Discovery of Four Apparently Cold Dusty Galaxies at z = 3.62–5.85 in the COSMOS Field: Direct Evidence of Cosmic Microwave Background Impact on High-redshift Galaxy Observables

We run extensive simulations to determine how to compute the chance probability of finding emission lines (with the same line-searching algorithm) as produced by noise fluctuations in a given spectral scan, depending on the S/N of the feature and its line width (broader lines are more unlikely to happen by chance because fewer independent realizations are available within the observed range; see also González-López et al. 2019), and also based on the full velocity range spanned by the data, the velocity sampling, and the range of acceptable line widths (from 100 to 1000 km s⁻¹ in this case). We assume well-behaved Gaussian noise in the data, as verified to hold for our ALMA spectra. We define the probability that a line detection is due to noise fluctuations as

$$\begin{eqnarray}&&{P}_{\mathrm{line}}=1-{P}_{0}^{{N}_{\mathrm{trials}}^{\mathrm{EFF}}},\end{eqnarray} \tag{ 1 }$$

where P₀ is the probability to have a line with S/N up to that observed in a single trial (approaching 1 for high-σ events), and the exponent ${N}_{\mathrm{trials}}^{\mathrm{EFF}}$ is the effective number of trials, which, based on our simulations, can be well approximated by

$$\begin{eqnarray}&&{N}_{\mathrm{trials}}^{\mathrm{EFF}}\sim 10\displaystyle \frac{{N}_{\mathrm{total},\mathrm{ch}}}{{N}_{\mathrm{line},\mathrm{ch}}}{N}_{\mathrm{line},\mathrm{ch}}^{0.58}\mathrm{log}\displaystyle \frac{{N}_{\mathrm{line},\mathrm{ch}}^{\max }}{{N}_{\mathrm{line},\mathrm{ch}}^{\min }},\end{eqnarray} \tag{ 2 }$$

where N_total,ch and N_line,ch are the number of channels in the entire spectrum scanned and in the recovered line, respectively, while ${N}_{\mathrm{line},\mathrm{ch}}^{\max }$ and ${N}_{\mathrm{line},\mathrm{ch}}^{\min }$ define the allowed velocity range of lines to be searched, expressed as number of channels. Notice that a naive treatment would account only for the first term (ratio of full velocity in spectrum to line velocity), while in reality the number of effective realizations is considerably larger. In fact, less than a full line-velocity shift is required to imply a new realization, and the number of effective realizations depends also on the range of velocity widths searched, as the line width is not known a priori. We emphasize that this recipe, and in particular Equation (2), holds only in the assumption that there is no active spatial search for the line position, as in our case, where the positions for the spectral extractions are all strongly constrained by the continuum detections. When one is blindly searching for emission lines, having also to determine their spatial position, the number of effective trials would be much larger, as Equation (2) would have to be multiplied by the number of effective spatial positions searched.¹⁴

We report the spurious probability for the strongest line in each spectrum in Table 1 (P_1st). They are below 0.1% in all cases except for the 4.7σ line of ID85000922, which has a 1.4% probability of being spurious, by itself.

Then, we searched for additional matching lines based on the redshift solutions determined by the first ones, using the same velocity range as determined by the primary line, assuming that they are CO transitions. We excluded, in fact, solutions in which the primary line was [C i] or H₂O, given that stronger accompanying CO lines would have been detected in the spectra in those cases. Meaningful second lines were detected in three galaxies with significance of 3.3σ–4.4σ. We compute chance probabilities to find matching second lines (P_2nd) based on the significance of each line and the number of search trials performed (N_2nd). The combined probability for the two-line match is in all cases ∼10⁻⁴ or lower; hence, we consider these three galaxies as reliable spectroscopic confirmations. The redshifts (and line identifications) are determined as CO (5−4)/CO (6−5) at z = 5.85, CO (4−3)/CO (5−4) at z = 4.44, and CO (4−3)/[C i] (1−0) at z = 3.62 (Figure 1). In the bottom panels of Figure 1, we present zoom-ins to the weighted averages of lines detected for each galaxy after continuum subtraction. Given that the CO (4−3) line of ID85001674 partially falls in a frequency gap, we ultimately calculate its redshift and velocity width from the [C i] (1−0) line. We note that the confirming/second line for ID85001929 is located at the edge of the spectral range and only partially covered by our data. Still, the available signal within the velocity range predetermined by the first line is strong enough that its spurious probability is at 0.013. We thus maintain that this z = 5.85 identification is secure.¹⁵ Therefore, ID85001929 at z = 5.85 is now the most distant known SMG in the COSMOS field, at a higher redshift than the recently found dusty galaxy CRLE at z = 5.67 (Pavesi et al. 2018).

