The Far-infrared Emission of the First Massive Galaxies

Our SED models in the FIR are based on those developed for primeval galaxies⁵ by De Rossi & Bromm (2017), hereafter DB17. According to their findings, the FIR SEDs of these galaxies are likely to be significantly warmer than typical FIR SEDs of galaxies in the 0 < z < 3 range. Our goal is to explore this transition between z = 3 and z ∼ 10. Such behavior likely arises in large part because of the high-energy density in young Pop II galaxies combined with the characteristics of the silicate-rich dust expected in systems that are dominated by Pop III enrichment. Silicates have relatively poor emission efficiency at wavelengths short of about 8 μm and longer than about 60 μm (Koike et al. 2003), accentuating the tendency from the high-energy density for relatively high temperatures for the dust. The transition in FIR SEDs may therefore depend not on the transition from Pop III to Pop II in itself, or on the duration of the Pop II burst of star formation, but directly on the transition away from silicate-rich dust. We will adapt the model of DB17 to conditions in the galaxies being detected in the FIR at redshifts of z ∼ 6 to explore this possibility.

The composition of Pop III dust is not well known (e.g., Schneider et al. 2016; Jaacks et al. 2018). As a specific example, Ji et al. (2014) considered silicon-based dust models taken from Cherchneff & Dwek (2010), and DB17 calculated the FIR SEDs for these different models. It is worth mentioning that to estimate the chemical composition of dust, Cherchneff & Dwek (2010) assumed non-equilibrium chemical kinetics for dust formation, while most steady-state models would predict a more significant carbon composition. The suppression of carbon in the former case is related to the hypothesis that carbon-rich regions in the supernova ejecta are microscopically mixed with helium ions. As in DB17, we follow these assumptions to probe how silicate-rich dust from Pop III/II stars can affect the infrared SED; we will return to the issue of carbon dust in Section 3.

By analyzing the different dust chemical models implemented in Ji et al. (2014), DB17 found that for galaxies with similar properties, the FIR intensities and general shapes of the predicted SEDs are similar, but the detailed spectral features show differences. As a result, the theoretical results can only be taken as representative; alternative mixtures of silicate-rich dust would alter the predicted SEDs, although the overall shape will remain controlled by the poor emission efficiency of this type of mineral outside the 8–60 μm range. Improving the correspondence between the predicted and an observed SED would require adjustments in the adopted silicate material that would have to be ad hoc in nature, given the lack of any other meaningful observational constraints.

Rather than attempting such adjustments, in the following section we will propose the galaxy Haro 11 as a kind of analog computer that can utilize empirically determined dust optical properties. As we discuss below, Haro 11 has an exceptionally high rate of star formation in a low-mass and (at least prior to the ongoing starburst) low-metallicity galaxy. These conditions are conducive to its interstellar dust being dominated by the outputs of very young and massive stars, as is the case for the high-redshift galaxies. The energy density in the dominant star-forming region in this galaxy is also similar to that in the high-redshift galaxies. We will show that the FIR SED of this galaxy generally resembles that predicted by the theoretical model. Lyu et al. (2016) have also suggested that the Haro 11 SED is a good match to the FIR SEDs of quasar host galaxies at z ∼ 6. We will therefore adopt its SED as a proxy for the theoretical one and use it in the remainder of the paper to compare with FIR observations of high-redshift galaxies.

2.2.1. Dust Composition

2.2.2. Sizes and SFR Surface Densities

2.2.3. Metallicity

The calculations of FIR SEDs by DB17 were for primeval galaxies dominated by Pop II stars, and assumed extremely low metallicities and compact star-forming complexes. Figure 1 shows the result of further developing their model in circumstances more applicable to the very luminous galaxies detectable at very high redshift, as described in Section 2.1. To increase the FIR outputs by 2–3 orders of magnitude to match the high luminosities of the galaxies, we (1) increased the virial masses (M_vir) of the modeled sources, (2) increased the gas metallicity to one-third of that of the Sun, and (3) assigned a dust-to-metal ratio of 0.02. In particular, the SEDs associated with different luminosities in Figure 1 correspond to sources with halo mass M_vir = 10¹⁰, 10¹¹, 10¹², and 10¹³ M_⊙. As in DB17, the stellar mass fraction is set to f_* = M_*/(M_* + M_gas) = 0.01.

