X-Ray Redshifts for Obscured AGN: A Case Study in the J1030 Deep Field

The J1030 field was observed by Chandra/ACIS-I with 10 different pointings between 2017 January and May, for a total exposure time of ∼479 ks and a field of view of 335 arcmin². This set of observations makes the J1030 field one of the deepest extragalactic X-ray surveys performed so far, allowing us to investigate the obscured AGN population up to high redshift and down to limiting fluxes of $\sim 3,0.6,2\times {10}^{-16}\,\mathrm{erg}\,{{\rm{s}}}^{-1}\,{\mathrm{cm}}^{-2}$ in the 0.5–7 (full), 0.5–2 (soft), and 2–7 (hard) keV bands, respectively. Different from the other Chandra deep/moderately deep fields (e.g., CDF-S, Giacconi et al. 2001; Luo et al. 2017; COSMOS-Legacy, Civano et al. 2012; Marchesi et al. 2016a), the J1030 field has not yet benefited from decades of spectroscopic follow-ups. The details of the observations and data reduction are given in Section 2 of Nanni et al. (2018). The source catalog (Nanni et al. 2020, hereafter N20) has been generated using wavdetect (Freeman et al. 2002) for the source detection and CIAO Acis Extract (Broos et al. 2010) for the source photometry and significance assessment. The final catalog contains 256 X-ray sources. These have then been matched with the available optical/infrared catalogs using a likelihood ratio technique (e.g., Ciliegi et al. 2003; Brusa et al. 2007; Luo et al. 2010); 252 of them have a counterpart in these wave bands (see N20 for further details).

In 2012, the J1030 field was observed with the Large Binocular Telescope (LBT) using the Large Binocular Camera (LBC) to obtain imaging of a $23^{\prime} \times 25^{\prime}$ area in the r, i, and z bands (Morselli et al. 2014) down to limiting AB magnitudes of 27.5, 25.5, and 25.2, respectively. In 2015, we performed a $24^{\prime} \times 24^{\prime}$ observation in the near-infrared Y and J bands (Balmaverde et al. 2017) at the Canada–France–Hawaii Telescope (CFHT) using WIRCam, with limiting AB magnitudes of 23.8 and 23.75, respectively.

The field is one of the four fields included in the Multi-wavelength Survey by Yale-Chile (MUSYC). Three MUSYC catalogs are available for J1030: the BVR catalog, obtained by selecting sources in the BVR stacked image down to 26.3 AB magnitudes (Gawiser et al. 2006), and the K-wide (Blanc et al. 2008) and K-deep (Quadri et al. 2007) catalogs, performed selecting sources in the K band down to K = 21 and 23 AB, respectively. As shown in Figure 1, the MUSYC BVR and K-wide data cover a $30^{\prime} \times 30^{\prime}$ area, while the MUSYC K-deep covers a smaller $10^{\prime} \times 10^{\prime}$ area. Furthermore, the field has been observed by the Spitzer Infrared Array Camera (IRAC) in the mid-infrared (MIR) at 3.6 and 4.5 μm. In this work, we used the available catalogs and images in the IRSA ⁸ archive that reach a depth of 22–23 AB magnitudes. All of the optical/infrared data sets and filters used in this paper are discussed in Section 4.

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From the X-ray catalog, we selected a sample of 54 obscured AGN candidates on the basis of their hardness ratio (HR), defined as

$$\begin{eqnarray}&&{\rm{HR}}=\displaystyle \frac{H-S}{H+S},\end{eqnarray} \tag{ 1 }$$

where H and S are the net count rates (i.e., background subtracted) in the hard and soft bands, respectively. Since the AGN X-ray emission is more absorbed at low energies, the HR value can be considered a good proxy of absorption for AGN with known redshift and simple (absorbed power-law) spectra (e.g., Mainieri et al. 2002).

However, the Chandra/ACIS-I photon collecting efficiency is rapidly decreasing in the soft band due to contamination (see the Chandra Proposers' Observatory Guide ⁹ ), making the HR a time-dependent quantity. This decrease has accelerated over the last few years, thus allowing only a qualitative comparison with previous works (e.g., Tozzi et al. 2001; Szokoly et al. 2004), as explained in Appendix A. Therefore, we performed X-ray spectral simulations based on the J1030 Chandra observations to reproduce the expected trends for our sample. The simulations were performed through XSPEC ¹⁰ v.12.9.1 (Arnaud 1996), assuming an absorbed power-law model at different redshifts (zphabs×powerlaw). The mean Galactic absorption at the J1030 field position, ${N}_{H}=2.6\times {10}^{20}\,{\mathrm{cm}}^{-2}$ (Kalberla et al. 2005), was also considered (phabs). In Figure 2, we show the HR values for typical AGN column densities (${10}^{21}\,{\mathrm{cm}}^{-2}\leqslant {N}_{H}\leqslant {10}^{24}\,{\mathrm{cm}}^{-2}$) as a function of redshift and with a canonical photon index ${\rm{\Gamma }}=1.9\pm 0.2$ (e.g., Nandra & Pounds 1994; Piconcelli et al. 2005; Lanzuisi et al. 2013a). It is worth mentioning that these curves are computed for simulated spectra with thousands of counts to reproduce the expected HR trends (e.g., Szokoly et al. 2004; Elvis et al. 2012). Based on our simulations, we chose a threshold of ${\rm{HR}}\gt -0.1$ (black horizontal line) to select obscured AGN. In fact, considering ${\rm{\Gamma }}=1.9$, this threshold allows the selection of obscured objects with ${N}_{H}\geqslant {10}^{22}\,{\mathrm{cm}}^{-2}$ up to $z\approx 0.5$, ${N}_{H}\geqslant {10}^{23}\,{\mathrm{cm}}^{-2}$ up to $z\approx 2.5$, and Compton-thick AGN (${N}_{H}\gtrsim {10}^{24}\,{\mathrm{cm}}^{-2}$) at all redshifts. If we instead consider a flatter (${\rm{\Gamma }}=1.7$) or steeper (${\rm{\Gamma }}=2.1$) power law, we obtain more positive or negative HR, respectively, but, in both cases, the chosen ${\rm{HR}}\gt -0.1$ threshold avoids the selection of unobscured sources at any redshift. Due to the limited photon statistics in our sample, we assumed a basic model to compute the different N_H curves shown in Figure 2. Despite this, if we consider more complex models, the chosen HR threshold remains valid (see Appendix B). In addition to the HR criterion, we selected sources detected in N20 with at least 50 net counts in the 0.5–7 keV band to allow an effective search for X-ray spectral features, such as the 6.4 keV Fe Kα emission line and the 7.1 keV Fe absorption edge. For sources not detected in one or two bands, N20 reported the 3σ net counts upper limits. Because we were looking for relatively hard objects, we considered an HR = 1 for sources undetected in the soft band and ${\rm{HR}}=-1$ for sources undetected in the hard band. We discarded 11 sources detected in the full band only. The final sample and adopted selection criteria are shown in Figure 3.

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The spectral extraction was made using the Chandra Interactive Analysis of Observations ¹¹ (CIAO) v.4.9 software. The choice of the extraction regions was performed by taking into account the source position on the detector, since the point-spread function (PSF) broadens as the off-axis angle (θ) increases. As the extraction radius, we use the 90% encircled energy radius (E = 1.49 keV) at the source position, and, to increase the signal-to-noise ratio of the faintest sources, we manually chose ad hoc slightly smaller radii. The regions used to extract the background spectra were selected next to each source and, to ensure a good background sampling, with an area at least 10 times larger. We extracted a spectrum from each observation covering a given source and then combined these spectra using the CIAO tool combine_spectra. The source spectra were grouped to a minimum of one count per energy bin to avoid empty channels. We checked the presence of at least one background count in each source energy bin and then adopted the modified C-statistic for direct background subtraction (Cash 1979; Wachter et al. 1979) or W-statistic to estimate the best-fit model parameters. The spectral analysis was performed using XSPEC v.12.9.1 (Arnaud 1996).

