Constraints on the X-ray Luminosity Function of AGN at = 5.7 6.4 with the Extragalactic Serendipitous Swift Survey

MNRAS 000, 1–10 (2022) Preprint 28 January 2022 Compiled using MNRAS LATEX style file v3.0

 Constraints on the X-ray Luminosity Function of AGN at = 5.7 − 6.4 with
 the Extragalactic Serendipitous Swift Survey
 C. L. Barlow-Hall,1★ J. Delaney,1 J. Aird,1,2 P. A. Evans,2 J. P. Osborne,2 and M. G. Watson2
 1 Institute for Astronomy, The University of Edinburgh, Royal Observatory, Blackford Hill, Edinburgh EH9 3HJ, UK
 2 School of Physics & Astronomy, University of Leicester, University Road, Leicester LE1 7RJ, UK
arXiv:2201.11139v1 [astro-ph.HE] 26 Jan 2022

 Submitted to MNRAS 2022-Jan-25

 ABSTRACT

 X-ray luminosity functions (XLFs) of Active Galactic Nuclei (AGN) trace the growth and evolution of supermassive black
 hole populations across cosmic time, however, current XLF models are poorly constrained at redshifts of > 6. In this work we
 constrain the bright-end of the XLF at = 5.7 − 6.4 using high-redshift AGN identified within the Extragalactic Serendipitous
 Swift Survey (ExSeSS) catalogue. Within ExSeSS we find one serendipitously detected X-ray selected > 6 AGN, ATLAS
 J025.6821-33.4627, with an X-ray luminosity of X = 8.47+3.40 −1
 −3.13 × 10 erg.s and = 6.31 ± 0.03 making it the highest redshift,
 44

 spectroscopically confirmed, serendipitously detected X-ray selected quasar known to date. We also calculate an upper limit on
 the space density at higher luminosities where no additional sources are found, enabling us to place constraints on the shape of
 the XLF. Our results are consistent with the rapid decline in the space densities of high-luminosity AGN toward high redshift
 as predicted by extrapolations of existing parametric models of the XLF. We also find that our X-ray based measurements are
 consistent with estimates of the bolometric quasar luminosity function based on UV measurements at & 6, although they
 require a large X-ray to bolometric correction factor (i.e. AGN that are relatively X-ray weak) at these high luminosities.
 Key words: galaxies: active – galaxies: evolution – galaxies: luminosity function, mass function – X-rays: galaxies.

 1 INTRODUCTION are Population III stellar remnants (e.g. Madau & Rees 2001) and
 Direct Collapse Black Holes (e.g. Volonteri & Begelman 2010),
 Most galaxies are thought to play host to a Supermassive Black Hole
 producing Black Hole seeds of masses 10 − 100M or 104 − 106 M
 (SMBHs), with SMBHs and galaxies thought to co-evolve (Kor-
 respectively. Thus, even for the case of direct collapse, a significant
 mendy & Ho 2013). When rapidly growing these SMBHs produce
 amount of growth must have occurred within the first few 100 Myrs
 strong emissions across a range of wavelengths, from radio to high-
 of cosmic time in order for these seed black holes to have attained the
 energy X-rays, powered by their accretion activity (e.g. see reviews of
 masses we observe. However, this growth remains poorly constrained,
 Reines & Comastri 2016; Padovani et al. 2017; Hickox & Alexander
 due to the lack of robust observational constraints on AGN within
 2018). These luminous SMBHs are known as Active Galactic Nu-
 the early Universe.
 clei (AGN) and form the basis for investigations of SMBHs beyond
 our local Universe, with AGN observed throughout the Universe and
 even out at redshifts of > 6. The growth of AGN populations across cosmic time and the evo-
 Large-scale optical and near-infrared imaging surveys have en- lution of AGN accretion rate is traced by the Quasar Luminosity
 abled the identification of luminous AGN out to = 7.54 (Bañados Function (QLF). The QLF describes the comoving space density of
 et al. 2018) and = 7.642 (Wang et al. 2021). Spectroscopic follow- AGN as a function of redshift and luminosity (e.g. Boyle et al. 2000;
 up observations not only confirm the redshifts of these sources but Page et al. 1996; Kalfountzou et al. 2014) and is measured using sur-
 also reveal them to have remarkably similar rest-frame UV spectra veys of AGN selected using optical, IR and X-ray data (Hopkins et al.
 to their lower redshift counterparts (e.g. Mortlock et al. 2011; De 2007; Ross et al. 2013). Many AGN have been discovered through
 Rosa et al. 2014; Shen et al. 2019). Applying single epoch scaling rest-frame optical/UV selection (e.g. Bañados et al. 2016), which
 relations indicates that they have masses of ∼ 106−8 M (see Table 1 is probed by optical and IR surveys that cover large areas of sky
 in Zubovas & King 2021), which are comparable to SMBH masses in identifying AGN out to very high redshifts (Matsuoka et al. 2019;
 the nearby Universe. This raises the question as to how these SMBHs McGreer et al. 2013). X-ray follow-up of high-redshift AGN samples
 formed and grew to these masses within the short time period of the identified through optical or IR surveys enable further investigations
 early Universe. the nature of these sources (Brandt et al. 2002; Vignali et al. 2001;
 The main seeding mechanisms theorised for the formation of AGN Vito et al. 2019). However, rest-frame optical/UV selection is biased
 towards the most luminous AGN sources, as these are more easily
 detectable by optical/UV telescopes above the emissions of the host
 ★ E-mail: c.barlow-hall@roe.ac.uk galaxy. Processes within the host galaxies can also contaminate the

