#  ALMA Lensing Cluster Survey:
Bright [C ii] 158 $\mu$m Lines from a Multiply Imaged Sub-$L^{\star}$ Galaxy
at $z=6.0719$

Seiji Fujimoto Cosmic Dawn Center (DAWN), Jagtvej 128, DK2200 Copenhagen N,
Denmark  Niels Bohr Institute, University of Copenhagen, Lyngbyvej 2, DK2100
Copenhagen Ø, Denmark  Masamune Oguri  Research Center for the Early Universe,
Graduate School of Science, The University of Tokyo, 7-3-1 Hongo, Bunkyo-ku,
Tokyo 113-0033, Japan  Department of Physics, The University of Tokyo, 7-3-1
Hongo, Bunkyo-ku, Tokyo 113-0033, Japan  Kavli Institute for the Physics and
Mathematics of the Universe (WPI), The University of Tokyo, 5-1-5 Kashiwanoha,
Kashiwa-shi, Chiba, 277-8583, Japan  Gabriel Brammer  Cosmic Dawn Center
(DAWN), Jagtvej 128, DK2200 Copenhagen N, Denmark  Niels Bohr Institute,
University of Copenhagen, Lyngbyvej 2, DK2100 Copenhagen Ø, Denmark  Yuki
Yoshimura  Department of Astronomy, Graduate School of Science, The University
of Tokyo, 7-3-1 Hongo, Bunkyo-ku, Tokyo 113-0033, Japan  Institute of
Astronomy, Graduate School of Science, The University of Tokyo, 2-21-1 Osawa,
Mitaka, Tokyo 181-0015, Japan  Nicolas Laporte  Kavli Institute for Cosmology,
University of Cambridge, Madingley Road, Cambridge CB3 0HA, UK  Cavendish
Laboratory, University of Cambridge, 19 JJ Thomson Avenue, Cambridge CB3 0HE,
UK  Jorge González-López  Núcleo de Astronomía de la Facultad de Ingeniería y
Ciencias, Universidad Diego Portales, Av. Ejército Libertador 441, Santiago,
Chile  Las Campanas Observatory, Carnegie Institution of Washington, Casilla
601, La Serena, Chile  Gabriel B. Caminha  Kapteyn Astronomical Institute,
University of Groningen, Postbus 800, 9700 AV Groningen, The Netherlands
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, School of Science, The University of Tokyo,
7-3-1 Hongo, Bunkyo-ku, Tokyo 113-0033, Japan  Adi Zitrin  Physics Department,
Ben-Gurion University of the Negev, P.O. Box 653, Be’er-sheva 8410501, Israel
Johan Richard  Univ Lyon, Univ Lyon1, Ens de Lyon, CNRS, Centre de Recherche
Astrophysique de Lyon UMR5574, F-69230, Saint-Genis-Laval,France  Masami Ouchi
National Astronomical Observatory of Japan, 2-21-1 Osawa, Mitaka, Tokyo
181-8588, Japan  Institute for Cosmic Ray Research, The University of Tokyo,
5-1-5 Kashiwanoha, Kashiwa, Chiba 277–8582, Japan  Kavli Institute for the
Physics and Mathematics of the Universe (WPI), The University of Tokyo, 5-1-5
Kashiwanoha, Kashiwa-shi, Chiba, 277-8583, Japan  Franz E. Bauer  Instituto de
Astrofısica, Facultad de Fısica, Pontificia Universidad Catolica de Chile Av.
Vicuna Mackenna 4860, 782-0436 Macul,Santiago, Chile  Millennium Institute of
Astrophysics (MAS), Nuncio Monse nor Santero Sanz 100, Providencia, Santiago,
Chile  Ian Smail  Centre for Extragalactic Astronomy, Department of Physics,
Durham University, South Road, Durham DH1 3LE, UK  Bunyo Hatsukade  Institute
of Astronomy, Graduate School of Science, The University of Tokyo, 2-21-1
Osawa, Mitaka, Tokyo 181-0015, Japan  Yoshiaki Ono  Institute for Cosmic Ray
Research, The University of Tokyo, 5-1-5 Kashiwanoha, Kashiwa, Chiba 277–8582,
Japan  Vasily Kokorev  Cosmic Dawn Center (DAWN), Jagtvej 128, DK2200
Copenhagen N, Denmark  Niels Bohr Institute, University of Copenhagen,
Lyngbyvej 2, DK2100 Copenhagen Ø, Denmark  Hideki Umehata  RIKEN Cluster for
Pioneering Research, 2-1 Hirosawa, Wako, Saitama 351-0198, Japan  Institute of
Astronomy, Graduate School of Science, The University of Tokyo, 2-21-1 Osawa,
Mitaka, Tokyo 181-0015, Japan  Daniel Schaerer  Observatoire de Genève,
Université de Genève, 51 Ch. des Maillettes, 1290 Versoix, Switzerland  CNRS,
IRAP, 14 Avenue E. Belin, 31400 Toulouse, France  Kirsten Knudsen  Department
of Space, Earth and Environment, Chalmers University of Technology, Onsala
Space Observatory, SE-43992 Onsala, Sweden  Fengwu Sun  Steward Observatory,
University of Arizona, 933 N. Cherry Ave, Tucson, AZ 85721, USA  Georgios
Magdis  Cosmic Dawn Center (DAWN), Jagtvej 128, DK2200 Copenhagen N, Denmark
Niels Bohr Institute, University of Copenhagen, Lyngbyvej 2, DK2100 Copenhagen
Ø, Denmark  DTU-Space, Technical University of Denmark, Elektrovej 327, DK2800
Kgs. Lyngby, Denmark  Francesco Valentino  Cosmic Dawn Center (DAWN), Jagtvej
128, DK2200 Copenhagen N, Denmark  Niels Bohr Institute, University of
Copenhagen, Lyngbyvej 2, DK2100 Copenhagen Ø, Denmark  Yiping Ao  Purple
Mountain Observatory and Key Laboratory for Radio Astronomy, Chinese Academy
of Sciences, Nanjing, China  Sune Toft  Cosmic Dawn Center (DAWN), Jagtvej
128, DK2200 Copenhagen N, Denmark  Niels Bohr Institute, University of
Copenhagen, Lyngbyvej 2, DK2100 Copenhagen Ø, Denmark  Miroslava Dessauges-
Zavadsky  Observatoire de Genève, Université de Genève, 51 Ch. des Maillettes,
1290 Versoix, Switzerland  Kazuhiro Shimasaku  Department of Astronomy,
Graduate School of Science, The University of Tokyo, 7-3-1 Hongo, Bunkyo-ku,
Tokyo 113-0033, 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  Karina Caputi  Kapteyn Astronomical Institute, University of Groningen,
P.O. Box 800, 9700AV Groningen, The Netherlands  Cosmic Dawn Center (DAWN),
Jagtvej 128, DK2200 Copenhagen N, Denmark  Haruka Kusakabe  Observatoire de
Genève, Université de Genève, 51 Ch. des Maillettes, 1290 Versoix, Switzerland
Kana Morokuma-Matsui  Institute of Astronomy, Graduate School of Science, The
University of Tokyo, 2-21-1 Osawa, Mitaka, Tokyo 181-0015, Japan  Kikuchihara
Shotaro  Department of Astronomy, Graduate School of Science, The University
of Tokyo, 7-3-1 Hongo, Bunkyo-ku, Tokyo 113-0033, Japan  Institute for Cosmic
Ray Research, The University of Tokyo, 5-1-5 Kashiwanoha, Kashiwa, Chiba
277–8582, Japan  Eiichi Egami  Steward Observatory, University of Arizona, 933
N. Cherry Ave, Tucson, AZ 85721, USA  Minju M. Lee  Max-Planck-Institut f ur
Extraterrestrische Physik (MPE), Giessenbachstr., D-85748, Garching, Germany.
Timothy Rawle  European Space Agency (ESA), ESA Office, Space Telescope
Science Institute, 3700 San Martin Drive, Baltimore, MD 21218, USA  Daniel
Espada  SKA Organisation, Lower Withington, Macclesfield, Cheshire SK11 9DL,
UK

(Received 2020 October 14; Revised 2020 December 14; Accepted 2020 December
29)

###### Abstract

We present bright [C ii] 158 $\mu$m line detections from a strongly magnified
and multiply-imaged ($\mu\sim 20$–160) sub–$L^{*}$ ($M_{\rm UV}$ =
$-19.75^{+0.55}_{-0.44}$) Lyman-break galaxy (LBG) at $z=6.0719\pm 0.0004$
from the ALMA Lensing Cluster Survey (ALCS). Emission lines are identified at
268.7 GHz at $\geq$ 8$\sigma$ exactly at positions of two multiple images of
the LBG behind the massive galaxy cluster RXCJ0600$-$2007\. Our lens models,
updated with the latest spectroscopy from VLT/MUSE, indicate that a sub region
of the LBG crosses the caustic and is lensed into a long ($\sim
6^{\prime\prime}$) arc with a local magnification of $\mu\sim 160$, for which
the [C ii] line is also significantly detected. The source-plane
reconstruction resolves the interstellar medium (ISM) structure, showing that
the [C ii] line is co-spatial with the rest-frame UV continuum at the scale of
$\sim$300 pc. The [C ii] line properties suggest that the LBG is a rotation-
dominated system whose velocity gradient explains a slight difference of
redshifts between the whole LBG and its sub region. The star formation rate
(SFR)–$L_{\rm[CII]}$ relations from the sub to the whole regions of the LBG
are consistent with those of local galaxies. We evaluate the lower limit of
the faint-end of the [C ii] luminosity function at $z=6$, and find that it is
consistent with predictions from semi analytical models and from the local
SFR–$L_{\rm[CII]}$ relation with a SFR function at $z=6$. These results imply
that the local SFR–$L_{\rm[CII]}$ relation is universal for a wide range of
scales including the spatially resolved ISM, the whole region of galaxy, and
the cosmic scale, even in the epoch of reionization.

galaxies: formation — galaxies: evolution — galaxies: high-redshift —
galaxies: ISM — galaxies: kinematics and dynamics — galaxies: luminosity
function

††journal: ApJ in press††software: casa (v5.4.0; McMullin et al. 2007), grizli
(Brammer et al. 2008), dendrogram (Goodman et al. 2009), galfit (Peng et al.
2010), eazy (Brammer et al. 2008), scarlet (Melchior et al. 2018), 3DBarolo(Di
Teodoro & Fraternali 2015), galpak3d (Bouché et al. 2015)

##  1 Introduction

Galaxy evolution is regulated by several key mechanisms in the interstellar
medium (ISM) such as disk formation, stellar and active galactic nuclei (AGN)
feedback, mass building via star formation and galaxy mergers, and clump
formations through disk instabilities. Resolving the ISM structure to study
local physical properties in high-redshift galaxies is thus essential in order
to understand the initial phase of galaxy formation and evolution.

During the past decades, hundreds of star-forming galaxies at $z>6$ have been
spectroscopically identified mainly with Ly$\alpha$ lines (e.g., Iye et al.
2006; Vanzella et al. 2011; Pentericci et al. 2011, 2014, 2018; Shibuya et al.
2012, 2018; Ono et al. 2012, 2018; Finkelstein et al. 2013; Oesch et al. 2015,
2016; Stark et al. 2017; Higuchi et al. 2019). The Atacama Large
Millimeter/submillimeter Array (ALMA) offers a rest-frame far-infrared (FIR)
spectroscopic window for these $z>6$ galaxies, especially with bright fine-
structure lines of [C ii] 158 $\mu$m and [O iii] 88 $\mu$m (e.g., Maiolino et
al. 2015; Inoue et al. 2016; Pentericci et al. 2016; Knudsen et al. 2016;
Matthee et al. 2017, 2019; Carniani et al. 2018; Smit et al. 2018; Bowler et
al. 2018; Hashimoto et al. 2018, 2019; Tamura et al. 2019; Fujimoto et al.
2019; Bakx et al. 2020). Since heavy elements produced in stars are returned
into the ISM, the metal gas properties traced by the fine-structure lines are
good probes of the star-formation history and related physical mechanisms
(Maiolino & Mannucci 2019). In fact, recent ALMA spatial and kinematic [C
ii]-line studies identify signatures of some key mechanisms, including disk
rotations (e.g., Jones et al. 2017; Smit et al. 2018), galaxy mergers (e.g.,
Hashimoto et al. 2019; Le Fèvre et al. 2020), and outflows (e.g., Gallerani et
al. 2018; Spilker et al. 2018; Fujimoto et al. 2019, 2020b; Ginolfi et al.
2020). In conjunction with other fine-structure lines of [O iii] and [N ii],
recent ALMA observations also allow us to perform multiple line diagnostics to
constrain the dominant ionization state of the ISM gas (e.g., Inoue et al.
2016; Pavesi et al. 2016; Laporte et al. 2019; Novak et al. 2019; Harikane et
al. 2020).

There are several challenges related to the FIR spectroscopy. The first is
sensitivity. While ALMA is the most sensitive mm/submm telescope and yielding
a large number of new findings about high-redshift galaxies, the detection of
FIR fine-structure lines from abundant, typical galaxies remains challenging.
For example, to observe a [C ii] line of $\sim 1\times$ 10${}^{8}\,L_{\odot}$
from $z=6$, about 2-hour observing time is required111Based on CASA Observing
Tool calculations to detect the [C ii] line of 1 $\times 10^{8}\,L_{\odot}$
with a line width of 200 km s-1 at $\geq 5\sigma$ in the velocity integrated
map.. However, such a source typically falls in the absolute UV magnitude
range of $M_{\rm UV}$ $\sim$ $-22.0$ – $-21.5$ mag (see e.g., Table 7 in
Hashimoto et al. 2019). This absolute UV magnitude range is $\sim$ 2–3 times
brighter than the characteristic luminosity $L^{*}$ in the UV luminosity
function at $z>6$ (e.g., Ono et al. 2018), indicating that $\gtrsim$ 10-hour
observing time is necessary to study the abundant, typical galaxies with
$L^{*}$ or sub-$L^{*}$ luminosities. The second challenge is high spatial
resolution observations towards these typical galaxies. Recent Hubble Space
Telescope (HST) studies report that the typical effective radius ($r_{\rm e}$)
in star-forming galaxies at $z>6$ is estimated to be $<$ 1 kpc ($\simeq
0\farcs 2$) (e.g., Holwerda et al. 2015; Shibuya et al. 2015; Bouwens et al.
2017; Kawamata et al. 2018). The ISM structure mostly comparable to the
$r_{\rm e}$ scale could be resolved by ALMA high-resolution observations down
to the $0\farcs 02$ scale. However, this requires even longer observing times
than $\gtrsim$ 10 hours estimated above just for the detection of the typical
galaxies. The third challenge is the requirement of prior spectroscopic
redshifts due to the narrow frequency coverage of ALMA (7.5-GHz coverage in a
single tuning), which may cause potential biases. In most cases, the prior
spectroscopic redshift is obtained from Ly$\alpha$ lines. While high-redshift
galaxies with Ly$\alpha$ spectroscopic redshifts show weak [C ii] lines at a
given star-formation rate (Carniani et al. 2018; Harikane et al. 2018, 2020;
cf. Schaerer et al. 2020), a recent study by Smit et al. (2018) indicates that
galaxies with no strong Ly$\alpha$ line may emit a strong [C ii] line. Because
the fraction of Ly$\alpha$ emitters (LAEs; e.g., equivalent width of
Ly$\alpha>$ 25 ${\rm\AA}$ ) is less than 30% among star-forming galaxies with
$M_{\rm UV}\sim-21.5$ mag at $z>6$ (e.g., Stark et al. 2011; Treu et al. 2013;
Tilvi et al. 2014; De Barros et al. 2017; Pentericci et al. 2018; Kusakabe et
al. 2020), follow-up observations of galaxies only with secure Ly$\alpha$
lines will systematically miss a majority of the representative population at
$z>6$. An ALMA blind line survey is one possible solution, but novel [C ii]
line emitters $z>6$ have not yet been identified due to the lack of
sufficiently deep and large survey volumes (e.g., Matsuda et al. 2015; Aravena
et al. 2016; Yamaguchi et al. 2017; Hayatsu et al. 2019; Yan et al. 2020;
Romano et al. 2020; Decarli et al. 2020).

