Spectral Classification and Ionized Gas Outflows in z ∼ 2 WISE-selected Hot Dust-obscured Galaxies

Our targets, listed in Table 1, were selected to probe the rest-frame UV-to-optical spectral properties of z ∼ 2 Hot DOGs, complementing our observation programs using the Keck and Gemini telescopes to investigate BH accretion using the Hα line (Wu et al. 2018). The XSHOOTER sample consists of 10 galaxies, some with existing redshifts at the time of writing the observation proposal (6/10). The remainder (4/10) were targeted with the added goal of securing a redshift. The color selection for heavily obscured, W1W2-dropout sources naturally prefers the rise in the near-to-mid-IR SED by dust-heated emission from the AGN located in the W3 band or longer wavelengths, corresponding to z ≳ 2 (see also Ricci et al. 2017a for biases against z ≲ 1).

To illustrate where our targets lie in obscuration and luminosity with respect to other populations of luminous, obscured AGNs, Figure 1 plots the AGN extinction derived from photometric SED fitting against the AGN luminosity.¹¹ The XSHOOTER target samples the E(B−V)–L_bol space, similar to the optical spectroscopic "full sample W1W2 dropouts" in Assef et al. (2015), spanning the highest luminosity and extinction values. Hot DOGs have E(B−V) values of a few to tens of mag and are up to an order of magnitude more obscured than nearly as luminous, heavily reddened type 1 quasars (Banerji et al. 2012, 2013, 2015; Temple et al. 2019), red type 1 quasars (Glikman et al. 2012), or extremely red type 1/2 quasars (Ross et al. 2015) with E(B−V) ≲ 1.5 mag. Estimating E(B−V) values for obscured AGNs can be quite sensitive to the method. Our values are derived from the photometric SED fitting as described in Assef et al. (2015). When compared to E(B−V) values measured by comparing modeled-to-observed WISE W2 flux ratios, the latter E(B−V) values are smaller by about a factor of 2 but do not change the main results of this work.

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The XSHOOTER (Vernet et al. 2011) instrument on the VLT was used to collect simultaneous 0.3–2.5 μm spectra of the targets, cross-dispersed along its three arms (UVB, VIS, and NIR). Due to the wide spectral coverage, XSHOOTER is capable of extracting multiple emission lines across the observed spectrum and obtaining a secure redshift. Slit widths of 1'', 12, and 12 were adopted for the UVB, VIS, and NIR arms, yielding spectral resolutions of R = 4290, 3360, and 3900, respectively. The data were taken under program 095.B-0507 in 2015 April through September. Each target was observed for either five or 10 frames (449 s in UVB, 600 s in VIS, 514 s in NIR per frame), depending on its H-band magnitude (Assef et al. 2015). We used an ABBA nodding mode. Calibration stars were observed for each target, as well as arcs for spatial and wavelength calibrations. Typical airmasses during the observations were 1–1.2, with seeing around 1'' and precipitable water vapor of 2–3 mm. The observations are summarized in Table 1. In addition to the XSHOOTER spectra, we include previously published Keck/MOSFIRE H- and K-band spectra (R ∼ 3600) for WISE J113634.29+423602.8 and Gemini/FLAMINGOS-2 JH-band (R ∼ 1000) spectra for WISE J221609.09+072353.3 from Wu et al. (2018).

Supplementing the spectra, we compiled rest-frame optical–to–mid-IR photometry-based measurements of L_bol from Assef et al. (2015), estimated based on the best-fit AGN template after removing host contamination and correcting for obscuration. For unobscured AGNs, the monochromatic 5100 Å bolometric correction has been widely used to estimate bolometric luminosities from a limited number of photometric bands (e.g., Richards et al. 2006). In the case of extremely obscured AGNs such as Hot DOGs, one could either (i) make a reddening correction from the photometric SED and apply a monochromatic bolometric correction (e.g., 5100 Å in Assef et al. 2015), (ii) directly integrate along the observed wavelengths without a bolometric correction (e.g., rest-frame UV to far-IR in Tsai et al. 2015), or (iii) make a compromised integration of the best-fit model (e.g., Z16). The first option provides a simple bolometric correction without requiring continuous coverage of the SED, while the second two options allow estimates based on the simple shape of the SED dominated by the mid-IR AGN torus emission: we adopt the Assef et al. (2015) formalism (option (i)) for a fair comparison to other studies.

The ESO Science Archive Facility provides phase-3 data products reduced with the Reflex pipeline (Freudling et al. 2013). The reduction steps include bias, dark, and flat corrections; wavelength and spatial scale calibrations; flux calibration; combination of frames with cosmic-ray rejection; and trace and 1D extraction. In addition to these standard steps, we performed a telluric absorption correction using Molecfit (Kausch et al. 2015; Smette et al. 2015). Using the standard stars selected close to the target in altitude and azimuth and observed before or after the science target with the same instrument configuration, we obtained the telluric correction. Visual inspection of the pipeline-processed spectra showed some strongly outlying pixel values outside the telluric bands. We removed these using an absolute signal-to-noise ratio (S/N) threshold of 30–300 times the median S/N, depending on target, but with an uncertainty still less than five times the median noise. These cuts removed 0.15% of the data, on average, while keeping the emission line profiles unchanged.

Our data, with five or 10 frames, suffer from an insufficient number of frames to fill multiples of four, the number used for a single nodding sequence by the standard ABBA nodding mode reduction pipeline. We therefore use the phase-3 data, which only keep 4/5 or 8/10 frames due to the pipeline limitations. Assuming that object photon noise follows a Poisson distribution, this is an 11% hit on the S/N. When we used eight frames, or twice the nodding sequence, we averaged the flux and propagated the errors from each sequence. When compiling the reduced data coming from adjacent arms, overlapping data near the dichroics (0.560 and 1.024 μm) were trimmed to where both sides of the arms have comparable S/Ns by constraining λ_UBV < 0.565 μm, 0.555 μm < λ_VIS < 1.03 μm, and 1.02 μm < λ_NIR < 2.40 μm. Lastly, we applied Galactic extinction corrections assuming R_V = 3.1 and E(B−V) values from the Schlafly & Finkbeiner (2011) extinction map.

In Figure 2, we plot the reduced, rest-frame 800–7000 Å spectra with available photometric fluxes and 2σ upper limits from Assef et al. (2015). Apart from the two spectra from Wu et al. (2018) missing the rest-UV wavelengths, the spectra cover major UV–optical emission lines. The average and 1σ scatter of the continuum S/N¹² are 0.9 ± 1.1. This is not a critical issue, as we focus on the brighter lines, and we apply S/N cuts when analyzing the lines in the forthcoming sections. We binned the spectra matched to the spectral resolution of each arm for analysis.

