# Family of New Binary Transition Metal Nitrides Superconductors

Zheng-Wei Liao  School of Physical Sciences, University of Chinese Academy of
Sciences, Beijing 100049, China  Xin-Wei Yi  School of Physical Sciences,
University of Chinese Academy of Sciences, Beijing 100049, China  Jing-Yang
You<EMAIL_ADDRESS>Department of Physics, National University of
Singapore, 2 Science Drive 3, Singapore 117551  Bo Gu<EMAIL_ADDRESS>Kavli
Institute for Theoretical Sciences, and CAS Center for Excellence in
Topological Quantum Computation, University of Chinese Academy of Sciences,
Beijng 100190, China  Gang Su<EMAIL_ADDRESS>School of Physical Sciences,
University of Chinese Academy of Sciences, Beijing 100049, China  Kavli
Institute for Theoretical Sciences, and CAS Center for Excellence in
Topological Quantum Computation, University of Chinese Academy of Sciences,
Beijng 100190, China

###### Abstract

Superconductivity in transition metal nitrides (TMNs) has been investigated
for a long time, such as zirconium nitride (ZrN) with a superconducting
transition temperature Tc of 10 K. Recently, a phase diagram has been revealed
in ZrNx with different nitrogen concentrations, which is very similar to that
of high-temperature copper oxide superconductors. Here, we study the TMNs with
face-centered cubic lattice, where ZrN and HfN have been experimentally
obtained, and predict eight new stable superconductors by the first-principle
calculations. We find that CuN has a high Tc of 39 K with a very strong
electron-phonon coupling (EPC) strength. In contrast to ZrN, CuN has softening
acoustic phonons at the high symmetry point $L$, which accounts for its much
stronger EPC. In addition, the highly symmetrical structure leads to
topological protected nodal points and lines, such as the hourglass Weyl loop
in $k_{x/y/z}=0$ plane and Weyl points in $k_{x/y/z}=2\pi/a$ plane, as well as
quadratic band touch at $\Gamma$ point. CuN could be a topological
superconductor. Our results expand the transition metal nitrides
superconductor family and would be helpful to guide the search for high
temperature topological superconductors.

Transition metal nitrides (TMNs), as a class of materials with excellent
physical properties, such as high hardness and strength, strong corrosion
resistance, high melting point, good chemical and thermal stability [1, 2, 3,
4, 5], have attracted extensive attention and wide applications in various
fields [6, 7, 8, 9, 10]. In addition, many TMNs exhibit superconductivity [11,
12]. Since the 1930s, many TMN superconductors have been discovered. Nitrogen
atoms in these superconductors provide strong bonding and large electron-
phonon coupling (EPC), resulting in superconductivity [13]. For instance, ZrN
is a typical TMN superconductor, which displays the highest superconducting
transition temperature Tc of 10.0 K among the IVB TMNs [14, 15]. As a hard
metal nitride superconductor, ZrN is suitable for applications under extreme
working conditions, while most materials with superior mechanical strength and
hardness are semiconductors or insulators, which lack metallicity and
superconductivity [16]. A recent theoretical study has shown that on the
deformation path of high-symmetry crystallography [001], Tc of ZrN can reach
17.1 K under tensile strain [17]. Several other TMN superconductors have been
reported including TiN with a maximal Tc of 6.0 K [18], HfN with a Tc of 8.8 K
[19], VN with a Tc of 8-9 K [20, 21], NbN with a Tc of 17-18 K [20, 22, 23],
TaN with a Tc of 10.8 K [24], and W2N with a Tc of 1.3 K [20]. MoN has been
predicted to have the high Tc since 1981 [11, 25, 26, 13, 27]. A high Tc of 30
K was theoretically predicted in MoN, but the low Tc of 5-14K were obtained in
the experiments [28, 29, 30, 31, 32], where the obtained MoN samples were not
pure and often contaminated with Mo2N, $\gamma$-Mo2N, Mo or even MoOx phases.

In addition, studies have shown that the doping have a great impact on the
superconductivity in TMNs. For example, the Tc of TiN0.995, TiN0.95, TiN0.8,
and TiN0.55 are 6.0, 1.7, 1.5, and 1.2 K, respectively [33]. Pure VN, 0.5% B
doped VN, 0.5% La doped VN, 0.2% B and La each co-doped VN, and 0.5% B and La
each co-doped VN have the Tc of 9.2, 8.0, 7.8, 8.2 and 5.6 K, respectively
[34, 35, 36, 37]. ZrNx with different nitrogen concentrations has been
synthesized recently. Although ZrNx is generally believed to obey the Bardeen-
Cooper-Schrieffer (BCS) superconducting mechanism, the phase diagram of ZrNx
is very similar to that of high-Tc copper oxide superconductors, indicating
the possible relation between high-temperature superconductivity and ZrNx
[38].

In this article, we study the face-centered cubic (FCC) TMN in which ZrN and
HfN have been experimentally reported. By substituting elements, eight new
stable superconductors have been predicted through first-principle
calculations. Interestingly, we found that CuN has a high Tc of 39 K with a
very strong EPC of 3.099. In contrast to ZrN, CuN has softening acoustic
phonons at the high symmetry point $L$, which accounts for its much stronger
EPC. In addition, the highly symmetrical structure leads to topological
protected nodal points and lines, such as the hourglass Weyl loop in
$k_{x/y/z}=0$ plane and Weyl points in $k_{x/y/z}=2\pi/a$ plane, as well as
quadratic band touch at $\Gamma$ point.

Figure 1: (a) The conventional cell of face-centered cubic lattice AB. (b) The
Fermi surface, (c) electronic band structure and projected density of states
(PDOS) of CuN.

As shown in Fig. 1(a), ZrN has a NaCl-type FCC lattice with the
$Fm\overline{3}m$ space group. By substituting Zr with other transition metal
elements and N with other VA group elements, we have obtained eight new
structurally stable transition metal pnictides superconductors by the density
functional theory (DFT) calculations, including CdN, CdP, CdAs, CuN, CuSb,
HfP, HgSb, and ZnSb. Using the MicMillan-Allen-Dynes approach [39, 40] based
on the BCS theory with a typical Coulomb repulsion value $\mu^{*}=0.1$, we
calculate the Tc, EPC stength $\lambda$, and logarithmic average frequency
$\omega_{log}$ of these superconductors as listed in Table 1. In addition, we
also calculate the Tc of two experimentally reported TMN superconductors ZrN
and HfN, which are 10.93 and 9.1 K, respectively, close to the experimental Tc
of 10 and 8.8 K. Among these stable FCC TMN superconductors, CuN has the
strongest EPC, and the highest Tc of 39 K among other TMNs. In the following,
we studied CuN as an example in details, and in the Supplemental Materials
(SM) we provide the structural information, electronic bands, phonon spectra,
density of states (DOS), and superconducting properties of other seven
transition metal pnictides superconductors [41].

The optimized lattice constant of CuN is 4.177 Å. Fig. 1(b) gives the Fermi
surface of CuN, where red and blue surfaces represent the different bands
crossing the Fermi surface, which obviously accords with the crystal symmetry
of this structure. Fig. 1(c) shows the band structure along the high-symmetry
paths $\Gamma$-$X$-$U$—$K$-$\Gamma$-$L$-$W$-$X$ and projected DOS of CuN,
which show the metallic property with the same contribution of Cu and N atoms
near the Fermi level.

Figure 2: (a) Shape of the Weyl loop in $k_{z}=0$ plane and (b) the distribution of Weyl points in $k_{y}=2\pi/a$ plane for CuN. (c) The isoenergy slice at -0.8 eV projected on (001) surface of CuN. (d) The quadratic band dispersion around $\Gamma$ point. Table 1: The calculated total EPC $\lambda$, $\omega_{log}$, Tc of our proposed new stable compounds as well as two experimentally synthesized materials HfN and ZrN for comparision. |  $\lambda$ |  $\omega_{log}$ (K) |  Tc (K) |  Tcexp. (K)
---|---|---|---|---
CdN |  1.314 |  199.5 |  19.86 |  -
CdP |  1.266 |  122.0 |  11.64 |  -
CdAs |  1.426 |  94.9 |  10.29 |  -
CuN |  3.099 |  124.2 |  39.29 |  -
CuSb |  1.016 |  84.5 |  6.03 |  -
HfN |  0.658 |  305.9 |  9.10 |  8.8 [19]
HfP |  0.670 |  151.8 |  4.74 |  -
HgSb |  0.956 |  85.9 |  5.57 |  -
ZnSb |  1.387 |  93.0 |  9.81 |  -
ZrN |  0.657 |  375.3 |  10.65 |  10.0 [14, 15]

Figure 3: The phonon spectra weighted by the magnitude of EPC
$\lambda_{q\nu}$, projected phonon density of states, Eliashberg spectral
function $\alpha^{2}F(\omega)$, and cumulative frequency-dependent EPC
$\lambda(\omega)$ of (a) CuN and (b) ZrN.

Because of the highly symmetric crystal structure, this material can exhibit
many interesting topological properties in the band structure [42]. Fig. 2(a)
shows the shape of the nodal loop obtained from the DFT calculations in the
$k_{z}=0$ plane within the Brillouin zone (BZ). Similarly, other two nodal
loops appearing on $k_{x}=0$ and $k_{y}=0$ planes can also be obtained by
symmetry. In Fig. 2(c), we plot the constant energy slice at -0.8 eV, which
cuts through the drumhead states, forming a few arcs. In Fig. 1(c), we can
find another band crossing point along $U$-$X$ path at about -1 eV. It is a
nontrivial band crossing, leading to a nodal Weyl point. Since the fourfold
rotation symmetry along $\Gamma$-$X$, there should exist another three Weyl
points in the $k_{y}=2\pi/a$ plane, which has been shown in Fig. 2(b). In
addition, the band dispersion around $\Gamma$ points is quadratic along all
three directions in $k$ space as shown in Fig. 2(d), indicating that CuN
possesses the quadratic contact point $\Gamma$ protected by the crystalline
symmetry [43].

In Fig. 3, we study the phonon spectra weighted by the magnitude of EPC
$\lambda_{q\nu}$, projected phonon DOS, Eliashberg spectral function
$\alpha^{2}F(\omega)$, and cumulative frequency-dependent EPC
$\lambda(\omega)$ for CuN and ZrN. For CuN, the vibration modes can be divided
into two parts: low-frequency modes (¡ 8 THz) contributed by both Cu and N
atoms and high-frequency modes (¿ 8 THz) dominated by vibration of N atoms.
For ZrN, the vibration modes can also be divided into two parts: low-frequency
modes and high-frequency modes dominated by Zr and N atoms, respectively.
Between the two frequency ranges of ZrN, there is a 4.5 THz phonon gap. It is
worth noting that at the high symmetry point L, the acoustic phonons of CuN
exhibit obvious softening, accounting for about 83% of the total EPC
($\lambda=3.099$) below 6 THz, while ZrN has no softening acoustic phonons,
and the vibration modes of Zr atom below 7.5 THz only account for 63% of the
total EPC ($\lambda=0.657$) below 7.5 THz. Softening acoustic phonons are the
main source for the large EPC in CuN.

Table 2: The DOS at Fermi level $N(E_{F})$ (in unit of states/spin/eV/cell), volume of primitive cell $V$ (Å3), $\omega_{log}$ (K), $\lambda$, and Tc (K) for CuN and ZrN as a function of pressure. |  $P$(GPa) |  $N(E_{F}$) |  $V$(Å3) |  $\omega_{log}$(K) |  $\lambda$ |  Tc(K)
---|---|---|---|---|---|---
CuN |  0 |  8.19 |  18.22 |  124.2 |  3.099 |  39.31
|  1 |  8.15 |  18.14 |  154.3 |  2.377 |  37.23
|  2 |  8.10 |  18.05 |  180.1 |  1.989 |  35.83
|  3 |  8.07 |  17.97 |  202.0 |  1.737 |  34.45
|  4 |  8.03 |  17.89 |  217.7 |  1.587 |  33.38
|  5 |  7.99 |  17.81 |  229.1 |  1.488 |  32.45
ZrN |  0 |  4.47 |  24.24 |  375.3 |  0.653 |  10.93
|  1 |  4.45 |  24.15 |  377.3 |  0.649 |  10.80
|  2 |  4.43 |  24.05 |  381.5 |  0.643 |  10.66
|  3 |  4.41 |  23.96 |  384.7 |  0.639 |  10.50
|  4 |  4.39 |  23.87 |  387.9 |  0.633 |  10.39
|  5 |  4.38 |  23.78 |  391.0 |  0.629 |  10.26

Fig. 4 gives pressure dependence of logarithmic Tc and EPC $\lambda$ of CuN
and ZrN, respectively. Their detailed calculation results are listed in Table
2. It can be seen that with increasing pressure, the EPC $\lambda$ of both CuN
and ZrN decrease, resulting in the decrease of Tc. The calculated $d$Tc/$dP$
of ZrN is -0.133 K/GPa, which is close to the experimental result of -0.17
K/GPa [44]. Based on the BCS theory, Tc($P$) obeys the following
relationship[45]:

$\frac{d\rm lnTc}{d{\rm ln}V}=-B\frac{d\rm
lnTc}{dP}=-\gamma+\Delta\\{\frac{d{\rm ln}\eta}{d{\rm ln}V}+2\gamma\\},$ (1)

where $B$ is the bulk modulus, $\gamma$ $\equiv$
-$d$ln$\langle\omega\rangle$/$d$ln$V$ is the Grüneisen parameter, $\eta$
$\equiv$ $N(E_{F})\langle I^{2}\rangle$ is the Hopfield parameter [46] with
$\langle I^{2}\rangle$ the square of the electron-phonon matrix element
averaged over the Fermi surface, and $\Delta$ $\equiv$
1.04$\lambda$(1+0.38$\mu^{*}$)[$\lambda$-$\mu^{*}$(1+0.62$\lambda$)]-2.
Because the first term on the right hand side of Eq. (1) is smaller than the
second term [47], the sign of $d$Tc/$dP$ is mainly determined by the relative
magnitude of the two terms in curly bracket in Eq. (1). The Grüneisen
parameter ($\gamma$) can be directly obtained from Table 2, and the Hopfield
term can be determined by Eq. (1). It can be observed that our calculated
results for ZrN are in perfect agreement with Eq. (1). For CuN, our calculated
$d$lnTc/$dP=-0.0304$ K/GPa from Eq. (1) is also comparable with the fitting
slope $-0.035$ K/GPa in Fig. 4(a).