In this paper, we report the blind detection of bright [C ii] 158-$\mu$m lines
from strongly lensed multiple images of a sub-$L^{*}$ galaxy at $z=6.0719$
behind the massive galaxy cluster RXCJ0600$-$2007, drawn from ALMA Lensing
Cluster Survey (ALCS). Making full use of large ancillary data sets, including
HST, Spitzer, and VLT and with help of gravitational lensing magnification, we
resolve the ISM structures and investigate the spatially resolved rest-frame
UV-to-FIR continuum and the [C ii] line properties down to a $\simeq$ 300 pc
scale. This is the first ALMA study to resolve the ISM properties in a
representative ($\simeq$ sub-$L^{*}$) galaxy in the epoch of reionization.

The structure of this paper is as follows. In Section 2, we overview the ALCS
survey and the data sets in RXCJ0600$-$2007 as well as strong lensing mass
models of the cluster. Section 3 outlines methods of the blind line
identification and optical–near infrared (NIR) properties of the two [C ii]
line emitters at $z=6.07$. In Section 4, we report and discuss intrinsic
characteristics of these two [C ii] line emitters with the correction of the
lensing magnification. A summary of this study is presented in Section 5.
Throughout this paper, we assume the Chabrier initial mass function (Chabrier
2003) and a flat universe with $\Omega_{\rm m}=0.3$, $\Omega_{\Lambda}=0.7$,
$\sigma_{8}=0.8$, and $H_{0}=70$ km s-1 Mpc-1. We use magnitudes in the AB
system (Oke & Gunn 1983).

Figure 1:  Left: False-color HST image of the cluster RXCJ0600$-$2007 (red:
F160W, green: F125W, blue: F814W). The white line denotes the critical curve
at $z=6.07$ estimated from our fiducial mass model. The green lines indicate
the ALCS area coverage in this cluster, within which the relative sensitivity
to the deepest part of the mosaic map is greater than 30%. The five multiple
image positions of RXCJ0600-$z$6 are marked with the yellow
$6^{\prime\prime}\times 6^{\prime\prime}$ squares. Middle: HST/F160W
$6^{\prime\prime}\times 6^{\prime\prime}$ cutouts for the multiple images of
$z$6.1/6.2, $z$6.3, and $z$6.4 from top to bottom. The red contours denote the
velocity-integrated [C ii] line intensity drawn at 1$\sigma$ intervals from
$\pm 2\sigma$ to $\pm$ 8$\sigma$. We use the natural-weighted map for
$z$6.1/6.2 and $z$6.3, while we use the $uv$-tapered ($1\farcs 8\times 1\farcs
8$) map for $z$6.4 to obtain the optimized S/N. The ALMA synthesized beams are
presented in the bottom left. The black bars indicate foreground sources. The
foreground source overlapping $z$6.1/6.2 is subtracted in our optical-NIR
analysis with galfit (see Section 3.2 and Appendix C. see also N. Laporte et
al. submitted). Right: [C ii] line spectra for $z$6.1/6.2, $z$6.3, and $z$6.4
from top to bottom that we obtain in the ALCS data cube. The yellow shade
indicates the [C ii] integration range for the velocity-integrated map whose
contours are shown in the middle panel. The blue curve is the best-fit single
Gaussian.

##  2 Data and Reduction

###  2.1 ALMA Lensing Cluster Survey

ALCS is a cycle-6 ALMA large program (Project ID: 2018.1.00035.L; PI: K.
Kohno) to map a total of 88-arcmin2 high-magnification regions in 33 massive
galaxy clusters at 1.2-mm in Band 6. The sample is selected from the best-
studied clusters drawn from HST treasury programs, i.e., the Cluster Lensing
And Supernova Survey with Hubble (CLASH; Postman et al. 2012), Hubble Frontier
Fields (HFF; Lotz et al. 2017), and the Reionization Lensing Cluster Survey
(RELICS; Coe et al. 2019). Observations were carried out between December 2018
and December 2019 in compact array configurations of C43-1 and C43-2 fine
tuned to recover strongly lensed (i.e., spatially elongated), low surface
brightness sources. The 1.2-mm mapping is accomplished with a 15-GHz wide
spectral scan in the ranges of 250.0–257.5 GHz and 265.0–272.5 GHz via two
frequency setups to enlarge the survey volume for line-emitting galaxies. The
spectral mode of Time Division Mode is used, which achieves the spectral
resolution of $\sim$28 km s-1 through these frequency setups. A full
description of the survey and of its main objectives will be presented in a
separate paper (in preparation).

###  2.2 RXCJ0600$-$2007

RXCJ0600$-$2007 is a massive ($\sim 10^{15}\,M_{\odot}$) galaxy cluster at
$z=0.43$ that is included in RELICS and was firstly identified in the Massive
Cluster Survey (MACS; Ebeling et al. 2001). As a part of ALCS, the ALMA
observations for RXCJ0600$-$2007 were performed in January 2019, mapping the
central area of $230^{\prime\prime}\times 90^{\prime\prime}$ in 105 pointings
with 46–49 12-m antennae providing baselines of 15–456 m under a precipitable
water vapor (PWV) of 0.6–1.3 mm. J0522-3627 was observed as a flux calibrator.
The bandpass and phase calibrations were performed with J0609-1542.

The ALMA data were reduced and calibrated with the Common Astronomy Software
Applications package version 5.4.0 (casa; McMullin et al. 2007) with the
pipeline script in the standard manner. With the CASA task tclean, continuum
maps were produced by utilizing all spectral windows. The tclean routines were
executed down to the 3$\sigma$ level. We adopted a pixel scale of $0\farcs 15$
and a common spectral channel bin of 30 km s-1. The natural-weighted map
achieved a synthesized beam FWHM of $1\farcs 22\times 0\farcs 95$ with
sensitivities in the continuum and the line in a 30-km s-1 width channel of
56.9 and 932 $\mu$Jy beam-1, respectively. We also produced several
$uv$-tapered maps in a parameter range of $0\farcs 8\times 0\farcs 8$ to
$1\farcs 8\times 1\farcs 8$ to obtain spatially integrated properties when
necessary. Throughout the paper, we used the natural-weighted map unless
mentioned otherwise.

HST/ACS–WFC3 and Spitzer/IRAC observations were carried out as a part of
RELICS (Coe et al. 2019) and Spitzer-RELICS (Strait et al. 2020) surveys,
respectively. HST images were obtained in the F606W (2180 s), F814W (3565 s),
F105W (1411 s), F125W (711 s), F140W (736 s), and F160W (1961 s) filters. The
IRAC channel 1 ($3.6~{}\mu\mathrm{m}$) and channel 2 ($4.5~{}\mu\mathrm{m}$)
integrations are approximately 10 hours each. We aligned all of the HST
exposures to sources in the PanSTARRS (DR1) catalog (Chambers et al. 2016;
Flewelling et al. 2016)—which we verified is consistent with the GAIA DR2
(Gaia Collaboration et al. 2018) astrometric frame— and created final mosaics
in a common pixel frame with 50 mas and 100 mas pixels for the ACS/WFC and
WFC3/IR filters respectively. We aligned the individual Spitzer exposures to
the same astrometric frame and generated final drizzled IRAC mosaics with a
pixel scale of $0\farcs 5$. Further details of the HST (Spitzer) image
processing with the grizli (golfir) software will be presented in Kokorev et
al. (in prep). In Figure 1, we present the false-color HST image of
RXCJ0600$-$2007.

Table 1: Observed FIR properties of bright [C ii] line emitters identified in
RXCJ0600$-$2007

Name |  $z$6.1/6.2 (arc)† |  $z$6.3 |  $z$6.4 |  $z$6.5‡
---|---|---|---|---
R.A. | 06:00:09.13 | 06:00:09.55 | 06:00:08.58 | 06:00:05.55
Dec. |  $-$20:08:26.49 |  $-$20:08:11.26 |  $-$20:08:12.54 |  $-$20:07:20.86
S/N | 9.2 | 8.0 | 3.0 | –
$\nu_{\rm center}$ [GHz] | 268.682 $\pm$ 0.011 | 268.744 $\pm$ 0.016 | (268.744)†† | –
FWHM [km s-1] | 169 $\pm$ 22 | 181 $\pm$ 34 | (181)†† | –
$z_{\rm[CII]}$ | 6.0736$\pm$0.0003 | 6.0719$\pm$0.0004 | (6.0719)†† | –
$S_{\rm[CII]}$ [Jy km s-1] | 4.83 $\pm$ 0.62 | 2.75 $\pm$ 0.20 | 0.44 $\pm$ 0.20 | –
$L_{\rm[CII]}$ [$\times 10^{9}\,L_{\odot}$] | 4.5 $\pm$ 0.40 | 2.3 $\pm$ 0.21 | 0.42 $\pm$ 0.19 | –
$f_{\rm 1.2mm}$ [mJy] | 0.35 $\pm$ 0.08 | 0.20 $\pm$ 0.08 |  $<$ 0.16 | –
[C ii] major-axis [′′] | 4.24 $\pm$ 0.82 | 1.17 $\pm$ 0.29 | –†† | –
[C ii] minor-axis [′′] | 0.63 $\pm$ 0.51 | 0.88 $\pm$ 0.43 | –†† | –
[C ii] position angle [∘] | 71 $\pm$ 5 | 8 $\pm$ 430 | –†† | –

† This source is called also as RXCJ0060-arc in N. Laporte et al. (submitted).
†† We do not perform any profile fitting to the spectrum and the 2D spatial
map of $z$6.4 due to its faintness. We adopt the FWHM and the peak frequency
based on $z$6.3 for calculating the velocity-integrated intensity of the line.
‡ $z$6.5 falls outside of the ALCS area coverage.

VLT/MUSE integral field spectroscopy of the RXCJ0600$-$2007 field was obtained
on 26th January 2018 (ESO program ID 0100.A-0792, P.I.: A. Edge). The 0.8-hour
observation was split in three exposures of $\rm 970~{}seconds$ each, centered
on the brightest cluster galaxy (BCG) covering $\rm 1~{}arcmin^{2}$ of the
cluster core. We use the standard MUSE reduction pipeline version 2.8.1
(Weilbacher et al. 2014) to create the final data-cube. In this process, we
used the self-calibration method based on the MUSE Python Data Analysis
Framework (Bacon et al. 2016; Piqueras et al. 2017) and implemented in this
version of the reduction pipeline. Finally, we applied the Zurich Atmosphere
Purge (ZAP, Soto et al. 2016) to remove the sky residuals that were not
completely removed by the MUSE pipeline.

We used the MUSE data cube to build our redshift catalog in two steps, similar
to Caminha et al. (2017, 2019). We first extracted the spectra of all sources
detected in the HST imaging, and in a second step, we performed a blind search
for faint-line emitters. This procedure allowed us to measure 76 secure
redshifts, of which 16 are emission from galaxies behind the cluster. This
redshift catalogue was used to identify cluster members and multiply imaged
galaxies that were used in strong lens mass modeling (see Section 3.3 for more
details). In Appendix A, we summarize the full spectroscopic sample from MUSE.

##  3 Data analysis

###  3.1 Line Identification

We conduct a blind line search in the ALMA data cubes with the channel widths
of 30 km s-1 and 60 km s-1. First, we produce three-dimensional signal-to-
noise ratio (S/N) cubes by dividing each channel with its standard deviation.
Here we use the ALMA data cubes before the primary beam correction. We then
search line candidates in the three-dimensional S/N cube by utilizing a
python-base software of dendrogram (Goodman et al. 2009) whose algorithm is
similar to clumpfind (Williams et al. 1994). In dendrogram, we obtain an
initial candidate catalog of line sources that meet the following criteria: at
least 10 pixels and/or channels with a pixel value of $\geq$ 2 (i.e., S/N
$\geq$ 2). Performing the same procedure in the negative peaks in the S/N
cubes under the assumption that the noise is Gaussian, dendrogram evaluates
the reliability of the initial line candidates based on the positive and
negative properties of the peak S/N histograms, spatially integrated pixel
values, and the channel width. This results in two reliable, bright line
emitters both at $\sim$ 268.7 GHz. We note that these two lines are also
robustly identified with an independent blind line search method of González-
López et al. (2017). Based on morphological, redshift, and gravitational lens
properties of these two line emitters obtained in detail analyses in the
following subsections (Section 3.2, 3.3, and 3.4), we refer to these two line
emitters as $z$6.1/6.2 (=$z6.1$ and $z$6.2) and $z$6.3 throughout this paper.

In Figure 1, we present the ALMA spectra and the velocity-integrated intensity
(i.e., moment 0) maps of $z$6.1/6.2 and $z$6.3. $z$6.1/6.2 shows an elongated
morphology with two peaks in the moment 0 map. Although there is a possibility
that a combination of the diffuse continuum and the noise fluctuation causes
multiple peaks (e.g., Hodge et al. 2016), we confirm in Appendix B that the
two peaks in $z$6.1/6.2 are not caused by this combination through a realistic
simulation. A single Gaussian fit to $z$6.1/6.2 and $z$6.3 in the line spectra
is summarized in Table 1. Although we obtain consistent full-width-at-half-
maximum (FWHM) values for the line widths between $z$6.1/6.2 and $z$6.3, their
frequency peaks are slightly different by 69 $\pm$ 22 km s-1. After
integrating over a velocity range of 1.5 $\times$ FWHM, $z$6.1/6.2 and $z$6.3
have S/N values of 9.2 and 8.0 at the peak pixels, respectively. A single
elliptical Gaussian fit over a spatial area of $6^{\prime\prime}\times
6^{\prime\prime}$ in the velocity-integrated maps with the CASA task of imfit
yields deconvolved spatial FWHM sizes of $4\farcs 24\times 0\farcs 82$ and
$1\farcs 17\times 0\farcs 29$ for $z$6.1/6.2 and $z$6.3, respectively. To
obtain the integrated property, here we use a $uv$-tapered ($1\farcs 0\times
1\farcs 0$) map for $z$6.1/6.2 in imfit. From line free channels, the
continuum is also detected in the $uv$-tapered map ($2\farcs 0\times 2\farcs
0$) at $4.5\sigma$ and $2.5\sigma$ level from $z$6.1/6.2 and $z$6.3,
respectively. We also summarize the imfit results and the continuum flux
density in Table 1. Further analyses for the continuum emission are presented
in N. Laporte et al. (submitted).

