FAR-INFRARED AND MOLECULAR CO EMISSION FROM THE HOST GALAXIES OF FAINT QUASARS AT z ∼ 6

There are 33 z ∼ 6 quasars that have published millimeter and radio observations from the literature (Bertoldi et al. 2003a; Petric et al. 2003; Carilli et al. 2004; Wang et al. 2007, 2008a; Willott et al. 2007). In this paper, we present new MAMBO observations of the 250 GHz dust continuum emission from eight z ∼ 6 quasars (Table 1) and molecular CO observations of three of them. Seven of the eight sources were discovered from the SDSS southern survey with optical magnitudes of 20.87 ⩽ z_AB ⩽ 22.16 (Jiang et al. 2008, 2009), and the other source, ULAS J131911.29+095051.4 (hereafter J1319+0950), was selected from the UKIRT Infrared Deep Sky Survey with z_AB = 19.99 (UKIDSS; Mortlock et al. 2009). Seven of the eight objects (all but SDSS J020332.39+001229.3, hereafter J0203+0012) have never been observed at millimeter wavelengths before. We also report new radio 1.4 GHz continuum observations for three of eight new MAMBO-observed objects and another z ∼ 6 quasar from CFHQS (Willott et al. 2007).

The new observations, together with the previous data, now provide us a sample of forty 250 GHz-observed z ∼ 6 quasars with optical magnitudes over a wide range, i.e., 18.74 ⩽ z_AB ⩽ 22.16, or quasar rest-frame 1450 Å AB magnitudes of 18.81 ⩽ m₁₄₅₀ ⩽ 22.28. Wang et al. (2008a) studied the dust emission from quasars at z ∼ 6, mainly based on the most luminous sample from the SDSS main survey. In this paper, we investigate the FIR and radio emission properties of the fainter quasars at z ∼ 6, i.e., a subsample of 18 objects with m₁₄₅₀ ⩾ 20.2 (Tables 1 and 2), and compare them to the bright z ∼ 6 quasars (m₁₄₅₀ < 20.2) discussed in previous papers (Bertoldi et al. 2003a; Petric et al. 2003; Wang et al. 2008a). We will discuss the optical measurements and quasar UV and bolometric luminosities based on m₁₄₅₀, which is derived from the SDSS z-band magnitudes and deep optical spectra (e.g., Fan et al. 2000, 2001, 2003, 2004, 2006; Jiang et al. 2008, 2009; Willott et al. 2007), or from infrared photometry in the cases in which significant dust reddening is presented in the optical spectra (e.g., McGreer et al. 2006).

Observations of the 250 GHz dust continuum from the eight z ∼ 6 quasars were carried out using the MAMBO-II 117 element array on the IRAM 30 m telescope (Kreysa et al. 1998) in the winters of 2008–2009 and 2009–2010. We adopted the standard on–off photometry mode with a chopping rate of 2 Hz by 32'' in azimuth (see also Wang et al. 2007, 2008a). The on-source observing time was about 1–3 hr for each object, to achieve 1σ rms sensitivities of ∼0.5–0.7 mJy. We used the MOPSIC pipeline (Zylka 1998) to reduce the data, and three sources, J1319+0950, SDSS J012958.51−003539.7 (hereafter J0129−0035), and J0203+0012, were detected at ⩾4σ. Wang et al. (2008a) observed J0203+0012 with MAMBO and did not detect the source with a 1σ rms of 0.7 mJy. Our new observations, together with the old data, improve the rms to 0.46 mJy for this object, and we detect the 250 GHz continuum at 4σ. We have also obtained a 2.4σ signal from another source, SDSS J023930.24−004505.4 (hereafter J0239−0045).

We also observed four z  ∼  6 quasars at 1.4 GHz in Winter 2008–2009, using the VLA in A configuration with a synthesized beam size of FWHM  ∼  14. Flux scales were measured using the standard VLA calibrators of 3C286 and 3C48, and phase calibrations were performed every 30 minutes on nearby phase calibrators (1254−116, 1507−168, 2136+006, and 2323−032). The typical rms noise level was ∼17 μJy beam⁻¹ in three hr of observing time for each source. We reduced the data with the standard VLA wide field data reduction software AIPS, and one source, J1319+0950, was detected at ∼3.8σ.