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We evaluated eight of the sets of dust optical constants from Ji et al. (2014) to sample the resulting variations in the SEDs; to suppress sharp features, we smoothed the optical constants with a logarithmic full width at half-maximum of 0.075 (i.e., a factor of 1.19). Specifically, for Figure 1 we used the dust composition UM-D-20 (see Table 1 in Ji et al. 2014), which is silicate rich with some alumina, and the "standard" size distribution from Ji et al. (2014). As we will show, the implementation of other silicon-based dust chemistries generally predicts similar shapes of the SEDs, but the detailed features exhibit some variations. The use of other size distributions for the dust grains can lead to moderate variations in the total luminosities, with the exact value depending on the dust chemical composition. As discussed in Ji et al. (2014), a variety of dust compositions and size distributions are possible; our calculations are illustrative of the FIR SEDs that might be expected from very young galaxies, but the detailed spectral features are a product of the specific grain composition assumed and should not be taken to be unique. For further details regarding these choices, we refer to DB17.

The models assume that the galaxy size is given by the virial radius. This has the effect that within a given family of models, such as those displayed in Figure 1, the mass density is similar, independent of luminosity. The luminosity density exhibits a modest dependence, increasing by an order of magnitude over the range of luminosities in Figure 1. As shown by DB17, moderate changes in the gas-phase and dust density profiles (at a fixed mass) would not generate significant changes in the SEDs, but an increase in dust-to-metal ratio or gas-phase metallicity drives an almost proportional increase of emission by dust without substantially changing the shape of the SED.

Figure 2 demonstrates how different assumptions affect the SEDs. All four SEDs in this figure are for a luminosity of ∼4 × 10¹⁰ L_⊙. In particular, we analyze the effects of extending the heating source by distributing individual stellar groups over a radius of 100 pc. The cases for an extended star cluster versus a single central source merge for wavelengths >5 μm, so the distribution of heating sources to first order has relatively little effect on the predicted mid- and FIR SED. The increase in stellar mass fraction (sometimes called star formation efficiency) is accompanied by a decrease in the virial radius by a factor of 0.45, so there is an increase by an order of magnitude in the luminosity density that manifests itself as a modestly "bluer" SED, shifted to shorter wavelengths by a factor of about two. Thus, the FIR SED changes with increasing luminosity density, but not dramatically. These results show that the emission efficiency of the silicate-rich dust has a strong influence on the FIR SED, making it relatively insensitive to changes in the architecture of the galaxy.

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We now consider why Haro 11 is a useful analog to luminous, high-redshift galaxies. The visible image of Haro 11 is complex, with three bright knots (A, B, and C) within a fainter surrounding, the general appearance of a complex merging system, and with a rich population of very young super star clusters (e.g., Adamo et al. 2010). However, the very luminous infrared emission is compact and centered on Knot B (e.g., Lyu et al. 2016; Chu et al. 2017); see Figure 3, and indicates an SFR of about 30 M_⊙ yr⁻¹ from this knot (Lyu et al. 2016). This knot has a higher metallicity than the rest of the galaxy (James et al. 2013), consistent with it being a site dominated by this ongoing starburst. Knot B has a radius of about 140 pc and a mass of about 2 × 10⁹ M_⊙, determined dynamically (Östlin et al. 2015). Its specific SFR is thus ∼15 Gyr⁻¹ and its SFR surface density is ∼500 M_⊙ yr⁻¹ kpc⁻²; both values are typical of luminous high-redshift galaxies.

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The stellar population powering the FIR emission appears to be very young, with a typical age of only 3.5 Myr (Adamo et al. 2010). A rough estimate of the total mass of very young stars is 3 × 10⁸ M_⊙, obtained by assuming the current level of SFR for 10⁷ years. A substantial fraction of the mass in Haro 11 arises from cold gas, which from its radial velocity resides primarily in Knot B (Cormier et al. 2014). Indeed, if the gas mass is estimated from the FIR emission, it accounts for most of the dynamical mass of 2 × 10⁹ M_⊙ (Östlin et al. 2015). Hence, the newly formed stars could be responsible for a very substantial fraction of the total stellar mass in Knot B; it is a region that is potentially young-star-dominated and hence a candidate to have silicate-rich dust.