Table 2: Observed HST and IRAC photometry of the multiple images of
RXCJ0600-$z$6

ID | F606W | F814W | F105W | F125W | F140W | F160W | 3.6$\mu$m | 4.5 $\mu$m
---|---|---|---|---|---|---|---|---
| ($\mu$Jy) | ($\mu$Jy) | ($\mu$Jy) | ($\mu$Jy) | ($\mu$Jy) | ($\mu$Jy) | ($\mu$Jy) | ($\mu$Jy)
$z$6.1/6.2 (arc) |  $<$ 0.07 | 0.32 $\pm$ 0.04 | 1.17 $\pm$ 0.07 | 1.41 $\pm$ 0.13 | 1.42 $\pm$ 0.11 | 1.34 $\pm$ 0.07 | 8.18 $\pm$ 0.42 | 6.02 $\pm$ 0.34
$z$6.3 | 0.05 $\pm$ 0.02 | 0.27 $\pm$ 0.02 | 0.99 $\pm$ 0.03 | 1.18 $\pm$ 0.06 | 1.17 $\pm$ 0.05 | 1.19 $\pm$ 0.03 | 5.46 $\pm$ 0.20 | 4.17 $\pm$ 0.17
$z$6.4 |  $<$ 0.02 | 0.07 $\pm$ 0.02 | 0.13 $\pm$ 0.03 | 0.14 $\pm$ 0.05 | 0.17 $\pm$ 0.04 | 0.13 $\pm$ 0.03 | $<0.59$ |  $<$ 0.46†
$z$6.5 |  $<$ 0.02 | 0.19 $\pm$ 0.02 | 0.51 $\pm$ 0.03 | 0.63 $\pm$ 0.05 | 0.66 $\pm$ 0.04 | 0.68 $\pm$ 0.03 | 0.40 $\pm$ 0.18 | 0.50 $\pm$ 0.15

$\dagger$ We obtain 0.34 $\pm$ 0.23 $\mu$Jy which we replace the 2$\sigma$
upper limit.

Figure 2:  Observed optical-NIR properties of $z$6.1/6.2, $z$6.3, $z$6.4, and
$z$6.5 that are predicted as multiple images of a background LBG at $z\sim 6$
consistently from different mass models (except for z6.5 whose identification
as a multiple image is tentative). Top: Cutouts of the HST
($3^{\prime\prime}\times 3^{\prime\prime}$, except for $z$6.1/6.2 with
$6^{\prime\prime}\times 6^{\prime\prime}$ ) and Spitzer
($8^{\prime\prime}\times 8^{\prime\prime}$) images. Some of the HST images are
integrated one. The filter name is presented at the top. Bottom: HST and
Spitzer photometry (black square) and the best-fit templates, where gray
triangles are the upper limits. The sum of individual eazy templates is shown
in the light blue curve. The yellow shaded region is the probability
distribution of the photometric redshift $p(z)$ from the SED fit. The red line
indicates the spectroscopic redshift from the [C ii] lines at $\sim$268.7 GHz.
A 3.6-$\mu$m excess feature in $z$6.1/6.2 and $z$6.3 is explained by the
contamination of the strong [O iii]$\lambda$5007 and H$\beta$ lines that are
often observed in $z\sim 6$ galaxies (e.g., Roberts-Borsani et al. 2016;
Harikane et al. 2018).

###  3.2 Optical-NIR Counterparts

The bright lines of $z$6.1/6.2 and $z$6.3 at $\sim$268.7 GHz could be CO or [C
ii] (e.g., Decarli et al. 2020). To determine which line corresponds to
$z$6.1/6.2 and $z$6.3, we investigate their optical to near-infrared (NIR)
properties. In the top panel of Figure 2, we show optical-NIR HST images
around $z$6.1/6.2 and $z$6.3. From the line peak positions, we identify clear
counterparts in the optical-NIR images within a spatial offset of $\sim
0\farcs 1$ for both $z$6.1/6.2 and $z$6.3. Both counterparts have a noticeable
dropout feature blueward of $\sim 1\,\mu$m, and the one near $z$6.1/6.2 shows
a highly elongated shape aligned with the elongated shape in the 268.684-GHz
line. In the highly elongated object near $z$6.1/6.2, we identify a compact
source at the center whose optical-NIR color is distinct from the other parts
of the elongated object and indicative of an overlapping foreground object by
chance. We carefully model and subtract this foreground object (see Appendix
C) to study the elongated object near $z$6.1/6.2 in the following analysis. In
the northeast from $z$6.3, we also identify a nearby compact object. This has
a photometric redshift of 0.50${}^{+0.05}_{-0.32}$ (Kokorev et al. in prep.),
presumably one of the member galaxies of RXCJ0600-2007 ($z=0.43$), but does
not affect the photometry of $z$6.3.

We conduct optical-NIR photometry and spectral energy distribution (SED)
analyses for these counterparts. We perform the aperture photometry and
summarize results in Table 2. The detail procedure of the aperture photometry
is described in Appendix C. With the aperture photometry results, we conduct
SED fitting using the eazy code (Brammer et al.
2008)222http://github.com/gbrammer/eazy-py. We fit the photometric flux
densities and their uncertainties with linear combinations of templates
derived following Brammer et al. (2008) but adopting Flexible Stellar
Population Synthesis models as the basis (Conroy et al. 2009; Conroy & Gunn
2010). We adopt the dust attenuation law of Kriek & Conroy (2013) with
$\delta=0$ (i.e., a Calzetti et al. 2000 shape with an additional 2175
${\rm\AA}$ dust feature).

In Figure 2, we show probability distributions of photometric redshifts for
the optical-NIR counterparts of $z$6.1/6.2 and $z$6.3. We find that $z$6.1/6.2
and $z$6.3 both have peak probabilities close to $z=6.0$, in excellent
agreement with the bright-line detection at $\sim$ 268.7 GHz if it is the [C
ii] 158 $\mu$m line at $z=6.07$. In this case, observed line luminosities
$L_{\rm line}$ (i.e., without the correction of the lensing magnification) are
estimated to be 4.5 $\pm$ 0.4 $\times 10^{9}\,L_{\odot}$ and 2.3 $\pm$ 0.2
$\times 10^{9}\,L_{\odot}$ for $z$6.1/6.2 and $z$6.3, respectively. With a
standard modified blackbody at $z=6$ with a peak dust temperature $T_{\rm d}=$
38 K (e.g., Faisst et al. 2020) and a dust emissivity index $\beta_{\rm
d}=1.8$ (e.g., Chapin et al. 2009; Planck Collaboration et al. 2011), we also
obtain the observed values of rest-frame FIR luminosities $L_{\rm FIR}$ to be
5.8 $\pm$ 1.3 $\times 10^{11}\,L_{\odot}$ and 3.3 $\pm$ 1.3 $\times
10^{11}\,L_{\odot}$ and subsequently line to rest-frame FIR luminosity ratios
to be 7.8 $\times 10^{-3}$ and 6.8 $\times 10^{-3}$, for $z$6.1/6.2 and
$z$6.3, respectively. These ratios are consistent with the typical range of
the [C ii] line $L_{\rm[CII]}$ and $L_{\rm FIR}$ ratio ($L_{\rm[CII]}/L_{\rm
FIR}$) among local galaxies (e.g., Brauher et al. 2008; Díaz-Santos et al.
2013), which also supports the bright lines at $\sim$ 268.7 GHz being the [C
ii] line. Based on the source redshift at $z=6.07$, we also confirm that star-
formation rate (SFR) estimates are consistent between the SED fitting with the
dust attenuation correction and the summation of the rest-frame UV ($L_{\rm
UV}$) and $L_{\rm FIR}$ following the work of Bell et al. (2005) scaled to the
Chabrier IMF,

${\rm SFR}\,[M_{\odot}\,{\rm yr^{-1}}]=1.09\times 10^{-10}(L_{\rm
FIR}+2.2L_{\rm UV}).$ (1)

Although the $z$6.3 solution shows a small non-zero probability of being at
$z\sim 1$, we also identify a 3.6-$\mu$m excess feature in both $z$6.1/6.2 and
$z$6.3 which is often observed in $z\sim 6$ galaxies due to the contamination
of the strong [O iii]$\lambda$5007 and H$\beta$ lines (e.g., Roberts-Borsani
et al. 2016; Harikane et al. 2018). Therefore, the high-$z$ solution at $z\sim
6$ is likely favored. Other possibilities for the bright line along with the
high-$z$ solution might be CO(16-15) at $z=5.85$ and CO(17-16) at $z=6.28$.
However, recent ALMA studies derive constraints on ratios of $L_{\rm
CO(16-15)}/L_{\rm FIR}$ and $L_{\rm CO(17-16)}/L_{\rm FIR}$ $\lesssim$
3$\times 10^{-4}$ among luminous quasars at similar redshifts (Carniani et al.
2019). This indicates that $L_{\rm line}/L_{\rm FIR}$ of $z$6.1/6.2 and $z$6.3
are nearly 1.5-dex higher than the typical range, strongly disfavoring the
possibilities of CO(16-15) and CO(17-16) lines. We thus conclude that
$z$6.1/6.2 and $z$6.3 are [C ii] line emitters at $z=6.07$. Note that we
confirm that the [C ii] line solution is further supported by the lens models,
intrinsic physical properties (see Section 3.4 and 3.5), and follow-up
Gemini/GMOS spectroscopy (N. Laporte et al. submitted).

Based on the redshift of $z=6.07$, we also examine the Ly$\alpha$ line in the
MUSE data cube around $z$6.1/6.2 and $z$6.3. We do not identify any Ly$\alpha$
features neither around $z$6.1/6.2 nor $z$6.3. With the rest-frame UV
luminosity, this provides 3$\sigma$ upper limits of the Ly$\alpha$ equivalent
width ($EW_{\rm Ly\alpha}$) at 4.4 ${\rm\AA}$ and 3.7 ${\rm\AA}$ for
$z$6.1/6.2 and $z$6.3, respectively. Given the dust continuum detection and
the redshift, the absence of the bright Ly$\alpha$ line would be ascribed to
dust and/or neutral hydrogen in interstellar and intergalactic media. This
emphasizes the importance of the ALMA blind line search which enable studies
of galaxies irrespective of their Ly$\alpha$ line properties in particular in
the epoch of reionization.

###  3.3 Mass Model

To study intrinsic physical properties of the [C ii] line emitters $z$6.1/6.2
and $z$6.3, we construct several mass models for the galaxy cluster
RXCJ0600–2007 ($z=0.430$), using independent algorithms including glafic
(Oguri 2010), Lenstool (Jullo et al. 2007), and Light-Traces-Mass (LTM; Zitrin
et al. 2015). Multiple images are selected based on the morphology and colors
of galaxies in the HST images taken with RELICS, guided by y mass models.
These models also exploit the MUSE spectroscopic redshift catalog (see Section
2.2) for redshift information of some multiple image systems as well as secure
identifications of cluster member galaxies. These models adopt nearly
identical sets of multiple image systems for constructing the mass models and
provide almost consistent predictions for multiple image positions and
magnification factors. A brief summary of these mass models is also presented
in N. Laporte et al. (submitted), while full details will be given in a
separate paper (in preparation). In this paper, we adopt the mass model of
glafic as a fiducial model for our analyses and here describe its construction
below, although we also use results of Lenstool and LTM models to evaluate
uncertainties in magnification factors.

We construct the mass model with glafic in the same manner as in Kawamata et
al. (2016). Our mass model consists of cluster-scale halos and cluster member
galaxies. We place the cluster-scale halos at the positions of the three
brightest cluster member galaxies in the core of the cluster. The position of
one of the three cluster-scale halos is treated as a free parameter, whereas
those of the other two cluster-scale halos are fixed to the galaxy positions.
The cluster-scale halos are modeled by an elliptical Navarro-Frenk-White (NFW;
e.g., Navarro et al. 1997) profile. The cluster member galaxies are selected
using both photometric redshifts of galaxies measured from HST images (Coe et
al. 2019) as well as galaxy colors. The position and shapes of the member
galaxies are fixed to those derived from the HST image and treat their
velocity dispersions and truncation radii using a pseudo-Jaffe ellipsoid as
model parameters assuming a scaling relation (see Kawamata et al. 2016, for
more details). In order to achieve a good fit, a member galaxy located at
(R.A., Dec.)=(06:00:10.664, $-20$:06:50.65) that produces multiple images is
treated as a separate component, again assuming a pseudo-Jaffe ellipsoid. An
external shear term, which provides a modest improvement of the mass modeling
result, is also included in the mass modeling of this cluster. After including
the multiple images presented in Section 3.4 that are confirmed with help of
our preliminary mass models, there are positions of 26 multiple images for
eight sets of multiple image systems (five multiple image sets with
spectroscopic redshifts) that we adopt as constraints. We optimize the
parameters of the mass model based on a standard $\chi^{2}$ minimization and
determine the best-fit mass model assuming a positional error of $0\farcs 6$
for each multiple image to account for perturbations by substructures in the
cluster that are not included in our mass model, and estimate the statistical
error using the Markov-chain Monte Carlo method. Our best fitting model has
$\chi^{2}=20.0$ for 17 degree of freedom. Interested readers are referred to
Kawamata et al. (2016) for more specific mass modeling procedures using
glafic.

Note that we do not include the foreground object overlapping $z$6.1/6.2 in
our mass models, because of the absence of its spectroscopic redshift. We will
discuss the potential contribution of the foreground object to the morphology
and magnification factor of $z$6.1/6.2 in Section 4.1.

###  3.4 Multiple Images

Figure 3:  Observed SEDs of the multiple images of RXCJ0600-$z$6 in the HST
bands normalized by the F125W band. The green, red, brown, and blue squares
present $z$6.1/6.2, $z$6.3, $z$6.4, and $z$6.5, respectively.

From all of our mass models, we consistently obtain the following two
predictions: i) $z$6.1/6.2 consists of a pair of two multiple images of a
galaxy at $z\sim 6$ behind RXCJ0600-2007, and ii) $z$6.1/6.2 and $z$6.3 are
also multiple images of the galaxy. The prediction of ii) is consistent with
the [C ii] morphology that has two close peaks. In fact, we confirm in
Appendix D that [C ii] line spectra produced at these two peaks show line
profiles consistent with each other. In addition, almost the same optical-NIR
SED shapes between $z$6.1/6.2 and $z$6.3 in Figure 2 support the prediction of
ii). Although we identify the slight velocity shift between $z$6.1/6.2 and
$z$6.3 by 69 $\pm$ 22 km s-1 (see Section 3.1), the offset is much smaller
than the typical FWHM range of the [C ii] line among $z\sim$ 4–6 galaxies
evaluated in the ALPINE survey ($\sim$120–380 km s-1; Béthermin et al. 2020),
suggesting that the slight velocity shift is explained by the differential
magnification at different regions of the galaxy. We thus interpret $z$6.1/6.2
and $z$6.3 as multiple images of the [C ii] line emission at $z=6.07$ from a
Lyman-break galaxy (LBG) behind RXCJ0600-2007. Hereafter we refer to the
background LBG as RXCJ0600-$z$6.