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The first goal for the XSHOOTER observations was to identify undetermined redshifts and confirm the redshifts of some sources with existing but uncertain optical spectroscopy (P. R. M. Eisenhardt et al. 2019, in preparation). We define quality "A" and "B" redshifts as those derived from sources with two or more lines with S/N ≥ 5 and ≥2, respectively. The systemic redshift is determined by the peak of the model fit around the prominent emission lines (from the common narrow component center of the Hα, Hβ, and [O ii] lines and that of Mg ii and [Ne v]), which should suffer less from systematic blueshifts (C iv) or blending issues (Lyα, N v, C iii], Hγ). We combined the UV and optical redshifts from S/N ≥ 3 detections and performed error-weighted averaging to calculate the systemic redshifts and uncertainty values. Comparing the systemic (redshift with median uncertainty of 0.00047) to the individual redshifts from the total profile of the stronger lines, we find that the latter lies mostly within three times its uncertainty to the systemic from the [O ii] line with S/N ≥ 3 (8/9) but not from the Hβ (3/8), [O iii] (1/11), and Hα (3/8) lines, where they often show clear blueshifts (Sections 4.4 and 4.5).

Out of the 10 XSHOOTER spectra, we determine two new redshifts (quality "A" for W0114–0812 and quality "B" for W2016–0041) and study eight with existing values (P. R. M. Eisenhardt et al., in preparation; including two redshifts added in Tsai et al. 2015). Six of these match within 1% to existing values (all quality "A" redshifts). However, two differ drastically: quality "A" z = 0.8301 for W0126–0529, previously z = 2.937, and quality "B" z = 2.958 for W2042–3245¹⁴ , previously z = 3.963. The two mismatches occur for redshifts previously determined from Gemini/GMOS observed-frame 4000–6500 Å spectra (Tsai et al. 2015). Both were quality "B" redshifts, derived by the detection but uncertain identification of a single feature. The broadband VLT spectra emphasize the importance of multiple line detections for confident redshift determinations, although our quality "B" redshifts could still be biased by noise, and we remove those sources from further analysis. As a side note, the two redshifts (W1136+4236 and W2216+0723) from Wu et al. (2018) were not flagged in their work but are both quality "A" by our definition, showing a clear detection of multiple lines.

We can obtain spectroscopic classifications of our targets, such as the Seyfert type¹⁵ (e.g., Osterbrock 1981), Baldwin, Phillips & Terlevich (BPT) diagram (e.g., Baldwin et al. 1981), [O ii]–[O iii] line ratio (e.g., Ferland & Osterbrock 1986), and Hα-to-Hβ ratio (e.g., Osterbrock 1989), to better differentiate between the AGN and star-forming properties of Hot DOGs. According to the narrow-to-broad-line flux ratios for Balmer lines (assuming the broad lines come from the broad-line region, not from outflows), all targets with sufficient emission line S/N (≥5) in Hα (N = 8) or Hβ (N = 4) show a clear narrow component with no pure type 1 (broad-line-dominated) classification. We calculate the Seyfert types using the total [O iii]/Hβ ratio and detectability of broad Balmer lines (Winkler 1992) using S/N ≥ 3 detections (and 3σ upper limits) for the [O iii]/Hβ ratio and S/N ≥ 5 detections for the detectability of broad Balmer lines. The Seyfert types shown in Table 6 range from type 1.5 (intermediate Hβ to [O iii], 2/8) to type 1.8 and higher (weak Hβ to [O iii], 6/8), apart from the undetermined type for W0126–0529 (which turns out not to be an AGN; Table 7) and a loosely determined type for W1719+0446 (type ≥1.2). Our type 1 fraction (2/8 counting up to type 1.5, but 3–5/8 when including type 1.8–1.9 sources as type 1), together with a majority of single broad Balmer line FWHM values of ≲3000 km s⁻¹ in Table 3, are marginally different from other red quasar spectra in the literature, which show type 1 fractions of ≳50%–57% (e.g., Glikman et al. 2012; Banerji et al. 2015; Ross et al. 2015) with a single broad-line FWHM  ≳ 3000 km s⁻¹. This difference likely arises because our targets have higher extinction (Section 2.1) and hence are obscuring more of the broad-line region.

Using the narrow-line (FWHM < 1200 km s⁻¹) ratios with S/N ≥ 3 detections or 3σ upper limits, we plot the [N ii], [S ii], and [O i] BPT diagrams in Figure 4. There are eight values in each panel and two lower limits on [O iii]/Hβ outside the plots based on the absence of [N ii]/Hα (W1719+0446, W2026+0716). For E(B−V) ≲ 1 (e.g., Table 8) and a Milky way extinction curve, the [N ii]/Hα, [S ii]/Hα, [O i]/Hα, and [O iii]/Hβ change by less than 0.004, 0.03, −0.06, and 0.06 dex if corrected for extinction, which is within the measurement uncertainties. We list each and the combined classifications based on the most overlaps in Table 7. Combined, seven are BPT AGNs, two are unconstrained (W0147–0923 and W1719+0446), and the remaining object is star-forming (W0126–0529). The latter source, the lowest-redshift galaxy in our sample, is not considered in the following when deducing and discussing the AGN properties of our targets. Considering the [S ii] λ6716, 6731 and [O i] λ6300 BPT diagrams, the remaining targets always favor the so-called Seyfert region of these diagrams over the LINER region, implying that Hot DOGs have a strong ionization source from the AGN.

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We independently check for signs of star formation activity in our sample using the [O ii]/[O iii] line ratio (e.g., Ho 2005; Kim et al. 2006), as the [O ii] emission usually originates from star formation rather than the AGN activity, and conversely for [O iii]. The [O ii]/[O iii] ratio values based on S/N ≥ 3 detections are listed in Table 8. Indeed, out of eight objects with both BPT classification and [O ii]/[O iii] ratio values, three that are classified exclusively as AGNs in all three BPT diagrams have [O ii]/[O iii] values of 0.1–0.4, and five with at least one potential H ii BPT classification have [O ii]/[O iii] values of 0.2–2.6. This is consistent with pure AGNs on the BPT diagrams showing stronger [O iii] to those with potentially mixed star formation with stronger [O ii]. The diverse range of [O ii]/[O iii] ratios is in line with a combination of AGN and star-forming galaxy templates explaining the photometric SEDs of Hot DOGs (e.g., Assef et al. 2015), but we caution against any conclusive interpretation of the AGN/star-forming nature of the oxygen lines, as the [O ii] line could also originate from the AGN activity (e.g., Yan et al. 2006; Maddox 2018; Section 4.5). Overall, the majority (7/8) of Hot DOGs are best explained as non-LINER AGNs on the BPT diagrams, and AGN activity dominates star formation in both the BPT and [O ii]/[O iii] diagnostics.