 © 2022 The Authors

2 C. L. Barlow-Hall et al.
AGN light at optical/UV and IR wavelengths, unlike X-ray selected and limits to model XLFs from previous studies in order to obtain
AGN samples. constraints on the XLF at high-redshift.
 X-ray selection is often used to identify samples of AGN without The source catalogue used in this study, ExSeSS, is introduced in
the strong bias toward the most luminous sources that affects opti- §2, while our process to identify high-redshift sources is outlined in
cal/UV selection. This lack of bias arises as few processes within §3. In §4.1 we compare predicted source yields based on extrapola-
galaxies produce significant X-ray emission and thus AGN easily tions of current XLF models to our observed sample, while in S4.2
outshine their host galaxies at X-ray wavelengths (see e.g. Padovani we use our data to place constraints on the XLF. We then compare the
et al. 2017). Furthermore, X-ray emission is much less susceptible constraints from our X-ray observations provide to existing estimates
to obscuration effects than the optical/UV light. Thus, AGN can of the bolometric Quasar Luminosity Function from rest-frame op-
be efficiently identified using X-ray surveys, with the accretion rate tical/UV samples (§5). We summarise our findings and present our
and thus growth of the central SMBH being reliably traced by the conclusions in §6. Throughout this paper we assume flat ΛCDM
X-ray emission. Thanks to their well-defined sensitivity and uncon- cosmology, with 0 = 70.0, Ω = 0.7 and Ω = 0.3, and all errors
taminated selection of AGN, X-ray surveys are especially useful for given are the 1 uncertainties on the values.
determining the QLF. The X-ray QLF, known as the X-ray Lumi-
nosity Function (XLF) can then be used to place constraints on the
activity of AGN across cosmic time and thus the rate of growth of
the early population of SMBHs.
 2 THE EXSESS CATALOGUE
 Prior studies of the XLF have shown that AGN populations evolve
substantially over cosmic time, increasing in both space density and The studies performed in this paper make use of the sample of X-ray
their typical luminosities between ∼ 0 and ∼ 2, where the overall sources from the Extragalactic Serendipitous Swift Survey (ExSeSS),
accretion rate density peaks (e.g. Ueda et al. 2014; Aird et al. 2015). as defined by Delaney et al. (prep). Here, we give a brief overview of
Toward higher redshifts ( & 3), the normalisation of the XLF is the sample construction and the vital step to define the area coverage
found to drop rapidly across all luminosities (e.g. Brusa et al. 2009; and sensitivity of the survey that enables our measurements of the
Vito et al. 2014; Georgakakis et al. 2015), placing constraints on the X-ray Luminosity Function (XLF).
extent of SMBH growth in the early Universe. However, the samples The X-ray Telescope on the Neil Gehrels Swift Observatory (here-
of X-ray selected AGN at & 5 remain extremely small: 2 with after the Swift-XRT Burrows et al. 2005) has performed both tar-
photometric redshifts in the ≈ 7Ms Chandra Deep Field South (Luo geted observations of X-ray sources and searches for unknown X-
et al. 2017), 2 with spectroscopic redshift (the highest at = 5.3) and ray counterparts to transient sources, often following the detection
7 with photometric redshifts (4 of which are at > 6 with the highest of Gamma-ray bursts by Swift’s own Burst Alert Telescope (Swift-
at = 6.85) in the Chandra COSMOS-Legacy survey (Marchesi et al. BAT), imaging nearly 4000 degrees2 of sky throughout its observing
2016). These small numbers are due to both the strong decline in the life (as of August 2018; Evans et al. 2020). The ExSeSS sample
XLF of AGN at high redshift, which can also be seen in the space was formed using the second Swift-XRT Point Source (2SXPS) cat-
density measured from optical/UV and IR surveys, and the depths alogue of Evans et al. (2020) that identified all point sources in the
required in order to detect even intrinsically luminous AGN at these full data set provided by Swift XRT observations between 2005 Jan-
extreme redshifts. Due to these small X-ray samples the parametric uary 01 and 2018 August 01. The effect of contamination by Galactic
models of the XLF are poorly constrained at high redshift by current sources and nearby, individually resolved sources in nearby galax-
observations. Yet even with samples of just a few AGN at these very ies was reduced by removing the areas of sky corresponding to the
high redshifts, we can begin to place important constraints on the Galactic plane (Galactic latitudes | | < 20◦ ), the Large and Small
XLF. Magellanic Clouds, M31 and M33, ensuring the sample is dominated
 With the launch of eROSITA—providing a new generation of by extragalactic sources. In addition to this only fields identified as
sensitive, wide-area X-ray surveys (Predehl et al. 2021)—there is ultra-clean (field flag=0) in 2SXPS are included in ExSeSS and only
the potential to discover many more of the rare, high-luminosity sources with a good detection flag are included in the sample.
X-ray selected AGN at & 6 and improve our constraints on the All target objects were removed along with any associated X-
XLF. Indeed, Khorunzhev et al. (2021) and Medvedev et al. (2020) ray detections using regions of the radius of the source (where an
have reported the discovery of highly luminous X-ray emission from extended counterpart could be identified, e.g. a host galaxy) and
quasars at ≈ 5.5 and = 6.18, respectively, in the early all-sky adopting a minimum radius of 2 arcminute, in order to prevent bias
eROSITA scans. Wolf et al. (2021) placed constraints on the XLF and create a truly serendipitous survey. Medium (1-2 keV), hard (2-
at ∼ 6 using a single X-ray detected quasar at = 5.81, found in 10 keV) and total (0.3-10 keV) energy band detections by Swift-XRT
the eROSITA Final Equatorial Depth Survey (eFEDS) that provides were taken, and wherever there are multiple observations of sky only
performance verification data in a ∼140 deg2 field at the depth of the the stacked images are used in order to maximise the exposure time.
final 4-year eROSITA all-sky surveys (Brunner et al. 2021). This process to create the ExSeSS sample is detailed in Delaney et al.
 In this paper, we present the results from a search for > 6 (prep).
X-ray AGN within the Extragalactic Serendipitous Swift Survey A key feature of ExSeSS is that the area of sky surveyed by Swift
(ExSeSS) catalogue (Delaney et al. prep). ExSeSS covers a to- to different exposure times is carefully tracked and well defined,
tal sky area of ∼2000 deg2 and reaches ultimate flux limits of enabling an accurate quantification of the “area curve”, giving the
 0.3−10keV ∼ 10−15 erg s−1 cm−2 for ∼ 0.1% of the area, which area of sky covered to different X-ray flux limits. Delaney et al.
are considerably deeper than the current eROSITA all-sky coverage. (prep) use simulations (from Evans et al. 2020) to determine the
We identify one X-ray source within ExSeSS that is associated with a area curves in the soft, hard and total energy bands. It is these area
previously known > 6 quasar with a spectroscopic redshift. Thanks curves that enable us to perform measurements of the observed X-ray
to the well-defined area coverage and sensitivity of ExSeSS, we are Luminosity Function based on the ExSeSS sample.
able to place direct observational constraints on the space density of The resulting ExSeSS catalogue is comprised of 79 152 X-ray
luminous X-ray AGN at > 6. We then compare the estimated XLF sources and covers 2086.6 degrees2 of sky.

MNRAS 000, 1–10 (2022)

Constraining the ∼ 6 XLF with ExSeSS 3

Figure 1. The high redshift AGN ATLAS J025.6821-33.4627, identified in ExSeSS. The total stacked X-ray image from Swift (left), smoothed by a Gaussian
of ∼ 2 pixels corresponding to half-energy width of the PSF of Swift (9 arcsec), and the z-band Atlas image (right) are shown. The optical position of the
source shown on the Swift image by the gold star. The radius of the solid red circles correspond to the 4.7 arcsec positional uncertainty of the source in Swift,
centred on the observed soft-band position. From these images the X-ray source can be seen, with no additional z-band counterparts lying within the positional
uncertainty, indicating that there is unlikely to be any contamination to the X-ray observation by other sources. The sky coordinates of the observed location in
optical and X-ray observations are shown on the corresponding image.