Table 3: Observed Physical Properties of the multiple images of RXCJ0600-$z$6

ID | $z_{\rm phot}$ | $z_{\rm spec}$ | $M_{\rm UV}$ | SFR | $M_{\rm star}$ | $A_{\rm v}$ | $\mu_{\rm whole}^{\dagger}$ | $\mu_{\rm local}^{\dagger}$
---|---|---|---|---|---|---|---|---
|  |  | (mag) | ($M_{\odot}$ yr-1) | ($\times 10^{9}\,M_{\odot}$) | (mag) |  |
$z$6.1/6.2 (arc) | 5.95${}^{+0.11}_{-0.13}$ | 6.0734 $\pm$ 0.0003 |  $-$23.23 $\pm$ 0.07 | 135${}^{+45}_{-23}$ | 41.9${}^{+0.4}_{-10.1}$ | 0.07${}^{+0.11}_{-0.05}$ | 29${}^{+4}_{-7}$ | 163${}^{+27}_{-13}$
$z$6.3 | 5.99${}^{+0.07}_{-0.09}$ | 6.0719 $\pm$ 0.0004 |  $-$23.06 $\pm$ 0.06 | 114${}^{+11}_{-26}$ | 20.1${}^{+1.8}_{-2.7}$ | 0.18${}^{+0.09}_{-0.11}$ | 21${}^{+14}_{-7}$ | –
$z$6.4 | 5.25${}^{+0.48}_{-4.24}$ | (6.0719) |  $-$21.02 $\pm$ 0.11 | 2.6${}^{+1.2}_{-0.1}$ | 0.23${}^{+0.12}_{-0.01}$ | 1.80${}^{+2.25}_{-1.78}$ | 3.3${}^{+2.4}_{-1.2}$ | –
$z$6.5 | 6.05${}^{+0.07}_{-0.07}$ | – |  $-$22.48 $\pm$ 0.05 | 8.7${}^{+0.7}_{-0.1}$ | 0.77${}^{+0.05}_{-0.01}$ | 0.03${}^{+0.12}_{-0.01}$ | 4.2${}^{+1.8}_{-1.3}$ | –

† We define $\mu_{\rm whole}$ and $\mu_{\rm local}$ as follows:
$\mu_{\rm whole}=$ (observed luminosity of the multiple image) / (overall
luminosity of the intrinsic galaxy)
$\mu_{\rm local}=$ (observed luminosity of the multiple image) / (local
luminosity of the strongly lensed, sub region near the caustic line),
where the sub region corresponds to the dashed rectangle area in Figure 4. The
errors are evaluated from the minimum to maximum range among our independent
mass models.

Subsequently, the different models also predict two additional multiple images
of RXCJ0600-$z$6, the positions of which we present in Figure 1, where we
identify corresponding optical-NIR objects. We refer to these potential
multiple images as $z$6.4 and $z$6.5. Note that different mass models predict
a consistent position for $z$6.4, while they predict different positions for
$z$6.5 with a scatter in a $\sim 12^{\prime\prime}$ scale. In this paper, we
focus an optical-NIR object as $z$6.5 predicted by one of our mass models, but
it should be regarded as tentative, as we will discuss below.

To investigate whether multi-wavelength properties of these potential multiple
images are similar to $z$6.1/6.2 and $z$6.3, we conduct the aperture
photometry for $z$6.4 and $z$6.5 in the optical-NIR bands. The detail
procedure of the aperture photometry is again described in Appendix C. In
Table 2 and Figure 3, we summarize the photometry results and the optical-NIR
colors normalized by the photometry at the F125W band, respectively. We find
that $z$6.4 and $z$6.5 have similar optical-NIR SEDs with $z$6.1/6.2, $z$6.3
within the errors, consistent with our mass model predictions as multiple
images at $z\sim 6$. We further perform the optical-NIR SED fitting to $z$6.4
and $z$6.5 in the same manner as $z$6.1/6.2 and $z$6.3. In Figure 2, we also
show the optical-NIR SED fitting results of $z$6.4 and $z$6.5. While the
$z$6.4 photometry also allows for a $z\sim 6$ solution, the possibility of
much lower redshifts cannot be excluded due to the large uncertainties from
its faint property and the potential contamination of the nearby BCG (see
Appendix C.3). $z$6.5 has a well-localized peak probability at $z\sim 6$,
though the HST-Spitzer color of $z$6.5 is much bluer than seen for the bright
images $z$6.1/6.2 and $z$6.3. In fact, an IRAC source at the location of
$z$6.5 should be easily detected if it has with the same color as those of the
other images.

Because $z$6.4 falls in the ALCS area coverage, we also examine whether the [C
ii] line emission is detected from $z$6.4 with a frequency consistent with
$z$6.1/6.2 and $z$6.3. In Figure 1, we also show the ALMA Band 6 spectrum of
$z$6.4 based on an optimized aperture with a radius of $1\farcs 5$. We find
that $z$6.4 has a tentative line detection (3.0$\sigma$) at the consistent
frequency with $z$6.1/6.2 and $z$6.3. Moreover, $z$6.4 has an asymmetry line
profile (the brighter peak at the higher frequency side) which is consistent
with the line profile of $z$6.3. These results strengthen the case that $z$6.4
is indeed one of the multiple images of the background LBG.

Based on these results, we find that the identification of $z$6.4 as one of
the multiple images is relatively secure from the consistent predictions of
the mass models as well as the line detection at the consistent frequency. On
the other hand, from the different predicted positions among different mass
models and the disagreements in the HST-Spitzer color with other multiple
images, the interpretation of $z$6.5 being another multiple image is not
secure and should be taken with caution until a spectroscopic redshift is
obtained in follow-up observations. We thus use the positions of $z$6.1/6.2,
$z$6.3, and $z$6.4 as constraints in deriving our best-fit mass models. We
present the critical curve at $z=6.07$ from the best-fit mass model of glafic
in the left panel of Figure 1. We summarize the [C ii] line properties and the
SED fitting results for all these multiple images in Table 1 and Table 3,
respectively.

###  3.5 Physical Properties of RXCJ0600-$z$6

Figure 4:  Left: The best-fit 2D Sérsic profile (effective radius in major axis = 1.2 kpc, axis ratio = 0.49, Sérsic index $n=$ 2.5) and coordinate (R.A., Decl. = 6:00:08.12, $-$20:07:39.55) of the lensed LBG RXCJ0600-$z$6 at $z=6.0719$ in the source plane based on the fiducial mass model. The fitting is performed based on the standard $\chi^{2}$ minimization only with the $1\farcs 7\times 1\farcs 7$ HST/F160W cutout of $z$6.3. The white dashed curves denote the caustic lines at the source redshift. Middle: The multiple images of RXCJ0600-$z$6 in the image plane, $z$6.1/6.2, $z$6.3, and $z$6.4, predicted by the fiducial mass model with the best-fit 2D Sérsic profile in the left panel. The black boxes show the $6^{\prime\prime}\times 6^{\prime\prime}$ areas around $z$6.3 and $z$6.4 at the same positions as the yellow boxes in Figure 1. The dashed black rectangle denotes the $6^{\prime\prime}\times 17^{\prime\prime}$ area around $z$6.1$z$6.2, which approximately corresponds to the dashed black rectangle shown in the left panel along the caustic line. The white line indicate the critical curve. Right: The HST/F160W image showing the multiple images of $z$6.1/6.2, $z$6.3, and $z$6.4. The image size is the same as the middle panel. The color and symbols are assigned in the same manner as the middle panel.  Table 4: Intrinsic physical properties of strongly lensed LBG of RXCJ0600-$z$6 Name | RXCJ0600-$z$6
---|---
Region | Whole | Sub
Counter image |  $z$6.3 |  $z$6.1/6.2
| (1) | (2)
RA | 06:00:08.11 | 06:00:08.13
Decl. |  $-$20:07:39.65 |  $-$20:07:39.53
$z_{\rm spec}$ | 6.0719$\pm$0.0004 | 6.0734$\pm$0.0003†
EW${}^{\rm rest}_{Ly\alpha}$ [${\rm\AA}$] | $<4.4$ | $<3.7$
$L_{\rm[CII]}$ [$\times 10^{8}\,L_{\odot}$] | 1.1${}^{+0.7}_{-0.5}$ | 0.3${}^{+0.1}_{-0.1}$
$M_{\rm UV}$ [mag] |  $-$19.75${}^{+0.55}_{-0.44}$ | $-17.70^{+0.17}_{-0.09}$
SFR [$M_{\odot}$ yr-1] | 5.4${}^{+4.5}_{-2.9}$ | 0.8${}^{+0.4}_{-0.2}$
$M_{\rm star}$ [$\times 10^{8}\,M_{\odot}$] | 9.6${}^{+6.0}_{-4.6}$ | 2.6${}^{+0.2}_{-1.0}$
$A_{\rm v}$ [mag] | 0.18${}^{+0.09}_{-0.11}$ | 0.07${}^{+0.11}_{-0.05}$
$r_{\rm e}$ [kpc] | 1.2 ${}^{+4.1}_{-0.1}$ | –
$n$ | 2.5${}^{+1.2}_{-0.1}$ | –
axis ratio | 0.49${}^{+0.03}_{-0.02}$ | –
PA [∘] | 84${}^{+2}_{-2}$ | –
$M_{\rm dyn}$ [$\times 10^{9}\,M_{\odot}$] | 3 $\pm$ 1 | –
$M_{\rm gas}$ [$\times 10^{9}\,M_{\odot}$] | 2 $\pm$ 1 | –
$f_{\rm gas}$ [%] |  $\sim$ 50–80 | –
Figure 5:  SFR–$M_{\rm star}$ relation. The red filled and open circles
indicate the relations before and after applying the correction of the lensing
magnification to the SED fitting results, respectively, for $z$6.1/6.2 and
$z$6.3. The errors include the uncertainty from the mass models (Tabel 3). The
red open squares indicate SFR estimates after the lensing magnification
correction based on the conversion from the UV and FIR luminosity of Bell et
al. (2005) scaled to the Chabrier IMF. The black line and the gray shaded
region denote the best-fit relation for $z\sim 6$ galaxies and its $1\sigma$
uncertainty evaluated in Iyer et al. (2018).

The configuration of the multiple images is helpful to obtain the precise
information about the source position and its surface brightness profile in
the source plane. Here we estimate the intrinsic two-dimensional (2D) surface
brightness profile by fitting the HST images assuming the fiducial mass model.
Specifically, we first produce a $1\farcs 7\times 1\farcs 7$ cutout HST/F160W
image of $z$6.3. With a single Sérsic profile model in the source plane, we
then obtain the best-fit effective radius $r_{\rm e}$ = 1.2${}^{+4.1}_{-0.1}$
kpc (major axis), axis ratio of 0.49${}^{+0.03}_{-0.02}$, position angle =
84${}^{+2\circ}_{-2}$, Sérsic index $n$ = 2.5${}^{+1.2}_{-0.1}$, and central
coordinate of (RA, Decl.)=(6:00:08.12, $-$20:07:39.55) based on standard
$\chi^{2}$ minimization. Because we find a degeneracy between $r_{\rm e}$ and
$n$, here we restrict the Sérsic index to the range of $1<n<4$ in the fitting.
We do not use $z$6.1/6.2 and $z$6.4 for the fitting due to the complicated
morphology and the contamination of the diffuse emission from the nearby BCG,
respectively. We note that here we ignore the clumpy structure of $z$6.3 for
the moment, which we will discuss later. We list these best-fit Sérsic profile
results in Table 4.

In Figure 4, we show the best-fit 2D Sérsic profile in the source plane and
its multiple images in the image plane. We find that a single Sérsic profile
well reproduces not only $z$6.3, but also $z$6.4 and $z$6.1/6.2 whose
elongated shape is interpreted as a result of the source crossing the caustic
line in the source plane and stretched over $\sim 6^{\prime\prime}$ scale in
the image plane (e.g., Vanzella et al. 2020). This interpretation is
consistent with the slight difference in the line peak frequencies and the
line profiles between $z$6.1/6.2 and $z$6.3 (Section 3.1), because the sub
region of the galaxy can have different kinematics properties compared to the
whole galaxy. By calculating the ratio of the spatial areas between the source
and image planes, the magnification factors for $z$6.1/6.2 and $z$6.3 in our
fiducial model (average of the three independent models) are estimated to be
$\sim$150 (163) and $\sim$35 (21), respectively. The observed luminosity of
$z$6.1/6.2 is 33 (29) times brighter than the intrinsic overall luminosity of
RXCJ0600-$z$6 due to the strong gravitational lensing effect near the caustic
line. By comparing physical properties of $z$6.3 and $z$6.4 that are both
tracing the whole region of the lensed galaxy, we confirm that our independent
mass models agree in the ratio of magnification factors between $z$6.3 and
$z$6.4 in the range of 6.1–6.7, which is consistent with the observed
$L_{\rm[CII]}$ ratio of 5.7 $\pm$ 2.7 between $z$6.3 and $z$6.4. These results
validates our best-fit mass models and 2D Sérsic profile in the source plane.

To be conservative, we adopt the average value of the magnification factors
and evaluate its uncertainty from the minimum to maximum values among our
independent mass models, when we estimate the intrinsic physical properties of
RXCJ0600-$z$6 in this paper. We list the average magnification factor and its
uncertainty in Table 3. Applying the average magnification factors to the FIR
(Section 3.1) and optical-NIR (Section 3.2) properties, we summarize the
intrinsic physical properties in whole and sub regions of RXCJ0600-$z$6 in
Table 4. Remarkably, we obtain the intrinsic absolute rest-frame UV magnitude
of $M_{\rm UV}=-19.75^{+0.55}_{-0.45}$, which is $\sim 3$ times fainter than
$L^{*}$ of the LBG luminosity function at $z=6$ ($M_{\rm
UV}=20.91^{+0.07}_{-0.06}$; Ono et al. 2018). In Figure 5, we show the SFR and
$M_{\rm star}$ relation of RXCJ0600-$z$6\. For comparison, we also present the
average relation among $z\sim 6$ galaxies estimated in Iyer et al. (2018)
(gray shaded region). We find that RXCJ0600-$z$6 falls on the average relation
from the sub to whole regions. We also find that the relation between the [C
ii] line width and luminosity in RXCJ0600-$z$6 agrees with the average value
among $z\sim 6$ galaxies and the theoretical prediction (see Figure 10 in
Kohandel et al. 2019). The circularized effective radius ($r_{\rm
e,circ}\equiv r_{\rm e}\times\sqrt{\rm axis\,ratio}$) of 0.84 kpc also falls
in a typical range among $z\sim 6$ galaxies with the similar UV luminosity
(see e.g., Figure 9 of Kawamata et al. 2018). These results indicate that
RXCJ0600-$z$6 is an abundant, representative sub-$L^{*}$ galaxy at this epoch.
We note that these intrinsic physical properties are consistent with
independent estimates in N. Laporte et al. (submitted) within the errors, even
though the SED fitting strategies are different due to the different scopes in
the paper.