Our sample of 54 objects has a median value of $\approx 80$ net counts in the 0.5–7 keV energy range and median fluxes of $6.8,1.0,5.4\times {10}^{-15}\,\mathrm{erg}\,{{\rm{s}}}^{-1}\,{\mathrm{cm}}^{-2}$ in the full, soft, and hard bands, respectively. Given the low photon statistics, we adopted a simple model based on a power law, an intrinsic N_H at the source redshift z, and Galactic absorption at the source position (phabs(zphabs×powerlaw)). The photon index Γ was fixed to 1.9, as commonly observed in AGN (e.g., Nandra & Pounds 1994; Lanzuisi et al. 2013a). We left the intrinsic N_H , z, and power-law normalization free to vary. To investigate the presence of emission lines, we included a redshifted Gaussian feature at 6.4 keV rest frame (zgauss) with a redshift parameter anchored to the corresponding absorption component, a free normalization, and a fixed line width $\sigma =10\,\mathrm{eV}$ (e.g., Nanni et al. 2018), which takes into account only the narrow component produced far away from the central SMBH by the absorbing cold medium, since the broad, relativistic component produced in the accretion disk is expected to be obscured (e.g., Risaliti & Elvis 2004). Using a simple absorbed power law plus a Gaussian line may not be a detailed description of the X-ray spectrum. Nevertheless, the limited count statistics did not allow an investigation of more complex spectral shapes (e.g., Lanzuisi et al. 2013a; Iwasawa et al. 2020). This choice is also justified by a few tests on more complex models, showed in Appendix B.2. In case of heavy obscuration, the main AGN features, such as the Fe Kα 6.4 keV line and the 7.1 keV Fe edge, become more prominent (e.g., Iwasawa et al. 2012) and easily recognizable, as shown in Figure 4. Once one of these features was identified, a redshift solution from the X-ray spectrum (hereafter z_X) was derived by evaluating the redshift-likelihood profile, computed with the steppar command. An example is reported in Figure 5. We considered to be reliable z_X those solutions where the difference in the C-statistic (${\rm{\Delta }}C$) between the global minimum (primary solution, i.e., the best-fit redshift) and its nearby maximum is at least 2.71, corresponding to a fit improvement at the 90% confidence level (see, e.g., Tozzi et al. 2006 and Brightman et al. 2014, who validated this threshold through simulations). We also investigated local minima (secondary solutions) where the above ${\rm{\Delta }}C$ criterion was satisfied, as for the case in Figure 5. Each selected z_X was then further investigated through simulations.

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Estimating the goodness of an X-ray spectral fit is not trivial in the case of low count statistics. We therefore built two different sets of spectral simulations to test the derived X-ray redshift solutions. The first set of simulations aims at verifying the significance of the candidate emission lines (Section 3.2.1), and the second aims at constraining for which photon statistics, as a function of N_H , z, and θ, we expect to obtain robust redshifts from X-ray data alone (Section 3.2.2). All simulated spectra were obtained using the XSPEC fakeit command.

3.2.1. Line Significance

The simulations reported in this paragraph refer to sources where the redshift estimate is driven by the iron Kα emission line. For each source in which the line was possibly detected, we established a significance criterion to deem a candidate emission line as reliable as follows (e.g., Lanzuisi et al. 2013b; Vignali et al. 2015). Because we do not know the redshift of the sources, we fitted the spectra using an absorbed power-law model (phabs×powerlaw; ${\rm{\Gamma }}=1.9$) with and without a Gaussian line (gauss; $\sigma =10$ eV). For the former, the line energy was fixed to the observed value. The best-fit parameters from the model without the line were then used to simulate 1000 spectra with the same characteristics (response matrices, ¹² exposure time, background and photon statistics) as the observed one. We fitted the simulated spectra using exactly the two models, fixing the continuum to the one derived from the model without the line and leaving the line energy free to vary, looking for all cases where

$$\begin{eqnarray}&&{\rm{\Delta }}{C}_{{\rm{sim}}}\geqslant {\rm{\Delta }}{C}_{{\rm{obs}}},\end{eqnarray} \tag{ 2 }$$

where ${\rm{\Delta }}{C}_{{\rm{obs}}}$ is the difference between the best-fit model with and without the possible line in the observed spectrum, while ${\rm{\Delta }}{C}_{{\rm{sim}}}$ is the difference between the same models in each simulated spectrum. The frequency at which Equation (2) occurs corresponds to the probability $P^{\prime}$ that the detected line is just a statistical fluctuation. Then

$$\begin{eqnarray}&&{P}_{{\rm{sim}}}=(1-P^{\prime} )\end{eqnarray} \tag{ 3 }$$

corresponds to the significance of the observed line. We considered an emission line reliable when ${P}_{{\rm{sim}}}\geqslant 90 \%$.

Since calculating P_sim is time-consuming, we proceeded as follows. The F-test is a fast method to evaluate the model improvement due to an additive component, but in the case of a Gaussian line, it is considered inappropriate (Protassov et al. 2002). However, allowing the line normalization to also be negative, ¹³ it can be used as an indication of the model improvement. After extensive testing on our data set, we found that, to obtain a reliable line (i.e., with ${P}_{{\rm{sim}}}\geqslant 90 \%$), an F-test probability (P_Ft)$\,\gt \,99 \%$ is required. We then decided to use the F-test as prescreening. When the P_Ft threshold is reached, the significance (P_sim) of the candidate lines is computed through the aforementioned simulations to provide a more solid evaluation. When a significant line has been found, it is used to derive an X-ray redshift solution, assuming that the detected line is the Fe Kα fluorescent emission line at 6.4 keV, which is the most probable emission line in the AGN X-ray spectrum (e.g., Fabian et al. 2000). Otherwise, the z_X determination is principally driven by the Fe 7.1 keV edge coupled with the photoelectric absorption cutoff.

3.2.2. Redshift Solutions as a Function of N_H , Net Counts, and Off-axis Angle

To verify the level of photon statistics allowing the redshifts to be derived from the X-ray analysis, we performed a second set of simulations. We simulated spectra not only with different AGN parameters but also using different responses and backgrounds, as these vary with the off-axis angle. This set of simulations aims to be global, i.e., reliable in every position of the observations, and it was used to further investigate the derived X-ray redshift solutions. To test the z_X driven by absorption features, we adopted an absorbed power-law model (zphabs×powerlaw) with a fixed intrinsic photon index ${\rm{\Gamma }}=1.9$, while for redshift solutions driven by the Fe Kα line, we also included a redshifted Gaussian line (zgauss) at 6.4 keV rest frame, with a width of $\sigma =10\,\mathrm{eV}$. We set different line normalizations to obtain a canonical range of rest-frame equivalent widths, between 10 eV and 2 keV, as a function of N_H (e.g., Ghisellini et al. 1994; Lanzuisi et al. 2015). In both models, we added an additional absorption component (phabs) with a fixed value of ${N}_{H}=2.6\times {10}^{20}\,{\mathrm{cm}}^{-2}$, corresponding to the mean Galactic absorption at the J1030 field position. To reproduce what is observed in deep X-ray surveys (e.g., Tozzi et al. 2006; Marchesi et al. 2016a), we simulated column densities from ${N}_{H}={10}^{21}$ to 10²⁴ cm^–2 with a logarithmic step of 0.5, redshifts up to 5 with a step of 0.5, and different power-law normalizations to obtain a number of full-band net counts in the range 10–1000. For each parameter combination, 500 spectra were simulated.

To simulate spectra that are as close as possible to those observed in the Chandra data, the response matrices of the real observation must be used. In general, the instrumental response drops as the off-axis angle increases, which therefore has to be taken into account. Assuming no azimuthal dependence of the instrument response, we extracted the ancillary response file (ARF) and the redistribution matrix file, for four relatively bright sources at different off-axis angles (θ ∼ 0', 32, 50, and 95) in order to reproduce the decrease of the effective area as a function of θ (Figure 6). In addition, it is necessary to evaluate the background spectra to be associated with the simulated source spectra. Assuming that the background only varies with the off-axis angle and has no azimuthal dependence, we extracted background spectra using a circular region of $r=2^{\prime}$ centered at the aimpoint and annuli of width 1' at a distance of 35, 55, and 95 from the aimpoint. To get a suitable background, the sources present in such regions were excluded. We used the CIAO tool dmcopy through the command exclude to remove X-ray sources identified by N20 in the J1030 field. The extracted background also needs to be rescaled by the source extraction area before being associated with it. Since the sources are simulated, they do not have a physical extraction region, so we rescaled the background to the width of the PSF (90% encircled energy radius at E = 1.49 keV) at specific off-axis angles.

