PRIMUS: INFRARED AND X-RAY AGN SELECTION TECHNIQUES AT 0.2 < z < 1.2

In the CDFS, ES1, and XMM fields, we use existing Spitzer IRAC imaging from Data Release 2 (DR2) of the Spitzer Wide-area Infrared Extragalactic Survey (SWIRE; Lonsdale et al. 2003). Within the shallow CDFS-SWIRE (orange outline in Figure 1), we include deeper source catalogs within the CDFS-SIMPLE region (solid red outline) and even deeper data in the CDFS-GOODS region (solid red polygon), both taken from version 3.0 of the Spitzer IRAC/MUSYC Public Legacy in E-CDFS (SIMPLE) survey data release (Damen et al. 2011). In the COSMOS field we use existing IRAC imaging from the S-COSMOS Survey (Sanders et al. 2007).

In fields with SWIRE coverage, we adopt source catalogs provided by the SWIRE team as part of their SWIRE DR2.¹² These catalogs were generated by performing source detection in each of the four IRAC channels and merging the source lists (see below for more details). For the CDFS-SIMPLE region with deeper IRAC data, we compared the measured fluxes for sources in both the SWIRE catalog and the SIMPLE catalog (Damen et al. 2011). Based on this comparison, we find that the SIMPLE fluxes need to be scaled by 2.02, 1.96, 1.69, and 1.56, in IRAC channels 1, 2, 3, and 4, respectively, to match the measured fluxes from the CDFS-SWIRE catalog. This multiplication factor accounts for the total flux correction applied by the SWIRE team compared to the 40 aperture used by the SIMPLE team and a small zero point difference. The errors are also multiplied by the above factors, and agree with the SWIRE errors after accounting for the difference in exposure times. The SIMPLE catalog includes our deepest IRAC data in the CDFS-GOODS region.

For the S-COSMOS IRAC data, we reproduce the SWIRE source detection procedure (as outlined in the SWIRE DR2 documentation) to ensure we measure robust fluxes and errors using a consistent technique. We downloaded the mosaic images in each IRAC channel from the NASA/IPAC Infrared Science Archive¹³ (May 2007 release). We use Astronomical Point source EXtractor pipeline in MOPEX to perform source detection in the mosaic for each channel on an individual basis. We combined the individual catalogs using the BANDMERGE procedure in MOPEX. The merged sources in the final catalog are required to be detected with signal-to-noise ratio (S/N) ⩾ 10 in channel 1 and S/N ⩾ 5 in channel 2. We follow the SWIRE handbook and apply a "coverage-S/N" flux limit, which is a factor proportional to the inverse of the square root of the coverage (C), the number of times that the object's location was scanned by IRAC in a given channel. This threshold limits flux measurements to where the flux of a detected object is large compared to the statistical fluctuations in the background sky in a given channel. Due to the deeper coverage of Spitzer IRAC scans in the COSMOS field, we require that the coverage threshold be 3, 3, 16, and 50 μJy $\times \sqrt{4/C}$ for channels 1, 2, 3, or 4, respectively, to match the SWIRE fields. For the majority of sources, our flux measurements are similar to those in the S-COSMOS public catalog, although the public catalog tends to have larger errors for all objects.

We combine our various fields and data sets into three "IR surveys depths" with data of comparable depth.

• 1.  
  GOODS, the deepest area of IRAC imaging within the CDFS field.
• 2.  
  COSMOS, including the S-COSMOS data and the SIMPLE coverage in the CDFS field that reaches comparable flux limits.
• 3.  
  SWIRE, which includes the entire area of the XMM and ES1 fields as well as the large area in the CDFS field with SWIRE coverage.

In Figure 2, we show the number density of sources for each survey depth as a function of 5.8 μm flux. We require a source to be detected in all four IRAC bands, which is necessary for our analysis in this paper. We set a 5.8 μm flux limit for each IR survey depth that is just above the point where the number counts start to turn over (vertical colored lines), indicating incompleteness in the IRAC-detected samples. Requiring a detection to the 5.8 μm flux limit guarantees an S/N above 2.4 and 2.1 in channels 3 and 4, respectively, with the median source having an S/N of approximately 10 in channels 3 and 4. These flux limits are given in Table 1. Our SWIRE IR survey depth includes the entire XMM, ES1, and CDFS fields (21.55 deg² in total); our COSMOS IR survey depth includes COSMOS and the SIMPLE region of the CDFS field (3.05 deg²). Our deepest IR survey depth (GOODS) includes only the small GOODS region of the CDFS field (0.06 deg²). In the rest of this paper, we adopt these limits and the corresponding area coverages when analyzing data to a given IR survey depth. The solid black line in Figure 2 shows the overall number density combining all of our fields that probe above the flux limit for each IR survey depth.

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PRIMUS (Coil et al. 2011) is the largest intermediate-redshift, faint-galaxy spectroscopic survey performed to date and covers ∼9 deg² in seven different fields with existing deep multi-wavelength imaging. We obtain low-resolution (R ∼ 40) spectra for ∼300,000 objects from the IMACS instrument on the Magellan I Baade 6.5 m telescope, targeting 80% of galaxies with i ≲ 22, with a statistically complete sample to a depth of i ∼ 23. By fitting galaxy, broad-line AGNs, and stellar spectral templates to low-resolution spectra and optical ground-based photometry, we have measured ∼120,000 robust redshifts with a precision of σ_z/(1 + z) < 0.5% and a catastrophic outlier rate of ≲3% (Δz/(1 + z) ⩾ 0.03). We classify objects as galaxies, broad-line AGNs, or stars based on the χ² of the best template fits. We derive K-corrections (Blanton & Roweis 2007) from the photometry. For further details of the survey design, targeting, and data, see Coil et al. (2011); for details of the data reduction, redshift confidence and precision, and completeness, see Cool et al. (2013).

Here, we use PRIMUS redshifts between 0.2 < z < 1.2 with high-confidence quality flags (Q ⩾ 3; see Coil et al. 2011). In the CDFS field, the PRIMUS science observations cover part of CDFS-SWIRE but are disjoint from the CDFS-SIMPLE and CDFS-GOODS area. We therefore also include the PRIMUS CDFS calibration field, which overlaps with the CDFS-GOODS and CDFS-SIMPLE areas, to increase the number of deep IRAC- and X-ray-detected objects with PRIMUS redshifts for Section 6. We thus have a total of 31,998 galaxies and broad-line AGNs with high-confidence PRIMUS redshifts in the COSMOS, CDFS, ES1, and XMM fields. Figure 1 shows the overlaps between the PRIMUS footprint and the IRAC and X-ray coverage. Table 2 gives the areas of the total and overlapping regions of each data set.

We have compiled published X-ray source catalogs based on existing Chandra and XMM-Newton X-ray data in the COSMOS, CDFS, ES1, and XMM fields (see Aird et al. 2012 for details). In all of the fields we use the likelihood ratio matching technique (e.g., Sutherland & Saunders 1992; Ciliegi et al. 2003; Brusa et al. 2007; Laird et al. 2009) to identify counterparts for each of the X-ray sources in the PRIMUS targeting optical photometry. We also apply the likelihood ratio technique to assign secure IRAC counterparts to the X-ray sources in the entire area with both X-ray and Spitzer IRAC coverage. Candidate counterparts are identified within 5'', although over 80% of likelihood ratio matched IRAC counterparts are within 2'' of their X-ray counterpart. The likelihood ratio matching technique accounts for optical, IR, and X-ray positional uncertainties, the probability of having a counterpart with a given magnitude, and the probability of a spurious match. Where multiple counterparts exist, we choose the match with the highest likelihood ratio and restrict to "secure" counterparts with likelihood ratios >0.5.

Within the CDFS field, we use the Luo et al. (2008) 2 Ms CDFS X-ray source catalog, one of the deepest Chandra surveys to date, and the four overlapping 250 ks Chandra point source catalogs of Lehmer et al. (2005). The 2 Ms catalog covers all of the CDFS-GOODS IRAC coverage (0.058 deg²), and part of CDFS-SIMPLE (0.25 deg²) and CDFS-SWIRE (0.31 deg²) coverage, to a depth of f_2-8 keV ∼ 5.5 × 10⁻¹⁷ erg s⁻¹ cm⁻². The 250 ks catalog surrounds the CDFS-GOODS field and covers more of the CDFS-SIMPLE (0.068 deg²) and CDFS-SWIRE (0.12 deg²) area to a depth of f_2-8 keV ∼ 6.7 × 10⁻¹⁶ erg s⁻¹ cm⁻².

The COSMOS field was observed with XMM-Newton (Hasinger et al. 2007) over the entire 2 deg² to a depth of f_2-10 keV ∼ 3 × 10⁻¹⁵ erg s⁻¹ cm⁻² and with much deeper Chandra data reaching f_2-10 keV ∼ 8 × 10⁻¹⁶ erg s⁻¹ cm⁻² for the central ∼0.9 deg² (Elvis et al. 2009). There are a total of 2769 X-ray sources in the area with both IR and X-ray coverage, and 2075 X-ray sources with X-ray, PRIMUS, and IR coverage; 383 of these have robust PRIMUS redshifts and classifications.

