# Little evolution of dust emissivity in bright infrared galaxies from 2 < _z_
< 6

B. A. Ward,${}^{\\!1}$ S. A. Eales,${}^{\\!1}$ R. J. Ivison2,3,4,5 and V.
Arumugam6

1Cardiff Hub for Astrophysics Research and Technology (CHART), School of
Physics and Astronomy, Cardiff University, The Parade, Cardiff CF24 3AA
2European Southern Observatory (ESO), Karl-Schwarzschild-Strasse 2, D-85748
Garching, Germany
3ARC Centre of Excellence for All Sky Astrophysics in 3 Dimensions (ASTRO 3D)
4School of Cosmic Physics, Dublin Institute for Advanced Studies, 31
Fitzwilliam Place, Dublin D02 XF86, Ireland
5Institute for Astronomy, University of Edinburgh, Royal Observatory,
Blackford Hill, Edinburgh EH9 3HJ
6Institut de Radioastronomie Millimétrique, 300 rue de la Piscine, Domaine
Universitaire, 38406 Saint Martin d’Hères, France E-mail<EMAIL_ADDRESS>
(Accepted XXX. Received YYY; in original form ZZZ)

###### Abstract

Variations in the dust emissivity index, $\beta$, within and between galaxies,
are evidence that the chemistry and physics of dust must vary on large scales,
although the nature of the physical and/or chemical variations is still
unknown. In this paper we estimate values of $\beta$ and dust temperature for
a sample of 109 dusty star-forming galaxies (DSFGs) over the range, $2<z<6$.
We compare the results obtained with both an optically-thin model and a
general opacity model, finding that our estimates of $\beta$ are similar
between the models but our estimates of dust temperature are not. We find no
evidence of a change in $\beta$ with redshift, with a median value of
$\beta=1.96$ for the optically-thin model with a confidence interval (16–84%)
of 1.67 to 2.35 for the population. Using simulations, we estimate the
measurement errors from our procedure and show that the variation of $\beta$
in the population results from intrinsic variations in the properties of the
dust in DSFGs. At a fixed far-infrared luminosity, we find no evidence for a
change in dust temperature, $T_{\textrm{dust}}$, with redshift. After allowing
for the effects of correlated measurement errors, we find an inverse
correlation between $\beta$ and $T_{\textrm{dust}}$ in DSFGs, for which there
is also evidence in low-redshift galaxies.

###### keywords:

submillimetre: galaxies – galaxies: evolution

††pubyear: 2023††pagerange: Little evolution of dust emissivity in bright
infrared galaxies from 2 < _z_ < 6–9

##  1 Introduction

Despite contributing only a small percentage ($\lesssim 1$ per cent) of the
mass of the interstellar medium (Galliano et al. 2018), interstellar dust
plays a pivotal role in the star-formation processes of galaxies. Dust
efficiently absorbs optical and ultraviolet (UV) light from young OB-type
stars and active galactic nuclei (AGN) and re-emits this radiation at longer
wavelengths, in the far-infrared (FIR) to millimetre-wave (mm) regimes (e.g.
Puget et al. 1996; Dwek et al. 1998; Fixsen et al. 1998; Dole et al. 2006;
Driver et al. 2008). This dust obscuration hides star-formation activity in
the most massive, dust-enshrouded galaxies in the early Universe. The most
intense star formation occurs in dusty star-forming galaxies (DSFGs) at high
redshifts ($z>1$) where stars were produced at rates of $\gtrsim$ 100–1000
$M_{\odot}$ yr-1, causing high infrared luminosities ($L_{\rm IR}$), often in
excess of $\sim 10^{12}$ L⊙ (Blain et al. 2002; Casey et al. 2014; Swinbank et
al. 2014; da Cunha et al. 2015, 2021; Dudzevičiūtė et al. 2020).

The first high-redshift DSFGs were discovered using the Sub-millimetre Common
User Bolometer Array (SCUBA; Holland et al. 1999) on the 15-m James Clerk
Maxwell Telescope (JCMT) at 850 µm (Smail et al. 1997; Barger et al. 1998;
Hughes et al. 1998; Eales et al. 1999). Since their discovery, many samples
have been selected at wavelengths between 450 µm – 2 mm using SCUBA’s
successor, SCUBA-2 (Holland et al. 2013), e.g. by Wang et al. (2017) and
Dudzevičiūtė et al. (2020), and by other single-dish FIR–mm facilities such as
the Herschel Space Observatory (e.g. Eales et al. 2010; Oliver et al. 2012),
AzTEC (e.g. Montaña et al. 2021) and the South Pole Telescope (SPT; e.g.
Vieira et al. 2010; Everett et al. 2020), aided by the negative $K$
correction, which means that flux densities decline very slowly for a fixed
$L_{\rm IR}$ at $z>1$ (e.g. Blain & Longair 1993).

The very large star-formation rates in these galaxies suggest they contain
very large masses of molecular gas. Since it is impossible to observe the
molecular hydrogen directly, the mass of molecular gas in the DSFGs has been
estimated by the standard method of making observations of a ‘tracer’ of the
hidden gas, then estimating the gas mass from the luminosity of the tracer and
a calibration factor linking the two. The tracers that have been used are the
CO molecule, dust grains and carbon atoms (e.g. Dunne et al. 2022). The dust
method (e.g. Magdis et al. 2012; Eales et al. 2012; Scoville et al. 2014) is
particularly useful for DSFGs because they are powerful sources of thermal
continuum emission. Studies of the DSFGs and other high-redshift galaxies have
shown that they contain a higher fraction of gas than observed in galaxies
today (e.g. Tacconi et al. 2010; Scoville et al. 2016; Scoville et al. 2017;
Millard et al. 2020).

In considering this result, however, it should be remembered that the
calibration factors are ultimately based on observations in our own Galaxy,
and therefore the method is based on the implicit assumption that the physics
and other properties of the tracer do not evolve with redshift. In the case of
the dust method, this boils down to the assumptions that the gas-to-dust ratio
and the physical and chemical properties of the dust do not change with
redshift. Recent observational (e.g. Shapley et al. 2020; Popping & Péroux
2022) and theoretical (e.g. Popping et al. 2017; Li et al. 2019) studies have
concluded that the dust-to-gas ratio does not appear to evolve with redshift.

One useful observational indicator of the physical and chemical properties of
the dust grains is the dust emissivity spectral index, $\beta$, which
describes the frequency dependence of the emissivity of the dust grains, with
the optical depth of the dust being approximated as a power-law,
$\tau\propto\nu^{\beta}$. Theoretical models of dust (e.g. Draine & Lee 1984)
predict that $\beta$ should range between 1–2 depending on the chemical
composition of the dust grains. The value for dust in the Galaxy is remarkably
uniform over the sky, with a value of $\beta=1.51\pm 0.01$ (Planck
Collaboration et al. 2015). Although the assumption necessary to use dust as a
tracer of the hidden molecular hydrogen is that the properties of dust are the
same in every galaxy, there is now plenty of evidence that this is not true.
For example, Lamperti et al. (2019) modelled the FIR SEDs of 192 nearby
galaxies from JINGLES (the JCMT dust and gas In Nearby Galaxies Legacy
Exploration Survey) and found $\beta$ values between 0.6 and 2.2. Also, there
is now evidence for radial variation of $\beta$ within galaxies (e.g. Smith et
al. 2012; Draine et al. 2014; Tabatabaei et al. 2014; Hunt et al. 2015;
Whitworth et al. 2019; Clark et al. 2023).

In this study, we test the assumption that the properties of dust are the same
at all cosmic epochs by investigating whether there is evidence for evolution
in the value of the dust emissivity index. We study bright FIR sources
detected by Herschel and SPT with spectroscopically determined redshifts
between $2\lesssim z_{\textrm{spec}}\lesssim 6$, and model their FIR dust
emission to determine the value of $\beta$ for each galaxy. We investigate
whether there is any difference between these values and those in galaxies
today, and also whether there is any change over this redshift range,
corresponding to the period between about a billion years after the Big Bang
and the epoch at which cosmic star formation was at its peak (e.g. Lilly et
al. 1996).

The paper is organised as follows. In Section 2 we describe the sample used
during this analysis. In Section 3 we derive the dust properties of our
sources by fitting their SEDs with modified blackbodies. In Section 4 we
consider the impact of the well-studied $\beta$–dust temperature degeneracy
and the extent to which this biases our results. We also discuss the
robustness of our estimated dust parameters by presenting simple simulations
that imitate the SED fitting routine on mock Herschel and SPT sources. Section
5 describes the results of our analysis and the suite of simulations used to
predict the accuracy of our estimated dust properties. We also discuss the
implications of our findings in the context of the existing literature for low
redshift galaxies. Finally, we provide a summary of our results in Section 6.
Throughout this paper we assume a $\Lambda$CDM cosmology with $H_{0}$ = 67.7
km s-1 Mpc-1 and $\Omega_{m}$ = 0.31 (Planck Collaboration et al. 2020).

##  2 The sample

The sample studied here is a combination of DSFGs from Herschel, specifically
from HerBS (Herschel Bright Sources; Bakx et al. 2018), and from the SPT
(Reuter et al. 2020).

###  2.1 HerBS sub-sample

The first sub-sample used in this study comes from the HerBS catalogue, a
selection of the brightest high-redshift sources detected during the Herschel-
Astrophysical Terahertz Large Area Survey (H-ATLAS; Eales et al. 2010). The
sample contains 209 galaxies with 500 µm flux densities above 80 mJy and
photometric redshift estimates in excess of 2.

Spectroscopic redshifts have been obtained for a selection of HerBS sources
within the South Galactic Pole field of H-ATLAS, as part of the Bright
Extragalactic ALMA Redshift Survey (BEARS; Urquhart et al. 2022, Bendo et al.
2023; Hagimoto et al. 2023). These redshifts were measured from CO spectral
lines detected with the band-3 receivers of the Atacama Compact Array (ACA),
which is part of the Atacama Large Millimetre Array (ALMA), and also from
spectral lines detected with the band-3 and -4 receivers of ALMA’s 12-m array.
The high spatial resolution of the ALMA band 4 images ($\sim 2"$) revealed
that many HerBS sources are composed of more than one source; only half of the
HerBS sources comprise a single source at ALMA’s spatial resolution. In total,
Urquhart et al. (2022) measured 71 spectroscopic redshifts for 62 entries in
the HerBS catalogue. Several HerBS sources comprise two or more ALMA sources
with spectroscopic redshifts that agree well, suggesting either that the
sources are physically associated galaxies or are multiple images caused by
gravitational lensing. For this study, we have retained the HerBS sources
which are multiples in the ALMA images, if the redshifts of the sources are
within 0.1 of each other (we have taken the average redshift as that of the
system). If there are multiple sources in the ALMA image within the Herschel
beam but a spectroscopic redshift for only one source, we have assumed the
sources are physical connected and used the redshift of the one source as the
redshift of the system.

In our study we have used all HerBS sources with spectroscopic redshifts and
photometry at 250, 350 and 500 µm (Herschel-SPIRE), at 850 µm (Bakx et al.
2018) and at 2 and 3 mm from ALMA (Bendo et al. 2023. We have also used
photometry in ALMA band 6 (1.1–1.4 mm) from the ALMARED survey (Jianhang Chen
et al., in preparation) of potentially ‘ultrared’ (see Ivison et al. 2016)
sources in H-ATLAS (project code 2018.1.00526.S). The minimum and maximum
redshifts are $z_{\textrm{min}}$ = 1.569 and $z_{\textrm{max}}$ = 4.509.

###  2.2 SPT sub-sample

Our second sub-sample comprises the 81 DSFGs selected using the SPT (Weiß et
al. 2013; Strandet et al. 2016; Reuter et al. 2020). For a source to be
included in this sub-sample, it must have a flux density above 25 mJy at 870
µm. Redshifts for all SPT sources were obtained via blind searches targeting
CO line emission (Weiß et al. 2013; Strandet et al. 2016; Reuter et al. 2020).
There are robust spectroscopic redshifts (from two or more CO lines) for all
except two sources. We have retained the two because their spectroscopic
redshifts are consistent with the redshifts estimated from photometry. The
sub-sample has redshifts over the range 1.867 < $z_{\textrm{spec}}$ < 6.901
and a median redshift of $3.9\pm 0.2$ (Reuter et al. 2020). The SPT redshift
distribution extends to higher redshifts than that of HerBS because the longer
selection wavelength modifies the effect of the negative $K$ correction. The
extreme apparent luminosities of the SPT sources implies that most of them are
gravitationally lensed, with lensing models based on ALMA observations giving
a median lensing magnification factor of $5.5\times$ (Spilker et al. 2016).