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For each parameter combination (N_H , z, net counts, and θ), 1000 spectra were simulated, for a total of 800,000. For each simulated spectrum, a fit was performed, and the best-fit redshift solution, z_X, was derived. We then computed the match percentage,

$$\begin{eqnarray}&&\mathrm{match}\ \% \ (z,{N}_{H},{\rm{counts}},\theta )=\displaystyle \frac{N({z}_{{\rm{X}}}\pm {\rm{\Delta }}z)}{N({z}_{{\rm{sim}}})},\end{eqnarray} \tag{ 4 }$$

that corresponds, for a specific range of redshift, N_H , number of net counts, and θ, to the number of simulated spectra in which z_X is consistent with the simulated redshift (z_sim) within a given tolerance ${\rm{\Delta }}z$, normalized to the total number of simulated spectra. Because of the low count statistics, the X-ray redshift solutions are sometimes poorly constrained. To determine if the redshift solutions are reliable, we rejected solutions with $| {\rm{\Delta }}z| \gt 0.15(1+{z}_{{\rm{sim}}})$, defined as outliers (this value has been found in previous works to be a reliable boundary for outliers; e.g., Hsu et al. 2014; Ananna et al. 2017; Luo et al. 2017; Simmonds et al. 2018). For a conservative approach, we discarded redshift solutions that are either upper or lower limits. We also checked the N_H values, rejecting solutions that are not consistent within the errors with the simulations, even if the redshift solutions are good (∼6%). An example of the simulations' performance is shown in Figure 7, where it is clear how column densities $\geqslant {10}^{22}\,{\mathrm{cm}}^{-2}$ are needed to obtain a reliable z_X for sources with a net count range as in our sample. Furthermore, the match percentage depends on redshift. If no emission lines are detected, the X-ray solutions are driven by the iron absorption edge coupled with the photoelectric cutoff, but, moving toward high redshift (z > 3–4), such an absorption complex ends up at ≲1.5 keV, where the effective area decreases dramatically. As a consequence, the match percentage decreases as the redshift increases. Besides showing how the main features of the X-ray spectrum become more prominent with increasing obscuration, and therefore more easily identifiable, the simulations give us an indication of the probability of deriving a correct redshift or not. We set a match percentage threshold of at least 50% to accept z_X solutions driven by absorption features. In this regard, we found reliable solutions down to a regime of ∼30 net counts for redshift ≲2.5 and ${N}_{H}\gt {10}^{23}\,{\mathrm{cm}}^{-2}$. For z_X solutions derived from the Fe Kα line instead, we considered this 50% threshold as an additional check, since it confirms the results obtained from Section 3.2.1. In our sample, where it was possible to derive an X-ray redshift solution (38 sources), the chosen threshold returns a mean match percentage of $\sim 70 \%$. The full procedure adopted for the z_X estimate is summarized in Figure 8.

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We used the available data sets in the optical and infrared bands to calculate photometric redshifts (hereafter z_phot) through an SED fitting procedure and test the X-ray redshift solutions. Photometric redshifts were obtained through the hyperz code (Bolzonella et al. 2000) using a variety of galaxy templates, detailed below, and a Calzetti et al. (2000) reddening law. The code finds the best-fit template through a standard ${\chi }^{2}$ minimization procedure, comparing template spectra to the observed SEDs as

$$\begin{eqnarray}&&{\chi }^{2}(z)=\displaystyle \sum _{i=1}^{{N}_{{\rm{filters}}}}{\left[\displaystyle \frac{{F}_{{\rm{obs}},i}-b\times {F}_{{\rm{temp}},i}(z)}{{\sigma }_{i}}\right]}^{2},\end{eqnarray} \tag{ 5 }$$

where ${F}_{{\rm{obs}},i}$ and ${F}_{{\rm{temp}},i}$ are the observed and template fluxes, ${\sigma }_{i}$ is the observed flux uncertainty, and b is a normalization constant. Here hyperz provides primary and secondary z_phot solutions, the best-fitting template spectrum, and the reduced ${\chi }^{2}$ for each given object. For a detailed description of the code, we refer to Bolzonella et al. (2000). An essential requirement for the SED fitting procedure is an extensive template library that covers the entire range of selected objects without adding unphysical degeneracies. Since the strong ONIR radiation from the accretion disk is heavily extinguished in obscured AGN, allowing the stellar emission of the host galaxy to dominate at these wavelengths (e.g., Merloni et al. 2014), photometric redshifts can be reliably estimated using standard galaxy templates. In this regard, we included the available photometry up to ∼5 μm. Above this wavelength, in fact, we expected that the rest-frame emission of the hot dusty torus may overcome the host galaxy stellar emission (e.g., Pozzi et al. 2012; Circosta et al. 2019), invalidating the choice of galaxy templates.

Following Ilbert et al. (2013), we used a library composed of 75 galaxy templates—19 empirical templates derived from the SWIRE library (Polletta et al. 2007), including both elliptical and spiral galaxies (S0, Sa, Sb, Sc, Sd, Sdm), plus 12 starburst templates and 44 additional red galaxy templates generated by Ilbert et al. (2009, 2013) through the Bruzual & Charlot (2003) stellar synthesis models—to better cover the color–redshift space. We built source SEDs using the catalogs described in Table 1; a total of 12 different filters are available from the blue optical to MIR wavelengths. When the same source is revealed in the same filter in more than one catalog, we used the magnitude value from the deepest observation, while if the source is either not detected in a specific band or detected with a signal-to-noise ratio $\lt 2$, we excluded the corresponding filter from the SED fitting procedure. We searched photometric redshift solutions from z = 0 to 7 with a step of ${\rm{\Delta }}z=0.05$. The photometric redshift accuracy also depends on the number of filters in which a specific source is detected. In this regard, S. Marchesi et al. (2021, in preparation) carried out a photometric redshift analysis of all the X-ray sources in the J1030 field, with a similar procedure and data set used in this section. Following this work, we set a threshold of at least five filters that corresponds to an rms $\approx 0.1$ when comparing the z_phot with the available spectroscopic redshift in the field.

In order to correctly reproduce the SED of each individual source, the photometry should sample the same physical region of the host galaxy in all bands. Therefore, the photometric redshift technique uses aperture-corrected magnitudes to account for the flux lost outside the fixed aperture due to the different seeing conditions in different observations. The photometry in the ONIR catalogs (LBT, WIRCam, and MUSYC) was obtained assuming a circular aperture of diameter $\sim 1\buildrel{\prime\prime}\over{.} 6$ (Gawiser et al. 2006; Quadri et al. 2007; Blanc et al. 2008; Morselli et al. 2014; Balmaverde et al. 2017). While in the LBC, WIRCam, and BVR MUSYC catalogs, the aperture correction was already considered, we estimated the aperture-correction terms in the MUSYC K-deep and K-wide catalogs as follows. For pointlike sources, the difference between total and aperture magnitudes is, by definition, the aperture correction. The mag_AUTO entries in the MUSYC K-deep and K-wide catalogs are a good approximation of the total magnitudes, and in Figure 9, we plotted the difference between them and the aperture magnitudes against the total magnitudes. It is evident that pointlike sources are arranged along a straight line whose value corresponds to the aperture-correction term to be applied to the catalog. We estimated and applied a correction of 0.45 and 0.62 for the MUSYC K-deep and K-wide catalogs, respectively. All of the ONIR catalogs were then matched together assuming a matching radius of 1'', and the resulting sources were then associated with the X-ray counterparts through a likelihood ratio (Sutherland & Saunders 1992) algorithm, ¹⁴ as widely discussed in N20.

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The IRAC channels 1 and 2, respectively at 3.6 and 4.5 μm, were introduced to improve the z_phot estimate. The inclusion of photometric points at longer wavelengths may increase the best-fit quality, but, due to the low angular resolution of the IRAC camera, separating the emission of close sources was not always possible. Therefore, we performed a visual check to associate the correct MIR counterpart with each X-ray source, and, to avoid any contamination from blended sources, we excluded ambiguous cases from the SED fitting procedure. Unlike the ONIR catalogs, whose aperture extraction diameters are similar, the IRAC fluxes are provided with aperture diameters of 38 and 58. We decided to use the fluxes with the smaller aperture to minimize blending effects. To take into account the larger aperture and different angular resolution, we built the SEDs adding in quadrature ${\rm{\Delta }}\mathrm{mag}=0.1$ to the IRAC magnitude error of each source in both channel 1 and channel 2.

The main sample contains 54 X-ray-selected obscured AGN candidates with $\approx 80$ median extracted net counts. Each source has been extensively analyzed in the X-rays and through SED fitting in the ONIR and MIR bands. To verify the goodness and quality of the X-ray redshift solutions, these are now compared with the obtained photometric redshifts. It was possible to estimate an X-ray redshift for 38 (∼70%) sources, down to ∼30 net counts, and a photometric redshift for 46 (∼85%) sources. We do not report X-ray redshift solutions for sources without significant spectral features or when we classified them as unreliable according to the spectral simulations, as discussed in Section 3. Sources without a photometric redshift estimate are those detected in less than five filters. We obtained both z_X and z_phot solutions for 33 (∼61%) sources, and for all but three there is a redshift estimate from at least one method. For XID 29, 130, and 135, no redshift could be found by any method. The latter two are very faint X-ray sources (only 26 and 39 full-band net counts were extracted, respectively) and lie at very large off-axis angles (100 and 69, respectively). Because of the poor spectral quality, no significant features were detected in their X-ray spectra. Though XID 29 has 133 net counts in the full band and lies at 29, no significant features were found in the X-ray spectrum. While XID 130 is detected only in IRAC, XID 29 and 135 do not reach the threshold on the number of filters, preventing the photometric redshift estimate. All of the obtained redshift solutions are summarized in Table 2, and they represent the first list of redshifts for the J1030 field.

