The X-ray and Mid-Infrared luminosities in Luminous Type 1 Quasars

TL;DR

This study analyzes a large sample of luminous type 1 quasars to clarify the intrinsic relationship between X-ray and mid-infrared luminosities, revealing a luminosity-dependent bilinear correlation that differs from local Seyfert galaxies.

Contribution

It demonstrates that observational biases, especially X-ray flux limits, significantly affect the observed $L_{X}-L_{MIR}$ relations and establishes a bilinear relation for luminous quasars.

Findings

The $L_{X}-L_{MIR}$ relation is bilinear with a shallower slope at higher luminosities.

X-ray flux limits are the primary factor causing discrepancies in previous studies.

Luminous quasars have a non-linear, luminosity-dependent $L_{X}-L_{MIR}$ relation.

Abstract

Several recent studies have reported different intrinsic correlations between the AGN mid-IR luminosity ($L_{MIR}$) and the rest-frame 2-10 keV luminosity ($L_{X}$) for luminous quasars. To understand the origin of the difference in the observed $L_{X}-L_{MIR}$ relations, we study a sample of 3,247 spectroscopically confirmed type 1 AGNs collected from Bo\"{o}tes, XMM-COSMOS, XMM-XXL-North, and the SDSS quasars in the Swift/XRT footprint spanning over four orders of magnitude in luminosity. We carefully examine how different observational constraints impact the observed $L_{X}-L_{MIR}$ relations, including the inclusion of X-ray non-detected objects, possible X-ray absorption in type 1 AGNs, X-ray flux limits, and star formation contamination. We find that the primary factor driving the different $L_{X}-L_{MIR}$ relations reported in the literature is the X-ray flux limits for different studies. When taking these effects into account, we find that the X-ray luminosity and mid-IR luminosity (measured at rest-frame $6\mu m$, or $L_{6\mu m}$) of our sample of type 1 AGNs follow a bilinear relation in the log-log plane: $\log L_X =(0.84\pm0.03)\times\log L_{6\mu m}/10^{45}{\rm erg\;s^{-1}} + (44.60\pm0.01)$ for $L_{6\mu m} < 10^{44.79}{\rm erg\;s^{-1}} $, and $\log L_X = (0.40\pm0.03)\times\log L_{6\mu m}/10^{45}{\rm erg\;s^{-1}} +(44.51\pm0.01)$ for $L_{6\mu m} \geq 10^{44.79}{\rm erg\;s^{-1}} $. This suggests that the luminous type 1 quasars have a shallower $L_{X}-L_{MIR}$ correlation than the approximately linear relations found in local Seyfert galaxies. This result is consistent with previous studies reporting a luminosity-dependent $L_{X}-L_{MIR}$ relation, and implies that assuming a linear $L_{X}-L_{MIR}$ relation to infer the neutral gas column density for X-ray absorption might overestimate the column densities in luminous quasars.

Figures

Tables

Table 1. Table 1: Survey Properties

	Boötes	XMM-COSMOS	XRT-SDSS (10 ks)	XRT-SDSS (5 ks)	XXL-North
X-ray survey area	9.3 deg2	2. deg2	$\cdots$ a	$\cdots$	25 deg2
(1) # of type 1 AGN	1410	322	241	362	1153
(2) z	0.14-4.58	0.10-4.25	0.08-3.68	$\cdots$	0.06-5.0
(3)	21.0	21.5	19.0	19.0	20.7
(4) X-ray energy range	0.5-7 (Chandra)	0.5-10 (XMM-Newton)	0.3-10 (Swift/XRT)	$\cdots$	0.5-10 (XMM-Newton)
(5) 0.5–7 keV X-ray flux limit	7.8	1.0	25.0	50.0	11.1
(6) X-ray detection fraction	$56\%$	$100\%$	$83\%$	$70\%$	$100\%$
(7) Median $L_X\left(2-10keV\right)$	$44.23$	$44.15$	$44.66$	$44.66$	$44.26$

Table 2. Table 2: Best-fit parameters for L X = α × L 6 ​ µm + β subscript 𝐿 X 𝛼 subscript 𝐿 6 µm 𝛽 L_{\rm X}=\alpha\times L_{6\micron}+\beta calculated using the Kelly ( 2007 ) method

Sample	Description		$N_{\mathrm{XD}}$	$N_{\mathrm{UL}}$	$\alpha$	$\beta$
			(1)	(2)	(3)	(4)
I. Best-fit parameters for luminous quasars (see §5.1)
Boötesa	Stacking		727	620b	$0.51\pm 0.06$	$44.23\pm 0.05$
Boötes	X-ray detected		$727$	$0$	$0.50\pm 0.02$	$44.36\pm 0.01$
Boötes	All		727	$620$	$0.49\pm 0.03$	$44.02\pm 0.02$
XMM-COSMOS	All, X-ray selected		293	0	$0.51\pm 0.03$	$44.30\pm 0.02$
Swift/SDSS (10 ks)	X-ray detected		198	0	$0.56\pm 0.05$	$44.31\pm 0.03$
Swift/SDSS (10 ks)	All		198	43	${0.56}_{-0.06}^{+0.05}$	$44.24\pm 0.03$
Swift/SDSS (5 ks)	X-ray detected		247	0	${0.58}_{-0.04}^{+0.05}$	$44.40\pm 0.03$
Swift/SDSS (5 ks)	All		247	115	$0.58\pm 0.05$	$44.24\pm 0.03$
XMM XXL-N	X-ray selected		1071	0	$0.59\pm 0.02$	$44.36\pm 0.01$
II. Effects of X-ray absorption (see §5.2)
Boötes (X-ray detected)	Assuming 20% of the sample are X-ray obscured		727	0	$0.49\pm 0.05$	$44.41\pm 0.01$
III. Effects of X-ray flux limit (see §5.3)
Mateos et al. 2015	Type 1 AGNs with $L_{6\mathrm{\mu m}}>{10}^{43.8}$ erg s-1		103	0	$0.81\pm 0.06$	$44.58\pm 0.04$
Boötes (X-ray detected)	$f_X>1.0\times {10}^{-13}$ [erg/s] (M15)		31	0	${0.80}_{-0.12}^{+0.11}$	$44.59\pm 0.07$
Boötes (X-ray detected)	$f_X>5\times {10}^{-14}$ [erg/s] (W12 5ks)		163	0	$0.58\pm 0.05$	$44.45\pm 0.03$
Boötes (X-ray detected)	$f_X>2.5\times {10}^{-14}$ [erg/s] (W12 10ks)		403	0	$0.52\pm 0.03$	$44.42\pm 0.02$
Boötes (X-ray detected)	$f_X>1.1\times {10}^{-14}$ [erg/s] (XXL-N)		665	0	$0.51\pm 0.02$	$44.38\pm 0.01$

Equations

$$\log L_{6\text{micron}}$$

$$\log L_X$$

$$\log L_{6\text{micron}}$$

$$\log L_X$$

$$\log L_{6\text{micron}}/{\mathrm{erg}\mathrm{s}}^{-1}$$

$$\log \frac{L_X}{{\mathrm{erg}\mathrm{s}}^{-1}}=(0.84\pm 0.03)$$

$$\log L_{6\text{micron}}/{\mathrm{erg}\mathrm{s}}^{-1}$$

$$\log \frac{L_X}{{\mathrm{erg}\mathrm{s}}^{-1}}=(0.40\pm 0.03)$$

Full text

galaxies: active

††slugcomment: ApJ accepted

1 Introduction

X-ray and mid-IR emission are both excellent tracers of supermassive black hole (SMBH) accretion activities. Since active galactic nucleus (AGN) emission at these wavelengths is less susceptible to the presence of obscuring material compared to optical wavelengths (e.g. Corrales et al., 2016), studying the correlation between the X-ray and mid-IR luminosities of AGNs is crucial for understanding the dust-enshrouded phase of galaxy-SMBH coevolution (e.g. Di Matteo et al., 2005; Hopkins et al., 2006; Gilli et al., 2007; Somerville et al., 2008; Treister et al., 2009). There are now a range of studies examining the correlation between AGN X-ray and mid-IR luminosities. Some of these works found that the mid-IR ($L_{\rm MIR}$) and X-ray luminosities ($L_{\rm X}$) follow an almost linear relation in low redshift, low luminosity AGNs (e.g. Lutz et al., 2004; Alexander et al., 2008; Gandhi et al., 2009; Lusso et al., 2011; Asmus et al., 2015). However, it is not clear whether such a linear relation holds for more luminous AGNs (i.e., quasars111We refer to AGNs with bolometric luminosity ($L_{\mathrm{bol}}$) more luminous than $10^{45}$ erg s*-1* as “quasars”.). Notably, studies comparing $L_{\rm X}$ to the UV luminosity for AGNs found out that the ratio between the X-ray and UV luminosities rapidly decreases with increasing UV luminosity for type 1 AGNs (e.g. Tananbaum et al., 1979; Strateva et al., 2005; Lusso et al., 2010). Since the rest-frame mid-IR emission of AGNs originates from the hot dust heated by the UV photons from the SMBH accretion disk, understanding the $L_{\rm X}-L_{\rm MIR}$ relation for luminous AGNs is also crucial for understanding the structure of the hot dust surrounding the central SMBH as well as the AGN accretion physics.