The SPT sources have photometry between 250 µm and 3 mm, corresponding to a
rest-frame range of 86 µm $\lesssim\lambda_{\textrm{rest}}\lesssim$ 380 µm,
meaning that the peak of the dust emission ($\sim$ 100 µm) and Rayleigh-Jeans
(R-J) tail are well sampled. The photometry includes flux densities measured
at 250, 350 and 500 µm (Herschel-SPIRE), at 870 µm (APEX-LABOCA), at 1.4 and 2
mm (SPT) and at 3 mm (ALMA). For 65 sources there are additional constraints
at 100 and 160 µm from Herschel-PACS.

###  2.3 Restrictions on sample for this study

The two sub-samples contain 143 galaxies with spectroscopic redshifts.
However, as we are interested in estimating the galaxy-integrated $\beta$ of
each source, which is characterised by the slope of the dust emission on the
R-J side of the Planck function, we restrict our final sample to those
galaxies that have at least two photometric measurements at observed
wavelengths beyond 1 mm. This reduces our sample to a total of 109 galaxies
(79 SPT DSFGs, plus 30 from HerBS/BEARS). While the fitting methods used in
this work are the same for both sub-samples, we treat the two separately to
look for effects that might have been caused by their different selection
wavelengths and flux density limits. In Fig. 1 we show the redshift
distributions of the two sub-samples. The photometry and redshifts of the
HerBS-BEARS sources are tabulated in Appendix LABEL:tab:data_herbs; the
corresponding data used in this work for the SPT sources can be found in
Reuter et al. (2020).

Figure 1: Spectroscopic redshift distributions for the SPT DSFGs (grey) and
those from HerBS/BEARS (red).

##  3 SED fitting

We model the FIR-to-millimetre spectra of our galaxies with a single modified
blackbody (MBB) model combined with a mid-IR power-law, where the MBB
represents the emission from the reservoir of cold dust, which represents most
of the dust in the ISM, and the power-law represents emission from any dust in
the vicinity of star-forming regions, young OB stars, or active galactic
nuclei (AGN). The observed flux density can be expressed in the following
form:

$S_{\nu,\textrm{obs}}=\begin{cases}\frac{\Omega}{(1+z)^{3}}(1-e^{-\tau_{\nu}})B_{\nu}(T_{\textrm{dust}}),&\nu\leq\nu_{\rm
c}\\\ N\nu^{-\alpha},&\nu>\nu_{\rm c}\\\ \end{cases}$ (1)

where $\Omega$ represents the solid angle subtended by the galaxy,
$\tau_{\nu}$ is the dust optical depth, $B_{\nu}(T_{\textrm{dust}})$ is the
Planck function assuming a characteristic dust temperature of the dust grains,
$T_{\textrm{dust}}$, $N$ is a normalisation for the power-law which is tied to
the normalisation of the blackbody, and $\alpha$ is the mid-IR power-law
index. The value of $\nu_{\rm c}$ is the frequency at which the gradient of
the MBB is equal to the value of $\alpha$.

We define the solid angle, $\Omega$, as $\frac{A(1+z)^{4}}{D_{\rm L}^{2}}$,
where $A$ is the area of the source and $D_{\rm L}$ represents the luminosity
distance. The optical depth, $\tau_{\nu}$, is defined as the product of dust
surface mass density, $\Sigma_{\textrm{dust}}=M_{\textrm{dust}}/A$, and the
dust opacity, $\kappa_{\nu}$, but is often assumed to take the form of a
power-law, ($\nu/\nu_{1})^{\beta}$, where $\nu_{1}$ represents the frequency
at which the optical depth equals unity, marking the frequency of the
transition between optically thin and optically thick dust. This parameter can
be reformulated in terms of intrinsic properties by equating the two forms of
$\tau$ such that $\nu_{1}=\nu_{0}(\kappa_{0}\Sigma_{\textrm{dust}})^{\beta}$.
In a similar fashion, $\kappa_{\nu}$ may be assumed to take a power-law form:
$\kappa_{\nu}=\kappa_{0}(\nu/\nu_{0})^{\beta}$, where $\kappa_{0}$ is the
emissivity of the dust grains per unit mass at a given reference frequency,
$\nu_{0}$. In the following, we assume $\kappa_{0}$ = 0.077 m2 kg-1 at
$\nu_{0}$ = 353 GHz (Dunne et al. 2000; James et al. 2002). Combining these,
we find the following equation for the observed flux density of a galaxy:

$S_{\nu,\textrm{obs}}=\begin{cases}\frac{\mu A(1+z)}{D_{\rm
L}^{2}}\Big{(}1-e^{-\frac{M_{\textrm{dust}}\kappa_{\nu}}{A}}\Big{)}B_{\nu}(T_{\textrm{dust}}),&\nu\leq\nu_{\rm
c}\\\ \mu N\nu^{-\alpha},&\nu>\nu_{\rm c}\\\ \end{cases}$ (2)

where we have included a term, $\mu$, to account for the possibility of
magnification due to gravitational lensing. For sources where there is no
evidence of gravitational lensing, we assume $\mu=1$.

As demonstrated by da Cunha et al. (2013), the heating of dust by the cosmic
microwave background (CMB) has a non-negligible effect on the shape of the FIR
SED when the temperature of the CMB is a significant fraction of the dust
temperature of a galaxy (see also Zhang et al. 2016 for the influence of the
CMB on the measurement of structural and dynamical properties). Including the
effects of the CMB on the observed dust continuum using the procedure outlined
in da Cunha et al. (2013), the CMB-adjusted MBB model is now given by:

$S_{\nu,\textrm{obs}}=\begin{cases}f_{\textrm{CMB}}\frac{\mu A(1+z)}{D_{\rm
L}^{2}}\Big{(}1-e^{-\frac{M_{\textrm{dust}}\kappa_{\nu}}{A}}\Big{)}B_{\nu}(T_{\textrm{dust}}(z)),&\nu\leq\nu_{\rm
c}\\\ f_{\textrm{CMB}}\mu N\nu^{-\alpha},&\nu>\nu_{\rm c}\\\ \end{cases}$ (3)

where we have made the following two changes: i) the prefactor,
$f_{\textrm{CMB}}$, denotes the fraction of the total dust emission that is
observed against the background caused by the hotter CMB at higher redshifts,
given by Equation 18 of da Cunha et al. (2013), i.e.
$f_{\textrm{CMB}}=\frac{S_{\nu}^{\textrm{observed}}}{S_{\nu}^{\textrm{intrinsic}}}=1-\frac{B_{\nu}[T_{\textrm{CMB}}(z)]}{B_{\nu}[T_{\textrm{dust}}(z)]}$;
and ii) we have re-defined the dust temperature to be a function of redshift
due to the influence on dust grains of the warmer CMB at higher redshifts. The
derived dust temperature is corrected for CMB effects using Equation 12 of da
Cunha et al. (2013), i.e.
$T_{\textrm{dust}}(z)=[T_{\textrm{dust},0}^{4+\beta}+T_{\textrm{CMB},0}^{4+\beta}((1+z)^{4+\beta}-1)]^{\frac{1}{4+\beta}}$,
where $T_{\textrm{dust},0}$ is the dust temperature as observed at $z=0$ and
$T_{\textrm{CMB},0}$ = 2.73 K is the CMB temperature at $z=0$. Note that all
further mention of the dust temperature refers to the CMB-corrected,
luminosity-weighted dust temperature as given by the model described above,
unless otherwise stated. Using this general form of the MBB, there are up to
four free parameters: the dust mass, the dust temperature,
$T_{\textrm{dust}}$, the dust emissivity index, $\beta$, and the frequency at
which the dust opacity reaches unity. If the physical area of the region
emitting the submillimetre emission is known, the opacity can be calculated
from the surface density of the dust.

We estimated the parameters of the dust using three different models. In the
first model, we assume the dust is optically thin at all wavelengths
($\tau_{\nu}\ll 1$), which eliminates the fourth of the free parameters in the
list above and modifies Eqn 3 to:

$S_{\nu,\textrm{obs}}=\begin{cases}f_{\textrm{CMB}}\frac{\mu(1+z)}{D_{\rm
L}^{2}}M_{\textrm{dust}}\kappa_{\nu}B_{\nu}(T_{\textrm{dust}}(z)),&\nu\leq\nu_{\rm
c}\\\ f_{\textrm{CMB}}\mu N\nu^{-\alpha},&\nu>\nu_{\rm c}\\\ \end{cases}$ (4)

However, since in DSFGs a large amount of dust is packed into a small region
(e.g. Ikarashi et al. 2017), it is possible that the dust is optically thick
at one or more of the wavelengths covered by our photometry (e.g. Conley et
al. 2011; Casey et al. 2019; Cortzen et al. 2020). We therefore also used two
models in which the dust becomes optically thick at wavelengths below
$\lambda_{1}$ = 100 µm or 200 µm.

Some of the SPT sources have measurements of the sizes of the submillimetre-
emitting region, typically $\sim 1$ kpc (Spilker et al. 2016). For this subset
of sources, we also fitted a modified MBB using the size of the emitting
region to calculate the opacity directly.

To estimate the mass and luminosity of the dust we require an estimate of the
lensing magnifications, which we have taken to be those quoted in the lens
modelling of Spilker et al. (2016) for the SPT sources and from the
correlation between CO luminosity and line width as determined by Urquhart et
al. (2022) for HerBS, building on the work of Harris et al. (2012). During SED
fitting we assume flat priors on all free parameters and take the best-fitting
SED to be the median value of the posterior distribution for each parameter.
The (1-$\sigma$) uncertainties are quoted at the 16th and 84th percentiles.
Calibration errors are added in quadrature with the flux density uncertainties
assuming absolute calibrations of 7 per cent (Herschel-PACS), 5.5 per cent
(Herschel-SPIRE), 5 per cent (SCUBA-2), 12 per cent (APEX-LABOCA), 7 per cent
(SPT) and 10 per cent (ALMA).

##  4 Results

###  4.1 Comparison between the optically thin and general opacity models

Fig. 2 shows the stacked posterior probability distributions for the dust
parameters derived for the two samples. The median values and the 16th and
84th percentiles of the probability distributions are listed in Table 1. The
median value of the main parameter of interest, the dust emissivity spectral
index, covers only a range of 0.2 for the three models, showing that it
doesn’t make much difference what one assumes about the opacity, a conclusion
also reached in a study of a different sample of DSFGs by Ismail et al.
(2023). We note, however, that the general opacity model with the highest
value of $\lambda_{1}$ does give the highest value of $\beta$, as observed
previously by McKay et al. (2023). Using a sample of 870 µm-selected galaxies
in GOODS-S, McKay et al. (2023) found good agreement in the median $\beta$
between an optically thin model (1.78) and an MBB model with $\lambda_{1}=100$
µm (1.80), but this increased to 2.02 for a model with $\lambda_{1}=200$ µm.
Some recent studies have suggested that $\beta>2$ is common among high-
redshift galaxies (Casey et al. 2019; Casey et al. 2021; Cooper et al. 2022),
but this may be due to these studies using a higher value for the wavelength
at which the dust becomes optically thick.

Figure 2: Stacked posterior distributions of log($\mu M_{\textrm{dust}}$), $T_{\textrm{dust}}$, $\beta$, log($\mu L_{\textrm{IR}}$) (rest-frame 8–1000 µm) and $\lambda_{\textrm{peak}}$ for SPT (top panels) and HerBS galaxies (bottom panels). The posterior distribution for each MBB model is illustrated as follows: optically thin (shaded); $\lambda_{1}=100$ µm (solid line); $\lambda_{1}=200$ µm (dashed line); fixed continuum size (hatched). Table 1: Median and 1$\sigma$ errors, estimated from the 16th, 50th and 84th percentiles of the stacked posterior distribution, for the parameters presented in Fig. 2. Parameter |  Optically |  $\lambda_{1}=$ |  $\lambda_{1}=$
---|---|---|---
|  thin |  100 µm |  200 µm
| SPT
log($\mu M_{\textrm{dust}}[$M${}_{\odot}]$) |  $9.92_{-0.31}^{+0.30}$ |  $9.77_{-0.34}^{+0.31}$ |  $9.49_{-0.41}^{+0.37}$
$T_{\textrm{dust}}$ [K] |  $32.07_{-8.44}^{+8.43}$ |  $40.77_{-11.77}^{+10.90}$ |  $55.63_{-9.28}^{+11.20}$
$\beta$ |  $1.98_{-0.39}^{+0.54}$ |  $1.91_{-0.32}^{+0.50}$ |  $2.18_{-0.30}^{+0.38}$
log($\mu L_{\textrm{IR}}[$L${}_{\odot}]$) |  $13.98_{-0.31}^{+0.19}$ |  $13.99_{-0.30}^{+0.19}$ |  $13.89_{-0.27}^{+0.23}$
$\lambda_{\textrm{peak}}$ [µm] |  $90.70_{-12.72}^{+17.33}$ |  $90.70_{-13.02}^{+18.36}$ |  $89.28_{-13.39}^{+15.21}$
| HerBS
log($\mu M_{\textrm{dust}}[$M${}_{\odot}]$) |  $9.62_{-0.22}^{+0.30}$ |  $9.49_{-0.22}^{+0.31}$ |  $9.30_{-0.26}^{+0.33}$
$T_{\textrm{dust}}$ [K] |  $32.73_{-5.92}^{+5.85}$ |  $41.16_{-8.20}^{+7.48}$ |  $55.18_{-9.23}^{+7.25}$
$\beta$ |  $1.92_{-0.32}^{+0.39}$ |  $1.85_{-0.25}^{+0.34}$ |  $2.08_{-0.25}^{+0.26}$
log($\mu L_{\textrm{IR}}[$L${}_{\odot}]$) |  $13.55_{-0.37}^{+0.27}$ |  $13.57_{-0.37}^{+0.26}$ |  $13.57_{-0.34}^{+0.25}$
$\lambda_{\textrm{peak}}$ [µm] |  $90.03_{-9.28}^{+12.70}$ |  $90.36_{-9.46}^{+13.42}$ |  $89.96_{-9.46}^{+14.92}$

The assumption that the dust is optically thin does, however, have a much
larger effect for some of the other parameters. There is a clear trend to
higher median dust temperatures with increasing $\lambda_{1}$, with increases
of $\sim$ 10 K and $\sim$ 20 K from the optically-thin model to the
$\lambda_{1}=100$ µm and to the $\lambda_{1}=200$ µm general opacity models,
respectively. The dust masses decrease by approximately 0.1 log($\mu$M⊙) and
0.4 log($\mu$M⊙) from the optically-thin model to the $\lambda_{1}=100$ µm and
the $\lambda_{1}=200$ µm general opacity models, respectively, though we note
that these decreases are smaller than the 1$\sigma$ widths of the probability
distributions. There is no significant difference in IR luminosities or peak
wavelengths between the models.

