al. (2011) Planck Collaboration, Abergel, A., Ade, P. A. R., et al. 2011, A&A, 536, A21, doi: 10.1051/0004-6361/201116455
  * Saintonge et al. (2017) Saintonge, A., Catinella, B., Tacconi, L. J., et al. 2017, ApJS, 233, 22, doi: 10.3847/1538-4365/aa97e0
  * Saito et al. (2015) Saito, T., Iono, D., Yun, M. S., et al. 2015, ApJ, 803, 60, doi: 10.1088/0004-637X/803/2/60
  * Saito et al. (2017) Saito, T., Iono, D., Espada, D., et al. 2017, ApJ, 834, 6, doi: 10.3847/1538-4357/834/1/6
  * Saito et al. (2020) Saito, T., Michiyama, T., Liu, D., et al. 2020, arXiv e-prints, arXiv:2007.08536.  https://arxiv.org/abs/2007.08536
  * Schreiber et al. (2018) Schreiber, C., Elbaz, D., Pannella, M., et al. 2018, A&A, 609, A30, doi: 10.1051/0004-6361/201731506
  * Scoville et al. (2014) Scoville, N., Aussel, H., Sheth, K., et al. 2014, ApJ, 783, 84, doi: 10.1088/0004-637X/783/2/84
  * Scoville et al. (2016) Scoville, N., Sheth, K., Aussel, H., et al. 2016, ApJ, 820, 83, doi: 10.3847/0004-637X/820/2/83
  * Serjeant (2012) Serjeant, S. 2012, MNRAS, 424, 2429, doi: 10.1111/j.1365-2966.2012.20761.x
  * Simons et al. (2017) Simons, R. C., Kassin, S. A., Weiner, B. J., et al. 2017, ApJ, 843, 46, doi: 10.3847/1538-4357/aa740c
  * Swinbank et al. (2012) Swinbank, A. M., Karim, A., Smail, I., et al. 2012, MNRAS, 427, 1066, doi: 10.1111/j.1365-2966.2012.22048.x
  * Swinbank et al. (2017) Swinbank, A. M., Harrison, C. M., Trayford, J., et al. 2017, MNRAS, 467, 3140, doi: 10.1093/mnras/stx201
  * Tacconi et al. (2020) Tacconi, L. J., Genzel, R., & Sternberg, A. 2020, arXiv e-prints, arXiv:2003.06245.  https://arxiv.org/abs/2003.06245
  * Tacconi et al. (2013) Tacconi, L. J., Neri, R., Genzel, R., et al. 2013, ApJ, 768, 74, doi: 10.1088/0004-637X/768/1/74
  * Tacconi et al. (2018) Tacconi, L. J., Genzel, R., Saintonge, A., et al. 2018, ApJ, 853, 179, doi: 10.3847/1538-4357/aaa4b4
  * Tadaki et al. (2019) Tadaki, K.-i., Iono, D., Hatsukade, B., et al. 2019, ApJ, 876, 1, doi: 10.3847/1538-4357/ab1415
  * Tamura et al. (2015) Tamura, Y., Oguri, M., Iono, D., et al. 2015, PASJ, 67, 72, doi: 10.1093/pasj/psv040
  * Tamura et al. (2014) Tamura, Y., Saito, T., Tsuru, T. G., et al. 2014, ApJ, 781, L39, doi: 10.1088/2041-8205/781/2/L39
  * Valentino et al. (2018) Valentino, F., Magdis, G. E., Daddi, E., et al. 2018, ApJ, 869, 27, doi: 10.3847/1538-4357/aaeb88
  * Walter et al. (2016) Walter, F., Decarli, R., Aravena, M., et al. 2016, ApJ, 833, 67, doi: 10.3847/1538-4357/833/1/67
  * Wardlow et al. (2018) Wardlow, J. L., Simpson, J. M., Smail, I., et al. 2018, MNRAS, 479, 3879, doi: 10.1093/mnras/sty1526
  * Yamaguchi et al. (2017) Yamaguchi, Y., Kohno, K., Tamura, Y., et al. 2017, ApJ, 845, 108, doi: 10.3847/1538-4357/aa80e0