Figure 4: Pressure dependent logarithmic Tc and $\lambda$ of (a) CuN and (b)
ZrN. The dash line is the fitting according to BCS theory.

In summary, we have found eight new stable transition metal pnictides
superconductors by the first-principle calculations. Among these
superconductors, CuN is found to have the strongest EPC of 3.099 and the
highest Tc of 39 K. Considering the high symmery crystal structure, we have
obtained the quadratic contact point at $\Gamma$, where the band dispersion is
quadratic along all three directions in $k$ space. Besides, Weyl loop in
$k_{x/y/z}=0$ plane and Weyl points in $k_{x/y/z}=2\pi/a$ plane have also been
uncovered, indicating the topological properties of CuN. In contrast to ZrN,
we find that the softening acoustic phonon models in CuN are responsible for
the much higher EPC strength and the much higher Tc. By studying the
relationship between Tc and pressure, we find that both ZrN and CuN match well
with the BCS superconducting mechanism. Our results not only predict CuN
having the highest Tc in TMN superconductors, but also offer new materials
with both superconductivity and novel topology in band structures that would
be helpful for studying majorana zero modes in topological quantum
computation.

This work is supported in part by the National Key R&D Program of China (Grant
No. 2018YFA0305800), the Strategic Priority Research Program of the Chinese
Academy of Sciences (Grant No. XDB28000000), the National Natural Science
Foundation of China (Grant No.11834014), and Beijing Municipal Science and
Technology Commission (Grant No. Z118100004218001). B.G. is supported in part
by the National Natural Science Foundation of China (Grants No. 12074378 and
No. Y81Z01A1A9 ), the Chinese Academy of Sciences (Grants No. YSBR-030, No.
Y929013EA2), the Strategic Priority Research Program of Chinese Academy of
Sciences (Grant No. XDB33000000), and the Beijing Natural Science
Foundation（(Grant No. Z190011).

## References

  * Jhi  _et al._ [1999] S.-H. Jhi, J. Ihm, S. G. Louie, and M. L. Cohen, Nature 399, 132 (1999).
  * Zerr  _et al._ [2003] A. Zerr, G. Miehe, and R. Riedel, Nat. Mater. 2, 185 (2003).
  * Chhowalla and Unalan [2005] M. Chhowalla and H. E. Unalan, Nat. Mater. 4, 317 (2005).
  * Crowhurst  _et al._ [2006] J. C. Crowhurst, A. F. Goncharov, B. Sadigh, C. L. Evans, P. G. Morrall, J. L. Ferreira, and A. J. Nelson, Science 311, 1275 (2006).
  * Jin  _et al._ [2020] W. Jin, W. Sun, X. Kuang, C. Lu, and L. Kou, J. Chem. Phys. 11, 9643 (2020).
  * Zou  _et al._ [2016] Y. Zou, X. Qi, C. Zhang, S. Ma, W. Zhang, Y. Li, T. Chen, X. Wang, Z. Chen, D. Welch, P. Zhu, B. Liu, Q. Li, T. Cui, and B. Li, Sci. Rep. 6, 1 (2016).
  * Chatelain  _et al._ [2020] L. Chatelain, E. Louyriac, I. Douair, E. Lu, F. Tuna, A. J. Wooles, B. M. Gardner, L. Maron, and S. T. Liddle, Nat Commun 11, 1 (2020).
  * Rivadulla  _et al._ [2009] F. Rivadulla, M. Bañobre-López, C. X. Quintela, A. Piñeiro, V. Pardo, D. Baldomir, M. A. López-Quintela, J. Rivas, C. A. Ramos, H. Salva, J.-S. Zhou, and J. B. Goodenough, Nat. Mater. 8, 947 (2009).
  * Wu  _et al._ [2012] M. Wu, X. Lin, Y. Wang, L. Wang, W. Guo, D. Qi, X. Peng, A. Hagfeldt, M. Grätzel, and T. Ma, J. Am. Chem. Soc. 134, 3419 (2012).
  * Ningthoujam and Gajbhiye [2015] R. Ningthoujam and N. Gajbhiye, Prog. Mater. Sci. 70, 50 (2015).
  * Papaconstantopoulos  _et al._ [1985] D. A. Papaconstantopoulos, W. E. Pickett, B. M. Klein, and L. L. Boyer, Phys. Rev. B 31, 752 (1985).
  * You  _et al._ [2021a] J.-Y. You, B. Gu, G. Su, and Y. P. Feng, Phys. Rev. B 103, 104503 (2021a).
  * Matthias and Hulm [1952] B. T. Matthias and J. K. Hulm, Phys. Rev. 87, 799 (1952).
  * Isaev  _et al._ [2005] E. I. Isaev, R. Ahuja, S. I. Simak, A. I. Lichtenstein, Y. K. Vekilov, B. Johansson, and I. A. Abrikosov, Phys. Rev. B 72, 064515 (2005).
  * Weber [1973] W. Weber, Phys. Rev. B 8, 5093 (1973).
  * Chen  _et al._ [2005] X.-J. Chen, V. V. Struzhkin, Z. Wu, M. Somayazulu, J. Qian, S. Kung, A. N. Christensen, Y. Zhao, R. E. Cohen, H. kwang Mao, and R. J. Hemley, Proc. Natl. Acad. Sci. 102, 3198 (2005).
  * Lu  _et al._ [2021] W. Lu, H. Zhai, Q. Li, and C. Chen, J. Phys. Chem. Lett. 12, 1985 (2021).
  * Johansson  _et al._ [1985] B. O. Johansson, J.-E. Sundgren, J. E. Greene, A. Rockett, and S. A. Barnett, J. VAC. SCI. TECHNOL. A 3, 303 (1985).
  * Roberts [1976] B. W. Roberts, J. Phys. Chem. Ref. Data 5, 581 (1976).
  * Toth [2014] L. Toth,  _Transition metal carbides and nitrides_ (Elsevier, 2014).
  * Toth  _et al._ [1966] L. Toth, C. Wang, and G. Yen, Acta. Metall. Mater. 14, 1403 (1966).
  * Gurvitch  _et al._ [1985] M. Gurvitch, J. Remeika, J. Rowell, J. Geerk, and W. Lowe, IEEE Transactions on Magnetics 21, 509 (1985).
  * Pessall  _et al._ [1968] N. Pessall, R. Gold, and H. Johansen, J. Phys. Chem. Solids 29, 19 (1968).
  * Reichelt  _et al._ [1978] K. Reichelt, W. Nellen, and G. Mair, J. Appl. Phys. 49, 5284 (1978).
  * Pickett  _et al._ [1981] W. Pickett, B. Klein, and D. Papaconstantopoulos, Physica B+ c 107, 667 (1981).
  * Lengauer [1988] W. Lengauer, J. Cryst. Growth 87, 295 (1988).
  * Zhao and Range [2000] X. Zhao and K.-J. Range, J. Alloy. Compd. 296, 72 (2000).
  * Tsai  _et al._ [1993] W. Tsai, M. Delfino, J. A. Fair, and D. Hodul, J. Appl. Phys. 73, 4462 (1993).
  * Cassinese  _et al._ [2000] A. Cassinese, M. Iavarone, R. Vaglio, M. Grimsditch, and S. Uran, Phys. Rev. B 62, 13915 (2000).
  * Nishi  _et al._ [1987] Y. Nishi, S. Tokunaga, and S. Moriya, J. Mater. Sci. Lett. 6, 1450 (1987).
  * Soignard  _et al._ [2003] E. Soignard, P. F. McMillan, T. D. Chaplin, S. M. Farag, C. L. Bull, M. S. Somayazulu, and K. Leinenweber, Phys. Rev. B 68, 132101 (2003).
  * Papaconstantopoulos  _et al._ [1984] D. Papaconstantopoulos, W. Pickett, B. Klein, and L. Boyer, Nature 308, 494 (1984).
  * Spengler  _et al._ [1978] W. Spengler, R. Kaiser, A. N. Christensen, and G. Müller-Vogt, Phys. Rev. B 17, 1095 (1978).
  * Ningthoujam  _et al._ [2002] R. S. Ningthoujam, N. Sudhakar, K. P. Rajeev, and N. S. Gajbhiye, J. Appl. Phys. 91, 6051 (2002).
  * Sudhakar  _et al._ [2004] N. Sudhakar, R. S. Ningthoujam, K. P. Rajeev, A. K. Nigam, J. Weissmüller, and N. S. Gajbhiye, J. Appl. Phys. 96, 688 (2004).
  * Ningthoujam [2004] R. S. Ningthoujam, Indian J. Phys. 78, 89 (2004).
  * Gajbhiye and Ningthoujam [2006] N. Gajbhiye and R. Ningthoujam, Mater. Res. Bull. 41, 1612 (2006).
  * Chen  _et al._ [2022] F. Chen, X. Bai, Y. Wang, T. Dong, J. Shi, Y. Zhang, X. Sun, Z. Wei, M. Qin, J. Yuan, _et al._ , arXiv preprint arXiv:2201.04340 (2022).
  * Allen and Dynes [1975] P. B. Allen and R. C. Dynes, Phys. Rev. B 12, 905 (1975).
  * McMillan [1968] W. L. McMillan, Phys. Rev. 167, 331 (1968).
  * [41] See Supplemental Material at [url] for calculation methods, results on structural information, electronic bands, phonon spectra, DOS, and superconducting properties of CdN, CdP, CdAs, CuSb, HfP, HgSb, and ZnSb, which includes Refs.[48, 49, 50, 51, 52, 53, 54, 55, 39].
  * You  _et al._ [2021b] J.-Y. You, X.-L. Sheng, and G. Su, Phys. Rev. B 103, 165143 (2021b).
  * You  _et al._ [2020] J.-Y. You, B. Gu, and G. Su, Phys. Rev. B 101, 184521 (2020).
  * Chen  _et al._ [2004] X.-J. Chen, V. Struzhkin, S. Kung, H. kwang Mao, R. Hemley, and A. Christensen, Phys. Rev. B 70, 014501 (2004).
  * Schilling [2007] J. S. Schilling, in  _Handbook of High-Temperature Superconductivity_ (Springer, 2007) pp. 427–462.
  * Hopfield [1971] J. Hopfield, Physica 55, 41 (1971).
  * Chen  _et al._ [2002] X. J. Chen, H. Zhang, and H.-U. Habermeier, Phys. Rev. B 65, 144514 (2002).
  * Kresse and Furthmüller [1996] G. Kresse and J. Furthmüller, Phys. Rev. B 54, 11169 (1996).
  * Giannozzi  _et al._ [2009] P. Giannozzi, S. Baroni, N. Bonini, M. Calandra, R. Car, C. Cavazzoni, D. Ceresoli, G. L. Chiarotti, M. Cococcioni, I. Dabo, A. D. Corso, S. de Gironcoli, S. Fabris, G. Fratesi, R. Gebauer, U. Gerstmann, C. Gougoussis, A. Kokalj, M. Lazzeri, L. Martin-Samos, N. Marzari, F. Mauri, R. Mazzarello, S. Paolini, A. Pasquarello, L. Paulatto, C. Sbraccia, S. Scandolo, G. Sclauzero, A. P. Seitsonen, A. Smogunov, P. Umari, and R. M. Wentzcovitch, J. Phys.: Condens. Matter 21, 395502 (2009).
  * Blöchl [1994] P. E. Blöchl, Phys. Rev. B 50, 17953 (1994).
  * Perdew  _et al._ [1996] J. P. Perdew, K. Burke, and M. Ernzerhof, Phys. Rev. Lett. 77, 3865 (1996).
  * Baroni  _et al._ [2001] S. Baroni, S. de Gironcoli, A. D. Corso, and P. Giannozzi, Rev. Mod. Phys. 73, 515 (2001).
  * Vanderbilt [1990] D. Vanderbilt, Phys. Rev. B 41, 7892 (1990).
  * Güntherodt [1980] G. Güntherodt, Le Journal de Physique Colloques 41, C5 (1980).
  * Giustino [2017] F. Giustino, Rev. Mod. Phys. 89, 015003 (2017).


# Dust Properties of 870 Micron Selected Galaxies in the GOODS-S

S. J. McKay Department of Physics, University of Wisconsin–Madison, 1150
University Avenue, Madison, WI 53706, USA  A. J. Barger Department of
Astronomy, University of Wisconsin–Madison, 475 N. Charter Street, Madison, WI
53706, USA  Department of Physics and Astronomy, University of Hawaii, 2505
Correa Road, Honolulu, HI 96822, USA  Institute for Astronomy, University of
Hawaii, 2680 Woodlawn Drive, Honolulu, HI 96822, USA  L. L. Cowie Institute
for Astronomy, University of Hawaii, 2680 Woodlawn Drive, Honolulu, HI 96822,
USA  F. E. Bauer Instituto de Astrofísica and Centro de Astroingeniería,
Facultad de Física, Pontificia Universidad Católica de Chile, Casilla 306,
Santiago 22, Chile  Millennium Institute of Astrophysics (MAS), Nuncio
Monseñor Sótero Sanz 100, Providencia, Santiago, Chile  Space Science
Institute, 4750 Walnut Street, Suite 205, Boulder, Colorado 80301, USA  M. J.
Nicandro Rosenthal Department of Astronomy, University of Wisconsin–Madison,
475 N. Charter Street, Madison, WI 53706, USA

(Accepted 03 May 2023)

###### Abstract

We analyze the dust properties of 57 870 $\mu$m selected dusty star-forming
galaxies in the GOODS-S using new deep ALMA 1.2 mm, 2 mm, and 3 mm continuum
imaging together with other far-infrared through millimeter data. We fit the
spectral energy distributions (SEDs) with optically thin modified blackbodies
to constrain the emissivity indices and effective dust temperatures, finding a
median emissivity index of $\beta=1.78^{+0.43}_{-0.25}$ and a median
temperature of $T_{d}=33.6^{+12.1}_{-5.4}$ K. We observe a negative
correlation between $\beta$ and $T_{d}$. By testing several SED models, we
determine that the derived emissivity indices can be influenced by opacity
assumptions. Our temperature measurements are consistent with no evolution in
dust temperature with redshift.

cosmology: observations — galaxies: distances and redshifts — galaxies:
evolution — galaxies: starburst

††journal: ApJ††facilities: ALMA, JCMT††software: astropy (Astropy
Collaboration et al., 2022), casa (McMullin et al., 2007), emcee (Foreman-
Mackey et al., 2013)

##  1 Introduction

Over the last several decades, dusty star-forming galaxies (DSFGs) have
emerged as a critical population at redshifts $z\gtrsim 1$. First discovered
with the Submillimeter Common User Bolometer Array (SCUBA) on the single-dish
James Clerk Maxwell Telescope (JCMT) (Smail et al., 1997; Barger et al., 1998;
Hughes et al., 1998; Eales et al., 1999), DSFGs boast some of the highest star
formation rates (SFRs) in the universe (up to several thousand
$\mathrm{M}_{\textnormal{$\odot$}}$ yr-1) and may be responsible for 25% to
80% of the star formation rate density between redshifts of $z=6$ to $z=$
2–2.5, respectively (Zavala et al., 2021).