We next investigate the level of obscuration within the narrow-line region using the Balmer decrements (narrow Hα/Hβ) listed in Table 8. Observationally, Kim et al. (2006) showed that type 1 AGNs generally have Balmer decrements close to the expected value, with a distribution sharply peaked at Hα/Hβ = 3.3, while Zakamska et al. (2003) showed that type 2 AGNs have higher values, with an average of 4.1 measured from their composite spectrum. Most (7/8) of our sources show Balmer decrements ≳4, indicating large amounts of dust obscuration. The translated E(B−V) values assuming the intrinsic Balmer decrements of 3.1 expected for AGN narrow-line regions (e.g., Osterbrock 1989) lie mostly around 0.3–0.7 mag, which is not only an order of magnitude higher than the type 1 AGN values (0.06 mag) in Kim et al. (2006) but also up to several times higher than that of the type 2 quasars at various redshifts (∼0.3 mag; Zakamska et al. 2003; Greene et al. 2014). Interestingly, the E(B−V) values from the broadband SED fitting (3–20 mag; Table 1) are another order of magnitude larger than the values inferred from the Balmer decrements (Table 8). This has already been seen (Zakamska et al. 2003, 2005; Greene et al. 2014) from similar measurements of candidate type 2 quasars, altogether suggesting a dense, stratified distribution of dust between the compact accretion disk and the extended narrow-line region in quasars. This is also consistent with significant amounts of scattered continuum light from unobscured lines of sight in some Hot DOGs or extremely red quasars (e.g., Assef et al. 2016; Hamann et al. 2017).

Hot DOGs, therefore, seem to require not only a dense source of extinction interior to the narrow-line region but also an extended distribution of gas or dust outside it. The overall dust temperature of Hot DOGs (50–120 K; e.g., Wu et al. 2012; Bridge et al. 2013) is marginally higher than that of starburst galaxies, implying that the dust may be associated with the AGN activity. One possibility is to consider host galaxy dust (e.g., Rigby et al. 2006; Polletta et al. 2008). As favored by merger-driven quasar fueling models and observations of Hot DOGs (e.g., Hopkins et al. 2008; Fan et al. 2016; Farrah et al. 2017), the obscured starburst galaxy will receive feedback from the quasar activity and become unobscured (see also Buchner & Bauer 2017; Gobat et al. 2018 for discussion of how the gas content in z ∼ 2 galaxies is higher than in local galaxies). We investigate whether the kiloparsec-scale obscuration is responsible for the reddening by estimating the spatial extent of the narrow-line region. We use the narrow-line region size–luminosity (R_NLR–L_{[O iii]}) relation at extinction-uncorrected ${L}_{[{\rm{O}}{\rm{iii}}]}\sim {10}^{43.5-45}\,\mathrm{erg}\,{{\rm{s}}}^{-1}$. This reaches the upper limit of ∼10 kpc extrapolated from lower luminosities (e.g., Husemann et al. 2013; Hainline et al. 2014; Liu et al. 2014; Storchi-Bergmann et al. 2018), comparable to or even larger than the size of the entire host galaxy. Thus, the extinction measured from the narrow-line region is more likely coming from the host galaxy, rather than from the central AGN.

Independent of the Balmer decrement–based E(B−V) estimates, we list the extinction-uncorrected, spectroscopically derived [O iii] rest-frame equivalent width (EW) values in Table 8. We compare the EW${}_{[{\rm{O}}{\rm{iii}}]}$ values for Hot DOGs to type 1 and 2 quasars at matched luminosities (${L}_{[{\rm{O}}{\rm{iii}}]}\gtrsim {10}^{43.5}\,\mathrm{erg}\,{{\rm{s}}}^{-1}$ from Greene et al. 2014; L₅₁₀₀ ≳ 10^46.5 erg s⁻¹ from Shen 2016). It appears that there is a large discrepancy in the observed values; the type 1 EW${}_{[{\rm{O}}{\rm{iii}}]}$ values are ∼100 Å in Greene et al. (2014), as opposed to ∼10 Å in Shen (2016). However, Shen (2016) noted the anticorrelation between EW${}_{[{\rm{O}}{\rm{iii}}]}$ and L₅₁₀₀ ([O iii] Baldwin effect; e.g., Baldwin 1977; Brotherton 1996), and as the data points in Greene et al. (2014) are from L₅₁₀₀ ≲ 10⁴⁶ erg s⁻¹ quasars from Shen et al. (2011), we use the value of 10 Å in Shen (2016) as a reference. Hot DOGs with a non-H ii BPT classification have EW${}_{[{\rm{O}}{\rm{iii}}]}$ values around 10–400 Å (median = 44 Å), with a corresponding E(B−V) between the lines of sight through the accretion disk and the [O iii] region of 0.03–1.14 mag (median 0.46 mag), assuming a Milky Way extinction curve. Note that we expect the intrinsic EW${}_{[{\rm{O}}{\rm{iii}}]}$ values (and thus the inferred extinction values) to be higher than measured, since these heavily obscured AGNs are expected to have nonnegligible host galaxy contributions to their continuum emission.

In Figure 5, we plot model profiles of the strongest lines,[O ii], Hβ/[O iii], and Hα, with their broad and narrow components highlighted. It is evident that most of the sample displays broadened or blueshifted [O iii] indicative of ionized gas outflows, often modeled by biconical motions where the blue wing is less affected by obscuration than the red, leading to an asymmetric line profile (e.g., Crenshaw et al. 2010; Zakamska & Greene 2014; Bae & Woo 2016). To quantify the presence of outflows as clear nongravitational motion to our line of sight, we first require the presence of a broad [O iii] component¹⁶ with ${\sigma }_{[{\rm{O}}{\rm{iii}}],\mathrm{broad}}\gt 400\,\mathrm{km}\,{{\rm{s}}}^{-1},$ to distinguish potential broadening by even the most massive galaxy potential. We also place an S/N ≥ 5 requirement on the [O iii] line to enable robust separation of the broad component from the narrow component (as we did in Section 4.2), since line modeling is susceptible to noise at low S/N. The fraction of [O iii] outflows is 8/9.