The oxygen abundance in Knot B is cited nominally as 12 + log(O/H) = 8.33 ± 0.01, where the quoted error does not allow for systematic errors (Guseva et al. 2012), or 8.25 ± 0.15 (James et al. 2013), values that are ∼36%–44% of solar.⁶ This abundance is a factor of about three greater than our estimate for the high-redshift (z ∼ 6) galaxies. The aromatic bands are very weak (Wu et al. 2006), consistent with a carbon-poor ISM. However, other explanations are possible since the excitation and destruction of the aromatic bands is a complex and not well-understood process.

In summary, the analogy of expected conditions in Haro 11 with those in luminous infrared galaxies at z ∼ 6 is reasonably good.

In Figure 4, we compare the SEDs from our theoretical models with that for Haro 11 (from Lyu et al. 2016, and references therein). As discussed previously, the model is conceptual—the lack of detailed information about the optical constants for the dust in silicate-rich galaxies means that the details of the SED are not significant. Nonetheless, some important trends are evident. First, the models do reflect the warmer temperature of Haro 11 reasonably well, and they all show the strong emission in the 10–40 μm range, characteristic of the very warm dust that is also seen in the high-redshift galaxies. The detailed behavior in this spectral range differs substantially from one set of optical constants to another. Even with much better constraints on the optical constants, a broad variety of behavior is also seen in spectra of local low-metallicity galaxies where the silicate bands appear anywhere from strong emission to absorption (Rémy-Ruyer et al. 2015). For these local galaxies, despite the complete sets of observations, modeling the silicate bands in a general way is still challenging; Rémy-Ruyer et al. (2015) had to introduce an additional spectral feature in the form of a warm modified blackbody to improve the fits to their observations. The selection of the appropriate optical constants among the possibilities for Pop III/II galaxies and detailed modeling of their spectra such as that for local galaxies by Rémy-Ruyer et al. (2015) is beyond the scope of this paper.

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However, the deficiency in output at wavelengths >100 μm relative to the behavior of Haro 11, present in all models, needs to be addressed. There is ongoing discussion of whether there is a modest amount of carbon dust in the earliest galaxies (e.g., Nozawa & Kozasa 2013; Marassi et al. 2015). In any case, the detection of the [C ii] 158 μm line in galaxies at z > 5 (e.g., Pentericci et al. 2016; Bradac et al. 2017; Decarli et al. 2017; Smit et al. 2018) demonstrates that there is some level of this element in typical ISMs at this redshift, indicating also the presence of carbon dust. We therefore computed the infrared SED expected for amorphous carbon, using optical constants from K. Misselt (2018, private communication), based on those of Zubko et al. (1996). It is not possible to determine the abundance of carbon in these galaxies accurately from the 158 μm line detections (Lagache et al. 2018). In addition, the emission efficiency of carbon depends strongly on its form (e.g., amorphous or graphite, see Rémy-Ruyer et al. 2015), so we do not show results as a function of relative densities of carbon and silicate dust. Instead, in Figure 5 we show the eight silicate models considered in Figure 4, but with 10% and 20% of the total luminosity contributed by carbon. It appears that the addition of a small amount of carbon dust can substantially improve the correspondence of the silicate-rich templates to the SED of Haro 11.

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Given the correspondence of the SED of Haro 11 to the model predictions, we adopt the Haro 11 SED as a proxy template to study the observed behavior of FIR SEDs with redshift. We provide the Haro 11 SED in electronic form in Appendix A.

At redshifts of 1–3, Rujopakarn et al. (2013) found that SFRs estimated from rest-frame 24 μm photometry on the basis of templates from Rieke et al. (2009) for log(L(TIR)) = 11.25–11.75 gave reasonably accurate results⁷ for all galaxies with infrared luminosities above ∼10¹¹ L_⊙. This conclusion indicates that these templates are a reasonably accurate representation of the FIR SED of these galaxies. We start with this result to provide the fiducial SED against which we will study the evolution with increasing redshift toward an FIR SED resembling more closely that of Haro 11.