Surveys of distant DSFGs have been performed on single-dish facilities, both
from the ground (JCMT, IRAM, the South Pole Telescope, and the Large
Millimeter Telescope (LMT)) and from space (the Herschel Space Observatory).
Single-dish observations sample large numbers of sources near the peaks of
their far-infrared (FIR) spectral energy distributions (SEDs). However, they
do not allow for accurate position measurements, and they are affected by
source blending due to poor spatial resolution (e.g., Biggs et al., 2011;
Barger et al., 2012). The new TolTEC camera on the LMT, with its fast mapping
speeds and high sensitivity (Wilson et al., 2020), may mitigate some of these
issues.

In contrast to single-dish imaging surveys, submillimeter/millimeter
interferometric surveys using NOEMA, the Submillimeter Array, and the Atacama
Large Millimeter/Submillimeter Array (ALMA) have the sensitivity to detect
faint sources (e.g., Ono et al., 2014; Aravena et al., 2016; Chen et al.,
2023). In addition, they provide accurate positions, are not affected by
source blending, and can resolve extended emission (Hodge & da Cunha, 2020).
The main drawback is that interferometric surveys are observationally
expensive. Currently, the most efficient strategy for studying large numbers
of DSFGs is to conduct single-dish surveys to identify sources and then to
follow them up with interferometry (e.g., Hodge et al., 2013; Cowie et al.,
2018; Stach et al., 2019).

Constraints on the dust and gas masses of DSFGs are important for determining
what is responsible for their high SFRs. The dust masses of DSFGs are the
highest of any known galaxy population ($\gtrsim
10^{8}~{}\mathrm{M}_{\textnormal{$\odot$}})$ (e.g., Swinbank et al., 2014; da
Cunha et al., 2015; Dudzevičiūtė et al., 2020). They have been used to
estimate the gas masses available to form stars through an assumed calibration
(e.g., Scoville et al., 2014, 2016; Suzuki et al., 2021). However, dust mass
measurements depend sensitively on the choice of dust emissivity spectral
index, $\beta$, which parameterizes how the dust emission varies with
wavelength (Blain et al., 2002; Casey et al., 2014). Since $\beta$ pertains to
the intrinsic makeup of the dust, it can vary across a galaxy (e.g., Planck
Collaboration et al., 2014).

In studies of the local universe, $\beta$ has been found to be between 1.5 and
2.0 (Dunne & Eales, 2001; Chapin et al., 2011; Clements et al., 2018). This
range is in line with theoretical predictions of $\beta$ between 1.0 and 2.5
(Draine & Lee, 1984). In the absence of direct measurements, values of $\beta$
from the Milky Way and local galaxies are often assumed for high-redshift
sources, with a common choice of $\beta=1.8$ for optically thin SED fits
(e.g., Scoville et al., 2016; Simpson et al., 2017; Dudzevičiūtė et al.,
2021). If this assumption is not valid, then it would systematically impact
measured dust and gas masses for the DSFG population.

Unfortunately, measuring $\beta$ directly for high-redshift galaxies is
difficult, requiring multiple observations spanning the
submillimeter/millimeter regime to break the degeneracy between the dust
temperature, $T_{d}$, and $\beta$. Most surveys provide data at one or two
wavelengths—often from low-resolution single-dish observations—and only a few
recent studies use ALMA data. In one such study, da Cunha et al. (2021) used
an optically thin modified blackbody and found a median $\beta=1.9\pm 0.4$ for
a sample of 27 DSFGs from the ALESS survey (Hodge et al., 2013; Karim et al.,
2013) with ALMA 870 $\mu$m and 2 mm data. In another study, Cooper et al.
(2022) used a combined general opacity blackbody and power law fit and found a
median $\beta=2.4\pm 0.3$ for a sample of 39 DSFGs in the SSA22 field (850
$\mu$m flux $>5.55$ mJy) with SCUBA-2 850 $\mu$m, AzTEC 1.1 mm, and ALMA 2 mm
data.

In this paper, we determine $\beta$ and $T_{d}$ for a large sample of DSFGs in
the GOODS-S that have observations in multiple ALMA bands. Our sample is taken
from the catalog of 75 ALMA sources detected at 870 $\mu$m ($>4.5\sigma$) by
Cowie et al. (2018), which were originally selected from SCUBA-2 850 $\mu$m
imaging.

We structure the paper as follows. In Section 2, we discuss the ALMA 870
$\mu$m catalog and our new longer-wavelength ALMA observations, along with
ancillary photometry and redshifts. In Section 3, we describe our SED fitting
methods and we constrain the dust properties of our sample. In Section 4, we
discuss the implications of our results in the context of other studies in the
literature. In Section 5, we summarize our results.

We assume a flat concordance $\Lambda$CDM cosmology throughout with
$\Omega_{m}=0.3$, $\Omega_{\Lambda}=0.7$, and $H_{0}=70.0$ km s-1 Mpc-1.

##  2 Data

###  2.1 Total Sample

Our main sample of galaxies comes from the SUPER GOODS program of Cowie et al.
(2018) (hereafter, C18). Using ALMA band 7 (central wavelength of 870 $\mu$m),
C18 followed up SCUBA-2 850 $\mu$m selected sources in the GOODS-S to obtain
accurate positions. We hereafter refer to the resulting 75 galaxies ($>$
4.5$\sigma$; their Table 4) as our total sample.

The total sample consists of sources ranging in 870 $\mu$m flux from 0.84 mJy
to 8.93 mJy. Of these 75 sources, 17 (23%) are at or below the SCUBA-2
4$\sigma$ confusion limit of $\sim$1.6 mJy at 850 $\mu$m (Cowie et al., 2017).
The optical/NIR counterparts to this sample vary from bright galaxies at lower
redshifts to sources that are very faint or undetected in the deep CANDELS HST
imaging.

###  2.2 ALMA Band 3, 4, and 6 Observations

In ALMA Project #2021.1.00024.S (PI: F. Bauer), we made ALMA spectral
linescans in band 6 (central wavelength of 1.24 mm), band 4 (1.98 mm), and
band 3 (3.07 mm) of 57 sources in the total sample, prioritizing those with
870 $\mu$m fluxes above 1.8 mJy and lacking a well-established spectroscopic
redshift.

The ALMA observation blocks were downloaded and calibrated using casa version
6.2.1-7 based on the associated PI scripts. The visibilities from individual
spectral setups in a given band were combined using concat. Dirty continuum
images were generated using tclean, adopting 0$\farcs$25 pixels, natural
weighting, and a “common” restoring beam. Mildly cleaned continuum images were
generated adopting multithreshold automasking with standard thresholds, 100
clean iterations with a 0.1 mJy threshold, pixel scales of 0, 5, and 10, and
robust=0.5. The resulting band 3, 4, and 6 images had central frequencies,
bandwidths, and beams of 97.662 GHz, 27.120 GHz, and $\theta_{\rm
beam}{=}$0$\farcs$99$\times$0$\farcs$83; 151.188 GHz, 23.370 GHz, and
$\theta_{\rm beam}{=}$1$\farcs$21$\times$1$\farcs$10; and 241.750 GHz, 23.250
GHz, and $\theta_{\rm beam}{=}$1$\farcs$36$\times$1$\farcs$08; respectively.

We measured peak fluxes and errors from the band 3, 4, and 6 cleaned continuum
images. C18 found that since the sources were resolved at 870 $\mu$m, the peak
fluxes (even those from the tapered images) underestimated the total fluxes.
They determined that taking a ratio of aperture flux measurements made over a
range of aperture radii gave a similar correction factor to estimates derived
by fitting the sources in the uv plane. We adopted the aperture method for the
current data as well, finding average correction factors of 1.3 for all three
bands (e.g., see Figure 2 of Cowie et al. 2023 for the band 4 data). Thus, we
use this correction factor to convert the band 6, 4, and 3 peak fluxes to
total fluxes (we hereafter refer to these as 1.2 mm, 2 mm, and 3 mm fluxes,
respectively).

We supplement our ALMA observations with the 1.13 mm observations from the
GOODS-ALMA survey (Gómez-Guijarro et al., 2022). These were also obtained with
ALMA in the band 6 frequency window, but their central frequency of 265 GHz is
sufficiently different from our central frequency of 242 GHz to make them
worth including in our SED fits. Of the 75 sources in the total sample, 39 are
also detected in the GOODS-ALMA sample. Although 8 of these 39 sources were
not targeted in our spectral program, we at least have the GOODS-ALMA 1.13 mm
measurements (hereafter, 1.1 mm).

In Table 1 (Appendix A), we list our ALMA 1.2 mm, 2 mm, and 3 mm peak fluxes
and errors in a table comprised of the 75 sources in the total sample. In
addition, we give the 1.1 mm total fluxes for the 39 sources that also appear
in the GOODS-ALMA catalog.

###  2.3 Ancillary Photometry and Redshifts

The GOODS-S is one of the most well-observed fields in the sky, with
comprehensive multiwavelength coverage from the X-ray to the radio regime. In
this section, we summarize the additional photometric and redshift data that
we use in our analysis.

Barger et al. (2022) presented deep SCUBA-2 450 $\mu$m observations of the
GOODS-S, including counterparts to some of the 870 $\mu$m sources described
here. Since that time, we have continued to deepen our SCUBA-2 450 $\mu$m
images of the GOODS-S (see Cowie et al. 2023). Our latest maps reach a central
rms noise of 1.67 mJy, which is about 10% deeper than the image used in Barger
et al. (2022). For our purposes here, the deeper images provide more robust
SCUBA-2 450 $\mu$m fluxes for our sample. These fluxes were obtained
sequentially by first measuring the peak SCUBA-2 flux for a given source
within $2^{\prime\prime}$ of the ALMA position, then removing that source from
the image to prevent blending in later measurements. However, we note that
this is not critical for SCUBA-2 450 $\mu$m data due to its higher resolution
and shallower depth than SCUBA-2 850 $\mu$m data. The 1$\sigma$ errors were
measured from the local rms noise map.

Spitzer/MIPS 24 and 70 $\mu$m and Herschel/PACS 100 and 160 $\mu$m
counterparts were obtained from the catalog of Elbaz et al. (2011) using a
1$\farcs$5 matching radius. 61 sources from the total sample have a 24 $\mu$m
counterpart.

We matched our sample within $4^{\prime\prime}$ to the HerMES DR3 catalog that
used Spitzer/MIPS prior positions for deblending (Oliver et al., 2012) to
obtain Herschel/SPIRE data at 250, 350, and 500 $\mu$m. We found 51 galaxies
with SPIRE counterparts that appear to be reliable based on the images. For 24
sources without a reasonable counterpart in either the PACS or SPIRE catalogs,
plus an additional 13 that did not appear in the PACS catalog, we measured the
fluxes ourselves from the images at the ALMA positions and normalized them to
the catalog fluxes. However, we do not use the SPIRE 500 $\mu$m fluxes in our
SED fits due to their source blending and lower spatial resolution.

Cowie et al. (2023) presented for the total sample the redshifts, both
photometric (hereafter, photz; these come from Straatman et al. 2016) and
spectroscopic (hereafter, specz), including five that were determined from our
ALMA data. In total there are 20 sources with speczs in the total sample
(27%).

The highest specz in our sample is for ALMA 68 (numbered as in Table 4 of C18)
at $z=5.58$, obtained in Oesch et al. (2023) using JWST NIRCam/grism spectra
from the FRESCO survey. A NIRCam F444W, F210M, F182M color image of this
source is shown in Figure 2. Although this source was undetected in the
CANDELS HST images, it is clearly visible in the NIRCam image.

The distribution of 870 $\mu$m fluxes and redshifts for our total sample is
shown in Figure 1. We list all the adopted redshifts in Table 1 (Appendix A),
including whether they are speczs or photzs (denoted by the number of decimal
places). For the six sources in the table with poor-quality photzs (quality
flag $Q>3$ in the Straatman et al. 2016 catalog), we put their photzs in
brackets.

Figure 1: Histograms of the (a) 870 $\mu$m flux densities and (b) redshifts
for the total sample (gray). Sources with speczs are shown in color in both
plots. The six sources in the total sample without redshifts are not shown in
(b), and the two with questionable $z_{\rm phot}>7$ are shown at a nominal
redshift of $z=7$.  Figure 2: Three-color JWST NIRCam image (red $=$ F444W,
green $=$ F210M, blue $=$ F182M) for ALMA 68 from the FRESCO survey (Oesch et
al., 2023). The green circle has a 1′′ radius and is centered on the ALMA 870
$\mu$m position.

###  2.4 Robust Subset Selection

Although, in principle, only three photometric data points are needed to break
the degeneracy between $\beta$ and $T_{d}$, the results are unlikely to be
reliable for individual galaxies unless there are well-constrained fluxes
sampling both the peak of the dust SED and the Rayleigh-Jeans (RJ) tail. For
example, da Cunha et al. (2021) found for their bright ALESS sample that
without both Herschel detections near the peak and ALMA 870 $\mu$m and 2 mm
fluxes on the tail, the derived parameters were poorly constrained.

