An ALMA CO(2–1) Survey of Nearby Palomar–Green Quasars

We use the channels above the 1σ level of the spectrum to generate the CO intensity (moment 0) map (Figure 2(a)). The integrated CO flux is measured from the intensity map by summing up the pixels within the 2σ contour of the source emission. We estimate the uncertainty from the standard deviation of 20 repeated off-source measurements using a circular aperture containing the same number of pixels as those within the 2σ contour of the source. The uncertainty of the absolute flux scale, ∼5%–10% (Fomalont et al. 2014; Bonato et al. 2018), is not included in our final flux uncertainty. The integrated CO spectrum, used to measure the line width (Section 3.2), is extracted from the line-emitting region above 2σ of the intensity map. CO(2–1) was previously detected in PG 0050+124 using the James Clerk Maxwell telescope (JCMT; 114 ± 23 Jy km s⁻¹; Papadopoulos et al. 2008) and in PG 1126−041 using Institut de Radioastronomie Millimétrique (IRAM) 30 m (24.7 ± 1.6 Jy km s⁻¹; Bertram et al. 2007).¹¹ Our line fluxes are reasonably consistent, with deviations ≲50%.

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Following Solomon & Vanden Bout (2005), the CO line luminosity is

$$\begin{eqnarray}&&{L}_{\mathrm{CO}}^{{\prime} }=3.25\times {10}^{7}\,{S}_{\mathrm{CO}}{\rm{\Delta }}\nu \,{\nu }_{\mathrm{obs}}^{-2}\,{D}_{L}^{2}\,{\left(1+z\right)}^{-3},\end{eqnarray} \tag{ 1 }$$

where ${L}_{\mathrm{CO}}^{{\prime} }$ is the CO line luminosity in units of ${\rm{K}}\,\mathrm{km}\,{{\rm{s}}}^{-1}\,{\mathrm{pc}}^{2}$, ${S}_{\mathrm{CO}}{\rm{\Delta }}\nu$ is the integrated line flux in units of $\mathrm{Jy}\,\mathrm{km}\,{{\rm{s}}}^{-1}$, ν_obs is the observed frequency of the CO(2–1) line in GHz, and D_L is the luminosity distance in Mpc. The factor ${\alpha }_{\mathrm{CO}}$ is needed to derive molecular gas masses from the CO luminosity (Bolatto et al. 2013, and references therein). Since ${\alpha }_{\mathrm{CO}}$ is usually quoted for the CO(1–0) line, we need to convert the line luminosity from ${L}_{\mathrm{CO}(2-1)}$ to ${L}_{\mathrm{CO}(1-0)}$ in order to derive the molecular gas mass. Fortunately, literature measurements of CO(1–0) are available for 15 of the objects in our ALMA sample; among them are eight detections (Section 3.3). We find a median value of ${R}_{21}={0.62}_{-0.07}^{+0.15}$ (Section 4.1). We adopt this median value of R₂₁ to convert all the new ALMA CO luminosities from ${L}_{\mathrm{CO}(2-1)}$ to ${L}_{\mathrm{CO}(1-0)}$ (Table 2). We do not consider the uncertainty on R₂₁, as it is hardly well-constrained by our data. However, if R₂₁ varies from 0.5 to 1.0, it could contribute to the final uncertainty of ${M}_{{{\rm{H}}}_{2}}$ as significantly as ${\alpha }_{\mathrm{CO}}$ (∼0.3 dex). It is reassuring that our estimated value of R₂₁ agrees well with values found in nearby galaxies and AGNs (Ocaña Flaquer et al. 2010; Sandstrom et al. 2013; Rosolowsky et al. 2015; Husemann et al. 2017; Saintonge et al. 2017 see Section 4.1).

