The Chandra Deep Wide-field Survey: A New Chandra Legacy Survey in the Boötes Field. I. X-Ray Point Source Catalog, Number Counts, and Multiwavelength Counterparts

The 281 Chandra observations were downloaded from the Chandra Data Archive²⁹ and reduced with CIAO 4.11 (Fruscione et al. 2006) and CALDB version 4.8.2.

After reprocessing the data with the task chandra_repro with the flag check_vf_pha=yes, the observations were aligned. First, we ran the source detection tool wavdetect on each observation, adopting a permissive threshold (using three scales of ($\sqrt{2}$, 2, 4) pixels and setting the number of allowed spurious sources per scale to be falsesrc = 3, i.e., allowing for ∼10 spurious sources per field). Then, we matched sources within $6^{\prime}$ of the aimpoint (where the Chandra point-spread function (PSF) is approximately circular) to the catalog of optical counterparts to the XBoötes sources of Brand et al. (2006)³⁰ using the task reproject_aspect. This task rotates and translates an observation to minimize the difference between the positions of the X-ray sources and the reference list of optical counterparts. The reproject_aspect task requires at least three sources in common between the X-ray and optical catalogs to solve for both a rotation and a translation. Six of our observations had fewer than three X-ray/optical sources in common. For these, we used method=trans instead of the default method. This flag forces reproject_aspect to use only translation transformations and requires a minimum of one common source. The data were then reprocessed using the new aspect solution files. The results of this calibration step are shown in Figure 1, which demonstrates how the calibration adjusts and narrows the source distribution: 68% of the data are matched within 07, 90% within 12, and 95% within 15. Of course, this approach relies upon the correct identification of optical counterparts to XBoötes sources; however, the width of the histograms in Figure 1 is consistent with being largely dominated by X-ray positional uncertainties. We tested that a pure X-ray alignment of the observations (i.e., not relying on optical counterparts but employing common X-ray sources detected in partially overlapping observations) requires the use of very off-axis sources due to the scarcity of overlap for many observations, hampering the accuracy of the astrometric correction. The large exposure time range for partially overlapping observations further complicates the feasibility of such an alignment on the whole 9.3 deg² area.

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The described procedure aligned the X-ray observations to the world coordinate system (WCS) of the NDWFS Boötes field, which is the USNO-A2 one. As noted by Cool (2007), the USNO-A2 WCS could have a systematic shift with respect to more recent ones. Thus, we decided to register all of the data and catalogs that we are going to use throughout this paper to the same astrometric reference, the most recent being the one provided by the Gaia mission (Gaia Collaboration et al. 2016). Cross-matching USNO-A2 with Gaia DR2 (Gaia Collaboration et al. 2018) in the same region of the sky, encompassing the whole Boötes field, we found a systematic shift of R.A._NDWFS−R.A._Gaia = 032 and decl._NDWFS−decl._Gaia = 020. After checking that the shift is consistent throughout the field, we applied the appropriate astrometric correction to all of our mosaics. On the other hand, the Spitzer data of the Boötes field that we are going to use in this paper did not show any coordinate shift with respect to Gaia's astrometry. We refer the interested reader to Appendix A for a detailed discussion of the astrometry and coordinate registration.

We cleaned our observations of time intervals with high background using the tool dmextract to create 0.5–7 keV band light curves and lc_sigma_clip to create a good time interval (GTI) file after applying 3σ clipping to the light curve. We then used dmcopy to filter our event files using the GTI files.

This procedure resulted in a minimal time loss of 31 ks over 3.4 Ms of data (less than 1%). We also note that, as reported by Murray et al. (2005), six XBoötes observations (i.e., ObsIDs 3601, 3607, 3617, 3625, 3641, and 3657) had a higher background compared to the others. These observations were not discarded, consistent with the procedure of Murray et al. (2005). In addition to these observations, the short (6.1 ks) ObsID 17423 suffered from flaring but was not discarded. The inclusion of these seven observations is expected to have a negligible impact on the detection of sources given their relatively short exposure.

Exposure maps were created with the fluximage task. In this work, we adopt the following Chandra bands: full or broad (F; 0.5–7.0 keV), soft (S; 0.5–2.0 keV), and hard (H; 2.0–7.0 keV). When run in its default mode, fluximage produces vignetting-corrected exposure maps in units of cm² s counts photon^–1, while if the tool is run with the parameter units = time, the effective area of the Chandra mirrors is ignored. To obtain vignetting-corrected exposure maps with units of seconds, we produced effective area maps running fluximage with units = area (i.e., units of cm² counts photon^–1) and then divided each default exposure map by the maximum value of its corresponding effective area map.

Since the exposure map is theoretically monochromatic, it is computed at a given effective energy. The effective energies adopted in this work are 2.3, 1.5, and 3.8 keV for the F, S, and H bands, respectively. The F-band exposure map of the whole field is shown in Figure 2. All of the mosaics used in this work were created with the CIAO task reproject_image and spatially rebinned by a factor of 4 to speed up the computation time. This means that each pixel of the 4 × 4 rebinned mosaics has a scale of 1968. A false-color image of the whole field is shown in Figure 3.