Table 2. Properties of Obscured AGN Candidates (${\rm{HR}}\gt -0.1$ and Net Counts $\geqslant 50$ from N20) in the J1030 Field

XID	Counts Full	HR	zphot	zX	NH	FX	LX	Feat.
(1)	(2)	(3)	(4)	(5)	(6)	(7)	(8)	(9)
2	${890}_{-30}^{+31}$	$-{0.01}_{-0.04}^{+0.04}$	${0.70}_{-0.02}^{+0.02}$	${0.59}_{-0.36}^{+0.12}$	${1.6}_{-0.2}^{+0.2}$	${28.39}_{-1.35}^{+1.25}$	${8.31}_{-0.39}^{+0.39}$	Edge
3	${141}_{-12}^{+13}$	$-{0.04}_{-0.08}^{+0.08}$	${0.98}_{-0.04}^{+0.04}$	−1	${2.4}_{-0.7}^{+0.8}$	${5.49}_{-0.53}^{+0.52}$	${3.65}_{-0.44}^{+0.43}$	⋯
8	${115}_{-11}^{+12}$	${0.07}_{-0.11}^{+0.11}$	${3.24}_{-0.04}^{+0.41}$	${2.80}_{-0.05}^{+0.05}$	${33.4}_{-9.2}^{+9.9}$	${3.38}_{-0.46}^{+0.41}$	${35.38}_{-5.95}^{+5.67}$	Line
11	${95}_{-10}^{+11}$	${0.05}_{-0.12}^{+0.12}$	${2.98}_{-0.71}^{+0.24}$	${1.94}_{-0.38}^{+1.05}$	${27.4}_{-6.9}^{+8.0}$	${3.67}_{-0.53}^{+0.40}$	${39.41}_{-6.57}^{+6.11}$	Edge
15	${407}_{-20}^{+21}$	${0.36}_{-0.05}^{+0.05}$	${0.94}_{-0.04}^{+0.11}$	−1	${6.4}_{-0.6}^{+0.6}$	${13.64}_{-0.84}^{+0.82}$	${9.19}_{-0.63}^{+0.62}$	⋯
16	${55}_{-8}^{+9}$	${0.03}_{-0.15}^{+0.16}$	${1.47}_{-0.05}^{+0.03}$	${1.22}_{-0.74}^{+0.95}$	${4.4}_{-1.9}^{+2.5}$	${1.85}_{-0.34}^{+0.23}$	${3.33}_{-0.68}^{+0.65}$	Edge
22	${65}_{-8}^{+9}$	${0.05}_{-0.16}^{+0.16}$	−1	${2.87}_{-0.06}^{+0.05}$	${32.3}_{-9.7}^{+12.3}$	${1.51}_{-0.27}^{+0.31}$	${17.36}_{-3.79}^{+3.73}$	Line
26	${261}_{-16}^{+17}$	$-{0.05}_{-0.07}^{+0.07}$	−1	${3.47}_{-0.21}^{+0.16}$	${23.8}_{-3.8}^{+6.0}$	${7.50}_{-0.62}^{+0.47}$	${108.52}_{-10.44}^{+10.28}$	Edge
28	${374}_{-19}^{+20}$	$-{0.07}_{-0.06}^{+0.06}$	${1.79}_{-0.08}^{+0.08}$	${1.32}_{-0.04}^{+0.04}$	$\lt 1.6$	${8.42}_{-0.64}^{+0.53}$	${10.98}_{-0.78}^{+0.77}$	Line
29	${133}_{-16}^{+17}$	${0.42}_{-0.09}^{+0.09}$	−1	−1	−1	${4.57}_{-0.43}^{+0.53}$	−1	⋯
30	${200}_{-14}^{+15}$	${0.32}_{-0.08}^{+0.08}$	${0.26}_{-0.02}^{+0.06}$	${0.27}_{-0.16}^{+0.70}$	${1.9}_{-0.3}^{+0.3}$	${5.95}_{-0.50}^{+0.48}$	${0.18}_{-0.02}^{+0.02}$	Edge
36	${204}_{-15}^{+16}$	${0.38}_{-0.07}^{+0.08}$	${0.96}_{-0.02}^{+0.04}$	${0.88}_{-0.54}^{+0.35}$	${8.1}_{-1.1}^{+1.2}$	${9.53}_{-0.87}^{+0.65}$	${6.84}_{-0.71}^{+0.69}$	Edge
37	${79}_{-9}^{+10}$	${0.18}_{-0.14}^{+0.14}$	${1.62}_{-0.08}^{+0.14}$	${1.65}_{-1.00}^{+0.43}$	${12.3}_{-2.8}^{+3.4}$	${2.46}_{-0.37}^{+0.34}$	${6.15}_{-1.07}^{+1.02}$	Edge
38	${119}_{-11}^{+12}$	$-{0.04}_{-0.10}^{+0.10}$	${1.16}_{-0.07}^{+0.05}$	${1.12}_{-0.68}^{+1.09}$	${3.6}_{-1.1}^{+1.2}$	${5.79}_{-0.63}^{+0.76}$	${5.98}_{-0.78}^{+0.76}$	Edge
39	${47}_{-7}^{+8}$	${0.03}_{-0.16}^{+0.17}$	${1.47}_{-0.04}^{+0.09}$	−1	$\lt 5.5$	${1.83}_{-0.38}^{+0.36}$	${3.18}_{-0.72}^{+0.69}$	⋯
40	${55}_{-8}^{+9}$	${0.21}_{-0.15}^{+0.15}$	${1.72}_{-0.26}^{+0.76}$	−1	${4.8}_{-2.5}^{+4.1}$	${2.35}_{-0.40}^{+0.30}$	${5.67}_{-1.14}^{+0.78}$	⋯
41	${118}_{-11}^{+12}$	${0.02}_{-0.11}^{+0.11}$	${2.28}_{-0.32}^{+0.23}$	${2.58}_{-0.49}^{+0.19}$	${16.8}_{-3.9}^{+5.1}$	${3.53}_{-0.45}^{+0.39}$	${20.08}_{-2.99}^{+3.02}$	Edge
46	${121}_{-11}^{+13}$	${0.06}_{-0.10}^{+0.10}$	${2.10}_{-0.09}^{+0.20}$	${1.85}_{-0.02}^{+0.03}$	${4.4}_{-2.0}^{+2.3}$	${5.15}_{-0.64}^{+0.59}$	${20.88}_{-2.94}^{+2.88}$	Line
47	${72}_{-9}^{+10}$	${0.55}_{-0.13}^{+0.13}$	${1.50}_{-0.15}^{+0.07}$	${1.38}_{-0.84}^{+0.32}$	${16.8}_{-3.6}^{+4.5}$	${4.40}_{-0.72}^{+0.63}$	${9.79}_{-1.88}^{+1.80}$	Edge
48	${44}_{-7}^{+9}$	${0.03}_{-0.16}^{+0.16}$	${1.71}_{-0.07}^{+0.06}$	−1	$\lt 6.8$	${1.94}_{-0.32}^{+0.31}$	${4.85}_{-1.05}^{+1.1}$	⋯
55	${52}_{-7}^{+9}$	${0.02}_{-0.14}^{+0.15}$	${0.69}_{-0.04}^{+0.02}$	−1	${3.9}_{-1.2}^{+1.4}$	${5.86}_{-1.03}^{+1.08}$	${1.80}_{-0.38}^{+0.35}$	⋯
56	${100}_{-11}^{+12}$	$-{0.02}_{-0.11}^{+0.12}$	${0.83}_{-0.02}^{+0.03}$	−1	${1.7}_{-0.7}^{+0.8}$	${3.82}_{-0.41}^{+0.60}$	${1.69}_{-0.26}^{+0.25}$	⋯
58	${28}_{-6}^{+7}$	${0.43}_{-0.15}^{+0.20}$	${2.65}_{-0.06}^{+0.10}$	−1	${17.6}_{-6.1}^{+14.5}$	${8.34}_{-2.80}^{+1.93}$	${61.96}_{-17.4}^{+21.57}$	⋯
59	${295}_{-18}^{+19}$	${0.61}_{-0.06}^{+0.08}$	${1.00}_{-0.03}^{+0.03}$	${0.79}_{-0.38}^{+0.21}$	${16.8}_{-1.6}^{+1.8}$	${16.63}_{-1.22}^{+1.02}$	${15.55}_{-1.37}^{+1.36}$	Edge
62	${55}_{-8}^{+9}$	${0.15}_{-0.17}^{+0.19}$	−1	${2.76}_{-0.24}^{+0.11}$	${47.0}_{-16.2}^{+-13.4}$	${4.40}_{-0.92}^{+0.72}$	${50.16}_{-12.13}^{+11.98}$	Edge
67	${205}_{-15}^{+16}$	${0.16}_{-0.08}^{+0.08}$	${0.35}_{-0.10}^{+0.06}$	${0.38}_{-0.01}^{+0.01}$	${1.3}_{-0.3}^{+0.3}$	${6.54}_{-0.62}^{+0.62}$	${0.44}_{-0.05}^{+0.05}$	Line