We use the Puccetti et al. (2006) point source catalog in the ES1 field from a mosaic of four partially overlapping XMM-Newton pointings which reached a depth of f_2-10 keV ∼ 2 × 10⁻¹⁵ erg s⁻¹ cm⁻² covering 0.52 deg² of the 0.9 deg² of PRIMUS and 2.16 deg² with IR coverage. Within the PRIMUS and IR areas (see Figure 1), there are 371 and 395 X-ray sources, respectively; there are 73 X-ray sources with secure PRIMUS redshifts.

The XMM X-ray data are from the Subaru/XMM-Newton Deep Survey (Ueda et al. 2008), which contains 7 deep XMM-Newton pointings, and the XMM-LSS X-ray survey (Pierre et al. 2007), which contains 45 pointings. The combined catalog contains X-ray sources to a depth of f_2-10 keV ∼ 2 × 10⁻¹⁵ erg s⁻¹ cm⁻² and covers 2.16 deg² of PRIMUS and 2.16 deg² of the IR area. There are 1840 and 4466 sources in the PRIMUS and IR overlap area, and 264 objects with secure redshifts.

Objects that are detected in all four IRAC bands are defined to be Stern et al. selected IR-AGNs if they have IRAC colors such that they lie within the following region in color–color space:

$$\begin{eqnarray} ({\rm [5.8]}-{\rm [8.0]}) &\,{>}\,& 0.6 \hbox{,} \end{eqnarray} \tag{ 1 }$$

$$\begin{eqnarray} ({\rm [3.6]}-{\rm [4.5]}) &\,{>}\,& 0.2\cdot ({\rm [5.8]}-{\rm [8.0]}) + 0.18 \hbox{, and} \end{eqnarray} \tag{ 2 }$$

$$\begin{eqnarray} ({\rm [3.6]}-{\rm [4.5]}) &\,{>}\,& 2.5\cdot ({\rm [5.8]}-{\rm [8.0]}) - 3.5. \end{eqnarray} \tag{ 3 }$$

We apply this selection technique for samples reaching our three different "IR survey depths" defined in Section 2.1 above. We identify a total of 1137, 10,343, and 6687 Stern et al. IR-AGNs in our GOODS, COSMOS, and SWIRE depth surveys, respectively. We show the number density of Stern et al. selected objects with X-ray coverage in Figure 2.

We also select IR-AGN samples using the IRAC color criteria presented by Donley et al. (2012). This color-selection technique was designed to limit contamination by star-forming galaxies, especially at high redshift, but still be both complete and reliable for the identification of luminous AGNs. We require that objects are detected in all four bands, and have IRAC colors such that they lie within the following region in IRAC color–color space:

$$\begin{eqnarray} &&x={\rm log_{10}}\left(\frac{f_{\rm 5.8\, {\mu }{\rm m}}}{f_{\rm 3.6\, {\mu }{\rm m}}} \right), \quad y={\rm log_{10}}\left(\frac{f_{\rm 8.0\, {\mu }{\rm m}}}{f_{\rm 4.5\, {\mu }{\rm m}}} \right) \end{eqnarray} \tag{ 4 }$$

$$\begin{eqnarray} x &\ge & 0.08 \hbox{ and } y \ge 0.15 \end{eqnarray} \tag{ 5 }$$

$$\begin{eqnarray} y &\ge & (1.21\times {x})-0.27 \end{eqnarray} \tag{ 6 }$$

$$\begin{eqnarray} y &\le & (1.21\times {x})+0.27 \end{eqnarray} \tag{ 7 }$$

$$\begin{eqnarray} f_{\rm 4.5\, {\mu }{\rm m}} &\,{>}\,& f_{\rm 3.6\, {\mu }{\rm m}} \hbox{ and } f_{\rm 5.8\, {\mu }{\rm m}} > f_{\rm 4.5\, {\mu }{\rm m}}, \hbox{ and } \end{eqnarray} \tag{ 8 }$$

$$\begin{eqnarray} f_{\rm 8.0\, {\mu }{\rm m}} &\,{>}\,& f_{\rm 5.8\, {\mu }{\rm m}}. \end{eqnarray} \tag{ 9 }$$

Like in Section 3.1, we apply this selection method to our three different IR survey depth samples. We identify a total of 212, 2657, and 4151 Donley et al. IR-AGNs in our GOODS, COSMOS, and SWIRE depth surveys, respectively. The number density of Donley et al. selected objects with X-ray coverage is shown in Figure 2.

We also investigated the identification of IR-AGN samples using a power-law selection technique (Alonso-Herrero et al. 2006; Polletta et al. 2006; Donley et al. 2007). This technique identifies IR-AGNs by fitting a power-law function to the IRAC fluxes, in contrast to the color selection techniques such as the Stern et al. and Donley et al. methods described above. Sources dominated by a power-law IR-AGN have a negative slope α < −0.5 such that their νF_ν flux increases with wavelength (Alonso-Herrero et al. 2006). Following Donley et al. (2007), we identify sources with four detected IRAC flux measurements as power-law AGNs when they are well fit by a power law of slope α ⩽ −0.5, where f_ν∝ν^α. We deem the power-law fit as good if P_χ > 0.1 (probability that the fit would have a χ² greater than the found χ²). We calculate the P_χ and fit parameters using the linfit procedure in IDL.

As in Sections 3.1 and 3.2, we apply the power-law selection technique to our SWIRE, COSMOS, and GOODS IR survey depths. To properly simulate a shallow SWIRE-like survey in our COSMOS depth fields (COSMOS and CDFS-SIMPLE), we not only apply the 5.8 μm flux limits but also scale up the errors in the flux measurements to accurately represent the characteristics of the shallower data. Specifically, we scale the errors in the fluxes by the square root of the differences in the exposure times for the COSMOS and SWIRE depth data. We verify that the median of our scaled errors matches the median error in the SWIRE depth fields.

In Figure 2, we show the number density of IR-AGNs identified using the power-law selection technique as a function of 5.8 μm flux. Unlike the Stern et al. and Donley et al. samples, the number density of the power-law sample is not a smooth function of IR flux. Donley et al. (2012) show that this is mainly due to the flux errors associated with each survey. While we have ensured similar data reduction and consistent error estimates for the different fields and samples, the uncertainty in a flux measurement will depend on the depth of the data. At a given flux, deeper IR surveys find fewer power-law AGNs as the smaller flux errors can result in larger χ² values for the power-law fit. Thus, a source identified as a power-law AGN in a shallow survey may not be identified in a deeper survey as the more precise photometry are found to deviate from a power law. Additionally, the power-law sample is a very small subsample of the total X-ray population, a major problem in selecting a large statistical population of AGNs.

Donley et al. (2012) suggest adopting a 10% uncertainty floor on all flux measurements. While this increases the number density of power-law-selected AGNs in our samples to that of Donley et al. (2012), the number density of these sources does not monotonically increase with increasing survey depth, due to the strong dependence of the power-law technique on the estimated flux uncertainty. The power-law technique appears to be a reliable method of selecting AGNs (in that a very high fraction have X-ray counterparts), however it selects a very small subsample of the full AGN population identified either with X-ray emission or with the Stern et al. or Donley et al. AGN selection techniques using SExtractor errors. Given the small sample size and the complex dependence on the depth of the IR data, we choose not to consider the power-law IR-AGN selection technique any further in the analysis of this paper.

We define samples of X-ray-selected AGNs based on detection in the hard (2–10 keV) energy band in our compiled catalogs (see Section 2.3). A hard-band-selected sample includes both X-ray unabsorbed and moderately absorbed AGNs, but misses Compton-thick sources. We do not include a soft-band-selected sample even though it probes to fainter X-ray fluxes (providing a larger sample) because it is more biased toward the detection of unabsorbed sources (with equivalent hydrogen column densities N_H  <  10²² cm⁻²).

Selecting in a single band also ensures we can accurately track (and correct for) the effects of variation in the X-ray sensitivity in our observations. Within an individual Chandra or XMM-Newton pointing, variations due to the vignetting and point-spread function of the telescope lead to variations in the X-ray flux limit that is reached across the field of view. Furthermore, the exposure times between different pointings can vary within an X-ray survey, leading to substantial variations in sensitivity. The minor differences in the X-ray spectral shapes at 2–10 keV have a minimal effect on the sensitivity and the differences are mainly due to the X-ray telescope. Furthermore, at the X-ray fluxes we are probing (f_2-10 keV ∼ ≳ × 10⁻¹⁶ erg s⁻¹ cm⁻²), the X-ray point source number counts are dominated by AGNs (Georgakakis et al. 2008; Lehmer et al. 2012). Thus, by comparing observed point source counts as a function of flux to the expected number based on the intrinsic log N−log S relation, we can estimate the fraction of our survey area that is sensitive to X-ray AGNs of a given X-ray flux.

In Figure 3, we show X-ray area curves (the area of a survey that is sensitive to sources above a given X-ray flux) for our various fields. We calculate these area curves using the ratio of the number of hard X-ray-detected sources in the parent X-ray source catalog and the predicted number of sources based on the X-ray log N−log S relation of Georgakakis et al. (2008), using bins of 0.2 dex width. We fix the area curve at the total area coverage above the lowest flux where ⩾90% of the predicted sources are detected to remove the effects of low source numbers at bright fluxes. To reduce the noise due to small numbers of sources at the faintest X-ray fluxes, we set a minimum flux threshold in each X-ray survey at the point where at least 10% of the expected number of sources are detected. These flux limits are shown in Figure 3 and given in Table 1.