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Precise background maps are crucial to having a reliable detection strategy and building accurate simulations. The quiescent (i.e., nonflaring) Chandra background can be explained, to first order, by the sum of an instrumental and an astrophysical component. The instrumental background is due to particles mimicking the signal of X-ray photons and to nonastrophysical X-ray photons, such as the ones produced by the interactions of particles with the spacecraft structure. The astrophysical background is the contribution from unresolved X-ray sources, mostly AGNs, that are fainter than the flux limit of each observation (i.e., the so-called cosmic X-ray background (CXB)). In addition, there is also diffuse emission of mainly soft X-rays from various local components, such as the Local Bubble and solar wind charge exchange mechanism (Markevitch et al. 2003; Slavin et al. 2013). Although the Chandra background is generally very low, the large dynamic range in exposure times over our field requires a careful disentangling of the two components for each observation; indeed, while the instrumental background is unvignetted, the astrophysical component is vignetted, band-dependent (the diffuse soft emission has a minor contribution in the hard band), and exposure-dependent (i.e., the fraction of unresolved sources, hence the CXB contribution, becomes smaller in deeper exposures).

We started by creating instrumental background maps following the procedure of Hickox & Markevitch (2006). Briefly, since the Chandra effective area rapidly decreases at E > 9 keV, the 9–12 keV band is vastly dominated by instrumental background. We then extracted the number of counts per pixel and per second in the 9–12 keV band with dmextract. The Chandra instrumental background is well known to be anticorrelated with the solar cycle (e.g., Markevitch et al. 2003)³¹ ; thanks to the large time span of our observations (∼15 yr), we were able to see this trend on more than one full solar cycle, as shown in Figure 4. This also confirmed the vastly nonastrophysical nature of the events in the 9–12 keV band; their number could thus be converted to instrumental background counts in other bands. Hickox & Markevitch (2006) estimated that the ratios between instrumental background in the F, S, and H bands and the one in 9–12 keV are 1.52, 0.4, and 1.12, respectively. Using these ratios, the total number of counts in the F, S, and H bands attributed to instrumental background could be obtained (B_Instr) and compared to the total number of counts extracted from the data (B_Data). To estimate the background directly from the data, for each observation, we extracted photons from a 6' radius circular area centered on the aimpoint, appropriately masking out point sources detected in a given band with significance >3.5σ and clusters from the catalog of Kenter et al. (2005). The counts extracted from the central 6' were then redistributed homogeneously over the whole field of view (FOV), rescaling by the ratio between the number of pixels of the ACIS-I FOV and those of the extraction area. Then, for each observation, we compared the number of background counts extracted from the data with the counts estimated from the instrumental background alone: if the difference B_Data–B_Instr was positively above 3σ (meaning that the data contained a significant number of excess counts of astrophysical nature), we subtracted the instrumental background from the total one, vignetted the resulting difference, and summed this component to the instrumental map to obtain the total background map (B_Tot). Otherwise, the background from the data was considered consistent within 3σ with the instrumental background map previously created.

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In Figure 5, we show the distribution of the differences (in σ) B_Data–B_Instr before applying any correction (orange dashed histogram) and B_Data–B_Tot after taking into account the CXB and soft diffuse emission (blue solid histogram), as well as their Gaussian fits (orange and blue dashed lines, respectively). As can be seen from Figure 5, the blue histograms are centered on zero and narrower than the orange ones. In the hard band, the distributions are very similar, implying that the background is mostly instrumental, and our extrapolation of the instrumental background from the 9–12 keV band was able to explain the background in our data in the majority of the observations.

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Chandra's PSF is known to be spatially and energy dependent. In particular, the farther away from the aimpoint and the higher the energy, the larger the size. The shape of the PSF also varies across the FOV with the azimuthal angle, significantly deviating from a circular shape and becoming more and more elliptical. However, the usual way to approximate this behavior is by defining the radius that encircles 90% of the energy, r₉₀. An approximate formula widely used in the literature (e.g., Hickox & Markevitch 2006) based on the trend of the PSF size with the off-axis angle θ shown in the Chandra Proposers' Observatory Guide³² is

$\begin{eqnarray}{r}_{90}\approx \left\{\begin{array}{ll}1^{\prime\prime} +10^{\prime\prime} {\left(\theta /10^{\prime} \right)}^{2}, & E=1.5\,\mathrm{keV}\\ 1\buildrel{\prime\prime}\over{.} 8+10^{\prime\prime} {\left(\theta /10^{\prime} \right)}^{2}, & E=6.4\,\mathrm{keV}.\end{array}\right.\end{eqnarray} \tag{ 1 }$

Using this approximate formula for r₉₀, we multiplied the computed value of r₉₀ in each pixel with the vignetting-corrected exposure map at the same pixel. We created maps of r₉₀ × t_Exp and ${r}_{90}^{2}\times {t}_{\mathrm{Exp}}$, then merged all of the observations together and divided the resulting mosaics for the total exposure map mosaic. This can be mathematically expressed (for r₉₀) as ${\displaystyle \sum }_{i=1}^{N}{r}_{90,i}{t}_{\mathrm{Exp},{\rm{i}}}/{\sum }_{i=1}^{N}{t}_{\mathrm{Exp},{\rm{i}}}$, where the sum is over the overlapping observations contributing to each pixel. This procedure effectively returned two mosaics of an exposure-weighted average of r₉₀ and ${r}_{90}^{2}$. While the former mosaic was used extensively in the following analysis to extract aperture photometry, the latter one was used to derive the sensitivity of the survey (Section 3.2).