The local, linear $L_{\rm X}-L_{\rm MIR}$ relation is illustrated by the results of Gandhi et al. (2009), who found that the spatially resolved nuclear $L_{\rm MIR}$ and $L_{\rm X}$ for local Seyfert galaxies are almost linearly correlated. Recently, Asmus et al. (2015) have extended this work to a number of more luminous AGNs from the 9-month Swift/BAT catalog (Tueller et al., 2008) and archival local AGNs with high spatial resolution mid-IR observations (Asmus et al., 2014). Asmus et al. (2015) found that the luminous AGNs in their sample have slightly more X-ray emission than the value predicted by the local linear relation between $L_{\rm MIR}$ and $L_{\rm X}$. While their result was only suggestive due to the limited size of their sample, it is supported by the study of higher redshift AGNs selected from the Bright Ultra-hard XMM–Newton survey (Mateos et al., 2015). However, some studies have also reported a luminosity dependent $L_{\rm X}-L_{\rm MIR}$ relation for luminous quasars, including the study of high redshift AGNs in COSMOS by Fiore et al. (2009) and the compilation of SDSS DR5 AGNs spanning a wide luminosity range studied by Stern (2015).

The lack of consensus on the universality of the $L_{\rm X}-L_{\rm MIR}$ correlations might be due to various observational limitations. In particular, for surveys such as COSMOS, the limited survey volumes restrict the number of rare AGN detected at the highest luminosities. On the other hand, wide-area surveys have shallower flux limits, making them less likely to detect fainter sources and higher redshift sources. Thus, the $L_{\rm X}-L_{\rm MIR}$ correlations could also be biased if the X-ray non-detected objects are not taken into account.

To understand whether such biases might affect the observed $L_{\rm X}-L_{\rm MIR}$ relations, we compile four different type 1 AGN samples spanning a wide range of survey areas and X-ray flux limits to investigate the intrinsic relationship between AGN mid-IR and X-ray emission over a wide dynamic range in luminosity.

To minimize the contamination from star formation related processes and the stellar emission in the host galaxy, we focus on luminous objects that are spectroscopically confirmed as type 1 AGNs. We use type 1 AGN samples from the AGN and Galaxy Evolution Survey (AGES, Kochanek et al., 2012) in the Boötes survey region, the publicly available AGN samples from the XMM-COSMOS survey (Lusso et al., 2010), the XMM-XXL North survey (Pierre et al., 2016; Menzel et al., 2016; Liu et al., 2016), and the SDSS DR5 quasars with serendipitous Swift/XRT observations (Wu et al., 2012a).

This paper is organized as follows. In §2 we describe the multi-wavelength data and the properties of each quasar catalog. In §3, we discuss the derivations of X-ray and mid-IR luminosities. In §4 and §5, we discuss the $L_{\rm X}-L_{6\micron}$ correlation and the possible biases that might affect the observed relations. A discussion and a summary are given in §6. Throughout the paper, we use the Vega magnitude system and assume a $\Lambda$CDM cosmology with $\Omega_{m}=0.3$, $\Omega_{\Lambda}=0.7$ and $H_{0}=70$ km s*-1* Mpc*-1*.

2 The Type 1 Quasar Samples

To investigate the correlation between X-ray luminosities and the mid-IR luminosities for type 1 AGNs with broad optical emission lines, we focus on extragalactic survey regions with X-ray observations and mid-IR observations from Spitzer or WISE. We select four different samples with a wide range of survey area and flux limits in order to understand the biases that might affect the observed $L_{\rm X}-L_{\rm MIR}$ relation.

2.1 Boötes type 1 quasar sample

One primary source of quasars for this study is the Boötes multiwavelength survey, which has a wide area ($9$ deg2) and excellent multiwavelength coverage. For this work, we use the 1,443 AGNs in the AGES catalog that are classified as “type 1” based on spectroscopic observations from the Hectospec instrument on the MMT observatory (i.e., sources that are best-fitted by the SDSS quasar template, see Kochanek et al., 2012, for details).

To ensure that the AGNs studied in this work have minimal impact from the radio-loud quasars that could have X-ray emission enhanced by the presence of relativistic jets (e.g., Zamorani et al., 1981; Wilkes & Elvis, 1987; Cappi et al., 1996; Brinkmann et al., 2000), we also use the Westerbork Synthesis Radio Telescope (WSRT) observations of the Boötes region to eliminate powerful radio AGNs. de Vries et al. (2002) surveyed the central $\approx 7$ deg2 of the NDWFS field at $1.4$ GHz to a limiting flux of $\approx 0.1$ mJy and beam size $13\arcsec\times 27\arcsec$. For the 46 matches (within 2$\arcsec$) between the WSRT radio sources and the AGES AGN catalog, we calculate their “radio-loudness” using a radio-loud definition of $R\geq 10$ (Kellermann et al., 1989). Radio-loudness is defined as the ratio between the 5 GHz and optical B-band (rest-frame) monochromatic luminosities, $R=L_{5\mathrm{GHz}}/L_{B}$. $L_{5\mathrm{GHz}}$ is derived from the WSRT observations at 1.4 GHz assuming a typical power-law spectrum, $S_{\nu}\propto\nu^{-0.7}$. The rest-frame $L_{B}$ is derived using the SED-fitting results described in §3.2. Of the 46 WSRT detected sources, 33 of them are radio-loud AGNs ($R>10$, Kellermann et al., 1989). We exclude the 33 radio-loud sources and focus on the remaining 1,410 radio-quiet AGNs in the following analysis.

Boötes is also covered by the XBoötes survey, a 9.3 deg2 mosaic of 126 short (5ks) Chandra ACIS-I images (Murray et al., 2005; Kenter et al., 2005) covering the entire AGES field. XBoötes contains 2,724 X-ray point sources with at least four counts in the AGES survey region. Of those, 790 X-ray point sources are far from bright stars and matched within $3.5^{\prime\prime}$ to the 1,410 type 1 AGNs with good spectroscopic redshifts from AGES at $0.14<z<3.61$ (Kenter et al., 2005; Hickox et al., 2009), yielding an X-ray detection fraction of $\sim 56\%$ for the type 1 AGNs. These X-ray point sources have 0.5-7 keV luminosities of $10^{42}<L_{X}<10^{45}$ erg s*-1* which are characteristic of moderate to luminous AGNs.

We also make use of the optical to near-IR broad-band photometry available in the Boötes field, which includes optical photometry from the NOAO Deep Wide Field Survey (Bw, R, I, Jannuzi & Dey, 1999), near-IR NEWFIRM (J, H, Ks, Gonzalez et al., 2010), mid-IR SDWFS (Spitzer IRAC, Ashby et al., 2009) and mid-IR observation at $24\;\micron$ from Spitzer MIPS (Rieke et al., 2004). An extensive description of the multiband photometry extraction can be found in Brown et al. (2007) and Chung et al. (2014).