Fig. 2 also shows the posterior probability distributions for the subset of
SPT galaxies for which we are able to estimate the area of the dust region
(hatched histograms). This distribution agrees well with the probability
distributions for the optically-thin model and for the general-opacity model
with $\lambda_{1}=100$ µm, but not for the one with $\lambda_{1}=200$ µm. For
this subset of galaxies we find a median value of $\lambda_{1}=88$ µm (16th
and 84th percentile range 44–224 µm). However, we note that this value is
inversely proportional to the value assumed for the dust opacity
($\kappa_{\nu}$) and since the opacity of interstellar dust is still very
uncertain (Clark et al. 2016), the frequency at which the opacity is unity is
also uncertain.

Fig. 3 shows the median values of the parameters estimated from the optically-
thin model versus those estimated from the general-opacity models. As we
discussed above, the estimates from the two types of model agree well for
$\beta$, $\lambda_{\textrm{peak}}$ and far-infrared luminosity, but there are
systematic difference for $T_{\textrm{dust}}$ and $M_{\textrm{dust}}$. The
probability distributions for the optically-thin model are a good match to
those for the subset of galaxies for which we know the size of the dust
region, and therefore the parameter estimates we discuss in the rest of this
paper have been derived using the optically-thin model. The panels in Fig. 3
show the size of the systematic errors on these estimates if the dust is
actually optically thick.

Figure 3: Comparison between the parameters derived from the optically-thin
and general-opacity models for the galaxies in the SPT (left column) and HerBS
(right column) sub-samples. The general opacity models are shown as filled and
open circles for $\lambda_{1}$ = 100 µm and 200 µm, respectively.

###  4.2 The $\beta$–dust temperature relationship

Where an MBB is fitted to an empirical SED it is well known that an artificial
anti-correlation can be produced between dust temperature and $\beta$ because
of correlated errors in the two parameters (Shetty et al. 2009; Kelly et al.
2012; Juvela & Ysard 2012). Fig. 4 shows that there is a negative correlation
between the measured values of $\beta$ and dust temperature for both sub-
samples when assuming that the dust is optically thin. The strength of these
correlations were tested with the Pearson correlation coefficient,
$r_{\textrm{Pearson}}$, and Spearman’s rank correlation coefficient,
$\rho_{\textrm{Spearman}}$. The values ($r_{\textrm{Pearson}}=-0.83$ and
$\rho_{\textrm{Spearman}}=-0.87$ for SPT; $r_{\textrm{Pearson}}=-0.89$ and
$\rho_{\textrm{Spearman}}=-0.89$ for HerBS) show that the SPT and HerBS
galaxies exhibit a strong negative $\beta$–$T_{\textrm{dust}}$ correlation and
the similarity between the two metrics suggests that few galaxies deviate from
the trend. In the following section, we address the extent to which the
observed $\beta$–$T_{\textrm{dust}}$ anti-correlation is a true relationship
between dust properties, reflecting an intrinsic change in the emissivity
properties of dust grains with temperature, and how much is due to the fitting
method.

Figure 4: Relationship between the dust temperature and the emissivity index
for the SPT (black) and HerBS (red) galaxies in the optically thin case.

###  4.3 Simulations

In order to assess how accurately our fitting routine derives a galaxy’s dust
parameters, we ran a suite of mock SEDs with known input parameters and
measured how precisely we recover the dust properties from our fits. We
generated these mock SEDs in the following way. We assumed that the dust
emission is described by the optically-thin model (Eqn 4). A catalogue of
models were produced with dust parameters randomly drawn from a uniform
distribution between the lower and upper bounds in Table 2, chosen to reflect
the width of the posterior distributions for the parameters derived for the
real sources. We created two samples of mock galaxies with similar properties
to the real sub-samples. To create the first sample, we placed each galaxy at
a random redshift between 2 and 4 and gave the galaxy the same photometric
coverage as for the HerBS sub-sample, omitting photometry at 1.2 mm as only a
few of the HerBS sources have photometry at this wavelength. To create the
second sample, we placed each galaxy at a random redshift between 2 and 6 and
gave the galaxy the same photometric coverage as the SPT sample. We created
observed SEDs of each mock galaxy by adding flux errors to the model SED,
drawing randomly from a normal distribution with a standard deviation equal to
the real observational error at that wavelength. We assumed a lensing
magnification of $5.5\times$ for all SPT galaxies and $5.3\times$ for all
HerBS galaxies, equal to the median lensing magnification for the two sub-
samples. Once we had calculated an observed SED for each mock galaxy, we
removed it from the sample if it fell below the detection limits of the real
samples (Section 2). For the sources that remained, we then derived the dust
parameters using the same modelling procedure we had applied to the real
sample.

Table 2: Upper and lower bounds assumed on the flat priors during the simulations described in Section 4.3. Parameter |  Bounds
---|---
log($\mu L_{\textrm{IR}}[$L${}_{\odot}]$) |  13–14 (HerBS)
|  13–14.5 (SPT)
$T_{\textrm{dust}}$ [K] |  20–50
$\beta$ |  0.5–4
$\alpha$ |  1–5

Fig. 5 shows the paramaters derived for the two artificial samples plotted
against the input values of these parameters. There is good agreement between
the input and output parameters. The r.m.s. error (RMSE) is calculated for
each dust parameter, showing the intrinsic scatter around the ‘true’ input
values. The RMSE indicate that our fitting recovers the input dust masses to
0.05 dex and 0.03 dex, the dust temperatures to 1.63 K and 1.61 K and $\beta$
to 0.13 and 0.09 of the input values for SPT and HerBS galaxies, respectively.
The accuracy of the output $T_{\textrm{dust}}$ decreases for higher
temperatures as a result of the SED shifting to shorter wavelengths, meaning
the peak is then less constrained by the Herschel photometry than at lower
temperatures.

The right-most panel of the two simulations in Fig. 5 shows the difference
between the input and output in the $T_{\textrm{dust}}$–$\beta$ plane. Since
there is no intrinsic correlation between $\beta$ and dust temperature in the
model, the anti-correlation in this panel must have been introduced by
correlated errors in the two parameters. The r.m.s. variations in $\beta$ and
$T_{\textrm{dust}}$ in the artificial dataset are 0.1 and 1.6 K, respectively.
The r.m.s. variations in $\beta$ and $T_{\textrm{dust}}$ in the real data
(Fig. 4) are 0.5 and 8 K, respectively, for the SPT galaxies, and 0.3 and 5 K,
respectively, for the HerBS galaxies. The large differences between the r.m.s.
values for the simulation and for the real samples imply there is a genuine
inverse correlation between $\beta$ and $T_{\textrm{dust}}$ which is not
caused by correlated measurement errors.

Figure 5: Comparison of the input and output (measured) values of dust mass,
dust temperature and dust emissivity index for the simulations described in
Section 4.3. The right-most panel shows the difference between the input and
output (measured) values of dust temperature against the difference between
the input and output (measured) values of $\beta$. The top row (black) shows
the results for the mock SPT galaxies and the bottom row (red) shows the
results for the mock HerBS galaxies.

##  5 Discussion

###  5.1 Evolution of $\beta$ with redshift

Fig. 6 shows the distribution of galaxies in the $\beta$–$z$ plane. The
average value of the emissivity spectral index of SPT and HerBS galaxies
($\beta=1.98$ and $1.91$) are slightly higher than the values typically
assumed for galaxies in the local Universe. It is significantly higher than
the value in our Galaxy, which is remarkably uniform over the sky:
$\beta=1.51\pm 0.01$ (Planck Collaboration et al. 2015). It is also
significantly higher than the average value of $\beta$ for a large sample of
nearby galaxies (Lamperti et al. 2019), although it is close to the values
measured for the most massive galaxies in this sample, which may be the
descendants of the DSFGs (Eales et al. 2023). Considering the samples
themselves, there is little evidence of any evolution in $\beta$ over the
redshift range $2<z<6$. The apparent negative redshift evolution within the
HerBS sample is mostly the effect of the five sources at $z\sim 4$. At this
redshift, the selection wavelength of HerBS, 500 µm, is close to the peak of
the MBB, thus biasing us towards higher temperatures. The intrinsic anti-
correlation between dust temperature and $\beta$ (Section 4.2) would then
explain the low values of $\beta$ for these galaxies.

Similar conclusions have been reached in two other recent studies. For an
independent sample of DSFGs selected from HerBS, Ismail et al. (2023) found no
evolution in $\beta$ over $2<z<3$. For a sample of 17 galaxies, Witstok et al.
(2023) found no evolution across $4<z<8$.

Figure 6: Distribution of $\beta$ for the HerBS (red) and SPT (black) galaxies
versus redshift, with the canonical values from JINGLES (green) and the Milky
Way (blue). The median value of the stacked posterior distributions are shown
as red and black lines for the HerBS and SPT sub-samples, respectively, with
the shaded regions showing the 16th to 84th percentiles.

###  5.2 Variation in $\beta$

To assess whether the variation of $\beta$ in Fig. 6 is due to genuine
differences in the properties of dust between DSFGs, or whether it is simply
due to measurement errors, we repeated the simulations of the previous
section. This time we used values of $\beta$ drawn from a uniform distribution
covering the $\pm 1\sigma$ range of the observed values in Fig. 6, generating
– as before – two samples of mock galaxies with observational data similar to
that of the SPT and HerBS sub-samples.

Fig. 7 shows a comparison of the input and output values of $\beta$ as a
function of redshift for both samples. There is no tendency for our fitting
method to over- or under-predict the true value of $\beta$ at any redshift.
The r.m.s. scatter in Fig. 7 is $\simeq$0.1, which is much less than the
r.m.s. scatter in $\beta$ in Fig. 6 ($\simeq$0.3–0.5), which shows that the
measurement errors are much smaller than the scatter in Fig. 6. Fig. 6
therefore shows that there are real differences in the properties of dust
between DSFGs, a phenomenon also seen in the low-redshift galaxy population
(Lamperti et al. 2019).

Figure 7: Difference between the input and output $\beta$ values,
$\Delta\beta=\beta_{\textrm{output}}-\beta_{\textrm{input}}$, as a function of
the input redshift for mock SPT galaxies (top panel) and mock Herschel
galaxies (bottom panel). Filled circles represent mock galaxies with SEDs that
would have been observed according to their respective survey limits. Dotted
vertical lines represent the minimum and maximum redshift observed in each
sub-sample. The thick solid lines represent the median in $\Delta\beta$.

In their study of an independent sample of DSFGs drawn from HerBS, Ismail et
al. (2023) reached similar conclusions. Across $2<z<4$, they found an average
value of $\beta=2.2\pm 0.2$ with a range of $\beta$ from 1.5 to 3.0, very
similar to our results for the combined HerBS/SPT sample. They, too, concluded
that the variation in $\beta$ is caused by intrinsic differences between the
dust in different DSFGs rather than by measurement errors. In their sample of
17 galaxies in the range $4<z<8$, Witstok et al. (2023) found a mean value of
$\beta=1.8\pm 0.3$, in agreement with our estimates, with a similar wide range
of $\beta$ from $\simeq 1.5$ to $\simeq 2.7$. Other recent studies of high-
redshift galaxies find similar values of $\beta$ (da Cunha et al. 2021; Cooper
et al. 2022).