3 IDENTIFYING X-RAY LUMINOUS HIGH REDSHIFT The total (0.3–10 keV) band X-ray flux of this source was estimated
 SOURCES from the total measured counts, observed by the Swift-XRT, using
 a conversion factor of 3.256 × 10−11 2 as in Delaney et al. (prep),
In order to apply constraints to the high redshift X-ray Luminosity
 with this source identified as a good detection in the total band. The
Function (XLF) with ExSeSS, we searched for all > 5.7 sources in
 rest-frame 2–10 keV X-ray luminosity was then determined from the
ExSeSS.
 observed 0.3–10 keV flux, assuming a power-law of photon index
 While a full statistical cross-matching between ExSeSS sources
 Γ = 1.9. Given the high redshift of the source, the observed 0.3–
and multiwavelength surveys, to identify robust counterparts, will be
 10 keV band probes high rest-frame energies and thus we do not apply
presented in a future work, here we adopt a simple cross-matching
 an additional correction for intrinsic absorption. The 0.3–10 keV band
to existing redshifts in the SIMBAD Database (Wenger et al. 2000).
 observed flux and rest-frame 2–10 keV band Luminosity, with the
We use a maximum separation on the sky of 9 arcsec, given by the
 sky coordinates of this high redshift X-ray source and spectroscopic
half-energy width of the point spread function of the Swift XRT with
 redshift of the counterpart, are given in Table 1.
> 90% of all ExSeSS sources having a positional uncertainty of less
 ATLAS J025.6821-33.4627 was originally identified as a candi-
than this value. 20 909 potential counterparts to the ExSeSS sources
 date high-redshift quasar by Carnall et al. (2015), based on it’s com-
with pre-existing redshifts in SIMBAD were identified.
 bined WISE and VST ATLAS colours, indicative of a = 5.7 − 6.5
 We note that the majority of ExSeSS sources do not have pre-
 source. Follow-up spectroscopy was obtained, by Carnall et al.
existing counterparts or redshift measurements. Nevertheless, fol-
 (2015), using the Low Dispersion Survey Spectrograph 3 on the
lowing this initial cross-matching we identified one high redshift
 Magellan-II telescope from which a redshift of = 6.31 ± 0.03 was
( > 6) counterpart to the ExSeSS X-ray sources, that of 2SXPS
 calculated based on the broad Lyman- line in the source’s spectrum.
J014243.7-332742, corresponding to the previously detected quasar
 The X-ray properties of ATLAS J025.6821-33.4627 from 2SXPS are
ATLAS J025.6821-33.4627. This source was then visually checked
 included in the compilation of known high-redshift quasars by Vito
to ensure there are no other potential counterparts to the X-ray source.
 et al. (2019) but we now identify this source as a truly serendipi-
At near-infrared wavelengths ATLAS J025.6821-33.4627 appears as
 tous detection with ExSeSS. Comparing to known X-ray detected
a point source, as can be seen in Figure 1, with a separation of
 high redshift AGN (see Khorunzhev et al. 2021, and references
2.3 arcsec between the ExSeSS source and the counterpart and no
 there in), as shown in figure 2, ATLAS J025.6821-33.4627 can be
other sources within the 4.7 arcsec uncertainty in the X-ray position.
In addition to this the estimated probability of only 0.03 1 that this is
a spurious alignment indicates it is most likely the same source.
 2 We adopt a fixed flux conversion that assumes a Galactic absorption with a
 column density of H = 2.50×1020 erg.s−1 .cm−2 (HI4PI Collaboration et al.
1 The spurious detection probability was calculated using the total area cov- 2016) and a photon index of Γ = 1.9 (e.g. Kalfountzou et al. 2014) calculated
ered by ExSeSS, the number of sources in the Ross & Cross (2020) catalogue using WebPIMMS, which matches the assumptions used to calculate the
and the approximate area covered by that catalogue. ExSeSS area curves.

 MNRAS 000, 1–10 (2022)

4 C. L. Barlow-Hall et al.

Table 1. The medium-band (1–2 keV) Swift position, observed Swift X-ray flux in the soft-band (0.5–2 keV), the rest-frame 2–10 keV luminosity given by this
soft-band flux and the z-band AB magnitude of the high redshift ExSeSS source.

 Object RA Dec 0.3−10keV X2−10keV z 
 erg.s−1 .cm−2 erg.s−1

 ATLAS J025.6821-33.4627 25.68153+0.0012
 −0.0010
 −33.46312+0.0015
 −0.0010
 6.31 ± 0.03 +1.84 × 10−15
 4.59−1.69 8.47+3.40
 −3.12
 × 1044 19.63 ± 0.06

 with X-ray luminosities high enough to be detectable by Swift-XRT
 at & 6.