Figure 6:  Left: $3^{\prime\prime}\times 3^{\prime\prime}$ and
$6^{\prime\prime}\times 6^{\prime\prime}$ HST/F160W cutouts of $z$6.3 (top)
and $z$6.1/6.2 (bottom) in the image plane. The small black crosses denote
bright clumps (marked $a$, $b$, and $c$) and an elongated structure towards
north east ($d$) in the rest-frame UV continuum. The dashed black cross
indicates the peak position of the rest-frame UV continuum after smoothing the
HST map to match the resolution with ALMA. The red and green contours
represent the [C ii] line and the rest-frame FIR continuum from ALMA drawn at
$1\sigma$ intervals from $2\sigma$ to $8\sigma$. The red and green squares
(triangles) in $z$6.3 ($z$6.1/6.2) show the emission peak pixel positions of
the [C ii] line and rest-frame FIR continuum with the 1$\sigma$ error bars,
respectively. The cyan contours is drawn at 2$\sigma$ for the rest-frame UV
continuum. The foreground galaxies are removed with galfit for the source
plane reconstruction. Here we use the natural-weighted map for the [C ii]
line, while the $uv$-tapered maps with $2\farcs 0\times 2\farcs 0$ and
$0\farcs 8\times 0\farcs 8$ for the rest-frame FIR continuum of $z6.3$ and
$z6.1/6.2$, respectively. Right: Source plane reconstruction of the [C ii]
line and rest-frame UV and FIR continuum of $z6.3$. The color and symbols
follow the same assignment as the left panel, where the cyan and red contours
show 10%, 30%, 50%, and 80% of the peak. To match the spatial resolutions of
ALMA and HST, the red contours are drawn from the source plane reconstruction
of the de-convolved [C ii] spatial distribution obtained with imfit (Section
3.1) that is smoothed with the HST PSF. The inset panel displays the source
plane reconstruction of $z6.1$/6.2. The cyan (red) contours show 1%, 3%, 5%,
and 10% (10%, 30%, and 50%) of the peak of the whole galaxy. The luminosity-
weighted center is marked with label $e$ which corresponds to the faint rest-
frame UV clump at the western part of the whole galaxy. The white ellipses
show the typical shape of the HST PSF reconstructed in the source plane. The
error bars of the red square, triangle, and green triangle incorporate the
average lensing magnification corrections and their uncertainties. Note that
two peaks (= triangles) in $z$6.1/6.2 in the image plane to be the multiple
pair (Section 3.3), which thus correspond to one peak (= triangle) in the
source plane.

##  4 [C ii] Views from ISM to Cosmic Scales

The uniquely and strongly lensed galaxy near the caustic line (Section 3.3)
allows us to study ISM properties from internal to whole scales of the host
galaxy. For an example of the whole view of the galaxy based on $z$6.3, the
spatial resolutions of the HST map of $\sim 0\farcs 2$ translate into $0\farcs
04$ (corresponding to $\sim$250 pc at $z=6.07$) after the correction of the
lensing magnification, providing sub-kpc scale ISM views. At the same time,
the blind aspect of the ALCS survey also allows us to statistically evaluate
the number density of the [C ii] line emitters at $z\sim 6$ in a cosmic scale
based on our successful identification of RXCJ0600-$z$6\. In conjunction with
the rest-frame UV and FIR continuum properties, we examine the [C ii] line
properties from the ISM to cosmic scales and discuss whether there is common
property or a large diversity among these multiple scales.

###  4.1 Spatial Distributions of UV, FIR, and [C ii] down to Sub-kpc Scale

Making full use of the gravitational lensing, we investigate spatial
distributions of the [C ii] line, rest-frame UV and FIR continuum on the
source plane and compare them. In the context of similar studies so far at
$z\sim$ 2–4 for bright dusty, starburst galaxies (e.g., Swinbank et al. 2010,
2015; Dye et al. 2015; Spilker et al. 2016; Rybak et al. 2015; Tamura et al.
2015; Hatsukade et al. 2015; Rybak et al. 2020; Rizzo et al. 2020) and less
massive galaxies (e.g., Dessauges-Zavadsky et al. 2017, 2019), this is a first
observation to resolve the ISM structure down to the sub-kpc scale for the
sub-$L^{*}$ galaxy in the epoch of reionization.

In the left panel of Figure 6, we present the rest-frame UV continuum maps for
$z$6.3 (i.e., whole region) and $z$6.1/6.2 (i.e., sub region) taken in the
HST/F160W band with the [C ii] line (red contour) and the rest-frame FIR
continuum (green contour) taken by ALMA. The emission peaks of the [C ii] line
and rest-frame FIR continuum are marked with the red and green squares
(triangles) for $z$6.3 ($z$6.1/6.2) with the 1$\sigma$ error bars333The error
is estimated by the approximate positional accuracy of the ALMA map $\Delta p$
in milliarcsec, given by $\Delta p$ = 70000/$(\nu*B*\sigma$), where $\sigma$
is the peak SNR in the map, $\nu$ is the observing frequency in GHz, and $B$
is the maximum baseline length in kilometers (see Section 10.5.2 in cycle 7
ALMA technical handbook), respectively. Here we do not examine the rest-frame
FIR continuum peak from $z$6.3 due to the poor significance at the $2.5\sigma$
level (see Section 3.1). In $z$6.3, the rest-frame UV continuum shows a clumpy
structure, and thus we mark these clumps with black crosses labeled with $a$,
$b$, and $c$, from brightest to faintest. The rest-frame UV continuum of
$z$6.3 also shows an elongated structure toward the north east, which we mark
with an additional black cross and label $d$. In $z$6.1/6.2, possible clumps
are more evident in the [C ii] line and the rest-frame FIR continuum with the
two-peak morphology. If RXCJ0600-$z$6 consists of a smooth disk, the
morphology of $z$6.1/6.2 would be a single smooth-arc shape as shown in the
middle panel of Figure 4. Therefore, the two-peak morphology of $z$6.1/6.2 may
imply that the ISM of RXCJ0600-$z$6 near the caustic line has a clumpy
structure in the source plane. An alternative possibility is that an
intrinsically smooth disk is stretched into the two-peak morphology in the
image plane due to the perturbation by the foreground object overlapping
$z$6.1/6.2 which is not included in our fiducial mass model. To check this
possibility, we include the foreground object in our mass model assuming that
it is a member galaxy of the cluster (see Appendix C.2) and find that the two-
peak structure can indeed be reproduced, if the mass associated with the
foreground object is comparable or larger than that expected from the scaling
relation of the luminosity and mass for cluster member galaxies constrained in
our mass modeling. We conclude that we need more follow-up data including the
spectroscopic redshift of the foreground object in order to discriminate these
two possibilities. We however confirm that both magnification factors of
$\mu_{\rm local}$ and $\mu_{\rm whole}$ for $z$6.1/6.2 are affected only by
$\sim$1–2 % even if we include the foreground object in the mass model as one
of the cluster members or outside of the cluster up to $z\sim 4$. The other
foreground object near $z6.3$ is classified as one of the member galaxies of
the cluster (Section 3.2) and predicted to produce the critical curve at the
southern east part of $z6.3$ in our mass model (see the middle panel of Figure
4). However, we find that the [C ii] morphology at the corresponding area is
not disturbed at all, suggesting that its lensing effect is negligible for
$z6.3$. We thus remove this foreground object from the mass model and the HST
map with galfit in the source plane reconstruction of $z6.3$.

In the right panel of Figure 6, we present the source plane reconstruction of
$z$6.3 (whole region), where the inset panel displays the source plane
reconstruction of $z$6.1/6.2 (sub region). To match the spatial resolution
between HST and ALMA, we create a [C ii] map from the de-convolved [C ii]
spatial distribution (Section 3.1), smooth it with the point spread function
(PSF) of the HST F160W band, and use this PSF-matched map for the source plane
reconstruction of the [C ii] line. In the right panel, the white ellipses
indicate the source plane reconstruction of the HST PSF whose FWHM is
decreased down to $\sim 200\times 100$ pc and $\sim 300\times 60$ pc around
$z6.3$ and $z6.1/6.2$, respectively. The other color and symbols follow the
same assignment as the left panel, where we apply the lens correction also to
the error bars. The error bar of the [C ii] line peak position in $z$6.3 (red
square) is decreased down to $\sim$300 pc. These results indicate that we are
able to map the ISM view down to a few hundred parsec scale. Note that our
independent mass models consistently suggest that the two-peak morphology of
$z$6.1/6.2 in the image plane are the pair of multiple images (Section 3.3)
regardless of whether there exists the foreground object or not, which thus
correspond to one peak in the source plane. We confirm that the entire
morphology in both whole and sub regions of the galaxy and the emission peak
positions does not change beyond the errors in the source plane whether we
include or not the foreground galaxy overlapping $z$6.1/6.2 in the mass model
as one of the member galaxies of RXCJ0600-2007.

Firstly from the reconstruction of $z$6.3, we find on the scale of the galaxy
that [C ii] line peak shows an offset of $\simeq$ 300 pc from the brightest
rest-frame UV clump of a, but they are consistent at the 1$\sigma$ error
level. With the axis ratio of 0.49 (Table 4), non-parametric measurements
directly on the surface brightness distributions in the source plane provide
$r_{\rm e}=$ 1.1 kpc and 2.6 kpc for the rest-frame UV continuum and the [C
ii] line emission, respectively, showing the spatially extended [C ii] gas
structure by a factor of $\sim$ 2.4. These results are consistent with the
recent ALMA results of Fujimoto et al. (2020b) for 23 individual normal star-
forming galaxies at $z\sim 4$–6, whereby generally the [C ii] line is
spatially more extended than the rest-frame UV continuum by factors of $\sim
2$–3 without a spatial offset beyond $\simeq$ a 1-kpc scale. The $r_{\rm e}$
value for the rest-frame UV continuum is also consistent with the Sérsic
profile fitting results of 1.2 kpc in the source plane presented in Section
3.5.

Secondly from the reconstruction of $z$6.1/6.2, we find in the sub region of
the galaxy that the [C ii] line is co-spatial with the rest-frame UV continuum
again, which is separated by $\sim 1.6$ kpc from the peak of the [C ii] line
and rest-frame UV continuum from the whole region of the galaxy. We mark the
luminosity-weighted center of the sub region with the black cross labeled $e$.
Remarkably, we find, in the independent rest-frame UV continuum map
reconstructed from $z6.3$, that the faint clump exists exactly at the position
of $e$ whose peak flux density is also consistent. These agreements in the
properties of the clump $e$ also support the robustness of our best-fit mass
models. We also find that the [C ii] and rest-frame FIR peaks observed in the
image plane are reconstructed in the source plane with a $\simeq$ 1 kpc offset
from the luminosity-weighted center of $e$. This indicates that the faint
diffuse emission or further faint clump near the caustic line is strongly
lensed and more prominently visible in the image plane than the clump $e$.
Given that RXCJ0600-$z$6 is quantified with $r_{\rm e}=1.2$ kpc in the rest-
frame UV continuum (Section 3.5), these results indicate that we are
witnessing very faint [C ii] and rest-frame FIR emitting region(s) near the
caustic line beyond the effective radius of the galaxy that is almost
invisible in other multiple images. Because of the poor significance level of
the rest-frame FIR continuum in $z6.3$, we cannot conclude whether the rest-
frame FIR continuum detected in $z6.1/6.2$ corresponds to the outskirt
emission of the whole galaxy or the localized emission at the sub region of
the galaxy.

Interestingly, the brightest peaks of the [C ii] line and the rest-frame FIR
continuum in $z$6.1/6.2 appear on opposite sides in the image plane (Left
bottom panel of Figure 6). In $z$6.1/6.2, the magnification factor is
generally the same on either side. The clear difference identified in the [C
ii] line strength at the high significance levels ($8.2\sigma$ and
$5.4\sigma$) suggests the existence of substructure of the mass distribution
along the line-of-sight of $z$6.1/6.2, which is so-called flux-ratio anomaly
(e.g., Mao & Schneider 1998). This is consistent with our interpretation that
the central compact object in the optical-NIR bands in $z$6.1/6.2 is the
foreground object which is responsible for this flux-ratio anomaly. However,
if the [C ii] and rest-frame FIR emitting regions are identical in the source
plane, the flux ratio should be the same between the [C ii] and rest-frame FIR
emission in the image plane. Although the current error bars of the spatial
positions are large, this independent observable of the flux ratio suggests
that the faint [C ii] and rest-frame FIR emitting regions are physically
offset in the sub region of RXCJ0600-$z$6\. This potential separation and the
detailed ISM structure in RXCJ0600-$z$6 must be addressed in future deeper and
higher-resolution observations.

Figure 7:  Kinematic properties of RXCJ0600-$z$6\. Left: Velocity-integrated
(i.e., intensity; top), velocity-weighted (middle), and velocity-dispersion
(bottom) maps of $z6.3$ are shown in the left column. The best-fit 3D model
with 3Dbarolo and the residual maps are presented in the middle and right
columns, respectively. The image size is $3^{\prime\prime}\times
3^{\prime\prime}$. Middle: Source-plane reconstruction of the intrinsic
velocity-weighted map of $z$6.3 that was obtained with galpak3D. We limit the
reconstruction up to the radius of $1\farcs 4$ from the [C ii] intensity peak
in the image plane. The grey dashed line is the caustic line. The red triangle
shows the luminosity-weighted center of the sub region of the galaxy. Right:
Observed radial velocity profile of $z6.3$ extracted from the three annuli
defined for the 3Dbarolo analysis (black circles) and the spatial and velocity
offsets of the sub region of the galaxy observed in $z6.1/6.2$ (red triangle).
The black line shows the best-fit, together with the associate $1\sigma$ error
(grey shade), of the radial velocity profile of $z6.3$ obtained from galpak3d.

###  4.2 Kinematics via [C ii]

We also examine the kinematics of RXCJ0600-$z$6 via the bright [C ii] line
emission. Here we focus on the [C ii] kinematics of $z6.3$ to characterize the
gas kinematics of the whole galaxy. In the left panel of Figure 7, we present
the velocity-integrated (top), velocity-weighted (middle), and velocity-
dispersion (bottom) maps of $z6.3$ in the image plane. We evaluate the root-
mean-square noise level from the data cube and create these maps with a three-
dimensional (3D) mask of all signal above the 2$\sigma$ level. We find that
the [C ii] line has the velocity gradient from $-$100 to +45 km s-1 in east
south to west north with its intensity extended up to a radius of $\sim
1\farcs 4$. Assuming the line width estimate of $z6.3$ (Table 1) and a
potential error of $\sim$30 km s-1 for the velocity gradient due to the
spectral resolution of our ALMA data cube (Section 2.1), we obtain $\Delta
v_{\rm obs}/2\sigma_{\rm tot}=0.94\pm 0.26$, where $\Delta v_{\rm obs}$ and
$\sigma_{\rm tot}$ are the full observed velocity gradient (uncorrected for
inclination) and the spatially-integrated velocity dispersion, respectively.
With an approximate diagnostic for the classification of rotation-dominated
and dispersion-dominated systems, $\Delta v_{\rm obs}/2\sigma_{\rm tot}=0.4$
(Förster Schreiber et al. 2009), we find that $z6.3$ is classified as the
rotation-dominated system. Note that the beam smearing effect generally makes
the velocity gradient [dispersion] underestimated [overestimated] in spatially
low-resolution maps (see e.g., Figure 7 of Di Teodoro & Fraternali 2015). This
strengthens the argument that $z6.3$ is the rotation-dominated system from the
increased $\Delta v_{\rm obs}/2\sigma_{\rm tot}$ value without the beam
smearing effect.