Some of the SPT galaxies in Fig. 6 have values of $\beta\simeq 3$, as do
galaxies in other high-redshift samples (Ismail et al. 2023; Witstok et al.
2023). Such values are higher than the global values seen in local galaxies
(Lamperti et al. 2019), although such high values have been seen within
Andromeda (Smith et al. 2012) and the Galaxy (Bracco et al. 2011). Are there
any methodological limitations which might explain these results?

For our sample, one possible explanation might be the 3-mm photometry for the
SPT sources. These flux densities were measured from ALMA images, so it is
possible that they are too low because of the possibility – as with all
interferometers – that some extended flux has been missed (the ‘zero baseline
problem’). We assessed how much would need to be lost by repeating the SED
fitting without the 3-mm photometry, with $\beta=2$. We did this for five
sources: SPT0112$-$55 ($\beta=3.62$), SPT0611$-$55 ($\beta=3.44$),
SPT2203$-$41 ($\beta=2.84$), SPT2349$-$52 ($\beta=3.62$) and SPT2357$-$51
($\beta=2.80$). Comparing the flux density at 3 mm predicted by the model with
the observed flux density, we find that the observed flux density would need
to be between 60–80 per cent lower than the true flux density to explain the
very high values of $\beta$. Although this is possible, given the structures
seen for high-redshift galaxies it seems unlikely, and it seems more likely to
us that some of these extreme values found by us and others are genuine.

###  5.3 The inverse $\beta$–$T_{\textrm{dust}}$ relationship

Our simulations imply that the inverse relationship seen between $\beta$ and
$T_{\textrm{dust}}$ is not the result of measurement errors. Another study of
an independent sample of HerBS sources reached a similar conclusion (Ismail et
al. 2023). A similar conclusion was also reached by a hierarchical Bayesian
study of nearby galaxies (Lamperti et al. 2019).

###  5.4 Evolution of $T_{\textrm{dust}}$ with redshift

Does the temperature of dust increase with redshift? This has been a matter of
intense debate in the literature. Some groups find evidence for an increase,
e.g. Magdis et al. (2012); Magnelli et al. (2014); Swinbank et al. (2014);
Béthermin et al. (2015); Ivison et al. (2016); Faisst et al. (2017); Schreiber
et al. (2018); Zavala et al. (2018); Liang et al. (2019); Ma et al. (2019);
Faisst et al. (2020); Bakx et al. (2021); Witstok et al. (2023), which might
not be surprising since both high specific star-formation rates and lower dust
abundances at high redshift might lead to higher dust temperatures. Others
report a fall, e.g. Symeonidis et al. (2013). An important caveat, noted by
many of those studies, is the correlation between dust temperature and
luminosity (Dunne et al. 2000) which – since galaxies at high redshift tend to
have high luminosities – might lead to a correlation between dust temperature
and redshift. In contrast, the works of Casey et al. (2018); Jin et al.
(2019); Lim et al. (2020); Dudzevičiūtė et al. (2020); Reuter et al. (2020);
Barger et al. (2022); Drew & Casey (2022) and Witstok et al. (2023), among
others, report little or no evolution of temperature with redshift at a given
fixed IR luminosity.

The dust temperature derived from SED fitting depends on the model assumed for
the dust opacity, whereas $\lambda_{\textrm{peak}}$ is the same for every
model. To avoid this dependence, we have derived $\lambda_{\textrm{peak}}$ for
each source and then calculated $T_{\textrm{dust}}$ from Wien’s displacement
law. Although this will not be the same as the true temperature of the dust,
it allows us to see whether there is any evidence for a change in
$T_{\textrm{dust}}$ with redshift.

The top panel of Fig. 8 shows the distribution of IR luminosity for the full
sample of galaxies as obtained from the fitting of Equation 4, as well as the
detection limits. In calculating the detection limits, we have assumed an SED
with a dust temperature of 32 K, the median temperature of the sample, and
$\beta=2$. To avoid the selection effect caused by the luminosity–temperature
relationship, we only consider sources selected in a small range of IR
luminosity, as illustrated by the boxed regions. The bottom panel of Fig. 8
shows the peak dust temperature ($T_{\textrm{peak}}=2.898\times 10^{3}$
[µmK]/$\lambda_{\textrm{peak}}$ [µm]) versus redshift. We have plotted the
observational relationships inferred by Schreiber et al. (2018), Bouwens et
al. (2020) and Viero et al. (2022). Bootstrap fitting with a linear model
gives an evolution rate of $5.20\pm 0.92$ and $-0.25\pm 0.52$
$T_{\textrm{peak}}/z$ for the HerBS and SPT galaxies, respectively. In a study
of an independent sample of HerBS galaxies, Ismail et al. (2023) found a rate
of evolution of $6.5\pm 0.5\,K/z$, similar to our value.

While there is statistical evidence within the HerBS sub-sample – both ours
and that of Ismail et al. (2023) – for the evolution of temperature with
redshift, there seems little evidence for any such evolution over the redshift
range covered by the combined HerBS and SPT sample.

We suggest there is a simple selection effect that explains the temperature
evolution seen in the HerBS sample. For the highest redshift sources in the
HerBS sub-sample, the selection wavelength in the rest frame is close to the
peak of the MBB, which leads to a selection bias towards sources with higher
temperatures. Given the lack of any obvious correlation when both sub-samples
are considered together, we conclude that there is no strong evidence for
temperature evolution. A plausible suggestion for the disagreement in the
literature was suggested by Drew & Casey (2022): the studies that have found
no evolution are those that looked for evolution at a fixed infrared
luminosity, as in this paper, whereas those that found evolution were
generally looking for evolution at a fixed stellar mass. The latter would be
expected to see evolution because of the gradual increase in specific star-
formation rate – and thus the increase in the intensity of the interstellar
radiation field – with redshift.

Figure 8: Top: distribution of HerBS (red) and SPT (black) galaxies in the
$\mu L_{\textrm{IR}}$–$z$ plane. Red and black lines represent the detection
limit of a source detected at $>80$ mJy (500 µm) and $>25$ mJy (870 µm) with
$\beta=2$ and $T_{\textrm{dust}}=32$ K. Boxed regions show the limits of the
unbiased, luminosity-limited set of sources from which we test evolutionary
trends in the dust temperature. Bottom: distribution of dust temperature with
redshift for the sources selected in our luminosity-limited subset. Observed
trends from Schreiber et al. (2018); Bouwens et al. (2020) and Viero et al.
(2022) are illustrated as cyan, orange and purple lines, respectively. The
median dust temperature of local sources from JINGLES is shown in green.

##  6 Conclusions

In this paper we estimate values of $\beta$ and dust temperature for a sample
of 109 dusty star-forming galaxies spanning $2<z<6$. We have obtained the
following results:

  * •

We have compared the results from fitting the emission from the galaxies with
an optically-thin model and with two general-opacity models, one in which the
dust becomes optically thick at a wavelength of 100 µm, the other where the
dust becomes optically thick at 200 µm. We find that our estimates of $\beta$,
the peak of the galaxy SED, and the far-infrared luminosity, are similar
between the models, although there is a change in $\beta$ of 0.2 between the
most extreme models. In contrast, our estimates of dust mass and dust
temperature depend strongly on the opacity assumptions.

  * •

For a sub-sample of 37 galaxies, we have measurements of the sizes of the
sources, which allows us to estimate the opacity directly. We find very
similar estimates of the galaxy parameters with this model to the optically-
thin model, which we use for the rest of the paper.

  * •

We find no evidence of any change in $\beta$ with redshift across $2<z<6$,
with a median value of $\beta=1.96$ for the optically-thin model and a 14–86%
confidence interval for the population of 1.65 to 2.38. Using simulations, we
estimate the measurement errors from our procedure and show that the variation
of $\beta$ in the population is the result of intrinsic variations in the
properties of the dust in DSFGs, the same found between galaxies in the low-
redshift Universe (Lamperti et al. 2019).

  * •

After allowing for the effect of correlated errors, we find an inverse
correlation between our estimates of $\beta$ and dust temperature.

  * •

We find no evidence at a fixed far-infrared luminosity for any change in dust
temperature with redshift.

The results in this and other recent papers show that the properties of dust
appear to differ between galaxies, both at high and low redshift. Ours and
other recent papers show that there appears to be intrinsic anti-correlation
between dust emissivity index and dust temperature. As far as we know, there
is no chemical/physical model of dust that explains either of these results.

## Acknowledgements

BW thanks the Engineering and Physical Sciences Research Council for a PhD
studentship. SAE thanks the Science and Technology Facilities Council
(consolidated grant ST/K000926/1) for the funds that supported this research.

## Data availability

The photometry for each galaxy and the results of fitting our models to the
photometry are listed in the appendix.