 4 OBSERVATIONAL CONSTRAINTS ON THE XLF AT
 >6
 Using the observed high-redshift X-ray source identified in ExSeSS
 we calculate limits on the AGN space density at the redshift of
 the sources, allowing us to provide constraints on the bright-end of
 the high-redshift XLF. In §4.1 we compare the number of sources
 observed with the number predicted by X-ray Luminosity Function
 models. We then present the constraints on the X-ray Luminosity
 function that are obtained from the ExSeSS source in §4.2.
 As there may be additional unidenitfied high redshift sources in
 ExSeSS, for which we do not have redshift data, the constraints
 obtained here are formally lower limits only. However, if we assume
 that any high-luminosity X-ray AGN would also be bright at rest-
Figure 2. The X-ray luminosity of our > 6 ExSeSS source with respect frame UV wavelengths (i.e. assuming they are unobscured) then
to redshift (black star). The distribution of known high redshift AGN with we would expect such sources would have been identified in the
spectroscopic redshifts, observed in X-rays (see Khorunzhev et al. 2021, and numerous searches for high- quasars and as such would have entered
references there in) is shown by the light blue points with the few serendip- our sample, if they exist. Under this assumption, we are able to place
itously detected high redshift sources from the eROSITA and COSMOS upper limits on the XLF at high luminosities based on the ExSeSS
surveys shown for comparison (red diamonds). Our source can be seen to
 survey.
be the highest redshift source identified serendipitously by X-ray surveys
with a well-defined area coverage, from which an XLF can be determined.
The sensitivity limits of eROSITA, for the equatorial region after 6 months 4.1 Predicted numbers of > 6 AGN
(eRASS:1), after 4 years (eRASS:8) and the Polar region after 4 years of the
survey are shown by the dashed line, the dot-dashed line and the dotted line The number of AGN X-ray sources, at different luminosities, that are
respectively, with our ExSeSS > 6 quasar lying along the limit of greatest expected to be observed in an X-ray survey, can be predicted using
sensitivity possible with eROSITA (Predehl et al. 2021; Sunyaev et al. 2021) XLF models and the area curve of the survey. As the area curve
and at the highest redshift of all serendipitously detected AGN. of the ExSeSS survey has been calculated we can perform such
 predictions of the expected number of AGN, at different redshifts
 and Luminosities, in ExSeSS.
 Using the parametrised XLF models of Pure Luminosity Evolu-
seen to lie just above the limit of the deepest sensitivity expected tion (PLE), Pure Density Evolution (PDE), Luminosity And Density
with eROSITA (that obtained at the Polar regions after 4 years of Evolution (LADE) and Luminosity Dependent Density Evolution
the survey; eRASS:8). Whilst there are many X-ray detected AGN (LDDE) from Georgakakis et al. (2015) we calculate the expected
few are serendipitously detected at redshifts of > 5 (as can be number of sources in ExSeSS. The XLF models are extrapolated
seen in figure 2). With a rest-frame 2–10 keV X-ray luminosity out to high redshifts, then the predicted number obtained using the
of X = 8.47+3.40
 −3.13
 × 1044 erg.s−1 , in ExSeSS, ATLAS J025.6821- integral
33.4627 is likely the highest redshift, spectroscopically confirmed,
serendipitously detected, X-ray selected quasar known to date.
 ∫ 2 ∫ log 
 During the cross-matching of sources within ExSeSS a second 2 co
 = ( , ) ( ( , )) log 
bright X-ray source was matched to the previously identified Quasar 1 log 1 
5C 2.183, with an spectroscopic redshift from the SDSS database of
 (1)
 = 6.16892 ± 0.00060 (Pâris et al. 2017). However, as this source
was detected in u, g, r, i, and z-bands of SDSS as well as the G-band where the XLF, ( , ), is the parametrised model (PLE, PDE,
of Gaia, not possible in a & 5 source, closer inspection of the SDSS LADE or LDDE from Georgakakis et al. 2015), co
 is the differ-
spectrum of the source was performed. From this spectrum it can be ential comoving volume and ( ( , )) is the sky area covered
seen that the redshift of the source is in fact = 0.714 as identified by ExSeSS to an observed flux, ( , ), corresponding to a given
by other studies (Machalski 1998; Pâris et al. 2018; Chen et al. 2018) 2–10 keV rest-frame luminosity, , and redshift, . The 1 uncer-
and thus is not an additional high redshift AGN within ExSeSS. tianties in these predicted numbers are obtained through Monte Carlo
 With this contaminant removed, and no other high-redshift sources simulations using the model parameter uncertainties of Georgakakis
identified, we can give a tentative number of high redshift sources et al. (2015).

MNRAS 000, 1–10 (2022)

Constraining the ∼ 6 XLF with ExSeSS 5

 Table 2. The predicted number of sources, at = 5.7 − 6.4, based on the four
 XLF models from Georgakakis et al. (2015) in the two luminosity bins where
 we place constraints on the number of high-redshift sources using ExSeSS.
 The 1 uncertainties on the model predictions are obtained through Monte
 Carlo error propagation (for the LADE model we give the 1 upper limit only
 given the large range). The / binned XLF estimates obtained
 from the observed ExSeSS sources, ( X , ), and the observed number of
 AGN in each luminosity bin are also given, with 1 limits based on the
 Poisson errors from Gehrels (1986) (see §4.2 for details).

 Model log X = 44.8 − 45.8 log X = 45.8 − 46.8
 erg.s−1 erg.s−1

 PLE 0.62+1.60
 −0.48
 +1.07
 0.30−0.25
 PDE 1.05+1.48
 −0.67
 +0.66
 0.31−0.22
 LADE < 0.051 < 0.015
 LDDE +10.8
 2.76−2.25 +16.8
 1.80−1.63

 ( X , ) 2.98+6.86
 −2.47
 × 10−9 < 2.22 × 10−10
 1/ max 2.98+6.85
 −2.46
 × 10−8 No Data

 +2.300
 1−0.827 < 1.814

Figure 3. The observed = 6−7 space density of AGN in the luminosity bins
of log X = 44.8 − 45.8 given by the high redshift ExSeSS sources (black
 only one high-redshift AGN found in ExSeSS we can place important
star). The space densities given by the Georgakakis et al. (2015) models are
 constraints on the space density of the high-redshift AGN population.
plotted, as a function of redshift, with the shaded regions showing the 1 
uncertainty regions. The model space densities can be seen to fall slightly Following the / method of Miyaji et al. (2001), we
above the observed space density, whilst remaining consistent within the 1 convert the observed number of sources in the luminosity bins of
uncertainties given by Gehrels (1986). log X = 44.8 − 45.8 and log X = 45.8 − 46.8 into a measurement
 of the AGN space density. The measured space density is calculated
 by scaling the predicted space density based on a given model of the
 XLF by the ratio of the observed number of sources, , to the
 We define the luminosity bins of our observed binned XLF values
 predicted number of sources, , given by,
to be 1 dex wide, with the minimum luminosity taken to be that
at which the area observable is 0.1% of the total observed area
of ExSeSS, in order to reduce the uncertainty inherent at the faint 
 
 
 Φ X , = Φ X , (2)
end of the area curve. This results in the lowest luminosity bin being 
log X = 44.8−45.8, in which the source, 2SXPS J014243.7-332742 where the predicted number, is estimated using the LDDE
(ATLAS J025.6821-33.4627), identified in ExSeSS falls. However, model (from Georgakakis et al. 2015), for a redshift range of =
we find no sources in the 1 dex brighter, log X = 45.8 − 46.8, bin 5.7 − 6.4 and in the luminosity
 bins given in table 2, and the model
or at higher luminosities, which we would expect to observe more space density, Φ X , , is given by the integral of the LDDE
easily. Thus we take an upper limit on the number of sources, given model of the XLF over the luminosity bin.
by the upper 1 Poisson uncertainty for a sample size of = 0 from The measurements of the space density of AGN based on the
Gehrels (1986), in this higher Luminosity bin. ExSeSS sources are shown in figure 3, compared to predicted space
 The XLF predictions for the number of sources observable, shown densities based on the four Georgakakis et al. (2015) XLF models for
in table 2, as given by the Georgakakis et al. (2015) models, are each of our luminosity bins. The 1 uncertainties on these binned
consistent with the observed number of sources, indicating the XLF XLF estimates are based on the Poisson uncertainties in the observed
models form reasonable predictions for the number of AGN, at these source numbers, as given by Gehrels (1986).
luminosities, in ExSeSS. However, the average of the Luminosity These observed space density estimate are generally in good agree-
And Density Evolution (LADE) model under predicts the number of ment with the extrapolated XLF models. The PLE and PDE models
observed sources, albeit with a very large 1 uncertainty. Nonethe- and the lower limit on the LDDE model agree within the 1 Poisson
less, the 1 upper limit based on the extrapolated uncertainty in the uncertainty on the observed space density, whilst the upper bound
LADE model remains below the 1 limit from our observed number; of the LADE model only agrees within the 3 Poisson uncertainty.
while not formally ruling out the LADE model, it is thus disfavoured Thus, our observed space density can be seen to show the expected
based on our measurements. decline in space density of AGN towards higher redshift, as seen in
 in the PLE, PDE and LDDE models, but this decline is not as rapid as
 some extrapolated models, such as the LADE model, would predict.
4.2 AGN space densities and estimates of the XLF Following a similar process, using the Miyaji et al. (2001)
 / , we then calculated the binned XLF estimates from
Extrapolating parametrised models of the AGN XLF, such as those the number of high redshift ExSeSS observations, given by
of Georgakakis et al. (2015); Aird et al. (2015); Ueda et al. (2014),
out to high redshift, provides insights into the AGN population in
 