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Similar to our IR observations, we group our X-ray fields into three different "X-ray survey depths."

• 1.  
  ∼1 Ms, which includes our deepest, 2 Ms X-ray data in the CDFS field.
• 2.  
  ∼100 ks, including the E-CDFS, COSMOS, and the deeper regions of the XMM field.
• 3.  
  ∼10 ks, including the ES1 field and the remainders of the COSMOS and XMM fields.

We give nominal hard-band flux limits for each field and survey depth in Table 1. While there are slight differences in the exact X-ray flux limit for each individual survey we show the surveys at one of the three "X-ray survey depths" for simplicity. However, we note that this flux limit is only reached over a small fraction (>10%) of the total area covered by our fields with a given "X-ray survey depth."

Finally, we calculate an "X-ray weight" for each hard-band X-ray-detected source based on the ratio of the total area coverage to the value of the area curve at the flux of the source. We use these X-ray weights to correct observed number densities of X-ray sources to the intrinsic number density. A faint X-ray source will be given a large weight to account for the small area that is sensitive to faint fluxes. For the total X-ray sample, the X-ray weights ensure we recover the intrinsic X-ray log N−log S as measured by Georgakakis et al. (2008) that was originally used to determine the area curves. However, the X-ray weights are also crucial when comparing between X-ray and IR-AGN samples (see Section 4).

In Section 6, we further limit our X-ray-detected sample to objects with an observed L_X  >  10⁴² erg s⁻¹, calculated from the hard-band flux assuming a photon index of Γ = 1.9. This conservative limit ensures our sample contains objects where the X-ray flux is dominated by emission due to an AGN, rather than star formation processes in the galaxy.

When comparing the X-ray AGN sample with the IRAC AGN samples we require that both samples lie within the overlapping window functions. Likewise for Section 6, we require that the objects fall within the region with IRAC, X-ray, and PRIMUS coverage. For these studies, we recompute our X-ray weights using the overlapping areas only.

Our first step is to examine the distribution of sources in the bivariate space of observed IRAC 5.8 μm flux and hard-band X-ray flux. In the center panel of Figure 4, we show contours (black) that trace the distribution of all sources (across our differing survey depths) with both X-ray and IRAC detections. The contours contain 30%, 50%, and 80% of these sources; the remaining 20% of sources outside these contours are shown by black squares. We also show the distribution of sources that are also identified as IR-AGNs using the Stern et al. (red) and Donley et al. (blue) selection techniques. Approximate IRAC and X-ray flux limits for the different survey depths are shown by the orange vertical and horizontal lines. We find that the majority of objects tend to be infrared bright when they are likewise X-ray bright. However, there is a population of objects that are infrared bright, but are X-ray faint, but very few that are X-ray bright but infrared faint. This is expected as it suggests that it is easier to have a Compton-thick source that obscures the X-rays, but much harder to obscure the infrared photons. Many of these infrared-bright, X-ray faint sources are not selected by either the Stern et al. or Donley et al. IR-AGN selection techniques and thus may not be AGNs or have dominate AGN components.

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The right panel of Figure 4 shows the number density of all IRAC- and X-ray-detected objects (black), Stern et al. IR-AGNs (red), and Donley et al. IR-AGNs (blue) as a function of their observed hard-band X-ray flux. Likewise the bottom panel shows the number density of the same samples as a function of their IRAC 5.8 μm flux. The number-density plots in each of the subpanels are limited to include selected sources above the flux limit in each field and normalized by the area that is sensitive to that flux. The number densities of either Stern et al. or Donley et al. IR-AGN sample do not increase toward fainter fluxes as steeply as the total number density of objects detected.

Our next goal is to determine how the number of sources identified as AGNs using a given selection technique varies as a function of IR flux. In the top panels of Figure 5, the solid black line shows the surface number density (number per square degree per logarithmic flux interval) of sources as a function of IRAC 5.8 μm flux for the Stern et al. (left) and Donley et al. (right) IR-AGN samples. At the brightest fluxes (f_5.8 μm  >  100 μJy), we combine all of the different fields to estimate the surface number density. At fainter fluxes we only use the fields with sufficient IR depth (see Table 1 for flux limits). For both the Stern et al. and Donley et al. IR-AGN samples the number densities increase rapidly with decreasing flux with an approximately power-law form, although the slope of the power law may flatten below f_5.8 μm  ∼  100 μJy for the Donley et al. sample. The dashed lines show the surface number densities of the Stern et al. or Donley et al. IR-AGN sample that is also detected in our X-ray data for our three different X-ray survey depths: ∼10 ks (red), ∼100 ks (green), and ∼1 Ms (blue). These surface number densities are corrected for the variation in X-ray flux limit over the field (the X-ray incompleteness) by applying the weights described in Section 3.4.

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The bottom panels show the fraction of the IRAC samples that are X-ray detected for each of the X-ray survey depths. These fractions vary significantly as a function of both IRAC flux and the depth of the X-ray data. At bright IR fluxes (f_5.8 μm  >  100 μJy), we find ≳70% of the IR-AGN samples are detected in X-ray data of ∼100–1000 ks depth, whereas with the shallow X-ray data (∼10 ks) this fraction is closer to 50%. At fainter IR fluxes the X-ray fraction decreases, although the falloff is highly dependent on the depth of the X-ray data. Figure 7 (left) directly compares the X-ray fractions of the Stern et al. and Donley et al. IR-AGN samples as a function of IR flux combining the different X-ray survey depths. At bright IR fluxes ∼60% of both IR-AGN samples are X-ray detected; at fainter IR fluxes a slightly higher fraction of the Donley et al. sources are identified in the X-ray data, although we note that the Donley et al. selection identifies a smaller sample of AGNs.

Our next step is to determine how the number of identified IR-AGN and X-ray AGN sources depends on the X-ray flux. In the top panels of Figure 6, we show the surface number density of X-ray-selected sources as a function of the X-ray flux. We only include fields that probe to sufficient X-ray depth (see Table 1 and Figure 2) in our estimates of the surface number density at any given X-ray flux. We apply our X-ray weights to account for the variation in X-ray depth within a field (see Section 3.4); only a limited fraction of the combined X-ray and IRAC area may be sensitive to a given X-ray flux, even above our nominal X-ray flux limits. The X-ray surface number densities follow the double power-law distributions presented by Georgakakis et al. (2008). The dashed lines show the surface number densities of X-ray sources that also satisfy the Stern et al. (left column) and Donley et al. (right column) IR-AGN selection criteria in fields reaching three different IR survey depths: SWIRE (red), COSMOS (green), and GOODS (blue). These number densities are also corrected for variation in X-ray sensitivity within a field. The bottom panels show the fractions of the X-ray sample that satisfy the IR selection criteria for the three IR survey depths for each of the IR-AGN samples. At bright X-ray fluxes (f_X  ∼  10⁻¹⁴ erg s⁻¹ cm⁻²), a fairly high fraction of the X-ray sources (∼60%) are identified as AGNs using either the Stern et al. or Donley et al. selection criteria, although substantially increasing the depth of the IR data does not appear to increase this fraction. At fainter X-ray fluxes, the fraction of X-ray sources identified with the IR criteria reduces. Increasing the depth of the IR data from SWIRE to COSMOS recovers a higher fraction, but further increasing from COSMOS to GOODS depths does not appear to significantly increase the IR fraction at a given X-ray flux, especially for the Donley et al. selection criteria. The more restrictive Donley et al. selection criteria generally identify a lower fraction of the X-ray sources as AGNs at any given X-ray flux compared to the Stern et al. selection (see Figure 7 (right panel) for a direct comparison of the fractions with COSMOS X-ray and IRAC depth data.

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With either of the IR-AGN selection techniques, the total number densities and the overlap between the IR-AGN and X-ray AGN samples depend on the survey depth. In the left panel of Figure 7, we show the X-ray-detected fraction of the Stern et al. (red) and Donley et al. (blue) IR-AGN samples as a function of the IRAC flux. For sources fainter than f_5.8 μm  ∼  100 μJy IRAC flux, the Stern et al. fraction is smaller than the Donley et al. fraction, and very similar for sources above f_5.8 μm  ∼  100 μJy. Below f_5.8 μm  ∼  100 μJy, the Stern et al. selection identifies a larger number of objects as AGNs than the Donley et al. selection (see Figure 8), which drives this fraction down. In the right panel of Figure 7, we show the Stern et al. (red) and Donley et al. (blue) IR-AGN fraction of the X-ray-detected sample. The Donley et al. IR-AGN selection identifies a lower fraction of X-ray-detected objects for most of the X-ray fluxes compared to the Stern et al. IR-AGN selection.