## References

  * Bakx et al. (2018) Bakx T. J. L. C., et al., 2018, MNRAS, 473, 1751
  * Bakx et al. (2021) Bakx T. J. L. C., et al., 2021, MNRAS, 508, L58
  * Barger et al. (1998) Barger A. J., Cowie L. L., Sanders D. B., Fulton E., Taniguchi Y., Sato Y., Kawara K., Okuda H., 1998, Nature, 394, 248
  * Barger et al. (2022) Barger A. J., Cowie L. L., Blair A. H., Jones L. H., 2022, ApJ, 934, 56
  * Bendo et al. (2023) Bendo G. J., et al., 2023, MNRAS, 522, 2995
  * Béthermin et al. (2015) Béthermin M., et al., 2015, A&A, 573, A113
  * Blain & Longair (1993) Blain A. W., Longair M. S., 1993, MNRAS, 264, 509
  * Blain et al. (2002) Blain A. W., Smail I., Ivison R. J., Kneib J. P., Frayer D. T., 2002, Phys. Rep., 369, 111
  * Bouwens et al. (2020) Bouwens R., et al., 2020, ApJ, 902, 112
  * Bracco et al. (2011) Bracco A., et al., 2011, MNRAS, 412, 1151
  * Casey et al. (2014) Casey C. M., Narayanan D., Cooray A., 2014, Phys. Rep., 541, 45
  * Casey et al. (2018) Casey C. M., et al., 2018, ApJ, 862, 77
  * Casey et al. (2019) Casey C. M., et al., 2019, ApJ, 887, 55
  * Casey et al. (2021) Casey C. M., et al., 2021, ApJ, 923, 215
  * Clark et al. (2016) Clark C. J. R., Schofield S. P., Gomez H. L., Davies J. I., 2016, MNRAS, 459, 1646
  * Clark et al. (2023) Clark C. J. R., Roman-Duval J. C., Gordon K. D., Bot C., Smith M. W. L., Hagen L. M. Z., 2023, ApJ, 946, 42
  * Conley et al. (2011) Conley A., et al., 2011, ApJ, 732, L35
  * Cooper et al. (2022) Cooper O. R., Casey C. M., Zavala J. A., Champagne J. B., da Cunha E., Long A. S., Spilker J. S., Staguhn J., 2022, ApJ, 930, 32
  * Cortzen et al. (2020) Cortzen I., et al., 2020, A&A, 634, L14
  * Dole et al. (2006) Dole H., et al., 2006, A&A, 451, 417
  * Draine & Lee (1984) Draine B. T., Lee H. M., 1984, ApJ, 285, 89
  * Draine et al. (2014) Draine B. T., et al., 2014, ApJ, 780, 172
  * Drew & Casey (2022) Drew P. M., Casey C. M., 2022, ApJ, 930, 142
  * Driver et al. (2008) Driver S. P., Popescu C. C., Tuffs R. J., Graham A. W., Liske J., Baldry I., 2008, ApJ, 678, L101
  * Dudzevičiūtė et al. (2020) Dudzevičiūtė U., et al., 2020, MNRAS, 494, 3828
  * Dunne et al. (2000) Dunne L., Eales S., Edmunds M., Ivison R., Alexander P., Clements D. L., 2000, MNRAS, 315, 115
  * Dunne et al. (2022) Dunne L., Maddox S. J., Papadopoulos P. P., Ivison R. J., Gomez H. L., 2022, MNRAS, 517, 962
  * Dwek et al. (1998) Dwek E., et al., 1998, ApJ, 508, 106
  * Eales et al. (1999) Eales S., Lilly S., Gear W., Dunne L., Bond J. R., Hammer F., Le Fèvre O., Crampton D., 1999, ApJ, 515, 518
  * Eales et al. (2010) Eales S., et al., 2010, PASP, 122, 499
  * Eales et al. (2012) Eales S., et al., 2012, ApJ, 761, 168
  * Eales et al. (2023) Eales S., Gomez H., Dunne L., Dye S., Smith M. W. L., 2023, arXiv e-prints, p. arXiv:2303.07376
  * Everett et al. (2020) Everett W. B., et al., 2020, ApJ, 900, 55
  * Faisst et al. (2017) Faisst A. L., et al., 2017, ApJ, 847, 21
  * Faisst et al. (2020) Faisst A. L., Fudamoto Y., Oesch P. A., Scoville N., Riechers D. A., Pavesi R., Capak P., 2020, MNRAS, 498, 4192
  * Fixsen et al. (1998) Fixsen D. J., Dwek E., Mather J. C., Bennett C. L., Shafer R. A., 1998, ApJ, 508, 123
  * Galliano et al. (2018) Galliano F., Galametz M., Jones A. P., 2018, ARA&A, 56, 673
  * Hagimoto et al. (2023) Hagimoto M., et al., 2023, MNRAS, 521, 5508
  * Harris et al. (2012) Harris A. I., et al., 2012, ApJ, 752, 152
  * Holland et al. (1999) Holland W. S., et al., 1999, MNRAS, 303, 659
  * Holland et al. (2013) Holland W. S., et al., 2013, MNRAS, 430, 2513
  * Hughes et al. (1998) Hughes D. H., et al., 1998, Nature, 394, 241
  * Hunt et al. (2015) Hunt L. K., et al., 2015, A&A, 576, A33
  * Ikarashi et al. (2017) Ikarashi S., et al., 2017, ApJ, 849, L36
  * Ismail et al. (2023) Ismail D., et al., 2023, arXiv e-prints, p. arXiv:2307.15747
  * Ivison et al. (2016) Ivison R. J., et al., 2016, ApJ, 832, 78
  * James et al. (2002) James A., Dunne L., Eales S., Edmunds M. G., 2002, MNRAS, 335, 753
  * Jin et al. (2019) Jin S., et al., 2019, ApJ, 887, 144
  * Juvela & Ysard (2012) Juvela M., Ysard N., 2012, A&A, 541, A33
  * Kelly et al. (2012) Kelly B. C., Shetty R., Stutz A. M., Kauffmann J., Goodman A. A., Launhardt R., 2012, ApJ, 752, 55
  * Lamperti et al. (2019) Lamperti I., et al., 2019, MNRAS, 489, 4389
  * Li et al. (2019) Li Q., Narayanan D., Davé R., 2019, MNRAS, 490, 1425
  * Liang et al. (2019) Liang L., et al., 2019, MNRAS, 489, 1397
  * Lilly et al. (1996) Lilly S. J., Le Fevre O., Hammer F., Crampton D., 1996, ApJ, 460, L1
  * Lim et al. (2020) Lim C.-F., et al., 2020, ApJ, 889, 80
  * Ma et al. (2019) Ma X., et al., 2019, MNRAS, 487, 1844
  * Magdis et al. (2012) Magdis G. E., et al., 2012, ApJ, 760, 6
  * Magnelli et al. (2014) Magnelli B., et al., 2014, A&A, 561, A86
  * McKay et al. (2023) McKay S. J., Barger A. J., Cowie L. L., Bauer F. E., Rosenthal M. J. N., 2023, ApJ, 951, 48
  * Millard et al. (2020) Millard J. S., et al., 2020, MNRAS, 494, 293
  * Montaña et al. (2021) Montaña A., et al., 2021, MNRAS, 505, 5260
  * Oliver et al. (2012) Oliver S. J., et al., 2012, MNRAS, 424, 1614
  * Planck Collaboration et al. (2015) Planck Collaboration et al., 2015, A&A, 576, A107
  * Planck Collaboration et al. (2020) Planck Collaboration et al., 2020, A&A, 641, A6
  * Popping & Péroux (2022) Popping G., Péroux C., 2022, MNRAS, 513, 1531
  * Popping et al. (2017) Popping G., Somerville R. S., Galametz M., 2017, MNRAS, 471, 3152
  * Puget et al. (1996) Puget J. L., Abergel A., Bernard J. P., Boulanger F., Burton W. B., Desert F. X., Hartmann D., 1996, A&A, 308, L5
  * Reuter et al. (2020) Reuter C., et al., 2020, ApJ, 902, 78
  * Schreiber et al. (2018) Schreiber C., Elbaz D., Pannella M., Ciesla L., Wang T., Franco M., 2018, A&A, 609, A30
  * Scoville et al. (2014) Scoville N., et al., 2014, ApJ, 783, 84
  * Scoville et al. (2016) Scoville N., et al., 2016, ApJ, 820, 83
  * Scoville et al. (2017) Scoville N., et al., 2017, ApJ, 837, 150
  * Shapley et al. (2020) Shapley A. E., Cullen F., Dunlop J. S., McLure R. J., Kriek M., Reddy N. A., Sanders R. L., 2020, ApJ, 903, L16
  * Shetty et al. (2009) Shetty R., Kauffmann J., Schnee S., Goodman A. A., 2009, ApJ, 696, 676
  * Smail et al. (1997) Smail I., Ivison R. J., Blain A. W., 1997, ApJ, 490, L5
  * Smith et al. (2012) Smith M. W. L., et al., 2012, ApJ, 756, 40
  * Spilker et al. (2016) Spilker J. S., et al., 2016, ApJ, 826, 112
  * Strandet et al. (2016) Strandet M. L., et al., 2016, ApJ, 822, 80
  * Swinbank et al. (2014) Swinbank A. M., et al., 2014, MNRAS, 438, 1267
  * Symeonidis et al. (2013) Symeonidis M., et al., 2013, MNRAS, 431, 2317
  * Tabatabaei et al. (2014) Tabatabaei F. S., et al., 2014, A&A, 561, A95
  * Tacconi et al. (2010) Tacconi L. J., et al., 2010, Nature, 463, 781
  * Urquhart et al. (2022) Urquhart S. A., et al., 2022, MNRAS, 511, 3017
  * Vieira et al. (2010) Vieira J. D., et al., 2010, ApJ, 719, 763
  * Viero et al. (2022) Viero M. P., Sun G., Chung D. T., Moncelsi L., Condon S. S., 2022, MNRAS, 516, L30
  * Wang et al. (2017) Wang W.-H., et al., 2017, ApJ, 850, 37
  * Weiß et al. (2013) Weiß A., et al., 2013, ApJ, 767, 88
  * Whitworth et al. (2019) Whitworth A. P., et al., 2019, MNRAS, 489, 5436
  * Witstok et al. (2023) Witstok J., Jones G. C., Maiolino R., Smit R., Schneider R., 2023, MNRAS, 523, 3119
  * Zavala et al. (2018) Zavala J. A., et al., 2018, MNRAS, 475, 5585
  * Zhang et al. (2016) Zhang Z.-Y., Papadopoulos P. P., Ivison R. J., Galametz M., Smith M. W. L., Xilouris E. M., 2016, Royal Society Open Science, 3, 160025
  * da Cunha et al. (2013) da Cunha E., et al., 2013, ApJ, 766, 13
  * da Cunha et al. (2015) da Cunha E., et al., 2015, ApJ, 806, 110
  * da Cunha et al. (2021) da Cunha E., et al., 2021, ApJ, 919, 30

##  Appendix A HerBS photometry

Table 3: Spectroscopic redshifts, lensing magnifications and photometry for HerBS sources. Source | Spec-z | $\mu$ | $S_{250\,\micron}$ | $S_{350\,\micron}$ | $S_{500\,\micron}$ | $S_{850\,\micron}$ | $S_{1.2\,\textrm{mm}}$ | $S_{2\,\textrm{mm}}$ | $S_{3\,\textrm{mm}}$
---|---|---|---|---|---|---|---|---|---
|  |  | (mJy) | (mJy) | (mJy) | (mJy) | (mJy) | (mJy) | (mJy)
HerBS-11 | 2.631 | 18.4 | 257.5$\pm$6.4 | 271.1$\pm$6.3 | 204.0$\pm$7.2 | 67.3$\pm$6.3 | – | 3.59$\pm$0.03 | 0.93$\pm$0.02
HerBS-14 | 3.782 | 36.4 | 116.3$\pm$6.1 | 177.0$\pm$6.3 | 179.3$\pm$7.5 | 77.9$\pm$6.4 | 28.9$\pm$0.6 | 7.31$\pm$0.04 | 1.50$\pm$0.02
HerBS-18 | 2.182 | 27.9 | 212.9$\pm$4.7 | 244.2$\pm$5.0 | 169.4$\pm$6.2 | 52.9$\pm$6.1 | – | 2.73$\pm$0.03 | 0.74$\pm$0.06
HerBS-21 | 3.323 | 6.1 | 125.8$\pm$5.5 | 185.5$\pm$5.8 | 155.1$\pm$7.4 | 51.3$\pm$6.3 | – | 3.92$\pm$0.03 | 0.94$\pm$0.10
HerBS-24 | 2.198 | 4.6 | 170.9$\pm$5.7 | 197.1$\pm$6.3 | 145.6$\pm$7.4 | 64.8$\pm$7.8 | – | 2.68$\pm$0.03 | 0.74$\pm$0.03
HerBS-25 | 2.912 | 49.8 | 112.5$\pm$5.0 | 148.0$\pm$5.4 | 143.4$\pm$6.5 | 49.2$\pm$5.7 | – | 3.52$\pm$0.03 | 0.81$\pm$0.11
HerBS-27 | 4.509 | 14.6 | 72.2$\pm$5.3 | 129.8$\pm$5.6 | 138.6$\pm$7.0 | 90.5$\pm$6.3 | 28.9$\pm$1.0 | 8.70$\pm$0.03 | 1.97$\pm$0.02
HerBS-28 | 3.925 | 5.7 | 79.4$\pm$5.8 | 135.4$\pm$6.0 | 140.0$\pm$7.4 | 79.4$\pm$7.8 | 25.5$\pm$1.2 | 5.56$\pm$0.03 | 1.66$\pm$0.10
HerBS-36 | 3.095 | 5.4 | 121.5$\pm$6.1 | 161.0$\pm$6.7 | 125.5$\pm$7.7 | 64.0$\pm$8.8 | – | 4.81$\pm$0.03 | 1.11$\pm$0.02
HerBS-39 | 3.229 | 5.1 | 118.3$\pm$5.1 | 141.2$\pm$5.5 | 119.7$\pm$6.8 | 36.5$\pm$7.0 | – | 3.01$\pm$0.03 | 0.64$\pm$0.03
HerBS-41 | 4.098 | 2.1 | 63.3$\pm$6.2 | 91.1$\pm$6.1 | 121.7$\pm$7.4 | 31.8$\pm$5.9 | 15.2$\pm$0.5 | 4.85$\pm$0.07 | –
HerBS-42 | 3.307 | 3.9 | 130.3$\pm$5.8 | 160.0$\pm$6.1 | 116.2$\pm$6.8 | 50.4$\pm$6.1 | – | 2.95$\pm$0.06 | 0.58$\pm$0.05
HerBS-49 | 2.727 | 15.3 | 76.8$\pm$6.0 | 110.9$\pm$6.2 | 110.4$\pm$7.3 | 31.9$\pm$8.5 | – | 1.69$\pm$0.04 | 1.13$\pm$0.09
HerBS-55 | 2.656 | 13.7 | 109.0$\pm$5.3 | 116.5$\pm$5.5 | 107.1$\pm$6.6 | 29.9$\pm$7.1 | – | 1.05$\pm$0.03 | 0.33$\pm$0.02
HerBS-57 | 3.265 | 15.0 | 118.1$\pm$4.9 | 147.3$\pm$5.2 | 105.4$\pm$6.4 | 60.7$\pm$8.5 | – | 3.05$\pm$0.02 | 0.50$\pm$0.02
HerBS-60 | 3.261 | 9.5 | 73.3$\pm$5.8 | 101.2$\pm$6.1 | 103.6$\pm$7.5 | 39.8$\pm$5.7 | 13.2$\pm$0.8 | 2.54$\pm$0.03 | 0.53$\pm$0.02
HerBS-68 | 2.719 | 10.5 | 139.1$\pm$5.3 | 144.8$\pm$5.4 | 100.5$\pm$6.6 | 46.7$\pm$7.7 | – | 1.62$\pm$0.04 | 0.32$\pm$0.05
HerBS-73 | 3.026 | 3.1 | 117.1$\pm$6.0 | 129.0$\pm$6.2 | 99.6$\pm$7.4 | 49.4$\pm$6.8 | – | 2.09$\pm$0.03 | 0.43$\pm$0.02
HerBS-86 | 2.564 | 4.8 | 77.4$\pm$5.6 | 90.7$\pm$5.8 | 96.0$\pm$7.4 | 33.8$\pm$6.1 | 8.1$\pm$0.6 | 1.56$\pm$0.02 | 0.25$\pm$0.02
HerBS-93 | 2.400 | 2.4 | 77.3$\pm$5.4 | 87.3$\pm$5.7 | 94.8$\pm$7.0 | 22.8$\pm$5.7 | 6.2$\pm$0.4 | 1.39$\pm$0.02 | 0.16$\pm$0.01
HerBS-103 | 2.942 | 4.3 | 126.1$\pm$5.3 | 131.2$\pm$5.7 | 93.5$\pm$7.0 | 33.7$\pm$7.6 | – | 1.40$\pm$0.03 | 0.42$\pm$0.02
HerBS-111 | 2.371 | 4.3 | 105.9$\pm$6.5 | 115.6$\pm$6.2 | 92.7$\pm$7.4 | 13.3$\pm$6.9 | – | 1.13$\pm$0.02 | 0.21$\pm$0.02
HerBS-132 | 2.473 | 4.8 | 86.7$\pm$5.8 | 102.6$\pm$6.0 | 90.6$\pm$7.8 | 16.6$\pm$7.4 | – | 0.86$\pm$0.02 | 0.16$\pm$0.02
HerBS-145 | 2.730 | 1.9 | 54.7$\pm$6.0 | 67.4$\pm$6.2 | 86.8$\pm$7.7 | 13.4$\pm$7.0 | 5.1$\pm$0.6 | 1.06$\pm$0.04 | –
HerBS-160 | 3.955 | 8.1 | 48.6$\pm$5.6 | 84.2$\pm$6.0 | 84.8$\pm$7.1 | 36.6$\pm$6.4 | 15.0$\pm$0.6 | 3.94$\pm$0.02 | 0.91$\pm$0.02
HerBS-182 | 2.227 | 1.1 | 89.0$\pm$5.7 | 109.1$\pm$6.2 | 82.3$\pm$7.9 | – | – | 1.04$\pm$0.03 | 0.21$\pm$0.02
HerBS-184 | 2.507 | 6.0 | 91.9$\pm$5.9 | 107.6$\pm$6.0 | 82.3$\pm$7.1 | – | – | 1.49$\pm$0.02 | 0.48$\pm$0.02
HerBS-200 | 2.151 | 0.4 | 107.1$\pm$6.1 | 109.7$\pm$6.0 | 80.5$\pm$7.5 | 11.3$\pm$6.6 | – | 0.78$\pm$0.02 | 0.25$\pm$0.01
HerBS-207 | 1.569 | 5.2 | 96.9$\pm$5.9 | 121.7$\pm$6.1 | 80.2$\pm$7.5 | 33.1$\pm$6.3 | – | 0.89$\pm$0.02 | 0.25$\pm$0.03
HerBS-208 | 2.481 | 1.4 | 69.4$\pm$5.1 | 91.9$\pm$5.5 | 80.1$\pm$6.6 | – | – | 1.40$\pm$0.05 | 0.32$\pm$0.04