Another advantage of the Boötes survey region is that it is covered by the Herschel Multi-tiered Extragalactic Survey (HerMES, Oliver et al., 2012). The inclusion of the far-IR photometry makes it possible to constrain the star formation rate even for luminous quasars (e.g., Netzer et al., 2007; Kirkpatrick et al., 2012; Mullaney et al., 2012; Chen et al., 2015), thus allowing for more accurate measurements of mid-IR AGN luminosities that could be contaminated by star formation processes. For this work, we adopt the SPIRE 250$\micron$ photometry from Alberts et al. (2014). For the 1,410 type 1 AGNs in AGES, $\approx 15\%$ of them are detected by SPIRE at $250\micron$. For these far-IR detected AGNs, we carefully examine the resulting SED fits in §3.2 and their $L_{\rm X}-L_{\rm MIR}$ relation in §5.4.

2.2 XMM-COSMOS X-ray AGN sample

Since XBoötes is a relatively shallow X-ray survey, we supplement it with the publicly available XMM-COSMOS catalog of X-ray selected type 1 AGNs from Lusso et al. (2010). The Lusso et al. (2010) catalog contains 545 X-ray AGNs of which 322 have secure spectroscopic redshift measurements and broad emission line width $>2,000$ km s*-1*. The 322 type 1 AGNs in the XMM-COSMOS sample were selected from a parent sample of 361 type 1 AGNs by excluding the 39 radio-loud AGNs identified using the same radio-loudness definition as described in §2.1.

As discussed in §3.2, we utilize broad-band multiwavelength photometry to determine the AGN contribution to the mid-IR luminosity of our AGNs. To this end, we also make use of the publicly available broad-band photometry in the COSMOS survey region culled from Capak et al. (2007), Sanders et al. (2007) and Elvis et al. (2012). In detail, we first obtain the optical positions by cross-correlating the XMM-COSMOS identification numbers (XIDs) of the Lusso et al. (2010) sources with those in the XMM-COSMOS multiwavelength catalog (Brusa et al., 2010), in which the optical to X-ray counterpart association is obtained based on a likelihood ratio technique (see §3 of Brusa et al., 2010, for details). We then cross-correlate the optical positions of the Lusso et al. (2010) sources to the Capak et al. (2007) and Sanders et al. (2007) catalogs. We use the broad-band photometry spanning optical to far-IR wavelengths that are comparable to the Boötes survey region for the SED fits. In detail, we use the Subaru optical photometry at u,g,r,i,z bands, and the near-IR J, H, K from Calar Alto, UH 88” and CFHT observatories, respectively. The mid-IR photometry comes from both the Spitzer IRAC and MIPS instruments, including 3.6, 4.5, 5.8, 8.0, 24, 70, 160 $\micron$. Similar to Boötes, the COSMOS survey region is also covered by HerMES. Therefore, we also match the Herschel SPIRE photometry to the 322 XMM-COSMOS AGNs with a $5\arcsec$ search radius. The detection fraction at SPIRE 250µm for the 322 XMM-COSMOS type 1 AGNs is also $\sim 15\%$. The $L_{\rm MIR}$ measurements and $L_{\rm X}-L_{\rm MIR}$ relation of these far-IR detected AGNs are also discussed in more detail in §3.2 and in §5.4.

2.3 XRT-SDSS: Optical AGN from SDSS and Swift/XRT

To investigate the $L_{\rm X}-L_{\rm MIR}$ relation, it is also important to consider the possible biases created by missing the most luminous sources due to the limited volume of surveys like Boötes and XMM-COSMOS. For this purpose we use the Swift/SDSS catalog from Wu et al. (2012a, W12 hereafter).

W12 matched all of the 77,429 optically selected SDSS DR5 quasars (Adelman-McCarthy & Collaboration, 2007; Schneider et al., 2007) to the Swift/XRT archive (Burrows et al., 2005) and found that there are 1,034 SDSS DR5 quasars within $20\arcmin$ of a Swift pointing. We refer to this catalog as the XRT-SDSS catalog throughout the rest of the paper. Due to the serendipitous nature of the XRT-SDSS catalog, the Swift/XRT exposure time ranges from $1-600$ ks. We follow the approach of W12 by focusing only on the 607 objects that are unambiguously identified as quasars and excluded objects that are radio-loud or obscured (see Table 9 of W12). W12 define a “clean sample” of quasars by enforcing a minimum XRT exposure time of $>10$ ks, which includes 241 objects with a $82\%$ X-ray detection rate. To maximize the sample size, we also consider a more liberal exposure time cut at $>5$ ks. The 5 ks XRT-SDSS sample includes 362 objects with a 70% X-ray detection rate.

We also make use of the photometry from the SDSS DR5 quasar catalog by cross-matching the SDSSID from the Table 7 of W12 to the SDSSID of the SDSS DR5 quasars catalog. All the quasars in the 10 ks and 5 ks samples have photometry in the u, g, r, i, z bands and $\approx 18\%$ of the W12 quasars have 2MASS J, H and Ks photometry.

To estimate the AGN mid-IR luminosity, we match the SDSS DR5 coordinates to the ALLWISE catalog using a matching radius of $2\arcsec$. We check the number of possibly misidentified sources by randomly shifting the positions of the XRT-SDSS sources by 1 arcmin and matching the shifted positions to the ALLWISE catalog. We find that $1.5\%$ of the randomly shifted positions have a WISE counterpart within $2\arcsec$, suggesting that the spurious matching rate between the XRT-SDSS sources and the WISE catalog is about $1.5\%$, which has a negligible effect on the $L_{\rm X}-L_{\rm MIR}$ relation. For the XRT-SDSS type 1 AGNs in the 10 ks and 5 ks samples, all of them have detections in at least three WISE bands.

2.4 XMM-XXL North X-ray AGN sample

The XMM-XXL-North survey (XXL-N hereafter) is the northern part of the XMM-XXL survey, which is comprised of two separate $\sim 25$ deg2 fields (Pierre et al., 2016). As part of the SDSS-III survey, X-ray sources matched to SDSS photometric objects with r$<22.5$ in XXL-N were all targeted by SDSS-III’s Baryon Oscillation Spectroscopic Survey (BOSS, Smee et al., 2013). The spectroscopic and photometric properties of the X-ray AGNs in XXL-N have recently been reported in Menzel et al. (2016, M16 hereafter). Of the 3,042 sources in the M16 catalog with BOSS spectra, 1,787 are classified as “broad-line” AGNs based on the presence of broad emission lines (H$\beta$, Mg II, C III, or C IV) with FWHM larger than 1,000 km s*-1*.

For this work, we directly use the SDSS photometry and WISE photometry provided by M16. The optical and mid-IR photometry in M16 is obtained by crossmatching the XMM positions with the SDSS or WISE positions using a likelihood-ratio matching method (see Georgakakis & Nandra 2011 and M16 for details). To maximize the photometric coverage of this dataset, we also obtain 2MASS photometry from the ALLWISE catalog, which provides the associations between the WISE source and the closest 2MASS source within a 3″ radius.

Notably, the M16 catalog includes X-ray sources from the XXL survey (Pierre et al., 2016) as well as the sources from the predecessor of XXL, the XMM-LSS survey. The XMM-LSS survey is a $\approx 4.5$ deg2 field at the center of XXL-N with deeper XMM-Newton coverage (10–40 ks). For this work, we consider only the sources with 0.5–2 keV fluxes above the “completeness limit” of the XXL survey, $5.0\times 10^{-15}$ ergs s*-1* cm*-2*. This flux limit is equivalent to $\approx 1.1\times 10^{-14}$ ergs s*-1* cm*-2* in the 0.5–7 keV band assuming a $\Gamma=1.8$ X-ray power-law spectrum. With this flux limit, the number of broad-line AGNs is reduced to 1,372. This approach also ensures that the XXL-N type 1 AGNs studied in this work have uniform X-ray coverage and high mid-IR detection fractions in the WISE bands ($\sim 87\%$ for 3.4, 4.6 and 12$\micron$ bands).

We also match the XXL-N type 1 AGNs to the VLA FIRST catalog (Becker et al., 1995) with a 2″radius. We calculate the radio-loudness of the 41 sources with FIRST counterparts using the same approaches described in previous subsections. Of the 41 sources with FIRST counterparts, 38 of them satisfy the same “radio-loud” definition and are excluded. Since our goal is to study the relation between X-ray and mid-IR luminosities of type 1 quasars, we focus only on the 1,153 X-ray detected type 1 AGNs that are not radio-loud and have a $>5\sigma$ detection significance in at least three WISE bands.