The velocity width of the integrated emission-line profiles of galaxies is commonly specified as the line width at 20% (W₂₀; e.g., Tully & Fisher 1977) or 50% (W₅₀; e.g., Tiley et al. 2016) of the peak intensity. For spectra with relatively low signal-to-noise ratio (S/N), we can obtain more accurate line widths by fitting a model line profile to the data instead of measuring them directly from the observed spectrum. With the aid of a suite of integrated spectra of simulated galaxies, Tiley et al. (2016) evaluated the effectiveness of various methods for fitting line profiles and concluded that the "double-peak" Gaussian function—a parabolic function bordered by a half-Gaussian symmetrically on either side—provides the most robust measure of W₅₀. The double-peak Gaussian function is defined as (Tiley et al. 2016)

$$\begin{eqnarray}f(v)=\left\{\begin{array}{ll}{A}_{{\rm{G}}}\times \exp \tfrac{-{\left[v-({v}_{0}-w)\right]}^{2}]}{2{\sigma }^{2}} & v\lt {v}_{0}-w\\ {A}_{{\rm{C}}}+a{\left(v-{v}_{0}\right)}^{2} & {v}_{0}-w\leqslant v\leqslant {v}_{0}+w\\ {A}_{{\rm{G}}}\times \exp \tfrac{-{\left[v-({v}_{0}+w)\right]}^{2}]}{2{\sigma }^{2}} & v\gt {v}_{0}-w\end{array}\right.,\end{eqnarray} \tag{ 2 }$$

where $-500\,\mathrm{km}\,{{\rm{s}}}^{-1}\lt {v}_{0}\lt 500\,\mathrm{km}\,{{\rm{s}}}^{-1}$ is the central velocity, w ($\gt 0\,\mathrm{km}\,{{\rm{s}}}^{-1}$) is the half width of the central parabola, σ ($\gt 0\,\mathrm{km}\,{{\rm{s}}}^{-1}$) is the width of the edge half-Gaussian profile, A_G > 0 is the peak flux of the half-Gaussian edges at ${v}_{0}\pm w$, A_C is the flux at the profile center, and $a=({A}_{{\rm{G}}}-{A}_{{\rm{C}}})/{w}^{2}$. Then, the two conventionally used line widths are given by

$$\begin{eqnarray}&&\begin{array}{l}{W}_{50}=2(w+\sqrt{2\mathrm{ln}2}\sigma ),\\ {W}_{20}=2(w+\sqrt{2\mathrm{ln}5}\sigma ).\end{array}\end{eqnarray} \tag{ 3 }$$

Equation (3) cannot accurately specify the width of strongly convex, single-peaked profiles. Under these circumstances, the data should be fit with a standard Gaussian function,¹² for which ${W}_{50}=2\sqrt{2\mathrm{ln}2}\sigma$ and ${W}_{20}=2\sqrt{2\mathrm{ln}5}\sigma$. Following Tiley et al. (2016), we adopt the standard Gaussian function when either of these two criteria holds: (1) the reduced chi-square of the standard Gaussian fit is closer to unity than that of the double-peak Gaussian function;¹³ (2) ${A}_{{\rm{G}}}/{A}_{{\rm{C}}}\lt 2/3$. We use a Markov Chain Monte Carlo method in the emcee package (Foreman-Mackey et al. 2013) to perform the fit.

An example of the profile-fitting method is shown in Figure 2(b). PG 0050+124 is one of the brightest objects in our sample. Appendix gives the data for the remaining 20 detected objects, all of which were successfully fit, apart from the tentative detection of PG 1341+258, which suffers from an exceptionally low S/N. Table 2 lists measurements of both W₅₀ and W₂₀, the latter because sometimes only this quantity is reported in the literature; we need to use our measured W₂₀/W₅₀ ratios to incorporate the published line widths into our analysis (Section 3.3). The systemic velocities of the CO line agree closely (<5% difference) with the optical redshifts, and for our final analysis, we simply adopt the latter.