69	${223}_{-16}^{+17}$	${0.15}_{-0.09}^{+0.09}$	−1	${1.50}_{-0.91}^{+0.90}$	${10.5}_{-1.8}^{+2.0}$	${8.29}_{-0.69}^{+0.65}$	${17.00}_{-1.82}^{+1.77}$	Edge
70	${215}_{-15}^{+16}$	${0.28}_{-0.07}^{+0.07}$	${0.72}_{-0.03}^{+0.04}$	${0.76}_{-0.07}^{+0.10}$	${5.1}_{-0.7}^{+0.7}$	${10.80}_{-0.76}^{+0.87}$	${3.74}_{-0.36}^{+0.35}$	Edge
75	${109}_{-12}^{+13}$	${0.60}_{-0.10}^{+0.14}$	${0.80}_{-0.01}^{+0.04}$	${1.44}_{-0.65}^{+0.28}$	${12.1}_{-2.0}^{+2.5}$	${10.70}_{-1.64}^{+1.24}$	${5.16}_{-0.80}^{+0.75}$	Edge
82	${123}_{-12}^{+13}$	${0.24}_{-0.11}^{+0.13}$	${4.06}_{-0.45}^{+0.36}$	−1	${53.4}_{-11.1}^{+12.1}$	${5.29}_{-0.76}^{+0.76}$	${154.38}_{-23.74}^{+22.2}$	⋯
87	${39}_{-7}^{+8}$	${0.51}_{-0.13}^{+0.18}$	${1.56}_{-0.05}^{+0.10}$	${1.54}_{-0.07}^{+0.13}$	${14.2}_{-3.5}^{+4.8}$	${19.23}_{-3.66}^{+3.63}$	${14.03}_{-3.64}^{+3.61}$	Edge
89	${39}_{-7}^{+8}$	${0.13}_{-0.20}^{+0.22}$	${1.01}_{-0.03}^{+0.05}$	${1.52}_{-0.92}^{+0.64}$	${6.9}_{-2.4}^{+3.0}$	${4.28}_{-0.99}^{+1.06}$	${3.27}_{-0.89}^{+0.79}$	Edge
91	${128}_{-12}^{+13}$	$-{0.10}_{-0.17}^{+0.16}$	${1.06}_{-0.04}^{+0.22}$	${0.92}_{-0.02}^{+0.03}$	${1.5}_{-0.8}^{+0.9}$	${18.78}_{-2.30}^{+2.18}$	${10.41}_{-1.47}^{+1.37}$	Line
94	${67}_{-9}^{+10}$	${0.52}_{-0.11}^{+0.12}$	${4.08}_{-0.69}^{+0.10}$	${2.72}_{-0.14}^{+0.23}$	${105.4}_{-20.5}^{+23.6}$	${6.50}_{-0.97}^{+1.08}$	${155.99}_{-28.57}^{+26.71}$	Edge
106	${107}_{-11}^{+12}$	${0.16}_{-0.12}^{+0.13}$	${0.53}_{-0.03}^{+0.02}$	${0.37}_{-0.02}^{+0.02}$	${1.7}_{-0.4}^{+0.5}$	${5.09}_{-0.63}^{+0.52}$	${0.33}_{-0.05}^{+0.05}$	Line
110	${120}_{-16}^{+17}$	${0.22}_{-0.14}^{+0.17}$	${0.98}_{-0.08}^{+0.04}$	${1.56}_{-0.94}^{+0.33}$	${6.4}_{-1.5}^{+1.8}$	${6.01}_{-0.91}^{+0.93}$	${5.24}_{-0.78}^{+0.71}$	Edge
115	${40}_{-7}^{+8}$	${0.52}_{-0.11}^{+0.14}$	${0.46}_{-0.03}^{+0.02}$	${0.66}_{-0.28}^{+0.56}$	${5.2}_{-1.4}^{+1.9}$	${5.76}_{-1.31}^{+1.00}$	${0.73}_{-0.18}^{+0.17}$	Edge
117	${33}_{-6}^{+7}$	${0.04}_{-0.19}^{+0.20}$	${3.10}_{-0.60}^{+0.80}$	${2.80}_{-1.68}^{+0.35}$	${36.0}_{-13.4}^{+16.3}$	${4.50}_{-1.00}^{+1.36}$	${54.26}_{-15.84}^{+14.63}$	Edge
119	${46}_{-7}^{+9}$	${0.30}_{-0.15}^{+0.17}$	${0.54}_{-0.02}^{+0.02}$	−1	${4.3}_{-1.2}^{+1.5}$	${4.31}_{-0.96}^{+0.76}$	${0.76}_{-0.17}^{+0.16}$	⋯
120	${19}_{-5}^{+6}$	$-{0.09}_{-0.21}^{+0.21}$	${0.95}_{-0.07}^{+0.09}$	−1	$\lt 2.1$	${4.12}_{-1.59}^{+1.19}$	${2.25}_{-0.52}^{+0.53}$	⋯
122	${82}_{-10}^{+11}$	${0.29}_{-0.12}^{+0.13}$	${1.29}_{-0.04}^{+0.02}$	${0.98}_{-0.60}^{+0.53}$	${9.6}_{-2.6}^{+2.9}$	${6.28}_{-0.85}^{+0.98}$	${9.03}_{-1.60}^{+1.51}$	Edge
127	${35}_{-6}^{+7}$	$-{0.02}_{-0.15}^{+0.16}$	${0.05}_{-0.02}^{+0.03}$	−1	$\lt 1.1$	${2.23}_{-0.51}^{+0.42}$	$({1.6}_{-0.3}^{+0.4})$ e-3	⋯
130	${26}_{-6}^{+7}$	$-{0.07}_{-0.32}^{+0.33}$	−1	−1	−1	${6.28}_{-1.80}^{+1.48}$	−1	⋯
131	${39}_{-7}^{+8}$	${0.05}_{-0.21}^{+0.23}$	${1.68}_{-0.15}^{+0.12}$	${1.53}_{-0.47}^{+0.71}$	${11.8}_{-4.6}^{+6.2}$	${4.09}_{-0.94}^{+0.91}$	${11.03}_{-3.00}^{+2.81}$	Edge
135	${34}_{-6}^{+8}$	${0.05}_{-0.18}^{+0.20}$	−1	−1	−1	${3.72}_{-0.76}^{+0.83}$	−1	⋯
137	${50}_{-8}^{+9}$	$-{0.04}_{-0.18}^{+0.19}$	${1.48}_{-0.76}^{+1.10}$	−1	$\lt 18.7$	${4.70}_{-0.87}^{+0.86}$	${8.96}_{-3.31}^{+2.58}$	⋯
140	${40}_{-7}^{+9}$	${0.02}_{-0.23}^{+0.24}$	${0.92}_{-0.04}^{+0.05}$	${0.82}_{-0.44}^{+0.74}$	$\lt 4.0$	${2.14}_{-0.45}^{+0.48}$	${1.06}_{-0.29}^{+0.29}$	Edge
192	${138}_{-15}^{+17}$	1.0	${1.76}_{-0.04}^{+0.06}$	${2.20}_{-0.28}^{+0.15}$	${77.4}_{-12.5}^{+15.5}$	${10.93}_{-1.41}^{+1.14}$	${60.94}_{-14.11}^{+13.66}$	Edge
196	${76}_{-9}^{+10}$	1.0	${0.54}_{-0.01}^{+0.02}$	${0.41}_{-0.25}^{+0.05}$	${28.9}_{-5.8}^{+10.2}$	${4.67}_{-0.67}^{+0.43}$	${1.67}_{-0.37}^{+0.48}$	Edge
200	${162}_{-15}^{+16}$	1.0	${1.42}_{-0.04}^{+0.03}$	${1.59}_{-0.54}^{+0.59}$	${3.4}_{-1.2}^{+1.4}$	${6.40}_{-0.75}^{+0.58}$	${10.48}_{-1.29}^{+1.23}$	Edge
201	${55}_{-8}^{+9}$	1.0	${0.86}_{-0.05}^{+0.02}$	${1.15}_{-0.21}^{+0.08}$	${34.1}_{-6.9}^{+10.5}$	${6.41}_{-1.15}^{+0.86}$	${6.12}_{-1.48}^{+1.63}$	Edge
207	${78}_{-10}^{+11}$	1.0	${0.14}_{-0.04}^{+0.03}$	${0.14}_{-0.09}^{+0.58}$	${6.3}_{-1.2}^{+1.9}$	${9.50}_{-1.50}^{+1.43}$	${0.10}_{-0.02}^{+0.02}$	Edge
220	${34}_{-7}^{+8}$	1.0	${1.57}_{-0.08}^{+0.06}$	${2.23}_{-0.34}^{+0.21}$	${60.5}_{-17.3}^{+27.9}$	${6.95}_{-1.41}^{+1.54}$	${26.87}_{-10.3}^{+10.58}$	Edge
221	${30}_{-6}^{+7}$	1.0	−1	${1.37}_{-0.28}^{+0.08}$	${58.8}_{-18.6}^{+25.1}$	${8.21}_{-1.95}^{+1.58}$	${25.15}_{-9.70}^{+9.71}$	Edge