* [HLZ16] Lingxiao Huang, Pinyan Lu, and Chihao Zhang.  Canonical paths for MCMC: From art to science.  In Proceedings of the 27th Annual ACM-SIAM Symposium on Discrete Algorithms (SODA), page 514–527, 2016.
  * [JS89] Mark Jerrum and Alistair Sinclair.  Approximating the permanent.  SIAM Journal on Computing, 18(6):1149–1178, 1989.
  * [JS93] Mark Jerrum and Alistair Sinclair.  Polynomial-time approximation algorithms for the Ising model.  SIAM Journal on Computing, 22(5):1087–1116, 1993.
  * [KW17] Jacob Korevaar and Jan Wiegerinck.  Several complex variables.  Korteweg-de Vries Institute for Mathematics, 2017.
  * [Liu21] Kuikui Liu.  From coupling to spectral independence and blackbox comparison with the down-up walk.  arXiv preprint arXiv:2103.11609, 2021.
  * [LLL14] Chengyu Lin, Jingcheng Liu, and Pinyan Lu.  A simple FPTAS for counting edge covers.  In Proceedings of the 25th Annual ACM-SIAM Symposium on Discrete Algorithms (SODA), pages 341–348, 2014.
  * [LLZ14] Jingcheng Liu, Pinyan Lu, and Chihao Zhang.  FPTAS for counting weighted edge covers.  In Proceedings of the 22nd Annual European Symposium on Algorithms (ESA), pages 654–665, 2014.
  * [LSS19] Jingcheng Liu, Alistair Sinclair, and Piyush Srivastava.  The Ising partition function: Zeros and deterministic approximation.  Journal of Statistical Physics, 174(2):287–315, 2019.
  * [LY52] Tsung-Dao Lee and Chen-Ning Yang.  Statistical theory of equations of state and phase transitions. II. Lattice gas and Ising model.  Physical Review, 87(3):410–419, 1952.
  * [McQ13] Colin McQuillan.  Approximating Holant problems by winding.  arXiv preprint arXiv:1301.2880, 2013.
  * [MS08] Igor L. Markov and Yaoyun Shi.  Simulating quantum computation by contracting tensor networks.  SIAM Journal on Computing, 38:963–981, 2008.
  * [O’D14] Ryan O’Donnell.  Analysis of Boolean Functions.  Cambridge University Press, USA, 2014.
  * [Orú14] Román Orús.  A practical introduction to tensor networks: Matrix product states and projected entangled pair states.  Annals of Physics, 349:117–158, 2014.
  * [PR17] Viresh Patel and Guus Regts.  Deterministic polynomial-time approximation algorithms for partition functions and graph polynomials.  SIAM Journal on Computing, 46(6):1893–1919, 2017.
  * [Reg18] Guus Regts.  Zero-free regions of partition functions with applications to algorithms and graph limits.  Combinatorica, 38:987–1015, 2018.
  * [Rue71] David Ruelle.  Extension of the Lee-Yang circle theorem.  Physical Review Letters, 26(6):303–304, 1971.
  * [Wag09] David G. Wagner.  Weighted enumeration of spanning subgraphs with degree constraints.  Journal of Combinatorial Theory, Series B, 99(2):347–357, 2009\.
  * [Wei06] Dror Weitz.  Counting independent sets up to the tree threshold.  In Proceedings of the 38th Annual ACM Symposium on Theory of Computing (STOC), pages 140–149, 2006.

##  Appendix A Proofs of Zero-Free Results

In this section, we supply proofs of the main zero-free statements used in
Section 5. As noted earlier, for technical reasons, we need straightforward
generalizations of prior results which do not make symmetry assumptions. We
manage to adapt previous arguments without much additional effort, which we
provide here for completeness.

The main idea in these zero-free proofs is to do induction by conditioning on
the assignment of fewer and fewer vertices (respectively, edges) for weighted
homomorphisms (respectively, tensor networks). However, one needs to
strengthen the inductive hypothesis beyond simple zero-freeness. To the best
of our knowledge, this type of argument was first pioneered by Barvinok, and
has had a wide range of applications; see [Bar15, Bar17a, BDP20, BB21] for
applications besides those discussed in this paper.

The crucial tool is the following geometric lemma, which provides a kind of
“reverse triangle inequality”. The version below is due to Boris Bukh; a
weaker version, with $\cos(\theta/2)$ replaced by $\sqrt{\cos\theta}$, was
known due to [Bar16b]. See [Bar16a] for a proof.

######  Lemma 47 (Angle Lemma).

Let $x_{1},\dots,x_{n}\in\mathbb{C}$ be nonzero complex numbers viewed as
vectors in $\mathbb{R}^{2}$. Suppose there is an angle $0\leq\theta<2\pi/3$
such that for all $i,j$, the angle between $x_{i},x_{j}$ is at most $\theta$.
Then we have the lower bound
$\left|\sum_{i=1}^{n}x_{i}\right|\geq\cos(\theta/2)\sum_{i=1}^{n}\left|x_{i}\right|$.

###  A.1 Proofs for Weighted Graph Homomorphisms

Our goal in this subsection is to prove Theorem 39, i.e. that the weighted
graph homomorphism partition function

$\displaystyle
Z_{\phi}^{S}(A)=\sum_{\begin{subarray}{c}\sigma:V\rightarrow[q]\\\
\sigma\mid_{S}=\phi\end{subarray}}\prod_{uv\in E}A^{uv}(\sigma(u),\sigma(v))$

is nonzero in a large polydisk around $1$, where $S\subseteq
V,\phi:S\rightarrow[q]$, and we view $Z_{\phi}^{S}(A)$ as a polynomial with
variables $\\{A^{uv}(j,k)\\}_{uv\in E,j,k\in[q]}$. For convenience, for a
$\delta>0$, define

$\displaystyle\mathcal{U}(\delta)=\\{A=\\{A^{uv}\\}_{uv\in
E}:|A^{uv}(j,k)-1|<\delta,\forall uv\in E,\forall j,k\in[q]\\}.$

Additionally, for a partial configuration $\phi:S\rightarrow[q]$, a vertex
$u\in V\setminus S$ and a spin $j\in[q]$, we write
$\phi_{u,j}:S\cup\\{u\\}\rightarrow[q]$ for the unique extension of $\phi$
with $\phi_{u,j}(u)=j$.