 
 
the very early Universe. However, due to the lack of observational X , = X , (3)
data at redshifts of > 6 with which XLF models can be fitted, 
these extrapolated XLF remain poorly constrained. Yet, even with where the predicted number, is given by the LDDE

 MNRAS 000, 1–10 (2022)

6 C. L. Barlow-Hall et al.

Figure 4. The observed binned XLF estimates, given by the / method (solid stars) plotted at the centre of the luminosity bins in which they were
calculated with the width of the luminosities bins shown by the error bars. The 1 uncertainties in the XLF estimates are shown. The XLF estimated given by
 Í
the 1/ method are also plotted (hollow star) for comparison. The observed XLF found by Wolf et al. (2021) is shown by the blue circle, and can be
seen to agree with our constraints to within 1 Left: The XLF models fitted by Georgakakis et al. (2015) plotted, with 1 uncertainties shown by the shaded
region. The observed XLF is consistent with the majority of the XLF models shown, confirming the relatively steep bright-end slope. Right: Ueda et al. (2014)
LDDE (green), Flexible Double Power-Law (FDPL) of Aird et al. (2015) (blue) and Gilli et al. (2007) LDDE (brown) XLF models plotted with the observed
XLF estimates. The 1 uncertainties in the Ueda et al. (2014) and Aird et al. (2015) models are shown by the shaded regions. The Ueda et al. (2014), Aird et al.
(2015) and Gilli et al. (2007) with high L decline (brown) models can be seen to predict the observed XLF whilst the Gilli et al. (2007) model without the
high L decline (light brown) predicts a higher space density than has been observed with the ExSeSS source and the upper limit at higher luminosities. Both
Gilli et al. (2007) models are shown with (dot-dashed) and without (dotted) compton thick AGN.

model as in equation
 
 3, is the observed number of AGN and From these plots we find that the majority of XLF models are
 X , is taken to be the value of the LDDE model at broadly consistent with what we observe, although may rule out
the centre of the adopted redshift and luminosity bins. The 1 un- more extreme classes of models with a very rapid decline in the
certainties in these binned XLF estimates are based on the Poisson space density to high redshift such as the Georgakakis et al. (2015)
uncertainties in the observed source numbers, as given by Gehrels LADE model. Thus, the distribution of accretion rates for AGN
(1986). populations at high-redshift is similar to that expected from models
 For comparison we also calculate the observed XLF estimates us- of lower redshift AGN populations. From these results it can be seen
ing the more commonly used 1/ method (given in table 1). The that at ∼ 6 we require very large survey areas to detect bright-end
1/ method is more biased than the / method, as AGN. However, these results do not provide constraints to the break
it does not account for a change in the XLF across a broad lumi- luminosity and the faint-end of the XLF, where the models begin
nosity range or with redshift and is thus more strongly affected by to diverge. Further observations of sources at the faint-end of the
source luminosity, in particular for low source numbers., however it XLF could allow us to differentiate between, and potentially rule
has no dependence on a fitted parametric model. We find that the out, certain XLF models.
 / estimate show negligible change depending on the As noted above, there may be other > 6 AGN within ExSeSS
XLF model that is used and agrees to well within 1 of the 1/ that we have not identified as they do not currently have redshift
method. estimates. Nevertheless, from the results presented here it can be
 seen that the ExSeSS observations provide important constraints on
 As can be expected from figure 3, the binned XLF estimates are the bright end of the high-redshift XLF.
found to be consistent with the fiducial values of the XLF models (see
figure 4). The PLE, PDE and LDDE models from Georgakakis et al.
(2015) agree with the observations within the 1 Poisson uncertainty,
whilst the LADE model in particular falls much lower than the binned
XLF estimate (see 4). In addition to the Georgakakis et al. (2015)
models we also plot the model XLF of Aird et al. (2015) and Ueda 5 OPTICAL PROPERTIES AND BOLOMETRIC QLF
et al. (2014), which agree to within 1 of the log X = 44.8 − 45.8 High redshift Active Galactic Nuclei (AGN) are often selected based
binned XLF estimate, and Gilli et al. (2007) which falls more than 3 on their Optical and UV properties. In order to compare our serendip-
above the binned XLF estimates, as shown in figure 4. The gradient itously selected X-ray AGN constraints to the observations of AGN
of the bright-end of these XLF models is consistent with the gradient populations performed using optical data we investigate the optical
indicated by the binned XLF estimates, with a value of & 0.367, properties of our high-redshift source (§5.1), using ATLAS data,
confirming the relatively steep bright-end slope of the XLF at & 6. and compare our X-ray based constraints on the Quasar Luminosity
 Plotting the result obtained from eFEDs data by Wolf et al. (2021) Function (QLF) at > 6 to prior estimates of the bolometric lumi-
we can see that these results are consistent to within 1 of the XLF nosity function, primarily based on optical samples at these redshifts
value given by a single source observed in only 140 degrees2 of sky. (§5.2).

MNRAS 000, 1–10 (2022)

Constraining the ∼ 6 XLF with ExSeSS 7

Table 3. The spectroscopic redshift of the ExSeSS source and its luminosity in the soft-band X-ray and z-band optical wavelengths. The monochromatic
luminosity at a wavelength of 2500Å is derived from the z-band luminosity, assuming a constant power-law relation of slope = −0.3, and the monochromatic
luminosity at an energy of 2 keV is obtained from the 2-10 keV Luminosity, assuming a power law spectrum. The final column gives the optical–to–X-ray slope,
 OX (see Equation 4).