Note. Columns are (1) X-ray ID; (2) net counts in the full (0.5–7 keV) band from the extracted spectra, where the errors were computed according to Gehrels (1986) and correspond to the 1σ level in Gaussian statistics; (3) HR calculated by N20, where we put 1.0 for sources where only hard counts were detected; (4) photometric redshifts obtained from the SED fitting procedure; (5) X-ray redshift solutions; (6) column density in units of 10²² cm⁻²; (7) observed 0.5–7 keV flux in units of 10⁻¹⁵ erg s⁻¹ cm⁻²; (8) intrinsic, absorption-corrected luminosity in the 2–10 keV rest-frame band in units of 10⁴³ erg s⁻²; and (9) X-ray feature on which the z_X is based, with "line" for the 6.4 keV Fe Kα line and "edge" for the 7.1 keV Fe absorption edge and associated photoelectric cutoff. Here "−1" indicates a nonderived quantity. Uncertainties are reported at the 1σ confidence level.

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In Figure 10, we show the comparison between the derived X-ray and photometric redshifts. The likelihood profiles of z_X and z_phot do not always provide a unique solution. In fact, especially for faint sources where the data quality is relatively poor, both distributions may have nonnegligible secondary solutions. If a primary X-ray solution matches (i.e., it is consistent within the errors) with a primary photometric solution (22/33 cases), we discard any secondary solutions. However, the remaining 11 cases (∼20% of the main sample) have secondary solutions, obtained with one method, that match with the primary and unique solutions derived with the other. In these cases, the primary solution from one method can constrain secondary solutions obtained with the other; hence, we selected the agreed redshift as the unique solution (open symbols). There are no cases with a clear match between secondary X-ray and photometric solutions. The blue dots refer to X-ray solutions driven only by absorption features, while the red squares indicate sources in which the 6.4 keV Fe Kα line has been detected. The majority (∼82%) of the solutions are driven by absorption features. As shown in the inset of the same figure, the uncertainties of the X-ray redshift solutions are larger for those sources in which it was not possible to identify any clear Fe Kα emission line. This is explained by the fact that the Fe Kα emission line is a very narrow feature compared to the Fe absorption edge. Thus, in the case of a detected Fe Kα line, the X-ray redshift probability sharply decreases before and after the best-fit value, resulting in a smaller uncertainty.

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Overall, there is a good correlation between z_X and z_phot despite a nonnegligible scatter. Considering a typical accuracy ¹⁵ of rms = 0.1 for photometric redshifts obtained with a similar number of filters (e.g., Capak et al. 2004; Zheng et al. 2004; S. Marchesi et al. 2021, in preparation) and rms = 0.1 for the X-ray simulations described in Section 3.2.2, we assumed a confidence region of $\pm 0.15(1+z)$, where 0.15 was computed by summing in quadrature the two rms terms. About 76% of the sources fall within the chosen confidence region (gray dotted lines). We defined as outliers those sources whose z_phot value does not lie within the z_X 1σ error bars, and the z_X value is outside the 0.15(1+z_phot) confidence region. The outlier fraction is then 9% (3/33 sources), and the total fraction of sources where we constrained a redshift solution with both methods is 56% (30/54). We achieved an accuracy of rms = 0.10, and, when considering only the primary solutions with both methods, the rms increases to ≈0.2. These rms values and the outlier fraction are comparable to those obtained in other X-ray redshift techniques. For instance, Simmonds et al. (2018) obtained an rms ≈ 0.2 and 8% outlier fraction when validating against reliable spectroscopic redshifts. This indicates that the assumptions adopted in our procedure can be considered appropriate.

We show one of the sample sources (XID 41) in Figure 11, for which there is a spectroscopic redshift measurement from Kriek et al. (2008), ${z}_{{spec}}=2.511$. This is the only spectroscopic redshift available so far for our sample. On the one side, we derived an X-ray redshift solution, ${z}_{X}={2.58}_{-0.49}^{+0.19}$, by fitting a prominent Fe 7.1 keV edge coupled with the photoelectric absorption cutoff. These features are produced by a heavily obscured yet Compton-thin AGN with ${N}_{H}\,={1.7}_{-0.4}^{+0.5}\times {10}^{23}\,{\mathrm{cm}}^{-2}$ (left panel). On the other side, the derived photometric redshift solution, ${z}_{{phot}}={2.28}_{-0.32}^{+0.23}$, is driven by a strong drop in the SED identified at $\sim 12500\,\mathring{\rm A}$ (right panel). This feature can be associated with a prominent 4000 Å break (e.g., Bruzual 1983; Kauffmann et al. 2003), which indicates that the host is a red and passive galaxy. Both of our solutions are consistent within the uncertainties with the spectroscopic redshift.

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Given that the good agreement between the two methods validates the obtained redshift solutions, it was possible to conduct a study of the physical and intrinsic X-ray properties of the selected obscured AGN sample. Photometric redshifts generally have smaller uncertainties than the X-ray ones, except for those cases where the Fe Kα line was detected. We therefore decided to use z_X when the Fe Kα line was identified and z_phot elsewhere. When none of the two above solutions was available, we used absorption-driven z_X, if estimated. The analysis was then feasible for 51 sources. The redshift distribution (Figure 12, top panel) spans from ∼0.1 to ∼4 with a median value of z = 1.3 and peaks between 0.5 and 1, in agreement with the obscured redshift distributions in other deep X-ray surveys (e.g., Fiore et al. 2008; Georgantopoulos et al. 2008).

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We performed the spectral analysis adopting an absorbed power-law model, as described in Section 3.1, but now fixing the redshift. We also carried out a few tests by independently fitting the source and background spectrum using a background model (see Marchesi et al. 2016a for details) and did not find significant differences in the derived best-fit source parameters.

The derived column density distribution (Figure 12, middle panel) ranges between 10²² and 10²⁴ cm⁻², with a mean value of ${N}_{H}=1.7\times {10}^{23}\,{\mathrm{cm}}^{-2}$ typical of Compton-thin AGN, plus one Compton-thick AGN candidate (${N}_{H}\sim 1.1\times {10}^{24}\,{\mathrm{cm}}^{-2}$) at $z\approx 4$. We show the obtained flux distributions in the full, soft, and hard bands in Figure 12 (bottom panel). The derived intrinsic, absorption-corrected, 2–10 keV rest-frame X-ray luminosity is in the range of $\sim {10}^{42}-{10}^{45}\,\mathrm{erg}\,{{\rm{s}}}^{-1}$, with a median value of ${L}_{2-10\mathrm{keV}}=8.3\times {10}^{43}\,\mathrm{erg}\,{{\rm{s}}}^{-1}$. We also report a very low (∼10⁴⁰ erg s⁻¹) luminosity object (XID 127) whose counterpart is extended in the optical images. It is identified as a very bright nearby galaxy (${z}_{{phot}}=0.05$, R = 18), and its X-ray emission may be produced by a very low luminosity AGN or stellar processes. Overall, we found that the level of obscuration increases as both redshift and luminosity increase (Figure 13). In particular, at ${L}_{2-10\mathrm{keV}}\lt {10}^{44}\,\mathrm{erg}\,{{\rm{s}}}^{-1}$ and $z\gt 1.5$, AGN with ${N}_{H}\gt {10}^{23}\,{\mathrm{cm}}^{-2}$ start dominating over AGN with column densities between 10²² and 10²³ cm⁻². However, we are biased toward the most obscured and luminous objects as the redshift increases. On the one hand, we lose ${N}_{H}\lt {10}^{23}\,{\mathrm{cm}}^{-2}$ sources at high redshift ($z\gtrsim 3$) because of the HR selection (see Figure 2). In addition, since the X-ray photoelectric absorption cutoff moves toward the lower boundary of the observational band (0.5 keV), the estimate of low N_H values becomes difficult (e.g., Tozzi et al. 2006; Marchesi et al. 2016a). On the other hand, we only see the brightest sources because the sample is flux-limited. We also point out that a fraction of obscured AGN at lower net count regimes (S. Marchesi et. al. 2021, in preparation) is probably missing. Our results are summarized in Table 2.