We searched for molecular CO (6–5) line emission from the three MAMBO detections using the 3 mm receiver on the PdBI. The source J1319+0950 was observed in 2009 Summer in C and D configurations (i.e., a spatial resolution of FWHM  ∼  35) with the narrowband correlator, which provides a bandwidth of 1 GHz in dual polarization. The observations of J0129−0035 and J0203+0012 were carried out using the new wide band correlator WideX with a bandwidth of 3.6 GHz in dual polarization, which covers well the Lyα-redshift uncertainty of Δz ∼ ±0.03 (Jiang et al. 2009). The observations were performed in 2010 Summer in D configuration with a synthesized beam size of FWHM ∼ 5''. Phase calibration was performed every 20 minutes with observations of phase calibrators, 1307+121 and 0106+013, and we observed MWC 349 as the flux calibrator. We reduced the data with the IRAM GILDAS software package (Guilloteau & Lucas 2000). The CO (6–5) line emission is detected from J1319+0950 at 4.8σ in 10.3 hr of on-source time with an rms sensitivity of 0.50 mJy beam⁻¹ per 70 km s⁻¹ binned channel, and from J0129−0035 at 5.3σ in 8.8 hr with an rms of 0.55 mJy beam⁻¹ per 100 km s⁻¹ channel. We also detect the continuum at >3σ at the CO (6–5) line frequency in both objects. The other source, J0203+0012, was undetected in both CO and 3 mm continuum in 13.9 hr of on-source time, with a sensitivity of 0.44 mJy beam⁻¹ per 100 km s⁻¹ channel.

We summarize the new millimeter and radio observations of the nine z ∼ 6 quasars in Table 1. Detailed measurements of the three new 250 GHz dust continuum/molecular CO (6–5) detections are listed below.

J1319+0950. The source was discovered in UKIDSS with m₁₄₅₀ = 19.65, which lies in the typical magnitude range of the optically bright z ∼ 6 quasars selected from the SDSS main survey (Mortlock et al. 2009). We detect the 250 GHz dust continuum emission from this object with a flux density of S_250 GHz = 4.20 ± 0.65 mJy, which is the third-strongest 250 GHz detection among the known z ∼ 6 quasars. The 1.4 GHz radio continuum is also detected from our VLA observation, with S_1.4 GHz = 64 ± 17 μJy (left panel of Figure 1). The radio emission appears unresolved with a peak at $13^{\rm h}19^{\rm m}11\mbox{$.\!\!^{\mathrm s}$}30$, 09°50'518, i.e., within 04 of the optical quasar position. We also see a 4σ peak (S_1.4 GHz = 68 ± 17 μJy) at $13^{\rm h}19^{\rm m}11\mbox{$.\!\!^{\mathrm s}$}65$, 09°50'532 on the radio map, which is 56 away from the quasar. For comparison, only 0.02 detections with S_1.4 GHz ⩾ 60 μJy are expected from a random circular sky area with a radius of 6'', according to the 1.4 GHz radio source counts from previous deep VLA observations (i.e., $N(>S_{1.4 \rm \,GHz})=(0.40\pm 0.04)(S_{1.4\rm \,GHz}/75\,{\rm \mu Jy})^{(-1.43\pm 0.13)}$; Fomalont et al. 2006). No optical counterpart has been found at the radio source position in the NASA/IPAC Extragalactic Database or the SDSS images.

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We observed the CO (6–5) line emission from this object with the PdBI. The line is detected at 4.8σ on the velocity-integrated map with a line flux of 0.43 ± 0.09 Jy km s⁻¹, 09 away from (i.e., consistent with) the quasar position (the upper panel of Figure 2). A Gaussian fit to the line spectrum yields a peak line flux density of 0.80 ± 0.15 mJy beam⁻¹ and a line width of FWHM = 537 ± 123 km s⁻¹, centered at the redshift of z = 6.1321 ± 0.0012. This is slightly higher than, but consistent with the Mg ii redshift of z = 6.127 ± 0.007, considering the measurement errors (±0.004; Mortlock et al. 2009) and the typical velocity shift and scatters 97 ± 269 km s⁻¹ (−0.002 ± 0.006 in redshift) between the broad Mg ii line emission and the quasar systemic redshift (Richards et al. 2002). We also detect the continuum under the CO (6–5) line emission with a peak value of S_97 GHz = 0.31 ± 0.08 mJy beam⁻¹ on the continuum map averaged over the line-free channels (right panel of Figure 1). We adopt this as the continuum flux density as the source is unresolved with the 5'' beam. We checked the position of the 1.4 GHz radio source 56 away from the quasar in the PdBI CO/97 GHz continuum images. There is no clear (⩾3σ) detection at this position and the measurement in the 97 GHz continuum image is 0.19 ± 0.08 mJy (right panel of Figure 1).