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In this section, we examine in more detail the extent of the overlap between the IR-AGN and X-ray AGN samples and how this depends on both the IR and X-ray survey depth. In Figure 8 (left panels), we show the fraction of the IR-AGN samples that are X-ray detected as a function of both the X-ray and IRAC flux limits for the different survey depths. The top-left panel shows the fraction of Stern et al. IR-AGNs that are X-ray detected; the bottom-left panel shows the fraction of Donley et al. IR-AGNs that are X-ray detected. These fractions vary smoothly over the range of IR and X-ray flux limits between 47% and 90% for both Stern et al. and Donley et al. IR-AGN samples. For comparable depth surveys in IR and X-ray (roughly the diagonal through these panels) ∼70%–80% of the IR-AGNs are detected at X-ray wavelengths. Increasing the depth of the IR data (moving to the left) reduces the fraction with X-ray detections; increasing the depth of the X-ray data (moving down) increases the fraction with X-ray detections. For our shallowest IR data, we find a very high fraction (90%) of the IR-AGNs are detected using our deepest X-ray data. Generally, the X-ray-detected fraction of the Donley et al. IR-AGN samples is higher than for the Stern et al. IR-AGN samples, particularly for deeper surveys.

In the two right-hand panels of Figure 8, we show the fraction of the X-ray AGN samples that are also identified as Stern et al. IR-AGNs (top-right panel) or Donley et al. IR-AGNs (bottom-right panel). For both selection techniques, the fraction varies smoothly between ∼4% and 53%, being smallest for shallow IR and deep X-ray samples (bottom right) and largest for deep IR and shallow X-ray samples (top left). The Donley et al. IR-AGN selection technique identifies a lower fraction of the X-ray sample than the Stern et al. IR-AGN selection technique. Comparing with Figure 6, we find that these low fractions are due to the large numbers of faint X-ray AGNs that are not selected by either IR-AGN technique, even with very deep IR data.

In Figure 9, we use Venn diagrams to examine the surface densities and overlap of the samples selected using our three AGN selection techniques as a function of both X-ray and IR survey depth. The number densities of the different subsamples are represented by circles located at the IR and X-ray flux limits for the different surveys. The size of the circles are proportional to the surface number densities; the overlaps between the samples are shown in Venn diagram form. The black cross indicates the X-ray and IR flux limits corresponding to the surveys used for each individual Venn diagram. The number density of X-ray AGNs (green circles) increases rapidly as the X-ray depth increases (moving down in Figure 9). Likewise, for deeper IRAC surveys we find a larger number density of both Stern et al. and Donley et al. IR-AGN samples. The overlap between the AGN samples varies substantially depending on the IR and X-ray survey depth. X-ray selection identifies a large population of sources at all survey depths that are not identified with the IR-AGN selection techniques, even with extremely deep IR data. The Donley et al. IR-AGN selection technique generally identifies a subset of the Stern et al. IR-AGN sample for all depths. At our shallowest IR survey depth (SWIRE: right-hand side of Figure 9), the IR-AGN samples overlap almost completely, while in deeper IR surveys an additional population is identified using the Stern et al. selection criteria. At the shallowest IR survey depth, a high fraction of the IR-AGNs are X-ray detected: up to 90% with our deepest X-ray data. Deeper IR data identify additional IR-AGNs, including populations that are not identified at X-ray wavelengths.

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In this section, we compare the redshift distributions of our three AGN samples to investigate whether the selection techniques identifying the same underlying AGN population. In Figure 10, we show the number per square degree of Stern et al. IR-AGN (red hatched), Donley et al. IR-AGN (blue hatched), and X-ray AGN (green solid) selected objects with PRIMUS redshifts. We restrict our samples to objects with secure PRIMUS redshifts between 0.2 < z < 1.2, where PRIMUS provides reliable redshifts for sources where the optical light is dominated by either the AGN or galaxy light and there is enough volume to construct a statistical sample. For the X-ray-detected sample, we ensure that the sample is not contaminated by X-ray-detected star-forming galaxies by restricting to sources with L_X  >  10⁴² erg s⁻¹. We apply the weights calculated in Section 3.4 to correct for the varying depth of the X-ray data but we do not correct for the fraction without PRIMUS redshifts. Requiring a redshift reduces our AGN sample sizes and imposes an apparent limiting optical magnitude brighter than i ∼ 23. The fraction of sources in our AGN samples that are brighter than this optical limit varies from ∼30% to ∼70%, depending on the depth of the corresponding IR or X-ray data. Of the sources targeted by PRIMUS, the fraction that have a high-quality redshift varies from ∼75% to ∼95%, depending on the depth of the IR or X-ray data, where the fraction is higher for shallower fields. This is comparable to the overall PRIMUS redshift success rate. We find comparably low catastrophic redshift outlier rates for the AGN samples compared to the full PRIMUS galaxy sample. The Stern et al. and Donley et al. distributions have been scaled up by a factor of four to show their relative redshift distributions compared to the completeness-corrected X-ray sample.

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The Donley et al. IR-AGN sample and X-ray AGN sample number-density distributions peak around z ∼ 0.6, which matches the peak of the entire PRIMUS galaxy sample, although the Donley et al. distribution does not fall off as quickly for higher redshifts compared to the X-ray AGN sample. We find that the redshift distribution of the Stern et al. IR-AGN sample has an unexpected shape with two peaks, at z ∼ 0.3 and z ∼ 1.1, which differs significantly from that of the Donley et al. IR-AGN and X-ray AGN samples or the overall PRIMUS galaxy sample. To investigate this unusual distribution, in Figure 11 we present IRAC color–color plots for the entire sample with IRAC detections in all four bands separated into four distinct redshift ranges. We overplot the Donley et al. IR-AGN (blue triangle) and X-ray AGN (green diamond) selected samples. Generally, the Donley et al. IR-AGN sample is a subset of the Stern et al. IR-AGN sample, whereas the X-ray AGN sample includes objects both inside and outside of the Stern et al. color wedge. The Stern et al. IR-AGN sample includes many other objects that are neither X-ray detected nor Donley et al. selected. We see that the majority of these objects enter in at low redshifts (z ∼ 0.3, leftmost panel) and high redshifts (z ∼ 1.0, rightmost panel). These objects mostly enter into the wedge at low ([3.6]−[4.5]) color and appear to be consistent with scatter from the larger galaxy population (the dark red points in Figure 11). We thus conclude that the Stern et al. IR-AGN sample suffers from significant contamination by normal, non-AGN galaxies at redshifts z ∼ 0.3 and z ∼ 1.1, resulting in the bimodal peaks in the redshift distribution seen in Figure 10.

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In this section, we investigate the trends in the previous section using empirical galaxy and AGN SED templates. Following Donley et al. (2008, 2012), we redshift the Polletta et al. (2008) IR templates and examine the changes to the observed IRAC colors. The Polletta et al. (2008) IR template set consists of four quiescent galaxy templates, eight star-forming galaxy templates, and six AGN templates, based on the observed SED of a representative sample of local galaxies and AGNs. We convolve the redshifted SED with each IRAC filter to predict the observed IRAC colors. In Figure 12, we show the predicted colors in four redshift bins for all the star-forming galaxy (blue dots), quiescent galaxy (red squares), and AGN (black crosses) templates. The lines track the change in color within the redshift bin for a given template.

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Generally, star-forming templates change the most in this color–color space, due to the polycyclic aromatic hydrocarbon (PAH) features in the SED. They start off with a very red ([5.8]−[8.0]) color that moves blueward due to the 8.0 μm PAH feature redshifting out of the 8.0 μm IRAC filter. The vertical ([3.6]−[4.5]) motion of star-forming templates is dominated by the 3.3 μm PAH feature moving from the 3.6 μm to the 4.5 μm IRAC filter.¹⁴ Together, both of these features can move the star-forming templates into the Stern et al. wedge at both low redshift (z ∼ 0.3) and high redshift (z ∼ 1.1). The star-forming templates that come into the Stern et al. wedge are not selected by the Donley et al. IR-AGN criteria.

For quiescent templates (red squares), we find a much simpler motion through the IRAC color–color space. Composed of older stars, these templates are generally dominated by the Rayleigh–Jeans tail with an IRAC power-law slope of α ∼ 2 located close to the origin. The SEDs are generally featureless, and the motion is dominated by the 1.6 μm stellar bump which redshifts into the 3.6 μm IRAC filter. These templates are generally excluded from either the Donley et al. or Stern et al. IR-AGN samples at these redshifts.

The predicted AGN template colors also show very little motion in this space. Their featureless, red power-law (α ∼ −0.5 to  − 3) SEDs tend to lie close to the power-law line. Both Donley et al. and Stern et al. IR-AGN selection techniques pick up these AGN/quasar (QSO) templates easily at z < 1.2.

The behaviors of the star-forming, quiescent, and AGN templates are consistent with the trends seen in the data in the previous section and explain the peaks in the Stern et al. IR-AGN redshift distribution. The Stern et al. IR-AGN sample is contaminated by star-forming galaxies at both low (z ∼ 0.3) and high (z ∼ 1.1) redshifts. The Donley et al. IR-AGN selection technique identifies AGN-dominated templates at z < 1.2.

The addition of a small AGN contribution (5%–10%) may move star-forming and quiescent SEDs into the Stern et al. wedge, but such sources are not selected by the Donley et al. criteria (e.g., Donley et al. 2012). Thus, the Stern et al. selection may identify a more complete sample of AGNs but it is hampered by a larger contamination by star-forming galaxies at particular redshifts.