The large time span of our observations across 15 yr of Chandra operation (between Chandra's Cycles 3 and 18), during which the spacecraft's effective area significantly changed, implies that the same intrinsic flux, spectral shape, and exposure time will result in a different number of counts detected in different positions on the field.

We used PIMMS as part of the Chandra Proposal Planning Toolkit³³ to obtain the energy conversion factors (ECFs) in the three energy bands adopted and for all of the Chandra cycles considered in this work (3, 4, 7, 8, 9, 10, 12, 14, 16, 17, and 18) and a range of photon indexes centered on ${\rm{\Gamma }}=1.4$, which is the average photon index of the populations of AGNs mostly making up the unresolved CXB spectrum (De Luca & Molendi 2004; Hickox & Markevitch 2006). Figure 6 graphically shows the significant evolution of the ECFs across the years, mainly in the soft band and seen regardless of the adopted spectral shape.

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To create an ECF mosaic of the field, we first created the single maps. For each observation, we considered the ECF given by its Chandra cycle, assuming a power law with photon index Γ = 1.4 and Galactic absorption ${N}_{{\rm{H}},\mathrm{Gal}}\,=1.04\times {10}^{20}$ cm⁻² (Kalberla et al. 2005), and convolved the appropriate ECF with the vignetting-corrected exposure map. Then, analogous to what was done for the PSF maps, we merged the observations to create a mosaic and divided the latter by the total exposure map mosaic.

As already mentioned, the list of sources returned by wavdetect was expected to be highly complete but, at the same time, include a significant number of spurious detections. Using a reasonable and justifiable probability threshold can drastically reduce the spurious fraction.

The total number of spurious sources detected in a simulation depends on Poisson statistics, the magnitude of the background, and the parameters of the detection algorithm (such as the sigmathresh and scales parameters of wavdetect). As long as the background used in the simulations is an accurate representation of the real background, and the detection process is the same as that adopted to detect sources in the real data, we can expect the number of spurious sources to be consistent within different realizations.

Spurious sources are, by definition, sources returned by wavdetect that do not correspond to a "real" input source. The fraction of spurious sources increases toward high probability (i.e., toward $\mathrm{log}P\sim 0$), with a tail of fewer spurious sources being assigned a lower probability. We can use our simulated sources to compare the relative fraction of matched (i.e., real) and unmatched (i.e., spurious) sources at each probability. We expect these normalized distributions to have different shapes, so that we can determine a probability threshold that maximizes the difference between the two. This is shown in Figure 7, where it can be seen that around $\mathrm{log}P\sim -4.5$, the difference between the two distributions is maximal. Due to statistical noise, the three sets of simulations (with different input sources) return slightly different peaks, as indicated by the yellow regions in Figure 7. Since setting an accurate probability threshold is crucial for this work, we computed the median of the three "Difference" curves in Figure 7 and fitted it with a third-degree polynomial function over the range $-6\lt \mathrm{log}P\lt -3$. The fits are shown in the insets in Figure 7. The final thresholds we determined to minimize the selection of spurious sources are $\mathrm{log}P=(-4.63,-4.57,-4.40)$ for the F, S, and H bands, respectively.

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Using the simulations, we could test how well we measure fluxes, comparing output and input fluxes, and compute the completeness of our sample. In particular, the comparison of the output and input fluxes shows the well-known Eddington bias (Eddington 1913), for which faint sources close to the flux limit of the survey tend to be brighter than their actual flux due to positive fluctuations being more likely to be detected than negative ones. This is exemplified in the top panel of Figure 8, where the points show (detected) single sources matched to their input counterparts. The Eddington bias effect is also clearly visible in the bottom panel of Figure 8.

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The ratio of the number of matched sources to the number of input ones as a function of (input) flux decreases toward fainter fluxes, reflecting the incompleteness of the survey. This ratio, rescaled by the total area of the survey, is in excellent agreement with the expected sensitivity curve obtained using an analytical calculation that includes Eddington bias (Georgakakis et al. 2008), as shown in Figure 9. To compute the expected sensitivity curve taking into account Eddington bias, we first multiplied each pixel of the background map mosaic with the exposure-averaged area of the PSF at that location, given by $\pi {r}_{90}^{2}$, so that ${B}^{* }=B\times \pi {r}_{90}^{2}$. Then we computed the minimum number of counts that, given the background value B* in that pixel, would result in a significant detection (i.e., a probability that counts are due to a background fluctuation lower than the imposed threshold). Then we defined an array of fluxes s, and we converted it into an array of expected number of counts, given by $T=s{ \mathcal C }{f}_{\mathrm{PSF}}{t}_{\mathrm{Exp}}+{B}^{* }$, where ${ \mathcal C }$ is the ECF from flux to count rate, f_PSF is the encircled energy fraction of the PSF (0.9 in our case), and t_Exp is the exposure time. For each pixel, we ended up with an array of probabilities $P(\mathrm{cts},T)$ that the minimum number of counts are observed, given the expected number of counts T. Finally, summing up all pixels at any given flux resulted in the expected sensitivity curve. The dips at high fluxes in Figure 9 are due to a handful of bright sources being undetected. Such sources mainly fall on the very edge of the mosaic and have a sufficiently large PSF that their effective surface brightness is lowered. In one case, a bright source was missed entirely because it lay very close to another (brighter) source and was detected and rejected by our code that removes duplicate detections in PSF wings. Representative values of flux limits at different levels of completeness are tabulated in Table 2.