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J0129−0035. This is the faintest of the SDSS z ∼ 6 quasars, with m₁₄₅₀ = 22.16 (Jiang et al. 2009). We detect 250 GHz dust continuum from the quasar host galaxy, with a flux density of S_250 GHz = 2.37 ± 0.49 mJy, comparable to the flux densities of the optically bright z ∼ 6 quasars with 250 GHz detections.

We detected the CO (6–5) line in this source with a flux of 0.37 ± 0.07 Jy km s⁻¹, i.e., at 5.3σ, integrated over a velocity range of 380 km s⁻¹ (the middle panel of Figure 2). The line width fitted with a single Gaussian is 283  ±  87 km s⁻¹, with a line peak of 1.2 ± 0.3 mJy. The host galaxy redshift measured with the CO line is z = 5.7794 ± 0.0008, which is in good agreement with the optical redshift of z_Lyα = 5.78 ± 0.03 measured with Lyα (Jiang et al. 2009). The underlying dust continuum is also detected with a flux density of S_102 GHz = 0.14 ± 0.04 mJy.

J0203+0012. This object is a broad absorption line quasar discovered by both the SDSS southern survey (Jiang et al. 2008) and the UKIDSS survey (Venemans et al. 2007; Mortlock et al. 2009) with m₁₄₅₀ = 20.97. The redshift measured with the quasar $\rm N\,\mathsc {v}$ line emission is z = 5.72 (Mortlock et al. 2009). Wang et al. (2008) detected the 1.4 GHz radio continuum with S_1.4 GHz = 195 ± 22 μJy (Wang et al. 2008). Our MAMBO observation yields a 250 GHz dust continuum flux density of S_250 GHz = 1.85 ± 0.46 mJy. We have searched for the CO (6–5) line emission from this object over a wide redshift range, from z = 5.60 to 5.84, but did not detect it in 14 hr of on-source time. Assuming a line width of 600 km s⁻¹, the 3σ upper limit of the CO flux is estimated to be 0.32 Jy km s⁻¹ (the lower panel of Figure 2).¹⁴ The dust continuum at the CO (6–5) line frequency is also undetected with a continuum sensitivity of 0.04 mJy beam⁻¹ and a 3σ upper limit of 0.12 mJy.

We calculate the FIR luminosities and upper limits for the eight z ∼ 6 quasars observed by MAMBO. We model the FIR emission with an optically thin graybody and normalize the model to the MAMBO-250 GHz and available PdBI 3 mm dust continuum flux densities/3σ upper limits (see also Wang et al. 2007, 2008). A dust temperature of T_d = 47 K and emissivity index of β = 1.6 are adopted in the model; these are typical values for the FIR and CO luminous quasars at lower redshifts (Beelen et al. 2006). We then integrate the model FIR SED from 42.5 μm to 122.5 μm to calculate the total FIR luminosities/upper limits. The FIR luminosity of the strongest MAMBO detection, J1319+0950, is 1.0 × 10¹³ L_☉, making it one of the most FIR-luminous objects among the z ∼ 6 quasar sample. The other two MAMBO-detected objects are from the SDSS southern survey sample and have FIR luminosities of 4 ∼ 5 × 10¹² L_☉. We also estimate the dust masses for the three MAMBO detections, using M_dust = L_FIR/4π∫κ_νB_νdν, where B_ν is the Planck function and κ_ν = κ₀(ν/ν₀)^β is the dust absorption coefficient with κ₀ = 18.75 cm² g⁻¹ at 125 μm (Hildebrand 1983). The estimated dust masses of the three sources are (2.5–5.7) ×  10⁸ M_☉. We finally calculate the 1.4 GHz radio continuum luminosity densities and upper limits for the four VLA observed objects. We assume a power-law continuum of f_ν ∼ ν^−0.75 (Condon 1992) to correct the observed frequency to the quasar rest frame. The results are summarized in Table 3; we also include the 1.4 GHz radio luminosity of J0203+0012 from Wang et al. (2008a).