##  Appendix B Best-fitting SED parameters

Table 4: Best-fitting SED parameters for all SPT sources. Source | Spec-z | log($M_{\textrm{dust}}$ [$M_{\odot}$]) |  $T_{\textrm{dust}}$ [K] | $\beta$ | $\chi^{2}_{\nu}$
---|---|---|---|---|---
SPT0002-52 | 2.351 | $8.95\begin{subarray}{c}+0.05\\\ -0.05\end{subarray}$ | $33.23\begin{subarray}{c}+2.19\\\ -1.80\end{subarray}$ | $2.08\begin{subarray}{c}+0.11\\\ -0.12\end{subarray}$ | 2.05
SPT0020-51 | 4.123 | $9.16\begin{subarray}{c}+0.06\\\ -0.06\end{subarray}$ | $33.49\begin{subarray}{c}+3.02\\\ -3.04\end{subarray}$ | $1.90\begin{subarray}{c}+0.18\\\ -0.17\end{subarray}$ | 0.57
SPT0027-50 | 3.444 | $9.29\begin{subarray}{c}+0.04\\\ -0.04\end{subarray}$ | $33.48\begin{subarray}{c}+1.35\\\ -1.25\end{subarray}$ | $2.03\begin{subarray}{c}+0.10\\\ -0.09\end{subarray}$ | 3.02
SPT0054-41 | 4.877 | $9.26\begin{subarray}{c}+0.08\\\ -0.06\end{subarray}$ | $31.67\begin{subarray}{c}+3.54\\\ -3.99\end{subarray}$ | $2.12\begin{subarray}{c}+0.25\\\ -0.22\end{subarray}$ | 0.84
SPT0103-45 | 3.090 | $9.57\begin{subarray}{c}+0.08\\\ -0.07\end{subarray}$ | $25.37\begin{subarray}{c}+2.84\\\ -2.88\end{subarray}$ | $2.14\begin{subarray}{c}+0.29\\\ -0.22\end{subarray}$ | 1.10
SPT0106-64 | 4.910 | $9.21\begin{subarray}{c}+0.05\\\ -0.05\end{subarray}$ | $41.39\begin{subarray}{c}+3.65\\\ -2.93\end{subarray}$ | $1.78\begin{subarray}{c}+0.14\\\ -0.15\end{subarray}$ | 3.65
SPT0109-47 | 3.614 | $8.78\begin{subarray}{c}+0.05\\\ -0.05\end{subarray}$ | $35.33\begin{subarray}{c}+2.12\\\ -3.32\end{subarray}$ | $1.96\begin{subarray}{c}+0.18\\\ -0.13\end{subarray}$ | 2.85
SPT0112-55 | 3.443 | $9.18\begin{subarray}{c}+0.24\\\ -0.35\end{subarray}$ | $14.35\begin{subarray}{c}+4.40\\\ -1.59\end{subarray}$ | $3.62\begin{subarray}{c}+0.25\\\ -0.50\end{subarray}$ | 6.53
SPT0113-46 | 4.233 | $8.84\begin{subarray}{c}+0.13\\\ -0.12\end{subarray}$ | $22.52\begin{subarray}{c}+5.08\\\ -3.33\end{subarray}$ | $2.35\begin{subarray}{c}+0.38\\\ -0.43\end{subarray}$ | 1.40
SPT0125-47 | 2.515 | $9.47\begin{subarray}{c}+0.05\\\ -0.05\end{subarray}$ | $38.81\begin{subarray}{c}+2.07\\\ -2.88\end{subarray}$ | $1.67\begin{subarray}{c}+0.14\\\ -0.11\end{subarray}$ | 2.69
SPT0125-50 | 3.957 | $8.77\begin{subarray}{c}+0.06\\\ -0.06\end{subarray}$ | $46.69\begin{subarray}{c}+3.35\\\ -2.78\end{subarray}$ | $1.32\begin{subarray}{c}+0.13\\\ -0.13\end{subarray}$ | 3.83
SPT0136-63 | 4.299 | $9.18\begin{subarray}{c}+0.07\\\ -0.06\end{subarray}$ | $28.24\begin{subarray}{c}+2.68\\\ -2.68\end{subarray}$ | $2.21\begin{subarray}{c}+0.22\\\ -0.20\end{subarray}$ | 0.42
SPT0147-64 | 4.803 | $9.30\begin{subarray}{c}+0.07\\\ -0.06\end{subarray}$ | $35.28\begin{subarray}{c}+3.98\\\ -5.08\end{subarray}$ | $1.70\begin{subarray}{c}+0.27\\\ -0.19\end{subarray}$ | 2.39
SPT0150-59 | 2.788 | $9.26\begin{subarray}{c}+0.05\\\ -0.05\end{subarray}$ | $33.29\begin{subarray}{c}+1.88\\\ -2.26\end{subarray}$ | $1.70\begin{subarray}{c}+0.13\\\ -0.11\end{subarray}$ | 2.63
SPT0155-62 | 4.349 | $10.01\begin{subarray}{c}+0.07\\\ -0.07\end{subarray}$ | $22.43\begin{subarray}{c}+2.40\\\ -1.78\end{subarray}$ | $2.21\begin{subarray}{c}+0.20\\\ -0.23\end{subarray}$ | 3.32
SPT0202-61 | 5.018 | $9.12\begin{subarray}{c}+0.04\\\ -0.04\end{subarray}$ | $57.61\begin{subarray}{c}+4.19\\\ -3.64\end{subarray}$ | $0.96\begin{subarray}{c}+0.11\\\ -0.11\end{subarray}$ | 6.46
SPT0226-45 | 3.233 | $9.01\begin{subarray}{c}+0.07\\\ -0.08\end{subarray}$ | $28.51\begin{subarray}{c}+4.58\\\ -2.15\end{subarray}$ | $2.38\begin{subarray}{c}+0.18\\\ -0.26\end{subarray}$ | 4.29
SPT0243-49 | 5.702 | $9.67\begin{subarray}{c}+0.11\\\ -0.09\end{subarray}$ | $26.80\begin{subarray}{c}+4.61\\\ -4.30\end{subarray}$ | $1.87\begin{subarray}{c}+0.39\\\ -0.38\end{subarray}$ | 0.71
SPT0245-63 | 5.626 | $9.84\begin{subarray}{c}+0.08\\\ -0.08\end{subarray}$ | $30.30\begin{subarray}{c}+4.99\\\ -3.47\end{subarray}$ | $2.09\begin{subarray}{c}+0.28\\\ -0.31\end{subarray}$ | 1.07
SPT0300-46 | 3.595 | $9.35\begin{subarray}{c}+0.07\\\ -0.07\end{subarray}$ | $31.26\begin{subarray}{c}+2.94\\\ -3.23\end{subarray}$ | $1.96\begin{subarray}{c}+0.26\\\ -0.20\end{subarray}$ | 0.96
SPT0311-58 | 6.901 | $9.28\begin{subarray}{c}+0.09\\\ -0.08\end{subarray}$ | $42.17\begin{subarray}{c}+10.20\\\ -7.33\end{subarray}$ | $1.66\begin{subarray}{c}+0.35\\\ -0.37\end{subarray}$ | 1.37
SPT0314-44 | 2.935 | $9.44\begin{subarray}{c}+0.04\\\ -0.04\end{subarray}$ | $31.57\begin{subarray}{c}+1.35\\\ -1.22\end{subarray}$ | $2.01\begin{subarray}{c}+0.10\\\ -0.09\end{subarray}$ | 2.96
SPT0319-47 | 4.510 | $9.38\begin{subarray}{c}+0.09\\\ -0.08\end{subarray}$ | $37.93\begin{subarray}{c}+5.07\\\ -5.60\end{subarray}$ | $1.66\begin{subarray}{c}+0.33\\\ -0.24\end{subarray}$ | 1.63
SPT0345-47 | 4.296 | $8.85\begin{subarray}{c}+0.06\\\ -0.06\end{subarray}$ | $52.45\begin{subarray}{c}+2.45\\\ -2.60\end{subarray}$ | $1.47\begin{subarray}{c}+0.12\\\ -0.11\end{subarray}$ | 1.22
SPT0346-52 | 5.655 | $9.24\begin{subarray}{c}+0.05\\\ -0.06\end{subarray}$ | $52.76\begin{subarray}{c}+4.10\\\ -7.17\end{subarray}$ | $1.31\begin{subarray}{c}+0.26\\\ -0.16\end{subarray}$ | 2.94
SPT0348-62 | 5.654 | $9.59\begin{subarray}{c}+0.14\\\ -0.12\end{subarray}$ | $26.19\begin{subarray}{c}+4.35\\\ -3.48\end{subarray}$ | $2.60\begin{subarray}{c}+0.30\\\ -0.33\end{subarray}$ | 0.21
SPT0402-45 | 2.683 | $9.46\begin{subarray}{c}+0.04\\\ -0.04\end{subarray}$ | $33.16\begin{subarray}{c}+1.60\\\ -1.62\end{subarray}$ | $2.07\begin{subarray}{c}+0.10\\\ -0.09\end{subarray}$ | 8.22
SPT0403-58 | 4.056 | $9.40\begin{subarray}{c}+0.07\\\ -0.07\end{subarray}$ | $36.14\begin{subarray}{c}+3.77\\\ -4.35\end{subarray}$ | $1.81\begin{subarray}{c}+0.23\\\ -0.19\end{subarray}$ | 2.28
SPT0418-47 | 4.225 | $8.30\begin{subarray}{c}+0.19\\\ -0.09\end{subarray}$ | $29.92\begin{subarray}{c}+2.89\\\ -9.65\end{subarray}$ | $2.43\begin{subarray}{c}+0.83\\\ -0.23\end{subarray}$ | 7.44
SPT0425-40 | 5.135 | $8.78\begin{subarray}{c}+0.05\\\ -0.05\end{subarray}$ | $44.68\begin{subarray}{c}+2.69\\\ -3.05\end{subarray}$ | $1.78\begin{subarray}{c}+0.15\\\ -0.12\end{subarray}$ | 4.65
SPT0436-40 | 3.852 | $9.20\begin{subarray}{c}+0.16\\\ -0.16\end{subarray}$ | $23.42\begin{subarray}{c}+7.81\\\ -3.87\end{subarray}$ | $2.59\begin{subarray}{c}+0.38\\\ -0.51\end{subarray}$ | 2.63
SPT0441-46 | 4.480 | $8.89\begin{subarray}{c}+0.13\\\ -0.13\end{subarray}$ | $28.48\begin{subarray}{c}+11.44\\\ -5.46\end{subarray}$ | $2.07\begin{subarray}{c}+0.50\\\ -0.55\end{subarray}$ | 3.66
SPT0452-50 | 2.011 | $10.07\begin{subarray}{c}+0.13\\\ -0.10\end{subarray}$ | $19.02\begin{subarray}{c}+2.76\\\ -2.34\end{subarray}$ | $2.02\begin{subarray}{c}+0.30\\\ -0.27\end{subarray}$ | 1.63
SPT0459-58 | 4.856 | $9.06\begin{subarray}{c}+0.11\\\ -0.09\end{subarray}$ | $31.52\begin{subarray}{c}+6.78\\\ -5.52\end{subarray}$ | $2.03\begin{subarray}{c}+0.42\\\ -0.37\end{subarray}$ | 0.68
SPT0459-59 | 4.799 | $9.48\begin{subarray}{c}+0.09\\\ -0.09\end{subarray}$ | $25.24\begin{subarray}{c}+3.93\\\ -2.83\end{subarray}$ | $2.29\begin{subarray}{c}+0.30\\\ -0.33\end{subarray}$ | 0.80
SPT0512-59 | 2.233 | $9.39\begin{subarray}{c}+0.04\\\ -0.05\end{subarray}$ | $29.06\begin{subarray}{c}+1.69\\\ -1.46\end{subarray}$ | $1.92\begin{subarray}{c}+0.14\\\ -0.13\end{subarray}$ | 0.83
SPT0516-59 | 3.404 | $8.68\begin{subarray}{c}+0.05\\\ -0.05\end{subarray}$ | $36.36\begin{subarray}{c}+2.21\\\ -2.30\end{subarray}$ | $1.97\begin{subarray}{c}+0.15\\\ -0.13\end{subarray}$ | 2.31