The ID20010161 has a single line detection at 95.2 GHz with 5.97σ significance and a spurious probability of ∼10⁻⁵ (Table 1); hence, it is highly secure. However, the redshift identification is less secure, as there are multiple solutions given the lack of a second line. We disfavor the single-line identification as CO (7−6) at z = 7.47 given the lack of [C i] (2−1) (the formal signal within the expected velocity range gives −2.7σ at 95.5 GHz) and H₂O (2₁₁–2₀₂) (0.07σ at 88.8 GHz) lines, since the H₂O line is typically 0.4–1.1 of the high-J CO flux in high-z lensed ULIRGs (Omont et al. 2013; Yang et al. 2016, 2019). We also exclude CO (6−5) at z = 6.26 because of the lack of detection of H₂O (2₁₁–2₀₂) (−1.075σ at 103.6 GHz). The solution with CO (4−3) at z = 3.84 is also less likely owing to the lack of any [C i] (1−0) signal (−0.7σ at 101.7 GHz), although it is not unconceivable that [C i] (1−0) might be intrinsically faint for this source, as it can reach ∼25% of the CO (4−3) flux (e.g., Walter et al. 2011; Valentino et al. 2018). This redshift is less secure given that no confirming lines are present in its spectrum. However, we consider the most likely identification of this line as CO (5−4) at z = 5.05, for which we would not expect other significant lines in the observed range. In the following we will use this redshift identification for this source. If the redshift should turn out to be instead z = 3.84, its dust temperature would be even lower and the conclusions derived from these sources even further strengthened.

These redshift identifications consistently confirm that no strong lines are expected in the 230 GHz spectral ranges of the Oteo data or in the 345 GHz spectral range of the Matsuda data, for either of the two galaxies for which 230 GHz or 345 GHz observations are available.

In Figure 2, we present ALMA (clean) images for this sample. We solidly detected continuum emission with peak significance of 9σ–14σ at 3 mm for the four sources, 13σ–19σ at 1 mm, and 29σ at 870 μm for those in which these are available (Table 2). We measured their sizes by fitting models in u-v space in the ALMA 1/0.8 mm and 3 mm collapsed data sets (also adding lines when available, but these provide only a modest contribution, being at substantially lower S/Ns). We combine the different tracers for a single galaxy in the u-v space, following the procedure presented in Puglisi et al. (2019). Only the lowest-redshift galaxy, ID85001674 at z = 3.62, is resolved with an FWHM size of 042 ± 004 (3.1 ± 0.3 kpc), implying an SFR surface density (Σ_SFR) of ${22}_{-9}^{+13}$ ${M}_{\odot }\,{\mathrm{yr}}^{-1}\,{\mathrm{kpc}}^{2}$. The remaining sources are unresolved; we show their 2σ size upper limits (and lower limits of SFR surface density Σ_SFR) in Table 3. ID85000922 at z = 4.4 has a size limit of <2.8 kpc, and the other two galaxies at z > 5 appear to be more compact with FWHM sizes <1.9 kpc. The measured sizes of our ALMA sources are significantly smaller than the 3 GHz sizes reported by Miettinen et al. (2017) for a sample of 115 known SMGs in the COSMOS field, with a median size of 4.6 ± 0.4 kpc. In contrast, our measured sizes are close to the ALMA sizes of high-z SMGs from Ikarashi et al. (2015) and Simpson et al. (2015), who report median sizes of 1.6 ± 0.14 kpc at 1.1 mm and 2.4 ± 0.2 kpc at 870 μm, respectively. Our ALMA sizes also agree with the 10 GHz selected sample in Murphy et al. (2017), which has a median size of 1.20 ± 0.28 kpc.

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Table 3.  CMB Impact on Observables

ID	LIR	${\beta }_{\mathrm{thin}}^{\mathrm{noCMB}}$	${\beta }_{\mathrm{thin}}^{\mathrm{CMB}}$	${\beta }_{\mathrm{thick}}^{\mathrm{CMB}}$	${T}_{\mathrm{dust},\mathrm{thin}}^{\mathrm{noCMB}}$	${T}_{\mathrm{dust},\mathrm{thin}}^{\mathrm{CMB}}$	${T}_{\mathrm{dust},\mathrm{thick}}^{\mathrm{CMB}}$	${M}_{\mathrm{dust},\mathrm{thin}}^{\mathrm{noCMB}}$	${M}_{\mathrm{dust},\mathrm{thin}}^{\mathrm{CMB}}$	${M}_{\mathrm{dust},\mathrm{thick}}^{\mathrm{CMB}}$	${L}_{\mathrm{CO}(5-4)}^{{\prime} \mathrm{obs}}$	${L}_{\mathrm{CO}(5-4)}^{{\prime} \mathrm{CMB}}$
	(1012 L⊙)				(K)	(K)	(K)	(108 M⊙)	(108 M⊙)	(108 M⊙)	(a)	(a)
85001929	12.8 ± 1.8	2.5 ± 0.2	2.2 ± 0.2	2.1 ± 0.2	40 ± 3	42 ± 3	61 ± 8	3.5 ± 0.6	4.3 ± 1.1	2.2 ± 0.9	2.0 ± 0.4b	2.9 ± 5.5b
20010161	6.2 ± 1.0	2.4 ± 0.2	2.0 ± 0.2	1.9 ± 0.2	32 ± 3	35 ± 4	40 ± 6	8.7 ± 2.5	10.4 ± 2.9	7.7 ± 2.1	1.3 ± 0.2	1.8 ± 0.3
85000922	5.1 ± 1.0	3.7 ± 0.6	2.3	2.1 ± 0.4	20 ± 4	30 ± 4	42 ± 6	18.3 ± 9.8	10.2 ± 1.3	7.0 ± 2.3	0.9 ± 0.2	1.3 ± 0.3
85001674	3.3 ± 0.8	2.7 ± 0.2	2.2 ± 0.2	2.1 ± 0.2	23 ± 2	24 ± 2	41 ± 5	29.2 ± 8.9	33.6 ± 8.1	13.0 ± 3.7	0.8 ± 0.1b	1.2 ± 0.2b