In general, well-sampled FIR SEDs are not available at very high redshifts, so our method is to collect all cases where there are measurements of reasonably high signal-to-noise ratio covering a sufficient wavelength range to be useful in constraining the shape of the FIR SED. We also require spectroscopic redshifts to avoid artificially broadening the SED due to redshift errors. Table 1 lists the galaxies and the sources for the measurements we have used. Most of the galaxies are selected at long wavelengths, although in the rest frame there are many examples bridging the peak of the FIR SED. Specifically, Wang et al. (2016) select at 160 μm, Casey et al. (2012) at 250–500 μm, Shu et al. (2016) at 500 μm, and the remaining cases in this redshift range are selected in the millimeter-range.

We shift the measurements to the rest frame and normalize them to the same value (relative to the assumed template) near the peak. This normalization uses χ² minimization to the template between rest wavelengths of 50 and 200 μm, i.e., χ² is minimized using all the measurements that fall within this spectral range. To support this step, we require at least three measurements with signal-to-noise ratios >3 in this spectral range. Since this range covers the peak of the SED, the normalization is roughly by FIR luminosity. The normalization depends only on the peak of the FIR SED, so it leaves the measurements at shorter and longer wavelengths to characterize the SED shape.

Figure 6 shows the result for 2 ≤ z < 4, along with the log(L(TIR)) = 11.25 and 11.75 templates (Rieke et al. 2009), as well as templates from Chary & Elbaz (2001) and Kirkpatrick et al. (2015). Some of the points are from combinations of data for a number of galaxies. The composite data points from Dunlop et al. (2017) are based on galaxies detected in a deep ALMA map of the Hubble Ultra-Deep Field. The red HIEROS from Wang et al. (2016) are massive, dusty, star-forming galaxies at redshifts of 2–3 (plus passive galaxies at higher redshifts that will not contribute to the FIR); the median redshift of these galaxies is z = 2.5 and the quoted errors are about the same size as the points. Béthermin et al. (2015) present stacked measurements in the COSMOS field; we show results from the bins between z = 2 and z = 3.5. Schreiber et al. (2018) stacked Herschel measurements of galaxies in the CANDELS fields; we show the results for 2.5 < z < 3.5. The remaining points are for individual galaxies.

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The values of χ² provide a quantitative evaluation of the validity of the templates. A few measurements have very high ratios of nominal signal-to-noise ratio, up to 20:1. To allow for effects such as confusion noise and at the same time to keep one or two measurements from dominating the results, we added a uniform additional noise of 15% of the measured total flux density value for the individual galaxies and 7.5% for the stacked results, in quadrature with the quoted noise. We also reject the 5 most discordant measurements (out of a total of 163), and evaluate the templates only between 11 and 700 μm. The log(L(TIR)) = 11.25 template yields ${\chi }_{\mathrm{red}}^{2}=1.87$. The agreement is significantly worse with the log(L(TIR)) = 11.75 template (${\chi }_{\mathrm{red}}^{2}$ = 2.91) and with the Chary & Elbaz (${\chi }_{\mathrm{red}}^{2}$ = 5.61), Kirkpatrick (${\chi }_{\mathrm{red}}^{2}$ = 4.06), and Schreiber (${\chi }_{\mathrm{red}}^{2}$ = 3.05) ones. The Magdis et al. (2012) z = 2.5 template is also somewhat worse, ${\chi }_{\mathrm{red}}^{2}=2.27$. These values of ${\chi }_{\mathrm{red}}^{2}$ could be reduced by assigning a value higher than 15% to the additional noise, but we have not done that because there are likely to be intrinsic variations among the galaxies relative to a single template, and they will contribute to the scatter. We will use the log(L(TIR)) = 11.25 template as the standard of comparison in the following sections to demonstrate any systematic changes. However, it appears to overestimate the strength of the aromatic bands modestly, consistent with the finding of Rujopakarn et al. (2013) that the rest ∼8 μm fluxes require templates with log(L(TIR)) = 11.25–11.75 for good fits. Since this paper is focused on wavelengths ≥11 μm, this is not an issue.

We now apply a similar approach to luminous infrared galaxies at $5\lt z\lt 7$. The measurements of individual galaxies with adequate FIR observations are listed in Table 1. We also used measurements of six quasars with adequate signal-to-noise ratio Herschel FIR measurements and a large excess attributed to their host galaxies, namely J0338+0021, J0756+4104, J0927+2001, J1202-3235, J1204-0021, and J1340+2813; flux densities and redshifts were taken from Leipski et al. (2014). We subtracted the quasar contribution using the "normal" quasar template (Xu et al. 2015), as opposed to dust-deficient quasar templates (see Lyu & Rieke 2017a), to derive the SEDs of the host galaxies. None of the fits in Lyu et al. (2016) would ascribe a significant fraction of the FIR emission to the quasar, a conclusion that is confirmed by arguments in Appendix B.