In this section, we extend the galaxy and AGN templates to higher redshifts (z ⩾ 1.2) beyond the range of PRIMUS to investigate the contamination of the IR-AGN samples. We track the templates out to z = 3 and compare to the full sample with IRAC detections thus including objects without spectroscopic redshifts. In Figure 13, we show the density of objects in IRAC color–color space for the Donley et al. IR-AGN (top-left panel), and X-ray AGN (top-right panel) samples, and for the entire sample (All) with IRAC detections (bottom-left panel). In the bottom-right panel, we show the behavior of the templates from z = 0 to z = 3.

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The Donley et al. IR-AGN sample (top-left panel) is centered around the power-law locus with the vast majority of the sample lying within the Stern et al. wedge. It spans the range of AGN/QSO templates (seen in the lower-right panel) over the entire redshift range (z = −3).

In comparison the X-ray AGN sample (top-right panel) not only spans the range of the Donley et al. sample, but also extends beyond the Donley et al. IR-AGN sample and the Stern et al. wedge. At ([3.5]−[4.5]) ∼ 0.75 and ([5.8]−[8.0]) > 0.5 (left of the Stern et al. wedge), there is a spur of X-ray sources that do not have a secure PRIMUS redshift and are not selected with the Donley et al. IR-AGN method. Comparing this spur with the template track panel, this region of objects appears to match both high-redshift (z ≳ 2.5) quiescent galaxy (square) and star-forming galaxies (cross). It is unclear whether these objects contain an X-ray bright AGN and are missed by the Donley et al. and Stern et al. methods, or are merely X-ray-detected high-redshift star-forming galaxy contaminants. If we assume that these objects have a z = 3 redshift, the hard-band X-ray luminosity would be distributed around L_X  >  10^44.6 erg s⁻¹, which is suggestive that they are highly luminous AGNs and not high-redshift star-forming galaxy contaminants. The objects below the Stern et al. wedge generally match quiescent galaxy templates with z < 1.5 and star-forming templates with 0.7 < z < 1.5. This population has been shown in Section 5.1 to be low-redshift galaxies with PRIMUS redshifts (z ∼ 0.7) containing genuine AGNs (L_X  >  10⁴² erg s⁻¹) that are not identified by either IR-AGN selection technique.

The small population of X-ray-detected sources with very red colors (([5.8]−[8.0]) > 2) to the bottom right of the Stern et al. wedge are generally identified as low-redshift (z ≲ 0.4) star-forming galaxies with low X-ray luminosities (L_X  <  10⁴² erg s⁻¹) and thus the X-ray emission may be attributed to star formation rather than AGN activity. This population is easily identified and not selected by either the Stern et al. or Donley et al. IR-AGN selection techniques.

For the entire sample of IRAC-detected objects (All, bottom-left panel), the bulk of the objects lie below the Stern et al. wedge, concentrating at colors of ([3.6]−[4.5]) < 0.0 and ([5.8]−[8.0]) > 0.1. The All sample contains four-band IRAC-detected objects from all of our fields. The colors of these objects correlate with predicted star-forming and quiescent galaxy template colors at 0.7 < z < 1.5. There is also a large density of sources to the left side of the Stern et al. wedge, like the z > 2.5 spur of X-ray-detected objects discussed earlier. The colors of these objects match both quiescent and star-forming galaxy SED templates at high redshift (z ∼ 2.5). A low fraction (∼5%) of these objects are high-redshift high-luminosity X-ray-detected AGNs, but the majority of these objects are not X-ray detected. These objects are not identified by the Donley et al. selection technique, but extend into the Stern et al. wedge contaminating it with high-redshift galaxies.

Some of these sources in the Stern et al. wedge could have a low-luminosity AGN, but the AGN signature is being masked in the MIR by the dominant galaxy light. Such AGNs could be identified using other techniques (e.g., X-ray, or via their optical spectra), but the true AGNs are probably a minority of the contaminating population. The dominant galaxy light suggests that they are not generally broad-line QSOs, but could host narrow-line AGNs.

In summary, the Stern et al. IR-AGN technique selects a large population of AGNs but is contaminated by low-, intermediate-, and high-redshift galaxies (z ∼ 0.3, z ∼ 1.1, and z ≳ 2.5). The Donley et al. IR-AGN selection technique generally is free of these galaxy contaminants, selecting just the AGN population. The X-ray-detected sample has colors that extend beyond the Stern et al. and Donley et al. selected samples, identifying a population of AGNs that are not detected by either IR selection technique and are not star-forming galaxy contaminants.

To investigate AGN and host galaxy rest-frame properties, we require a PRIMUS spectroscopic redshift. This reduces our AGN sample sizes and imposes an optical apparent magnitude limit. To be targeted by PRIMUS an object must have an optical magnitude brighter than i ∼ 23. The fraction of sources in our AGN samples that are brighter than this optical limit varies from ∼30% to ∼70% depending on the depth of the corresponding IR or X-ray data. Of these sources that are brighter than the PRIMUS targeting limit, ∼20%–60% were targeted in PRIMUS (not all objects to the apparent magnitude limit can be targeted due to slit collisions). The fraction of targeted AGNs that have a high-quality redshift varies from ∼75% to ∼95% depending on the depth of the IR or X-ray data, where the fraction is higher for shallower fields. This is comparable to the overall PRIMUS redshift success rate. Some targeted sources for which we fail to measure a redshift lie at z > 1.2 where the Balmer break falls beyond the wavelength coverage of PRIMUS. Our redshift success range in the 0.2 < z < 1.0 range is likely to be higher. The fraction of our IR- and X-ray-selected AGNs with PRIMUS redshifts is ∼20%–70% depending on depth of the data. We find that the subsample with PRIMUS redshifts spans the full range of IR and X-ray fluxes in our IR and X-ray AGN samples. In the rest of this section, we use these subsamples to compare the rest-frame properties of the IR- and X-ray-selected AGNs to an optical limit of i ∼ 23.

In this section, we investigate rest-frame properties related to the AGNs themselves. In Figure 14, we compare the IR and X-ray luminosities for the AGN samples. IR luminosities are calculated at a rest-frame wavelength of 3.6 μm by interpolating the log of the fluxes between the two nearest IRAC channels. IR luminosities are calculated at a rest-frame wavelength of 3.6 μm in two manners: (1) interpolating the log of the fluxes using the two nearest IRAC channels, and (2) using template fit synthesized luminosities from K-correct (Blanton & Roweis 2007). We find no significant differences in the luminosity distributions between these two methods and use the former for the remainder of the paper. The solid green circles indicate PRIMUS AGNs that have IR and X-ray detections and are above L_X  >  10⁴² erg s⁻¹. X-ray luminosities are calculated from the 2–10 keV fluxes, using an unobscured power law with a photon index of Γ = 1.9. The light green arrows indicate upper limits on the 3.6 μm IR luminosity for X-ray-detected AGNs that lack IR detections. The IR luminosity upper limits are derived from the IR depth of the survey and the redshift of the object. The X-ray luminosity upper limits for the Stern et al. or Donley et al. IR-AGNs that are not X-ray detected are estimated from the X-ray depth of each field. We conservatively use the X-ray flux limit corresponding to the flux at which 50% of expected X-ray sources are detected in that field. For a more conservative 90% detection limit, the upper limits increase by roughly a factor of 2–4. Objects that are selected by either Stern et al. or Donley et al. IR-AGN techniques are indicated by red squares or blue triangles, respectively. Generally, sources that are detected in both IR and X-ray have correlated luminosities, which roughly follows the Richards et al. (2006) mean QSO SED shown as a solid orange line. The correlation does contain significant scatter with more sources scattering to much lower X-ray luminosities. The sources that scatter to lower X-ray luminosities are found two orders of magnitude below the mean QSO SED.

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We find that both IR-AGN color selection techniques tend to select sources with higher IR and X-ray luminosities compared to the entire X-ray-selected AGN sample. A weighted K-S test of the distributions of the X-ray luminosities, as shown in the top-right subpanel of Figure 14, rejects the null hypothesis that either the Stern et al. or Donley et al. IR-AGNs are drawn from the same distribution as the X-ray-selected AGNs at the >99.9% confidence level. X-ray selection identifies a large population of AGNs with low luminosities (below L_X  <  10^43.5 erg s⁻¹) that are not identified as either Stern et al. or Donley et al. IR-AGNs.

These trends are also shown in Figure 15(a), which plots the subset of X-ray AGNs detected in all four IRAC bands in IRAC color–color space, where the color of the points scales with X-ray luminosity. In this figure we do not limit sources to be above L_X  >  10⁴² erg s⁻¹. Sources below this limit reside in the region below the Stern et al. or Donley et al. IR-AGN selection regions and are easily distinguished from IR-selected AGNs. Low X-ray luminosity (L_X  <  10^43.5 erg s⁻¹) AGNs are scattered throughout the IRAC color–color space, whereas high X-ray luminosity (L_X  >  10^43.5 erg s⁻¹) AGNs cluster near the power-law line within the Stern et al. wedge. For high X-ray luminosity (L_X  >  10^43.5 erg s⁻¹) AGNs with IRAC detections, ∼46% ± 6% to ∼79% ± 4% lie within the spaces defined by the Donley et al. and Stern et al. IR-AGN selections, respectively.