To study the rotation kinematics, we analyze our data in the image plane with
softwares of 3Dbarolo (Di Teodoro & Fraternali 2015) and galpak3d (Bouché et
al. 2015) that are tools for fitting 3D models to emission-line data cubes. In
the left panel of Figure 7, we also show the best-fit 3D model and residual
maps with 3Dbarolo by assuming three annuli for its tilted ring fitting
algorithm. We find an excellent agreement on the intensity map and that the
residual velocities in velocity-weighted and velocity-dispersion maps are
generally less than the spectral resolution of our ALMA data cube ($\sim$28 km
s-1; Section 2.1). Although the residual in the velocity dispersion is
relatively large near the edge of the mask, this is likely because the faint
outskirt emission near the edge is masked in some velocity channels and the
observed velocity dispersion is underestimated. These results suggest that the
[C ii] kinematics of $z6.3$ is well reproduced by the best-fit 3D model. We
confirm that an independent 3D modeling of a single exponential disk with
galpak3d also provides the best-fit values of the rotation velocity, the
velocity dispersion, and the inclination fully consistent within errors with
the 3Dbarolo results. We summarize the details for the 3D modeling and the
results in Appendix E.

In the middle panel, we present the velocity-weighted map of $z6.3$ in the
source plane via the reconstruction in the same manner as Section 4.1. To
understand the intrinsic picture without the beam smearing effect, here we use
the best-fit intrinsic (i.e., resolution free) map obtained from galpak3d for
the reconstruction. In the right panel, we also show the [C ii] radial
velocity extracted from the three annuli with 3Dbarolo (black circle) as well
as the best-fit (black line) and the 1$\sigma$ error (gray shade) of the
rotation curve in the tanh formalization obtained from galpak3d. The
correction the lensing magnification is applied to the radius scale. For
comparison, the spatial and velocity offsets of $z6.1/6.2$ are shown with the
red triangle in both middle and right panels. We find that $z$6.1/6.2 agrees
with the velocity gradient of $z$6.3 within the errors, which is consistent
with our interpretation that $z$6.1/6.2 is the sub region of RXCJ0600-$z6$.
This suggests that the clump $e$ in the sub region of the galaxy (Section 4.1)
is likely a small star-forming region within the rotation disk of the host
galaxy.

For the rotation-dominated system, we obtain the dynamical mass $M_{\rm dyn}$
of $(3\pm 1)\times 10^{9}\,M_{\odot}$ based on an assumption of the disk-like
gas potential distribution, following the equation (4) in Dessauges-Zavadsky
et al. (2020)

$\displaystyle\left(\frac{M_{\rm dyn}}{M_{\odot}}\right)$ $\displaystyle=$
$\displaystyle 1.16\times 10^{5}\left(\frac{v_{\rm rot}}{\rm
km\,s^{-1}}\right)^{2}\left(\frac{r_{\rm e}}{\rm kpc}\right),$ (2)

where $v_{\rm rot}$ is the rotation velocity of the gaseous disk after the
inclination correction. We calculate the inclination from the axis ratio of
the best-fit surface brightness profile results for the rest-frame UV
continuum (Section 3.5), assuming that the higher-resolution map provides a
better constrain for the inclination. We adopt $r_{\rm e}$ and $v_{\rm rot}$
from the source plane reconstruction of the [C ii] line (Section 4.1) and the
galpak3d results, respectively. We caution that the uncertainty of the
inclination could remain by $\sim 30\%$ even in the spatially resolved
analysis (e.g., Rizzo et al. 2020), and thus the uncertainty in the above
$M_{\rm dyn}$ estimate could be even larger. Given the negligible contribution
of the dark matter halo in the galactic scale, we estimate the molecular gas
mass $M_{\rm gas}$ to be $\sim$ 1–2 $\times 10^{9}\,M_{\odot}$ by subtracting
$M_{\rm star}$ (Section 3.5) from $M_{\rm dyn.}$. It is worth noting that this
$M_{\rm gas}$ range agrees with another estimate based on an empirically
calibrated method in Zanella et al. (2018), given by

$\displaystyle\left(\frac{L_{\rm[CII]}}{L_{\odot}}\right)=10^{-1.28\,(\pm
0.21)}\times\left(\frac{M_{\rm gas}}{M_{\odot}}\right)^{0.98\,(\pm 0.02)},$
(3)

which suggests $M_{\rm gas}=3^{+2}_{-1}\times 10^{9}\,M_{\odot}$, despite the
potentially large uncertainty of the inclination. These results indicate that
RXCJ0600-$z6$ is a gas-rich galaxy with a high gas fraction of $f_{\rm gas}$
($\equiv M_{\rm gas}/(M_{\rm star}+M_{\rm gas}$)) $\sim$ 50–80 %. This is
consistent with recent ALPINE results that [C ii]-detected ALPINE galaxies
with $M_{\rm star}\sim 1\times 10^{9}\,M_{\odot}$ have $f_{\rm gas}$ $\sim$
60–90 % (see Figure 8 in Dessauges-Zavadsky et al. 2020). These $M_{\rm dyn}$,
$M_{\rm gas}$, and $f_{\rm gas}$ estimates are also listed in Table 4.

Note that we cannot rule out the possibility that the velocity gradient is
originally caused by complex dynamics with interacting, merging galaxies.
Future higher resolution observations will confirm the smooth rotation of the
disk or break the complex dynamics into the multiple components.

Figure 8:  Left: $L_{\rm[CII]}$–SFR relation. The red filled and open circles
indicate $z$6.1/6.2 and $z$6.3 before and after the correction of the lensing
magnification, respectively. The errors include the uncertainty from the mass
models (Tabel 3). The red open squares indicate the SFR estimates after the
correction of the lensing magnification based on the conversion from the UV
and FIR luminosity of Bell et al. (2005) scaled to the Chabrier IMF. The
previous results from local to high-$z$ star-forming galaxies are shown with
black circles (local LIRGs; Díaz-Santos et al. 2013), black squares (local
dwarfs; De Looze et al. 2014), black triangles (local spirals; Malhotra et al.
2001), black inverse triangles ($z\sim 0.3$ (U)LIRGs; Magdis et al. 2014),
green circles ($z\sim 4$–6 star-forming galaxies from ALPINE; Schaerer et al.
2020), magenta pentagons ($z\sim 5$ submillimeter galaxies (SMGs); Cooke et
al. 2018), magenta circles ($z\sim$ 2–6 lensed SMGs; Spilker et al. 2016), and
blue squares (compilation of recent ALMA results for $z>5$ star-forming
galaxies; Matthee et al. 2020 and Harikane et al. 2020). We adopt the average
relation of the low-$z$ Hii-galaxy/starburst sample from De Looze et al.
(2014), which is adjusted to the Chabrier IMF by reducing the SFR by a factor
of 1.06 in the same manner as Schaerer et al. (2020). The arrow indicates the
3$\sigma$ upper limit. The $L_{\rm FIR}$ value in the literature is firstly
converted into a total IR luminosity $L_{\rm TIR}$ (8–1000 $\mu$m), and then
we calculate SFR by using the calibration of Murphy et al. (2011). The
spatially resolved results ($\Sigma$SFR and $\Sigma L_{\rm[CII]}$) for local
galaxies are also presented with open triangles (Herrera-Camus et al. 2015) by
assuming the area of 1 kpc2. The dashed line and gray shade denote the
$L_{\rm[CII]}$–SFR relation obtained from local star-forming galaxies (De
Looze et al. 2014) and its dispersion, respectively. Right:
$L_{\rm[CII]}$/SFR–$\Sigma$SFR relation. The color assignments on the symbols
are the same as the left panel. We define the star-forming area by a circular
area of the rest-frame UV emission with a radius of $\sqrt{2}\times r_{\rm
e,circ}$ for the $\Sigma$ SFR estimate. We use the factor of $\sqrt{2}$ in
accordance to Spilker et al. (2016). For $z$6.1/6.2 and $z$6.3, we evaluate
the rest-frame UV size by reducing the [C ii] size measurements with imfit
(Table 1) by a factor of 2 (Section 4.1). For the ALPINE sources, we use the
rest-frame UV size measurement results in Fujimoto et al. (2020b). For local
LIRGs and lensed SMGs, we use the rest-frame FIR size measurement results in
Spilker et al. (2016) by assuming that the star-forming activity is dominated
in the rest-frame FIR emitting regions in these objects.

###  4.3 SFR and $L_{\rm[CII]}$ Relation

In the left panel of Figure 8, we show the relation between SFR and
$L_{\rm[CII]}$ for $z$6.1/6.2 and $z$6.3. For comparison, we also show local
and high-redshift galaxy results taken from the literature (Malhotra et al.
2001; Díaz-Santos et al. 2013; Magdis et al. 2014; De Looze et al. 2014;
Herrera-Camus et al. 2015; Spilker et al. 2016; Cooke et al. 2018; Harikane et
al. 2020; Matthee et al. 2019; Schaerer et al. 2020) and the
$L_{\rm[CII]}$–SFR relation obtained from local star-forming galaxies in De
Looze et al. (2014). The observed $L_{\rm[CII]}$ of both $z$6.3 and $z$6.1/6.2
fall on the most luminous $L_{\rm[CII]}$ regime among typical (e.g., SFR
$\lesssim$ 100 $M_{\odot}\,{\rm yr}^{-1}$) high-$z$ star-forming galaxies,
demonstrating the power of the gravitational lensing. After the correction of
the lensing magnification, we find that both $z$6.3 and $z$6.1/6.2 fall
slightly above, but still likely follow the SFR–$L_{\rm[CII]}$ relation of the
local galaxies within the dispersion. This is consistent with recent ALMA
results that the average SFR–$L_{\rm[CII]}$ relation among high-redshift star-
forming galaxies at $z\sim 4$–9 is well within the intrinsic dispersion of the
local relation (Carniani et al. 2018, 2020; Schaerer et al. 2020). Given that
RXCJ0600-$z$6 is consistent with being an abundant, sub-$L^{*}$ galaxy at
$z=6$ (see Section 3.5), these results may suggest that the SFR–$L_{\rm[CII]}$
relation, defined by local galaxies, holds from the spatially resolved sub-kpc
ISM to the whole scales in abundant galaxies even up to the epoch of
reionization.

To further study the $L_{\rm[CII]}$–SFR relation, the right panel of Figure 8
presents $L_{\rm[CII]}$/SFR and SFR surface density ($\Sigma$SFR). This
relation or another relation between $L_{\rm[CII]}/L_{\rm FIR}$ and $L_{\rm
FIR}$ surface density ($\Sigma L_{\rm FIR}$) are known to have tight anti-
correlations where the deficit of the [C ii] line is explained by the high
ionization state in the ISM around regions with high $\Sigma$SFR or $\Sigma
L_{\rm FIR}$ (e.g., Díaz-Santos et al. 2013; Spilker et al. 2016; Gullberg et
al. 2018; Ferrara et al. 2019). Importantly, these relations are not affected
by the lensing magnification, because the same magnification factor applies to
all these values. We find that $z$6.1/6.2 shows a higher $L_{\rm[CII]}$/SFR
ratio, while both $z$6.1/6.2 and $z$6.3 are consistent with the trend of the
anti-correlation. This indicates that the difference of $\Sigma$SFR causes the
difference of the [C ii] line luminosity at a given SFR between $z$6.1/6.2 and
$z$6.3, which is likely consistent with the source plane reconstruction
results in Section 4.1: the faint [C ii]-emitting region of $z$6.1/6.2 is
separated from the bright rest-frame UV clumps by $\sim$1.6 kpc in the source
plane, where the ionization state of the local ISM is thought to be moderate.

We note that recent ALMA observations show non-detection results of the [C ii]
line from similarly star-forming galaxies at $z\sim 4$–9 at the same time (see
upper limits in Figure 8), indicative of the existence of galaxies whose
$L_{\rm[CII]}$–SFR relations are different from those of the local galaxies.
Given the requirement of prior spectroscopic redshift with the Ly$\alpha$ line
in most cases (Section 3.2), those non-detections might be related to recent
reports of the potential anti-correlation between $L_{\rm[CII]}$/SFR and EWLyα
(Harikane et al. 2018, 2020; Carniani et al. 2018). In contrast to the most
cases, RXCJ0600-$z$6 is identified in the blind survey and its physical
properties (rest-frame EWLyα $<$ 4.4 ${\rm\AA}$ ) agrees with the potential
anti-correlation reported in Harikane et al. (2018). Another lensed galaxy at
$z=6.15$ (Calura et al. 2021) also follows the similar trend with relatively
large rest-frame EWLyα (60 $\pm$ 8 ${\rm\AA}$) and small $L_{\rm[CII]}$/SFR
($\sim 2\times 10^{5}$). A caution still remains that Schaerer et al. (2020)
report a weak dependence of $L_{\rm[CII]}$/SFR of EWLyα. Since the [C ii] line
emissivity depends on the ISM properties such as the ionization state,
metallicity, and gas density (e.g., Vallini et al. 2015), the different
$L_{\rm[CII]}$–SFR relations could be alternatively explained by a larger
dispersion of the ISM properties in high-$z$ galaxies than in local galaxies.
The uncertainties of the SFR estimates might contribute to the large
dispersion in high-$z$ galaxies due to assumptions of the star-formation
history, the dust-attenuation curve, and the stellar population age as
discussed in Carniani et al. (2020) and Schaerer et al. (2020). Another recent
reports of the extended [C ii] line morphology up to a radius of $\sim$10 kpc
(e.g., Fujimoto et al. 2019, 2020b; Ginolfi et al. 2020; Novak et al. 2020)
might be also related to some of those non-detections, because the surface
brightness of the extended emission is significantly decreased in relatively
high-resolution maps (Carniani et al. 2020). Based on the visibility-based
stacking, the secondary extended component up to the $10$-kpc scale is
estimated to have the average contribution to the total line luminosity of
$\sim$ 50 % around star-forming galaxies (Fujimoto et al. 2019) and $\sim
20\%$ around quasars (Novak et al. 2020) at $z\sim 6$. These non-negligible
contributions could matter if the request sensitivity is close to the
detection limit around the 5$\sigma$ level. However, this is not the case if
the carbon in the extended [C ii] gas is ionized by such as the gravitational
energy in the cold stream, the shock heating in the outflow and/or inflow gas,
and the AGN feedback, instead of the photoionization powered from the star-
forming regions (see e.g., Section 5 of Fujimoto et al. 2019).