We will need the following lemmas to implement an inductive approach.

######  Lemma 48 (Lemma 3.3 from [BS17]).

Let $\tau,\delta>0$, and suppose $A\in\mathcal{U}(\delta)$. Let $S\subseteq
V$, $\phi:S\rightarrow[q]$, $u\in V\setminus S$ be arbitrary. Assume the
following hold:

  1. (1)

$Z_{\phi_{u,j}}^{S\cup\\{u\\}}(A)\neq 0$ for every $u\in V\setminus S$ and
every $j\in[q]$;

  2. (2)

For every $u\in V\setminus S$ and every $j\in[q]$, we have

$|Z_{\phi_{u,j}}^{S\cup\\{u\\}}(A)|\geq\frac{\tau}{\Delta}\sum_{v\sim
u}\sum_{k\in[q]}|A^{uv}(j,k)|\cdot\left|\frac{\partial}{\partial
A^{uv}(j,k)}Z_{\phi_{u,j}}^{S\cup\\{u\\}}(A)\right|.$

Then for every $u\in V\setminus S$ and every $j,k\in[q]$, the angle between
$Z_{\phi_{u,j}}^{S\cup\\{u\\}}(A)$ and $Z_{\phi_{u,k}}^{S\cup\\{u\\}}(A)$ in
$\mathbb{C}$ is at most $\frac{2\delta\Delta}{\tau(1-\delta)}$.

###### Proof.

By assumption (1), the relevant partition functions are nonzero, and so the
logarithm is well-defined when applied to these partition functions and we may
bound the angle between $Z_{\phi_{u,j}}^{S\cup\\{u\\}}(A)$ and
$Z_{\phi_{u,k}}^{S\cup\\{u\\}}(A)$ by

$\displaystyle\left|\log Z_{\phi_{u,j}}^{S\cup\\{u\\}}(A)-\log
Z_{\phi_{u,k}}^{S\cup\\{u\\}}(A)\right|.$ (26)

The strategy is to write $Z_{\phi_{u,k}}^{S\cup\\{u\\}}(A)$ as
$Z_{\phi_{u,j}}^{S\cup\\{u\\}}(\tilde{A})$ for some
$\tilde{A}\in\mathcal{U}(\delta)$ which differs from $A$ by a small number of
coordinates, and then apply the Fundamental Theorem of Calculus and assumption
(2). For every $v\sim u$, we set $\tilde{A}^{uv}(j,c)=A^{uv}(k,c)$ for every
$c\in[q]$, and $\tilde{A}^{uv}(\ell,c)=A^{uv}(\ell,c)$ for all $\ell\neq j$.
For all other edges $vw\in E$, we set $\tilde{A}^{vw}=A^{vw}$.

It is clear that
$Z_{\phi_{u,k}}^{S\cup\\{u\\}}(A)=Z_{\phi_{u,j}}^{S\cup\\{u\\}}(\tilde{A})$.
By the Fundamental Theorem of Calculus, we may upper bound Eq. 26 by

$\displaystyle\max_{B\in\mathcal{U}(\delta)}\sum_{v\sim
u}\sum_{c\in[q]}\left|\frac{\partial}{\partial A^{uv}(j,c)}\log
Z_{\phi_{u,j}}^{S\cup\\{u\\}}(B)\right|\cdot\underset{\leq 2\delta\text{ since
}A,\tilde{A}\in\mathcal{U}(\delta)}{\underbrace{|A^{uv}(j,c)-\tilde{A}^{uv}(j,c)|}}$
$\displaystyle\leq{}$
$\displaystyle\frac{2\delta}{1-\delta}\max_{B\in\mathcal{U}(\delta)}\underset{\leq\Delta/\tau\text{
by assumption (2)}}{\underbrace{\sum_{v\sim
u}\sum_{c\in[q]}|A^{uv}(j,c)|\cdot\frac{1}{|Z_{\phi_{u,j}}^{S\cup\\{u\\}}(B)|}\cdot\left|\frac{\partial}{\partial
A^{uv}(j,c)}Z_{\phi_{u,j}}^{S\cup\\{u\\}}(B)\right|}}$ $\displaystyle\leq{}$
$\displaystyle\frac{2\delta\Delta}{\tau(1-\delta)}.\qed$

######  Lemma 49 (Lemma 3.4 from [BS17]).

Let $0\leq\theta<2\pi/3$, $\delta>0$, and suppose $A\in\mathcal{U}(\delta)$.
Let $S\subseteq V$, $\phi:S\rightarrow[q]$ be arbitrary. Assume the following
hold:

  1. (1)

$Z_{\phi_{u,j}}^{S\cup\\{u\\}}(A)\neq 0$ for every $u\in V\setminus S$ and
every $j\in[q]$;

  2. (2)

The angle between $Z_{\phi_{u,j}}^{S\cup\\{u\\}}(A)$ and
$Z_{\phi_{u,k}}^{S\cup\\{u\\}}(A)$ in $\mathbb{C}$ is at most $\theta$, for
every $u\in V\setminus S$ and every $j,k\in[q]$.