To date, CO(1–0) measurements have been published for 32 PG quasars, as summarized in Shangguan et al. (2018). Among them, 15 objects were included in our ALMA program and hence now have both CO transitions observed (Table 3), leaving 17 remaining that only have CO(1–0) data (Table 4). The published CO fluxes were converted to luminosities according to our adopted cosmological parameters. Line widths were reported as either W₂₀ or W₅₀, usually with no uncertainties specified. We homogenize the line widths adopting W₂₀/W₅₀ = 1.17 ± 0.19, the median ratio measured in our ALMA sample. Of the 15 quasars with both CO(1–0) and CO(2–1) observations, 8 are detected in both lines.¹⁴ These objects provide valuable insight on R₂₁ (Section 4.1), which is needed to convert the line luminosity from CO(2–1) to CO(1–0), as discussed in Section 3.1.

Table 3.  CO(2–1)/CO(1–0) Line Ratio

Object	$\mathrm{log}\,{L}_{\mathrm{CO}(1-0)}^{{\prime} }$	$\mathrm{log}\,{L}_{\mathrm{CO}(2-1)}^{{\prime} }$	R21	$\langle U\rangle$	Reference
	(${\rm{K}}\,\mathrm{km}\,{{\rm{s}}}^{-1}\,{\mathrm{pc}}^{2}$)	(${\rm{K}}\,\mathrm{km}\,{{\rm{s}}}^{-1}\,{\mathrm{pc}}^{2}$)
(1)	(2)	(3)	(4)	(5)	(6)
PG 0003+199	<8.87	7.05 ± 0.05	>0.02	15.16	1a
PG 0007+106	<9.07	8.45 ± 0.05	>0.24	7.95	2
PG 0050+124	9.74 ± 0.02	9.54 ± 0.01	0.63 ± 0.02	10.07	3b
PG 0934+013	<8.34	8.31 ± 0.03	>0.93	7.20	4
PG 1011−040	9.05 ± 0.04	8.83 ± 0.01	0.60 ± 0.05	5.10	4
PG 1119+120	8.50 ± 0.06	8.35 ± 0.02	0.71 ± 0.11	17.57	3
PG 1126−041	9.12 ± 0.04	8.85 ± 0.02	0.53 ± 0.05	15.36	4
PG 1211+143	<8.73	7.76 ± 0.03	>0.11	2.00	5
PG 1229+204	8.69 ± 0.11	8.38 ± 0.03	0.49 ± 0.13	4.02	5
PG 1310−108	<8.16	7.76 ± 0.02	>0.40	3.98	4
PG 1404+226	8.98 ± 0.11	8.76 ± 0.02	0.60 ± 0.15	2.39	5
PG 1426+015	9.12 ± 0.07	9.02 ± 0.02	0.79 ± 0.14	8.32	5
PG 1501+106	<9.24	7.31 ± 0.04	>0.01	25.48	1
PG 2130+099	8.85 ± 0.06	8.81 ± 0.01	0.90 ± 0.12	11.04	3
PG 2214+139	<8.55	7.84 ± 0.05	>0.19	2.55	5a

Notes. (1) Object name. (2) CO(1–0) line luminosity from the literature. (3) CO(2–1) line luminosity from our ALMA observations (see Table 2). (4) CO line luminosity ratio, ${R}_{21}\equiv {L}_{\mathrm{CO}(2-1)}^{{\prime} }/{L}_{\mathrm{CO}(1-0)}^{{\prime} }$. (5) Mean interstellar radiation field intensity derived from the IR spectral energy distribution of the quasar (Shangguan et al. 2018). (6) References: (1) Maiolino et al. (1997); (2) Evans et al. (2001); (3) Evans et al. (2006); (4) Bertram et al. (2007); (5) Scoville et al. (2003).

^a ${L}_{\mathrm{CO}(1-0)}^{{\prime} }$ is considered an upper limit; the archival measurement has a poor S/N. ^bThe line flux and FWHM of PG 0050+124 from Evans et al. (2006) are entirely consistent with those reported recently by Tan et al. (2019).