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It is well known that outflows traced by [O iii] line widths of ${\sigma }_{[{\rm{O}}{\rm{iii}}],\mathrm{broad}}\gt 400\,\mathrm{km}\,{{\rm{s}}}^{-1}$ are prevalent in average luminosity, type 1/2 quasars from the Sloan Digital Sky Survey (SDSS; e.g., Mullaney et al. 2013; Woo et al. 2016; Rakshit & Woo 2018), but we test whether the [O iii] kinematics of Hot DOGs indicate higher ${\sigma }_{[{\rm{O}}{\rm{iii}}],\mathrm{broad}}$ values at their high luminosities. In Figure 6, we plot the broad [O iii] line width, shift to the systemic redshift as a function of the observed (extinction-uncorrected) total [O iii] luminosity for our targets, and compare them to other quasars in the literature. Among ${L}_{[{\rm{O}}{\rm{iii}}]}\gt {10}^{42}$ erg s⁻¹ quasars in the figure, 74%–79% have ${\sigma }_{[{\rm{O}}{\rm{iii}}],\mathrm{broad}}\gt 400\,\mathrm{km}\,{{\rm{s}}}^{-1}$, consistent with the majority of local AGNs showing outflows irrespective of type at ${L}_{[{\rm{O}}{\rm{iii}}]}\gt {10}^{42}$ erg s⁻¹ (e.g., Woo et al. 2016; Rakshit & Woo 2018). However, the eight Hot DOGs with [O iii] outflows not only have ${\sigma }_{[{\rm{O}}{\rm{iii}}],\mathrm{broad}}$ = 700–2200 km s⁻¹ (median 1100 km s⁻¹), but also ${\sigma }_{[{\rm{O}}{\rm{iii}}],\mathrm{total}}$ = 600–2000 km s⁻¹ (median 1100 km s⁻¹), and line shifts of ${\rm{\Delta }}{v}_{[{\rm{O}}{\rm{iii}}],\mathrm{broad}}$ = −2000–100 km s⁻¹ (median −1100 km s⁻¹). These are much larger than the observed limits, ${\sigma }_{[{\rm{O}}{\rm{iii}}]}$ = 500–600 and Δv_{[O iii]} = −(500–600) km s⁻¹, irrespective of whether we use the broad or total [O iii] profile from L_{[O iii]} < 10⁴³ erg s⁻¹ AGNs in Figure 6 or from the literature (e.g., Woo et al. 2016; Rakshit & Woo 2018).

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In Figure 6, we further compare [O iii] outflow kinematics at high luminosity among heavily obscured (U12; B15; Z16 ) type 2 quasars (V11; Harrison et al. 2014; Harrison et al. 2016; hereafter H14; H16) and unobscured quasars (H16; Shen 2016; hereafter S16; Jun et al. 2017; hereafter J17; Vietri et al. 2018 including the data from Bischetti et al. 2017; hereafter V18). It is noticeable that Hot DOGs, together with reddened quasars from Z16, show comparable or even stronger [O iii] broadening or blueshift than type 1 quasars at the highest luminosity. If a simple, smooth, and toroidal geometry of the obscuring structure aligned with the broad-line region were to explain most of the obscuration (e.g., Antonucci 1993; Urry & Padovani 1995), we would expect Hot DOGs, with their high levels of reddening, to appear as type 2 AGNs seen very close to edge-on with a high line-of-sight column density. Indeed, most of the measured Seyfert types are close to type 2 (≳1.8; Table 6), favoring an edge-on orientation of the AGN structure if the obscuration is well explained by geometry. Assuming the biconical outflows are aligned perpendicular to the dusty torus (e.g., Fischer et al. 2014; Marin 2016), the edge-on geometry implies a low line-of-sight velocity of the outflowing material, minimizing the observed Doppler shift and broadening of the (blueshifted/redshifted cone and thus the total) [O iii] profile (e.g., Bae & Woo 2016). The observed [O iii] kinematics in Figure 6 do not follow this expectation, however, but are more consistent with intrinsically higher extinction, expected to produce large blueshifts (e.g., Bae & Woo 2016). Thus, we require a different source of obscuration beyond the simple torus (e.g., Hönig et al. 2013; Asmus et al. 2016 for polar dust geometry and, e.g., Buchner & Bauer 2017 for extended obscuration within the host galaxy) or a complex velocity structure within the outflowing [O iii] region (e.g., outflows with a large bicone opening angle or the spherical outflows seen in [C ii] observations of the Hot DOG W2246–0526 from Díaz-Santos et al. 2016). Although there is a large scatter in the data, we find that the outflow kinematics in Figure 6 show stronger broadening and blueshift as a function of luminosity or reddening, whereas the differences between type 1 and 2 AGN at a given luminosity are relatively minor. This suggests that the obscured quasar phase is related to the production of strong ionized gas outflows, irrespective of possible inclination effects.

Using the outflow kinematics of the broad [O iii] profile under the assumption of a uniform, filled spherical/biconical outflow geometry (e.g., Maiolino et al. 2012), we estimate the outflow quantities for Hot DOGs—mass outflow rate (${\dot{M}}_{\mathrm{out}}$), energy injection rate (${\dot{E}}_{\mathrm{out}}$), and momentum flux (${\dot{P}}_{\mathrm{out}}$)—as follows:

$$\begin{eqnarray}\begin{array}{rcl}{\dot{M}}_{\mathrm{out}} & = & \displaystyle \frac{3{M}_{\mathrm{gas}}{v}_{\mathrm{out}}}{{R}_{\mathrm{out}}}\\ {\dot{E}}_{\mathrm{out}} & = & \displaystyle \frac{1}{2}{\dot{M}}_{\mathrm{out}}{v}_{\mathrm{out}}^{2}=\displaystyle \frac{3{M}_{\mathrm{gas}}{v}_{\mathrm{out}}^{3}}{2{R}_{\mathrm{out}}}\\ {\dot{P}}_{\mathrm{out}} & = & {\dot{M}}_{\mathrm{out}}{v}_{\mathrm{out}}=\displaystyle \frac{3{M}_{\mathrm{gas}}{v}_{\mathrm{out}}^{2}}{{R}_{\mathrm{out}}}\end{array},\end{eqnarray} \tag{ 1 }$$

Ionized gas mass (M_gas), outflow size (R_out), and extinction-/projection-corrected outflow velocity (v_out) are derived as follows:

$$\begin{eqnarray}\begin{array}{rcl}{M}_{\mathrm{gas}} & = & 4.0\times {10}^{7}{M}_{\odot }\\ & & \times \ \left(\displaystyle \frac{C}{{10}^{[{\rm{O}}/{\rm{H}}]}}\right)\left(\displaystyle \frac{{L}_{[{\rm{O}}{\rm{iii}}],\mathrm{broad}}}{{10}^{44}\,\mathrm{erg}\,{{\rm{s}}}^{-1}}\right){\left(\displaystyle \frac{\langle {n}_{{\rm{e}}}\rangle }{{10}^{3}{\mathrm{cm}}^{-3}}\right)}^{-1}\\ {R}_{\mathrm{out}} & = & {R}_{\mathrm{out}}({L}_{[{\rm{O}}{\rm{iii}}]}\gt {10}^{43}\,\mathrm{erg}\,{{\rm{s}}}^{-1})=3\,\mathrm{kpc}\\ {v}_{\mathrm{out}} & = & 2{\sigma }_{0}=2\sqrt{{\sigma }_{[{\rm{O}}\,{\rm{iii}}],\mathrm{broad}}^{2}+{v}_{[{\rm{O}}\,{\rm{iii}}],\mathrm{broad}}^{2}},\end{array}\end{eqnarray} \tag{ 2 }$$

where $C=\langle {n}_{{\rm{e}}}{\rangle }^{2}/\langle {n}_{{\rm{e}}}^{2}\rangle$, n_e is the electron density, and [O/H] is the metallicity of the gas in solar units.