SPT0520-53 | 3.779 | $9.09\begin{subarray}{c}+0.08\\\ -0.07\end{subarray}$ | $32.54\begin{subarray}{c}+3.22\\\ -4.43\end{subarray}$ | $1.83\begin{subarray}{c}+0.27\\\ -0.18\end{subarray}$ | 1.54
SPT0528-53 | 4.737 | $8.69\begin{subarray}{c}+0.12\\\ -0.08\end{subarray}$ | $36.91\begin{subarray}{c}+6.77\\\ -7.38\end{subarray}$ | $1.76\begin{subarray}{c}+0.36\\\ -0.30\end{subarray}$ | 1.14
SPT0529-54 | 3.368 | $9.38\begin{subarray}{c}+0.09\\\ -0.11\end{subarray}$ | $19.07\begin{subarray}{c}+3.29\\\ -2.03\end{subarray}$ | $2.62\begin{subarray}{c}+0.29\\\ -0.36\end{subarray}$ | 0.80
SPT0532-50 | 3.399 | $9.33\begin{subarray}{c}+0.04\\\ -0.04\end{subarray}$ | $38.56\begin{subarray}{c}+2.15\\\ -1.94\end{subarray}$ | $1.38\begin{subarray}{c}+0.13\\\ -0.12\end{subarray}$ | 0.86
SPT0538-50 | 2.786 | $8.98\begin{subarray}{c}+0.05\\\ -0.06\end{subarray}$ | $33.59\begin{subarray}{c}+1.98\\\ -1.75\end{subarray}$ | $1.71\begin{subarray}{c}+0.15\\\ -0.16\end{subarray}$ | 1.36
SPT0544-40 | 4.269 | $9.14\begin{subarray}{c}+0.05\\\ -0.05\end{subarray}$ | $38.11\begin{subarray}{c}+2.45\\\ -2.89\end{subarray}$ | $1.65\begin{subarray}{c}+0.15\\\ -0.14\end{subarray}$ | 2.75
SPT0550-53 | 3.128 | $9.20\begin{subarray}{c}+0.08\\\ -0.09\end{subarray}$ | $25.27\begin{subarray}{c}+3.60\\\ -2.46\end{subarray}$ | $2.17\begin{subarray}{c}+0.26\\\ -0.28\end{subarray}$ | 1.92
SPT0551-48 | 2.583 | $9.54\begin{subarray}{c}+0.06\\\ -0.06\end{subarray}$ | $39.01\begin{subarray}{c}+2.94\\\ -2.66\end{subarray}$ | $1.52\begin{subarray}{c}+0.18\\\ -0.16\end{subarray}$ | 2.27
SPT0551-50 | 3.164 | $9.24\begin{subarray}{c}+0.05\\\ -0.05\end{subarray}$ | $32.99\begin{subarray}{c}+1.71\\\ -1.58\end{subarray}$ | $1.83\begin{subarray}{c}+0.13\\\ -0.12\end{subarray}$ | 2.83
SPT0552-42 | 4.438 | $9.25\begin{subarray}{c}+0.10\\\ -0.09\end{subarray}$ | $27.06\begin{subarray}{c}+3.98\\\ -3.29\end{subarray}$ | $2.03\begin{subarray}{c}+0.25\\\ -0.25\end{subarray}$ | 0.88
SPT0553-50 | 5.323 | $8.89\begin{subarray}{c}+0.07\\\ -0.07\end{subarray}$ | $39.50\begin{subarray}{c}+5.27\\\ -4.90\end{subarray}$ | $1.65\begin{subarray}{c}+0.28\\\ -0.25\end{subarray}$ | 0.90
SPT0555-62 | 4.815 | $9.01\begin{subarray}{c}+0.10\\\ -0.10\end{subarray}$ | $30.00\begin{subarray}{c}+8.01\\\ -3.66\end{subarray}$ | $2.22\begin{subarray}{c}+0.27\\\ -0.40\end{subarray}$ | 4.84
SPT0604-64 | 2.481 | $9.52\begin{subarray}{c}+0.04\\\ -0.04\end{subarray}$ | $28.98\begin{subarray}{c}+1.14\\\ -1.11\end{subarray}$ | $2.13\begin{subarray}{c}+0.09\\\ -0.08\end{subarray}$ | 4.53
SPT0611-55 | 2.026 | $8.92\begin{subarray}{c}+0.07\\\ -0.07\end{subarray}$ | $19.45\begin{subarray}{c}+1.33\\\ -2.26\end{subarray}$ | $3.44\begin{subarray}{c}+0.34\\\ -0.22\end{subarray}$ | 4.45
SPT0625-58 | 2.727 | $9.48\begin{subarray}{c}+0.04\\\ -0.04\end{subarray}$ | $32.41\begin{subarray}{c}+1.44\\\ -1.32\end{subarray}$ | $1.80\begin{subarray}{c}+0.09\\\ -0.09\end{subarray}$ | 2.85
SPT0652-55 | 3.347 | $9.65\begin{subarray}{c}+0.14\\\ -0.08\end{subarray}$ | $24.80\begin{subarray}{c}+2.77\\\ -4.72\end{subarray}$ | $2.35\begin{subarray}{c}+0.39\\\ -0.24\end{subarray}$ | 4.53
SPT2031-51 | 2.452 | $9.40\begin{subarray}{c}+0.04\\\ -0.04\end{subarray}$ | $29.27\begin{subarray}{c}+1.37\\\ -1.26\end{subarray}$ | $2.00\begin{subarray}{c}+0.10\\\ -0.10\end{subarray}$ | 2.18
SPT2037-65 | 3.998 | $9.89\begin{subarray}{c}+0.04\\\ -0.04\end{subarray}$ | $46.72\begin{subarray}{c}+3.06\\\ -3.15\end{subarray}$ | $0.70\begin{subarray}{c}+0.13\\\ -0.11\end{subarray}$ | 2.16
SPT2048-55 | 4.090 | $9.24\begin{subarray}{c}+0.05\\\ -0.06\end{subarray}$ | $42.17\begin{subarray}{c}+3.71\\\ -3.35\end{subarray}$ | $1.11\begin{subarray}{c}+0.15\\\ -0.14\end{subarray}$ | 1.07
SPT2101-60 | 3.155 | $9.11\begin{subarray}{c}+0.04\\\ -0.04\end{subarray}$ | $33.91\begin{subarray}{c}+1.73\\\ -1.70\end{subarray}$ | $1.93\begin{subarray}{c}+0.12\\\ -0.11\end{subarray}$ | 1.76
SPT2103-60 | 4.436 | $8.65\begin{subarray}{c}+0.10\\\ -0.09\end{subarray}$ | $22.20\begin{subarray}{c}+2.88\\\ -2.33\end{subarray}$ | $2.68\begin{subarray}{c}+0.28\\\ -0.29\end{subarray}$ | 2.44
SPT2129-57 | 3.260 | $9.04\begin{subarray}{c}+0.05\\\ -0.05\end{subarray}$ | $37.26\begin{subarray}{c}+2.08\\\ -1.88\end{subarray}$ | $1.75\begin{subarray}{c}+0.11\\\ -0.11\end{subarray}$ | 3.51
SPT2132-58 | 4.768 | $9.24\begin{subarray}{c}+0.09\\\ -0.10\end{subarray}$ | $29.82\begin{subarray}{c}+9.38\\\ -4.40\end{subarray}$ | $1.91\begin{subarray}{c}+0.36\\\ -0.48\end{subarray}$ | 1.90
SPT2134-50 | 2.780 | $8.67\begin{subarray}{c}+0.05\\\ -0.06\end{subarray}$ | $37.39\begin{subarray}{c}+3.60\\\ -2.13\end{subarray}$ | $1.70\begin{subarray}{c}+0.14\\\ -0.15\end{subarray}$ | 3.51
SPT2146-55 | 4.567 | $9.03\begin{subarray}{c}+0.10\\\ -0.10\end{subarray}$ | $33.32\begin{subarray}{c}+8.78\\\ -5.89\end{subarray}$ | $1.80\begin{subarray}{c}+0.39\\\ -0.39\end{subarray}$ | 1.01
SPT2147-50 | 3.760 | $9.01\begin{subarray}{c}+0.07\\\ -0.07\end{subarray}$ | $30.72\begin{subarray}{c}+3.25\\\ -2.49\end{subarray}$ | $2.05\begin{subarray}{c}+0.21\\\ -0.20\end{subarray}$ | 2.62
SPT2152-40 | 3.851 | $9.29\begin{subarray}{c}+0.22\\\ -0.11\end{subarray}$ | $30.11\begin{subarray}{c}+6.51\\\ -9.32\end{subarray}$ | $1.92\begin{subarray}{c}+0.69\\\ -0.38\end{subarray}$ | 4.47
SPT2203-41 | 5.194 | $9.23\begin{subarray}{c}+0.15\\\ -0.12\end{subarray}$ | $21.94\begin{subarray}{c}+2.92\\\ -2.43\end{subarray}$ | $2.84\begin{subarray}{c}+0.24\\\ -0.27\end{subarray}$ | 1.89
SPT2232-61 | 2.894 | $9.18\begin{subarray}{c}+0.06\\\ -0.06\end{subarray}$ | $27.91\begin{subarray}{c}+2.27\\\ -2.59\end{subarray}$ | $2.24\begin{subarray}{c}+0.19\\\ -0.15\end{subarray}$ | 2.93
SPT2311-45 | 2.507 | $9.18\begin{subarray}{c}+0.05\\\ -0.05\end{subarray}$ | $31.51\begin{subarray}{c}+2.45\\\ -1.74\end{subarray}$ | $1.78\begin{subarray}{c}+0.12\\\ -0.14\end{subarray}$ | 2.71
SPT2311-54 | 4.280 | $9.18\begin{subarray}{c}+0.06\\\ -0.06\end{subarray}$ | $40.77\begin{subarray}{c}+3.49\\\ -4.39\end{subarray}$ | $1.83\begin{subarray}{c}+0.20\\\ -0.16\end{subarray}$ | 4.16
SPT2316-50 | 3.141 | $8.71\begin{subarray}{c}+0.13\\\ -0.11\end{subarray}$ | $23.11\begin{subarray}{c}+3.44\\\ -3.33\end{subarray}$ | $2.75\begin{subarray}{c}+0.38\\\ -0.31\end{subarray}$ | 1.04
SPT2319-55 | 5.293 | $8.74\begin{subarray}{c}+0.10\\\ -0.10\end{subarray}$ | $29.34\begin{subarray}{c}+6.12\\\ -3.78\end{subarray}$ | $2.21\begin{subarray}{c}+0.29\\\ -0.33\end{subarray}$ | 1.10
SPT2332-53 | 2.726 | $9.18\begin{subarray}{c}+0.04\\\ -0.05\end{subarray}$ | $39.12\begin{subarray}{c}+1.82\\\ -1.68\end{subarray}$ | $1.86\begin{subarray}{c}+0.11\\\ -0.11\end{subarray}$ | 6.82
SPT2335-53 | 4.756 | $8.46\begin{subarray}{c}+0.11\\\ -0.09\end{subarray}$ | $30.55\begin{subarray}{c}+4.68\\\ -5.40\end{subarray}$ | $2.68\begin{subarray}{c}+0.38\\\ -0.31\end{subarray}$ | 1.93
SPT2340-59 | 3.862 | $8.84\begin{subarray}{c}+0.14\\\ -0.11\end{subarray}$ | $26.28\begin{subarray}{c}+3.79\\\ -4.89\end{subarray}$ | $2.71\begin{subarray}{c}+0.41\\\ -0.29\end{subarray}$ | 3.62
SPT2349-50 | 2.876 | $9.37\begin{subarray}{c}+0.07\\\ -0.07\end{subarray}$ | $30.69\begin{subarray}{c}+3.06\\\ -4.05\end{subarray}$ | $2.06\begin{subarray}{c}+0.24\\\ -0.18\end{subarray}$ | 3.26
SPT2349-52 | 3.900 | $8.58\begin{subarray}{c}+0.22\\\ -0.17\end{subarray}$ | $19.49\begin{subarray}{c}+4.23\\\ -2.99\end{subarray}$ | $3.62\begin{subarray}{c}+0.37\\\ -0.34\end{subarray}$ | 2.62