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Table 4.  Literature Sample

Object	z	$\mathrm{log}\,\lambda {L}_{\lambda }$(5100 Å)	$\mathrm{log}\,{M}_{\mathrm{BH}}$	$\mathrm{log}\,{M}_{* }$	q	Reference	${S}_{\mathrm{CO}}{\rm{\Delta }}\nu$	$\mathrm{log}\,{L}_{\mathrm{CO}}^{{\prime} }$	$\mathrm{log}\,{M}_{{\rm{H}}2}$	W50	W20	Reference	$\mathrm{log}\,{L}_{\mathrm{IR}}$	$\mathrm{log}\,{M}_{\mathrm{gas}}$
		$(\mathrm{erg}\,{{\rm{s}}}^{-1})$	$({M}_{\odot })$	$({M}_{\odot })$			$(\mathrm{Jy}\,\mathrm{km}\,{{\rm{s}}}^{-1})$	$({\rm{K}}\,\mathrm{km}\,{{\rm{s}}}^{-1}\,{\mathrm{pc}}^{2})$	$({M}_{\odot })$	$(\mathrm{km}\,{{\rm{s}}}^{-1})$	$(\mathrm{km}\,{{\rm{s}}}^{-1})$		$(\mathrm{erg}\,{{\rm{s}}}^{-1})$	$({M}_{\odot })$
(1)	(2)	(3)	(4)	(5)	(6)	(7)	(8)	(9)	(10)	(11)	(12)	(13)	(14)	(15)
PG 0052+251	0.155	45.00	8.99	11.05	0.55	2	2.0	9.39	9.88	429	500a	3	${44.51}_{-0.02}^{+0.02}$	10.29 ± 0.20
PG 0157+001	0.164	44.95	8.31	11.53	0.60	1	5.5 ± 0.5	9.88 ± 0.04	10.37 ± 0.30	270a	315	4	${45.85}_{-0.05}^{+0.03}$	10.80 ± 0.20
PG 0804+761	0.100	45.03	8.55	10.64	0.65	2	2.0 ± 0.5	9.00 ± 0.11	9.49 ± 0.32	755	881a	5	${43.83}_{-0.05}^{+0.07}$	8.79 ± 0.21
PG 0838+770	0.131	44.70	8.29	11.14	⋯	⋯	2.5 ± 0.4	9.34 ± 0.07	9.83 ± 0.31	60a	70	4	${44.72}_{-0.04}^{+0.03}$	10.21 ± 0.20
PG 0844+349	0.064	44.46	8.03	10.69	0.39	1	<1.5	<8.48	<8.97	⋯	⋯	5	${43.61}_{-0.02}^{+0.02}$	10.01 ± 0.21
PG 1202+281	0.165	44.57	8.74	10.86	0.92	2	<2.4	<9.53	<10.02	⋯	⋯	6	${44.53}_{-0.03}^{+0.03}$	9.51  ± 0.20
PG 1226+023	0.158	45.99	9.18	11.51	0.65	2	1.82 ± 0.02	9.37 ± 0.01	9.86 ± 0.30	490	572	8	${42.58}_{-0.57}^{+0.47}$	9.11  ± 0.57
PG 1309+355	0.184	44.98	8.48	11.22	⋯	1	<0.6	<9.02	<9.51	⋯	⋯	3	${44.41}_{-0.04}^{+0.04}$	10.40 ± 0.23
PG 1351+640	0.087	44.81	8.97	10.63	0.98	2	2.7 ± 0.5	9.01 ± 0.08	9.50 ± 0.31	260a	303	4	${44.78}_{-0.05}^{+0.04}$	9.67 ± 0.20
PG 1402+261	0.164	44.95	8.08	10.86	0.45	1	2.0	9.44	9.93	⋯	⋯	3	${45.01}_{-0.04}^{+0.04}$	9.93 ± 0.20
PG 1411+442	0.089	44.60	8.20	10.84	0.71	1	<1.8	<8.85	<9.34	⋯	⋯	5	${44.14}_{-0.03}^{+0.03}$	9.90 ± 0.21
PG 1415+451	0.114	44.53	8.14	>10.53	⋯	⋯	2.1 ± 0.3	9.14 ± 0.06	9.63 ± 0.31	90a	105	4	${44.40}_{-0.01}^{+0.02}$	9.73 ± 0.20
PG 1440+356	0.077	44.52	7.60	11.05	0.66	1	6.6 ± 0.6	9.29 ± 0.04	9.78 ± 0.30	310a	362	4	${44.77}_{-0.01}^{+0.02}$	9.95  ± 0.20
PG 1444+407	0.267	45.17	8.44	11.15	0.78	1	0.7	9.42	9.91	257	300a	3	${44.97}_{-0.05}^{+0.05}$	9.51 ± 0.24
PG 1545+210	0.266	45.40	9.47	11.15	⋯	1	<1.0	<9.57	<10.07	⋯	⋯	3	<44.03	<10.38
PG 1613+658	0.129	44.81	9.32	11.46	⋯	1	8.0 ± 0.6	9.83 ± 0.03	10.32 ± 0.30	400a	467	4	${45.39}_{-0.02}^{+0.02}$	10.56 ± 0.20
PG 1700+518	0.282	45.69	8.61	11.39	0.49	1	3.9 ± 0.7	10.22 ± 0.08	10.71 ± 0.31	260a	303	7	${45.81}_{-0.05}^{+0.02}$	10.64  ± 0.20