To look for changes in FIR SEDs between z ≈ 3 and 6, in Figure 7(a) we compare all of the available measurements with adequate signal-to-noise ratio at the latter redshift to the local log(L(TIR)) = 11.25 template that gave the best fit for 2 < z < 4. The correspondence is poor, largely because the SED is too cool. The poor fit would also apply to the approach of Magdis et al. (2012), whose template resembles the log(L(TIR)) = 11.25 one, and who argue for little SED evolution at z > 3. The local log(L(TIR)) = 12.25 template has a warmer SED, but nonetheless, it has significant divergence at both short and long wavelengths. Schreiber et al. (2018) extrapolate the temperature relation derived for z < 4 to deduce an SED at z = 6 with a temperature about 10 K warmer than at z = 3.5. Figure 7(a) shows their template for z = 6; the correspondence with the data is improved but still not fully satisfactory. To render these impressions quantitative, we follow the procedures in Section 4.1 to determine χ² for the templates in Figure 7(a). We found a ${\chi }_{\mathrm{red}}^{2}$ of 6.1 for the log(L(TIR)) = 11.25 template, 4.5 for the log(L(TIR)) = 12.25 one, and 3.25 for the one from Schreiber et al. (2018). None of them fits well because the SEDs are too narrow. The measurements indicate the need for a broader template.

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Figure 8 is similar to Figure 7, but for 4 ≤ z < 5. It appears that the SEDs at z ≈ 4.5 are a bridge from the local templates to the broader one needed to fit the data at z > 5. Figure 7(b) shows the measurements in Figure 7(a), fitted by the Haro 11 SED by χ² minimization, using the same procedures as in Section 4.1. The fit has a ${\chi }_{\mathrm{red}}^{2}$ of 1.57. There appears to be no systematic difference between the points derived from quasar host galaxies and those from galaxies without luminous AGN. However, the points between 10 and 50 μm are mostly derived from the host galaxies because only they have useful measurements in the 100 and 160 μm Herschel bands. Appendix B demonstrates that it is very unlikely that these values are significantly contaminated by quasar emission. For comparison, we also show the relevant template from Schreiber et al. (2018). The Haro 11 SED is broader, and this results in a better fit to the measurements, as indicated by its lower χ², 1.57 versus 3.25, when computed in identical ways. To convert the relative ${\chi }_{\mathrm{red}}^{2}$ values into probabilities, we increased the additional noise component from 15% to 22% of the measured flux, to set ${\chi }_{\mathrm{red}}^{2}$ for the Haro 11 template to 1. The probability to obtain by chance the resulting ${\chi }_{\mathrm{red}}^{2}$ for the Schreiber template, ${\chi }_{\mathrm{red}}^{2}$ = 2.10, is then ∼10⁻⁸. This probability is reduced as the additional noise term is reduced from 22% toward 15%.

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We conclude that the Haro 11 SED provides a significantly better template for galaxies at z > 5 compared with templates derived at lower redshift. Other than the quasar host galaxies, the galaxies in Figure 7 are nearly all selected in the millimeter-waveband, corresponding to rest wavelengths around 150 μm, which could produce a mild bias toward "cold" SEDs. Nonetheless, their SEDs are significantly brighter in the mid-infrared (10–40 μm) and hence "warmer" than is indicated by templates derived at lower redshift, as shown in Figure 7.

The slope of the Haro 11 SED at wavelengths longer than 100 μm may be slightly steeper than those of other templates. To set an upper limit for the CMB contribution to the dust temperature, we estimated the emissivity index β (where the departure from gray emissivity goes as λ^−β) just for wavelengths ≥100 μm, obtaining β = 1.2. A more conventional estimate gives β ≈ 1.9 (Lyu et al. 2016), compared with values of 1.5–2.0 for typical high-metallicity main-sequence luminous galaxies (e.g., Elbaz et al. 2011; Magdis et al. 2012; da Cunha et al. 2013). Any differences in the submillimeter SEDs would be subtle, making it difficult to distinguish the two cases in that spectral range. Therefore, this potential bias does not undermine the conclusion that the Haro 11 template is the better template to use at these redshifts.