We calculate the CO (6–5) line luminosities and upper limits for the three CO-observed z ∼ 6 quasars, using L'_CO(6–5) = 3.25 × 10⁷IΔv_CO(6–5)ν_obs⁻²D_L²(1 + z)⁻³ (K km s⁻¹ pc²), where IΔv_CO(6–5) is the line flux or 3σ upper limit in Jy km s⁻¹ presented in Table 1, ν_obs is the observed frequency of the CO (6–5) line in GHz, D_L is the luminosity distance in Mpc, and z is the quasar redshift (Solomon & Vanden Bout 2005). We then assume a CO excitation ladder similar to that of J1148+5251 (Riechers et al. 2009), for which L'_CO(6–5)/L'_CO(1–0) ≈ 0.78, allowing us to calculate the CO (1–0) luminosities. The FIR-to-CO(1–0) luminosity ratios of J1319+0950 and J0129−0035 are 520 and 340 L_☉ K⁻¹ km s⁻¹ pc², respectively, which are similar to the values found for other z ∼ 6 CO-detected quasars (Wang et al. 2010). We then calculate the molecular gas masses (M_gas = M[H₂ + He]), adopting a CO luminosity-to-gas mass conversion factor of α = 0.8 M_☉ (K km s⁻¹ pc²)⁻¹ seen in low-redshift ultra-luminous infrared galaxies (ULIRGs; Solomon et al. 1997; Downes & Solomon 1998). The molecular gas masses for the two new CO detections are all of order 10¹⁰ M_☉ (Table 3).

We calculate the average 250 GHz and 1.4 GHz continuum for the 18 faint z ∼ 6 quasars. The mean flux densities are derived for (1) all sources, (2) MAMBO detections, and (3) MAMBO non-detections, using the MAMBO and VLA measurements weighted by inverse variance (see also Wang et al. 2008a). We also present the average values for the 22 bright quasars (m₁₄₅₀ < 20.2) for comparison. The average emission of this bright quasar sample is consistent with the results presented in Wang et al. (2008a). We list all the results in Table 4.

Table 4. Average FIR and Radio Emission of the z ∼ 6 Quasars

Group	Numbera	〈L1450〉b	〈f250 GHz〉	〈LFIR〉	$\frac{\left\langle L_{\rm FIR}\right\rangle }{\left\langle L_{1450}\right\rangle }$	Numberc	〈f1.4 GHz〉	〈L1.4 GHz〉d	q
		(1012 L☉)	(mJy)	(1012 L☉)			(μJy)	(L☉ Hz−1)
(1)	(2)	(3)	(4)	(5)	(6)	(7)	(8)	(9)	(10)
Optically faint z ∼ 6 quasars with m1450 ⩾ 20.2
All objects	18	4.3	0.85 ± 0.12	2.0 ± 0.3	0.46	14	19 ± 6	0.012 ± 0.003	1.63 ± 0.15
(radio quiet)	16	4.0	0.79 ± 0.13	1.9 ± 0.3	0.46	12	2 ± 6	<0.013	>1.57
250 GHz detections	5	4.3	1.85 ± 0.20	4.3 ± 0.5	1.01	4	45 ± 10	0.030 ± 0.07	1.59 ± 0.11
(radio quiet)	4	4.2	1.86 ± 0.23	4.3 ± 0.5	1.03	3	1 ± 12	<0.024	>1.68
250 GHz non-detections	13	4.3	0.33 ± 0.15	<1.8	<0.42	10	7 ± 7	<0.019	...
(radio quiet)	12	3.9	0.32 ± 0.15	<1.8	<0.46	9	2 ± 7	<0.015	...
Optically bright z ∼ 6 quasars with m1450 < 20.2
All objects	22	17.6	1.64 ± 0.13	3.8 ± 0.3	0.22	22	50 ± 3	0.033 ± 0.002	1.49 ± 0.04
(radio quiet)	21	16.9	1.68 ± 0.13	3.9 ± 0.3	0.23	21	38 ± 3	0.026 ± 0.002	1.61 ± 0.05
250 GHz detections	8	19.1	2.71 ± 0.18	6.3 ± 0.4	0.33	8	43 ± 4	0.029 ± 0.003	1.77 ± 0.05
250 GHz non-detections	14	16.8	0.55 ± 0.18	1.3 ± 0.4	0.08	14	59 ± 5	0.040 ± 0.003	0.94 ± 0.15
(radio quiet)	13	15.5	0.58 ± 0.18	1.4 ± 0.4	0.09	13	32 ± 5	0.021 ± 0.003	1.23 ± 0.15