Thus, to better determine the dust properties of our sample, we restrict much
of our analysis to sources with observations in at least two ALMA bands, i.e.,
in addition to the ALMA 870 $\mu$m measurement, at least one ALMA measurement
at wavelengths longer than 1 mm. We also require the sources to have a
redshift estimate. This is satisfied by 52 out of the 57 sources with ALMA
spectral observations (the other 5 lack redshifts), and all 8 of the
additional sources with GOODS-ALMA 1.1 mm measurements (see Section 2.2).
However, we exclude one source (ALMA 58) that was not detected in the SPIRE
images or SCUBA-2 450 $\mu$m image, and we remove two others (ALMA 45 and ALMA
54) with questionable $z_{\rm phot}>7$ (their FIR colors suggest lower
redshift solutions). After excluding these 3 sources, there are 57 sources
which we keep in our analysis; we refer to these as the robust subset. The 18
sources not in the robust subset are marked with brackets around their source
numbers in Table 1.

Within the robust subset, there are 19 sources with speczs (33%). The median
redshift for the robust subset is $z=2.37$, and the 870 $\mu$m flux range is
0.93–8.93 mJy with a median of 2.54 mJy. Each source in the sample has at
least two ALMA measurements by our selection criteria, but 44 (77%) have four
or more ALMA measurements (2 have 3 ALMA bands and the remaining 11 have just
2 ALMA bands). Thus, the FIR SEDs of our sample are generally better sampled
than others in the literature (e.g., da Cunha et al., 2021; Cooper et al.,
2022).

##  3  SED Analysis and Dust Properties

The main goal of this work is to measure the dust properties of our sample of
DSFGs as accurately as possible. We do this by fitting the SEDs using simple
isothermal models. We fit the SEDs for 69 sources—the 57 in the robust subset
plus the remaining 12 in the total sample with redshifts.

We fit the photometry of each source with a single-temperature modified
blackbody (MBB), for which the flux density, $S_{\nu}$, for rest-frame
frequency, $\nu$, is given as $S_{\nu}\propto\kappa(\nu)B_{\nu}(T_{d})$. Here
$B_{\nu}(T_{d})$ is the Planck distribution and $\kappa(\nu)$ is the
frequency-dependent dust opacity. Although the dust consists of components at
a range of different temperatures, an effective $T_{d}$ description is often
used because it serves as a good trade-off between number of model parameters
and quality of fit to the FIR SED.

Additionally, we make the assumption that the emission is optically thin at
the wavelengths we are fitting, such that $\kappa(\nu)\propto\nu^{\beta}$,
where $\beta$ is the dust emissivity spectral index. At the resolution of our
observations, this $\beta$ represents a galaxy-averaged value; it manifests
primarily in the observed slope of the RJ fall-off.

The optically thin, isothermal MBB has been widely used as a successful model
for the FIR/(sub)millimeter dust emission of galaxies (e.g., Kovács et al.,
2006; Magdis et al., 2012; Jin et al., 2019; Dudzevičiūtė et al., 2020; da
Cunha et al., 2021; Barger et al., 2022), so by choosing it, we enable
comparisons of our derived parameters with those from the literature.

Figure 3: Difference in median likelihood estimates with and without CMB
corrections for $\beta$ (top) and for $T_{d}$ (bottom) vs. redshift, for the
robust subset. Models with the CMB included find slightly higher $T_{d}$ and
slightly lower $\beta$ on average, but the effect is minimal.

We use the Python-based Markov chain Monte Carlo (MCMC) package emcee
(Foreman-Mackey et al., 2013) to fit the MBB to our data and recover the
posterior likelihood distributions for the parameters. The free parameters in
these fits are the overall normalization, $\beta$, and $T_{d}$. We use flat
priors on $\beta$ between 0.8 and 4.0 and on $T_{d}$ between 10 K and 90 K. We
choose this Bayesian approach over a simple least-squares fitting algorithm
because least-squares fitting has been shown to introduce an artificial
correlation between $T_{d}$ and $\beta$ (Shetty et al., 2009a, b; Kelly et
al., 2012).

In the fits, we only consider the photometry at rest-frame wavelengths higher
than 50 $\mu$m. We add a 5% error in quadrature to the uncertainties to
correct for differences in the absolute flux calibration across
FIR/(sub)millimeter bands. We perform all of the fits at our adopted redshifts
(see Table 1) without allowing redshifts to vary.

We include the corrections to the SED from the CMB as outlined in da Cunha et
al. (2013). However, these are expected to be small for sources at $z<5$. We
check this by fitting MBBs without the CMB correction to the robust subset and
comparing the median likelihood values for $\beta$ and $T_{d}$ to those which
include the CMB effects. The results are shown in Figure 3. We find small
offsets in both $\beta$ and $T_{d}$, but conclude that the effects of the CMB
are fairly negligible for our sample.

Three sources, ALMA 54, ALMA 61, and ALMA 69, none of which is in the robust
subset, had 3 or fewer photometric data points included in the fit. Since the
number of data points must always be greater than the number of free
parameters, for these sources we only fit $T_{d}$ and not $\beta$.

Figure 4: Top row: Best-fit MBB (black curve) and 16th to 84th percentile
range from the posterior distribution of the MCMC models (blue shaded region)
for ALMA 43 (left), a source in the robust subset with five ALMA measurements;
ALMA 48 (center), also in the robust subset but with only two ALMA bands; and
ALMA 60 (right), a source not in the robust subset due to the lack of
observations at wavelengths longer than 870 $\mu$m. Photometry: Red
circles—ALMA 1.1 mm, 1.2 mm, 2 mm, and 3 mm, maroon pentagon—ALMA 870 $\mu$m,
green square—SCUBA-2 450 $\mu$m, dark red stars—Herschel/PACS 100 and 160
$\mu$m and SPIRE 250 and 350 $\mu$m. The MBB is only fit to the data at rest-
frame wavelengths $\geq$50 $\mu$m (points not included in the fits are marked
with black squares). The legend lists the parameters for the MBB fit and the
adopted redshift from Table 1. Bottom row: Marginalized likelihood
distributions (histograms) and joint likelihood distribution (contours) of
$\beta$ and $T_{d}$ for each of these sources. While the fits to the data are
good in all cases, the dust parameters $\beta$ and $T_{d}$ can only be well
constrained for ALMA 43 and ALMA 48, which have millimeter observations.

For most of our sources, the breadth of wavelength coverage allows us to
constrain the dust parameters tightly (we report the median likelihood
estimate of $\beta$ and $T_{d}$ as the measured value here and throughout the
paper). This can be quantified by the uncertainties on the measured
parameters, for which we use the 16th to 84th percentile range of the
posterior likelihood distribution. For the 44 sources in the robust subsample
with 3 or more ALMA measurements, we find a median 16th to 84th percentile
range for $\beta$ of 0.45 and a median 16th to 84th percentile range for
$T_{d}$ of 7.0 K. For the 13 sources with only 2 or 3 ALMA bands available,
the parameters become more uncertain: the median range on $\beta$ is 1.05 and
the median range on $T_{d}$ is 16.5 K. The errors are generally even higher
for sources not in the robust subset.

We illustrate this for a concrete example in Figure 4, where we compare the
fits for ALMA 43, which has 5 ALMA measurements; ALMA 48, which has just 2;
and ALMA 60, which is not included in the robust subset due to it only having
ALMA 870 $\mu$m and shorter wavelength data. We also show the marginalized
likelihood distributions (histograms) and joint likelihood distributions
(contour plot) for $\beta$ and $T_{d}$ for each source. For ALMA 43, the
parameters are tightly constrained, with $\beta=1.54^{+0.27}_{-0.26}$ and
$T_{d}=37.0^{+6.1}_{-4.7}$ K. The elongated shape of the joint likelihood
distribution reflects the intrinsic degeneracy between $\beta$ and $T_{d}$,
though in this case the parameters are well-constrained regardless. For ALMA
48, we find $\beta=2.54^{+0.65}_{-0.65}$ and $T_{d}=32.2^{+10.4}_{-6.1}$ K;
the parameters are still well-constrained with slightly higher errors. In
contrast, for ALMA 60, though the peak of the SED is sampled by the SPIRE and
submillimeter data, the lack of millimeter data means that a much larger range
of emissivities are consistent with the available data (clearly apparent in
the posterior likelihood distribution for $\beta$). We find
$\beta=1.72^{+1.20}_{-0.68}$ and $T_{d}=36.0^{+14.4}_{-11.1}$ K for this
source. These errors are 2–4$\times$ higher than those for ALMA 43, which is
also reflected in the 16th to 84th percentile distribution of models shown in
the SED fits and the relatively wide joint likelihood distribution for $\beta$
and $T_{d}$.

Figure 5: Histograms of the (a) dust emissivity indices and (b) effective dust
temperatures measured for the 57 sources in the robust subset (hatched and red
in (a); hatched and blue in (b)) and the 12 remaining sources in the total
sample with redshifts (gray). Three of the latter sources do not have a
measured $\beta$ due to limited photometry and hence are not shown in (a). We
only allowed $\beta$ to range between 0.8 and 4.0 and $T_{d}$ to range between
10 K and 90 K.

In Figure 5, we show histograms of (a) $\beta$ and (b) $T_{d}$ for the 57
sources in the robust subset (red and hatched) and the remaining 12 sources in
the total sample with redshifts (gray). For the robust subset in (a), we find
a wide distribution of $\beta$ that peaks in the range $\beta=$ 1.6–2.0 and
has a median $\beta=1.78^{+0.43}_{-0.25}$, where the uncertainties are the
16th to 84th percentile range. The error on the median derived from
bootstrapping the sample is $\pm 0.06$. Our median $\beta$ is consistent with
the frequently-assumed value for high-redshift studies of $\beta=1.8$ and with
recent measurements by da Cunha et al. (2021), as discussed in the
introduction. Considering only the 19 robust subset sources with speczs, we
find a median $\beta=1.74^{+0.28}_{-0.22}$.

We note that just six sources in the robust subset had measured emissivity
indices of $\beta\gtrsim 2.8$. It is possible that these sources do in fact
have very steep emissivities, or this could be an artifact of the fitting
driven by relatively high uncertainties on the 2 mm and/or 3 mm data. Five of
the sources only have photzs, so there is some possibility that an incorrect
redshift could affect these fits as well. We show the SED for one of the six
sources, ALMA 36, in Figure 6, along with the posterior likelihood
distribution for $T_{d}$ and $\beta$. All six of these sources have large
errors on $\beta$ that would make them consistent with $\beta\sim 2.8$ or
below.

Figure 6: Same as Figure 4, but for ALMA 36.

For the robust subset in Figure 5(b) (blue and hatched), we find 15 K
$<T_{d}<$ 70 K with a median $T_{d}=33.6^{+12.1}_{-5.4}$ K, where the error
refers to the 16th to 84th percentile range of the sample. The error on the
median derived from bootstrapping is $\pm 1.0$ K. This median $T_{d}$ is
similar to the median $T_{d}=30.4_{-4.7}^{+6.9}$ K found by Dudzevičiūtė et
al. (2020) for their ALMA SCUBA-2 UDS sample and to the median
$T_{d}=30^{+14}_{-8}$ K found by da Cunha et al. (2021) (again, uncertainties
are the 16th to 84th percentile range). Both of these results were derived
using similar methods to ours, though Dudzevičiūtė et al. (2020) assumed a
fixed $\beta=1.8$. Considering only the 19 robust subset sources with speczs,
we find a median $T_{d}=32.8^{+13.0}_{-1.5}$ K.

We also integrate the best-fit MBB from rest-frame 8 $\mu$m to 1000 $\mu$m for
each member of the robust subset to measure the FIR luminosities of our
sources and to help in investigating selection biases. We then multiply the
result by a correction factor of 1.35 (as was done in, for example, Jin et al.
2019) to account for the fact that MBB models for DSFGs typically
underestimate the MIR flux that results from warmer dust components. The
resulting luminosities range from $L_{\text{IR}}=3.9\times 10^{11}$
$\mathrm{L}_{\textnormal{$\odot$}}$to $L_{\text{IR}}=1.4\times 10^{13}$
$\mathrm{L}_{\textnormal{$\odot$}}$. We discuss the relation between dust
temperature and FIR luminosity in Section 4.3.

In Figure 7, we show the relationship between $\beta$ and $T_{d}$ that we
measure for the robust subset, with colors denoting redshifts. We find a
general negative correlation, with Pearson coefficient $r=-0.67$, $p$ value
$=1\times 10^{-10}$. This relationship is expected, both from other studies of
high-redshift galaxies (e.g., da Cunha et al., 2021) and from various
observations of molecular clouds within the Galactic plane (e.g., Paradis et
al., 2010). Laboratory tests of silicate grains have also shown an intrinsic
negative correlation of emissivity with temperature (Agladze et al., 1996;
Boudet et al., 2005; Inoue et al., 2020).

Although the intrinsic degeneracy between $\beta$ and $T_{d}$ can play a role
in a negative correlation (e.g., Galliano et al., 2018), our SEDs are very
well-sampled and thus we expect that our data are sufficient to break the
degeneracy. In Figure 7 we overplot the stacked joint likelihood distribution
for the robust subset (shown as contours) and find that it traces the median
likelihood estimates for $\beta$ and $T_{d}$ well. This implies that the
observed negative correlation is robust. da Cunha et al. (2021) discuss this
correlation and several possible selection effects that could artificially
produce this result. However, we find the negative correlation is only
minimally affected when considering only sources with $L_{\rm IR}>2\times
10^{12}$ $\mathrm{L}_{\textnormal{$\odot$}}$and $z<3.5$, where we expect a
greater degree of completeness. This suggests that selection effects are
unlikely to bias our result.

In general, one would expect $\beta$ to correlate with a ratio of fluxes on
the RJ tail of the blackbody, such as the 1.2 mm/2 mm ratio, although a spread
of temperatures and redshifts may introduce scatter into this relation. In
Figure 8, we check for this correlation between $\beta$ and the 1.2 mm/2 mm
flux ratio for all robust subset sources with a 2 mm measurement. Although
there is scatter in the trend as expected, we find a significant correlation
($r=0.50$, $p=7\times 10^{-4}$. This correlation remains when we only consider
sources with 2 mm flux $>$ 0.16 mJy, where the sample is expected to be
essentially complete (Cowie et al., 2023). This cut excludes most of the
sources with $\beta\gtrsim 2.8$ and the one source with 1.2 mm/2 mm ratio
$>20$ (due to a low signal-to-noise detection at 2 mm).