For Equation (2), we adopt the M_gas equations from Nesvadba et al. (2011) and Carniani et al. (2015) and assume C/10^[O/H] = 1, where the equations from both works become identical. As for R_out, the R_NLR–L_{[O iii]} relation saturates at around 10 kpc beyond L_{[O iii]} ≳ 10⁴³ erg s⁻¹, as noted in Section 4.3, but the radius where the outflows are effective can be smaller than the maximal extent of the outflowing [O iii] line-emitting region. Observations of luminous quasars at L_{[O iii]} ≳ 10⁴³ erg s⁻¹ range between ∼1 and 10 kpc and differ upon using the spatial offset (∼1 kpc; e.g., Carniani et al. 2015) or the spatial extent (∼1 kpc when flux weighted, e.g., Kang & Woo 2018; ∼5–10 kpc when measured kinematically or above an S/N threshold, e.g., Cano-Díaz et al. 2012; Cresci et al. 2015; Perna et al. 2015; Kang & Woo 2018) of the broad/blueshifted [O iii] component. We use a representative value, 3 kpc, considering the diversity in the measurement methods, and we are unable to spatially resolve the outflow size with our observations. For the objects on which both of the [S ii] doublet lines are detected with S/N ≥ 3, namely W1136+4236 and W2136–1631, we measure [S ii] λ6716/[S ii] λ6731 ratios of 0.82 ± 0.19 and 1.02 ± 0.21, respectively. Assuming a 10,000 K temperature, the line ratios correspond to electron densities n_e ∼ 1100 and ∼600 cm⁻³, respectively (Osterbrock 1989). These values are on the higher end but within the range of measurements for various AGNs (100–1000 cm⁻³; e.g., Holt et al. 2006; Nesvadba et al. 2006; Perna et al. 2015; Karouzos et al. 2016; Rakshit et al. 2017; but see also Baron & Netzer 2019). We fix $\langle {n}_{e}\rangle =300\,{\mathrm{cm}}^{-3}$ in between the boundary of the reported values due to the limited number of our measurements, but this may underestimate the outflow quantities by a factor of 2–4 if the Hot DOGs turn out to have n_e ∼ 600−1000 cm⁻³. Lastly, v_out is adopted from Bae & Woo (2016) and Bae et al. (2017) as a combination of the velocity dispersion and the offset to correct for dust extinction and projection effects.

There are several systematic uncertainties in using Equations (1) and (2) to derive [O iii] outflow quantities (e.g., Harrison et al. 2018). First, we assume the narrow [O iii] comes from a nonoutflowing region and use only the broad [O iii] to measure outflow quantities. This assumption appears valid for our sample, as most of the narrow Gaussian components have offsets within ∼100 km s⁻¹ to the systemic redshift (with exceptions for W1103–1826 and W2026+0716) for non-H ii BPT sources with line S/N ≥ 5 (Table 3 and Figure 5). There are counterexamples of narrow emission lines altogether drifting against the stellar absorption line–based redshift for local AGNs (e.g., Bae & Woo 2016), but the occurrence is rare, supporting the fact that the narrow component of the [O iii] is less likely to be outflowing. Still, there are 3/9 BPT non-H ii sources with [O iii] S/N ≥ 5 and narrow component widths σ_{[O iii],narrow} > 400 km s⁻¹. This is partly due to our narrow-line FWHM limit allowing up to 1200 km s⁻¹, which is an arbitrary division between a broad and a narrow line (Section 3), and we check how much the outflow quantities change if we substitute the broad [O iii] component properties in Equation (2) into those of the total. We find that M_gas changes by a factor of 1.30–1.61 and v_out changes by a factor of 0.81–0.89. Here ${\dot{M}}_{\mathrm{out}}$, ${\dot{E}}_{\mathrm{out}}$, and ${\dot{P}}_{\mathrm{out}}$ change by factors of 1.16–1.30, 0.84–0.92, and 1.02–1.05, respectively. The overall changes are relatively insensitive to the choice of the broad or total [O iii] profile, as the change in M_gas cancels out with that of v_out.

Second, we are using measured, i.e., extinction-uncorrected, [O iii] luminosities to derive the outflow quantities. Though we take this effect into consideration for the outflow velocity in Equation (2), the stratified distribution of obscuring material between the lines of sight through the AGN center and the narrow-line region (Tables 1 and 8) complicates the extinction correction. As the narrow-line region is more extended than the region responsible for the majority of the dust obscuration (see Section 4.3), we adopt the Balmer decrement–based E(B−V) values to estimate the extinction correction for L_{[O iii],broad}, assuming a similar size of the outflowing [O iii] line region as the narrow Balmer line region. Hot DOGs typically show E(B−V) ∼ 0.3–0.7 mag (Table 8), corresponding to the [O iii] extinctions by factors of ∼3–9 for the Milky Way extinction curve. This reduces the M_gas by the same amount. As the objects with Balmer decrement values are limited, we use L_{[O iii],broad} observed values when deriving M_gas and the outflow quantities dependent on M_gas but interpret the values considering that they are likely underestimated. The outflow quantities normalized by the same extinction-uncorrected luminosity, or the outflow efficiencies, are more reliable under the effect of extinction.

Third, we saw an order-of-magnitude range in the measurement of both the outflow size and the electron density due to the distribution of measurements and dependence on the measurement method. Assuming our Hot DOGs span the full range of distribution in R_out and $\langle {n}_{{\rm{e}}}\rangle$ at our probed luminosities, the uncertainty in the average value of R_out and $\langle {n}_{{\rm{e}}}\rangle$ dividing the range by the square root of the number of objects, i.e., 10/$\sqrt{8}$, will be a few times. We thus estimate the uncertainty in the average outflow quantity (proportional to 1/${R}_{\mathrm{out}}\langle {n}_{{\rm{e}}}\rangle$) to be an order of magnitude.