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When a source was detected in more than one band, we used the coordinates of the band with the most significant detection as the final X-ray position, and we computed the positional error as ${r}_{50}/\sqrt{N}$ (Puccetti et al. 2009), where r₅₀ was estimated from r₉₀ as r₅₀ = 5/9 × r₉₀,⁴¹ and N is the net counts (within r₅₀) in the band with the most significant detection. When r₅₀ was smaller than a 4 × 4 pixel area of our mosaic, we used r₉₀ instead (this was necessary for ∼2% of the sources). The distribution of the positional errors derived in this way is shown in Figure 10 and has a median value of ∼08. Positional errors formally smaller that 01 (occurring for 19 very bright sources) have been conservatively set to 01, following Civano et al. (2016) and Puccetti et al. (2009). The cumulative distribution shows that 90% of the sources have a positional error of <15 and 99% less than 25, while 71% of them have a positional error smaller than 1''. These somewhat nonoptimal values (compared, for example, with the values reported by Civano et al. 2016, for the Chandra COSMOS Legacy survey, in which 85% of the sources have a positional error of <1'') could be attributed to the large fraction of observations overlapping with a significantly different roll angle, which ultimately results in a slightly larger PSF size on some parts of the field, and the lower median net counts of the CDWFS sources (15, 11, and 13 compared to 30, 20, and 22 in Chandra COSMOS Legacy in the F, S, and H bands, respectively). Our positional errors are consistent with the errors derived using 68% confidence level formulae as reported by Kim et al. (2007a). Hence, we consider our positional errors as 1σ uncertainties on the position of detected sources.

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Once a unique set of coordinates was derived for each source, we reextracted aperture photometry for all bands at the same position, analogous to what was done for the simulations. We briefly recall the procedure here. Using the exposure-weighted average r₉₀ to extract counts, background counts, and exposure, we computed the net counts and converted them to a count rate. The flux was computed correcting the count rate for the encircled energy fraction of the PSF within r₉₀ and converted to flux using the exposure-weighted average ECF (assuming Γ = 1.4). Using a single set of coordinates for each source, the most significant detection had exactly the same aperture photometry as before, while the other two bands were adjusted and realigned with the new coordinates.

For the bands in which each source was not significantly detected, this procedure assigned a new probability of being spurious. If this new probability was higher than our imposed detection threshold, we considered the source to be detected in that band (although originally missed by wavdetect) and computed its count rate and flux as outlined above. If, instead, the source had a probability of being spurious higher than our adopted threshold, we computed 3σ upper limits on the net counts, count rate, and flux following Gehrels (1986).

The distributions of fluxes measured for sources significantly detected in each of the three bands are shown in Figure 11; for display purposes, the brightest source in the field, i.e., the variable star KT Boötes (F-band flux of F_X ∼ 3 ×10⁻¹² erg cm⁻² s⁻¹), is not included.

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As a final step, we cross-matched our catalog with that of Kenter et al. (2005), which contains 3293 X-ray point sources detected with more than 4 counts.⁴² We found that 447 out of 3293 XBoötes sources (∼14%) do not have an entry (within 1.1 × r₉₀) in the CDWFS catalog. Such a high number of missing sources cannot be simply explained by the spurious fraction of the XBoötes survey, expected to be around 1% (Kenter et al. 2005). The large majority (95%) of the missing XBoötes sources have ≤6 counts in the Kenter et al. (2005) catalog. Of the 21 missing sources with seven or more counts, half of them are spread over such a large PSF area that the background is nonzero, and their significance drops. A few other sources are genuinely drowned out by the background with deeper data. For others, there is a count mismatch; i.e., unrealistically large PSF radii are required to match the number of counts reported in the XBoötes catalog. These latter cases imply that the updated data reduction and/or different light-curve filtering may have introduced some subtle differences among the CDWFS and XBoötes data.

Table 3 shows the breakdown of the missing sources, split into sources absent from the candidate list of sources in output from wavdetect and those additionally missed due to our reliability cuts (i.e., detected as potential sources by wavdetect and later rejected). It can be seen from Table 3 that the majority of the missing sources (283/447 ∼ 63%) are already missing from the wavdetect output of candidates. While two-thirds of the total area of the field is covered by additional data compared to the original XBoötes survey, one-third of the area (∼3 deg²) comprises the very same data. A rough distinction between sources lying on the same XBoötes data and those falling where new observations have been done is obtained in Table 3 by splitting the missing sources by exposure: sources with t_Exp < 10 ks lie mostly over the original XBoötes edge of the field.