Then for every $u\in S$, we have the lower bound

$\displaystyle|Z_{\phi}^{S}(A)|\geq\frac{\cos(\theta/2)}{\Delta}\sum_{v\sim
u}\sum_{k\in[q]}|A^{uv}(\phi(u),k)|\cdot\left|\frac{\partial}{\partial
A^{uv}(\phi(u),k)}Z_{\phi}^{S}(A)\right|.$

###### Proof.

If $v\in S$ as well, then there is a unique $k\in[q]$ for which
$\frac{\partial}{\partial A^{uv}(\phi(u),k)}Z_{\phi}^{S}(A)\neq 0$, namely
$k=\phi(v)$. In this case, $A^{uv}(\phi(u),k)\cdot\frac{\partial}{\partial
A^{uv}(\phi(u),k)}Z_{\phi}^{S}(A)=Z_{\phi}^{S}(A)$. Otherwise, $v\notin S$ and
$\frac{\partial}{\partial
A^{uv}(\phi(u),k)}Z_{\phi}^{S}(A)=\frac{1}{A^{uv}(\phi(u),k)}\cdot
Z_{\phi_{v,k}}^{S\cup\\{v\\}}(A)$, where $\phi_{v,k}$ is the unique extension
of $\phi$ mapping $v$ to $k$.

Combining these two observations, we obtain

$\displaystyle\sum_{v\sim
u}\sum_{k\in[q]}|A^{uv}(\phi(u),k)|\cdot\left|\frac{\partial}{\partial
A^{uv}(\phi(u),k)}Z_{\phi}^{S}(A)\right|$ $\displaystyle={}$
$\displaystyle|N(u)\cap S|\cdot|Z_{\phi}^{S}(A)|+\sum_{v\sim u:v\notin
S}\sum_{k\in[q]}|Z_{\phi_{v,k}}^{S\cup\\{v\\}}(A)|$ $\displaystyle\leq{}$
$\displaystyle|N(u)\cap
S|\cdot|Z_{\phi}^{S}(A)|+\frac{1}{\cos(\theta/2)}\underset{=|N(u)\setminus
S|\cdot|Z_{\phi}^{S}(A)|}{\underbrace{\left|\sum_{v\sim u:v\notin
S}\sum_{k\in[q]}Z_{\phi_{v,k}}^{S\cup\\{v\\}}(A)\right|}}$ (Lemma 47)
$\displaystyle\leq{}$
$\displaystyle\frac{\Delta}{\cos(\theta/2)}\cdot|Z_{\phi}^{S}(A)|.$

Rearranging yields the desired result. ∎

With these lemmas in hand, we can now prove the main zero-free result.

###### Proof of Theorem 39.

Let $0<\theta<2\pi/3$ be a parameter to be determined later, set
$\tau=\cos(\theta/2)$, and let $\delta>0$ satisfy
$\theta=\frac{2\delta\Delta}{\tau(1-\delta)}$; in particular,
$\delta=\frac{\frac{1}{2\Delta}\theta\cos(\theta/2)}{1+\frac{1}{2\Delta}\theta\cos(\theta/2)}$.
We show by descending induction on $|S|$ that the following three statements
are all true:

  1. (i)

For every $S\subseteq V$, $\phi:S\rightarrow[q]$ and
$A\in\mathcal{U}(\delta)$, we have $Z_{\phi}^{S}(A)\neq 0$.

  2. (ii)

For every $S\subseteq V$, $u\in V\setminus S$, $\phi:S\rightarrow[q]$,
$A\in\mathcal{U}(\delta)$ and $j,k\in[q]$, the angle between
$Z_{\phi_{u,j}}^{S\cup\\{u\\}}(A)$ and $Z_{\phi_{u,k}}^{S\cup\\{u\\}}(A)$ in
$\mathbb{C}$ is at most $\theta$.

  3. (iii)

For every $S\subseteq V$, $u\in S$, $A\in\mathcal{U}(\delta)$, we have the
inequality

$\displaystyle|Z_{\phi}^{S}(A)|\geq\frac{\cos(\theta/2)}{\Delta}\sum_{v\sim
u}\sum_{k\in[q]}|A^{uv}(\phi(u),k)|\cdot\left|\frac{\partial}{\partial
A^{uv}(\phi(u),k)}Z_{\phi}^{S}(A)\right|.$

The base case $S=V$ is easily verified since $Z_{\phi}^{S}(A)=\prod_{uv\in
E}A^{uv}(\phi(u),\phi(v))$, a product of nonzero complex numbers.