###  4.4 [C ii] Luminosity Function

Figure 9:  Cumulative [C ii] luminosity function at $z=6$ with recent [C ii]
line studies at $z>4$ . The red circle shows the number density of the [C ii]
line emitter based on our successful detection from the strongly lensed LBG
with the effective survey volume of the full ALCS data cubes composed of 33
galaxy clusters. The lower limit is estimated from the Poisson uncertainty at
the single-sided confidence level of 84.13% presented in Gehrels (1986).
Recent ALMA blind line survey results are presented with blue triangle (243
archival data cubes; Matsuda et al. 2015), blue inverse triangle (four massive
galaxy clusters; Yamaguchi et al. 2017), blue square (ASPECS; Decarli et al.
2020), and blue cross (SSA22; Hayatsu et al. 2017, 2019). The green circle
presents the ALPINE results (Loiacono et al. 2020; Yan et al. 2020). Here we
show only the estimate from the serendipitous [C ii] line detection at $z\sim
5$ whose redshift is sufficiently separated from (i.e., not associated with)
the central ALPINE targets. The magenta square and pentagon show the
serendipitous [C ii] line detection from bright SMGs at $z\sim 5$ reported in
Swinbank et al. (2012) and Cooke et al. (2018), respectively. The red shade
indicates the current constraints based on our and previous results so far
obtained. For comparison, we also present semi-analytical model results
(Popping et al. 2016; Lagache et al. 2018) and SFR function (SFRF; Smit et al.
2016), including the dust correction based on the SMC extinction law, whose
SFR value is converted into $L_{\rm[CII]}$ with the local [C ii]–SFR relation
(De Looze et al. 2014).

A key goal of ALCS is to constrain the number density of the line emitters.
Although the complete blind line survey results with all 33 fields will be
presented in a separate paper (in preparation), we can evaluate a lower limit
of the [C ii] luminosity function at $z\sim 6$ with our [C ii] line detection
from the strongly lensed LBG at $z=6.0719$.

To do this, we first measure the effective survey area using mass models for
all 33 ALCS clusters at $z\sim 6$ constructed in the same manner as described
in Section 3.3. After the correction of the lensing magnification, we obtain
an effective survey area of $\sim 49$ (2) arcmin2 at $L_{\rm[CII]}=1.0\times
10^{9}$ (108) $\,{L_{\odot}}$, assuming the line width of FWHM=200 km s-1 with
the 5$\sigma$ detection limit. We then convert the effective survey area to
the survey volume, based on the frequency setup in the ALCS observations
covering the [C ii] line emission at $z=5.974$–6.172 and 6.381–6.602, and
derive a lower limit of the [C ii] luminosity function at $z=6$.

In Figure 9, we present the number density of [C ii] line emitters at $z=6$,
including recent [C ii] line studies at $z>4$ (Swinbank et al. 2012; Matsuda
et al. 2015; Yamaguchi et al. 2017; Cooke et al. 2018; Hayatsu et al. 2019;
Yan et al. 2020; Decarli et al. 2020). For comparison, we also present [C ii]
luminosity functions from semi-analytical models (Popping et al. 2016; Lagache
et al. 2018) and from the observed SFR function (SFRF; Smit et al. 2016) of
optically-selected galaxies. For the conversion from SFRF to [C ii] luminosity
function, we adopt the [C ii]–SFR relation of the local star-forming galaxies
estimated in De Looze et al. (2014). We find that our lower limit estimate is
consistent with both the semi-analytical results and SFRF. Note that we do not
apply any completeness corrections to our lower limit estimate. The
incompleteness for strongly lensed sources with large spatial sizes is
generally significant due to its low surface brightness (e.g., Bouwens et al.
2017; Kawamata et al. 2018; Fujimoto et al. 2017). Although the incompleteness
largely depends on the assumption of the intrinsic source size, this may
indicate that the lower limit could be placed still higher and that the faint-
end of the [C ii] luminosity function might be close to SFRF. Indeed, other
constraints from the recent [C ii] line studies are also consistent with SFRF
at the bright regime ($L_{\rm[CII]}\gtrsim 10^{8.5}\,L_{\odot}$). Although the
brightest-end ($L_{\rm[CII]}\gtrsim 10^{9.2}\,L_{\odot}$ ) of SFRF is smaller
than the constraints obtained from the SMG studies (Cooke et al. 2018), this
is explained by the absence of such dusty-obscured galaxies in the SFRF based
on the optically-selected galaxies. Therefore, the constraints of the [C ii]
luminosity function so far obtained are likely consistent with the prediction
from the local SFR–$L_{\rm[CII]}$ relation and the SFRF at $z=6$.

###  4.5 From ISM to Cosmic Scales

The source plane reconstruction in Section 4.1 unveils the ISM structure down
to a few hundred parsec scales, where we find that [C ii] line is not
displaced beyond a $\sim$ 300-pc from the rest-frame UV continuum from the
spatially resolved ISM to the whole galaxy. In Section 4.3, we find that the
SFR–$L_{\rm[CII]}$ relations from the spatially resolved ISM to the whole
galaxy are consistent with those of local galaxies. In Section 4.4, we obtain
the lower limit at the faintest regime of the $z=6$ [C ii] luminosity
function. We find that the prediction from the $z=6$ SFR function and the
SFR–$L_{\rm[CII]}$ relation of the local galaxies is consistent with our and
previous constraints on the $z=6$ [C ii] luminosity function in the wide
$L_{\rm[CII]}$ range. Given the unbiased aspect of the ALCS survey and indeed
the representative physical properties of RXCJ0600-$z$6 among the abundant
population of the low-mass regime of $z\sim 6$ star-forming galaxies (Section
3.5), our results may imply that the SFR–$L_{\rm[CII]}$ relation of local
star-forming galaxies is universal for a wide range of scales including the
spatially resolved ISM, the whole region of the galaxy, and the cosmic scale.

##  5 Summary

In this paper, we present the blind detection of a multiply-imaged line
emitter behind the massive galaxy cluster RXCJ0600$-$2007 in a cycle-6 ALMA
large project of ALMA Lensing Cluster Survey (ALCS). The optical–NIR property
and our lens model analyses suggest that the emission line is the [C ii] 158
$\mu$m line from a Lyman-break galaxy (LBG) at $z=6.0719\pm 0.0004$ behind
RXCJ0600-2007. We study the relation between the star-formation rate (SFR) and
[C ii] line luminosity ($L_{\rm[CII]}$), the morphology, and the kinematics in
the spatially resolved interstellar medium (ISM) as well as the whole scale of
the LBG, and provide a lower limit at the faint-end of the [C ii] luminosity
function at $z\sim 6$, with help of the gravitational lensing magnification.
The main findings of this paper are summarized as follows:

  1. 1.

We perform blind line search for the ALCS data cube in RXCJ0600-2007 and
identify two bright lines at $\geq 8\sigma$ levels at 268.682 $\pm$ 0.011 GHz
and 268.744 $\pm$ 0.016 GHz, one of which shows a strongly lensed arc shape.
Both lines have optical–NIR counterparts with clear Lyman-break feature at
$\sim$9000 ${\rm\AA}$, indicative of the lines corresponding to the [C ii] 158
$\mu$m at $z=6.07$. The optical–NIR spectral energy distribution (SED)
analysis shows that probability distributions of their photometric redshifts
are in excellent agreement with the [C ii] line redshift, while other possible
FIR lines at $z\sim 6$ are hard to explain the luminosity ratio between the
line and continuum. We thus conclude that these two lines are [C ii] lines.

  2. 2.

Our lens models, updated with the latest spectroscopic follow-up results with
VLT/MUSE, suggest that these lines arise from a strongly magnified and
multiply imaged ($\mu\simeq 20-160$) Lyman-break galaxy (LBG) at $z=6.0719$
with a circularized effective radius of $\sim$0.8 kpc and an intrinsic
luminosity in the rest-frame UV $\sim 3$ times fainter ($M_{\rm UV}$ =
$-19.7^{+0.5}_{-0.4}$) than the characteristic luminosity at this epoch. A sub
region of the LBG crosses the caustic line in the source plane and thus
stretched into an arc over $\sim 6^{\prime\prime}$ in the image plane, for
which the [C ii] line is also significantly detected. Our lens models also
predict another two multiple images in this field. We identify the sources at
the predicted positions and find that their optical–NIR colors agree with the
other multiple images of the LBGs. One of them falls in the ALCS area
coverage, where we detect a tentative [C ii] line ($3.0\sigma$) at the same
frequency as the other multiple images.

  3. 3.

After the correction of the lensing magnification, the whole of the LBG and
its sub region are characterized with $L_{\rm[CII]}$ of 1.1${}^{+0.7}_{-0.5}$
$\times 10^{8}$ and 0.3${}^{+0.1}_{-0.1}$ $\times 10^{8}\,L_{\odot}$, SFR of
5.4${}^{+4.5}_{-2.9}$ and 0.8${}^{+0.4}_{-0.2}$ $M_{\odot}$ yr-1, and stellar
mass ($M_{\rm star}$) of 9.6${}^{+6.0}_{-4.6}$ $\times 10^{8}$ and
2.6${}^{+0.2}_{-1.0}$ $\times 10^{8}\,M_{\odot}$, respectively. From the whole
to sub regions of the LBG, the SFR and $M_{\rm star}$ values falls on the
average relation among $z\sim 6$ galaxies, indicating that the LBG is an
abundant, representative galaxy at this epoch.

  4. 4.

The source plane reconstruction resolves the ISM down to $\sim$100–300 pc. The
[C ii] line from the whole region of the LBG is co-spatial with the rest-frame
UV continuum, while the sub region of the LBG is placed $\sim$1.6 kpc away
from the galactic center and bright rest-frame UV clumps. The two-peak
morphology observed in the [C ii] line and rest-frame FIR continuum in the arc
show a $\sim$1 kpc offset from the luminosity-weighted center of the sub
region of the LBG, which likely consists either of a clumpy structure or a
smooth disk but stretched into the two-peak morphology due to the perturbation
by a foreground galaxy. In these two peaks, the [C ii] line and the rest-frame
FIR continuum exhibit the flux ratio anomaly differently, which suggests that
the faint [C ii]- and FIR-emitting regions are displaced near the caustic.

  5. 5.

We find that our results in both whole and sub regions of the LBG fall on the
SFR–$L_{\rm[CII]}$ and surface density of SFR ($\Sigma$SFR)–$L_{\rm[CII]}$/SFR
relations obtained in local star-forming galaxies. The sub region of the
galaxy has a lower $\Sigma$SFR and a higher $L_{\rm[CII]}$/SFR value. This is
consistent with the absence of the bright rest-frame UV clumps around the sub
region of the LBG that is placed $\sim$1.6 kpc away from the galactic center,
where $\Sigma$SFR is expected to be low.

  6. 6.

We find that the LBG is classified as a rotation-dominated system based on the
full observed velocity gradient and the velocity dispersion of the LBG via the
bright [C ii] line emission. The 3D modeling with 3DBarolo and galpak3D
provide consistent results for the rotation kinematics that explains the
spatial and velocity offsets of the sub region of the LBG. We estimate the
dynamical mass of $M_{\rm dyn}=(3\pm 1)\times 10^{9}\,M_{\odot}$ and obtain
the gas fraction of $\sim 50$–80%.

  7. 7.

We derive a lower limit on the [C ii] luminosity function at $z=6$. We find
that it is consistent with current semi-analytical model predictions. In
conjunction with previous ALMA results, we also find that constraints on the
[C ii] luminosity function at $z=6$ so far obtained agree with the prediction
from the SFR–$L_{\rm[CII]}$ relation of local star-forming galaxies and the
SFR function at $z=6$.

  8. 8.

With the blind aspect of the ALCS survey and the SFR–$L_{\rm[CII]}$ relations
from the sub to whole regions of the LBG, our results may imply that the local
SFR–$L_{\rm[CII]}$ relation is universal for a wide range of scales including
the spatially resolved ISM, the whole region of the galaxy, and the cosmic
scale even up to $z=6$, which we derive in an unbiased manner.

We thank the anonymous referee for the careful review and valuable comments
that improved the clarity of the paper. We thank Justin Spilker and Tanio
Díaz-Santos for sharing their measurements. We also thank John R. Weaver and
Yuchi Harikane for useful comments on the paper and Francesca Rizzo for
helpful comments for the kinematic analysis. This paper makes use of the ALMA
data: ADS/JAO. ALMA #2018.1.00035.L. ALMA is a partnership of the 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 the ESO, AUI/NRAO, and NAOJ. This work is based on observations and
archival data made with the Spitzer Space Telescope, which is operated by the
Jet Propulsion Laboratory, California Institute of Technology, under a
contract with NASA along with archival data from the NASA/ESA Hubble Space
Telescope. This research made also use of the NASA/IPAC Infrared Science
Archive (IRSA), which is operated by the Jet Propulsion Laboratory, California
Institute of Technology, under contract with the National Aeronautics and
Space Administration. This work was supported in part by World Premier
International Research Center Initiative (WPI Initiative), MEXT, Japan, and
JSPS KAKENHI Grant Number JP18K03693. S.F. acknowledges support from the
European Research Council (ERC) Consolidator Grant funding scheme (project
ConTExt, grant No. 648179) and Independent Research Fund Denmark grant
DFF–7014-00017. The Cosmic Dawn Center is funded by the Danish National
Research Foundation under grant No. 140. NL acknowledges the Kavli Fundation.
GBC and KIC acknowledge funding from the European Research Council through the
Consolidator Grant ID 681627-BUILDUP. F.E.B acknowledges supports from ANID
grants CATA-Basal AFB-170002, FONDECYT Regular 1190818, and 1200495, and
Millennium Science Initiative ICN12_009. IRS acknowledges support from STFC
(ST/T000244/1). KK acknowledges support from the Swedish Research Council and
the Knut and Alice Wallenberg Foundation.

, glafic (Oguri 2010), lenstool (Jullo et al. 2007), ltm (Zitrin et al. 2015)

##  Appendix A MUSE Spectroscopic Catalog

In Table LABEL:tab:muse_list, we summarize the spectroscopic sample from
VLT/MUSE (ESO program ID 0100.A-0792, PI: A. Edge) which we use for
constraining our lens mass models.