Notes. We calculate the average flux densities using 〈f_ν〉 = ∑w_if_{ν, i}/∑w_i, w_i = 1/σ²_{ν, i}, and 〈σ_ν〉 = (∑w_i)^−0.5 (Omont et al. 2003), where f_{ν, i} and σ_{ν, i} are the measurement and 1σ rms for each object at 250 GHz or 1.4 GHz. For results with 〈f_ν〉 less than 3〈σ_ν〉, we take the stacking average value plus 3〈σ_ν〉 as the upper limits of the average flux densities and calculate the corresponding upper limits for the luminosities. ^aNumber of sources observed at 250 GHz. ^bAverage luminosity at rest-frame 1450 Å, derived from m₁₄₅₀. ^cNumber of sources observed at 1.4 GHz. ^dA radio spectral index of −0.75 (Condon 1992) is adopted here to calculate the rest frame 1.4 GHz lunimosity.

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The mean 250 GHz flux density is 0.85 ± 0.12 mJy for all 18 faint quasars, and 1.85 ± 0.20 mJy for the five MAMBO-detected sources, which are about 50% and 70% of the values found for the bright quasars, respectively. The results are about the same when the two radio-loud quasars are excluded. The stacking measurement for the 13 MAMBO-undetected sources in the faint sample is 0.33 ± 0.15 mJy, which places an upper limit of <0.78 mJy (stacking average plus the 3σ rms) for the average millimeter dust continuum emission.

We convert the average 250 GHz flux densities to 42.5–122.5 μm FIR luminosities, adopting the assumptions described in Section 3.2, and calculate the average FIR-to-1450 Å luminosity ratios (〈L_FIR〉/〈L₁₄₅₀〉) for each sample (see Table 4). The average luminosity ratios for all the sources and the MAMBO detections in the faint-quasar sample are two and three times higher than the corresponding values found with bright quasars. This is consistent with the nonlinear FIR-to-active galactic nucleus (AGN) luminosity relationships of L_FIR ∼ L^α_AGN, where α < 1 found with the AGN starburst at low and high redshifts (Beelen 2004; Hao et al. 2005, 2008; Wang et al. 2008a; Serjeant & Hatziminaoglou 2009; Lutz et al. 2010), as we will discuss further in Section 4.2. However, the different average luminosity ratios found with the MAMBO detections in the bright and faint quasar samples may be partly due to the selection effect that the detection limit of the MAMBO observations is about the same for the two samples, while the AGN luminosities are different by design.

The average 1.4 GHz continuum flux density, derived from 14 VLA-observed sources in the faint quasar sample, is 19 ± 6 μJy, and 2 ± 6 μJy when the two radio-loud sources are excluded. This suggests an upper limit of 20 μJy for the radio-quiet and UV/optically faint z ∼ 6 quasars. We estimate the average rest-frame 1.4 GHz luminosities/upper limits for the two samples and list the results in Table 4. We plot L_FIR versus L_{ν, 1.4 GHz} of all the MAMBO-detected z ∼ 6 quasars together with the average luminosities of the MAMBO-detections and non-detections in Figure 3, and compare them to sample of IRAS-selected local star-forming galaxies (Yun et al. 2001). The local star-forming galaxies with FIR and radio emission powered by the active star formation have an average luminosity ratio of q = log (L_FIR/3.75 × 10¹² L_☉) − log (L_1.4 GHz/L_☉ · Hz⁻¹) = 2.34. The range 1.6 < q < 3.0 contains 98% of the IRAS galaxies (Yun et al. 2001). The average luminosities/upper limits of the MAMBO-detected z ∼ 6 quasars, though lower than the mean value of q = 2.34, are all within this range defined by local star forming galaxies. This argues that, in addition to the AGN power, star formation also contributes significantly to the FIR-to-radio emission of the MAMBO-detected z ∼ 6 quasars. The average ratio of the MAMBO-undetected bright z ∼ 6 quasars is offset from this range.