Now, let $S\subseteq V$ with $|S|<|V|$.

  * Proof of (i)

Let $u\in V\setminus S$, which exists since $|S|<|V|$. It follows that (i)
holds for $S\cup\\{u\\}$ by the inductive hypothesis. Since
$Z_{\phi}^{S}(A)=\sum_{k\in[q]}Z_{\phi_{u,k}}^{S\cup\\{u\\}}(A)$, Lemma 47
applied to $Z_{\phi}^{S}(A)$ yields (i) assuming that (ii) holds. We prove
(ii) below.

  * Proof of (ii)

Let $u\in V\setminus S$, which exists since $|S|<|V|$. Then (i) and (iii) hold
for $S\cup\\{u\\}$ by the inductive hypothesis. (ii) then follows by Lemma 48.

  * Proof of (iii)

Let $u\in S$. Then (i) holds for $S\cup\\{u\\}$ by the inductive hypothesis.
Since (ii) holds for $S$ (as proved earlier), we may then apply Lemma 49,
yielding (iii) for $S$.

Now, we choose $0<\theta<2\pi/3$. As we wish to maximize the size of our zero-
free region, i.e. $\delta$, we need to maximize $\theta\cos(\theta/2)$. As
shown in [Reg18], the maximum is attained when $2/\theta=\tan(\theta/2)$,
which has solution $\theta^{*}\approx 1.72067$ and has objective value
$x^{*}=\theta^{*}\cos(\theta^{*}/2)\approx 1.12219$. This yields
$\delta=\frac{\frac{x^{*}}{2}}{\Delta+\frac{x^{*}}{2}}$ as claimed. ∎

###  A.2 Proofs for Tensor Network Contractions

Our goal in this subsection is to prove Theorem 41, i.e. that the tensor
network partition function

$\displaystyle
Z_{\phi}^{F}(h)=\sum_{\begin{subarray}{c}\sigma:E\rightarrow[q]\\\
\sigma\mid_{F}=\phi\end{subarray}}\prod_{v\in V}h_{v}(\sigma\mid_{E(v)})$

is nonzero in a large polydisk around $1$, where $F\subseteq
E,\phi:F\rightarrow[q]$, and we view $Z_{\phi}^{F}(\cdot)$ as a polynomial
with variables $\\{h_{v}(\alpha)\\}_{v,\alpha}$. We prove the following
stronger result.

######  Theorem 50 (Generalization of Theorem 6 from [Reg18]).

Let $G=(V,E)$ be a graph of maximum degree $\leq\Delta$. Then for every
$F\subseteq E$, $\phi:F\rightarrow[q]$, $\eta>0$, and $0\leq\theta<2\pi/3$,
the function $Z_{\phi}^{F}(h)$ is nonzero whenever $h\in\prod_{v\in
V}S_{v}(\delta,\eta)$, where

$\displaystyle
S_{v}(\delta,\eta)=\left\\{h_{v}:[q]^{E(v)}\rightarrow\mathbb{C}:\begin{subarray}{c}|h_{v}(\alpha)-h_{v}(\beta)|<\delta,\forall\alpha,\beta:E(v)\rightarrow[q],\\\
|h_{v}(\alpha)|\geq\eta,\forall\alpha:E(v)\rightarrow[q]\end{subarray}\right\\}$

and
$\delta=\eta\cdot\min\left\\{1,\frac{\theta\cos(\theta/2)}{\Delta+1}\right\\}$.

Before we prove this result, let us see how this gives Theorem 41.

###### Proof of Theorem 41.

Observe that $S_{v}(\delta,\eta)$ contains a disk around $1$ of radius
$\min\\{\delta/2,1-\eta\\}$. Using Theorem 50 and given that
$\delta=\eta\cdot\min\left\\{1,\frac{\theta\cos(\theta/2)}{\Delta+1}\right\\}$,
where $0<\theta<2\pi/3$, our goal is to maximize $\theta\cos(\theta/2)$ over
$0<\theta<2\pi/3$ to obtain the largest zero-free disk. As shown in [Reg18],
this maximum is attained when $2/\theta=\tan(\theta/2)$, which has solution
$\theta^{*}\approx 1.72067$ and has objective value
$x^{*}=\theta^{*}\cos(\theta^{*}/2)\approx 1.12219$. Given this, to obtain the
largest possible radius disk, we equalize $1-\eta$ and
$\delta/2=\eta\cdot\frac{x^{*}}{2(\Delta+1)}$. Solving, we obtain
$\eta=\frac{1}{1+\frac{x^{*}}{2(\Delta+1)}}$, yielding radius
$\frac{\frac{x^{*}}{2(\Delta+1)}}{1+\frac{x^{*}}{2(\Delta+1)}}$ as desired. ∎

It remains to prove Theorem 50. We will need the following lemmas to implement
an inductive approach.