Notes. L_IR: IR luminosity at 8–1000 μm corrected for CMB effect (dashed curve in Figure 3); ${\beta }_{\mathrm{thin}}^{\mathrm{noCMB}}$, ${T}_{\mathrm{dust},\mathrm{thin}}^{\mathrm{noCMB}}$, and ${M}_{\mathrm{dust},\mathrm{thin}}^{\mathrm{noCMB}}$: RJ slope, dust temperature, and dust mass from optically thin MBB fits without accounting for CMB; ${\beta }_{\mathrm{thin}}^{\mathrm{CMB}}$, ${T}_{\mathrm{dust},\mathrm{thin}}^{\mathrm{CMB}}$, and ${M}_{\mathrm{dust},\mathrm{thin}}^{\mathrm{CMB}}$: RJ slope, dust temperature, and dust mass from optically thin MBB fits accounting for CMB effect; ${\beta }_{\mathrm{thick}}^{\mathrm{CMB}}$, ${T}_{\mathrm{dust},\mathrm{thick}}^{\mathrm{CMB}}$, and ${M}_{\mathrm{dust},\mathrm{thick}}^{\mathrm{CMB}}$: RJ slope, dust temperature, and dust mass from optically thick MBB fits accounting for CMB effect; ${L}_{\mathrm{CO}(5-4)}^{{\prime} \mathrm{CMB}}$: CO (5−4) luminosities corrected for CMB effect assuming ${T}_{\mathrm{exc}}={T}_{\mathrm{dust},\mathrm{thin}}^{\mathrm{CMB}}$ in LTE condition.

^aUnits of ${10}^{10}\,{\rm{K}}\,\mathrm{km}\,{{\rm{s}}}^{-1}\,{\mathrm{pc}}^{2}$. ^bThe observed CO (5−4) luminosities for these two sources are extrapolated as $1.2\times {L}_{\mathrm{CO}(4-3)}^{{\prime} }$ and $0.8\times {L}_{\mathrm{CO}(6-5)}^{{\prime} }$, respectively.

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The high compactness of this sample is unlikely to be caused by AGN activity, because the emission from any dusty torus would be negligible at FIR/(sub)millimeter wavelength, and it cannot dominate the 3 mm emission (rest frame 440–650 μm for this sample). In any case, we do not see any evidence for an AGN component in their cold SEDs (see Section 5.2), and none are X-ray detected from Jin et al. (2018). Therefore, we speculate that the compact morphology in this very highly star-forming sample is a consequence of ongoing mergers, similarly to local ULIRGs (Soifer et al. 2000; Juneau et al. 2009) and high-z compact starbursts (Puglisi et al. 2017; Marrone et al. 2018; Calabrò et al. 2019).

In the bottom panels of Figure 2, we show K_s+SPLASH color images for this sample. All sources have a red counterpart in SPLASH images quite consistent with the ALMA imaging; however, only two of them are entirely consistent in position and extension with the ALMA detections. The SPLASH counterpart of the z = 5.85 galaxy appears to be more extended than its ALMA morphology, the IRAC peak being located about 08 to the south, where some faint extension is seen at 1 mm in the ALMA imaging as well (Figure 2, top left panel). ID85000922 at z = 4.44 has a very faint SPLASH IRAC 3.6 μm counterpart at its ALMA position, while the blue source with ∼20 offset appears to be unrelated, having a well-constrained z_phot = 0.86 (Laigle et al. 2016).

The ID20010161 and ID85001674 galaxies are detected in the 3 GHz image with 5.0σ and 3.6σ (Daddi et al. 2017; Smolčić et al. 2017), showing infrared-to-1.4 GHz radio luminosity ratio q_IR = 2.3 and 2.4, which agree well with the evolution of ${q}_{\mathrm{IR}}{(z)=(2.88\pm 0.03)(1+z)}^{0.19\pm 0.01}$ for star-forming galaxies in Delhaize et al. (2017), while the other sources are either marginally or not detected in the radio (<2.8σ). No signal is observed in the UltraVISTA K_s band (or any shorter wavelength) for our sources, which thus could be qualified as K_s-band dropouts. No photometric redshift is, of course, available for these sources in near-IR or optically selected catalogs.