2.5 Key properties of samples

We list the key properties of the samples used in this work in Table 1. A common feature of the sources selected from these four catalogs is that the sources are all optically confirmed as broad-line AGNs, which ensures our $L_{\rm X}-L_{\rm MIR}$ measurements should have a minimal impact due to obscuration. The median r-band magnitudes of these samples are 21.0, 21.5, 19.0, 20.7 mag for AGES, XMM-COSMOS, XRT-SDSS, and XXL-N, respectively.

Both the AGES and XMM-COSMOS catalogs have heterogeneous spectroscopic depths. AGES specifically targeted sources identified as an AGN at other wavelengths down to i $<22.5$, while for other galaxies the limiting magnitude is i $<20$ (see Kochanek et al., 2012, for details). For the XMM-COSMOS sample, the spectroscopic data come from existing SDSS spectra, the magnitude-limited zCOSMOS catalog (i $<22.5$, Lilly et al., 2009), and spectroscopic observations with MMT and IMACS/Magellan down to i $\approx 25$. For the SDSS DR5 quasar catalog, the spectroscopic depth is brighter ($i\lesssim 19.1$ for low-redshift quasars and $i<20.2$ for higher redshift quasars, see Schneider et al. 2007, for details.) As for the XXL-N X-ray AGN catalog, the spectroscopic depth of BOSS is r$<22.5$, which is deeper than the SDSS DR5 quasar catalog and similar to Boötes and XMM-COSMOS.

For the Boötes and XRT-SDSS samples, the mid-IR observations are from either Spitzer or WISE, and are complete for these optically luminous AGNs222Only for the three shorter wavelength WISE bands., thus these samples are only flux-limited in the optical and X-ray bands. Notably, the X-ray non-detected objects in the AGES and XRT-SDSS samples are still covered by X-ray observations, which allows us to take the X-ray non-detected sources into account when measuring the $L_{\rm X}-L_{\rm MIR}$ relation. Both the XMM-COSMOS and XXL-N samples are X-ray selected, so these catalogs are flux-limited in the X-ray and optical wavelengths. As for mid-IR observations, the Spitzer observations are complete for the XMM-COSMOS sample and the WISE coverage for the XXL-N sample is also highly complete ($\sim 87\%$).

3 Luminosities in the X-ray and mid-IR

In this section, we briefly describe the methods used to calculate the rest-frame $2-10$ keV luminosity and the mid-IR luminosity for the AGNs in each catalog. For comparison, we plot the redshift, X-ray luminosity at rest-frame $2-10$ keV ($L_{\rm X}$), and the luminosity of the AGN component at $6\;\micron$ ($L_{6\micron}$) in Figure 1. The $L_{\mathrm{X}}$ and $L_{6\micron}$ histograms are also shown in Figure 1.

3.1 X-ray luminosity

For the Boötes sample, the X-ray photon count rates in the 0.5-7 keV band are converted to a flux using a conversion factor of $7.8\times 10^{-15}$ ergs cm*-2* s*-1* for an object with 4 counts in a 5 ks exposure. This is derived based on the assumption of an unabsorbed X-ray spectrum with a photon index of $\Gamma=1.8$ (see Kenter et al., 2005; Murray et al., 2005, for a complete discussion). We then converted the 0.5-7 keV luminosity to the rest-frame 2-10 keV luminosity with a k-correction using the same spectral index. We note that Galactic absorption column density for the XBoötes survey is negligible ($\approx 10^{20}$ cm*-2*), but we still take it into account in our calculation for consistency with other samples.

For the XMM-COSMOS sample, the rest-frame $2-10$ keV X-ray luminosity comes from Lusso et al. (2010), in which the X-ray count rates in the $0.5-2$ keV and $2-10$ keV bands are converted into rest-frame $2-10$ keV luminosities using a Galactic column density $N_{\rm H}=2.5\times 10^{20}$ cm*-2* (see Cappelluti et al., 2009), and assuming photon indicies of $\Gamma=2$ and $\Gamma=1.7$ for the soft and hard bands, respectively. For consistency, we revise the 2 - 10 keV X-ray luminosities using the same approach and assumptions as applied to the XBoötes sample. This causes changes of $\sim 8\%$ relative to the values in the Lusso et al. (2010) due to the different assumptions regarding the intrinsic photon-index of the X-ray power-law spectrum.

For the XRT-SDSS sample, the estimate of X-ray luminosity varies depending on the quality of available X-ray data. We briefly describe the approach taken by W12 and refer the readers to §2 in Wu et al. (2012a) for details. In the XRT-SDSS sample, the observed-frame $0.3-10$ keV flux for each source with XRT counts $>100$ is derived by fitting the observed counts to estimate the X-ray power-law index and intrinsic absorption. For sources with between 30 and 100 XRT counts, the intrinsic absorption is fixed to zero while the spectral index is still a free parameter. For sources with XRT counts less than 30, the flux is obtained by assuming a fixed spectral index of $\Gamma=2$ and a fixed zero intrinsic absorption. For consistency, we again modify the derived $2-10$ keV $L_{\rm X}$ for the XRT-SDSS sample to account for the different choices in the AGN intrinsic spectral index. This causes changes of $\sim 15\%$ relative to the values reported in W12.

For the XXL-N sample, the X-ray luminosities come from Liu et al. (2016) who estimated the intrinsic X-ray luminosity by jointly fitting the XMM-Newton PN and MOS data with the Bayesian X-ray Analysis package (BXA, Buchner et al., 2014)333We note that one of the 1,153 AGNs is only detected at the 0.5–2 keV band. For this object we calculate its 2–10 keV $L_{\rm X}$ based on the 0.5–2 keV flux assuming a power-law X-ray spectrum with a $\Gamma=1.8$ photon index. The model used to fit the data is a combination of three different models that take the intrinsic power-law continuum, absorption, Compton scattering features, and a soft scattering component into account (see §4.1 of Liu et al., 2016, for details). To broadly assess if the BXA-based $L_{\rm X}$ are comparable to our other estimates of $L_{\rm X}$, we re-calculate the X-ray luminosities for the XXL-N sample by converting the 0.5–8 keV photon count rates reported in Liu et al. (2016) to rest-frame 2–10 keV $L_{\rm X}$ assuming a power-law X-ray spectrum with a $\Gamma=1.8$ photon index. The average difference from the BXA estimates is only 0.03 dex. This is not surprising, as type 1 AGNs X-ray AGNs have been found to have little to no absorption (e.g. Hickox et al., 2007). For this work, we adopt the BXA-based intrinsic $L_{X}$ to minimize the possible bias on the measured $L_{\rm X}-L_{\rm MIR}$ relation due to any X-ray absorption.

For the majority of type 1 quasars in AGES, XMM-COSMOS, and XRT-SDSS samples, the X-ray absorption correction to the X-ray luminosity is not available from spectral fitting due to the limited photon counts. Therefore, the uncertainties in their X-ray luminosities were estimated based on Poisson noises of the count rates calculated using the Gehrels (1986) method. For these luminous type 1 AGNs, the intrinsic X-ray absorption could be considered negligible. In particular, an X-ray stacking analysis has shown that the hardness ratios for type 1 AGNs in Boötes are consistent with little to no absorption ($N_{\rm H}\sim 10^{20}$ cm*-2*, Hickox et al., 2007). We discuss the possible effects of X-ray absorption on the observed $L_{\rm X}-L_{6\micron}$ relation further in §5.2.

3.2 SED fitting analysis and mid-IR luminosity

To estimate the contamination from the host galaxy to the AGN mid-IR luminosity, we use SED-fits to calculating the intrinsic, de-absorbed AGN mid-IR luminosity, including Herschel far-IR photometry when available.

We follow the approach described in Chen et al. (2015) by fitting the photometry with three different components: an AGN spanning from near-UV to far-IR, a stellar population in the host galaxy and a model for dust emission from reprocessed starlight.