While Table 3 reveals that the sources missed by wavdetect are homogeneously scattered across the field regardless of exposure time, the reasons for missing sources in areas of different exposure are likely different. On one hand, when the exposure time is increased by coadding observations, the Eddington bias effect becomes less significant, source variability can play a role, and genuinely spurious sources are less likely to be detected again. Among these factors, the first one is likely the most effective in this case: our simulations show that ∼16% of simulated sources with t_Exp < 8 ks are detected with an output flux more than twice the input one due to Eddington bias. The effect drastically reduces to ∼3% for sources with t_Exp > 40 ks. A possible additional role could be played by the degrading sensitivity of Chandra with time, resulting in diluting the genuine signal of faint sources with the increasing exposure (and hence background).

On the other hand, XBoötes sources missing from wavdetect output on the very same XBoötes data require a more careful treatment and a different explanation. As discussed in Section 3, wavdetect was run on the whole field mosaic, with additional inputs such as the exposure map and PSF mosaics.⁴³ Murray et al. (2005) and Kenter et al. (2005) did not mention if any additional information was fed to wavdetect, but it is likely they did not use exposure or PSF maps when detecting sources over each of the 126 XBoötes observations. In the left panel of Figure 12 can be seen that our inclusion of the PSF information is very likely playing a role. The median r₉₀ of the sources lying on the low-exposure part of the field and missed by wavdetect is almost twice the median r₉₀ of the CDWFS catalog.⁴⁴ Smoothing the low counts of such sources on a large PSF area likely dilutes their contrast with the background and lowers their significance. This emphasizes a caveat regarding using a simple circular area of radius r₉₀ to detect sources and extract aperture photometry. The sampling of the PSF is indeed not homogeneous, and counts (especially in a low-exposure, low-counts regime) will generally not distribute uniformly across the circular area. A more rigorous approach would require taking into account the actual shape and substructure of the PSF when detecting sources and extracting aperture photometry, which is outside the scope of this work.

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The fact that sources missed by wavdetect have generally larger off-axis angles is corroborated by the distribution of offsets between XBoötes sources and their optical NDWFS counterparts as defined by Brand et al. (2006). In the right panel of Figure 12, it is clear that sources missing in CDWFS show larger-than-average offsets to their matched optical counterparts, likely a combination of large positional errors (deriving from large PSF sizes and low X-ray counts) and faint optical counterparts. Indeed, considering the whole Brand et al. (2006) catalog of XBoötes optical counterparts, we find that ∼62% have an entry in the AGN and Galaxy Evolution Survey (AGES) optical spectroscopic catalog of Kochanek et al. (2012). However, when considering the subset of 447 XBoötes sources missing from CDWFS, only ∼38% have an entry in the AGES catalog. Once again, this confirms that these sources are likely very faint and with low significance.

Thus, unsurprisingly, even if all 447 sources would have been picked up by wavdetect, ∼88% of them would have been rejected by our imposed thresholds. The remainder of the sources (∼12%, 53) that satisfy our reliability thresholds have been added to our catalog. For the newly added sources, aperture photometry was computed for each band at the position of the source in the Kenter et al. (2005) catalog, extracting counts from the full-resolution mosaic, analogous to what was done for the other sources. As can be seen in Table 3, the majority of these significant sources (40) were genuinely missed by wavdetect, while a few of them were missed due to reliability cuts. These latter sources are few, rare cases in which the position returned by wavdetect is slightly offset with respect to the position in the Kenter et al. (2005) catalog. This tiny offset (often around one native Chandra pixel) is enough to make the source significance fluctuate across the thresholds.

After detecting sources independently in the F, S, and H bands, merging the three catalogs together into a list of 6838 sources, and adding 53 significant XBoötes sources, the final number of unique, X-ray point sources in our catalog is 6891.

Estimating the final spurious fraction of the CDWFS catalog is not trivial, since our simulations treat each band independently (we recall that we expect, on average, 58, 48.5, and 52 spurious sources from simulations in the F, S, and H bands, respectively) and we added some sources from a previously published catalog. In a best-case scenario in which all spurious sources of the F band are made up by the spurious sources of the S and H bands, which are instead detected independently, and none of the added XBoötes sources is spurious, we would end up with an estimated spurious fraction of 100.5/6891 = 1.5%. In a worst-case scenario in which the spurious sources in the F, S, and H bands are all different sources, and 1% of the added XBoötes sources are also spurious, we would end up with a spurious fraction of 159/6891 = 2.3%. We estimated the most likely true spurious fraction by computing how many F-band spurious sources could also be detected in the S and H bands by splitting the F-band counts into the S and H bands assuming Γ = 1.4 and through Poisson statistics. On average, 14% and 41% of the F-band spurious sources would satisfy our reliability thresholds in the S and H bands, respectively. Furthermore, we believe the 53 XBoötes sources that satisfy our reliability thresholds to be real. Hence, the total number of spurious sources in the final catalog is expected to be ∼127/6891 = 1.8%.

Along with the list of the excluded 394 XBoötes sources (447 − 53) that do not satisfy our thresholds, we also release the full electronic CDWFS catalog. An extract of the first 10 sources of both catalogs can be found in Appendix B (Table B1 and B2, respectively), while the full lists are available online as electronic catalogs.