We collected super-deblended FIR/(sub)millimeter photometry together with the ALMA 0.8/1 mm and 3 mm continuum measurements for deriving their full dust SEDs. We start by fitting these SEDs using modified blackbody (MBB) models from Casey et al. (2012) with free T_dust and β, not accounting for the effect of the CMB. We find that these galaxies are fitted with very cold dust temperatures T_dust = 20–41 K and abnormally steep Rayleigh–Jeans (RJ) slopes β = 2.4–3.7, which appear to be well determined with relatively small errors, particularly for the two objects with highly accurate 1 mm and 3 mm ALMA continuum measurements in the RJ tail. Such steep slopes had not been reported in literature previously. However, galaxies are always observed against the CMB, and the CMB effect on the continuum would be nonnegligible on dusty SEDs, especially for our systems that appear to be fairly cold for their high z (da Cunha et al. 2013).

We thus use alternative optically thin MBB SEDs (Magdis et al. 2012) accounting for the CMB effect on dust continuum following the prescriptions by da Cunha et al. (2013). MBB models are not ideal to describe the Wien part of galaxies SEDs but are appropriate here, given the photometric sampling available for our galaxies. We convert the luminosity of MBB models L_MBB to L_{IR,8–1000 μm} by multiplying by a constant of 1.35, a median value for starburst samples (Magdis et al. 2012) based on comparison with Draine & Li (2007). In Figure 3, we show the best fit to the observed photometry with solid curves, while dashed curves indicate the intrinsic SEDs before accounting for the CMB effect on the continuum (green curve). We can clearly see that the CMB depresses the dust continuum at longer wavelengths, severely impacting its level at 3 mm (see Section 4.4), thus making the observed RJ slope steeper. In Table 3, we list derived parameters for fits with and without accounting for the CMB. In both cases we find consistently cold dust temperature T_dust = 20–42 K, which are also consistent with those derived by fitting Casey et al. (2012) models. The consistence of T_dust,obs and T_{dust,thin+CMB} shows that accounting for the CMB effect does not alter T_dust for this sample, implying that the effects of the T_dust–β degeneracy are negligible for our galaxies, within the uncertainties. This is because the CMB is affecting the observed continuum at long enough wavelengths, while it has barely any effect on the peak of the dust SED where T_dust is measured. We verified that the impact of the CMB on the derived T_dust is negligible out to z = 5; even for a galaxy with intrinsic T_dust = 18 K, the marginal overestimate of T_dust would be less than 1 K in that case. Strikingly, the SEDs corrected for CMB effect yield β ∼ 2–2.2, consistent with typical β values found in the literature for star-forming galaxies, within the uncertainties (of about 0.2 in β).

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Have we thus really directly detected the effect of the CMB on galaxy SEDs, for the first time (as far as we know)? This appears to be the case, as we argue that the β values of order 2.4–3 are not plausible and the CMB is indeed required to obtain a physically meaningful interpretation of the SEDs of our targets. In fact, large Herschel surveys have been used to constrain the slopes of local galaxies, finding β ≈ 1.5 and up to a maximum of 2 (Boselli et al. 2012; Bianchi 2013). None of the local galaxies with an apparent best fitting β > 2 are at more than 1σ from β = 2 (Rémy-Ruyer et al. 2013). Also, although our high-redshift sample might be somewhat unusual because of particularly cold dust for its redshift (as discussed in more detail later in this paper), there is no evidence for steeper β slope in cold galaxies in literature. For example, our MW (T_dust = 19 K) is colder than this sample but has no steep β slope, where the analysis of the full MW dust SED yields β ≈ 1.8 ± 0.2 (Planck Collaboration et al. 2011). Without accounting for the CMB, the best-constrained galaxies in our sample (not considering ID85000922 given its poor SED¹⁶ ) have β > 2 at significance ranging from 2σ to 3.5σ. Taken together, their average SED is steeper than β = 2 at about the 5σ level. Therefore, we conclude that the steep observed β slopes are unambiguous evidence that the CMB is having a measurable effect on our target galaxies, which arises from a severe reduction of the dust continuum observed at 3 mm, or beyond about 500 μm in the rest frame.