We list the estimated outflow quantities for the eight Hot DOGs with [O iii] S/N ≥ 5 and σ_{[O iii],broad} > 400 km s⁻¹ in Table 9. Extinction-uncorrected mass outflow rates are 60–4860 M_⊙ yr⁻¹ (median 970 M_⊙ yr⁻¹). The estimated star formation rates (SFRs) for a sample of Hot DOGs based on SED fitting are ≲300–600 M_⊙ yr⁻¹ (e.g., Eisenhardt et al. 2012; Jones et al. 2014; Díaz-Santos et al. 2018), comparable to the starbursts in SMGs at similar redshifts, which show values between 100 and 1000 M_⊙ yr⁻¹ (e.g., Magnelli et al. 2012). This implies that Hot DOGs have comparably higher mass outflow rates than SFRs, even when uncorrected for extinction.

Table 9.  Outflow Quantities

Name	$\mathrm{log}{M}_{\mathrm{gas}}$	vout	${\dot{M}}_{\mathrm{out}}$	$\mathrm{log}{\dot{E}}_{\mathrm{out}}$	$\mathrm{log}{\dot{P}}_{\mathrm{out}}$	${\dot{M}}_{\mathrm{out}}/{\dot{M}}_{\mathrm{acc}}$	${\dot{E}}_{\mathrm{out}}/{L}_{\mathrm{bol}}$	${\dot{P}}_{\mathrm{out}}c/{L}_{\mathrm{bol}}$
	(M⊙)	(km s−1)	(M⊙ yr−1)	(erg s−1)	(dyn)		(%)
W0114–0812	8.12 ± 0.12	2967 ± 331	404 ± 119	45.05 ± 0.19	36.88 ± 0.15	1.82 ± 0.55	0.09 ± 0.04	0.18 ± 0.07
W0226+0514	7.53 ± 0.13	1816 ± 482	63 ± 25	43.81 ± 0.37	35.86 ± 0.26	3.37 ± 1.76	0.06 ± 0.06	0.20 ± 0.14
W1103–1826	8.28 ± 0.09	5008 ± 556	968 ± 231	45.88 ± 0.17	37.48 ± 0.13	9.61 ± 2.65	1.34 ± 0.56	1.61 ± 0.54
W1136+4236	8.61 ± 0.03	3169 ± 98	1306 ± 105	45.62 ± 0.05	37.42 ± 0.04	5.71 ± 0.51	0.32 ± 0.04	0.60 ± 0.06
W1719+0446	8.06 ± 0.45	4283 ± 225	507 ± 521	45.47 ± 0.45	37.14 ± 0.45	6.44 ± 6.78	0.66 ± 0.70	0.92 ± 0.97
W2026+0716	9.02 ± 0.04	4502 ± 149	4864 ± 465	46.49 ± 0.06	38.14 ± 0.05	7.65 ± 0.80	0.86 ± 0.12	1.15 ± 0.14
W2136–1631	8.74 ± 0.01	2051 ± 32	1143 ± 39	45.18 ± 0.02	37.17 ± 0.02	2.48 ± 0.09	0.06 ± 0.00	0.17 ± 0.01
W2216+0723	8.47 ± 0.10	3223 ± 322	971 ± 242	45.50 ± 0.16	37.29 ± 0.13	4.41 ± 1.33	0.26 ± 0.11	0.47 ± 0.17

Note. Outflow quantities are shown for [O iii] lines with S/N ≥ 5 having a broad component (σ_{[O iii],broad} > 400 km s⁻¹). Here M_gas is the ionized outflowing gas mass estimated using the broad [O iii] profile without extinction correction on L_{[O iii],broad}, v_out is the outflow velocity with extinction-/projection-correction, ${\dot{M}}_{\mathrm{out}}$ is the mass outflow rate, ${\dot{E}}_{\mathrm{out}}$ is the energy injection rate, ${\dot{P}}_{\mathrm{out}}$ is the momentum flux, ${\dot{M}}_{\mathrm{acc}}$ is the mass accretion rate (calculated as L_bol/ηc² with radiative efficiency η = 0.1), and L_bol is the bolometric luminosity using an [O iii]-to-bolometric correction factor of 3500 (Heckman et al. 2004).

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We also obtain the mass-loading factor between the mass outflow and accretion, i.e., ${\dot{M}}_{\mathrm{out}}/{\dot{M}}_{\mathrm{acc}}$, with a mass accretion rate ${\dot{M}}_{\mathrm{acc}}={L}_{\mathrm{bol}}/\eta {c}^{2}$ and radiative efficiency η = 0.1 (e.g., Soltan 1982). Using the [O iii]-to-bolometric correction factor (Heckman et al. 2004) removes the extinction dependence on L_{[O iii]} in ${\dot{M}}_{\mathrm{out}}$ when dividing by L_bol. The ${\dot{M}}_{\mathrm{out}}/{\dot{M}}_{\mathrm{acc}}$ values are 1.8–9.6 (median 5.7) so that the amount of gas outflowing is larger than that accreted, assuming Heckman et al. (2004) values are appropriate for Hot DOGs.¹⁷ Combined, we find the mass outflow rate of Hot DOGs, with an uncertainty of an order of magnitude or more on the average value, to be marginally greater than the gas consumption due to star formation or fueling the AGN itself, demonstrating the role of ionized gas outflows in depleting the ISM around the AGN and competing against gas cooling to form additional stars.

Energy injection rates range from ${\dot{E}}_{\mathrm{out}}=6.5\times {10}^{43}$ to 3.1 × 10⁴⁶ erg s⁻¹ (median 3.2 × 10⁴⁵ erg s⁻¹) without extinction correction, also being 0.058%–1.3% (median 0.32%) of L_bol, independent of the extinction effect. The fraction of the bolometric luminosity transferred to kinetic energy, or the feedback efficiency, is consistent with observations of luminous, obscured quasars (e.g., Z16); theoretical models and simulations of ∼10³ km s⁻¹ winds, including kinetic energy (e.g., King 2003; Choi et al. 2012; Roth et al. 2012); or a weak outflow-induced cloud expansion (Hopkins & Elvis 2010). Also, the bolometric luminosity and the fraction of it injected into the ISM are near the limit where it can blow out the gas (e.g., see discussion in Díaz-Santos et al. 2016). Our feedback efficiencies are one to two orders of magnitude smaller than the thermal efficiencies required to match the normalization of the local BH mass–stellar velocity dispersion (M_BH–σ_*) relation (5%; Di Matteo et al. 2005) or those used in prescriptions for hydrodynamic cosmological simulations (>10%; e.g., Dubois et al. 2014; Schaye et al. 2015; Weinberger et al. 2017), but these works have a thermal feedback mode alone or involve subgrid prescriptions in the simulations due to limited resolution. Momentum flux values are in the range ${\dot{P}}_{\mathrm{out}}=7.2\times {10}^{35}$–1.4 × 10³⁸ dyn (median 2.0 × 10³⁷ dyn), being 0.17–1.6 (median 0.60) times L_bol/c. Outflows are classified as energy-driven if they retain their thermal energy and momentum-driven if they radiate away. Following Tombesi et al. (2015) and Feruglio et al. (2015) for energy-conserving outflows observed in local ULIRGs (IRAS F11119+3257 and Mrk 231), the sub-L_bol/c momentum flux values for Hot DOGs are comparable to or lower than those for ULIRGs at v_out ∼ 10³ km s⁻¹, indicating that the ionized outflows in Hot DOGs are marginally more likely to be momentum-driven than energy-driven. We summarize our findings in Figure 7. Hot DOGs show among the strongest outflow quantities (${\dot{M}}_{\mathrm{out}}$, ${\dot{E}}_{\mathrm{out}}$, ${\dot{P}}_{\mathrm{out}}$) and outflow efficiencies (${\dot{M}}_{\mathrm{out}}/{\dot{M}}_{\mathrm{acc}}$, ${\dot{E}}_{\mathrm{out}}/{L}_{\mathrm{bol}}$, ${\dot{P}}_{\mathrm{out}}c/{L}_{\mathrm{bol}}$) compared to the AGN samples in the literature.