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In Gilli et al. (2019), we reported the discovery of a galaxy overdensity at $z\approx 1.7$. The structure has eight members confirmed by secure ONIR spectroscopic redshifts (VLT/MUSE and LBT/LUCI; Gilli et al. 2019), plus three members confirmed by Atacama Large Millimeter/submillimeter Array (ALMA) spectra (D'Amato et al. 2020). Ten of them are star-forming galaxies, and one, located at the center of the overdensity, is a Compton-thick (${N}_{H}={1.5}_{-0.5}^{+0.6}\times {10}^{24}\,{\mathrm{cm}}^{-2}$) Fanaroff–Riley type II (FRII) radio galaxy. This source is the only one detected in the X-rays (XID 189 in N20). Due to the limited area (∼1'–15) covered around the FRII core by ONIR spectroscopic and ALMA observations, we were able to estimate an overdensity projected size of at least ∼800 kpc. In this work, we found six sources (XID 37, 40, 48, 69, 131, and 137) whose redshift solutions are consistent with z = 1.7 and, therefore, may be new overdensity members. In particular, XID 37 and 131 have both photometric and X-ray redshift estimates, while for XID 69 and the remaining sources, we derived only the X-ray and photometric redshift solutions, respectively. Using an angular scale of 8.5 kpc arcsec⁻¹, valid for the adopted cosmology at z = 1.7, the projected distances between these sources and the FRII core are in the range 2–4 Mpc, suggesting that the structure may be more extended than previously estimated. The proposed technique can hence be used to identify possible AGN members of cosmological structures, such as galaxy overdensities and protoclusters, which may be extended up to several megaparsecs (e.g., Gilli et al. 2003; Overzier 2016).

In Figure 14, we compare the observed HRs as a function of the derived redshifts and column densities with the expected simulated trends. There is a clear distinction between red (${N}_{H}\lt {10}^{23}\,{\mathrm{cm}}^{-2}$) and light blue (${10}^{23}\lt {N}_{H}\lt {10}^{24}\,{\mathrm{cm}}^{-2}$) points. The only candidate at ${N}_{H}\gt {10}^{24}\,{\mathrm{cm}}^{-2}$ (dark blue point) also lies in the corresponding region. This confirms that an HR threshold, carefully calibrated on the data, can be a good proxy for the selection of obscured AGN, as discussed before. However, other than a dependence on the instrument effective area, the expected HR is also a function of the AGN spectral shape, as described in Section 2.3 for typical Γ values.

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In the following, we discuss how the HR trends may be influenced by commonly observed AGN spectral features: soft excess and reflection. The soft excess is possibly produced by scattered radiation into the line of sight (e.g., Ueda et al. 2007; Brightman & Nandra 2012) and is emitted over the primary AGN continuum at rest-frame energies below ∼1−2 keV. Especially for high obscuration levels (${N}_{H}\gt {10}^{23}\,{\mathrm{cm}}^{-2}$), where the main power-law radiation is strongly extinguished at soft rest-frame energies, the soft excess may give a nonnegligible contribution to the observed soft band, decreasing the HR value for sources up to z ∼ 2−3. However, its emission is observed to be $\lt 10 \%$ of the primary component (e.g., Lanzuisi et al. 2015), with typical values of 1%–3% (e.g., Gilli et al. 2007; Ricci et al. 2017), and its contribution to the observed HR is diluted because of the strong decrease of the Chandra collecting efficiency at $E\lesssim 1\,\mathrm{keV}$. The combination of these factors keeps the ${\rm{HR}}=-0.1$ threshold valid (see Appendix B.1). The reflected emission is produced by the reprocessing of the primary X-ray continuum by circumnuclear material, and it may also influence the HR for sources with high N_H . In general, the reflection component peaks at rest-frame energies of ∼30 keV (e.g., Ajello et al. 2008), with a small to moderate contribution in the observed soft band even at a redshift of ∼3−4. As a consequence, we do not expect to miss obscured AGN with the adopted HR selection criteria. A detailed modeling of these spectral components is shown in Appendix B.

In the following, we discuss the main differences and similarities between our procedure and other X-ray redshift techniques adopted in AGN surveys (e.g., Civano et al. 2005; Iwasawa et al. 2012; Simmonds et al. 2018; Iwasawa et al. 2020).

• (i)  
  We constrained X-ray redshifts down to a lower net count regime, ∼30 net counts for particular cases (see Section 3.2.2). This is similar to Simmonds et al. (2018) but markedly different from the hundreds of counts required in other Kα-based redshift derivations. Because the procedure has strong dependencies on the number of net counts, and considering that it is calibrated on the instrumental response, we expect this net count threshold to be a reliable lower limit for other Chandra deep observations.
• (ii)  
  The significance of each X-ray redshift solution is analyzed through extensive simulations based on both local (Section 3.2.1) and, different from any other method, global (Section 3.2.2) backgrounds and responses. For the latter, we directly extracted the background as a function of the off-axis angle to get a realistic background estimate. Then, we rescaled and associated it with the simulated source spectra by taking into account the PSF size and the instrumental response at the source position. Taking into account the off-axis angle effects is crucial to maximize and properly interpret the X-ray information, especially for pointed fields. A different solution for the background handling is to model it. For example, Simmonds et al. (2018) sampled a single representative background for each instrument using archival observations. We refer to that paper for additional information and to Buchner et al. (2014) for details about background modeling in X-ray data.
• (iii)  
  As discussed in Section 5.1, there is a nonnegligible fraction (∼20%) of sources in which a secondary redshift solution was constrained by a unique z_phot or z_X. This suggests that the combined use of photometric and X-ray solutions can solve redshift degeneracies that often arise in both methods and may provide better redshift estimates (see, e.g., Vignali et al. 2015; Simmonds et al. 2018; Iwasawa et al. 2020). For this reason, photometric redshifts are included in the procedure, and, for these particular cases, the final redshift solutions are driven by the combination of the two methods.

Considering the above points, the proposed procedure can be used in current X-ray deep fields, where there is still a fraction (∼5%) of X-ray sources without a solid redshift estimate (e.g., Salvato et al. 2009; Marchesi et al. 2016b; Masini et al. 2020). For example, taking sources with at least 50 net counts and ${\rm{HR}}\gt -0.1$ in the 7 Ms CDF-S (Luo et al. 2017) and AEGIS-XD (Nandra et al. 2015) surveys, the fraction of objects without any redshift estimate is 1%–2% (11 in both fields), and 3%–4% of sources (37 and 26, respectively) are associated only with photometric redshifts that have nonnegligible secondary solutions. Moreover, 11% of sources in the CDF-S are single-line spectroscopic redshifts (e.g., Szokoly et al. 2004), which are often not reliable (e.g., Simmonds et al. 2018; Barger et al. 2019), and the AEGIS-XD survey has 12% of sources marked with a poor quality flag. These cases are optimal for the proposed procedure, which may estimate reliable redshifts from the X-ray spectra alone and/or compare them with the available redshifts to better evaluate secondary photometric solutions and single-line spectroscopic redshift. These sources are potentially high-redshift obscured AGN, which are very challenging to reveal with the current methods (e.g., Cowie et al. 2020) and may contribute to the high-redshift obscured AGN fraction, which is still uncertain and subject to debate (Georgakakis et al. 2017; Vito et al. 2018).