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In the bottom panel of Figure 14, we plot the residual X-ray to IR luminosity ratio from the Richards et al. (2006) mean quasar SED shown in the upper panel (solid orange line). In this panel, we have removed the AGN not detected in the IR. We find that there is a tail of sources with negative residuals such that their X-ray to IR luminosity ratio is below the Richards et al. (2006) mean quasar SED (solid orange) line. There are between 3% ± 1% and 6% ± 3% of Stern et al. and Donley et al. sources with residuals less than −2.0; these sources are particularly X-ray underluminous. These sources are likely obscured AGNs where the observed X-ray luminosity is suppressed due to absorption in the 2–10 keV X-ray band. We note that the IR luminosity L_3.6 μm may be systematically inflated by galaxy light for lower luminosity AGNs. In the IR, the galaxy light dominates over the AGN light above an accretion rate of 1% Eddington (see Section 6.3). However, the sources that are underluminous in X-rays are very luminous in the IR (L_3.6 μm  >  10⁴⁴erg s⁻¹) such that the galaxy contribution is small relative to the AGN.

We also examine the obscuration of the X-ray AGN using HRs. We convert the hard- and soft-band fluxes to equivalent on-axis Chandra count rates and calculate the HRs as (H−S)/(H+S), where H and S are the 2–10 keV and 0.5–2 keV count rates, respectively. Large HR values (HR ⩾ 0.2) indicate significant absorption in the hard X-ray band (N_H  ≳ 3 ×  10²¹ cm⁻²) at z ∼ 0.6 (Hasinger 2008). In Figure 15(b), we show the HRs for the subset of X-ray AGNs detected in all four IRAC bands in IRAC color–color space and find that lower HR (unobscured) AGNs tend to fall within the Stern et al. wedge. This is consistent with IR-AGN samples identifying primarily more luminous AGNs, which tend to be unobscured. Obscured AGNs lie throughout the IRAC color–color space, with 48% ± 3% in the Stern et al. wedge and the remainder in the region where the galaxy light dominates the IR colors.

We compare the HRs of the IR-AGN and X-ray AGN samples in two X-ray luminosity bins to limit the large number of luminous IR-AGN sources from dominating the comparison. In the right panel of Figure 16, we show the distributions of HRs for X-ray luminous sources (L_X  >  10⁴³ erg s⁻¹). The Stern et al. IR-AGN, Donley et al. IR-AGN, and X-ray-detected AGN samples all span a wide range of HRs. The fraction of obscured sources (HR > 0.2) is comparable for all three samples. The fraction in the X-ray AGN sample (28% ± 7%) is slightly larger than the fraction in the Stern et al. (18% ± 10%) or Donley et al. (20% ± 17%), although the differences are not statistically significant.

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In the left panel of Figure 16, we show the distributions of HRs for lower luminosity X-ray sources (L_X  <  10⁴³ erg s⁻¹). The fraction of obscured sources (HR > 0.2) for the X-ray AGN sample (28% ± 5%) is slightly smaller than the fraction in the Stern et al. (36% ± 15%) or Donley et al. (67% ± 27%) AGN samples. Unlike at high X-ray (L_X  >  10⁴³ erg s⁻¹) luminosities, there is a hint at low X-ray luminosities that the IR-AGN samples identify a more obscured population for lower luminosity AGNs; however, there are not statistically significant differences between the HR distributions in either luminosity range. Larger deep-IR samples are needed to confirm if this small population of obscured sources statistically differs from the X-ray sample.

Finally, in Figure 17 we show the 3.6 μm IR luminosity number-density distribution for the IR-AGN samples that are divided into X-ray detected (solid histograms) and not X-ray detected (hatched histograms). We also show a subset in green of the X-ray AGN sample that is detected in all IRAC four bands. For both the Stern et al. and Donley et al. AGN samples, we do not find any significant differences in the L_IR distributions for the sources with X-ray detections to those without X-ray detections. These latter sources could be missed in X-ray due to heavy obscuration, but nevertheless they have a similar distribution of IR luminosities and do not outnumber the subsample of the X-ray AGN sample detected in IRAC. The low number density of IR-AGN sources without X-ray detections may partially be due to having relatively deeper X-ray survey data compared to the IRAC survey data. Using the 3.6 μm and X-ray luminosity from the Richards et al. (2006) mean quasar SED, we find that the depth of the X-ray data in our fields is typically 2–3 times deeper than the IRAC depth, for light that is dominated by luminous AGNs. Thus, while IR-AGN samples are efficient at identifying luminous AGNs, it remains unclear if they identify a large population of obscured sources relative to X-ray AGN samples due to the differences in survey depths.

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In this section, we investigate the optical rest-frame colors and stellar masses of the host galaxies for our IR-AGN and X-ray AGN samples with PRIMUS redshifts. For broad-line AGNs, the optical light is dominated by the AGN rather than the host galaxy light; this prevents an accurate estimate of the host galaxy properties. Therefore, in this section we exclude sources classified as broad-line AGNs, where the PRIMUS spectrum is better fit with χ² of at least 50 by a broad-line AGN template compared to any galaxy template. These broad-line AGNs are a sizable fraction of the X-ray AGN (12% ± 1%), Stern et al. IR-AGN (16% ± 2%), and Donley et al. IR-AGN (32% ± 3%) samples.

In the top panels of Figure 18, we plot the rest-frame optical (u − g) colors and stellar masses for the X-ray AGN (left), Stern et al. IR-AGN (center), and Donley et al. IR-AGN (right column) samples. We show the relevant AGN sample with black squares and PRIMUS galaxies with gray contours (30%, 50%, and 80% contours) and gray points for individual galaxies outside of the 80% contour. In the left panel, we show the X-ray AGNs that are selected by either of the Stern et al. IR-AGN or Donley et al. IR-AGN selection techniques with green diamonds. In the center and right panels, we show IR-AGNs that are also X-ray detected in green diamonds. The stellar masses are estimated using the SED fitting code iSEDfit (see Moustakas et al. 2013, for complete details). iSEDfit is a Bayesian fitting program that compares broadband photometry against large Monte Carlo grids of SED models that span a wide range of parameters (e.g., ages, metallicities, star formation histories, starbursts, and dust content). We construct our stellar mass models using the Bruzual & Charlot (2003) stellar population synthesis models assuming the Chabrier (2003) initial mass function from 0.1 to 100$\mathcal {M}_*$. We assume a uniform stellar metallicity prior in the range 0.004 < Z < 0.04, and smooth, exponentially declining star formation histories, ψ(t)∝e^−γt, with γ drawn uniformly from the interval [0.01, 1] Gyr⁻¹, and we include bursts of star formation. In the bottom panels of Figure 18, we calculate the fraction of all PRIMUS galaxies that are detected in one of the AGN selection methods as a function of stellar mass for red and blue sources. We define each source as blue or red using the magnitude and redshift dependent cut from Aird et al. (2012)

$$\begin{equation} C = (u-g) - (0.671 - 0.031{M_g} - 0.065{z}), \end{equation} \tag{ 10 }$$

where M_g is the absolute g-band magnitude and z is the redshift. To calculate the number density and account for the variation in survey depth, we use the X-ray completeness corrections (X-ray sensitive area for a given X-ray flux) for the sources that are X-ray detected and the IR–optical overlap area for all other sources.

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We find that the fraction of galaxies with detected X-ray AGNs increases with stellar mass. The fraction of blue galaxies with X-ray AGNs is a factor ∼2 higher than for red galaxies (across all stellar masses). These trends are in agreement with Aird et al. (2012), where the increase with stellar mass is attributed to a selection effect. AGNs in massive galaxies appear more luminous (for the same specific accretion rate scaled relative to the host stellar mass) and so are easier to detect. AGNs selected by either Stern et al. IR-AGN or Donley et al. IR-AGN techniques follow the same basic trend, although both techniques generally identify a lower fraction of AGNs in both red and blue galaxies at all stellar masses. The rate that the fraction of red galaxies with Stern et al. or Donley et al. AGNs increases with stellar mass is much slower than for red galaxies with X-ray AGNs. This lower rate of change for the IR-AGN selections suggests that they are less efficient at identifying AGNs in red, massive galaxies ($\mathcal {M}_*$$\;{\gtrsim }\;{10}^{10.5}\, \mathcal {M}_{\odot }$). The fraction of galaxies with Stern et al. AGNs is higher than the X-ray fraction at low masses, which may be due to contamination of Stern et al. IR-AGNs by star-forming galaxies.