We created ad hoc AGN templates by combining the near-UV to near-IR empirical AGN template from Assef et al. (2010) with the infrared AGN SEDs from Mullaney et al. (2011) and Netzer et al. (2007). For each AGN template, we create a grid of AGN templates with $0<E(B-V)<10$ using a hybrid extinction curve combining an SMC-like (Small Magellanic Cloud) extinction curve at $\lambda<3300$ Å (Gordon & Clayton, 1998) with a Galactic extinction curve at longer wavelengths (Cardelli et al., 1989), with $R_{V}=3.1$ for both (see Assef et al., 2010, for details).

For the host galaxy templates, we consider two different components: the contribution from the stellar population of the galaxy, which accounts for the optical to near-IR emission; and a starburst component, which represents the mid- to far-IR dust emission from re-processed stellar light. For the stellar population component, we adopt the three empirical galaxy templates from Assef et al. (2010) representing starburst (Im), continuous star-forming (Sbc) and old stars (elliptical), respectively. We follow the approach described in Chen et al. (2015) by replacing the $>4.9\;\micron$ hot dust components of the Sbc and Im templates assuming a Rayleigh-Jeans tail identical to the elliptical galaxy to create empirical stellar population templates without dust emission. For the starburst component, we use a total of 171 starburst templates from Chary & Elbaz (2001), Dale & Helou (2002), and Kirkpatrick et al. (2012) to accommodate a wide range of spectral shapes of star-forming galaxies.

Given the SED templates, we fit the observed photometry using a $\chi^{2}$ minimization algorithm to find these best-fit SED for each object. From the best-fit SEDs, we calculate the monochromatic luminosities of the AGN component at $6\;\micron$. To account for uncertainties in the derived $L_{6\micron}$ due to both the uncertainties in the flux measurements and any degeneracy between the AGN and host galaxy components, we employ a bootstrapping approach. For each source, we randomly scatter the original photometry in every band with their $1\sigma$ uncertainties and redo the SED-fits. We repeat this process 500 times for each source. For the Boötes, XMM-COSMOS , XRT-SDSS , and XXL-N samples, the median $L_{6\micron}$ uncertainties are $0.08$, $0.11$, $0.12$ and $0.13$ dex, respectively.

For type 1 AGNs in our sample, the majority of the sources have mid-IR SEDs dominated by the AGN component. The average AGN fraction (the absorbed AGN component contribution at $6\;\micron$) for the Boötes  XMM-COSMOS and XRT-SDSS  samples are $89\%$, $95\%$, $92\%$, and $81\%$, respectively. For $98\%$ of the AGNs in our final sample, the AGN component dominates (AGN fraction $>50\%$) at $6\;\micron$. However, there is a caveat when estimating the host galaxy contamination at mid-IR wavelengths with the SED fitting approach for the XRT-SDSS sample. Unlike the Boötes and the XMM-COSMOS samples that include far-IR observations, the XRT-SDSS sample relies on a very small number of WISE photometric data points to constrain the host galaxy contribution in the mid-IR. Since starburst activity can also produce strong mid-IR emission, we use the 228 far-IR detected sources in Boötes and XMM-COSMOS to examine whether the exclusion of far-IR photometry could affect the observed AGN $6\;\micron$ luminosity. We redid the SED-fits excluding the photometry at observed wavelengths longer than $24\;\micron$ and then compared the estimates for $L_{6\micron}$. We found a median difference of only $\pm 0.03$ dex, which is not surprising since the AGNs studied in this work are luminous optical quasars. This 0.03 dex uncertainty was added in quadrature to the measured $L_{6\micron}$ for all objects lacking far-IR photometry.

4 The Correlation between X-ray and mid-IR luminosities for X-ray detected quasars

In Figure 2, we show the $L_{\rm X}-L_{6\micron}$ distributions of sources with X-ray detections for all four catalogs. For comparison, we also show the Gandhi et al. (2009), Mateos et al. (2015)444We note that for the five most X-ray luminous quasars in this sample, one of them is a lensed quasar and three of them are radio-loud. Therefore, we do not include these objects in this plot. and Stern (2015) samples, along with the $L_{\rm X}-L_{6\micron}$ relation for local Seyfert galaxies from Gandhi et al. (2009). Clearly, the four catalogs studied in this work show a $L_{\rm X}-L_{6\micron}$ distribution departing from the Gandhi et al. (2009) relation and other roughly linear relations suggested by studies of local active galaxies (e.g., Lutz et al., 2004; Maiolino et al., 2007; Asmus et al., 2015). The $L_{\rm X}-L_{6\micron}$ distributions for our samples are in broad agreement with the Fiore et al. (2009) and Stern (2015) luminosity-dependent $L_{\rm X}-L_{6\micron}$ relations for luminous X-ray AGNs. We have converted the monochromatic luminosities measured at different wavelengths for these comparison samples (e.g. $5.8\micron$ and $12\micron$) to $L_{6\micron}$ using the Assef et al. (2010) AGN template.

To obtain a simple parametrized $L_{\rm X}-L_{6\micron}$  relation for AGNs spanning a wide range of AGN luminosity, we fit the combined Boötes, XMM-COSMOS, XRT-SDSS, and XXL-N samples assuming that their $L_{\rm X}-L_{6\micron}$ relation follows the bilinear equation

[TABLE]

discussed by Fiore et al. (2009). Here $L_{6\micron}^{\star}$ is the “breaking luminosity”, $(m_{1},b_{1})$ and $(m_{2},b_{2})$ stand for the slope and intercept for each segment of the bilinear relation. This equation is identical to the assumption that $L_{\mathrm{X}}$ and $L_{6\micron}$ follow a broken power-law relation in the linear space. Since the broken power law relation assumes continuity on the breaking point, the number of free parameters is 3, because $b_{2}=\log L_{6\micron}^{\star}\times(m_{1}-m_{2})+b_{1}$. We next fit the data using an iterative $\chi^{2}$ minimization algorithm (Levenberg–Marquardt) based on the MPFIT package in IDL. The best-fitting parameters for ($\log L_{6\micron}^{\star}$, $m_{1}$, $m_{2}$, $b_{1}$, $b_{2}$) are $(44.79,0.84,0.40,44.60,44.51)$, and the corresponding uncertainties are $(0.11,0.03,0.03,0.01,0.01)$. We show this best-fit broken power-law relation in Figure 3. The break luminosity of our bilinear relation is significantly higher than that found by Fiore et al. (2009). This is not surprising, as there are very few objects in the Fiore et al. (2009) sample with mid-IR luminosity smaller than their break luminosity. For our combined sample of X-ray detected type 1 AGNs, there are 1,301 sources with $L_{6\micron}$ smaller than our best-fit break luminosity, $L_{6\micron}=6.2\times 10^{44}$ ergs s*-1*. We also fit the data with a simple linear relation, $\log L_{X}=\alpha\times\log\frac{L_{6\micron}}{10^{45}\mathrm{ergs^{-1}}}+\beta$, and an f-test rejects this model over the bilinear model with a $>99.9\%$ confidence level according to the f-test probability. It is also important to note that the location of the break luminosity might depend strongly on how the sample populates the $L_{\rm X}-L_{6\micron}$ parameter space. To assess how the sparse distribution of our sample in $L_{\rm X}-L_{6\micron}$ affects the results, we divide our sample into six $L_{6\micron}$ bins of approximately equal size. We then weight the total $\chi^{2}$ of each bin by its source number such that $L_{6\micron}$ bins with smaller source numbers have similar statistical power to the $L_{6\micron}$ bins with larger source numbers. We find that the break luminosity increases by 0.5 dex with this approach.

In practice, the result of fitting a bilinear $L_{\rm X}-L_{6\micron}$ relation to an unevenly distributed AGN sample will not only depend on the intrinsic $L_{\rm X}-L_{6\micron}$ slopes, but also the relative numbers of low-luminosity AGNs and luminous quasars. Due to the volume and flux-limited nature of extragalactic surveys, it is extremely difficult to construct a sample that could populate the $L_{\rm X}-L_{6\micron}$ parameter space as evenly as our simple weighted $\chi^{2}$ minimization exercise effectively does. Therefore, we consider the result of this simple exercise an “upper limit” on the break luminosity of a bilinear $L_{\rm X}-L_{6\micron}$ relation, and conclude that the linear $L_{\rm X}-L_{6\micron}$ relation for lower luminosity AGNs cannot be extended to quasars that are more luminous than $L_{6\micron}=1.4\times 10^{45}$ ergs s*-1*.