 Object X2−10keV 2keV zAB OX
 2500Å
 erg.s−1 erg.s−1 .Hz−1 erg.s−1 .Hz−1 erg.s−1 .Hz−1

 ATLAS J025.6821-33.4627 6.31 ± 0.03 8.47+3.40
 −3.12
 × 1044 +0.80 × 1027
 2.01−0.74 1.67+0.10
 −0.09
 × 1033 2.08+0.12
 −0.11
 × 1033 +0.08
 −2.31−0.05

 relation of Nanni et al. (2017) (as shown in figure 5). Thus,
 2500Å
 although this is a relatively optically bright and X-ray weak source,
 such an optical luminosity is within the range expected given its
 X-ray luminosity. This indicates that the physics of accretion in this
 high redshift source is the same as seen in lower redshift AGN, as
 it follows the same optical-to-X-ray slope, and the source detection
 is not a consequence of being relatively X-ray luminous. However,
 as the optical-to-X-ray slope determined here falls below the average
 expected for this optical luminosity, this source is X-ray weak relative
 to its UV luminosity, although variability in the X-ray emissions
 could have an impact on our measurement. If the X-ray weak nature
 of this source is typical of high-redshift sources this could explain
 the steep drop in space density with increasing redshift. Given its
 X-ray weakness and steep OX , in order to calculate the bolometric
 luminosity of ATLAS J025.6821-33.4627 from its X-ray luminosity,
 a large correction factor will be required.

Figure 5. The optical-to-X-ray slope as a function of the 2500Å monochro- 5.2 The Bolometric QLF
matic luminosity of the serendipitously detected ExSeSS source ATLAS
 More high redshift AGN have been identified through rest frame
J025.6821-33.4627 (black star). For comparison the OX values of samples
from Pons et al. (2020); Nanni et al. (2017) are shown by the yellow and UV selection than X-ray selection, due to the limited survey areas
turquiose points, respectively. The standard relation for the optical-to-X-ray covered with current X-ray telescopes compared to the efficiency and
slope, as a function of the 2500Å monochromatic Luminosity, determined availability of large-scale optical/near-infrared imaging campaigns.
by Nanni et al. (2017), is shown (dashed line) with shaded regions showing In order to determine how well our observed XLF estimate compares
the 1 (grey) and 3 (light grey) uncertainties. Our high redshift ExSeSS to the Luminosity Functions given by Optical/UV selected AGN, we
source can be seen to have a relatively low OX , whilst still falling within 3 compare our X-ray binned XLF estimates to the Bolometric QLF.
of the expected relation. We convert the X-ray Luminosity bins and binned XLF estimates
 into bolometric terms using a bolometric correction factor, adopt-
 ing the luminosity-dependent corrections determined by Shen et al.
5.1 Optical to X-ray Properties
 (2020), where bol = bol ( bol ) × X−ray and bol ( bol ) is the
While ATLAS J025.6821-33.4627 is a highly luminous source at op- X-ray bolometric correction factor, which depends on bolometric
tical wavelengths, we have shown that it is identified serendipitously luminosiy, bol . For ATLAS 025.6821-33.4627 we estimate a bolo-
based on X-ray selection as part of ExSeSS. In order to investigate metric correction factor based on the Shen et al. (2020) prescription.
its nature and the relation between the emissions of its accretion disk The binned XLF estimates, ( X−ray , ), are converted to bolometric
and X-ray corona, we determine the optical-to-X-ray relation of the QLF values, ( bol , ), using the equation
source, as in Vito et al. (2019), given by
 ! log X−ray
 2keV ( bol , ) = ( X−ray , ) (5)
 OX = 0.3838 log (4) log bol
 
 2500Å where the X-ray luminosity, X−ray , is directly related to the
where the optical-to-X-ray slope, OX , is given by the ratio of the bolometric luminosity, bol , by the bolometric conversion factor
monochromatic X-ray luminosity of the source at a rest-frame energy bol ( bol ). The resulting binned estimate is shown in figure 6) and
of 2 keV, 2 keV , and the optical luminosity of the source at a rest- compared to the Shen et al. (2020) bolometric QLF model, which
frame wavelength of 2500Å, . We determine the value of was based on a fit to a combination of bolometrically corrected
 2500Å
the optical luminosity using the observed z-band luminosity and UV/optical, IR and X-ray luminosity functions spanning a wide range
assuming a power-law continuum of ∝ , with = −0.3 (see of redshifts. At ∼ 6, the Shen et al. (2020) model is primarily
Bañados et al. 2016; Pons et al. 2020, and references therein), as constrained by prior estimates of the QLF based on rest-frame UV
detailed in appendix A. The 2 keV luminosity is determined from the selected data; the original measurements, compiled from the litera-
2–10 keV band luminosity assuming a photon index of Γ = 1.9 (see ture, are shown by the purple points in figure 6 (see Shen et al. 2020,
equation A3 in the appendix). The values used are shown in table 3. and references therein).
 We find the optical-to-X-ray slope of ATLAS J025.6821-33.4627 From figure 6 it can be seen that adopting a bolometric correc-
is −2.31+0.08
 −0.05
 , which agrees to within 3 of the expected OX − tion factor of bol = 182.4, as inferred by the Shen et al. (2020)

 MNRAS 000, 1–10 (2022)

8 C. L. Barlow-Hall et al.

Table 4. Estimates of the bolometric luminosity for our source, calcualted from the soft-band X-ray and 1450Å luminosities. The optical luminosity at wavelength
of 1450Å is derived from the z-band luminosities, assuming a constant power-law relation of slope = −0.3, the bolometric luminosities are then calculated,
from the 2-10keV band and 1450Å wavelength luminosities, using the conversion method of Shen et al. (2020).

 Object bol from 2−10keV 
 1450Å bol from 1450Å
 −1
 erg.s−1 erg s−1 Å erg s−1

 ATLAS J025.6821-33.4627 1.54+0.91
 −0.72
 × 1047 +0.15 × 1045
 2.52−0.14 +0.07 × 1048
 1.16−0.06

 to the Optical/UV luminosity. Thus such a large X-ray bolometric
 correction factor is therefore warranted.
 Nonetheless, larger X-ray selected samples are needed to deter-
 mine if such large correction factors are generally appropriate for
 AGN in this luminosity–redshift regime and enable a more detailed
 comparison of X-ray and rest-frame UV based constraints on the
 QLF.