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Having seen the occurrence and strength of [O iii] outflows in our luminous, obscured AGN sample, we now assess the usefulness of broad emission lines in measuring M_BH for highly obscured AGNs. We overplot in Figure 8 the normalized profiles of prominent lines for the sources with potential outflows in addition to the [O iii] line. The figure includes the 5/7 BPT non-H ii Hot DOGs with significantly broad Hα (${\sigma }_{{\rm{H}}\alpha ,\mathrm{broad}/\mathrm{narrow}}\gt 400\,\mathrm{km}\,{{\rm{s}}}^{-1}$) with S/N > 5 from Table 3 (W0114–0812, W1103–1826, W1136+4236, W2136–1631, and W2216+0723). The normalized broad Balmer line components are weaker than the broad [O iii] but are blueshifted for all five objects, with absolute values comparable to or smaller than the broad [O iii] blueshift with a fraction of 0.53–1.23 times (median 0.65). This trend is also seen in the first moment of the Balmer lines (Δv in Table 3) being blueshifted, but not as much as that of the [O iii].

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The broadening of the Balmer line might come from the broad-line region, with blueshifts indicating outflows within the broad-line region (e.g., Vietri et al. 2018) or the broad-line region simply being highly obscured. We check this by calculating the Balmer decrement of objects having S/N ≥ 5 in Hα and showing a broad component in both Hα/Hβ (W1103–1826, W1136+4236, W2136–1631, and W2216+0723). The values are 1.93 ± 1.02, 4.28 ± 0.87, 1.93 ± 0.22, and 5.95 ± 2.17, respectively, all comparable to or smaller than the narrow Balmer decrements in Table 8. This is inconsistent with the expected higher extinction (and thereby higher Balmer decrements) toward the broad-line region compared to the narrow-line region, assuming that the intrinsic Balmer decrements of the broad-line region for Hot DOGs are comparable to the observed values for luminous type 1 quasars at z ∼ 2 (median and scatter of 3.2 ± 1.4; Shen & Liu 2012), which themselves are comparable to the intrinsic values for the narrow lines (3.1; e.g., Osterbrock 1989). Therefore, we find it more likely that broad and blueshifted Balmer lines originate outside the broad-line region and that they are due to outflows within the narrow-line region. This is also consistent with the objects not showing a broad Hα (2/7) still having ${\sigma }_{{\rm{H}}\alpha ,\mathrm{broad}/\mathrm{narrow}}\gt 300\,\mathrm{km}\,{{\rm{s}}}^{-1}$, which is on the broader side to be explained by nonoutflowing motion unless the host galaxies of Hot DOGs have grown to sufficiently massive bulges at z ∼ 2.

Following Wu et al. (2018), we check if the Balmer line outflows are as strong as the [O iii] outflows by fitting the rest-frame 3500–7000 Å spectra with a common broad-line center and width and calculating the F-distribution probability of the reduced χ² from the fit having a statistically significant improvement. Out of the S/N ≥ 5 spectra, 1/9 (W0147–0923) has an improvement (95% or higher F-distribution probability) such that the broad Balmer lines are indistinguishable from the outflowing [O iii] component, but 7/9 (including all of the broad Hα sources in Figure 8) yield significantly improved fits when the broad [O iii] center and width are detached from the rest of the lines. We thus can rule out the case where all of the broad Balmer lines are showing outflows as strong as the [O iii], in accord with Wu et al. (2018) and consistent with weaker but similar Balmer line kinematics as the [O iii] found in SDSS type 2 quasars (e.g., Kang et al. 2017).

In addition to the likelihood of outflows in the narrow Balmer line region for some Hot DOGs, we plot in the remaining panels of Figure 8 the 3/6 BPT non-H ii sources with significantly broad [O ii] (${\sigma }_{[{\rm{O}}\mathrm{II}],\mathrm{broad}/\mathrm{narrow}}\gt 400\,\mathrm{km}\,{{\rm{s}}}^{-1}$; W1719+0446, W2026+0716, W2216+0723) with S/N > 5. Additionally, 1/3 of the remaining (3/6) sources (W0114–0812) shows ${\sigma }_{[{\rm{O}}\mathrm{II}],\mathrm{broad}/\mathrm{narrow}}\gt 300\,\mathrm{km}\,{{\rm{s}}}^{-1}$. This can be explained by either strong AGN outflows broadening the [O ii] line profile or the kinematics of the ISM in the star-forming region being highly disturbed by AGN outflows and/or galaxy merging (e.g., Díaz-Santos et al. 2016, 2018). As [O ii] is a forbidden line, a fraction of them being broad adds support for the presence of outflows and hints at an ∼kiloparsec-scale, narrow-line region origin for at least some of the broadened Balmer lines.

The presence of outflowing material within the Balmer or even the [O ii] lines can cause serious misinterpretation when using the broad emission line width to measure M_BH. Although we are limited to a single object (W2136–1631) in Figure 8 covering all of the major UV/optical broad emission lines, the hydrogen lines (Lyα, Hβ, Hα) and Mg ii profiles are indistinguishable from [O iii], with C iv showing a stronger broad/blueshifted wing component. This suggests that not only the Balmer lines but also the rest-frame UV broad emission lines could be contaminated by outflows within the narrow-line region. Therefore, not only the M_BH measurements reported for Hot DOGs (e.g., Ricci et al. 2017a; Tsai et al. 2018; Wu et al. 2018) but also any M_BH measurement for a highly obscured AGN with a broad and blueshifted permitted emission line should be carefully tested for the presence of outflows through signatures of broad and blueshifted forbidden lines (e.g., Alexander et al. 2008; Ricci et al. 2017c), especially at L_{[O iii]} > 10⁴² erg s⁻¹ (Figure 6). On the other hand, we expect that the M_BH values for Hot DOGs from emission lines likely coming from scattered light (e.g., Assef et al. 2016, 2019) are unlikely to be affected by outflows.