In Figure 3, we compare our bilinear regression fit with the Gandhi et al. (2009) linear relation for local Seyferts and the second order polynomial fit of Stern (2015). While our bilinear relation is largely consistent with the second order polynomial fit of Stern (2015), it is not clear which regression model is the best option. Therefore, in addition to assuming that the $L_{\rm X}-L_{6\micron}$ distribution follows a specific functional form, it is also useful to adopt a non-parametric approach to visualize the relation between $L_{\mathrm{X}}$ and $L_{6\micron}$ of our sample.

We use the Gaussian process regression algorithm from Gpy555http://sheffieldml.github.io/GPy/with a polynomial kernel function to fit all of the X-ray detected sources studied in this work. The Gaussian process regression algorithm assumes that the dependent variable (the $L_{\rm X}$ in this work) of a sample of finite size could be described by a multivariate Gaussian function with dimensions equal to the sample size. With an assumed kernel function (i.e. the covariance function of the Gaussian process), the Gaussian process algorithm also takes the measurement uncertainties into account and analytically finds the best-fit multivariate Gaussian function. We can then compute the non-parametric prediction using the posterior probability function based on the Gaussian process regression results.

In Figure 3, we also show this non-parametric fit to the data with 1$\sigma$ uncertainty as the gray shaded region. We find that both our bilinear regression fit and the second order polynomial fit of Stern (2015) follow the non-parametric prediction closely, which demonstrates that a change of the $L_{\rm X}-L_{6\micron}$ slope occurs at $L_{6\micron}\sim 10^{44}$–$10^{45}$ $\textmd{erg}\;\textmd{s}^{-1}$.

5 Possible biases and comparison to the literature

In this section, we explore how the different observational constraints affect the estimate of the intrinsic $L_{\rm X}-L_{6\micron}$ relation for AGNs. For simplicity, we do not consider the broken power-law regression when addressing various observational constraints. Instead, we focus on how X-ray absorption, X-ray non-detections, and X-ray flux limits affect the linear $L_{\rm X}-L_{6\micron}$ relation of the more luminous objects in our sample (i.e., the quasars with $\log L_{6\micron}/$erg s${}^{-1}>43.8$, which corresponds to a bolometric luminosity of $10^{45}$ erg s*-1* assuming the Hopkins et al. 2007 bolometric correction factors).

5.1 Accounting for X-ray non-detected type 1 quasars

For the shallow flux limits of the Boötes and XRT-SDSS samples, a significant fraction of sources are not detected in the X-rays. For the Boötes type 1 AGNs, Hickox et al. (2007) used an X-ray stacking analysis to show that the X-ray properties of the X-ray undetected type 1 mid-IR quasars are consistent with luminous X-ray AGNs with little or no absorption. Therefore, it is important to take into account the average X-ray contribution for those optically unobscured quasars without a direct X-ray detection when deriving the $L_{\rm X}-L_{6\micron}$  correlation since their average X-ray luminosity could be non-negligible.

For the Boötes quasars, we used an X-ray stacking analysis to account for sources not individually detected in X-rays. We divide these quasars into five bins of $L_{6\micron}$ and calculated their stacked X-ray luminosity. We define the stacked X-ray counts as the average number of background-subtracted photons detected within the 90% point-spread function (PSF) energy encircled radius at 1.5 keV, $r_{90}$, where $r_{90}=1^{\prime\prime}+10^{\prime\prime}(\theta/10^{\prime})^{2}$ and $\theta$ is the angle from the Chandra optical axis666Chandra Proposers’ Observatory Guide (POG), available at http://cxc.harvard.edu/proposer/POG.. We only include sources within $\theta<6\arcmin$ ($r_{90}<4\farcs 6$) in the analysis. We adopt background surface brightnesses of 3.0 and 5.0 counts s*-1* deg*-2* for the $0.5-2$ keV and $2-7$ keV bands, based on the estimates of the diffuse background from Hickox et al. (2007). We convert count rates (counts s*-1*) to flux (ergs s*-1* cm*-2*) using the conversion factors $6.0\times 10^{-12}$ ergs cm*-2* count*-1* in the 0.5-2 keV band and $1.9\times 10^{-11}$ ergs cm*-2* count*-1* in the 2-7 keV band. To estimate the average X-ray stacking luminosity from the X-ray flux, we assume that all galaxies in each bin of $L_{6\micron}$ reside at the average luminosity distance for the galaxies in that bin. More details of the stacking procedure are described in §5.1 of Hickox et al. (2007).

To derive the $L_{\rm X}-L_{6\micron}$ relation including both the X-ray detected and non-detected sources, we calculate the average $L_{\mathrm{X}}$ in bins of $L_{6\micron}$ by taking the weighted average of the individually detected sources and the stacking luminosity. We find that the average $L_{X}$ of the entire Boötes sample has an L_{\rm X}\ -L_{6\micron}\ slope similar to that of the X-ray detected sources derived in §4 (also see Figure 3) with a smaller intercept: $\log L_{X}=(0.51\pm 0.06)\times\log L_{6\micron}/10^{45}\mathrm{erg\;s^{-1}}+(44.23\pm 0.05)$.

Another useful approach to take the X-ray non-detected sources into account when deriving the linear relation between $L_{\mathrm{X}}$ and $L_{6\micron}$ is the Bayesian maximum likelihood method presented by Kelly (2007, K07 hereafter). The K07 method determines the best-fit linear relation by sampling the $L_{\mathrm{X}}$ values of non-detected sources from the prior provided by the detected sources and the value of the upper limits. For the Boötes and the XRT-SDSS samples, the $L_{\rm X}$ upper limits for the X-ray non-detected sources were calculated using the flux limits and the corresponding redshift for each source. We then use the K07 method to recalculate the best-fitting parameters for the complete Boötes and XRT-SDSS samples, respectively. In detail, we use K07’s method to perform Markov Chain Monte Carlo simulation using a Metropolis–-Hastings algorithm sampler with 10,000 iterations to obtain the posterior probability distribution of the linear regression parameters. The best-fitting parameters are determined as the median of the posterior probability distributions of the model parameters. We adopt the 1$\sigma$ (68%) uncertainties as the 16 and 84 percentiles of the posterior probability distributions.

In the first part of Table 2, we list the best-fitting parameters for different samples with and without the consideration of X-ray non-detections calculated using the K07 method. For comparison, we also list the best-fit minimum $\chi^{2}$ regression result for the data from Boötes stacking analysis. We also list the best-fit parameters for XMM-COSMOS and XXL-N samples calculated using the K07 method. For the XRT-SDSS sample, due to the luminous nature of the SDSS DR5 catalog and the 10ks X-ray exposure time cut, the X-ray detection fraction is $>83\%$. This leads to a similar $L_{\rm X}-L_{6\micron}$  relation regardless of the treatment of the X-ray non-detections. For the Boötes sample, we find that the inclusion of X-ray non-detected sources does lead to a best-fit with a smaller $L_{\mathrm{X}}$ throughout the $L_{6\micron}$ range. This might be due to Eddington bias, as the XBoötes sources have as few as 4 counts in the 5 ks XBoötes Chandra observations. The slopes of the $L_{\rm X}-L_{6\micron}$ relation also do not change significantly when compared to the best-fit result for the X-ray detected sources. The slopes derived from the stacked average are also consistent with the results using the K07 regression analysis on samples with and without the inclusion of X-ray non-detections.

The results suggest that the inclusion of X-ray non-detected objects does not alter the $L_{\rm X}-L_{6\micron}$ relation significantly given the relatively high X-ray detection fraction of the Boötes and Swift/SDSS samples.

5.2 Effects of X-ray absorption on $L_{\rm X}-L_{6\micron}$

As we mentioned at the end of §3.1, a large fraction of our type 1 quasars do not have sufficient X-ray photon counts to properly correct for attenuation of the observed $L_{\mathrm{X}}$ due to the small gas absorption. The recent study of $L_{\rm X}-L_{6\micron}$ (Stern, 2015) using the type 1 quasars from Just et al. (2007) also assumed that type 1 quasars have negligible X-ray absorption column densities. While the average hardness ratio for type 1 AGNs does support this assumption (Hickox et al., 2007), a recent study by Merloni et al. (2014) that also focused on the XMM-COSMOS type 1 AGNs has also shown that a non-negligible fraction ($\sim 20\%$) of optical type 1 AGNs have a hardness ratio consistent with $N_{\rm H}>10^{22}$cm*-2* (see Figure 4 of Merloni et al., 2014).