Notes. (1) Source name. (2) Redshift. (3) AGN monochromatic luminosity of the continuum at 5100 Å. (4) BH mass. (5) Stellar mass of the host galaxy. The uncertainty of the direct measurements is ∼0.3 dex. The lower limits come from bulge masses estimated from the BH mass using the ${M}_{\mathrm{BH}}\mbox{--}{M}_{\mathrm{bulge}}$ relation. See Table 1 of Shangguan et al. (2018) for more details. (6) Axial ratio, derived from GALFIT modeling of the quasar host galaxies. (7) References of the axial ratio. (8) Integrated line flux of CO(1–0) emission. (9) CO(1–0) line luminosity. (10) Molecular gas mass derived from CO line luminosity, assuming ${\alpha }_{\mathrm{CO}}=3.1\,{M}_{\odot }\,{({\rm{K}}\mathrm{km}{{\rm{s}}}^{-1}{\mathrm{pc}}^{2})}^{-1}$. (11) Width of the CO integrated profile at 50% of its maximum. (12) Width of the CO integrated profile at 20% of its maximum. The flagged line widths were originally provided in the literature; other values are converted assuming ${W}_{20}/{W}_{50}=1.17$. (13) References for the CO(1–0) measurements. (14) IR luminosity of the host galaxy from spectral energy distribution decomposition by Shangguan et al. (2018). (15) Total gas mass derived from the dust mass. Columns (2)–(5) and (15) are collected from Table 1 of Shangguan et al. (2018). References: (1) Kim et al. (2017); (2) Y. Zhao et al. (2020, in preparation); (3) Casoli & Loinard (2001); (4) Evans et al. (2006); (5) Scoville et al. (2003); (6) Evans et al. (2001); (7) Evans et al. (2009); (8) Husemann et al. (2019).

^aThe line width was originally provided in the literature.

