followed the same procedure for Ly$\alpha$ measurements, however due to there
only being $\sim 300$ spectra with $z>3.4$ in the Euclid Collaboration: Lusso
et al. (2024) sample we do not separate by template class. Instead, we take
the median Ly$\alpha$ luminosity and $\sigma_{\rm MAD}$ of the full $z>3.4$
sub-sample.

In our EWS sample, there were 1.0 $\times$ 107 candidate unobscured AGN in the
H$\alpha$$+$[N ii] range $0.01\leq z\leq 2.05$. We found 4644 (0.05%) of these
AGN became newly observable with the inclusion of H$\alpha$$+$[N ii]. There
were 1.9 $\times$ 106 unobscured AGN at $3.4\leq z\leq 7.0$ of which 119
(0.006%) became observable with the inclusion of Ly$\alpha$. In total a lower
limit of 4763 out of 1.2 $\times$ 107 (0.04%) unobscured AGN could have been
observed with the inclusion of broad emission lines in our unobscured AGN
template.

For the EDS sample the inclusion of H$\alpha$$+$[N ii] resulted in an
additional 60 detectable unobscured AGN out of 3.3 $\times$ 104 at $0.01\leq
z\leq 2.05$, a gain of 0.2%. There were 5.9 $\times$ 103 unobscured AGN at
$z>3.4$ of which only two (0.03%) became observable by Euclid with Ly$\alpha$
considered. In total 62 of 3.9 $\times$ 104 (0.16%) unobscured AGN could be
observed with the inclusion of broad emission lines in our unobscured AGN
template.

The enhanced proportion of newly detected AGN from H$\alpha$$+$[N ii] in the
EDS is due to the deeper apparent magnitudes (i.e., fluxes) probed in the
survey. As the 5$\sigma$ flux threshold is deeper, a small change in the
source flux provided by emission lines is more likely to allow an AGN to
breach this threshold and become detectable with Euclid. In both surveys the
inclusion of Ly$\alpha$ made a smaller difference to detectability compared to
H$\alpha$$+$[N ii]. This is due to the higher redshift range where Ly$\alpha$
occupies the Euclid bands. The additive Ly$\alpha$ flux excess in this regime
is comparatively smaller and so has a lower impact on detectability than with
H$\alpha$$+$[N ii]. We observe from Figs. 8 and 9 that whilst there is a high
density of AGN near the detection threshold at $0.01\leq z\leq 2.05$, the
parameter space is more sparse at $z>3.4$. This means there are less AGN that
could become observable due to a flux perturbation from emission lines at
higher redshifts.

Despite these numbers amounting to a lower limit, the derived number of newly
observed AGN are a negligible percentage of the total observable sources and
fall well within the Poisson error ($\sigma=\sqrt{N}$) of our data sets. We
therefore conclude that the omission of AGN emission lines contributes a
minimal effect and uncertainty on our total AGN number estimates in the Euclid
photometric surveys.

####  5.1.5 Uncertainty comparison

Here we compare the uncertainties determined for different models and
assumptions in this work. In Fig. 12 we present a collation of the relative
uncertainties quantified in this section on the number of AGN with a detection
in at least one Euclid band. The blue shaded region signifies the expected
relative Poisson noise ($\sigma=\sqrt{N}$) for our EDS detectable AGN.

Figure 12: Relative 1$\sigma$ uncertainty of the number of AGN detectable in
at least one Euclid band ($N_{\rm AGN}$) for each of the quantified sources of
uncertainty introduced in this work. Displayed in this plot are uncertainties
introduced by the X-ray luminosity function (XLF), infrared bolometric
correction dispersion (IRBC), X-ray bolometric correction dispersion (XBC) and
unobscured AGN emission lines (EL) for which a lower limit was quantified
using the Ly$\alpha$ emission line and H$\alpha$$+$[N ii] complex. The
expected relative Poisson noise for the EDS, calculated as
$\sigma=\sqrt{N_{\rm AGN}}$, is indicated by the shaded blue region.

The dominant source of uncertainty in our analysis is the uncertainty in the
XLF. The XLF relative uncertainty is several orders of magnitude greater than
any other source of uncertainty quantified here and is the only source that
generates an uncertainty greater than the derived Poisson noise for our EDS
sample. Amounting to 12.5% for the EDS and 6.7% for the EWS, we may consider
this the overall uncertainty of our detectable AGN estimates. Well within the
EDS Poisson noise, the second largest uncertainty is driven by the IR and
X-ray bolometric correction dispersion. These relative uncertainties are
already negligible compared to that of the XLF.

We were unable to quantify the impact that extrapolating the Merloni et al.
(2014) optically obscured AGN fraction model has on our results. Although we
have established our extrapolation agrees with the general trends of
literature results, the ground truth of the redshift evolution and X-ray
luminosity dependence of the optically obscured AGN fraction remains
unresolved. Despite the complexity in constraining such dependencies we hope
that the extension of such models in the redshift and luminosity domain can be
an area of investigation with next-generation facilities.

In a scenario where our predictions in this analysis are proven to be in
tension with Euclid observations, we can back-propagate through our assumed
models and framework to identify where disparities lie between our current
understanding and the ground truth. This will in turn serve to inform the
community of areas where there are gaps and inconsistencies in our current
modelling of AGN properties and demographics.