######  Lemma 51 (Lemma 8 from [Reg18]).

Let $\tau>0$, $F\subseteq E$, $\phi:F\rightarrow[q]$ and $u\in V$ be
arbitrary. Suppose for all $h\in\prod_{v\in V}S_{v}(\delta,\eta)$ and all
$\psi:F\cup E(u)\rightarrow[q]$ extending $\phi$, the following hold:

  1. (1)

$Z_{\psi}^{F\cup E(u)}(h)\neq 0$;

  2. (2)

For all $v\in N(u)\cup\\{u\\}$, we have

$\displaystyle|Z_{\psi}^{F\cup
E(u)}(h)|\geq\tau\sum_{\begin{subarray}{c}\alpha:E(v)\rightarrow[q]\\\
\text{compatible with
}\psi\end{subarray}}|h_{v}(\alpha)|\cdot\left|\frac{\partial}{\partial
h_{v}(\alpha)}Z_{\psi}^{F\cup E(u)}(h)\right|.$

Then for all extensions $\psi,\tilde{\psi}:F\cup E(u)\rightarrow\mathbb{C}$ of
$\phi$, the angle between $Z_{\psi}^{F\cup E(u)}$ and $Z_{\tilde{\psi}}^{F\cup
E(u)}(h)$ in $\mathbb{C}$ is at most $\frac{\delta(\Delta+1)}{\tau\eta}$.

###### Proof.

By assumption (1), the relevant partition functions are nonzero, and so the
logarithm is well-defined when applied to these partition functions and we may
bound the angle between $Z_{\psi}^{F\cup E(u)}(h)$ and
$Z_{\tilde{\psi}}^{F\cup E(u)}(h)$ by

$\displaystyle\left|\log Z_{\psi}^{F\cup E(u)}(h)-\log Z_{\tilde{\psi}}^{F\cup
E(u)}(h)\right|.$ (27)

The strategy is to write $Z_{\tilde{\psi}}^{F\cup E(u)}(h)$ as
$Z_{\psi}^{F\cup E(u)}(\tilde{h})$ for some $\tilde{h}\in\prod_{v\in
V}S_{v}(\delta,\eta)$ which differs from $h$ by a small number of coordinates,
and then apply the Fundamental Theorem of Calculus and assumption (2). Let
$v\in V$. We consider three cases.

  * •

$\mathbf{v\notin N(u)\cup\\{u\\}}$: In this case, $\psi,\tilde{\psi}$ agree on
$E(v)$ and so we may simply take $h_{v}=\tilde{h}_{v}$.

  * •

$\mathbf{v\in N(u)}$: In this case, $\psi,\tilde{\psi}$ differ only on the
single edge $uv$. If $\alpha:E(v)\rightarrow[q]$ agrees with $\psi$ on $uv$,
then let $\alpha^{\prime}:E(v)\rightarrow[q]$ be given by replacing
$\alpha(uv)=\psi(uv)$ with $\tilde{\psi}(uv)$, and take
$\tilde{h}_{v}(\alpha)=h_{v}(\alpha^{\prime})$. Otherwise, just set
$\tilde{h}_{v}(\alpha)=h_{v}(\alpha)$. (Note that it does not really matter
what we set $\tilde{h}_{v}(\alpha)$ to since $Z_{\psi}^{F\cup E(u)}(h)$ only
has the term $h_{v}(\alpha)$ when $\alpha$ agrees with $\psi$ on $uv$.
However, we wish to minimize the number of coordinates in which $h,\tilde{h}$
differ.)

  * •

$\mathbf{v=u}$: In this case, just set
$\tilde{h}_{v}(\psi\mid_{E(v)})=h_{v}(\tilde{\psi}\mid_{E(v)})$ and
$\tilde{h}_{v}(\alpha)=h_{v}(\alpha)$ for all $\alpha\neq\psi\mid_{E(v)}$.

It is clear that $Z_{\tilde{\psi}}^{F\cup E(u)}(h)=Z_{\psi}^{F\cup
E(u)}(\tilde{h})$. By the Fundamental Theorem of Calculus , we may upper bound
Eq. 27 by

$\displaystyle\max_{x\in\prod_{v\in V}S_{v}(\delta,\eta)}\sum_{v\in
N(u)\cup\\{u\\}}\sum_{\begin{subarray}{c}\alpha:E(v)\rightarrow[q]\\\
\text{compatible with }\psi\end{subarray}}\left|\frac{\partial}{\partial
h_{v}(\alpha)}\log Z_{\psi}^{F\cup
E(u)}(x)\right|\cdot\left|h_{v}(\alpha)-\tilde{h}_{v}(\alpha)\right|$
$\displaystyle\leq{}$ $\displaystyle\frac{\delta}{\eta}\max_{x\in\prod_{v\in
V}S_{v}(\delta,\eta)}\underset{\leq\Delta+1}{\underbrace{\sum_{v\in
N(u)\cup\\{u\\}}}}\underset{\leq 1/\tau\text{ by assumption
}(2)}{\underbrace{\sum_{\begin{subarray}{c}\alpha:E(v)\rightarrow[q]\\\
\text{compatible with
}\psi\end{subarray}}|h_{v}(\alpha)|\cdot\frac{1}{|Z_{\psi}^{F\cup
E(u)}(x)|}\cdot\left|\frac{\partial}{\partial h_{v}(\alpha)}Z_{\psi}^{F\cup
E(u)}(x)\right|}}$ (Definition of $S_{v}(\delta,\eta)$) $\displaystyle\leq{}$
$\displaystyle\frac{\delta(\Delta+1)}{\tau\eta}.\qed$