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As listed in Table 3, the line ratios of the eight quasars in our study with both lines detected span ${R}_{21}=0.49\mbox{--}0.90$. The ${50}_{-25}^{+25}\mathrm{th}$ percentile value, calculated with the Kaplan–Meier product-limit estimator kmestimate in IRAF.ASURV (Feigelson & Nelson 1985; Lavalley et al. 1992), is ${R}_{21}={0.62}_{-0.07}^{+0.15}$. If the CO emission is thermalized and optically thick, the intrinsic brightness temperature and the luminosity of the line are independent of J and the rest frequency, and R₂₁ = 1. Indeed, a value of R₂₁ ≈ 1 is observed in the inner parts of spiral galaxies (Braine & Combes 1992), local luminous IR galaxies (Papadopoulos et al. 2012), and high-redshift galaxies (Carilli & Walter 2013; Daddi et al. 2015). However, recent studies find lower values of R₂₁ on the global scales of nearby galactic disks (R₂₁ ≲ 0.8; Leroy et al. 2013; Rosolowsky et al. 2015; Saintonge et al. 2017). Ocaña Flaquer et al. (2010) report R₂₁ ≈ 0.6 for nearby radio galaxies, and some low-redshift quasars can reach ${R}_{21}\approx 0.5$ (Husemann et al. 2017), while in IR-luminous quasars, R₂₁ ≈ 0.4–1.2, with a mean value of ∼0.8 (Xia et al. 2012). Therefore, our low-redshift quasars exhibit R₂₁ values fully consistent with those derived from global measurements of nearby inactive and active galaxies. In contrast, high-redshift quasars show low-J CO line ratios suggestive of optically thick, thermally excited emission, indicating that the molecular gas emission comes from a compact region in the centers of the host galaxies (Carilli & Walter 2013).

The relative spatial coverage of CO(1–0) and CO(2–1) introduces additional uncertainties into the interpretation of R₂₁, especially for single-beam observations of nearby galaxies when both lines are observed with the same telescope¹⁵ or interferometer configuration. To alleviate such complications, it is customary to scale down the flux of CO(1–0) to match that of CO(2–1), often limiting the measurement of the line ratio to the central part of the galaxy. For example, Husemann et al. (2017) use Hα emission to estimate the spatial distribution of CO(1–0) and scale down the flux of CO(1–0) to match that of CO(2–1). Saintonge et al. (2017), in contrast, observe CO(2–1) using the Atacama Pathfinder Experiment (APEX) 12 m telescope, whose 230 GHz beam of 27'' better matches the 22'' beam of the IRAM 30 m telescope for CO(1–0). Fortunately, the relatively large distances of our sources obviate these complications. At $z\gtrsim 0.05$, the CO(1–0) emission of our objects should be mostly captured by the beam of the IRAM 30 m telescope (Evans et al. 2006; Bertram et al. 2007), while all of the CO(2–1) emission should be contained within the maximum recoverable scale (∼29'' at 230 GHz) of ACA, which is confirmed by our 15'' tapered measurements (Section 2). Meanwhile, CO(1–0) emission may still be underestimated to some extent, when the emission size is comparable to the beam size but its spatial distribution is unknown from the single-dish observation. This may also contribute to the uncertainty of R₂₁.

The low values of R₂₁ for our quasars indicate that the molecular gas is optically thick but either is subthermally excited (Ocaña Flaquer et al. 2010; Husemann et al. 2017) or has a low temperature (≲10 K; Braine & Combes 1992). Motivated by Daddi et al. (2015), who found a significant sublinear correlation between the mean intensity of the interstellar radiation field of the galaxy ($\langle U\rangle$; Draine & Li 2007) and the CO(5–4)/CO(2–1) ratio, we checked but failed to find a clear correlation between R₂₁ and $\langle U\rangle$ for the quasar host galaxies (Figure 3(a)). Unfortunately, the number of objects with statistically meaningful measurements is too small to perform a formal statistical test. $\langle U\rangle$ comes from the study of IR spectral energy distributions of PG quasars (Shangguan et al. 2018). The dust temperatures of the quasar host galaxies, however, are ≳20 K (Shangguan et al. 2018), which are not entirely consistent with the molecular gas temperature of ≲10 K expected from R₂₁ ≈ 0.6 if the gas is thermally excited (see Figure 1 of Braine & Combes 1992). Although a detailed discussion is beyond the scope of this paper, we note that the continuum emission of almost all of the quasars are unresolved with our ACA measurements, while CO(2–1) of nearly half of the quasars is resolved.¹⁶ This suggests that the dust emission from far-IR to submillimeter is predominantly powered by an AGN or nuclear starburst. We do not know whether the quasar affects the excitation of the low-J CO lines, as R₂₁ seems unrelated to the AGN luminosity (Figure 3(b)). We conclude that, similar as low-z galaxies and AGNs, the low-J CO emission of our quasars is subthermally excited.