The final detailed breakdown of the number of X-ray sources detected in each combination of the three bands is reported in Table 4.

We first explored whether we could accurately recover the input logN–logS of the simulations.

We applied both methods and compared their robustness in recovering the input logN–logS in our simulations. As shown in Figure 13, both methods are generally able to recover the input logN–logS. As expected, the nonstandard method (the set of colder colors in Figure 13) better recovers the input shape. It is worth noting that the lower panels of Figure 13 show the ratio between output/input logN–logS where the input logN–logS corresponds to the analytical shape of the curve, not the actual Poissonian realization effectively used for the different sets of simulations. The spread of the measured number counts at bright fluxes is due to the Poisson fluctuations introduced by this process, depicted using gray shading in Figure 13.

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As can be seen in Figure 13, in the hard band, one simulation set overshoots the faint end of the input logN–logS by a factor of ≲1.5. However, this happens only in the hard band, in one set out of three simulations, and only when extrapolating to fluxes below the flux limit (which is depicted in Figure 13 by the warm colors, as the standard method extends over the range of actual fluxes of the detected sources). In summary, we are confident that both methods fairly recover the underlying distribution of the number counts, and the nonstandard method is more accurate and reliable over the range of fluxes covered by our sources.

Motivated by the previous analysis, we applied both methods to our X-ray catalog to derive robust number counts, exploiting the large number of sources detected in each band. As reported in the second column of Table 5, all sources significantly detected in each band (excluding the brightest source in the field, which, as stated previously, is a star) were used in the following computation. The differential number counts were fit with a broken power law of the form

$\begin{eqnarray}{dN}/{dS}=\left\{\begin{array}{ll}K{\left({S}_{x}/{S}_{\mathrm{ref}}\right)}^{{\beta }_{1}}, & {S}_{x}\leqslant {S}_{b}\\ K{\left({S}_{b}/{S}_{\mathrm{ref}}\right)}^{{\beta }_{1}-{\beta }_{2}}{\left({S}_{x}/{S}_{\mathrm{ref}}\right)}^{{\beta }_{2}}, & {S}_{x}\gt {S}_{b}\end{array}\right.,\end{eqnarray} \tag{ 4 }$

where S_ref = 10⁻¹⁴ erg cm⁻² s⁻¹ and S_b is the break flux; the results are shown in Table 5. The uncertainties were estimated through bootstrapping. We note that in this analysis, we did not separate AGNs from galaxies and stars. However, we expect galaxies to have a negligible impact on the number counts at the relatively bright fluxes probed here, with stars even less significant (see Lehmer et al. 2012). Having computed the differential number counts, we integrated them to obtain logN–logS, which is shown in Figure 14 for the nonstandard method. Note that the results using the standard method are fully consistent with those in the figure apart from some fluctuations due to the different treatment of Eddington bias.

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Figure 14 also shows the number counts from other Chandra surveys.⁴⁵ Since CDWFS updates the previous XBoötes survey, it is particularly informative to consider the best-fit logN–logS from Kenter et al. (2005) in the soft and hard bands (solid green lines in Figure 14). In the soft band, Kenter et al. (2005) fit their number counts with a broken power law; our results are consistent with these results, and we can better constrain the parameters of the fit. On the other hand, Kenter et al. (2005) fit the hard number counts with a single power law and obtained a significantly steeper slope, while the results we obtain for CDWFS are in much better agreement with other Chandra surveys, at least at the bright end of the distribution. In particular, the faint end of the hard-band logN–logS appears to be steeper than the other measurements, even before extrapolating. This higher density is also found with logN–logS computed using the standard method. Wang et al. (2004) also noted this higher density and attributed it to an overdensity of faint hard X-ray sources in the deepest region of the field (i.e., the LaLa survey; Wang et al. 2004).

The best-fit parameters of the differential number counts can also be used to infer the resolved fraction of the CXB; this is trivially done by computing the integral

$\begin{eqnarray}&&{\int }_{{S}_{\min }}^{{S}_{\max }}\displaystyle \frac{{dN}}{{dS}}{SdS},\end{eqnarray} \tag{ 5 }$

where S_min and S_max are the minimum and maximum flux of the detected sources. The result of this integral gives the total X-ray flux per square degree resolved into single sources. Thus, the fraction of resolved CXB is obtained by comparing the resolved flux to the total CXB intensity in a given energy band. Converting the values reported by Hickox & Markevitch (2006) to match our energy bands assuming Γ = 1.4, we obtained a resolved fraction of 65.0% ± 12.8% and 81.0% ± 11.5% for the soft and hard bands, respectively. The uncertainties were estimated assuming a normal PDF for the parameters of the dN/dS and solving the integral 5000 times, each time randomly picking the parameters according to their PDFs. The uncertainty on the total intensity of the CXB as reported by Hickox & Markevitch (2006) was propagated to obtain the error on the resolved fraction. Since in this paper, we detected only point sources, the resolved flux does not take into account the contribution from extended sources (such as galaxy clusters or groups), which is expected to be larger in the soft band. The resolved fraction of the CXB is consistent within the uncertainties with that reported by Kim et al. (2007b) for the ChaMP survey in both the S and H bands, at comparable depth and total area with CDWFS; also, in the H band, with that reported by Hickox & Markevitch (2006).