Figure 7: For the robust subset, emissivity index vs. dust temperature, both
measured from MBB fits. The data are color-coded by adopted redshift (right-
hand scale). Error bars represent 16th to 84th percentile ranges from the
likelihood distributions. The stacked joint likelihood distribution for the
robusts subset is shown as contours (levels are 39%, 86.4%, 98.8%, i.e., the
1$\sigma$, 2$\sigma$, 3$\sigma$, levels of a 2D Gaussian). Figure 8:
Emissivity index measured from MBB fits vs. the 1.2 mm/2 mm flux ratio, for
the robust subset with both ALMA 1.2 mm and 2 mm measurements. The data are
color-coded by adopted redshift (right-hand scale). The error bars on $\beta$
are the 16th to 84th percentile range of the likelihood distribution, while
the errors on the flux ratio are derived from the respective photometric
errors from Table 1. ALMA 34 is shown with a rightward-facing triangle since
its flux ratio is off the plot at 26.9, due to its low-significance detection
at 2 mm.

In Table 2, we summarize the median likelihood estimates for the dust
parameters obtained from the MBB fits for the robust subset. In Appendix B, we
show the photometry and best-fit MBB for all 69 sources that we fit.

##  4 Discussion

###  4.1 Interpretation of the Measured Emissivity Index

Our median $\beta=1.78^{+0.43}_{-0.25}$ is broadly consistent with theoretical
predictions for the interstellar medium (ISM), which give $\beta\sim 2$
(Draine & Lee, 1984). Measurements of the Milky Way’s ISM have yielded
$\beta=1.5$ (Paradis et al., 2009) and $\beta=1.8$ (Planck Collaboration et
al., 2011). These values are within the range spanned by our sample, though
our results suggest that $\beta=1.5$ may not be appropriate as an assumption
for high-redshift galaxies as a whole.

However, as we mentioned in Section 3, the emissivities that we measure from
our SED fits are galaxy-integrated values that characterize the slope of the
overall dust emission spectrum on the RJ side. The connection to the intrinsic
emissivity of the dust composition of the galaxy is difficult to make and
relies on assumptions about the relative proportions of dust grains of
different sizes and structures and the ISM geometry, among other factors.
Higher values of $\beta$ could suggest a greater proportion of larger
crystalline dust grains that are at generally lower temperatures (Agladze et
al., 1996). However, a range of temperatures in the dust components would, in
general, flatten the effective RJ slope and hence reduce the measured $\beta$.

Regardless of the physical interpretation, we have shown that isothermal MBBs
with $\beta\approx$ 1.8 can be used to fit the FIR SEDs of high-redshift dusty
galaxies spanning a range of submillimeter/millimeter fluxes, redshifts, and
observed wavelengths.

###  4.2 Comparisons with General Opacity Models

While many authors assume an optically thin MBB model as a good trade-off
between number of model parameters and quality of fit to the FIR SED, there is
no guarantee that the emission is optically thin at shorter wavelengths. Some
works (e.g., Conley et al. 2011; Casey et al. 2019) suggest that the dust may
remain optically thick out to rest-frame wavelengths $\lambda_{0}\sim$ 200–300
$\mu$m. Accordingly, several recent studies of the dust properties of DSFGs
have instead used general opacity factors in their MBB models (e.g., Casey et
al., 2021; Cooper et al., 2022).

Figure 9: General opacity + power law MBB fit (red dot-dashed curve), with MIR
power law slope $\alpha=2.0$, compared with optically thin MBB fit (black
curve) for ALMA 24. The 16th to 84th percentile range from the posterior
distribution of the MCMC models is shaded red for the opacity + power law
model and shaded blue for the optically thin model. The optically thin model
is fit to all points at rest-frame wavelengths greater than 50 $\mu$m (points
below this are marked with black squares) while the opacity + power law fit is
fit to all data. The measured dust temperature and emissivity for each are
given in the figure legend, showing a slightly steeper $\beta$ and a higher
$T_{d}$ for the opacity + power law model.

To make a direct comparison with these studies, we next fit the SEDs of the
robust subset with a general opacity MBB model of the form
$S(\nu)\propto(1-e^{-\tau})B_{\nu}(T)$, where $\tau$ is the optical depth
described by the power law $\tau\propto(\nu/\nu_{0})^{\beta}$. Here $\nu_{0}$
is the turnover frequency at which the optical depth equals 1. We assume a
turnover wavelength $\lambda_{0}=200$ $\mu$m, and we include a MIR power law
with $\alpha=2.0$, following the prescription of Casey (2012).

With the inclusion of the power law on the short-wavelength side of the FIR
peak, we fit the photometry down to a rest-frame wavelength of 10 $\mu$m. The
MIR power law is included in the SED fits of both Casey et al. (2021) and
Cooper et al. (2022), though the former vary the slope $\alpha$ in their fits,
where possible, and generally find steep slopes ($\alpha\approx$ 3.5–7). We
choose to leave $\alpha$ fixed at 2.0 so that our model is identical to that
of Cooper et al. (2022).

In Figure 9, we show a comparison of this opacity + power law model and the
optically thin model fits for ALMA 24. Both fit the data well but with
somewhat different median likelihood values. The optically thin fit gives
$\beta=1.44$ and $T_{d}=42.0$ K, while the opacity + power law fit gives
$\beta=1.76$ and $T_{d}=59.0$ K. We also note that for this galaxy, the
$\alpha=2.0$ power law slope does a good job of fitting the measured MIPS 70
$\mu$m point, which is not included in our optically thin fit.

Figure 10: Histogram of errors in our $\beta$ values for the robust subset
opacity + power law MBB fits (blue) with the errors from Cooper et al. (2022)
(hatched and red). The errors shown are half of the 16th to 84th percentile
range determined for each individual measurement.

We find median $\beta=2.06^{+0.56}_{-0.37}$ and $T_{d}=57.5^{+12.6}_{-12.2}$ K
for the robust subset using the opacity + power law model. This median $\beta$
is slightly lower than those reported in Casey et al. (2021) and Cooper et al.
(2022), who find median $\beta=2.2^{+0.5}_{-0.4}$ and
$\beta=2.4^{+0.3}_{-0.3}$, respectively. The errors here are the 16th to 84th
percentile range of each sample and thus reflect the spread in measured
$\beta$ values rather than the errors on the individual $\beta$ values.

Thus, in Figure 10, we compare the individual errors on our $\beta$ values
with the errors from Cooper et al. (2022), who only had ALMA data in the 2 mm
band. Since we and Cooper et al. (2022) report both upper and lower errors
determined from the likelihood distributions, in the figure we take the
average of the upper and lower errors for each source (i.e., half of the 16th
to 84th percentile range; for a Gaussian posterior this would be the 1$\sigma$
error) for a rough comparison of errors. We see that the addition of more ALMA
bands in our case reduces the errors on the individual $\beta$ measurements.

A Mann-Whitney test gives a 0.05% probability that our sample comes from the
same underlying distribution as that of Cooper et al. (2022). This may be due
to their combining single-dish observations with ALMA 2 mm data, or to the
brighter flux limit of their sample (SCUBA-2 850 $\mu$m flux $>$ 5.55 mJy).
However, the 7 sources in our sample that meet their selection criteria have
median $\beta=1.89$ from the opacity + power law fits, suggesting that the
discrepancy reflects a problem with mixed single-dish and interferometric
measurements.

The median temperature we measure for the opacity + power law model is
$\sim$24 K lower and the median emissivity is 0.3 higher than the results for
the optically thin MBB. For comparison, da Cunha et al. (2021) found dust
temperatures that were $\sim$10 K lower for their optically thin model than
for their general opacity model but found that emissivity was robust against
different opacity assumptions—though they did not include a MIR power law and
they allowed $\lambda_{0}$ to vary between 60–140 $\mu$m. Meanwhile, for
simulated galaxies with optically thick emission out to $\sim$200 $\mu$m,
Hayward et al. (2011) found that dust temperatures derived from an optically
thin model could underpredict its optically thick counterpart by $\sim$20 K,
which is closer to the deviation we measure.

To test whether our discrepancies in $\beta$ and $T_{d}$ were related to the
assumption of $\lambda_{0}=200$ $\mu$m and/or the inclusion of the MIR power
law, we fit the SEDs of the robust subset again with three other general
opacity models: a general opacity MBB with $\lambda_{0}$ fixed to 200 $\mu$m
with no MIR power law, a general opacity MBB with $\lambda_{0}$ fixed to 100
$\mu$m with a MIR power law, and a general opacity MBB with $\lambda_{0}$
fixed to 100 $\mu$m but no power law included (to make a better comparison
with the general opacity model used in da Cunha et al. 2021).

Figure 11: For the robust subset, (a) Dust temperature vs. FIR luminosity, (b)
FIR luminosity vs. redshift, (c) Dust temperature vs. redshift, and (d) Same
as (c) but only for the sources with a FIR luminosity $>2\times 10^{12}$
$\mathrm{L}_{\textnormal{$\odot$}}$and $z<3.5$, where our sample is more
likely to be complete. The dashed line shows the median $T_{d}=33.6$ K, and
the shaded region denotes the 16th to 84th percentile range of the robust
subset. In all panels, sources with speczs are shown as red pentagons while
those with photzs are shown as blue squares.

For the $\lambda_{0}=100$ $\mu$m MBB model with no power law, we find median
$\beta=1.80$ and median $T_{d}=42$ K. The results are only minimally affected
by the inclusion of the MIR power law: for the $\lambda_{0}=100$ $\mu$m MBB
model with the power law we find median $\beta=1.78$ and median $T_{d}=45$ K.
The median $\beta$ for these models is consistent with the optically thin
case, and the median $T_{d}$ is larger by about $\sim$10 K. These results are
consistent with those of da Cunha et al. (2021), who used a similar model.
However, for the $\lambda_{0}=200$ $\mu$m MBB model with the power law
included, we measure a median $\beta=2.02$ and median $T_{d}=56$ K. Thus, we
observe that the assumption of the turnover wavelength has a significant
effect on the derived parameters, and the emissivity is not necessarily robust
against opacity assumptions if the turnover wavelength is high enough. This
may also help to explain why models with $\lambda_{0}=200$ $\mu$m such as that
of Cooper et al. (2022) find a higher median $\beta$ for their sample than
that of da Cunha et al. (2021), though as we have already discussed, we find
$\beta$ values inconsistent with Cooper et al. (2022) even under identical
modeling assumptions.

Finally, we note that the general opacity models (with or without a power law)
did not necessarily provide a better fit to the FIR data than our optically
thin model, aside from being able account for shorter wavelength fluxes in the
cases where the power law was included.

Even if we cannot constrain with certainty the underlying dust properties or
opacity of the sources, we can conclude that our range of dust parameters are
consistent with theory and with recent studies of the local and high-redshift
universe. This is true whether we assume a general opacity or optically thin
model, though we caution that—as many authors have pointed out—making direct
comparisons between parameters derived using different models is not
straightforward. Allowing for differences in the assumed models, we confirm
that $\beta$ between 1.8 and 2.0 is appropriate for DSFGs at high redshift.

###  4.3 Dust Temperature Variation with Redshift

Although a number of studies have found that dust temperature increases with
redshift (e.g., Magdis et al., 2012; Magnelli et al., 2014; Béthermin et al.,
2015; Schreiber et al., 2018; Zavala et al., 2018; Sommovigo et al., 2022),
there is debate over whether this is simply a selection bias due to picking
out higher luminosity sources at higher redshifts. Other recent studies find
little to no evidence of temperature evolution with redshift when the
luminosity dependence is taken into account (Lim et al., 2020; Dudzevičiūtė et
al., 2020; Barger et al., 2022; Drew & Casey, 2022). The luminosity ranges of
these studies overlap with ours in part or in full, although Drew & Casey
(2022) consider a lower-redshift selection of $0<z<2$ galaxies with a
luminosity range that extends down to $L_{\text{IR}}\sim 10^{10}$
$\mathrm{L}_{\textnormal{$\odot$}}$.

Our robust subset is also consistent with no evolution. In Figure 11(a), we
show $T_{d}$ from our optically thin MBB fits plotted against FIR luminosity.
We see an increase in dust temperature with luminosity. From Figure 11(b), we
also see a general increase in $T_{d}$ with redshift. However, when we
restrict our analysis to sources with $L_{\text{IR}}>2\times 10^{12}$
$\mathrm{L}_{\textnormal{$\odot$}}$and $z<3.5$, where the sample is less
likely to suffer from incompleteness, we find the Pearson coefficient for the
temperature versus redshift relation is $r=0.10$, with a $p$ value of 0.61,
indicating no statistically significant correlation. This is shown in Figure
11(c); the lack of visible trend is clear. We note that since we employ photzs
for some of our sources and redshift and dust temperature are degenerate in
MBB SED fits, incorrect redshifts could affect our results. Thus, it is
possible we have underestimated the errors on the dust temperatures of some
sources.

##  5 Summary

We analyzed the dust properties of a large sample of GOODS-S galaxies selected
at 870 $\mu$m using new ALMA continuum observations at millimeter wavelengths.
Compared to other large samples of DSFGs, the ALMA observations, which probe
the RJ side of the FIR SED, reach some of the deepest depths at these
wavelengths. Here we summarize our main results:

  1. 1.

Using optically thin, isothermal MBB fits, we measured a median
$T_{d}=33.6^{+12.1}_{-5.4}$ K and a median $\beta=1.78^{+0.43}_{-0.25}$ for
our robust subset of 57 sources.

  2. 2.

We observed a negative correlation between $\beta$ and $T_{d}$ for our robust
subset. Since our FIR SEDs are relatively well-sampled (up to nine photometric
points from rest-frame 50–1500 $\mu$m) and based on the stacked likelihood
distributions, this relationship appears robust. We also confirm that it is
unlikely to have been produced by selection effects.

  3. 3.

We determined that the opacity assumptions used in the MBB fits can affect the
measured values for $\beta$ as well as $T_{d}$. We found that a general
opacity MBB with $\lambda_{0}=100$ $\mu$m gave similar values of $\beta$ to
optically thin fits, while a general opacity MBB with $\lambda_{0}=200$ $\mu$m
gave higher values of $\beta$. In all cases $T_{d}$ was higher for the general
opacity MBB fits.

  4. 4.