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The CO line luminosity correlates strongly with the IR luminosity of galaxies, both active and inactive, at low and high redshifts (e.g., Sanders & Mirabel 1985; Solomon & Vanden Bout 2005; Genzel et al. 2010; Saintonge et al. 2011; Xia et al. 2012; Carilli & Walter 2013). Sensitive to the Lyman continuum emission absorbed and reprocessed by dust (Kennicutt 1998b), the IR luminosity provides an excellent tracer of the star formation rate in star-forming galaxies. Therefore, the ratio of L_IR to ${L}_{\mathrm{CO}(1-0)}^{{\prime} }$ reflects the global star formation efficiency of the molecular gas. In quasar host galaxies, however, emission from hot dust, heated by BH accretion, may dominate the IR luminosity and contribute a significant fraction of the emission, even up to ∼100 μm (Lani et al. 2017; Lyu & Rieke 2017; Zhuang et al. 2018). We calculate the 8–1000 μm IR luminosity from the cold dust emission decomposed from the integrated spectral energy distributions of Shangguan et al. (2018).¹⁷

Figure 4 compares the L_IR–${L}_{\mathrm{CO}(1-0)}^{{\prime} }$ relation of PG quasars with those of star-forming galaxies and starburst systems triggered by galaxy mergers (Genzel et al. 2010). Starburst galaxies are typically ≳0.4 dex above the so-called main sequence of the star-forming galaxies (e.g., Elbaz et al. 2018; Shangguan et al. 2019). As with other types of galaxies, the host galaxies of quasars clearly also exhibit a strong correlation. We fit the relation of the PG quasars with Linmix (Kelly 2007),¹⁸ accounting for the upper limits in ${L}_{\mathrm{CO}(1-0)}^{{\prime} }$. The best fit,

$$\begin{eqnarray}&&\mathrm{log}\,{L}_{\mathrm{CO}(1-0)}^{{\prime} }=0.94\left({}_{-0.08}^{+0.08}\right)\mathrm{log}\,{L}_{\mathrm{IR}}-32.90\left({}_{-3.40}^{+3.35}\right),\end{eqnarray} \tag{ 4 }$$

is consistent with a linear relation between L_IR and ${L}_{\mathrm{CO}(1-0)}^{{\prime} }$. Both the slope and the zero-point are consistent with the relation for starburst galaxies. The total scatter of the relation (∼0.3 dex) is dominated by an intrinsic scatter of ${0.29}_{-0.04}^{+0.05}$ dex. PG 1226+023 is excluded from the fit because the IR luminosity of its host galaxy is very uncertain (see Table 4 and Shangguan et al. 2018), but the fit results do not depend on this choice. We searched for, but failed to find, a statistically significant partial correlation of the L_IR–${L}_{\mathrm{CO}(1-0)}^{{\prime} }$ relation with any plausible third variable (e.g., the AGN luminosity).