Our results show that the SEDs of infrared galaxies continue to evolve toward warmer temperature going from z ∼ 3 to z ∼ 6, in agreement with the extrapolation in Schreiber et al. (2018), but contrary to the assumptions in Elbaz et al. (2011) and Magdis et al. (2012). The extent of the potential change in shape of the FIR SEDs for z ≳ 5 is not anticipated in any previous sets of templates.

The influence of template choice on the deduced FIR luminosity with Herschel measurements is not uniquely determined; it depends on the bands measured and the method used to normalize the templates to the measurements (i.e., there is often an additional error term above the nominal measurement errors). For a specific example, however, we have assumed a galaxy at z = 5, 6, or 7 with measurements at 250, 350, and 500 μm (typical for such data on galaxies at z ∼ 6) and equal weights. We then find that the luminosity deduced with the Rieke et al. (2009) LIRG and ULIRG templates and the Schreiber et al. (2018) one for z = 5 would all be about 77% lower than with the Haro 11 one. At z = 6, the results would be 99%, 68%, and 77% of that from the Haro 11 template, respectively. At z = 7, they are 121%, 79%, and 82%, respectively. These values are only illustrative, given the differences that would result from different measurement weights.

Single-band ALMA measurements near 1 mm are being used to estimate infrared luminosities and SFRs at z ≈ 6 (e.g., Wang et al. 2013; Willott et al. 2017; Decarli et al. 2018). One consequence of the FIR SED derived here for high-redshift galaxies is that such an approach could systematically underestimate the total infrared luminosities and hence the SFRs. Figure 9 compares a number of relevant SEDs to illustrate the issue. The LIRG spectrum (log(L(TIR)) = 11.25) has been shown to be typical of even the most luminous galaxies at 1 < z < 3 (Rujopakarn et al. 2013, this paper). The modified blackbody (T_d = 47 K, β = 1.6) is commonly used to interpret the ALMA 1 mm measurements (e.g., Wang et al. 2007, 2008; Willott et al. 2013, 2017; Leipski et al. 2014). The horizontal black line extends from 100 to 400 μm, roughly the relevant rest spectral range for the interpretation of the ALMA bands 6 and 7 results at z ∼ 1–10. The various possible choices for the SED result in significant differences in the total infrared luminosities that would be deduced.

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This issue is illustrated graphically in Figure 10, which shows the conversion from flux densities measured in ALMA Band 7 (870 μm) to galaxy luminosities in units of 10¹² L_⊙, as in Schreiber et al. (2018). The differences in the conversion factors for objects at 1 < z ≤ 4 can be understood from Figure 6, which compares the different templates with observed photometry and confirms that the Rieke et al. (2009) template is a good overall fit to the available measurements at these redshifts. The Kirkpatrick et al. (2015) template agrees closely with the Rieke et al. (2009) template at wavelengths <100 μm, which account for much of the total IR luminosity, but is significantly brighter at longer wavelengths. Consequently, a given luminosity yields a larger signal at the long wavelengths probed by ALMA Band 7 over this range of redshift, corresponding to observed wavelengths of 400–170 μm. In other words, a given signal in mJy needs a smaller multiplication factor to be converted into the total infrared luminosity using the Kirkpatrick template compared with the Rieke one. In contrast, the template from Chary & Elbaz (2001) is higher at λ < 100 μm and lower at longer wavelengths compared to the Rieke et al. (2009) template, such that a given signal in Band 7 requires a substantially higher total infrared luminosity. As shown in Figure 9, the modified blackbody is also higher at λ < 100 μm and lower longward of this wavelength compared to the Rieke et al. (2009) template, resulting in a similar behavior. The differences between the Chary & Elbaz (2001) and Kirkpatrick et al. (2015) templates are a factor of 3–4 over this redshift range. The Schreiber et al. (2018) results are based on templates that evolve in shape with redshift, and hence the behavior differs from the rest. None of these templates are constrained by data at z > 4, so we show the results at higher redshifts as dotted lines.