SPT2351-57 | 5.811 | $8.65\begin{subarray}{c}+0.07\\\ -0.06\end{subarray}$ | $39.58\begin{subarray}{c}+4.69\\\ -4.72\end{subarray}$ | $1.98\begin{subarray}{c}+0.23\\\ -0.21\end{subarray}$ | 3.76
SPT2353-50 | 5.578 | $8.83\begin{subarray}{c}+0.11\\\ -0.10\end{subarray}$ | $35.44\begin{subarray}{c}+8.18\\\ -6.36\end{subarray}$ | $1.94\begin{subarray}{c}+0.37\\\ -0.34\end{subarray}$ | 1.85
SPT2354-58 | 1.867 | $9.09\begin{subarray}{c}+0.05\\\ -0.07\end{subarray}$ | $31.95\begin{subarray}{c}+3.77\\\ -2.00\end{subarray}$ | $2.05\begin{subarray}{c}+0.12\\\ -0.16\end{subarray}$ | 1.73
SPT2357-51 | 3.070 | $9.39\begin{subarray}{c}+0.16\\\ -0.09\end{subarray}$ | $21.63\begin{subarray}{c}+2.26\\\ -3.65\end{subarray}$ | $2.80\begin{subarray}{c}+0.36\\\ -0.24\end{subarray}$ | 3.82
Table 5: Best-fitting SED parameters for all HerBS sources. Source | Spec-z | log($M_{\textrm{dust}}$ [$M_{\odot}$]) |  $T_{\textrm{dust}}$ [K] | $\beta$ | $\chi^{2}_{\nu}$
---|---|---|---|---|---
HerBS-11 | 2.631 | $8.63\begin{subarray}{c}+0.03\\\ -0.04\end{subarray}$ | $34.01\begin{subarray}{c}+1.78\\\ -1.58\end{subarray}$ | $1.84\begin{subarray}{c}+0.10\\\ -0.10\end{subarray}$ | 1.63
HerBS-14 | 3.782 | $8.42\begin{subarray}{c}+0.04\\\ -0.03\end{subarray}$ | $37.51\begin{subarray}{c}+1.77\\\ -2.08\end{subarray}$ | $1.64\begin{subarray}{c}+0.12\\\ -0.10\end{subarray}$ | 1.45
HerBS-18 | 2.182 | $8.47\begin{subarray}{c}+0.03\\\ -0.04\end{subarray}$ | $29.05\begin{subarray}{c}+1.45\\\ -1.29\end{subarray}$ | $1.95\begin{subarray}{c}+0.11\\\ -0.11\end{subarray}$ | 0.94
HerBS-21 | 3.323 | $9.01\begin{subarray}{c}+0.04\\\ -0.04\end{subarray}$ | $35.63\begin{subarray}{c}+1.73\\\ -1.70\end{subarray}$ | $1.81\begin{subarray}{c}+0.12\\\ -0.11\end{subarray}$ | 0.18
HerBS-24 | 2.198 | $9.27\begin{subarray}{c}+0.04\\\ -0.04\end{subarray}$ | $28.70\begin{subarray}{c}+1.65\\\ -1.43\end{subarray}$ | $1.87\begin{subarray}{c}+0.11\\\ -0.11\end{subarray}$ | 3.60
HerBS-25 | 2.912 | $8.18\begin{subarray}{c}+0.05\\\ -0.05\end{subarray}$ | $31.53\begin{subarray}{c}+1.91\\\ -2.51\end{subarray}$ | $1.84\begin{subarray}{c}+0.17\\\ -0.14\end{subarray}$ | 1.56
HerBS-27 | 4.509 | $8.84\begin{subarray}{c}+0.04\\\ -0.04\end{subarray}$ | $40.87\begin{subarray}{c}+2.16\\\ -2.49\end{subarray}$ | $1.46\begin{subarray}{c}+0.13\\\ -0.12\end{subarray}$ | 3.56
HerBS-28 | 3.925 | $9.20\begin{subarray}{c}+0.04\\\ -0.04\end{subarray}$ | $37.69\begin{subarray}{c}+2.01\\\ -1.99\end{subarray}$ | $1.57\begin{subarray}{c}+0.13\\\ -0.12\end{subarray}$ | 1.70
HerBS-36 | 3.095 | $9.20\begin{subarray}{c}+0.04\\\ -0.04\end{subarray}$ | $36.81\begin{subarray}{c}+2.29\\\ -2.13\end{subarray}$ | $1.50\begin{subarray}{c}+0.12\\\ -0.11\end{subarray}$ | 1.80
HerBS-39 | 3.229 | $8.93\begin{subarray}{c}+0.04\\\ -0.04\end{subarray}$ | $37.62\begin{subarray}{c}+2.30\\\ -2.29\end{subarray}$ | $1.75\begin{subarray}{c}+0.13\\\ -0.12\end{subarray}$ | 1.06
HerBS-41 | 4.098 | $9.44\begin{subarray}{c}+0.06\\\ -0.07\end{subarray}$ | $44.29\begin{subarray}{c}+3.95\\\ -3.40\end{subarray}$ | $1.31\begin{subarray}{c}+0.20\\\ -0.20\end{subarray}$ | 8.36
HerBS-42 | 3.307 | $9.01\begin{subarray}{c}+0.04\\\ -0.04\end{subarray}$ | $37.83\begin{subarray}{c}+2.24\\\ -2.00\end{subarray}$ | $1.82\begin{subarray}{c}+0.11\\\ -0.11\end{subarray}$ | 1.99
HerBS-49 | 2.727 | $8.48\begin{subarray}{c}+0.05\\\ -0.05\end{subarray}$ | $28.93\begin{subarray}{c}+2.14\\\ -2.08\end{subarray}$ | $2.06\begin{subarray}{c}+0.19\\\ -0.18\end{subarray}$ | 14.77
HerBS-55 | 2.656 | $8.26\begin{subarray}{c}+0.04\\\ -0.04\end{subarray}$ | $30.86\begin{subarray}{c}+2.17\\\ -2.09\end{subarray}$ | $2.22\begin{subarray}{c}+0.16\\\ -0.15\end{subarray}$ | 7.48
HerBS-57 | 3.265 | $8.40\begin{subarray}{c}+0.05\\\ -0.04\end{subarray}$ | $36.51\begin{subarray}{c}+2.04\\\ -2.33\end{subarray}$ | $1.89\begin{subarray}{c}+0.13\\\ -0.11\end{subarray}$ | 6.40
HerBS-60 | 3.261 | $8.66\begin{subarray}{c}+0.04\\\ -0.04\end{subarray}$ | $32.10\begin{subarray}{c}+2.16\\\ -2.58\end{subarray}$ | $1.97\begin{subarray}{c}+0.17\\\ -0.14\end{subarray}$ | 1.09
HerBS-68 | 2.719 | $8.44\begin{subarray}{c}+0.04\\\ -0.04\end{subarray}$ | $34.16\begin{subarray}{c}+2.22\\\ -1.99\end{subarray}$ | $2.01\begin{subarray}{c}+0.14\\\ -0.13\end{subarray}$ | 2.97
HerBS-73 | 3.026 | $9.04\begin{subarray}{c}+0.05\\\ -0.04\end{subarray}$ | $34.99\begin{subarray}{c}+2.41\\\ -3.22\end{subarray}$ | $1.93\begin{subarray}{c}+0.18\\\ -0.13\end{subarray}$ | 3.95
HerBS-86 | 2.564 | $8.88\begin{subarray}{c}+0.06\\\ -0.05\end{subarray}$ | $26.11\begin{subarray}{c}+2.13\\\ -2.72\end{subarray}$ | $2.32\begin{subarray}{c}+0.22\\\ -0.17\end{subarray}$ | 3.73
HerBS-93 | 2.400 | $9.07\begin{subarray}{c}+0.06\\\ -0.05\end{subarray}$ | $24.88\begin{subarray}{c}+1.79\\\ -2.12\end{subarray}$ | $2.50\begin{subarray}{c}+0.18\\\ -0.15\end{subarray}$ | 6.77
HerBS-103 | 2.942 | $8.75\begin{subarray}{c}+0.04\\\ -0.04\end{subarray}$ | $38.66\begin{subarray}{c}+2.67\\\ -2.46\end{subarray}$ | $1.85\begin{subarray}{c}+0.13\\\ -0.13\end{subarray}$ | 2.53
HerBS-111 | 2.371 | $8.79\begin{subarray}{c}+0.05\\\ -0.05\end{subarray}$ | $28.77\begin{subarray}{c}+1.92\\\ -1.75\end{subarray}$ | $2.24\begin{subarray}{c}+0.14\\\ -0.14\end{subarray}$ | 2.00
HerBS-132 | 2.473 | $8.64\begin{subarray}{c}+0.05\\\ -0.05\end{subarray}$ | $26.67\begin{subarray}{c}+2.04\\\ -2.06\end{subarray}$ | $2.52\begin{subarray}{c}+0.19\\\ -0.17\end{subarray}$ | 1.69
HerBS-145 | 2.730 | $9.04\begin{subarray}{c}+0.05\\\ -0.06\end{subarray}$ | $28.69\begin{subarray}{c}+2.79\\\ -2.87\end{subarray}$ | $2.24\begin{subarray}{c}+0.29\\\ -0.24\end{subarray}$ | 7.24
HerBS-160 | 3.955 | $8.81\begin{subarray}{c}+0.04\\\ -0.04\end{subarray}$ | $39.28\begin{subarray}{c}+2.46\\\ -2.36\end{subarray}$ | $1.50\begin{subarray}{c}+0.13\\\ -0.13\end{subarray}$ | 0.52
HerBS-182 | 2.227 | $9.44\begin{subarray}{c}+0.05\\\ -0.05\end{subarray}$ | $26.41\begin{subarray}{c}+1.84\\\ -1.77\end{subarray}$ | $2.29\begin{subarray}{c}+0.16\\\ -0.15\end{subarray}$ | 0.39
HerBS-184 | 2.507 | $8.80\begin{subarray}{c}+0.04\\\ -0.05\end{subarray}$ | $33.57\begin{subarray}{c}+2.73\\\ -2.37\end{subarray}$ | $1.71\begin{subarray}{c}+0.15\\\ -0.14\end{subarray}$ | 4.62
HerBS-200 | 2.151 | $9.76\begin{subarray}{c}+0.05\\\ -0.05\end{subarray}$ | $30.84\begin{subarray}{c}+2.68\\\ -2.21\end{subarray}$ | $2.04\begin{subarray}{c}+0.15\\\ -0.15\end{subarray}$ | 5.58
HerBS-207 | 1.569 | $8.99\begin{subarray}{c}+0.05\\\ -0.05\end{subarray}$ | $21.10\begin{subarray}{c}+1.39\\\ -1.30\end{subarray}$ | $2.36\begin{subarray}{c}+0.15\\\ -0.15\end{subarray}$ | 3.43
HerBS-208 | 2.481 | $9.45\begin{subarray}{c}+0.06\\\ -0.05\end{subarray}$ | $27.87\begin{subarray}{c}+2.11\\\ -2.08\end{subarray}$ | $2.07\begin{subarray}{c}+0.18\\\ -0.16\end{subarray}$ | 0.96

##  Appendix C Example Posterior Distributions

Figure 9: The posterior distributions for SPT 0002-52 and HerBS-11 obtained
from FIR SED fitting with an optically thin dust model (shaded contours), and
two general opacity models, one with $\lambda_{1}=100\,\mu$m (solid contours),
the other with $\lambda_{1}=200\,\mu$m (dashed contours). Parameters shown are
the dust mass $M_{\textrm{dust}}$, the dust temperature $T_{\textrm{dust}}$,
the dust emissivity index $\beta$, IR luminosity $L_{\textrm{IR}}$ and the
peak wavelength $\lambda_{\textrm{peak}}$.