To obtain spectroscopic redshifts, we matched the CDWFS catalog with the AGES catalog (Kochanek et al. 2012) using the NDWFS I-band coordinates (or the SDWFS ones, for the 810 sources with IR-only counterparts) and adopting a matching radius of 05. We shifted the whole AGES catalog in order to correct its astrometry with respect to Gaia (see Appendix A). In total, we had 2346 valid spectroscopic redshifts. Then, we exploited the recent release of hybrid photometric redshifts (template + machine learning; see Duncan et al. 2018a, 2018b) for the upcoming data release of the LOFAR Two-meter Sky Survey Deep Fields (Duncan et al. 2020, submitted). Analogous to that done for the spectroscopic redshifts, we matched our CDWFS sources with a matching radius of 05 and chose the closest counterpart when multiple I-band entries were found within this distance. We ended up with a total of 6447 photometric redshifts. The overall quality of the photometric redshifts compared to the spectroscopic ones is shown in Figure 16, and it is generally very good. A handful of outliers show a photometric redshift higher than 6 but are all actually associated with much lower spectroscopic redshifts. We thus suggest caution when focusing on the 30 CDWFS sources with z_Phot ≳ 6. The normalized median absolute deviation, ${\sigma }_{\mathrm{NMAD}}=1.48\times \mathrm{median}(| {z}_{\mathrm{ph}}-{z}_{\mathrm{sp}}| /(1+{z}_{\mathrm{sp}}))$, is 0.054, while the fraction of outliers, defined as the fraction of sources where $| {z}_{\mathrm{ph}}-{z}_{\mathrm{sp}}| /(1+{z}_{\mathrm{sp}})\gt 0.15$, is 13.23%. These already excellent values and likely will soon be further improved by the upcoming Subaru HSC data coverage of the Boötes field. Table 6 summarizes the number of counterparts and redshifts for the whole X-ray catalog with p_any > 0 and the subset for which p_any > 0.29.

Once the redshift of a given X-ray source is known, a rest-frame luminosity can be computed from the observed flux. Although very penetrating, X-rays suffer an absorption bias depending on the amount of gas column density along the line of sight (usually denoted with N_H). This absorption effect is band-dependent and mostly affects soft X-rays. To derive statistically reliable absorption-corrected rest-frame luminosities, we first estimated the gas column density N_H for any given source. We used a standard method, which estimates the column density using the combination of the HR (defined as HR = (H−S)/(H+S), where H and S are the hard and soft counts, respectively) and the redshift information. The HR was computed for each X-ray source using the Bayesian estimator for hardness ratio (BEHR; Park et al. 2006) and taking into account the slightly larger area from which the hard counts have been extracted during the aperture photometry step due to the larger PSF size in the H band.

The HR encodes the observed spectral shape of the source; since the absorption bias is band-dependent, the redshift has to be known to properly estimate the intrinsic column density at the location of the source. We therefore created a grid of theoretical curves of HR as a function of redshift for a large range of column density using XSPEC (Arnaud 1996) and a simple power law with photon index Γ = 1.8, consistent with the average, absorption-corrected value of the whole AGN population (e.g., Ricci et al. 2017). We modified the assumed power law to account for absorption by a fixed Galactic column density N_H,Gal = 1.04 × 10²⁰ cm⁻² (Kalberla et al. 2005), estimated approximately at the center of the Boötes field, and an additional amount of column density in the range log(N_H/cm⁻²) = [20, 25] with a step of 0.02 dex.⁴⁷

Once the correlated parameters HR and z were known for a given source, the closest N_H-dependent theoretical curve in our grid was used to estimate the column density that, for a typical power law of Γ = 1.8 at redshift z, resulted in the observed HR (and, ultimately, the observed S- and H-band counts).

It is worth stressing that the spectral shape of a local heavily obscured AGN is much more complex than what is defined by a simple obscured power law. First, the assumed simple model does not include the effect of Compton scattering, which becomes important at the highest column densities, close to the Compton-thick threshold (i.e., N_H ≳ 10²⁴ cm⁻²); ignoring it leads to an underestimation of the intrinsic luminosity for such highly obscured AGNs (e.g., Li et al. 2019). Moreover, the possible presence of a reflection component would harden the spectrum, resulting in an overestimation of the obscuration (e.g., Wilkes et al. 2013). In addition, prominent soft emission, likely arising from photons that have scattered off clouds on a scale much larger than the nuclear one (where most of the obscuring material lies), dominates the soft X-ray spectrum of heavily obscured AGNs at E ≲ 2–3 keV (Bianchi & Guainazzi 2007; Ricci et al. 2017). This implies that, for high column densities approaching the Compton-thick level and for low-to-intermediate redshifts, the HR estimated from a more realistic model can be much softer than what would be expected from a simple obscured power law. However, if we try to incorporate a more physical model when creating a grid of theoretical HR(z) tracks, many tracks end up crossing each other, invalidating our method of using the HR–z plane to uniquely estimate individual source properties. The effect of such a simplification on estimates of column density is that a fraction of AGNs appearing as unobscured and faint will, in reality, be obscured and intrinsically more luminous (Lambrides et al. 2020). A careful assessment of the obscuration state of our sample is outside the scope of the present paper and requires a combination of multiwavelength diagnostics, e.g., using rest-frame MIR luminosities and X-ray spectral analysis, to disentangle genuine, faint, and unobscured AGNs from luminous obscured ones.