 6 CONCLUSIONS
 Luminosity functions provide a tracer of the AGN populations across
 cosmic time. However, at high redshift these luminosity functions
 remain poorly constrained.
 In this paper, we identified one X-ray selected AGN at > 6 within
 the carefully constructed sample of serendipitous X-ray sources from
 Swift-XRT observations that form ExSeSS (Delaney et al. prep). The
 serendipitous X-ray source, 2SXPS J014243.7-332742, is matched
Figure 6. The observed binned estimate of the bolometric QLF, converted with the optically bright = 6.31 ± 0.03 quasar ATLAS J025.6821-
from the binned XLF estimate found in §4.2 from the single source detection 33.4627, making it he highest redshift serendipitously detected X-ray
at log X = 44.8 − 45.8. The observed estimate is calculated using the selected quasar to date. With this detection, we are able to determine
conversion factor of Shen et al. (2020) (black star) for the ExSeSS source, the space density of AGN and place constraints on the X-ray lumi-
which agrees within 2 of the QLF model of Shen et al. (2020), shown in nosity function at ∼ 6.
blue. The Far UV data, converted to bolometric values, to which the QLF
 Our conclusions are as follows:
model was fitted is shown by the violet points.
 • The space density of AGN given by the high redshift ExSeSS
 source shows the steep decline of AGN with increasing redshift. This
luminosity-dependent conversions, our X-ray based estimates of the observed decline is consistent with the expected exponential decline
bolometric QLF are consistent within 2 of the Shen et al. (2020) in the space density of luminous AGN with increasing redshift.
model and prior UV/optical constraints. • We place constraints on the shape of the ∼ 6 XLF by com-
 In §5.1, we found that our serendipitously detected AGN is rela- bining our single detection and an upper limit at higher luminosities.
tively X-ray weak, when compared with its rest-frame UV luminosity, These constraints indicate a relatively steep bright-end slope of the
and thus likely requires a large X-ray–to–bolometric correction fac- XLF, with & 0.367, as found at lower redshifts. Our binned XLF
tor. Thus we also estimate the 1450Å luminosity (based on the z-band estimates are broadly consistent with the extrapolation of parametric
magnitude of the source and assuming a power-law UV spectrum, XLF models, based on fits to lower redshift data. However the con-
as in §5.1), which we use to provide an alternative estimate of the straints we have obtained here rule out the most extreme declines in
bolometric luminosity of our source, for comparison. This is done the bright-end of the XLF indicated by some model extrapolations
by adopting the luminosity-dependent optical-to-bolometric correc- (e.g. the LADE model of Georgakakis et al. 2015).
tions from Shen et al. (2020) as for the X-ray luminosity above. The • Our XLF constraints are consistent (within ∼ 1 ) with the
various luminosity estimates are provided in table 4. prior measurements by Wolf et al. (2021), which were based on the
 The bolometric luminosity inferred from the rest-frame UV light identification of a single = 5.81 X-ray selected AGN in the 140 deg2
is +0.15 × 1048 erg s−1 , which whilst this is not
 = 2.52−0.14 eFEDS field observed by eROSITA.
 1450Å to bol
consistent with the bolometric luminosity inferred from the X-ray, a • We find that the Optical-to-X-ray slope, OX , of our serendip-
value of X−ray to bol = 1.54+0.91
 −0.72
 × 1047 erg s−1 obtained using the itously detected source is relatively low, indicating this source is
large X-ray bolometric correction factor of bol = 182.4, it is within relatively X-ray weak compared to its optical luminosity. However,
the luminosity bin of our QLF measurement. This discrepancy in this source still lies within 3 of the expected OX -L relation,
 2500Å
the bolometric values is likely due to the assumptions made when indicating the accretion process in this high redshift source operates
calcualting and the differences in the Shen et al. (2020) in a similar manner to AGN at lower redshift. As this source is X-
 1450Å to bol
bolometric correction factors for UV and X-ray. We also note that ray weak in comparison to its optical luminosity, a high correction
systematic uncertainties from the bolometric correction factor are factor is required to converting its X-ray Luminosity to a bolometric
not included in the errors for our estimates of the bolometric lumi- estimate.
nosity. In addition to this the optical-to-X-ray slope investigated in • Applying a bolometric correction to the X-ray luminosity of our
§5.1 shows that the X-ray luminosity is relatively low in comparison serendipitously detected source, we find a good agreement between

MNRAS 000, 1–10 (2022)