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To the best of our knowledge, there has never been a formal study of the CO-to-H₂ conversion factor (${\alpha }_{\mathrm{CO}}$) of galaxies hosting AGNs powerful enough to qualify as quasars. Here, we use our new CO measurements, in combination with previous total gas measurements estimated from dust content (Shangguan et al. 2018), to put a rough constraint on ${\alpha }_{\mathrm{CO}}$ in quasar host galaxies. As a starting point, we adopt ${\alpha }_{\mathrm{CO}}$ = 3.1 ${M}_{\odot }\,{({\rm{K}}\mathrm{km}{{\rm{s}}}^{-1}{\mathrm{pc}}^{2})}^{-1}$ with 0.3 dex uncertainty, as recommended by Sandstrom et al. (2013), who, as in Leroy et al. (2011), simultaneously solved for ${\alpha }_{\mathrm{CO}}$ and the gas-to-dust ratio for 26 nearby star-forming galaxies for the first time beyond the Local Group. This value of ${\alpha }_{\mathrm{CO}}$ is slightly lower than, but consistent with, the canonical Milky Way value of 4.3 ${M}_{\odot }\,{({\rm{K}}\mathrm{km}{{\rm{s}}}^{-1}{\mathrm{pc}}^{2})}^{-1}$ (Bolatto et al. 2013), and it does not appear to depend strongly on metallicity for galaxies with metallicities similar to and above that of the Milky Way. While nuclear activity potentially can affect the molecular gas of the host (e.g., Krips et al. 2008), there is no clear evidence that the presence of an AGN influences ${\alpha }_{\mathrm{CO}}$ (Sandstrom et al. 2013), even when AGN feedback is in principle powerful enough to be effective (Rosario et al. 2018).

Figure 5 plots the variation of the molecular gas fraction (${M}_{{{\rm{H}}}_{2}}/{M}_{\mathrm{gas}}$) as a function of stellar mass (M_⋆), where M_gas is the total mass of the cold interstellar medium (${M}_{{\rm{H}}{\rm{I}}}+{M}_{{{\rm{H}}}_{2}}$) inferred from the dust mass, as described in Shangguan et al. (2018). The host galaxies of PG quasars,¹⁹ accounting for the censored data, have a 50 ± 25 percentile molecular gas fraction of 40% ± 24% and a stellar mass of ${10}^{10.89\pm 0.22}\,{M}_{\odot }$.²⁰ The quasars gathered from the literature on average have a higher molecular gas fraction than those newly observed using ALMA. This is an obvious observational selection effect. If we limit ourselves to the unbiased ALMA sample, the 50 ± 25 percentile molecular gas fraction becomes 32% ± 18% for a stellar mass of 10^10.86±0.19 M_⊙. The molecular gas fraction of the quasars are in rough agreement with, but slightly more elevated than, that of inactive galaxies of the similar stellar mass (Catinella et al. 2018; the blue line in Figure 5).²¹ This is not unexpected. AGNs, in general, and quasars, in particular, reside preferentially in bulge-dominated galaxies (Ho et al. 1997; Ho 2008; Kim et al. 2017; Zhao et al. 2019), and bulge-dominated systems tend to have higher molecular gas fractions (Catinella et al. 2018). In other words, at any given stellar mass, AGN hosts, by virtue of their earlier type morphologies, should have higher molecular gas fractions, as observed.

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The above analysis, while far from a rigorous derivation, does suggest that the host galaxies of low-redshift quasars have an ${\alpha }_{\mathrm{CO}}$ value not too dissimilar from that of ordinary star-forming galaxies and lower luminosity AGNs. We do not believe that the CO-to-H₂ conversion factor of PG quasars can be as low as ${\alpha }_{\mathrm{CO}}$ = 0.8 ${M}_{\odot }\,{({\rm{K}}\mathrm{km}{{\rm{s}}}^{-1}{\mathrm{pc}}^{2})}^{-1}$, a value commonly advocated for ultraluminous IR galaxies (ULIRGs; Downes & Solomon 1998). Such a low value of ${\alpha }_{\mathrm{CO}}$ would result in molecular gas mass fractions substantially lower than those of star-forming galaxies (the orange pentagon in Figure 5). This seems improbable. As shown in Section 4.2, quasar host galaxies follow nearly the same L_IR–${L}_{\mathrm{CO}(1-0)}^{{\prime} }$ relation as starburst galaxies, suggesting that they have similarly high star formation efficiencies.