In general, the impact of CMB on galaxy SEDs can be expressed as a function of the contrast between dust and CMB temperatures, with the highest impact when these temperatures are close. We recall that what counts is the apparent temperature, as encoded in the SED shape and rest-frame peak rather than the intrinsic temperature (which could differ in the case of high optical depths). It is therefore relevant to evaluate how the T_dust for our galaxies relates to the cosmic evolution of temperatures in galaxies and the CMB. In the left panel of Figure 4, we compare dust temperature T_dust for our sample to various objects taken from the literature (Magdis et al. 2011; Riechers et al. 2013, 2014, 2017; Ciesla et al. 2014; Béthermin et al. 2015; Pavesi et al. 2018; Schreiber et al. 2018), and to the evolving CMB temperature, versus redshift. We also fitted the available photometry for the z = 6.9 SPT0311-58 galaxy (Strandet et al. 2017; Marrone et al. 2018) and obtained ${T}_{\mathrm{dust}}=41\pm 2\,{\rm{K}}$ using the same MBB models including the CMB effect. In the right panel of Figure 4, we present the comparison in terms of the intensity of the radiation field $\left\langle U\right\rangle$: all values are consistently and homogeneously measured using the DL07 (Draine & Li 2007) models, eliminating any systematics in the T_dust–z diagram coming from different assumptions of β and fitting techniques of the MBB models. In terms of both T_dust and $\left\langle U\right\rangle$, this sample has significant cold dust content with respect to both z < 4 MS galaxies and z > 5 SMGs.

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The z = 5.85 galaxy has the highest T_dust and $\left\langle U\right\rangle$ in this sample, though its T_dust is still lower than the extrapolation of the T_dust–z correlation for MS galaxies, and its $\left\langle U\right\rangle$ is lower than the extrapolation from a broken fundamental metallicity relation (FMR) as in Béthermin et al. (2015). The z = 5.05 source shows a dust temperature similar to the z = 4 starburst GN20 (Daddi et al. 2009). The z = 4.44 and z = 3.62 ones have the coldest T_dust in this sample, which are just moderately above the CMB temperature by factors of 1.5–2.0. Not counting this study, among previously known galaxies, only GN20 is robustly colder than MS galaxies, while CRLE is lower than the MS at only ∼1.2σ significance. The cold dust in SPT0311-58 at z = 6.9 had not been reported previously. A significant population of cold galaxies is now clearly detected.

Please note that the increase of T_dust with redshift within our sample of four targets is simply a selection effect: we had originally selected our targets to be at z_phot > 6 based on photometric redshifts derived using an HFLS3 template. As a result, the lower the redshift with respect to z = 6, the intrinsically colder the temperature necessarily is (and the larger the implied dust masses). We show the T_dust–z degeneracy trends for each galaxy in our sample in Figure 4 (and 7): any redshift trend that our four targets might define is spurious as linked to this degeneracy and affected by selection. If we had used a cold dust template like GN20, we would have obtained z_phot = 4.0–5.5 for our targets, which appear to be closer to the observations.

We review and summarize here how CMB impacts various observables for this sample, as a necessary step before proceeding to infer their global properties for understanding their nature, and also as a reference for designing and conducting future observations of high-z dusty galaxies.

Continuum: The CMB acts as a nonnegligible background for cold systems against which the line emission and continuum emission are measured (Combes et al. 1999; Papadopoulos et al. 2000; da Cunha et al. 2013). According to Equation (18) in da Cunha et al. (2013), given the observed dust temperatures in our galaxies, the ratios of intrinsic/observed continuum are in the range of 1.32–1.45 at 3 mm and 1.06–1.09 at 1 mm. These ratios are monotonically increasing toward lower redshifts for our sample, an effect due to the fact that lower-redshift objects are colder and closer in relative terms to the CMB temperature at each redshift. The induced underestimate on the intrinsic dust continuum is more severe at 3 mm, and this has to be carefully accounted for when designing and conducting observations for high-redshift galaxies. The IR luminosities of the galaxies are also somewhat affected by the CMB, with a small underestimate by a factor of ≈1.05.

CO emission: A higher CMB temperature enhances the population of the high-J CO levels and also corresponds to a higher background against which the lines must be detected (Combes et al. 1999; Papadopoulos et al. 2000; Obreschkow et al. 2009). Ultimately for fundamental thermodynamic reasons, T_CMB <= T_excitation <= T_kinetic for any collisionally excited transition. Then, when the kinetic temperature of a cold gas reservoir gets closer to the CMB temperature, its low-J CO line brightness diminishes by contrast to the background (Papadopoulos et al. 2000; da Cunha et al. 2013). Given the limited range of transitions observed for our galaxies and the degeneracy between temperature and density in the analysis of CO spectral line energy distributions (SLEDs; e.g., Weiß et al. 2005; Dannerbauer et al. 2009; Daddi et al. 2015), it is not possible to properly estimate the expected effect for our galaxies. Nevertheless, one could obtain some guidance by assuming, for example, that the kinetic temperature of the gas in the CO transitions we observed is close to the dust temperature in the same galaxies, as expected for local thermodynamic equilibrium (LTE). This would imply observed CO fluxes at the level of 70% of the intrinsic ones according to Equation (32) in da Cunha et al. (2013). This seems to be reasonably consistent with observations. In Figure 5 we present the CO (5−4) luminosities of this sample in the ${L}_{\mathrm{CO}}^{{\prime} }\mbox{--}{L}_{\mathrm{IR}}$ diagram. The average ratio of the observed CO (5−4) luminosity (blue data points) to L_IR is 0.7 times lower than the best fit and 0.5 times lower than the linear fit in Liu et al. (2015a). Thus, the CMB effect provides a reasonable explanation for the seemingly low CO fluxes observed in our sample, although we cannot fully demonstrate this, lacking a measurement of the gas kinetic temperature. This is nevertheless interesting information for planning follow-up searches for CO emission in high-redshift galaxies, as reduction factors of up to ×2 in fluxes, due to the CMB, might not be uncommon to be encountered, particularly for galaxies with low CO excitation temperatures.