In Figure 19(a), we estimate the specific accretion rate (λ) of the X-ray-detected sample as a function of the host stellar mass. Following Aird et al. (2012), we define the specific accretion rate from the bolometric luminosity derived from the X-ray luminosity using the Hopkins et al. (2007) quasar bolometric corrections.¹⁵ For the hard X-ray-selected sample, this is dominated by the emission from the AGN with possibly only minor contribution from the host galaxy. From the bolometric luminosity, we calculate the specific accretion rate

$$\begin{equation} \lambda = \frac{L_{{\rm bol}}}{L_{{\rm Edd}}} = \frac{L_{{\rm bol}}}{1.3\times 10^{38}\;\mathrm{erg\;s^{-1}}\times 0.002 \frac{\mathcal {M}_*}{\mathcal {M}_{\odot }}}, \end{equation} \tag{ 11 }$$

where $\mathcal {M}_*$ is the host galaxy stellar mass and L_bol is the bolometric luminosity. The X-ray sources with IRAC detections are solid green circles and the sources without IRAC detections are open green circles. Sources with IRAC detections that are selected by either the Stern et al. or Donley et al. IR-AGN selection technique are outlined with red squares or blue diamonds. The vast majority of Donley et al. IR-AGNs also satisfy the Stern et al. IR-AGN selection criteria. This figure shows two populations of X-ray sources that are not identified as IR-AGNs: those that are not detected by IRAC (see Section 4.3) and those that are detected by IRAC but do not meet either the Stern et al. or Donley et al. criteria. The former population consists primarily of sources at low stellar masses and high specific accretion rates. These sources are primarily found in the fields that have shallower IRAC data compared to their X-ray data (ES1, XMM, and COSMOS$_{\tt Shallow}$).

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The majority of the X-ray AGNs that are not Stern et al. or Donley et al. selected (but do have IRAC detections: green solid symbols in Figure 19(a)) have large stellar masses ($\mathcal {M}_*$$\;{> 3\times }\;{10}^{10}\, \mathcal {M}_{\odot }$), low to moderate X-ray luminosities (L_X  <  10⁴³ erg s⁻¹), and low specific accretion rates (λ  <  1%). Additionally, many of these sources have moderate to large MIR to X-ray luminosity ratios (L_3.6 μm/L_X > 1). In fact, their L_3.6 μm/L_X ratios are larger than typically found for the IR-AGN population and for optically selected quasars (e.g., Richards et al. 2006). This suggests that stellar light from their massive host galaxies is dominating the emission in the IRAC bands. Indeed, for a stellar mass-to-light ratio of order unity (e.g., Bell & de Jong 2001) and L_3.6 μm/L_X for a pure AGN SED (Richards et al. 2006), a galaxy with a specific accretion rate of λ  =  1% would have comparable emission at 3.6 μm from stellar light and from the central AGN. At lower specific accretion rates, the host galaxy would then dominate the emission in the IRAC bands. This provides a natural explanation for why many X-ray AGNs at lower specific accretion rates are not selected as IR-AGNs. These findings are in good agreement with Donley et al. (2012), who show that the average SED of luminous AGNs is missed due to a noticeable 1.6 μm stellar bump prevalent in older, massive galaxies.

In Figure 19(b), we estimate the specific accretion rate based on MIR luminosity. The specific accretion rates for these sources are less well constrained due to the possible addition of galaxy light to the observed MIR flux. To ensure that the bolometric luminosity distributions are similar for the X-ray and IRAC overlapping sample, we first use the Richards et al. (2006) mean quasar SED to estimate the L_X luminosity from the L_3.6 μm luminosity (see the orange line in Figure 14). We apply the Hopkins et al. (2007) hard X-ray bolometric correction to the equivalent X-ray luminosity to calculate the specific accretion rate for these sources. We do not use the Hopkins et al. (2007) bolometric corrections for the IR luminosity due to the overprediction of the bolometric luminosity for IR-AGN sources that are also X-ray detected. This method for estimating specific accretion rates illustrates the floor near 10⁻² < λ < 10⁻³, where galaxy light dominates the MIR luminosity.

In Figure 19(c), we show the specific accretion rate distribution for Stern et al. IR-AGNs, Donley et al. IR-AGNs, and X-ray AGNs. We restrict this comparison to AGNs in host galaxies with stellar mass $\mathcal {M}_*$$\;{> 3\times }\;{10}^{10}\, \mathcal {M}_{\odot }$, due to incompleteness at lower stellar masses. For all AGNs with an X-ray detection, we use the specific accretion rate derived from the X-ray luminosity. For the IR-AGNs that are not detected in the X-rays we use their estimated specific accretion rate from their MIR luminosity. Some of the IR-AGNs with X-ray detections have significant absorption in the X-rays (e.g., Figure 14). For IR-AGNs with L_3.6 μm/L_X > 10, we use the specific accretion rates derived from their MIR luminosities. We note that this does not have a significant effect on the lambda distributions in Figure 19(c). Both the Stern et al. and Donley et al. IR-AGN samples have specific accretion rate distributions that do not match the X-ray distribution at the >99.7% confidence level from using a two-sided weighted K-S test. In particular, the X-ray sample includes a significant number of sources at low to moderate specific accretion rates that are not identified as IR-AGNs. These low to moderate specific accretion rate sources are preferentially found in red, high-mass galaxies, which explains the larger fraction of red host galaxies identified with X-ray AGNs compared to IR-AGN selection techniques.

We first consider the fraction of IR-AGNs that are detected in X-rays, as a function of both the IR and X-ray data depth. Generally, a higher fraction of Donley et al. IR-AGNs are X-ray detected than in the Stern et al. IR-AGN sample, consistent with our findings above that the Stern et al. IR-AGN selection can suffer from contamination. At the relatively shallow IR depth of the SWIRE survey, 60% of IR-AGNs are X-ray detected when the X-ray depth is comparable to the IR depth. Using the deepest IR and X-ray data, the fraction of IR-AGNs that are X-ray detected rises to ∼70%–80%. However, this fraction rises to ∼90% when combining shallow IR and deep X-ray data, indicating that the vast majority of IR-AGNs are indeed X-ray emitting AGNs.

As discussed above, IR-AGN selection methods identify additional AGNs not detected in X-rays, when the IR and X-ray data are of comparable depth. At shallow IR depths, the Stern et al. and Donley et al. techniques recover very similar number densities of AGNs with a large overlap between the populations. As IR depth increases, both the Stern et al. and Donley et al. selections identify a larger population of AGNs. However, the number density recovered by the Stern et al. selection increases at a greater rate—in our deepest IR data Stern et al. recover approximately twice the number density of AGNs as the Donley et al. IR-AGN selection. As discussed in Section 5, much of the relative increase is likely due to contamination. For any depth IR and X-ray data, the addition of the Stern et al. IR-AGN sample to the X-ray AGN sample increases the total AGN sample by 7%−27%, depending on the depth of the IR and X-ray data. In comparison, addition of Donley et al. IR-AGN selection to X-ray AGN selection increases the total AGN sample by 4%−20%. Both IR-AGN selection methods increase the total AGN samples more at shallower IR and X-ray depths. At all IR and X-ray depths studied here, X-ray AGN selection identifies a higher number density of AGNs than either IR-AGN selection method.

It is also extremely important to account for the well-known and easily characterized incompleteness of X-ray selection. The variation in X-ray sensitivity over a field can substantially reduce the number of IR-AGNs with observed X-ray detections. It is vital to account for this effect to understand any true, underlying differences between AGN populations.

At all IR and X-ray depths studied here we find a much higher fraction of IR-AGNs are detected in X-rays, compared to values in the literature. This is due to the X-ray completeness corrections that we make, to account for the varying X-ray sensitivity within a field. If we do not apply these completeness weights, we find similar fractions to those in the literature. For example, in the XBoötes survey with shallow IR (f_5.8 μm  ∼  51 μJy) and shallow X-ray (f_X  ∼  10⁻¹⁵ erg s⁻¹ cm⁻²) data, Hickox et al. (2009) find that 38% of their Stern et al. IR-AGNs are X-ray detected. Using similar depth data if we do not apply X-ray completeness corrections we find that 37% of Stern et al. IR-AGNs are X-ray detected; however, this fraction rises to 63% after applying the completeness corrections. Donley et al. (2007) use deep IR (f_5.8 μm  ∼  14.5 μJy) and medium-depth X-ray (f_X  ∼  10⁻¹⁵ erg s⁻¹ cm⁻²) data in the CDFN field to find that 33% of the Stern et al. IR-AGNs are X-ray detected. Using similar depth data if we do not apply X-ray completeness corrections we find a similar fraction of 35% of Stern et al. IR-AGNs are X-ray detected; however, this fraction rises to 57% after applying the completeness corrections. Park et al. (2010) use very deep IR data (f_5.8 μm  ∼  6.3 μJy) and medium-depth X-ray (f_X  ∼  10⁻¹⁶ erg s⁻¹ cm⁻²) data in the EGS field and find that a comparatively low fraction of Stern et al. IR-AGNs are X-ray detected. As their X-ray data are not as deep as their IR data, it is not surprising that they find a low fraction. We do not have comparably deep IR data; however, at a similar X-ray depth using the deepest IR data we have, we find that at most 35% of IR-AGNs should be X-ray detected, if X-ray completeness is not accounted for. This upper limit is consistent with their value.

Thus, we find that the wide range of X-ray-detected IR-AGN fractions reported in the literature can be accounted for by the varying depths of both the IR and X-ray data used in these studies. We also find that accounting for X-ray incompleteness, which is generally not done in the literature, increases this fraction by a factor ∼2, which results in 25% more of the IR-AGNs being counted (statistically) as X-ray detected. This illustrates how differing depths of data can lead to very different conclusions in prior studies of IR and X-ray AGNs.