After restricting to sources in our robust subset with $L_{\text{IR}}>2\times
10^{12}$ $\mathrm{L}_{\textnormal{$\odot$}}$and $z<3.5$, we find no evidence
for temperature evolution from $z=1$ to $z=3.5$.

This work is one of only a few to directly measure the emissivity index and
dust temperature of individual DSFGs using deep ALMA millimeter imaging. We
find that the dust emission of DSFGs is well represented by modified
blackbodies with $\beta\sim 1.8$. Future observations of larger and fainter
samples of DSFGs using ALMA and TolTEC will confirm whether the dust
characteristics of DSFGs differ from local galaxies and help to disentangle
competing theories about the origins of their extreme dust masses and star
formation rates.

We thank the anonymous referee for constructive comments that helped us to
improve the manuscript. We gratefully acknowledge support for this research
from the William F. Vilas Estate (S. J. M.), a Kellett Mid-Career Award and a
WARF Named Professorship from the University of Wisconsin-Madison Office of
the Vice Chancellor for Research and Graduate Education with funding from the
Wisconsin Alumni Research Foundation (A. J. B.), NASA grant 80NSSC22K0483 (L.
L. C.), the Millennium Science Initiative Program – ICN12_009 (F. E. B), CATA-
Basal – FB210003 (F. E. B), and FONDECYT Regular – 1190818 (F. E. B) and
1200495 (F. E. B). The National Radio Astronomy Observatory is a facility of
the National Science Foundation operated under cooperative agreement by
Associated Universities, Inc. This paper makes use of the following ALMA data:
ADS/JAO.ALMA#2021.1.00024.S.
ALMA is a partnership of ESO (representing its member states), NSF (USA), and
NINS (Japan), together with NRC (Canada), MOST and ASIAA (Taiwan), and KASI
(Republic of Korea), in cooperation with the Republic of Chile. The Joint ALMA
Observatory is operated by ESO, AUI/NRAO, and NAOJ. The James Clerk Maxwell
Telescope is operated by the East Asian Observatory on behalf of The National
Astronomical Observatory of Japan, Academia Sinica Institute of Astronomy and
Astrophysics, the Korea Astronomy and Space Science Institute, the National
Astronomical Observatories of China and the Chinese Academy of Sciences (grant
No. XDB09000000), with additional funding support from the Science and
Technology Facilities Council of the United Kingdom and participating
universities in the United Kingdom and Canada. We wish to recognize and
acknowledge the very significant cultural role and reverence that the summit
of Maunakea has always had within the indigenous Hawaiian community. We are
most fortunate to have the opportunity to conduct observations from this
mountain.

##  Appendix A Flux Densities and Dust Properties

In Table 1, we list the positions and redshifts of the total sample, along
with the ALMA 1.2 mm, 2 mm, and 3 mm fluxes from the present work and the 1.1
mm fluxes from Gómez-Guijarro et al. (2022). In Table 2, we give the median
likelihood $\beta$ and $T_{d}$ values and the errors from the posterior
likelihood distributions for our optically thin MBB fits, as well as the FIR
luminosities and errors measured from the fits.

Table 1:  Total Sample Redshifts and Fluxes

C18 |  |  |  | Total | Peak | Peak | Peak
---|---|---|---|---|---|---|---
No. | R.A. | Decl. | $z$ | $f_{1.13\,{\rm mm}}$ | Error | $f_{1.24\,{\rm mm}}$ | Error | $f_{2\,{\rm mm}}$ | Error | $f_{3\,{\rm mm}}$ | Error
| J2000.0 | J2000.0 |  | (mJy) | (mJy) | (mJy) | (mJy)
1 | 53.030373 | -27.855804 | 2.574 |  $\cdots$ |  $\cdots$ | 2.48 | 0.08 | 0.61 | 0.02 | 0.146 | 0.019
2 | 53.047211 | -27.870001 | 3.690 |  $\cdots$ |  $\cdots$ | 2.71 | 0.09 | 0.54 | 0.02 | 0.119 | 0.019
3 | 53.063877 | -27.843779 | 2.648 | 2.06 | 0.10 | 1.55 | 0.1 | 0.33 | 0.02 | 0.078 | 0.02
4 | 53.020374 | -27.779917 | 2.252 | 2.75 | 0.09 | 1.87 | 0.08 | 0.27 | 0.04 | 0.076 | 0.04
5 | 53.118790 | -27.782888 | 2.309 | 2.15 | 0.11 |  $\cdots$ |  $\cdots$ | 0.26 | 0.03 | 0.114 | 0.043
[6] | 53.195126 | -27.855804 |  $\cdots$ |  $\cdots$ |  $\cdots$ | 1.41 | 0.08 | 0.41 | 0.02 | 0.039 | 0.021
7 | 53.158371 | -27.733612 | 3.672 | 2.15 | 0.12 | 1.36 | 0.06 | 0.38 | 0.02 | 0.081 | 0.019
8 | 53.105247 | -27.875195 | 2.69 (2.62-2.73) |  $\cdots$ |  $\cdots$ | 1.41 | 0.1 | 0.25 | 0.02 | 0.05 | 0.02
9 | 53.148876 | -27.821167 | 2.322 | 2.11 | 0.11 |  $\cdots$ |  $\cdots$ | 0.32 | 0.04 | 0.052 | 0.018
10 | 53.082085 | -27.767279 | 2.41 (2.34-2.46) | 1.34 | 0.11 | 0.73 | 0.1 | 0.26 | 0.02 | 0.032 | 0.022
[11] | 53.079376 | -27.870806 |  $\cdots$ | 2.05 | 0.12 | 0.99 | 0.1 | 0.28 | 0.02 | 0.024 | 0.018
12 | 53.142792 | -27.827888 | 3.764 | 2.25 | 0.12 | 1.26 | 0.08 |  $\cdots$ |  $\cdots$ | 0.055 | 0.02
13 | 53.074837 | -27.875916 | [2.73 (2.68-4.43)] | 1.67 | 0.10 | 0.7 | 0.06 | 0.33 | 0.02 | 0.033 | 0.018
14 | 53.092335 | -27.826834 | 2.73 (2.63-2.80) | 1.07 | 0.09 | 0.8 | 0.1 | 0.18 | 0.02 | 0.012 | 0.019
15 | 53.024292 | -27.805695 | 2.14 (2.06-2.23) |  $\cdots$ |  $\cdots$ | 0.47 | 0.07 | 0.17 | 0.02 | 0.006 | 0.02
16 | 53.082752 | -27.866585 | 3.37 (3.17-3.41) | 1.40 | 0.12 | 0.92 | 0.11 | 0.28 | 0.03 |  $\cdots$ |  $\cdots$
17 | 53.146629 | -27.871029 | 3.57 (3.45-3.81) |  $\cdots$ |  $\cdots$ | 0.81 | 0.08 | 0.21 | 0.02 | 0.069 | 0.02
18 | 53.092834 | -27.801332 | 3.847 | 1.39 | 0.10 | 0.9 | 0.08 | 0.23 | 0.02 | 0.084 | 0.019
19 | 53.108795 | -27.869028 | [4.47 (4.07-5.65)] | 1.24 | 0.10 | 1.17 | 0.09 | 0.24 | 0.02 | 0.08 | 0.018
20 | 53.198292 | -27.747889 | 1.93 (1.89-1.96) | 0.65 | 0.12 | 0.43 | 0.1 | 0.16 | 0.02 |  $\cdots$ | 0.018
21 | 53.178333 | -27.870222 | 3.78 (3.70-3.83) |  $\cdots$ |  $\cdots$ | 0.78 | 0.09 | 0.18 | 0.02 | 0.06 | 0.016
22 | 53.183460 | -27.776638 | 2.698 | 1.49 | 0.13 |  $\cdots$ |  $\cdots$ |  $\cdots$ |  $\cdots$ |  $\cdots$ |  $\cdots$
23 | 53.157207 | -27.833500 | 1.58 (1.56-1.61) | 1.36 | 0.12 | 0.47 | 0.07 | 0.14 | 0.02 | 0.03 | 0.018
24 | 53.102791 | -27.892860 | 1.96 (1.90-1.99) |  $\cdots$ |  $\cdots$ | 0.61 | 0.06 | 0.18 | 0.02 | 0.044 | 0.017
25 | 53.181377 | -27.777557 | 2.696 | 1.41 | 0.12 |  $\cdots$ |  $\cdots$ |  $\cdots$ |  $\cdots$ |  $\cdots$ |  $\cdots$
26 | 53.070251 | -27.845612 | 3.78 (3.70-3.82) | 0.83 | 0.09 | 0.51 | 0.08 | 0.19 | 0.02 | 0.066 | 0.02
27 | 53.014584 | -27.844389 | [1.78 (1.67-2.73)] |  $\cdots$ |  $\cdots$ | 0.52 | 0.08 | 0.16 | 0.02 | 0.053 | 0.018
28 | 53.139290 | -27.890722 | [3.33 (3.20-4.38)] |  $\cdots$ |  $\cdots$ | 0.56 | 0.08 | 0.13 | 0.02 | 0.05 | 0.02
[29] | 53.137127 | -27.761389 |  $\cdots$ | 0.92 | 0.10 | 0.68 | 0.09 | 0.13 | 0.02 | 0.031 | 0.019
30 | 53.071709 | -27.843693 | 1.86 (1.81-1.92) | 0.83 | 0.10 | 0.46 | 0.08 | 0.09 | 0.02 | 0.02 | 0.019
31 | 53.077377 | -27.859612 | 1.95 (1.89-2.00) | 0.51 | 0.10 | 0.44 | 0.07 | 0.07 | 0.02 | 0.011 | 0.022
32 | 53.049751 | -27.770971 | 2.75 (2.57-2.78) | 0.93 | 0.10 | 0.47 | 0.07 | 0.12 | 0.02 | 0.042 | 0.017
33 | 53.072708 | -27.834278 | 1.58 (1.53-1.61) | 0.69 | 0.11 | 0.49 | 0.08 | 0.08 | 0.02 | 0.057 | 0.019
34 | 53.090752 | -27.782473 | 1.95 (1.86-1.97) | 0.76 | 0.09 | 0.35 | 0.07 | 0.01 | 0.03 |  $\cdots$ | 0.022
35 | 53.091747 | -27.712166 | 1.612 |  $\cdots$ |  $\cdots$ | 0.35 | 0.09 | 0.1 | 0.02 | 0.042 | 0.018
36 | 53.086586 | -27.810249 | 2.37 (2.28-2.42) | 0.49 | 0.10 | 0.44 | 0.07 | 0.04 | 0.02 | 0.025 | 0.018
37 | 53.146378 | -27.888807 | 2.96 (2.87-3.05) |  $\cdots$ |  $\cdots$ | 0.43 | 0.09 | 0.1 | 0.03 | 0.011 | 0.018
38 | 53.092335 | -27.803223 | 2.31 (2.26-2.38) | 0.85 | 0.11 | 0.8 | 0.09 | 0.12 | 0.02 |  $\cdots$ | 0.019
39 | 53.124332 | -27.882696 | 3.04 (2.99-3.21) |  $\cdots$ |  $\cdots$ | 0.35 | 0.08 | 0.04 | 0.02 | 0.032 | 0.018
40 | 53.131123 | -27.773195 | 2.223 | 0.72 | 0.11 | 0.56 | 0.05 | 0.12 | 0.03 | 0.015 | 0.026
41 | 53.172832 | -27.858860 | [4.13 (3.45-4.46)] |  $\cdots$ |  $\cdots$ | 0.66 | 0.11 | 0.16 | 0.02 | 0.039 | 0.019
42 | 53.091629 | -27.853390 | 2.34 (2.30-2.42) | 0.81 | 0.10 | 0.41 | 0.09 | 0.12 | 0.02 | 0.029 | 0.02
43 | 53.068874 | -27.879723 | 2.39 (2.32-2.62) | 1.03 | 0.11 | 0.83 | 0.07 | 0.16 | 0.03 | 0.034 | 0.021
[44] | 53.087166 | -27.840195 |  $\cdots$ | 0.85 | 0.11 | 0.73 | 0.09 | 0.18 | 0.02 | 0.07 | 0.017
[45] | 53.041084 | -27.837721 | [7.62 (7.15-7.93)] |  $\cdots$ |  $\cdots$ | 0.54 | 0.08 | 0.14 | 0.02 | 0.022 | 0.018
46 | 53.104912 | -27.705305 | 1.613 |  $\cdots$ |  $\cdots$ |  $\cdots$ |  $\cdots$ |  $\cdots$ |  $\cdots$ | 0.049 | 0.024
47 | 53.163540 | -27.890556 | 2.19 (2.12-2.22) |  $\cdots$ |  $\cdots$ | 0.5 | 0.1 | 0.07 | 0.02 | 0.018 | 0.019
48 | 53.160664 | -27.776251 | 2.543 | 0.99 | 0.12 |  $\cdots$ |  $\cdots$ |  $\cdots$ |  $\cdots$ |  $\cdots$ |  $\cdots$
49 | 53.053669 | -27.869278 | 1.87 (1.84-1.93) |  $\cdots$ |  $\cdots$ | 0.5 | 0.11 | 0.07 | 0.02 |  $\cdots$ | 0.019
50 | 53.089542 | -27.711666 | 1.69 (1.64-1.70) |  $\cdots$ |  $\cdots$ | 0.3 | 0.08 | 0.07 | 0.03 | 0.012 | 0.022
51 | 53.067833 | -27.728889 | 2.32 (2.29-2.43) |  $\cdots$ |  $\cdots$ | 0.39 | 0.08 | 0.09 | 0.02 | 0.028 | 0.019
52 | 53.064793 | -27.862638 | [4.78 (4.35-5.10)] | 0.54 | 0.10 |  $\cdots$ |  $\cdots$ | 0.14 | 0.03 | 0.002 | 0.017
53 | 53.198875 | -27.843945 | 1.56 (1.50-1.60) | 1.01 | 0.12 |  $\cdots$ |  $\cdots$ |  $\cdots$ |  $\cdots$ |  $\cdots$ |  $\cdots$
[54] | 53.181995 | -27.814196 | [9.42 (9.35-9.83)] |  $\cdots$ |  $\cdots$ |  $\cdots$ |  $\cdots$ | 0.06 | 0.02 | -0.02 | 0.02
[55] | 53.048378 | -27.770306 |  $\cdots$ | 0.71 | 0.11 | 0.24 | 0.1 | 0.06 | 0.02 | 0.01 | 0.019
56 | 53.107044 | -27.718334 | 2.299 |  $\cdots$ |  $\cdots$ |  $\cdots$ |  $\cdots$ | 0.1 | 0.03 |  $\cdots$ |  $\cdots$
[57] | 53.033127 | -27.816778 | 3.08 (3.00-3.68) |  $\cdots$ |  $\cdots$ |  $\cdots$ |  $\cdots$ |  $\cdots$ |  $\cdots$ |  $\cdots$ |  $\cdots$
[58] | 53.183666 | -27.836500 | [4.73 (4.39-4.90)] | 1.23 | 0.12 |  $\cdots$ |  $\cdots$ | 0.19 | 0.03 | 0.042 | 0.018
59 | 53.094044 | -27.804195 | 2.325 |  $\cdots$ |  $\cdots$ | 0.37 | 0.09 | 0.07 | 0.03 | 0.01 | 0.02
[60] | 53.124584 | -27.893305 | 2.53 (2.41-2.60) |  $\cdots$ |  $\cdots$ |  $\cdots$ |  $\cdots$ |  $\cdots$ |  $\cdots$ |  $\cdots$ |  $\cdots$
[61] | 53.132751 | -27.720278 | [4.67 (4.48-5.23)] |  $\cdots$ |  $\cdots$ |  $\cdots$ |  $\cdots$ |  $\cdots$ |  $\cdots$ |  $\cdots$ |  $\cdots$
[62] | 53.080669 | -27.720861 | 2.94 (2.88-3.03) |  $\cdots$ |  $\cdots$ |  $\cdots$ |  $\cdots$ |  $\cdots$ |  $\cdots$ |  $\cdots$ |  $\cdots$
63 | 53.120041 | -27.808277 | 1.83 (1.78-1.88) | 0.55 | 0.12 |  $\cdots$ |  $\cdots$ |  $\cdots$ |  $\cdots$ |  $\cdots$ |  $\cdots$
[64] | 53.117085 | -27.874918 | 3.26 (3.20-3.40) |  $\cdots$ |  $\cdots$ |  $\cdots$ |  $\cdots$ |  $\cdots$ |  $\cdots$ |  $\cdots$ |  $\cdots$
65 | 53.131458 | -27.841389 | 1.58 (1.56-1.62) | 0.74 | 0.10 |  $\cdots$ |  $\cdots$ |  $\cdots$ |  $\cdots$ |  $\cdots$ |  $\cdots$
[66] | 53.044708 | -27.802027 | 0.653 |  $\cdots$ |  $\cdots$ |  $\cdots$ |  $\cdots$ |  $\cdots$ |  $\cdots$ |  $\cdots$ |  $\cdots$
[67] | 53.072002 | -27.819000 | 1.69 (1.66-1.84) |  $\cdots$ |  $\cdots$ |  $\cdots$ |  $\cdots$ |  $\cdots$ |  $\cdots$ |  $\cdots$ |  $\cdots$
68 | 53.120461 | -27.742083 | 5.58 | 0.95 | 0.12 |  $\cdots$ |  $\cdots$ |  $\cdots$ |  $\cdots$ |  $\cdots$ |  $\cdots$
[69] | 53.113125 | -27.886639 | 2.55 (2.47-2.64) |  $\cdots$ |  $\cdots$ |  $\cdots$ |  $\cdots$ |  $\cdots$ |  $\cdots$ |  $\cdots$ |  $\cdots$
70 | 53.141251 | -27.872860 | 3.14 (3.06-3.34) |  $\cdots$ |  $\cdots$ |  $\cdots$ | 0.06 | 0.04 | 0.02 |  $\cdots$ | 0.019
[71] | 53.056873 | -27.798389 | 1.71 (1.63-1.72) |  $\cdots$ |  $\cdots$ |  $\cdots$ |  $\cdots$ |  $\cdots$ |  $\cdots$ |  $\cdots$ |  $\cdots$
72 | 53.119957 | -27.743137 | [3.76 (3.47-4.31)] | 0.71 | 0.12 |  $\cdots$ |  $\cdots$ |  $\cdots$ |  $\cdots$ |  $\cdots$ |  $\cdots$
73 | 53.142872 | -27.874084 | 2.19 (2.09-2.22) |  $\cdots$ |  $\cdots$ | 0.04 | 0.08 | 0.01 | 0.02 | -0.01 | 0.015
74 | 53.093666 | -27.826445 | 0.732 |  $\cdots$ |  $\cdots$ | 0.19 | 0.1 | 0.04 | 0.03 | 0.017 | 0.018
[75] | 53.074837 | -27.787111 |  $\cdots$ |  $\cdots$ |  $\cdots$ |  $\cdots$ |  $\cdots$ |  $\cdots$ |  $\cdots$ |  $\cdots$ |  $\cdots$