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Size measurement: The CMB might also affect the 3 mm continuum sizes of our sources, in the case in which T_dust gradients are present, similarly to local spiral galaxies in which the outskirts are colder. In that case, the outskirt surface brightness would be impacted more strongly by the CMB than the hotter centers, resulting in apparently smaller half-light radii. Taking as examples the temperature gradients found in some typical local galaxies M31, M33, and NGC 628, Zhang et al. (2016) computed that this effect could be at the level of 10%–20% at z = 4–5 for 3 mm bands, while a negligible size bias is expected at ∼1 mm. Given that this effect is in any case not very strong (even for surface brightness it would be at most 40% for the local examples in Zhang et al. 2016), and that our sources are very compact and possibly merging driven starbursts where we consider it unlikely that they might have strong T_dust gradients, we conclude that the CMB should not impact strongly the size measurements for this sample (the situation could be different for extended cold reservoirs in high-z disks). We can verify this directly for the galaxy ID85001674 at z = 3.623. This object is the coldest and closest in relative terms of the CMB temperature and shows a combined size 042 ± 004 from 3 mm+1 mm+0.8 mm observations. The sizes measured from 1 mm/0.8 mm data alone are 032 ± 007 and 038 ± 004, both about 1σ–1.5σ lower than the combined size. This is contrary to what would be expected if CMB were affecting sizes, showing that indeed any CMB-induced size bias should be quite small.

Line widths and dynamical masses: Together with sizes, velocity widths are used to estimate dynamical masses of galaxies, as we report in Table 3 (M_dyn,2Re), following the relations given in Daddi et al. (2010) and Coogan et al. (2018). The CMB could also impact the observed CO line width V_FWHM, in the presence of gas kinetic temperature gradients, as described above for the sizes. The magnitude of this effect has not been quantified so far, but we could expect that it should be roughly comparable to the one on sizes, i.e., probably small. The CO spectra of the galaxies in this work are still relatively low in S/N, and thus we assume that any CMB effect on line widths would likely be in the noise for our sources. In any case, the net effect of the CMB on derivation of dynamical mass from observables could imply that they are somewhat underestimated given ${M}_{\mathrm{dyn},2\mathrm{Re}}\propto {V}_{\mathrm{FWHM}}^{2}\times {R}_{{\rm{e}}}$.

Being much less affected by CMB and with higher intrinsic luminosities, [C ii] 158 μm observations would be particularly valuable for this and other high-redshift samples. They would provide robust measurements on both size and dynamics, with minimal effects from the CMB. However, see discussion later in Section 5.2 for other reasons why [C ii] 158 μm might also be affected in these galaxies.

SFR surface density: We measure the SFR surface densities by converting the L_IR estimates into SFRs (Kennicutt 1998) and dividing by the area ${{\rm{\Sigma }}}_{\mathrm{SFR}}=\mathrm{SFR}/(2\pi {{R}_{{\rm{e}}}}^{2})$. We compared the SFR surface density to data from the literature, particularly taking samples shown in an analogous figure from Simpson et al. (2017). In Figure 6, our sample shows a trend of increasing luminosity surface density with dust temperature, consistent with trends in the lower-z sample of Simpson et al. (2017) and Hodge et al. (2019). Meanwhile, our sample shows a higher-luminosity surface density than any lower-z sample. Our values are at the limit of the Stefan–Boltzmann law, consistent with a single emitting, homogeneous starburst core. Our sample shows comparable SFR surface density with respect to local (U)LIRGs (Liu et al. 2015b), which are 2–3 orders of magnitude higher Σ_SFR than normal spirals. The coldest galaxy (z = 3.623) in this sample appears to be even colder than local IR-detected spirals from Liu et al. (2015b), while it has higher Σ_SFR by 2–3 orders of magnitude. The high SFR surface densities suggest that these galaxies are more efficient starbursts than low-efficiency disks. Their extreme compactnesses indeed suggest mergers. In fact, using the evolutionary trend found by van der Wel et al. (2014), we would expect a disk-like galaxy at z = 5 with M_* ∼ 5 × 10¹⁰ M_⊙ to have an FWHM size of ∼4.5 kpc, substantially larger than the sizes we have measured for our objects (Table 4).