Constraining the ∼ 6 XLF with ExSeSS 9
our observed bolometric QLF and the parametric model QLF of Georgakakis A., et al., 2015, MNRAS, 453, 1946
Shen et al. (2020). The observed bolometric QLF we have calculated Gilli R., Comastri A., Hasinger G., 2007, A&A, 463, 79
is consistent with UV/Optically selected quasars within the redshift HI4PI Collaboration et al., 2016, A&A, 594, A116
range of = 5.7 − 6.4. Hickox R. C., Alexander D. M., 2018, ARA&A, 56, 625
 Hopkins P. F., Richards G. T., Hernquist L., 2007, ApJ, 654, 731
 The measurements presented in this paper provide important con- Hunter J. D., 2007, Computing in Science Engineering, 9, 90
straints on the extent of supermassive black hole growth within the Kalfountzou E., Civano F., Elvis M., Trichas M., Green P., 2014, MNRAS,
early Universe. Our estimates, based on the 2086.6 degrees2 cov- 445, 1430
ered by ExSeSS, indicate that the ongoing all-sky surveys being Khorunzhev G. A., et al., 2021, Astronomy Letters, 47, 123
performed by eROSITA will identify a few tens to a few hundred Kormendy J., Ho L. C., 2013, ARA&A, 51, 511
 Luo B., et al., 2017, ApJS, 228, 2
high-luminosity AGN at > 6, once sufficient depth is achieved
 Machalski J., 1998, A&AS, 128, 153
(i.e. by eRASS:8 all-sky depth), enabling further investigation of the Madau P., Rees M. J., 2001, ApJ, 551, L27
growth of SMBHs within the early Universe. However, much deeper Marchesi S., et al., 2016, ApJ, 827, 150
surveys will be required in order to constrain faint end of the X-ray Matsuoka Y., et al., 2019, ApJ, 883, 183
Luminosity Function, which will become possible within the next McGreer I. D., et al., 2013, ApJ, 768, 105
decade with new telescopes such as the Athena X-ray Observatory. Medvedev P., et al., 2020, MNRAS, 497, 1842
 Miyaji T., Hasinger G., Schmidt M., 2001, A&A, 369, 49
 Mortlock D. J., et al., 2011, Nature, 474, 616
 Mukai K., 1993, Legacy, 3, 21
ACKNOWLEDGEMENTS
 Nanni R., Vignali C., Gilli R., Moretti A., Brandt W. N., 2017, A&A, 603,
CLBH and JD acknowledge support from STFC PhD studentships. A128
JA acknowledges support from a UKRI Future Leaders Fellowship Padovani P., et al., 2017, A&ARv, 25, 2
 Page M. J., et al., 1996, MNRAS, 281, 579
(grant code: MR/T020989/1). This research has made used of Swift-
 Pâris I., et al., 2017, A&A, 597, A79
XRT data from the Neil Gehrels Swift Observatory and benefited
 Pâris I., et al., 2018, A&A, 613, A51
greatly from the work by the Swift Team and the 2SXPS catalogue Pons E., McMahon R. G., Banerji M., Reed S. L., 2020, MNRAS, 491, 3884
of Evans et al. (2020). This research has made use of the SIMBAD Predehl P., et al., 2021, A&A, 647, A1
database, operated at CDS, Strasbourg, France. We have benefited Reines A. E., Comastri A., 2016, Publ. Astron. Soc. Australia, 33, e054
from the publicly available programming language Python, includ- Ross N. P., Cross N. J. G., 2020, MNRAS, 494, 789
ing NumPy & SciPy (van der Walt et al. 2011; Virtanen et al. 2020), Ross N. P., et al., 2013, ApJ, 773, 14
Matplotlib (Hunter 2007), Astropy (Astropy Collaboration et al. Selsing J., Fynbo J. P. U., Christensen L., Krogager J. K., 2016, A&A, 585,
2013) and the Topcat analysis program (Taylor 2013). This research A87
has made use of the online count rate simulator WebPIMMs (Mukai Shen Y., et al., 2019, ApJ, 873, 35
 Shen X., Hopkins P. F., Faucher-Giguère C.-A., Alexander D. M., Richards
1993).
 G. T., Ross N. P., Hickox R. C., 2020, MNRAS, 495, 3252
 Sunyaev R., et al., 2021, A&A, 656, A132
 Taylor M., 2013, Starlink User Note, 253
DATA AVAILABILITY Ueda Y., Akiyama M., Hasinger G., Miyaji T., Watson M. G., 2014, ApJ, 786,
 104
The ExSeSS catalogue and area curve used in this work is provided by Vignali C., Brandt W. N., Fan X., Gunn J. E., Kaspi S., Schneider D. P.,
Delaney et al. (prep) which is based on the 2SXPS catalogue (Evans Strauss M. A., 2001, AJ, 122, 2143
et al. 2020) available at https://www.swift.ac.uk/2SXPS/ and Virtanen P., et al., 2020, Nature Methods, 17, 261
provides the full X-ray data. ATLAS observations are available Vito F., Gilli R., Vignali C., Comastri A., Brusa M., Cappelluti N., Iwasawa
through the VST ATLAS public survey http://osa.roe.ac.uk/ K., 2014, MNRAS, 445, 3557
#dboverview_div. Vito F., et al., 2019, A&A, 630, A118
 Volonteri M., Begelman M. C., 2010, MNRAS, 409, 1022
 Wang F., et al., 2021, ApJ, 907, L1
 Wenger M., et al., 2000, A&AS, 143, 9
REFERENCES Wolf J., et al., 2021, A&A, 647, A5
Aird J., Coil A. L., Georgakakis A., Nandra K., Barro G., Pérez-González Zubovas K., King A., 2021, MNRAS, 501, 4289
 P. G., 2015, MNRAS, 451, 1892 van der Walt S., Colbert S. C., Varoquaux G., 2011, Computing in Science
Astropy Collaboration et al., 2013, A&A, 558, A33 and Engineering, 13, 22
Bañados E., et al., 2016, ApJS, 227, 11
Bañados E., et al., 2018, Nature, 553, 473
Boyle B. J., Shanks T., Croom S. M., Smith R. J., Miller L., Loaring N.,
 APPENDIX A: DERIVED LUMINOSITY CALCULATIONS
 Heymans C., 2000, MNRAS, 317, 1014
Brandt W. N., et al., 2002, ApJ, 569, L5 Assuming the flux of the AGN follows a power law relation of ∝
Brunner H., et al., 2021, arXiv e-prints, p. arXiv:2106.14517 (see e.g. Pons et al. 2020; Bañados et al. 2016; Selsing et al.
Brusa M., et al., 2009, ApJ, 693, 8
 2016), with = −0.3, the flux in the z-band can be converted to a
Burrows D. N., et al., 2005, Space Sci. Rev., 120, 165
 different wavelength flux using
Carnall A. C., et al., 2015, MNRAS, 451, L16
Chen Z.-F., Pan D.-S., Pang T.-T., Huang Y., 2018, ApJS, 234, 16
  
 z−band 
   
De Rosa G., et al., 2014, ApJ, 790, 145 obs
Delaney J., Aird J., Evans P. A., Barlow-Hall C., Watson M. G., in prep, = = (A1)
 z−band z−band rest (1 + )
 ExSeSS, in prep 2021
Evans P. A., et al., 2020, ApJS, 247, 54 where is the monochromatic flux at a rest frame frequency 
Gehrels N., 1986, ApJ, 303, 336 and z−band is the flux in the z-band (in units of erg.s−1 .cm−2 .Hz−1 ),

 MNRAS 000, 1–10 (2022)

10 C. L. Barlow-Hall et al.
given by the observed z-band apparent magnitude. The ratio of these
fluxes is given by the frequency of the z-band, z−band , and the
observed frequency at which to determine the monochromatic flux,
 obs , or by the wavelength of the z-band, z−band , and the rest frame
wavelength at which to determine the monochromatic flux, rest . For
ATLAS the z-band central wavelength, used in this calculation, is
8780.0Å.
 From the monochromatic flux, found using equation A1, the
 −1
monochromatic luminosity (in units of erg.s−1 .cm−2 .Å ) is given
by the equation

   OX
 z−band
 rest = 4 2 (1 + ) 2 z−band (A2)
 rest rest (1 + )
 where the monochromatic luminosity, rest , at a rest frame wave-
length of rest is related to the monochromatic flux observed in the
z-band, z−band , by the luminosity distance , the rest-frame wave-
length, the central wavelength of the z-band z−band and the power
 . is the redshift of the source and is the speed of light.
 The monochromatic luminosity at an energy of 2 keV, 2 keV , is
given by the equation

 (2 − Γ) 2−10keV
 = ( ) = 1−Γ (A3)
 (10.0keV2−Γ − 2.0keV2−Γ )
 where the X-ray spectrum is assumed to be given by a power-law
 ( ) ∝ −Γ , with a photon index of Γ = 1.9, and the measured
hard-band luminosity is given by the total (0.3-10 keV) band flux.
The resulting monochromatic luminosity can then be converted from
units of erg.s−1 .keV−1 to erg.s−1 .Hz−1 by multiplying the luminos-
ity by a factor of h. For the calculation of OX , we calculate this
monochromatic luminosity at an energy = 2 keV

This paper has been typeset from a TEX/LATEX file prepared by the author.

MNRAS 000, 1–10 (2022)