We applied the X-ray redshift procedure to the J1030 field, which lacks the massive spectroscopic coverage of the other X-ray deep fields. Simmonds et al. (2018) showed through simulations that X-ray redshifts may be derived from observations of existing missions (Chandra, XMM-Newton, NuSTAR, Swift/XRT). Similarly, the proposed method can be applied to future deep Chandra and XMM-Newton observations, the forthcoming eRASS surveys (e.g., Merloni et al. 2012), the future Athena observatory (e.g., Aird et al. 2013) and, hopefully, missions under study such as Lynx (e.g., Gaskin et al. 2019) or AXIS (e.g., Mushotzky et al. 2019), which will observe faint, heavily obscured objects with no or extremely faint ONIR counterparts. While eROSITA's soft response and short exposures limit the applicability of X-ray redshift methods (Simmonds et al. 2018), large-area, high-resolution missions such as Athena will likely benefit tremendously from X-ray redshifts. In deep Athena, Lynx, and AXIS surveys, obtaining adequately deep photometric data and/or ONIR spectroscopy will be very costly. Thus, by applying a method similar to the one described in this work and tuning the HR and number of counts thresholds to take into account the effective area of future missions, it will be possible to provide reliable X-ray redshift solutions for obscured AGN.

To validate our ${\rm{HR}}\gt -0.1$ selection criterion for obscured AGN, we tested how adopting different spectral shapes affects the HR-redshift curves for different N_H values. We recall that the model used in this paper is a simple absorbed power law (hereafter M0; Figure 2). The simulations were performed as discussed in Section 2.3. In each model, the mean Galactic absorption at the J1030 field position (${N}_{H}=2.6\times {10}^{20}\,{\mathrm{cm}}^{-2}$) was considered.

We modeled the soft excess emission, adding a secondary redshifted power law (zpowerlw) to M0. The redshift parameter was linked between the components, and no intrinsic absorption was applied to the secondary power law, assuming it is scattered emission into the line of sight. The photon index Γ of the two power laws was fixed to 1.9. The soft excess contribution to the main continuum is observed to be <10%, with typical values of 1%–3% (e.g., Gilli et al. 2007; Ricci et al. 2017), and it was considered by adding a multiplicative 3% constant to the secondary power law. The results for this model (hereafter M1) are shown in Figure B1 (top panel). Compared to the M0 curves, there is a strong HR decrease for logN_H  = 24 at $z\lt 2$ (up to ΔHR ∼ 0.5–1 at $z\lt 0.5$), a small decrease for logN_H  = 23 at $z\lt 0.5$ (ΔHR ∼ 0.1–0.2), and no significant effects for logN_H  = 22–21. This is due to the fact that, as the N_H increases, the soft rest-frame main emission is more and more depressed by the absorption, and therefore the soft excess becomes dominant at these energies. This effect is diluted at high redshift, when the primary hard rest-frame emission is redshifted enough to cover the observed-frame soft band.

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The reflection is modeled by adding a pexrav component (Magdziarz & Zdziarski 1995) to M1. The R value was fixed to –1 (e.g., Marchesi et al. 2016a) to simulate pure reflection, while the photon index and normalization were linked to the main power-law component. The inclination angle and high-energy cutoff were set to the default values $\theta =60^\circ$ and E = 100 keV, respectively. The results for this model (hereafter M2) are shown in Figure B1 (middle panel). There are no major changes in the HR values as a function of redshift compared to the double power-law model. For logN_H  = 21, 22, and 23, the HR slightly increases at $z\gtrsim 1$ (ΔHR < 0.1), while for logN_H  = 24, there is an increase at $z\lt 0.5$ and a decrease at higher redshift (ΔHR ≲ 0.1). The pexrav model adds flux in both the soft and hard rest-frame bands. Therefore, as a function of redshift and N_H , there is a combination of reflection and soft excess that subtly changes the HR values.

For comparison with M2, in Figure B1 (bottom panel), we show the results obtained with the MYTorus model (Murphy & Yaqoob 2009). It adopts an azimuthally symmetric toroidal shape for the obscuring material, with a fixed half-opening angle of $60^\circ$. The torus material is assumed to be neutral, cold, and uniform. The reflection is included within this model but with a more physically motivated treatment, while the soft excess is modeled with a secondary redshifted power law. We refer to this model as M3. We fixed ${\rm{\Gamma }}=1.9$ for all of the components and the inclination angle between the observer and the torus axis to $\theta =75^\circ$. In MYTorus, logN_H varies in the range 22–25, so the logN_H  = 21 curve is not reported. The curve for logN_H  = 22 is very similar to those of M0 and M1. The logN_H  = 23 trend is similar to that of M1, and for logN_H  = 24, the HR values are lower than those of all other methods (e.g., ΔHR$\,\lesssim \,0.1$ compared to M2), with a larger spread for the chosen Γ values. Interestingly, the HR increase at $z\gtrsim 1$ for logN_H  = 22 and 23 is not the same as what we observe for M2, due to the different reflection treatment; see also Marchesi et al. (2020) for a comparison between the different shapes of pexmon (Nandra et al. 2007), which includes the pexrav model, and BORUS (Baloković et al. 2018), a physically self consistent torus model.

In summary, as shown in Figure B1, the ${\rm{HR}}=-0.1$ threshold (black solid line in each panel) remains valid. In fact, with an ${\rm{HR}}\gt -0.1$, we avoid unobscured sources at any redshift.

We intend to demonstrate here that a simple absorbed power law (M0) is a reasonable model to obtain reliable X-ray redshift estimates. To prove it, we performed several runs of simulations similar to those described in Section 3.2.2 using the M3 model described in Appendix B.1. The explored parameter space is logN_H  = 22, 23, and 24 and z = 0.5, 1, 2, 3, and 4. We varied the power-law normalization to obtain sources with net counts in the ranges 10–100 (low regime) and 100–1000 (high regime). For each parameter combination, 100 spectra were simulated. The simulated spectra were then fitted with M0 and the three models presented in Appendix B.1: M1 accounts for the soft excess, and M2 and M3 also include the reflection. Compared to M0, treated as in Section 3.2.2, the additional free parameters in M1–M3 were the secondary power-law normalization, constrained to be $\lt 10 \%$ of the primary power law, and the scattering normalization in M3.

In Figure B2, we show the difference in the match percentage (Equation (4)) between the results obtained with M0 and the three complex models. Positive values indicate that M0 is more effective in recovering the simulated source redshift, while negative values indicate that complex models are more efficient. No clear trends are found for logN_H  = 22. For logN_H  = 23 and 24, there are clear trends as a function of redshift. When compared to M1–M3, M0 is penalized below z ∼ 1–2, while at higher redshift, it is more effective. This is because the contribution of the secondary power law, present in all of the complex models, influences the fit for sources at low redshift, especially where the primary continuum is strongly extinguished (log${N}_{H}\gt 22$) in the soft band. At higher redshift, instead, this contribution is diluted by the redshifted main power law. The reason for these discrepancies is attributed to the increased number of parameters in M1–M3. In particular, the secondary power-law normalization plays an important role, even if constrained to observed values. In fact, the spectral fit may interpret a noise fluctuation as a real spectral shape, fitting a wrong normalization and then a wrong redshift. The more complex the model spectral shape, the higher the risk of misinterpretation. These differences are more evident for higher N_H , as discussed in Appendix B.1, and for the low count regime because of the poor spectral quality.

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In general, it is clear that for logN_H  = 22, there are no significant differences between the models ($\lt 2 \%$), as well as for logN_H  = 23 in the high count regime ($\lt 5 \%$). For logN_H  = 24 and logN_H  = 23 in the low count regime, the differences may instead be nonnegligible. For logN_H  = 23 in the low count regime, the differences are <10%, while for logN_H  = 24, they are <10% and <15% in the high and low count regimes, respectively. Overall, M0 gives better results above z ∼ 1–2 for all models and is preferable against M2 for logN_H  = 24 at any redshift. Below z ∼ 1–2, M0 is penalized for the remaining cases, with differences within ∼10%, except for M3 in the low count regime, logN_H  = 24, and $z\lt 1$, where there is an ∼15% difference. However, for M3, there may be a bias introduced by the simulated model, which is the same as the fitted one, that may produce a larger difference in the match percentage against M0.

Given these results, we can safely say that using an absorbed power-law model is a reasonable assumption for the redshift estimate of obscured AGN. The main spectral features, such as the Fe 6.4 keV emission line, the Fe 7.1 keV absorption edge, and the photoelectric absorption cutoff, are included in the absorbed power-law model and are those that drive the redshift solutions. Therefore, for a limited photon statistics, introducing more complex models not only does not substantially improve the results, but it may also introduce degeneracies due to the possible misidentification of complex features.