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Table 4.  Physical Quantities

ID	FWHM Size	M*	${M}_{\mathrm{dyn},2\mathrm{Re}}$	ΣSFR	${M}_{\mathrm{gas},\mathrm{dyn}}$	${M}_{\mathrm{gas},{\rm{C}}{\rm{I}}}$	${M}_{\mathrm{gas},\mathrm{FMR}}$
	(arcsec/kpc)	(1010 M⊙)	(1010 M⊙)	(M⊙ yr−1 kpc2)	(1010 M⊙)	(1010 M⊙)	(1010 M⊙)
85001929	<0.33 (1.90)	11.6 ± 5.8	<15.9	>196	<8.2	⋯	5.7 ± 1.3
20010161	<0.25 (1.55)	9.0 ± 5.2	<12.0	>135	<6.8	⋯	12.5 ± 2.8
85000922	<0.43 (2.83)	1.8 ± 1.6	<19.4	>31	<16.9	⋯	25.1 ± 8.6
85001674	0.42 ± 0.04 (3.10 ± 0.30)	3.9 ± 2.0	17.6 ± 2.6	${22}_{-9}^{+13}$	10.2 ± 4.2	9.7 ± 2.2	48.3 ± 12.1

Note. M_*: stellar mass; ${M}_{\mathrm{dyn},2\mathrm{Re}}$: dynamic mass within FWHM size; SFR surface density ${{\rm{\Sigma }}}_{\mathrm{SFR}}\equiv \mathrm{SFR}/(2\pi {{R}_{e}}^{2})$, where R_e is half of the FWHM size.

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Dust and stellar masses: MBB dust masses are listed in Table 4. We calculated dust mass from DL07 templates, where we use the intrinsic SED (i.e., corrected for the CMB effect) of each galaxy as recovered from the MBB, take six data points (from 80 to 3000 μm in rest frame), and fit those data points with DL07. The MBB dust masses are consistent within 20% with the mass scaled to DL07; both result in a massive dust content for this sample. To constrain their stellar masses, we collected IRAC photometry in literature (Sanders et al. 2007) for the z = 5.05 source and obtained SPLASH photometry for the others by PSF fitting the SPLASH images on ALMA positions. We fit stellar SEDs using a 200 Myr template allowing for dust attenuation, following Liu et al. (2018). We list stellar masses in Table 4. Despite the IRAC detections well in the optical rest frame, we expect uncertainties at least at the level of ×2 for our galaxies, given the lack of detections of age-dependent features in their optical SEDs. In some cases the dust-to-stellar-mass ratios can reach 10% levels (Figure 7), much higher than what is found for typical MS galaxies and consistent with merger-driven systems like bright SMGs (see also Tan et al. 2014; Rujopakarn et al. 2019).

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Starburstiness: Defined by the ratio of the specific SFR for a given galaxy to the same specific SFR of an average MS galaxy at that redshift. At redshifts 4 < z < 6 the MS value is not very securely determined, so this classification is uncertain. However, taking the extrapolation of the trends defined in Schreiber et al. (2015), we found that the z = 4.44 galaxy has a starburstiness SFR/SFR_MS > 4–60 and appears thus to be a secure starburst galaxy despite its large uncertainty on the stellar mass. The remaining galaxies have fairly high masses and starburstiness of 1.5–2.7, which are all above the MS extrapolation at z > 3.6–6 but not by large factors. In fact, they could be considered MS galaxies in the definition by Rodighiero et al. 2011. However, as shown by several works (e.g., Elbaz et al. 2018; Puglisi et al. 2019), a substantial number of compact, probably merger-driven galaxies are observed even within the MS boundary at high masses, already at intermediate redshift 1 < z < 3.

Gas masses: Inferring gas masses would be the most direct way to establish whether these objects are high- or low-efficiency star formers. We constrain the total gas mass (molecular plus any atomic gas) of our sample using three different methods: (1) based on the dynamical mass (three of which are only upper limits), after subtracting stellar mass using the Equation (3) in Daddi et al. (2010), assuming 20% of dark matter; (2) based on dust mass scaled to DL07, using FMR, FMR MS, elliptical, and solar-metallicity-based G/D conversion factors, respectively; (3) for the z = 3.62 galaxy with a [C i] (1−0) detection, given the tight correlations between [C i] and low-J CO in local and high-z galaxies (Jiao et al. 2017, 2019; Valentino et al. 2018), we estimated its gas mass using the relation in Valentino et al. (2018) assuming an excitation temperature of T_ex = 30 K and show the resulting ${M}_{\mathrm{gas},[{\rm{C}}{\rm{I}}]}$ in Table 4. We cannot estimate gas masses from CO, as the lowest-J transition we detect is CO (4−3), whose conversion to CO (1−0) is plagued by too large and uncertain excitation corrections. In Figure 8, we compared the three different derivations of gas masses in the Schmidt-Kennicutt plane. Comparing to the relations for SBs and MS galaxies taken from Sargent et al. (2014), one can see that most values tend to be toward the high SFEs typical of SBs, with quite some large range and large uncertainties as exemplified by the scatter in the derived gas masses from different methods and assumptions. However, it is interesting to consider the z = 3.62 source, which is the coldest in absolute terms and closest to the CMB temperature and has also alternative derivations of gas mass from the [C i] luminosity and the dynamical mass. The two latter estimates agree well between them but are lower than the estimate based on M_dust by factors of 4–5, suggesting that M_dust are probably overestimated.