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At z ∼ 6, the Haro 11 SED appears to be a better fit to the data than any of the templates derived at lower redshift, and total luminosities derived using it on the basis of ALMA Band 7 measurements will be a factor of 2–4 higher than those from the extrapolations of the other templates. At redshifts of z ∼ 4–5, it is possible that there is a mixture of behavior in the FIR, making the total infrared luminosity estimation for a typical case ambiguous.

Nonetheless, it is worth considering in more detail the possibility that the warm FIR spectral component that produces the deviation from "local" templates in the 10–50 μm range might be dust heated by the quasar. Lyu & Rieke (2017b) advanced a number of arguments that show that the intrinsic SEDs of Type 1 quasars drop at wavelengths longer than 20–30 μm. Some local AGNs have an additional emission component due to polar dust that boosts the emission in this region (e.g., Asmus et al. 2016; Lyu & Rieke 2018), but this component is largely missing in more luminous AGN, probably because of the higher radiation pressure in their polar directions that tends to eject material rapidly. Hypothesizing that this warm component is present in all six of the high-redshift quasars used to derive the stellar-powered SED in this paper requires that their SEDs differ significantly from those of lower-redshift quasars, which would be contrary to all the evidence that the SEDs are identical, including in the infrared (e.g., Fan 2009; Lyu & Rieke 2017a).

Figure 11 illustrates the issue specifically for these six quasars. It shows that the strongest excesses above the quasar continuum template in the 10–50 μm range are associated with powerful excesses at the longer FIR wavelengths. These latter excesses almost certainly arise from dust heated by star formation in the host galaxies. This relationship is not expected if the quasar itself is responsible for the 10–50 μm emission, in which case it would be expected to be independent of the output of the host galaxies.

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The hypothesis that this emission arises from heating by the quasar can also be tested for these specific objects by energy balance: is the expected optical-to-UV luminosity of the quasar sufficiently higher than the infrared luminosity to make it plausible that the infrared arises through heating of dust by the quasar? In general, the value of the ratio of these quantities should reveal the covering fraction of dusty structures around the AGN central engine. We have applied this test to the six quasars in this study. We used the log(L(TIR)) = 12.25 template to represent the star formation in the host galaxies, since it provides a rough fit at the long wavelengths. To this, we added the output of a single temperature graybody with a wavelength-dependent emissivity proportional to ${\lambda }^{-\beta }$, with β = 2. This fit should produce a lower limit to the luminosity of the warm component since it produces an SED peaked relatively sharply at the critical wavelengths; addition of components over a range of temperatures would increase the derived luminosity, for example, since they would add emission at wavelengths that are not well constrained by the data. We computed the energy balance assuming that all the luminosity at wavelengths short of 1.1 μm was available to heat the warm dust and that all the luminosity at longer wavelengths was from such heated dust, except for the contribution from the ULIRG template. The rationale for this approach can be found in Lyu & Rieke (2017b). The results are in Table 3, shown as lower limits because of the potential underestimate of the warm-component luminosity due to the assumption of a single temperature.

With the simple assumption of isotropic emission, the first three quasars listed are already in violation of the expectations from energy balance. A more rigorous comparison needs to take account of the expected anisotropies in the emission by the central engine of an AGN. This case has been modeled by Stalevski et al. (2016), who studied the relation between the covering fraction and the reradiated accretion disk emission with a torus that is optically thick in the mid-IR. Their simulations indicate how the luminosity changes with different parameter values for the torus. For the cases where the predicted SED matches that observed, the upper limit to the ratio of reradiated to central engine luminosity is ∼0.75 (see the further discussion in Lyu & Rieke 2017b). All six quasars are above this upper limit (see Table 2).

The quasars in Table 3 constitute about half of those in Leipski et al. (2014) with useful FIR measurements, i.e., detections at 3:1 or higher signal-to-noise ratio in multiple bands. It is unlikely from an energy balance perspective that half of the quasars with useful Herschel data both have some unique circum-nuclear structure to emit in the mid-IR and are also pushing the limits of energy balance. Of course, the energy balance argument could be circumvented if the ultraviolet spectra of these quasars were anomalously bright, but then we need to invoke two departures from typical quasar behavior (UV and mid-IR), when all the evidence to date is that the high-redshift quasars are identical in all observables to their lower-redshift counterparts (e.g., Fan 2009).