We alternatively estimate M_BH and the Eddington ratios (f_Edd) shown in Table 10 using the M_BH–σ_* relation and the width of the common narrow component of [O ii]/Balmer lines, noting that σ_* is similar to narrow emission line widths but with a large uncertainty (a factor of ∼2; e.g., Greene & Ho 2005a; Zakamska & Greene 2014; Bennert et al. 2018). Here we select the narrow-line width from [O ii]/Balmer lines instead of [O iii] to minimize blending from outflows. Considering the dispersion of the M_BH–σ_* relation itself (a factor of 2–3; e.g., Kormendy & Ho 2013; Woo et al. 2013) and the possibility of a systematic offset for various types of AGNs compared to the local inactive galaxies on the BH–galaxy scaling relation (up to a factor of several; e.g., Kim et al. 2008; U12; Sexton et al. 2019), the M_BH estimates for Hot DOGs could be inaccurate by up to an order of magnitude. Having in mind those uncertainties, we find most of the M_BH values to be on the order of 10⁹ M_⊙, while those associated with σ_narrow > 400 km s⁻¹ are given with upper limits and show values around ∼10¹⁰ M_⊙. We convert the M_BH estimates and use the L_bol from the extinction-corrected 5100 Å luminosity with a bolometric correction (e.g., Figure 1) into the Eddington ratio f_Edd = L_bol/L_Edd, which ranges on the order of unity to 10. This can be interpreted as the AGN accretion being highly effective near or even beyond its Eddington limit, but there are possibilities of a systematic offset in f_Edd for Hot DOGs.

Table 10.  Accretion Rates

Name	$\mathrm{log}{M}_{\mathrm{BH}}$	fEdd	fEdd(MBH < 1010 M⊙)
	(M⊙)
W0114–0812	9.34 ± 0.15	2.19 ± 0.77	>0.48
W0147–0923	9.20 ± 0.35	9.02 ± 7.25	>1.44
W0226+0514	8.95 ± 0.28	8.08 ± 5.24	>0.71
W1103–1826	9.18 ± 0.30	3.60 ± 2.49	>0.54
W1136+4236	<9.96 ± 0.13	>0.28 ± 0.09	>0.26
W1719+0446	<10.01 ± 0.46	>0.81 ± 0.86	>0.83
W2026+0716	<10.01 ± 0.27	>0.43 ± 0.27	>0.45
W2136–1631	8.08 ± 0.07	15.98 ± 2.40	>0.19
W2216+0723	<10.02 ± 0.17	>0.23 ± 0.09	>0.24

Note. Here ${M}_{\mathrm{BH}}$ is the mass assuming the local M_BH–σ_* relation and using σ_narrow as a substitute for σ_* from [O ii], Hβ, or Hα lines with S/N ≥ 5. Here f_Edd is the Eddington ratio, L_bol/L_Edd, where L_bol is derived from the SED fit (e.g., Figure 1), and L_Edd is the Eddington luminosity. The f_Edd values are derived using $\mathrm{log}{M}_{\mathrm{BH}}/{M}_{\odot }=8.49+4.377\mathrm{log}\{{\sigma }_{* }\,(\mathrm{km}\,{{\rm{s}}}^{-1})/200\}$ (Kormendy & Ho 2013 and a 1–1 relation between σ_* and σ_narrow; e.g., Bennert et al. 2018) and an observed upper limit of M_BH ∼ 10¹⁰ M_⊙ for the most massive SDSS quasars (Jun et al. 2017).

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To better tell whether the super-Eddington accretion seen in some of the Hot DOGs is real or due to systematic overestimation, we estimate the f_Edd values again from considering an upper limit of M_BH ∼ 10¹⁰ M_⊙ for the most massive z ∼ 2 quasars (Jun et al. 2017). Listed in Table 10, the corresponding lower limits on f_Edd values are around 0.2–1.4 (mean 0.6), indicating near-Eddington-limited accretion even if M_BH reaches ∼10¹⁰ M_⊙ and super-Eddington accretion if M_BH ≲ 5 × 10⁹ M_⊙. In any case, we find without any measurement of M_BH that Hot DOGs are likely accreting near or beyond the Eddington limit. If we trust the f_Edd values from the M_BH–σ_* relation, however, 4/9 sources have f_Edd exceeding 3 (but with large measurement uncertainties), which is hard to explain with models allowing accretion modes up to several times the Eddington limit (e.g., see discussion in Tsai et al. 2018). We thus find it likely that some of our f_Edd values are largely overestimated due to either an underestimated M_BH or an overestimated L_bol.

Under the merger-driven AGN triggering mechanism (e.g., Hopkins et al. 2008; Hickox et al. 2009), starbursts in the merging system will be followed by obscured BH growth. Using a simple BH–galaxy mass scaling relation, M_BH values for Hot DOGs are more likely to be higher relative to the central host mass due to the time delay between BH and galaxy growth (e.g., U12) and the bulge formation lagging the BH growth (e.g., Peng et al. 2006; Jun et al. 2017) at z ∼ 2. Alternative explanations for extremely high f_Edd values for Hot DOGs could be due to L_bol being overestimated by anisotropic radiation (e.g., Abramowicz et al. 1988; Wang et al. 2014), but this is less likely to be significant, as the SED peaking in the IR is relatively isotropic. Gravitational lensing is also a less likely explanation given the morphologies of Hot DOGs imaged by the Hubble Space Telescope (e.g., Tsai et al. 2015).

To summarize, our sample of Hot DOGs shows high f_Edd values, with lower limits often near the Eddington limit. This may be produced by underestimated M_BH (σ_*) values, but with large potential uncertainties in both the M_BH and L_bol, it is hard to tell from our data whether super-Eddington accretion is occurring. The efficient accretion probed by our objects is consistent with scenarios with BH feeding in luminous, obscured AGNs triggering strong ionized gas outflows. Our sample is also in line with the more X-ray-luminous population of SMGs showing a stronger, near-Eddington-limited accretion than the X-ray-weaker counterpart (Alexander et al. 2008), following the evolutionary model predictions. However, it is in tension with the absence of a hard X-ray-selected local AGN population with high f_Edd and E(B−V) (Ricci et al. 2017c), suggesting that the local population might simply lack the rapidly accreting, obscured AGN seen at z ∼ 2 (H. D. Jun et al. 2019, in preparation).