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The behavior displayed by the galaxy ID85001674, as discussed in the previous subsection, suggests an alternative explanation for the nature of these cold galaxies, which could solve a number of problems with them, particularly the coldest ones. In fact, the dust-to-stellar-mass ratios of the coldest galaxies are very large, reaching 10%, which seems very high (Figures 7 and 8). Given that the mass in dust can be taken as a proxy of the mass in metals, this would imply that the metallicity in these objects could be truly extreme. Similarly, also the gas-to-stellar-mass fraction would be seemingly ×10 or higher, as is only expected for primordial galaxies, which would be in contradiction with the high metallicities (Figure 8). All these values are derived from M_dust estimates, and in all cases assuming models with optically thin dust emission. These measurements are not supported by a dynamical estimate of the gas masses, nor by derivation from [C i] (1−0), a transition known to be in most cases optically thin given the low fractional abundance of carbon in the neutral form (Valentino et al. 2018). It could well be that the dust continuum emission is instead optically thick in these systems, even at the peak of the emission and possibly somewhat beyond, similar to the optically thick SB nuclei found in Arp 220 (Papadopoulos et al. 2010; Wilson et al. 2014; Scoville et al. 2017). This is supported by the extreme compactness of our sources and their very high SFR surface densities. A simple calculation for ID85001674 assuming a spherical geometry with constant density of dust ρ within the measured radius R_e, using the measured dust mass from the thin models, and using κ from the Jones et al. (2013) opacity models would imply an optical depth (τ = κρR_e) at 100 μm rest (the observed SED peak) toward the center of about 1. Of course, there are many implicit assumptions and unknowns in this calculation,¹⁷ but certainly the case for optically thick dust until the observed SED peak is a plausible one for these galaxies (and for the others, which are likely even more compact).

If the dust emission is optically thick around 100 μm (and possibly beyond), the suppression of the Wien emission would make the observed SEDs seem cold, while the intrinsic temperatures would be much higher (e.g., Scoville & Kwan 1976; Condon et al. 1991; Conley et al. 2011; Scoville 2012). Conversely, together with the underestimate in T_dust, we would be overestimating M_dust by some factor that would depend ultimately on the optical depth (i.e., from the ratio between the intrinsic T_dust and the real T_dust). In all cases L_IR (hence SFR) derivations would be unaffected.

The standard treatment of optically thick emission assumes a constant T_dust at all wavelengths (e.g., Casey et al. 2012) and is unlikely to be realistic, given that radial gradients of T_dust are expected (e.g., Scoville 2012). Explicitly accounting for such effects is difficult, and we hope to present calculations of more physical thick models in future works. Nevertheless, we have fitted our targets using the more standard approach of optically thick sources with constant temperature and accounting for the effects of the CMB. Results are reported in Table 3, along with the other estimates. The empirical evidence discussed above for ID85001674 requires that we might be overestimating dust masses by factors of up to 4–5 for our sample. These thick models indeed already account for a ×3 reduction of M_dust for ID85001674 and ≈×2 for the other galaxies. True corrections might be even larger.

If these sources are really optically thick, then the ultimate reason why CMB is so strongly affecting these galaxies is not because their physical dust temperatures are cold but because they seem cold. Similarly, this would also explain why these galaxies seem to be colder than MS galaxies, when they might not be in reality. T_dust values in Table 3 derived for the thick fits are already comparable to or larger than MS galaxies at the same redshifts. In this case their CO kinetic temperatures would also be higher, rendering CMB effects on their line fluxes negligible: the reduction of CO fluxes might be instead due to high optical depths (see also Papadopoulos et al. 2010).

A strong reduction of M_dust in SBs would, finally, also solve the problematic observations that their dust-to-stellar-mass ratios are ∼5× larger than those of MS galaxies (Béthermin et al. 2015), while dynamical estimates suggest that the gas-to-stellar-mass ratios are near unity (Silverman et al. 2015, 2018a, 2018b). The case should be thus seriously considered that many of the merging driven SBs are actually optically thick in the FIR, a case previously proposed for local ULIRGs by Papadopoulos et al. (2010).

More observations would be needed to confirm the optically thick case. One interesting path would be comparing to [C i]-derived excitation temperatures, given that [C i] lines should remain thin. Another possibility would be to use [C ii] 158 μm as a direct tracer of the gas mass (Zanella et al. 2018) and from there more realistic M_dust estimates via assumptions of G/D ratios: while [C ii] could be in principle heavily affected by optical depth effects, it seems to provide fair gas mass estimates in local ULIRGs (perhaps by lucky coincidence). More observations are also needed to establish whether these cold (i.e., with long peak rest wavelengths) SEDs are prevalent in the distant universe, and if the optically thick case could provide a general explanation for the odd inversion that seems to take place at high redshifts, with SB galaxies becoming colder than MS galaxies.