The HR–z plane, together with the relative redshift and HR distributions of our sample, is shown in the left panel of Figure 17. As can be seen from the top plot of this panel, ∼90% of our sample lies within z < 3. The HR distribution of the whole sample, shown in the right plot, demonstrates that the median HR of the whole (HR-constrained) sample ($\overline{\mathrm{HR}}=-0.16$) is similar to the expected HR for an unobscured power law with photon index Γ = 1.8 but shifted to a harder (i.e., larger) HR. Indeed, the whole HR distribution looks asymmetric and is well fit by a double Gaussian function (e.g., Civano et al. 2016), with (μ, σ) = (0.22, 0.29) and (−0.27, 0.21) representing a mix of the unobscured and obscured populations of AGNs.

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Since the early stages of X-ray surveys, AGNs have been known to occupy a well-defined region of the X-ray and optical flux parameter space. In particular, the ratio between the X-ray and optical flux, defined as X-ray–to–optical flux ratio (X/O) $=\ \mathrm{log}{f}_{x}+m/2.5+C$ (where C is a constant dependent on the photometric band considered; e.g., Tananbaum et al. 1979; Maccacaro et al. 1988), is generally in the range [−1, 1]. Moreover, correlations have been found between X/O and the X-ray luminosity, mainly for obscured AGNs (Fiore et al. 2003; Marchesi et al. 2016); at any given X-ray luminosity, obscured AGNs are expected to have their UV/optical emission fainter than their unobscured counterparts, resulting in a higher X/O. Hence, we can further check the broad correspondence between HR and obscuration by investigating the behavior of the X/O as a function of hard-band luminosity, using the H-band intrinsic luminosity and i-band magnitude to compute X/O; we estimated C = 5.03 for the NDWFS i-band filter, although its exact value is not important, as it just shifts the whole figure to higher or lower values of X/O. In the right panel of Figure 17, we color-code sources based on their HR and visually demonstrate through their density contours that, in general, hard sources fall in the region of the plane where obscured AGNs are expected to be. Likewise, soft sources are more spread out, in particular at L_X > 10⁴³ erg s⁻¹.

Having estimated N_H values using the HR–z plane, we computed the correction factor k, defined as the ratio between the obscured and intrinsic flux k = f_obs/f_int, where f_int is estimated with an unobscured power law with Γ = 1.8 following Marchesi et al. (2016). Since our N_H range is limited to log(N_H/cm⁻²) ≥ 20, we assigned k = 1 to those sources with log(N_H/cm⁻²) ≤ 20. We translated 1σ uncertainties on the HR into uncertainties on the column density, which in turn constrained the minimum and maximum values of k for the lower and higher N_H limits, respectively.

Once we have calculated k for each band, we compute the rest-frame intrinsic flux ${f}_{\mathrm{int}}^{\mathrm{RF}}={f}_{\mathrm{int}}{\left(1+z\right)}^{{\rm{\Gamma }}-2}$ (with Γ = 1.8) and then, finally, the intrinsic luminosity at a given redshift using ${L}_{\mathrm{int}}^{\mathrm{RF}}=4\pi {D}_{L}^{2}{f}_{\mathrm{int}}^{\mathrm{RF}}$, where D_L is the luminosity distance.

A plot of the rest-frame intrinsic luminosity in the 2–10 keV band (converted from the 2–7 keV luminosity assuming Γ = 1.8) as a function of redshift is shown in the left panel of Figure 18. This panel demonstrates that the range of exposure times for observations in the Boötes field allows a relatively broad range of the luminosity–redshift plane to be explored. In particular, thanks to its combination of wide area and deep exposures, the CDWFS has sufficient statistical power to probe both above and below the knee of the luminosity function (L_*, as parameterized by Aird et al. 2015) in the redshift range z ∼ 0.5–3. This is crucial to probe the full distribution of mass accretion rates for the AGN population over a large redshift range. This is even more evident when the distribution of rest-frame 2–10 keV luminosities for CDWFS, COSMOS Legacy (Civano et al. 2016; Marchesi et al. 2016), and Stripe82X (LaMassa et al. 2016) are compared, as demonstrated in the right panel of Figure 18. We show the 2–10 keV band luminosity, since it is less affected by obscuration corrections, but the situation is the same for the soft band. The combination of the wide area and the depth accumulated through 15 yr of Chandra observations makes CDWFS unique in terms of coverage. The CDWFS has more sources than the COSMOS Legacy and Stripe82X surveys combined at any luminosity. Furthermore, the distribution of CDWFS luminosities bridges the gap between COSMOS Legacy and Stripe82X, adding a significant number of sources in the luminosity range log(L/erg s⁻¹) ∼44–45. This is crucial to probe the typical and most luminous accreting black holes at the peak epoch of AGN and galaxy coevolution.

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