Note. — Columns: (1) Source number from Table 4 of C18 (brackets refer to
sources not in the robust subset), (2) and (3) ALMA 870 $\mu$m R.A. and decl.,
(4) adopted redshift taken from the compilation in Cowie et al. (2023) (three
digits after the decimal point for spectroscopic redshifts—except for the JWST
NIRSpec redshift for source 68 from Oesch et al. (2023)—and two digits after
the decimal point for photometric redshifts, plus 68% confidence ranges from
Straatman et al. 2016 for photometric redshifts, given in parentheses), (5)
and (6) total ALMA 1.13 mm flux and error from Gómez-Guijarro et al. (2022),
(7) and (8) measured peak ALMA 1.24 mm flux and error from this work, (9) and
(10) measured peak ALMA 2 mm flux and error from this work, (11) and (12)
measured peak ALMA 3 mm flux and error from this work. In Column (4), values
in brackets refer to photzs which had quality flag $Q>3$ in the catalog of
Straatman et al. (2016).

Table 2:  Robust Subset Dust Properties

No. | $z$ | $\beta$ |  $T_{d}$/K | $\log(L_{\rm IR}/\mathrm{L}_{\textnormal{$\odot$}})$
---|---|---|---|---
1 | 2.574 | 1.53${}^{+0.1}_{-0.1}$ | 32.8${}^{+1.9}_{-1.7}$ | 12.79${}^{+0.04}_{-0.05}$
2 | 3.69 | 1.84${}^{+0.12}_{-0.12}$ | 34.5${}^{+2.1}_{-2.0}$ | 12.98${}^{+0.05}_{-0.06}$
3 | 2.648 | 1.73${}^{+0.14}_{-0.14}$ | 36.1${}^{+2.6}_{-2.3}$ | 12.84${}^{+0.05}_{-0.05}$
4 | 2.252 | 2.01${}^{+0.17}_{-0.16}$ | 31.4${}^{+2.4}_{-2.1}$ | 12.84${}^{+0.05}_{-0.05}$
5 | 2.309 | 1.75${}^{+0.16}_{-0.15}$ | 32.7${}^{+2.9}_{-2.5}$ | 12.66${}^{+0.05}_{-0.06}$
7 | 3.672 | 1.45${}^{+0.14}_{-0.14}$ | 43.4${}^{+3.7}_{-3.4}$ | 12.88${}^{+0.08}_{-0.07}$
8 | 2.69 | 2.21${}^{+0.2}_{-0.18}$ | 26.6${}^{+2.7}_{-2.4}$ | 12.53${}^{+0.09}_{-0.09}$
9 | 2.322 | 1.71${}^{+0.16}_{-0.16}$ | 32.8${}^{+2.7}_{-2.4}$ | 12.63${}^{+0.07}_{-0.07}$
10 | 2.41 | 1.63${}^{+0.21}_{-0.2}$ | 37.1${}^{+4.5}_{-3.7}$ | 12.64${}^{+0.08}_{-0.08}$
12 | 3.76 | 1.86${}^{+0.25}_{-0.23}$ | 38.4${}^{+4.0}_{-3.6}$ | 12.86${}^{+0.08}_{-0.07}$
13 | 2.73 | 0.99${}^{+0.16}_{-0.12}$ | 49.7${}^{+5.8}_{-5.7}$ | 12.61${}^{+0.12}_{-0.12}$
14 | 2.73 | 1.96${}^{+0.21}_{-0.2}$ | 45.4${}^{+6.1}_{-4.9}$ | 13.14${}^{+0.1}_{-0.1}$
15 | 2.14 | 2.84${}^{+0.44}_{-0.38}$ | 20.6${}^{+3.2}_{-2.9}$ | 12.19${}^{+0.1}_{-0.13}$
16 | 3.37 | 1.71${}^{+0.28}_{-0.25}$ | 36.2${}^{+5.7}_{-4.9}$ | 12.62${}^{+0.14}_{-0.13}$
17 | 3.57 | 2.12${}^{+0.27}_{-0.25}$ | 27.9${}^{+3.6}_{-3.2}$ | 12.36${}^{+0.11}_{-0.1}$
18 | 3.847 | 1.57${}^{+0.22}_{-0.21}$ | 49.4${}^{+7.5}_{-6.0}$ | 12.99${}^{+0.11}_{-0.12}$
19 | 4.47 | 1.44${}^{+0.25}_{-0.22}$ | 48.0${}^{+12.1}_{-9.0}$ | 12.82${}^{+0.21}_{-0.22}$
20 | 1.93 | 1.55${}^{+0.16}_{-0.16}$ | 42.8${}^{+3.3}_{-2.8}$ | 12.65${}^{+0.05}_{-0.04}$
21 | 3.78 | 1.91${}^{+0.26}_{-0.24}$ | 37.0${}^{+5.7}_{-5.1}$ | 12.67${}^{+0.16}_{-0.16}$
22 | 2.698 | 1.37${}^{+0.55}_{-0.4}$ | 45.7${}^{+13.6}_{-10.8}$ | 12.76${}^{+0.17}_{-0.2}$
23 | 1.58 | 1.78${}^{+0.14}_{-0.14}$ | 33.8${}^{+1.9}_{-1.7}$ | 12.44${}^{+0.04}_{-0.04}$
24 | 1.96 | 1.44${}^{+0.13}_{-0.12}$ | 42.0${}^{+2.6}_{-2.3}$ | 12.63${}^{+0.04}_{-0.04}$
25 | 2.696 | 1.75${}^{+0.62}_{-0.55}$ | 33.1${}^{+8.5}_{-6.2}$ | 12.44${}^{+0.1}_{-0.13}$
26 | 3.78 | 1.03${}^{+0.23}_{-0.15}$ | 69.2${}^{+12.1}_{-12.9}$ | 12.98${}^{+0.18}_{-0.23}$
27 | 1.78 | 1.90${}^{+0.35}_{-0.31}$ | 22.3${}^{+3.4}_{-3.1}$ | 11.81${}^{+0.1}_{-0.14}$
28 | 3.33 | 1.78${}^{+0.39}_{-0.36}$ | 36.4${}^{+8.0}_{-6.1}$ | 12.44${}^{+0.14}_{-0.18}$
30 | 1.86 | 2.00${}^{+0.2}_{-0.19}$ | 30.2${}^{+2.3}_{-2.1}$ | 12.31${}^{+0.06}_{-0.06}$
31 | 1.95 | 2.19${}^{+0.23}_{-0.23}$ | 31.7${}^{+2.5}_{-2.2}$ | 12.40${}^{+0.05}_{-0.05}$
32 | 2.75 | 1.76${}^{+0.3}_{-0.29}$ | 45.7${}^{+10.1}_{-7.1}$ | 12.82${}^{+0.18}_{-0.14}$
33 | 1.58 | 2.15${}^{+0.17}_{-0.17}$ | 30.2${}^{+1.8}_{-1.6}$ | 12.41${}^{+0.04}_{-0.04}$
34 | 1.95 | 2.09${}^{+0.23}_{-0.23}$ | 33.6${}^{+2.7}_{-2.3}$ | 12.42${}^{+0.05}_{-0.05}$
35 | 1.612 | 1.74${}^{+0.18}_{-0.17}$ | 32.3${}^{+2.5}_{-2.2}$ | 12.25${}^{+0.05}_{-0.05}$
36 | 2.37 | 3.32${}^{+0.43}_{-0.49}$ | 20.1${}^{+3.3}_{-2.3}$ | 12.09${}^{+0.09}_{-0.1}$
37 | 2.96 | 2.79${}^{+0.58}_{-0.52}$ | 20.9${}^{+4.9}_{-4.1}$ | 11.94${}^{+0.23}_{-0.24}$
38 | 2.31 | 1.78${}^{+0.23}_{-0.22}$ | 33.8${}^{+4.3}_{-3.6}$ | 12.39${}^{+0.1}_{-0.1}$
39 | 3.04 | 2.92${}^{+0.48}_{-0.44}$ | 31.5${}^{+6.0}_{-4.7}$ | 12.71${}^{+0.14}_{-0.16}$
40 | 2.223 | 1.93${}^{+0.33}_{-0.31}$ | 30.9${}^{+5.2}_{-4.0}$ | 12.28${}^{+0.12}_{-0.11}$
41 | 4.13 | 1.92${}^{+0.55}_{-0.43}$ | 29.2${}^{+11.2}_{-8.4}$ | 12.15${}^{+0.35}_{-0.25}$
42 | 2.34 | 1.88${}^{+0.37}_{-0.33}$ | 35.3${}^{+6.8}_{-5.2}$ | 12.46${}^{+0.12}_{-0.1}$
43 | 2.39 | 1.54${}^{+0.27}_{-0.26}$ | 37.0${}^{+6.1}_{-4.7}$ | 12.45${}^{+0.13}_{-0.1}$
46 | 1.613 | 1.02${}^{+0.19}_{-0.14}$ | 47.0${}^{+3.7}_{-4.1}$ | 12.44${}^{+0.05}_{-0.05}$
47 | 2.19 | 2.17${}^{+0.35}_{-0.33}$ | 28.3${}^{+4.8}_{-3.9}$ | 12.20${}^{+0.12}_{-0.12}$
48 | 2.543 | 2.54${}^{+0.65}_{-0.65}$ | 32.2${}^{+10.4}_{-6.1}$ | 12.67${}^{+0.17}_{-0.11}$
49 | 1.87 | 1.51${}^{+0.29}_{-0.27}$ | 34.6${}^{+4.6}_{-4.0}$ | 12.11${}^{+0.08}_{-0.1}$
50 | 1.69 | 2.22${}^{+0.33}_{-0.3}$ | 29.4${}^{+3.6}_{-2.8}$ | 12.21${}^{+0.07}_{-0.1}$
51 | 2.32 | 1.99${}^{+0.47}_{-0.42}$ | 28.2${}^{+6.6}_{-4.9}$ | 12.07${}^{+0.15}_{-0.16}$
52 | 4.78 | 1.69${}^{+0.59}_{-0.41}$ | 48.1${}^{+20.0}_{-15.1}$ | 12.62${}^{+0.31}_{-0.36}$
53 | 1.56 | 1.15${}^{+0.35}_{-0.24}$ | 38.7${}^{+4.3}_{-4.3}$ | 12.12${}^{+0.06}_{-0.07}$
56 | 2.299 | 1.62${}^{+0.59}_{-0.46}$ | 32.8${}^{+11.1}_{-7.2}$ | 12.04${}^{+0.21}_{-0.23}$
59 | 2.325 | 3.09${}^{+0.54}_{-0.55}$ | 16.1${}^{+3.6}_{-2.7}$ | 11.59${}^{+0.15}_{-0.18}$
63 | 1.83 | 1.79${}^{+0.42}_{-0.41}$ | 33.5${}^{+5.4}_{-3.9}$ | 12.16${}^{+0.07}_{-0.08}$
65 | 1.58 | 1.66${}^{+0.41}_{-0.4}$ | 30.7${}^{+4.7}_{-3.6}$ | 11.98${}^{+0.08}_{-0.08}$
68 | 5.58 | 1.59${}^{+0.69}_{-0.47}$ | 69.6${}^{+14.4}_{-25.0}$ | 13.07${}^{+0.27}_{-0.53}$
70 | 3.14 | 2.49${}^{+0.79}_{-0.71}$ | 37.9${}^{+18.0}_{-9.6}$ | 12.55${}^{+0.25}_{-0.26}$
72 | 3.76 | 1.58${}^{+0.86}_{-0.44}$ | 64.2${}^{+17.3}_{-20.2}$ | 12.97${}^{+0.24}_{-0.27}$
73 | 2.19 | 3.49${}^{+0.38}_{-0.73}$ | 22.3${}^{+6.5}_{-2.9}$ | 12.03${}^{+0.18}_{-0.34}$
74 | 0.7323 | 2.06${}^{+0.15}_{-0.14}$ | 26.9${}^{+0.9}_{-0.9}$ | 11.79${}^{+0.02}_{-0.02}$