Although the effect of absorption is less significant for high redshift quasars because the observed X-rays correspond to harder rest-frame energies, it is still important to understand the impact of gas absorption on the observed $L_{\rm X}-L_{6\micron}$ relation. We estimate the effect of the bias caused by the possible presence of X-ray obscuration by conservatively assuming that $20\%$ of the sources have $N_{\rm H}=10^{23}$ cm*-2*. We note that this assumption is an extreme case in which many of the optical type 1 AGNs are heavily obscured in the X-ray. For the XMM-COSMOS sample studied by Merloni et al. (2014), less than $5\%$ of the optical type 1 AGNs have hardness ratios consistent with $N_{\rm H}>10^{23}$ cm*-2*. We estimate the correction on the $L_{\rm X}-L_{6\micron}$ relations using the following steps: (1) Randomly select $20\%$ of the X-ray detected sources. (2) Assume these objects have an absorption column density of $N_{\rm H}=10^{23}$ cm*-2*, and calculate the “de-absorbed” X-ray luminosity with an intrinsic X-ray spectral index of $\Gamma=1.8$. (3) Recalculate the best-fit $L_{\rm X}-L_{6\micron}$ relations using the de-absorbed X-ray luminosities and the K07 method. (4) Repeat (1) through (3) for 1,000 times. The result of this bootstrapping analysis suggests that even for the unlikely case in which $20\%$ of the optical type 1 AGNs have heavy absorbing column densities, the slope of $L_{\rm X}-L_{6\micron}$ is only $\approx 0.01$ lower than the result neglecting absorption. The median values and standard deviations of the slopes and intercepts from the bootstrapping analysis are listed in Table 2.

A larger part of this is that Compton-thin absorption has little effect on X-ray luminosity estimates for higher redshift sources (e.g., see Just et al. 2007 and Stern 2015). At $z\sim 1$, even for an AGN obscured by $N_{\rm H}\sim 10^{23}$ cm*-2* column densities, its observed-frame 0.5-7 keV flux would only be attenuated by $\sim 50\%$. Therefore, it is not surprising that the $L_{\rm X}-L_{\rm MIR}$ relation does not change significantly even for the case in which $20\%$ of the type 1 AGNs are X-ray absorbed. Since gas absorption only has a small effect on the slope of the $L_{\rm X}-L_{\rm MIR}$ relation, the difference between the flattened $L_{\rm X}-L_{6\micron}$ relation found in our study (and Fiore et al. 2009, Stern 2015) and the linear relation that extends to luminous AGNs reported by Mateos et al. (2015) and Asmus et al. (2015) must be caused by factors other than the possible presence of absorption in optical type 1 AGNs.

5.3 Effects of X-ray flux limits

As we have shown in §5.1, the exclusion of X-ray non-detected sources could result in a biased $L_{\rm X}-L_{6\micron}$ relation, but the effect is within the uncertainty for the Boötes and XRT-SDSS samples due to their high X-ray detection fractions. In §5.2, we also show that X-ray absorption should have little effects on the $L_{\rm X}-L_{6\micron}$ relation for luminous quasars. Here we examine if the X-ray flux limits are the primary factor that drives the various $L_{\rm X}-L_{6\micron}$ relations reported in the literature. We note that the mid-IR survey flux limits for luminous quasars are more homogeneous across different studies than the X-ray flux limits, as almost every luminous optical quasar in studies of the $L_{\rm X}-L_{6\micron}$ relation have clear detections at mid-IR wavelengths.

Recently, Mateos et al. (2015, M15 hereafter) reported a $L_{X}-L_{6\micron}$ relation for the hard X-ray ($4.5-10$ keV) sample selected from Bright Ultra-hard XMM-Newton Survey (BUXS). They found an approximately linear $L_{X}-L_{6\micron}$ relation even for X-ray luminous quasars with $L_{\rm X}$ up to $\approx 10^{46}$ erg s*-1*, in disagreement with our results and the results from Fiore et al. (2009); Stern (2015).777M15 fitted the SEDs of their sources with separate AGN accretion disk and AGN torus components. Thus the $L_{6\micron}$ in M15 is inevitably lower than the $L_{6\micron}$ of our work, which is derived by decomposing the AGN and host galaxy components instead of separating the AGN accretion disk and AGN torus components. However, we note that the difference is small as the typical AGN accretion disk contribution is very small at $6\;\micron$ ($\sim 9\%$, according to M15). In this work, we compare the $L_{X}-L_{6\micron}$ relation from M15 with our results by adding the average $9\%$ AGN accretion disk emission back to the $L_{6\micron}$ reported in that work. The M15 sample has a hard ($4.5-10$ keV) X-ray flux limit of $6\times 10^{-14}$ erg s*-1cm-2*. To convert the flux limit to an energy range comparable to the $0.5-7$ keV of the Chandra observations, we assume a simple power-law X-ray spectrum with a photon index $\Gamma=1.8$. Thus, the flux limit for the BUXS is equivalent to $\sim 1.6\times 10^{-13}$ erg s*-1cm-2* at $0.5-7$ keV, which is approximately 2 dex shallower than the XMM-COSMOS sample and 1 dex shallower than the Boötes and the XRT-SDSS samples.

To test whether the different slope observed in M15 is due to the shallow X-ray flux limit, we apply several different X-ray flux limits to the Boötes sample and examine their effect on the slope of the derived $L_{\rm X}-L_{6\micron}$ relation. We first apply the XRT-SDSS 10ks and 5ks flux limits and the XXL-N flux limit to the Boötes sample, and find that the $L_{X}/L_{6\micron}$ slopes for the Boötes sample decrease for lower flux limits.

We next apply the converted BUXS flux limits to the Boötes sample and recalculate the $L_{\rm X}-L_{6\micron}$ relation using the K07 method. Due to the shallowness of the BUXS flux limit, there are only 38 sources in Boötes with an X-ray flux larger than $1\times 10^{-13}$ erg s*-1cm-2*. Considering the difference in survey area, the number of sources is consistent with the M15 type 1 sample. For the Boötes type 1 AGNs, only 4% of the sources would have a flux limit higher than that of the M15 sample, suggesting that surveys with shallow X-ray flux limits wll produce a biased $L_{\rm X}-L_{6\micron}$ relation because they miss the vast majority of the AGN population of similar $L_{\rm MIR}$.

The best-fit $L_{\rm X}-L_{6\micron}$  slope estimated using the K07 method for the Boötes subsample with the $1\times 10^{-13}$ erg s*-1cm-2* cut is significantly steeper than the original Boötes sample. This highlights the necessity of deep X-ray observations in order to reveal the intrinsic $L_{\rm X}-L_{6\micron}$ relation when the X-ray survey flux limits are too shallow. Figure 4 shows the effect of survey X-ray flux limits on the slope of the $L_{\rm X}-L_{6\micron}$ relation.

It is also interesting that the best-fit slope for the XXL-N sample is $0.59\pm 0.02$, which is higher than that of the X-ray detected AGNs in Boötes, $0.50\pm 0.02$ despite their similar X-ray flux limits. This is likely due to the fact that XXL-N has more extremely luminous sources due to its larger survey volume. It is also possible that a small number of high redshift radio-loud quasars in XXL-N were not identified in the shallow VLA FIRST catalog ($\sim 1$ mJy). Nonetheless, the best-fit $L_{\rm X}-L_{6\micron}$ slopes for the four samples studied in this work are still much smaller than the results reported in previous studies with much shallower flux limits, suggesting that the intrinsic $L_{\rm X}-L_{\rm MIR}$ relation could only be recovered by considering samples selected from both deep and wide X-ray surveys.

5.4 Host galaxy contamination at mid-IR wavelengths

In contrast to X-ray flux limits that might cause a steeper $L_{X}/L_{6\micron}$ relation, contamination from host galaxies at mid-IR wavelengths could cause the observed $L_{X}/L_{6\micron}$ relation to be shallower than its intrinsic value (e.g., Lusso et al., 2013).