######  Lemma 52 (Lemma 9 from [Reg18]).

Let $0\leq\theta<2\pi/3$, $u\in V$, $F\subseteq E$ satisfying $F\supseteq
E(u)$, and $\phi:F\rightarrow[q]$. Suppose for all $v\in N(u)\cup\\{u\\}$, all
$h\in\prod_{v\in V}S_{v}(\delta,\eta)$, and all extensions
$\psi,\tilde{\psi}:F\cup E(v)\rightarrow[q]$ of $\phi$, the following hold:

  1. (1)

$Z_{\psi}^{F\cup E(v)}(h)\neq 0$;

  2. (2)

The angle between $Z_{\psi}^{F\cup E(v)}(h)$ and $Z_{\tilde{\psi}}^{F\cup
E(v)}(h)$ in $\mathbb{C}$ is at most $\theta$.

Then for all $v\in N(u)\cup\\{u\\}$ and all $h\in\prod_{v\in
V}S_{v}(\delta,\eta)$, we have

$\displaystyle|Z_{\phi}^{F}(h)|\geq\cos(\theta/2)\sum_{\begin{subarray}{c}\alpha:E(v)\rightarrow[q]\\\
\text{compatible with
}\phi\end{subarray}}|h_{v}(\alpha)|\cdot\left|\frac{\partial}{\partial
h_{v}(\alpha)}Z_{\phi}^{F}(h)\right|.$

###### Proof.

The conclusion is trivially true if $v=u$, since by the assumption
$E(u)\subseteq F$, there is only one $\alpha:E(v)\rightarrow[q]$ compatible
with $\phi$, namely $\phi\mid_{E(u)}$ itself. In this case, $h_{v}(\alpha)$
divides $Z_{\phi}^{F}(h)$ and we can replace $\cos(\theta/2)$ by $1$.

Suppose $v\in N(u)$. Since
$Z_{\phi}^{F}(h)=\sum_{\begin{subarray}{c}\psi:F\cup E(v)\rightarrow[q]\\\
\psi\mid_{F}=\phi\end{subarray}}Z_{\psi}^{F\cup E(v)}(h)$, assumptions (1) and
(2) make Lemma 47 applicable, yielding

$\displaystyle|Z_{\phi}^{F}(h)|$
$\displaystyle\geq\cos(\theta/2)\sum_{\begin{subarray}{c}\psi:F\cup
E(v)\rightarrow[q]\\\ \psi\mid_{F}=\phi\end{subarray}}|Z_{\psi}^{F\cup
E(v)}(h)|$
$\displaystyle=\cos(\theta/2)\sum_{\begin{subarray}{c}\alpha:E(v)\rightarrow[q]\\\
\text{compatible with
}\psi\end{subarray}}|h_{v}(\alpha)|\cdot\left|\frac{\partial}{\partial
h_{v}(\alpha)}Z_{\phi}^{F}(h)\right|$

as desired. ∎

With these lemmas in hand, we may now proceed with the proof of Theorem 50.

###### Proof of Theorem 50.

Let $\eta>0$ and $0\leq\theta<2\pi/3$ be arbitrary, and take
$\tau=\cos(\theta/2)$,
$\delta=\eta\cdot\min\left\\{1,\frac{\theta\tau}{\Delta+1}\right\\}$. We show
by descending induction on $|F|$ that the following three statements are all
true:

  1. (i)

For every $F\subseteq E$, $\phi:F\rightarrow[q]$ and $h\in\prod_{v\in
V}S_{v}(\delta,\eta)$, we have $Z_{\phi}^{F}(h)\neq 0$.

  2. (ii)

For every $F\subseteq E$, $u\in V$, $\phi:F\rightarrow[q]$, $h\in\prod_{v\in
V}S_{v}(\delta,\eta)$ and $\psi,\tilde{\psi}:F\cup E(u)\rightarrow[q]$
extending $\phi$, the angle between $Z_{\psi}^{F\cup E(u)}(h)$ and
$Z_{\tilde{\psi}}^{F\cup E(u)}(h)$ in $\mathbb{C}$ is at most $\theta$.

  3. (iii)

For every $F\subseteq E$, $u\in V$ satisfying $E(u)\subseteq F$,
$\phi:F\rightarrow[q]$, $h\in\prod_{v\in V}S_{v}(\delta,\eta)$ and $v\in
N(u)\cup\\{u\\}$, we have the inequality

$\displaystyle|Z_{\phi}^{F}(h)|\geq\cos(\theta/2)\sum_{\begin{subarray}{c}\alpha:E(v)\rightarrow[q]\\\
\text{compatible with
}\phi\end{subarray}}|h_{v}(\alpha)|\cdot\left|\frac{\partial}{\partial
h_{v}(\alpha)}Z_{\phi}^{F}(h)\right|.$

The base case $F=E$ is easily verified since $Z_{\phi}^{F}(h)=\prod_{v\in
V}h_{v}(\phi\mid_{E(v)})$, a product of nonzero complex numbers.