Note. — Columns: (1) Source number from Table 4 of C18, (2) adopted redshift
taken from the compilation in Cowie et al. (2023) and used in MBB fits, (3)
median likelihood emissivity index and error (16th to 84th percentile range
from the likelihood distribution) from the optically thin MBB fits, (4) median
likelihood dust temperature and error (16th to 84th percentile range from the
likelihood distribution) from the optically thin MBB fits, (3) FIR luminosity
measured from the median likelihood model and error (16th to 84th percentile
range from the distribution of accepted MCMC models) from the optically thin
MBB fits. Speczs in Column (4) are listed with 3 decimal places—except for the
JWST NIRSpec redshift for source 68 from Oesch et al. (2023)—and photzs with 2
decimal places; values in brackets refer to photzs which had quality flag
$Q>3$ in the catalog of Straatman et al. (2016).

##  Appendix B SED Fits

In Figure 12, we show the MBB SED fits for the 69 sources from the total
sample that have redshifts.

Figure 12: Optically thin MBB SED fits (black curves) and 16th to 84th
percentile ranges of the accepted MCMC models (blue shaded regions) for the 69
sources with redshifts in Table 1. Photometry: Red circles—ALMA 1.1 mm, 1.2
mm, 2 mm, and 3 mm, maroon pentagon—ALMA 870 $\mu$m, green square—SCUBA-2 450
$\mu$m, dark red stars—Herschel/PACS 100 and 160 $\mu$m and SPIRE 250 and 350
$\mu$m, blue triangles—Spitzer/MIPS 70 $\mu$m. The fits are made only to the
data at wavelengths greater or equal to rest-frame 50 $\mu$m (points not
included in the fits are marked with black squares).
The entire figure set (69 images) is available in the online journal.

Fig. Set12. MBB SED Fits

## References

  * Agladze et al. (1996) Agladze, N. I., Sievers, A. J., Jones, S. A., Burlitch, J. M., & Beckwith, S. V. W. 1996, ApJ, 462, 1026
  * Aravena et al. (2016) Aravena, M., Decarli, R., Walter, F., et al. 2016, ApJ, 833, 68
  * Astropy Collaboration et al. (2022) Astropy Collaboration, Price-Whelan, A. M., Lim, P. L., et al. 2022, ApJ, 935, 167
  * Barger et al. (2022) Barger, A. J., Cowie, L. L., Blair, A. H., & Jones, L. H. 2022, ApJ, 934, 56
  * Barger et al. (1998) Barger, A. J., Cowie, L. L., Sanders, D. B., et al. 1998, Nature, 394, 248
  * Barger et al. (2012) Barger, A. J., Wang, W. H., Cowie, L. L., et al. 2012, ApJ, 761, 89
  * Béthermin et al. (2015) Béthermin, M., Daddi, E., Magdis, G., et al. 2015, A&A, 573, A113
  * Biggs et al. (2011) Biggs, A. D., Ivison, R. J., Ibar, E., et al. 2011, MNRAS, 413, 2314
  * Blain et al. (2002) Blain, A. W., Smail, I., Ivison, R. J., Kneib, J. P., & Frayer, D. T. 2002, Phys. Rep., 369, 111
  * Boudet et al. (2005) Boudet, N., Mutschke, H., Nayral, C., et al. 2005, ApJ, 633, 272
  * Casey (2012) Casey, C. M. 2012, MNRAS, 425, 3094
  * Casey et al. (2014) Casey, C. M., Narayanan, D., & Cooray, A. 2014, Phys. Rep., 541, 45
  * Casey et al. (2019) Casey, C. M., Zavala, J. A., Aravena, M., et al. 2019, ApJ, 887, 55
  * Casey et al. (2021) Casey, C. M., Zavala, J. A., Manning, S. M., et al. 2021, ApJ, 923, 215
  * Chapin et al. (2011) Chapin, E. L., Chapman, S. C., Coppin, K. E., et al. 2011, MNRAS, 411, 505
  * Chen et al. (2023) Chen, J., Ivison, R. J., Zwaan, M. A., et al. 2023, MNRAS, 518, 1378
  * Clements et al. (2018) Clements, D. L., Pearson, C., Farrah, D., et al. 2018, MNRAS, 475, 2097
  * Conley et al. (2011) Conley, A., Cooray, A., Vieira, J. D., et al. 2011, ApJ, 732, L35
  * Cooper et al. (2022) Cooper, O. R., Casey, C. M., Zavala, J. A., et al. 2022, ApJ, 930, 32
  * Cowie et al. (2023) Cowie, L. L., Barger, A. J., & Bauer, F. E. 2023, ApJ, submitted
  * Cowie et al. (2017) Cowie, L. L., Barger, A. J., Hsu, L. Y., et al. 2017, ApJ, 837, 139
  * Cowie et al. (2018) Cowie, L. L., González-López, J., Barger, A. J., et al. 2018, ApJ, 865, 106
  * da Cunha et al. (2013) da Cunha, E., Groves, B., Walter, F., et al. 2013, ApJ, 766, 13
  * da Cunha et al. (2015) da Cunha, E., Walter, F., Smail, I. R., et al. 2015, ApJ, 806, 110
  * da Cunha et al. (2021) da Cunha, E., Hodge, J. A., Casey, C. M., et al. 2021, ApJ, 919, 30
  * Draine & Lee (1984) Draine, B. T., & Lee, H. M. 1984, ApJ, 285, 89
  * Drew & Casey (2022) Drew, P. M., & Casey, C. M. 2022, ApJ, 930, 142
  * Dudzevičiūtė et al. (2020) Dudzevičiūtė, U., Smail, I., Swinbank, A. M., et al. 2020, MNRAS, 494, 3828
  * Dudzevičiūtė et al. (2021) —. 2021, MNRAS, 500, 942
  * Dunne & Eales (2001) Dunne, L., & Eales, S. A. 2001, MNRAS, 327, 697
  * Eales et al. (1999) Eales, S., Lilly, S., Gear, W., et al. 1999, ApJ, 515, 518
  * Elbaz et al. (2011) Elbaz, D., Dickinson, M., Hwang, H. S., et al. 2011, A&A, 533, A119
  * Foreman-Mackey et al. (2013) Foreman-Mackey, D., Hogg, D. W., Lang, D., & Goodman, J. 2013, PASP, 125, 306
  * Galliano et al. (2018) Galliano, F., Galametz, M., & Jones, A. P. 2018, ARA&A, 56, 673
  * Gómez-Guijarro et al. (2022) Gómez-Guijarro, C., Elbaz, D., Xiao, M., et al. 2022, A&A, 658, A43
  * Hayward et al. (2011) Hayward, C. C., Kereš, D., Jonsson, P., et al. 2011, ApJ, 743, 159
  * Hodge & da Cunha (2020) Hodge, J. A., & da Cunha, E. 2020, Royal Society Open Science, 7, 200556
  * Hodge et al. (2013) Hodge, J. A., Karim, A., Smail, I., et al. 2013, ApJ, 768, 91
  * Hughes et al. (1998) Hughes, D. H., Serjeant, S., Dunlop, J., et al. 1998, Nature, 394, 241
  * Inoue et al. (2020) Inoue, A. K., Hashimoto, T., Chihara, H., & Koike, C. 2020, MNRAS, 495, 1577
  * Jin et al. (2019) Jin, S., Daddi, E., Magdis, G. E., et al. 2019, ApJ, 887, 144
  * Karim et al. (2013) Karim, A., Swinbank, A. M., Hodge, J. A., et al. 2013, MNRAS, 432, 2
  * Kelly et al. (2012) Kelly, B. C., Shetty, R., Stutz, A. M., et al. 2012, ApJ, 752, 55
  * Kovács et al. (2006) Kovács, A., Chapman, S. C., Dowell, C. D., et al. 2006, ApJ, 650, 592
  * Lim et al. (2020) Lim, C.-F., Wang, W.-H., Smail, I., et al. 2020, ApJ, 889, 80
  * Magdis et al. (2012) Magdis, G. E., Daddi, E., Béthermin, M., et al. 2012, ApJ, 760, 6
  * Magnelli et al. (2014) Magnelli, B., Lutz, D., Saintonge, A., et al. 2014, A&A, 561, A86
  * McMullin et al. (2007) McMullin, J. P., Waters, B., Schiebel, D., Young, W., & Golap, K. 2007, in Astronomical Society of the Pacific Conference Series, Vol. 376, Astronomical Data Analysis Software and Systems XVI, ed. R. A. Shaw, F. Hill, & D. J. Bell, 127
  * Oesch et al. (2023) Oesch, P. A., Brammer, G., Naidu, R. P., et al. 2023, arXiv e-prints, arXiv:2304.02026.  https://arxiv.org/abs/2304.02026
  * Oliver et al. (2012) Oliver, S. J., Bock, J., Altieri, B., et al. 2012, MNRAS, 424, 1614
  * Ono et al. (2014) Ono, Y., Ouchi, M., Kurono, Y., & Momose, R. 2014, ApJ, 795, 5
  * Paradis et al. (2009) Paradis, D., Bernard, J. P., & Mény, C. 2009, A&A, 506, 745
  * Paradis et al. (2010) Paradis, D., Veneziani, M., Noriega-Crespo, A., et al. 2010, A&A, 520, L8
  * Planck Collaboration et al. (2011) Planck Collaboration, Abergel, A., Ade, P. A. R., et al. 2011, A&A, 536, A25
  * Planck Collaboration et al. (2014) —. 2014, A&A, 571, A11
  * Schreiber et al. (2018) Schreiber, C., Elbaz, D., Pannella, M., et al. 2018, A&A, 609, A30
  * Scoville et al. (2014) Scoville, N., Aussel, H., Sheth, K., et al. 2014, ApJ, 783, 84
  * Scoville et al. (2016) Scoville, N., Sheth, K., Aussel, H., et al. 2016, ApJ, 820, 83
  * Shetty et al. (2009a) Shetty, R., Kauffmann, J., Schnee, S., & Goodman, A. A. 2009a, ApJ, 696, 676
  * Shetty et al. (2009b) Shetty, R., Kauffmann, J., Schnee, S., Goodman, A. A., & Ercolano, B. 2009b, ApJ, 696, 2234
  * Simpson et al. (2017) Simpson, J. M., Smail, I., Swinbank, A. M., et al. 2017, ApJ, 839, 58
  * Smail et al. (1997) Smail, I., Ivison, R. J., & Blain, A. W. 1997, ApJ, 490, L5
  * Sommovigo et al. (2022) Sommovigo, L., Ferrara, A., Pallottini, A., et al. 2022, MNRAS, 513, 3122
  * Stach et al. (2019) Stach, S. M., Dudzevičiūtė, U., Smail, I., et al. 2019, MNRAS, 487, 4648
  * Straatman et al. (2016) Straatman, C. M. S., Spitler, L. R., Quadri, R. F., et al. 2016, ApJ, 830, 51
  * Suzuki et al. (2021) Suzuki, T. L., Onodera, M., Kodama, T., et al. 2021, ApJ, 908, 15
  * Swinbank et al. (2014) Swinbank, A. M., Simpson, J. M., Smail, I., et al. 2014, MNRAS, 438, 1267
  * Wilson et al. (2020) Wilson, G. W., Abi-Saad, S., Ade, P., et al. 2020, in Society of Photo-Optical Instrumentation Engineers (SPIE) Conference Series, Vol. 11453, Society of Photo-Optical Instrumentation Engineers (SPIE) Conference Series, 1145302
  * Zavala et al. (2018) Zavala, J. A., Aretxaga, I., Dunlop, J. S., et al. 2018, MNRAS, 475, 5585
  * Zavala et al. (2021) Zavala, J. A., Casey, C. M., Manning, S. M., et al. 2021, ApJ, 909, 165