We have carefully modeled the possible cool dust contamination using the strong constraints provided by Herschel observations at far-IR wavelengths. Here we further scrutinize the SEDs for the X-ray non-detected, mid-IR bright type 1 quasars in our Boötes and XMM-COSMOS samples. We find that they have a median AGN fraction of $88\%$ at rest-frame $6\;\micron$, suggesting that their mid-IR SEDs are almost entirely dominated by the AGN component. For the high $L_{6\micron}$ sources in our sample, the W3 and MIPS $24\;\micron$ bands still show no signs of strong PAH emission or silicon absorption. While it is possible for strong nuclear starbursts to produce mid-IR continuum with spectra similar to that of the AGN template (e.g., Ballantyne, 2008), local nuclear starburst galaxies are often hosted by less massive galaxies with moderate luminosity AGNs with $L_{\rm X}$ less than the average $L_{\rm X}$ of our mid-IR bright quasars. In §3.2, we also demonstrated that the host galaxy contamination is small even for far-IR luminous objects in Boötes and XMM-Newton. Thus, we argue that for our sample of luminous quasars, it is unlikely that the high $L_{6\micron}$ derived from the best-fitting AGN template is due to substantial contamination from their host galaxies. Since the mid-IR luminosity in our type 1 quasars is indeed powered by the AGN, the shallower $L_{\rm X}-L_{6\micron}$ slope suggests that the AGN X-ray luminosity does not trace the AGN mid-IR luminosity in the same fashion as seen in local Seyfert galaxies.

6 Discussion and conclusion

In this paper, we study the possible origins of the disparity between the different $L_{X}-L_{6\micron}$ relations found in the literature. We assemble samples of spectroscopically confirmed broad-line AGNs (type 1) across a wide range of X-ray survey areas and depths to investigate the AGN intrinsic $L_{X}-L_{6\micron}$ relation for luminous quasars. We test several observational constraints that could bias the observed $L_{\rm X}-L_{6\micron}$ relation, including intrinsic X-ray absorption, host galaxy contamination at mid-IR wavelengths, and X-ray survey flux limit. We argue that the most important factor that differentiates the $L_{\rm X}-L_{6\micron}$ relations from different studies is the X-ray survey flux limits (i.e., the Eddington bias), as we find that other factors do not affect the $L_{\rm X}-L_{6\micron}$ relation of luminous quasars significantly.

For the 2,509 X-ray detected AGNs in our sample, we find that their $L_{\rm X}-L_{6\micron}$ relation could be well-described by the bilinear function

[TABLE]

where the break luminosity is $\log L_{6\micron}/{\rm erg\;s}^{-1}\approx 44.79\pm 0.11$. For luminous quasars, the slope of their $L_{\rm X}-L_{6\micron}$ relation is significantly flatter than the approximately linear relation observed in low- to moderate-luminosity AGNs (Gandhi et al., 2009; Asmus et al., 2015), which supports studies that suggest type 1 quasars have higher $L_{\rm MIR}$ to $L_{\rm X}$ ratios than their local Seyfert counterparts (Fiore et al., 2009; Stern, 2015). The fit in Equation (2) does not take the X-ray non-detected AGNs into account, but we also show that the inclusion of X-ray non-detected AGNs does not affect the $L_{\rm X}-L_{6\micron}$ slope significantly given the high X-ray detection fractions of our samples.

Since the rest-frame mid-IR emission in AGNs originates from hot dust heated by UV photons from the accretion disk, it is natural to consider the well-studied ratio between X-ray and UV monochromatic luminosities ($\alpha_{ox}=0.38(\log L_{2\mathrm{keV}}/L_{2500\mathrm{\AA\ }})$) to explain the luminosity-dependent $L_{\rm X}-L_{6\micron}$ relation. As pointed out by a number of studies (e.g., Tananbaum et al., 1979; Strateva et al., 2005), the $\alpha_{\rm OX}$ to AGN UV luminosity relation suggests that UV-luminous AGNs have relatively weak X-ray emission compared to their less-luminous counterparts. On the other hand, the radiation mechanism of AGN rest-frame mid-IR emission is driven by the UV photons from the accretion disk and the geometry of the dusty torus itself. While several dusty torus models and observations have described the effect of increasing AGN UV luminosity on the geometry of the dust distribution (Lawrence, 1991; Lusso et al., 2013), these models only predict a luminosity-dependent AGN obscured fraction and no drastic change of AGN UV to mid-IR spectral shapes is suggested. In fact, observational studies by Treister et al. (2008) and Lusso et al. (2010) have shown that the AGN mid-IR to bolometric luminosity ratio is only weakly dependent on the AGN bolometric luminosity. Studies of average SDSS quasar SEDs with different infrared luminosities have also found marginal variation in the average UV to mid-IR SEDs (Richards et al., 2006; Assef et al., 2011).

The relatively weak X-ray emission in luminous quasars has also been found in studies of X-ray bolometric corrections (e.g., Hopkins et al., 2007; Vasudevan & Fabian, 2007; Lusso et al., 2013), which suggest that X-ray bolometric correction factors increases for AGNs with higher accretion efficiency (i.e., with higher Eddington rates). The increase of X-ray bolometric correction factor in luminous quasars has also been implicitly suggested by the luminosity-dependent density evolution of the AGN X-ray luminosity function (Aird et al., 2010), as the rapid drop of the X-ray luminosity function implies that the most luminous X-ray AGNs are extremely rare. On the other hand, the AGN mid-IR luminosity function does not drop as significantly (e.g. Brown et al., 2006).

If AGN X-ray and mid-IR luminosities followed the tight $L_{\rm X}-L_{6\micron}$ correlation seen in the local Seyfert galaxies, there would be as many as 40 type 1 quasars with $L_{X}>10^{44.7}$ erg s*-1* in the survey volume of XBoötes . However, according to the recent Aird et al. (2010) X-ray luminosity function measured using deep X-ray surveys, the number of AGNs with $L_{X}>10^{44.7}$ erg s*-1* should only be 22.3 in the volume and redshift range of the Boötes dataset used in this work. In Boötes, there are only 18 sources more luminous than $L_{X}>10^{44.7}$ erg s*-1*. Indeed, recent discoveries of the most infrared luminous AGNs in the universe, the hot dust-obscured galaxies (hot-DOGs, e.g. Eisenhardt et al., 2012; Wu et al., 2012b; Tsai et al., 2015) have also found that their intrinsic X-ray luminosity is more consistent with the luminosity-dependent $L_{\rm X}-L_{6\micron}$ relations (Stern et al., 2014; Assef et al., 2016; Ricci et al., 2016).

In conclusion, we have shown that for type 1 quasars, the relationship between AGN mid-IR and X-ray luminosities is not a simple power law, as has been observed for nearby Seyfert-luminosity AGNs. This result is crucial for the studies of AGN-galaxy coevolution, as the dynamical range of X-ray luminosities is considerably smaller than the dynamical range of AGN mid-IR luminosities.

Much of this work was in progress at the time of Steve Murray’s untimely death in 2015. This work would not be possible without his generous GTO contribution to the XBoötes dataset and his strong support of Boötes multiwavelength surveys. He will be greatly missed by all of us. We thank the referee for a careful reading and constructive suggestions that have improved the manuscript. C.-T.J.C was supported by the William H. Neukom 1964 Institute for Computational Science. R.C.H. acknowledges support from the National Science Foundation through AST award number 1515364 and from an Alfred P. Sloan Research Fellowship. R.J.A was supported by FONDECYT grant number 1151408. This publication makes use of data products from the Wide-field Infrared Survey Explorer, which is a joint project of the University of California, Los Angeles, and the Jet Propulsion Laboratory/California Institute of Technology, funded by the National Aeronautics and Space Administration. This publication also makes use of data products from the Two Micron All Sky Survey, which is a joint project of the University of Massachusetts and the Infrared Processing and Analysis Center/California Institute of Technology, funded by the National Aeronautics and Space Administration and the National Science Foundation. Funding for SDSS-III has been provided by the Alfred P. Sloan Foundation, the Participating Institutions, the National Science Foundation, and the U.S. Department of Energy Office of Science. The SDSS-III website is http://www.sdss3.org/.