Table 5: MUSE Spectroscopic Catalog RELICS ID | R.A. | Dec. | $z_{\rm spec}$ | flag
---|---|---|---|---
| deg | deg |  |
(1) | (2) | (3) | (4) | (5)
467 | 90.0386151 | $-20.1280873$ | 0.0 | 4
474 | 90.0261297 | $-20.1283111$ | 0.4366 | 2
490 | 90.0410860 | $-20.1282533$ | 0.8943 | 3
508 | 90.0352780 | $-20.1293624$ | 0.0 | 4
510 | 90.0366252 | $-20.1292593$ | 0.4230 | 3
514 | 90.0389765 | $-20.1292234$ | 0.4293 | 3
524 | 90.0333696 | $-20.1293114$ | 0.5614 | 3
538 | 90.0258085 | $-20.1298490$ | 1.0270 | 9
539 | 90.0260218 | $-20.1303021$ | 0.4448 | 3
543 | 90.0333594 | $-20.1313998$ | 0.4299 | 3
545 | 90.0346490 | $-20.1312617$ | 0.4316 | 3
547 | 90.0322874 | $-20.1302526$ | 0.4234 | 3
571 | 90.0288919 | $-20.1307349$ | 0.8751 | 3
606 | 90.0317981 | $-20.1320082$ | 0.2662 | 3
620 | 90.0264851 | $-20.1330304$ | 0.3284 | 3
624 | 90.0292841 | $-20.1331843$ | 0.4164 | 3
625 | 90.0286343 | $-20.1325697$ | 0.3449 | 3
626 | 90.0288471 | $-20.1326848$ | 0.3448 | 3
647 | 90.0377894 | $-20.1333237$ | 0.4176 | 3
660 | 90.0400373 | $-20.1375945$ | 0.3843 | 3
662 | 90.0412169 | $-20.1343049$ | 0.384 | 3
684 | 90.0410355 | $-20.1343563$ | 0.3838 | 3
685 | 90.0359084 | $-20.1336945$ | 0.4190 | 2
687 | 90.0328670 | $-20.1339062$ | 0.2298 | 3
697 | 90.0359808 | $-20.1344771$ | 0.4215 | 3
699 | 90.0321831 | $-20.1357916$ | 0.4245 | 2
705 | 90.0332455 | $-20.1367707$ | 0.0 | 4
711 | 90.0349938 | $-20.1360342$ | 0.4332 | 3
724 | 90.0265792 | $-20.1350258$ | 0.7360 | 3
735 | 90.0274334 | $-20.1354704$ | 0.4280 | 3
736 | 90.0257819 | $-20.1357459$ | 0.4294 | 3
737 | 90.0267125 | $-20.1360004$ | 0.4300 | 3
742 | 90.0340260 | $-20.1357916$ | 0.4266 | 3
743 | 90.0376604 | $-20.1357714$ | 0.4369 | 3
754 | 90.0337757 | $-20.1341905$ | 0.4233 | 2
772 | 90.0278737 | $-20.1374801$ | 0.4307 | 3
779 | 90.0326653 | $-20.1350609$ | 0.4305 | 3
786 | 90.0429917 | $-20.1367709$ | 5.4589 | 9
791 | 90.0281193 | $-20.1381123$ | 0.5089 | 3
792 | 90.0335009 | $-20.1349955$ | 0.4304 | 3
801 | 90.0394961 | $-20.1371561$ | 4.5043 | 3
802 | 90.0348279 | $-20.1356744$ | 0.4319 | 3
806 | 90.0364247 | $-20.1377419$ | 0.4295 | 3
814 | 90.0414618 | $-20.1358813$ | 0.0 | 4
823 | 90.0317133 | $-20.1382886$ | 0.4276 | 3
836 | 90.0264868 | $-20.1381907$ | 0.0 | 4
860 | 90.0365781 | $-20.1393201$ | 0.4177 | 3
862 | 90.0279194 | $-20.1393703$ | 0.4321 | 3
863 | 90.0278743 | $-20.1397608$ | 0.4296 | 3
870 | 90.0430009 | $-20.1395918$ | 0.4392 | 3
871 | 90.0425305 | $-20.1399135$ | 0.3825 | 3
886 | 90.0419909 | $-20.1399491$ | 0.4462 | 2
887 | 90.0340961 | $-20.1398346$ | 0.4315 | 3
899 | 90.0295615 | $-20.1407949$ | 0.4317 | 3
900 | 90.0296018 | $-20.1404579$ | 0.4323 | 3
938 | 90.0430509 | $-20.1413250$ | 0.4197 | 3
941 | 90.0343581 | $-20.1391201$ | 0.4195 | 3
956 | 90.0355945 | $-20.1419747$ | 0.4240 | 2
957 | 90.0387747 | $-20.1421929$ | 0.4314 | 3
962 | 90.0306151 | $-20.1429810$ | 0.0866 | 3
963 | 90.0303906 | $-20.1430867$ | 0.0866 | 3
964 | 90.0304180 | $-20.1428506$ | 0.0866 | 3
965 | 90.0304371 | $-20.1427616$ | 0.0866 | 3
966 | 90.0299297 | $-20.1421244$ | 0.0866 | 3
967 | 90.0300655 | $-20.1424500$ | 0.0866 | 3
973 | 90.0349638 | $-20.1425005$ | 0.4255 | 3
974 | 90.0350788 | $-20.1429515$ | 0.4313 | 3
990 | 90.0258089 | $-20.1429104$ | 0.5491 | 3
1000 | 90.0296362 | $-20.1430670$ | 0.4305 | 3
1003 | 90.0384290 | $-20.1430473$ | 0.5479 | 9
1024 | 90.0337998 | $-20.1436577$ | 2.7722 | 3
1025 | 90.0340880 | $-20.1437083$ | 2.7723 | 3
1027 | 90.0351249 | $-20.1438250$ | 2.7723 | 9
1029 | 90.0356487 | $-20.1438539$ | 2.7725 | 3
900001 | 90.0424970 | $-20.1364998$ | 3.5238 | 3
900003 | 90.0283847 | $-20.1376962$ | 5.4067 | 9

(1) ID from the RELICS public catalogue of
hlsp_relics_hst_wfc3ir_rxc0600-20_multi_v1_cat.txt444https://relics.stsci.edu/.
IDs starting with 900 are MUSE detections with no counterpart in the mentioned
catalogue. (2) Observed right ascension in degrees. (3) Observed declination
in degrees. (4) MUSE spectroscopic redshift. (5) Redshift quality flag. 2:
likely, 3: secure measurement, 9: single line measurement, and 4: field stars.

##  Appendix B Two-peak Morphology in $z$6.1/6.2

To check the possibility that the two-peak morphology of the [C ii] line in
$z$6.1/6.2 is caused by the noise fluctuation boosted by the underlying
diffuse emission (e.g., Hodge et al. 2016), we perform a mock observation with
the CASA task simobserve towards $z$6.1/6.2 in the same manner as Fujimoto et
al. (2020a). Here we assume the single elliptical Gaussian for the [C ii] line
surface brightness distribution of $z$6.1/6.2 based on the imfit results in
the $uv$-tapered map (Section 3.1). We then obtain the visibility data set
through simobserve and produce the natural-weighted velocity-integrated map of
the [C ii] line. We repeat the mock observation to producing the map 1,000
times. Given that the spatial offset of $\sim 2\farcs 0$ and the significance
levels of 8.2$\sigma$ and 5.4$\sigma$ between the two peaks in $z$6.1/6.2, we
then search multiple positive peaks that are located with spatial offsets of
less than $2\farcs 5$ and detected at $\geq 5.4\sigma$ levels, utilizing
SExtractor version 2.5.0 (Bertin & Arnouts 1996). We identify 7 out of 1,000
maps have the multiple peaks that meet the above criteria. These results
indicate that the two-peak morphology of the [C ii] line in $z$6.1/6.2 might
be caused by the noise fluctuation with a probability of $\sim$ 0.7%. Note
that we find that all multiple peaks identified in the 7 maps show their flux
ratios almost identical, which is different from the two peaks observed in
$z$6.1/6.2 (ratio $\sim$ 8:5). This indicates that the close separation as
well as the flux ratio of the two peaks observed in $z$6.1/6.2 is hardly
explained by the noise fluctuation. In fact, we identify only 1 out of 1,000
maps that has a flux ratio of multiple peaks similar to the two peaks in
$z$6.1/6.2, but with the spatial offset of $4\farcs 7$. Therefore, we conclude
that the possibility of the noise fluctuation is negligible in the two-peak [C
ii] line morphology of $z$6.1/6.2.

##  Appendix C Optical–NIR Photometry

We adopt separate strategies for extracting robust photometry for the four
lensed images as described below to account for the crowded cluster field and
varying degrees of extended source morphology. In general, we model the full
IRAC mosaics using a strategy similar to that of Merlin et al. (2015), where
we use image thumbnails of each source and neighbors taken from the high-
resolution HST/WFC3 F160W image and knowledge of the WFC3 and IRAC point
spread functions (PSFs) to model the low-resolution IRAC image.

###  C.1 Images $z$6.3 and $z$6.5

The sources of interest in these images are relatively bright and fairly well
separated from their nearest bright (projected) neighbors (Figure 1). We
measure aperture flux densities in each of the HST filters using fixed
$D=0\farcs 7$ apertures centered on the source of interest to define the
colors. To determine the overall flux normalization, we model the source
morphology of the lensed image and nearby neighbors using the non-parametric
morphological fitting code Scarlet (Melchior et al. 2018). All of the WFC3/IR
images (and their PSFs) are used to constrain the Scarlet morphological model.
We scale all of the HST aperture measurements $F_{ap,\mathrm{i}}$ by the
aperture correction $F_{S,\mathrm{F160W}}/F_{ap,\mathrm{F160W}}$, where
$F_{S,\mathrm{F160W}}$ is the integral of the Scarlet model evaluated in the
F160W filter and $F_{S,\mathrm{F160W}}$ is the aperture measurement in that
filter. The photometric uncertainties are measured in the same apertures on
the inverse variance image in each filter. For the IRAC flux densities of
these images, we subtract all modeled sources other than the source of
interest and perform aperture photometry on this cleaned image using
$D=3\farcs 0$ apertures, which we correct to the same “total” scale as for HST
using aperture corrections of 1.6 and 1.7 for channels 1 and 2, respectively,
that were derived from a separate bright, isolated source in the field.

###  C.2 Extended arc image $z$6.1/6.2

This image is a highly elongated arc extending over $\approx$6 arcsec
coincident with a foreground compact source in the center (Figure 1). Here, we
model both overlapping sources in the F160W image as parametric Sersic
profiles using the galfit software (Peng et al. 2010). For the photometry of
the lensed arc and foreground image in the WFC3/IR filters, we fit for the
relative normalizations of the two Sersic components convolved with the
appropriate PSFs. For IRAC, we convolve the model Sersic profiles with the
IRAC PSF and fit for the normalization of the source of interest and all
neighboring sources in the least-squares optimization. As for HST, the
normalization of the scaled morphological components is adopted as the
photometric measurement without additional aperture corrections. For the
optical images where the arc is not readily visible, we measure an aperture
flux density and its associated uncertainty within a large rectangle aperture
approximately $1\farcs 2\times 3\farcs 0$.

Note that the de-blended color of the foreground object is similar to the
color of cluster members, and the best-fit SED shows the photometric redshift
at $0.57^{+0.14}_{-0.17}$ which is close to the cluster redshift at $z=0.43$
(see also Laporte et al. submitted). Although this suggests the foreground to
be one of the cluster members, we do not include it in our fiducial mass model
due to potential systematics in the de-blending process. The detail
contribution of the foreground object to morphology and magnification factors
of $z$6.1/6.2 (Section 4.1) must be investigated after we obtain the
spectroscopic redshift of the foreground object.

###  C.3 Faint image $z$6.4

The final faint image of $z$6.4 is close to the cluster core and the BCG.
Although it is not deblended as a separate source in our original photometric
catalog (and associated IRAC model), a source is readily apparent in the F160W
image (Figure 1). We estimate photometry of this image by placing fixed
$D=0\farcs 7$ and $D=3\farcs 0$ apertures centered on the F160W position in
the HST and IRAC filter mosaics, respectively, and scale these measurements by
aperture corrections derived for point sources.

##  Appendix D [C ii] Spectra of $z6.1$ and $z6.2$

Figure 10:  Zoom-in [C ii] line spectra of $z6.1$ and $z$6.2 as a function of
velocity with respect to the frequency peak of $z$6.3. The inset panel shows
the same image cutout as the middle panel of Figure 1 for $z6.1$/6.2. The red
and blue squares denote apertures that are used to produce the [C ii] spectra
for $z6.1$ and $z6.2$, respectively. The black line and the grey shade
indicate the integrated [C ii] line spectrum of $z6.1$/6.2 and the [C
ii]-detected channels, respectively. The red and blue lines present the [C ii]
spectra for $z6.1$ and $z6.2$, respectively, that are normalized to the peak
of the integrated one.

As a sanity check of our interpretation that one of the [C ii] line emitters
consists of a pair of multiple images of $z6.1$ and $z6.2$, we compare [C ii]
spectra between $z6.1$ and $z6.2$. In Figure 10, we show the [C ii] spectra of
$z6.1$ (red line) and $z6.2$ (blue line) that are normalized to the peak of
$z$6.1/6.2 (black line). We find that $z6.1$ and $z6.2$ have [C ii] line
profiles consistent with each other within the errors, which agrees with our
interpretation of $z6.1$ and $z6.2$ being a pair of multiple images.

##  Appendix E [C ii] rotation modeling

In Section 4.2, we find that $z6.3$ is likely a rotation-dominated system. In
Table 6, we summarize the 3D modeling results for the [C ii] line around
$z6.3$ with 3Dbarolo (Di Teodoro & Fraternali 2015) and galpak3d (Bouché et
al. 2015). For 3Dbarolo, because the ALMA beam has a half-width-at-half-
maximum (HWHM) of $\sim 0\farcs 45$ at the [C ii] line frequency along the
orientation of the velocity gradient, we adopt three ($\sim$ 1.4/0.45) annuli
with the width of $0.^{\prime\prime}45$ for the tilted ring fitting algorithm.
We use the THRESHOLD mask with the 2$\sigma$ limit for the data cube, and the
spatial center, systemic velocity, rotation velocity ($v_{\rm rot}$), velocity
dispersion ($\sigma_{\rm vel}$), position angle (PA), and inclination (incl.)
are used as free parameters in the fitting. The errors are estimated based on
the minimization algorithm in a Monte Carlo approach. For galpak3d, we adopt
the exponential-disk for the flux profile, the Gaussian for the thickness
profile, and the $\tanh$ formalization of $V_{\rm max}\times\tanh(r/r_{\rm
V})$ for the rotation curve, where $V_{\rm max}$ and $r_{\rm V}$ are the
maximum velocity and the turnover radius, respectively. In place of the mask,
we use a cutout data cube by $3\farcs 8\times 3\farcs 8$ and [$-120$: +120] km
s-1 for the fitting. We set the maximum iteration number of 20,000. The
spatial center, systemic velocity, flux, $r_{\rm e}$, PA, incl., $r_{\rm V}$,
$V_{\rm max}$, and $\sigma_{\rm vel}$ are used as free parameters. The errors
are evaluated based on a Markov chain Monte Carlo approach.

Table 6: 3D Modeling Results

|  3Dbarolo | galpak3d
---|---|---
| Ring1 | Ring2 | Ring3 | Exponential disk
radiusa [′′] | 0–0.45 | 0.45–0.9 | 0.9–1.35 |  0.55 $\pm$ 0.02
$v_{\rm LOS}$b [km s-1] |  22${}^{+5}_{-4}$ |  76${}^{+22}_{-22}$ |  62${}^{+4}_{-3}$ |  62 $\pm$ 4
$\sigma_{\rm vel}$ [km s-1] |  46${}^{+9}_{-8}$ |  54${}^{+11}_{-11}$ |  59${}^{+7}_{-7}$ |  55 $\pm$ 3
incl. [deg] | 56 | 57 | 59 |  57 $\pm$ 3
PAc [∘] | 36 | 32 | 27 |  21 $\pm$ 3

$a$ Inner and outer radii of the tilted rings for 3Dbarolo and effective
radius for galpak3d. The turnover radius is estimated to be $0\farcs 62\pm
0\farcs 11$ in the $\tanh$ formalization for the rotation curve in galpak3d.
$b$ Line-of-sight projected rotation velocity. We present $V_{\rm max}$ for
galpak3d.
$c$ We follow the definition of PA as the orientation from the y-axis in anti-
clockwise, which is different from that of the original output of 3Dbarolo.

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