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The correlations between FIR luminosities and AGN UV, optical, and bolometric luminosities have been studied widely with samples of infrared and optically selected quasars to understand the origin of quasar FIR emission. Study of samples of the Palomar-Green quasars and Seyfert galaxies shows a relationship of L_FIR ∼ L^{0.7 ∼ 0.8}_AGN for typical optically-selected AGN/quasars in the local universe (Hao et al. 2005; Netzer et al. 2007; Lutz et al. 2010). A shallower relationship of L_FIR ∼ L^0.4_AGN was found for the local IR-luminous quasars hosted by starburst ULIRGs (Hao et al. 2005) and high-redshift FIR-and-CO luminous quasars (Beelen 2004; Hao et al. 2008; Wang et al. 2007; Lutz et al. 2010), which suggests excess FIR emission powered by extreme starburst in these systems. Wang et al. (2008a) reported that the millimeter-detected bright z ∼ 6 quasars follow the shallow luminosity trend defined by the local IR quasars.

In this work, we investigate the FIR-to-AGN luminosity correlation of the full sample z ∼ 6 quasars, and compare with the samples of MAMBO-250 GHz or SCUBA-350 GHz observed optically bright quasars at z ∼ 2–5 (Omont et al. 2001, 2003; Carilli et al. 2001; Priddey et al. 2003a), and local IR quasars (Zheng et al. 2002; Hao et al. 2005). We convert the UV or optical luminosities of all objects to AGN bolometric luminosities assuming a UV-to-optical power-law continuum of f_ν ∼ ν^−0.5, and an optical-to-AGN bolometric luminosity conversion factor of L_bol = 10.4 νL_{ν, 4400 Å} (Richards et al. 2006). In Figure 4, we plot all the z ∼ 6 quasars, as well as the local IR quasar sample and the average luminosities of the submillimeter/millimeter ((sub)mm)-observed z ∼ 2–5 quasar samples (Omont et al. 2003; Hao et al. 2008). The average FIR luminosities of the z ∼ 2–5 quasars are derived with the mean flux densities at 250 GHz or 350 GHz (Omont et al. 2001, 2003; Carilli et al. 2001; Priddey et al. 2003a), adopting the same assumptions of dust temperature and emissivity index used for the z ∼ 6 quasars (see Section 3.2). We fit a line to the average luminosities of all these (sub)mm-observed quasar samples at z ∼ 2 to 6 using the Ordinary Least Square method (Isobe et al. 1990). This yields an relationship of

$$\begin{equation} \log \,\left(\frac{L_{\rm FIR}}{L_{\odot }}\right)=(0.62 \,\pm\, 0.09)\log \left(\frac{L_{\rm bol}}{L_{\odot }}\right)\,+\,(3.90\,\pm\, 1.29)\quad \end{equation} \tag{ 1 }$$

between the average FIR and AGN emission in these high-redshift optically selected quasars, and the nonlinear slope of 0.62 might suggest contributions from both host galaxy star formation and AGN power to the FIR emission (Beelen 2004; Netzer et al. 2007; Hao et al. 2005, 2008; Wang et al. 2008a; Serjeant & Hatziminaoglou 2009; Lutz et al. 2010).

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The MAMBO-250 GHz observations at sub-mJy sensitivity have detected the most FIR luminous objects among the z ∼ 6 quasars (Bertoldi et al. 2003a; Petric et al. 2003; Wang et al. 2007, 2008; this work). Most of these objects follow the shallower luminosity correlation trend of the local IR quasars and the (sub)mm-detected quasars at z ∼ 2 to 5. In particular, the five MAMBO-detected z ∼ 6 quasars with m₁₄₅₀ ⩾ 20.2 (Willott et al. 2007; Wang et al. 2008a; this work) overlap with the high luminosity end of the IR quasars. This may suggest a starburst-dominant FIR emission in these most FIR luminous quasars at z ∼ 6, similar to that found in the local IR quasars. A fit to all the (sub)mm detections in the high-z quasar sample and the local IR quasars gives a relationship of

$$\begin{equation} \log \,\left(\frac{L_{\rm FIR}}{L_{\odot }}\right)=(0.45\pm 0.03)\log \left(\frac{L_{\rm bol}}{L_{\odot }}\right)+(6.62\pm 0.39).\quad \end{equation} \tag{ 2 }$$