Much, if not most, of the difference between the Stern et al. IR-AGN sample and Donley et al. IR-AGN sample is due to contamination by galaxies that do not host an AGN (or at least are not dominated by a luminous AGN in the MIR). Using PRIMUS redshifts, we study the evolution of the distribution of Stern et al. and Donley et al. IR-AGNs and X-ray-detected AGNs in IRAC color–color space from z ∼ 0.2 to z ∼ 1.2 and compare our results to the predicted evolution of star-forming and quiescent galaxy and AGN SED templates in this space. We find that the Stern et al. IR-AGN sample is contaminated at z ∼ 0.3 by star-forming galaxies and at z ∼ 1.1 by quiescent galaxies. This is reflected in the fraction of Stern et al. IR-AGNs that are not detected in X-rays; for the full Stern et al. IR-AGN sample with PRIMUS redshifts, spanning 0.2 < z < 1.2, this fraction is 27% ± 4%, while at 0.2 < z < 0.45 it is 76% ± 10% and at 1.0 < z < 1.2 it is 86% ± 13% when using the X-ray completeness weights.

Extending our analysis of the galaxy and AGN templates to higher redshift, beyond the reach of PRIMUS, we find that the Stern et al. IR-AGN samples will be contaminated by high-redshift star-forming galaxies. We do not find any galaxy contamination in the Donley et al. IR-AGN sample; this is reflected by the lack of evolution in the fraction of Donley et al. IR-AGNs that are not X-ray detected. Using the Donley et al. IR-AGN sample as a baseline, we find that galaxy contamination in the Stern et al. IR-AGN selection is significant only for IR surveys deeper than the SWIRE limit of f_5.8 μm  ∼  100 μJy.

Overall, we find that the Donley et al. IR-AGN selection is less complete than the Stern et al. IR-AGN selection. This is demonstrated by the fact that a smaller fraction of X-ray AGNs are also Donley et al. IR-AGNs. Additionally, there are correctly identified AGNs within the Stern et al. IR-AGN sample that are not found by the Donley et al. IR-AGN selection. However, the Donley et al. IR-AGN selection is more reliable than the Stern et al. IR-AGN selection, in that a higher fraction of Donley et al. IR-AGNs are also X-ray detected, and the Donley et al. IR-AGN selection does not suffer from galaxy contamination. The Stern et al. and Donley et al. IR-AGN samples demonstrates the importance of finding a balance between minimizing both contamination and incompleteness.

Both the Donley et al. and Stern et al. IR-AGN selections appear to be biased, however, in that they select high-luminosity AGNs. This is seen in the X-ray luminosity distributions of the IR-AGN samples compared to the X-ray AGN sample, where the IR-AGN selection techniques preferentially sample the population of L_X  ≳  10^43.5 erg s⁻¹ AGNs. These high-luminosity sources have IRAC colors similar to power-law AGNs and are easily detected with shallow IR surveys.

Even with a relatively deep X-ray survey and shallow IR data, we find that there is at least ∼10% of the IR-AGN population which is not detected in X-rays even after correcting for the variable X-ray sensitivity. For samples of comparable X-ray and IR depth, this fraction is typically ∼20%–30%. This reflects the uniqueness of IR-AGN selection, in that these are sources that are only identified in the IR and would typically be missed in X-ray surveys. What kind of AGNs does IR selection uniquely identify? These AGNs could in theory be missed by X-ray selection either because they are intrinsically less X-ray luminous, due to a lower accretion rate onto the supermassive black hole, or they could be obscured by a high column density of gas and dust, limiting the ability of even deep X-ray surveys to detect them. For the population of IR-AGNs that are X-ray detected, as discussed above we find that IR-AGN selection techniques preferentially select a population of luminous AGNs with L_X  ≳  10^43.5 erg s⁻¹. However, there is a small fraction (∼8%) of IR-AGNs with very high IR to X-ray luminosity ratios, implying heavy obscuration. Therefore, up to ∼20% of IR-AGNs could be sources with moderate to heavy obscuration.

In comparing the HR distributions of IR- and X-ray-selected AGNs, we find no statistically significant difference in the distributions for X-ray AGNs with higher or lower X-ray luminosity. This comparison does not account for the ∼10% of IR-AGN sources that are not X-ray detected. Thus, while the majority of IR-AGNs are also X-ray AGNs, the IR selection techniques adds a small but important population of obscured sources that are missed even with the deepest X-ray surveys. Additionally, the 10% of IR-AGN sources that are not X-ray detected may be Compton-thick and thus consistent with X-ray background synthesis models (e.g., Gilli et al. 2007).

With the exception of broad-line sources, the optical SEDs of the IR-AGN and X-ray AGN samples are dominated by light from the host galaxy, which allows us to determine galaxy properties for these sources. Generally, the Stern et al. and Donley et al. IR-AGN samples follow the same trends as the X-ray population, showing that IR-AGNs and X-ray AGNs do not populate vastly different host galaxy populations. The galaxy hosts span a similar range of stellar masses (${10}^{10} < \mathcal {M}_*/ \mathcal {M}_{\odot }< {10}^{11.5}$) and optical colors compared to the PRIMUS parent galaxy sample. The Donley et al. IR-AGN and X-ray AGN samples identify similar fractions of AGNs in blue, star-forming hosts and red, quiescent hosts. These fractions of red and blue AGN hosts are similar to the red and blue fractions for all PRIMUS galaxies.

Hickox et al. (2009) find that X-ray AGNs are preferentially found in galaxy hosts in the green valley, the minimum of the optical color bimodality between the red sequence and the blue cloud (Martin et al. 2007), whereas Stern et al. IR-AGNs are typically in blue host galaxies. We do not agree with these results, which did not correct for X-ray incompleteness and contamination in the Stern et al. sample. Most recent papers (e.g., Silverman et al. 2009; Xue et al. 2010; Aird et al. 2012) find that the X-ray AGN fraction is highly stellar mass dependent. Similar to Nandra et al. (2007), Silverman et al. (2009), and Aird et al. (2012) we find that at a given stellar mass the fraction of galaxies with X-ray AGNs is higher in the blue cloud. In particular for the green valley, Nandra et al. (2007) use the ∼200 ks X-ray data in the EGS to find that 13% of X-ray-detected AGNs have green host galaxies, which is consistent with our X-ray AGN sample (9%) using a similar green valley definition. If X-ray completeness correction weights are not applied, the fraction of green host galaxies with X-ray AGNs rises only slightly, from 9% to 11%.

For the Stern et al. IR-AGNs, we generally find equal fractions of sources with blue or red host galaxies. There is a slightly larger fraction with blue hosts when compared using stellar-mass matched samples, but this is mainly due to blue star-forming host galaxies contaminating the sample rather than differences in the underlying population. The Donley et al. IR-AGN sample is more evenly distributed between red and blue host galaxies at fixed stellar mass.

We do find that the fraction of galaxies hosting an AGN increases with galaxy stellar mass. The fraction of massive ($\mathcal {M}_*$$\;{> }\;{10}^{11}\, \mathcal {M}_{\odot }$) galaxy hosts is larger for the X-ray AGN sample (78% ± 2%) compared to the Stern et al. IR-AGN (52% ± 2%) or Donley et al. IR-AGN sample (69% ± 4%). Conversely, the fraction of less massive (${10}^{9} < \mathcal {M}_*/ \mathcal {M}_{\odot }< {10}^{10}$) galaxy hosts for the X-ray AGN sample (7% ± 1%) is smaller than either the Stern et al. IR-AGN sample (21% ± 2%) or the Donley et al. IR-AGN sample (11% ± 3%). However, these differences are minor compared to the similarity of the host galaxy stellar mass distribution between any of the AGN selection techniques.

From the host galaxy stellar masses and the estimated AGN bolometric luminosities we infer the specific accretion rates, which provide a valuable measure of the intrinsic differences of the selected AGN populations. While IR-AGN samples are generally found in similar host galaxies as X-ray AGNs, we find that IR-AGNs have relatively high specific accretion rates. X-ray AGNs that are not identified using IR-AGN techniques include both sources that are not detected by IRAC and have high accretion rates and low stellar masses and sources that are detected by IRAC that have low to moderate specific accretion rates and high stellar masses. Some of the sources that are not IRAC detected are from fields (ES1, XMM, and COSMOS$_{\tt Shallow}$) with shallow IRAC data. For these luminous sources, we do not find any significant difference in the HRs of the X-ray AGN and IR-AGN samples, suggesting similar numbers of obscured sources. The X-ray sources with IRAC detections that are not IR-AGN selected have lower specific accretion rates and reside in red, massive galaxies and have large IR to X-ray luminosities. This suggests that they are not being identified by the IR-AGN techniques due to a bright galaxy component in the MIR due to the 1.6 μm stellar bump which is dominating the AGN light. They contribute to the large rise in the fraction of red massive galaxies that have an X-ray AGN. This accounts in part for the significant population of X-ray AGNs that is not identified by either IR-AGN technique.