Now, let $F\subseteq E$ with $|F|<|E|$.

  1. Proof of (i)

Let $v\in V$ with $E(v)\not\subseteq F$. Since $|F\cup E(v)|>|F|$, (i) holds
for $F\cup E(v)$ by the inductive hypothesis. Since
$Z_{\phi}^{F}(h)=\sum_{\begin{subarray}{c}\psi:F\cup E(v)\rightarrow[q]\\\
\psi\mid_{F}=\phi\end{subarray}}Z_{\psi}^{F\cup E(v)}(h)$, Lemma 47 applied to
$Z_{\psi}^{F\cup E(v)}(h)$ yields (i) assuming that (ii) holds. We prove (ii)
below.

  2. Proof of (ii)

Let $u\in V$ and $\phi:F\rightarrow[q]$. If $E(u)\subseteq F$, then the claim
is trivially true since $\psi=\tilde{\psi}=\phi$. Otherwise, assume
$E(u)\not\subseteq F$ and let $\psi,\tilde{\psi}:F\cup E(u)\rightarrow[q]$
extend $\phi$. Since $|F\cup E(u)|>|F|$, (i) and (iii) hold for $F\cup E(u)$
by the inductive hypothesis. Applying Lemma 51 to $F\cup E(u)$ then yields
(ii).

  3. Proof of (iii)

Let $u\in V$ with $E(u)\subseteq F$. Without loss of generality, we may assume
such an $u$ exists since otherwise, there is nothing to prove. Let $v\in
N(u)\cup\\{u\\}$. If $E(v)\subseteq F$, then (iii) trivially holds with
$\cos(\theta/2)$ replaced by $1$, since there is only one term in the
summation, namely $\alpha=\phi\mid_{E(v)}$. Hence, assume $E(v)\not\subseteq
F$. In this case, $|F\cup E(v)|>|F|$ and so (i) holds for $F\cup E(v)$ by the
inductive hypothesis. Since (ii) for $F$ holds (as proved earlier), we may
then apply Lemma 52, yielding (iii) for $F$. ∎

##  Appendix B Proofs of Technical Lemmas

###### Proof of Lemma 34.

It was shown in [GLLZ21] that

$\Gamma=\left(-\overline{\mathbb{H}}_{\varepsilon}^{2}\right)^{\mathsf{c}}=\left\\{\rho
e^{i\theta}:\rho<\frac{2\varepsilon^{2}}{1-\cos\theta},\,0<\theta<2\pi\right\\}.$

To make this more interpretable, we rewrite the set in Cartesian coordinates.
If $z=\rho e^{i\theta}$, then by Euler’s formula we may write $z=x+iy$ where
$x=\rho\cos\theta$ and $y=\rho\sin\theta$. We then obtain

$\displaystyle\rho<\frac{2\varepsilon^{2}}{1-\cos\theta}$
$\displaystyle\Longleftrightarrow\qquad$
$\displaystyle\rho(1-\cos\theta)<2\varepsilon^{2}$
$\displaystyle\Longleftrightarrow\qquad$
$\displaystyle\rho<x+2\varepsilon^{2}$
$\displaystyle\Longleftrightarrow\qquad$ $\displaystyle
x^{2}+y^{2}<(x+2\varepsilon^{2})^{2}$ $\displaystyle\Longleftrightarrow\qquad$
$\displaystyle y^{2}<4\varepsilon^{2}(x+\varepsilon^{2}).$

Therefore, we see that

$\Gamma=\left\\{x+iy:y^{2}<4\varepsilon^{2}(x+\varepsilon^{2})\right\\},$

which clearly contains $\mathbb{R}_{+}$.

Furthermore, for $\lambda\in\mathbb{R}_{+}$ we have

$\displaystyle\mathrm{dist}(\lambda,\partial\Gamma)$
$\displaystyle=\inf_{z\in\partial\Gamma}|z-\lambda|$
$\displaystyle=\inf_{(x,y)\in\mathbb{R}^{2}:\,y^{2}=4\varepsilon^{2}(x+\varepsilon^{2})}\sqrt{(x-\lambda)^{2}+y^{2}}$
$\displaystyle=\inf_{x\in[-\varepsilon^{2},\infty)}\sqrt{(x-\lambda)^{2}+4\varepsilon^{2}(x+\varepsilon^{2})}$
$\displaystyle=\begin{cases}\lambda+\varepsilon^{2},&\lambda\in(0,\varepsilon^{2});\\\
2\varepsilon\sqrt{\lambda},&\lambda\in[\varepsilon^{2},\infty).\end{cases}$

This establishes the lemma. ∎