###  5.2 Photometric AGN selection in Euclid

Optical to MIR colour-colour cuts are often invoked to select samples of AGN
from photometric data sets (e.g. Stern et al. 2005; Donley et al. 2012; Assef
et al. 2018). The colours used in AGN selection exploit spectral features
unique to AGN, such as the MIR excess, to separate AGN from inactive galaxies
and stars in the colour-colour parameter space. Because they can be quickly
and cheaply applied to large data sets, simple colour selection criteria are
used in many cases as an initial selection for candidates of a population
before applying more complex identification methods. This will likely be the
use-case for colour selection of AGN in Euclid.

Only a fraction of the AGN detected by Euclid, i.e., present in at least one
image, will actually be selected as AGN based on Euclid photometry alone.
Indeed, half of the obscured AGN SED classes we assign in Sect. 3.3.3 (PASS,
SFG, SB) are entirely free from AGN spectral features in the Euclid wavelength
range. This means that without external data these AGN will appear
indistinguishable from non-active galaxies. Therefore, it will not be possible
to identify all of our Euclid detectable AGN using photometric criteria, with
Euclid data alone or otherwise.

Colour selection criteria for AGN using Euclid photometry has been explored in
depth in Bisigello et al. (in prep). Optimal selection criteria were derived
for the EWS and EDS based on Euclid photometry exclusively and with the
inclusion of ancillary photometric observations, such as with Spitzer/IRAC
(Fazio et al. 2004) and Rubin/LSST. Each selection was defined to target
either unobscured AGN or all AGN including obscured and composite sources,
with optimal criteria derived to maximise the F1-score, i.e. the harmonic mean
of sample completeness ($C$; fraction of selected AGN with respect to the
total AGN population) and purity ($P$; fraction of selected sample that are
AGN, not contaminants). Each criterion exhibits a selection function with a
unique redshift dependence. In the following, we have applied each of the
optimal colour selection criteria to our samples of simulated AGN to assess
how many Euclid detectable AGN we can expect to select using Euclid colours.
We verified that the AGN colours derived with our models are consistent with
those derived in Bisigello et al. (in prep) over the relevant ranges for AGN
selection.

We consider only AGN in our samples with 5$\sigma$ detections in the
constituent bands for each colour criterion. For AGN selection schemes
incorporating Rubin/LSST bands we consider the 5$\sigma$ point-source final
co-added depths reported in Ivezić et al. (2019); (u, g, r, i, z, Y) = (25.6,
26.9, 26.9, 26.4, 25.6, 24.8). Detailed breakdowns of the photometric
selection criteria and the results of applying each to our data are provided
in Appendix F. We additionally present density plots depicting our AGN sample
with each colour selection criterion considered in this section (Figs. 20,
21).

Table 6: Performance of Euclid photometry AGN selection criteria (Bisigello et al., in prep.) applied to our samples of EWS and EDS AGN. We consider only AGN that are detected above the 5$\sigma$ limiting magnitude for the four constituent filters in each colour criterion. For each criterion we report the total number of selected AGN, the surface density of selected AGN in deg-2, the completeness ($C$) at the 5$\sigma$ level in the relevant filters, and the expected purity ($P$) for the criterion, as estimated in Bisigello et al. (in prep.). The final row for each survey denotes the union ($\cup$) of all considered selection criteria. Survey | Target Class | Photometry |  Total Selected AGN |  Surface Density (deg-2) |  Unobscured AGN |  Obscured AGN | $C$ | $P$
---|---|---|---|---|---|---|---|---
EWS | Unobscured AGN | Euclid | 4.8 $\times$ 106 | 331 | 4.4 $\times$ 106 | 3.7 $\times$ 105 | 0.23 | 0.17
EWS | Unobscured AGN |  Euclid, Rubin/LSST | 5.7 $\times$ 106 | 393 | 5.5 $\times$ 106 | 1.8 $\times$ 105 | 0.45 | 0.92
EWS | All AGN |  Euclid, Rubin/LSST | 6.0 $\times$ 106 | 413 | 4.3 $\times$ 106 | 1.7 $\times$ 106 | 0.51 | 0.19
EWS | $\cup$ |  Euclid, Rubin/LSST | 8.1 $\times$ 106 | 556 | 6.0 $\times$ 106 | 2.1 $\times$ 106 | 0.37 | -
EDS | Unobscured AGN | Euclid | 1.7 $\times$ 104 | 346 | 1.6 $\times$ 104 | 400 | 0.11 | 0.23
EDS | Unobscured AGN |  Euclid, Rubin/LSST | 2.0 $\times$ 104 | 392 | 1.9 $\times$ 104 | 390 | 0.45 | 0.92
EDS | All AGN |  Euclid, Rubin/LSST | 2.9 $\times$ 104 | 579 | 2.0 $\times$ 104 | 9.5 $\times$ 103 | 0.32 | 0.58
EDS | $\cup$ |  Euclid, Rubin/LSST | 3.5 $\times$ 104 | 692 | 2.5 $\times$ 104 | 1.0 $\times$ 104 | 0.22 | -

Table 6 provides a summary of the performance of each photometric selection
criterion applied to our AGN sample. Absolute expected numbers of selected AGN
and surface densities are presented as well as an assessment of the sample
completeness. In all circumstances the quoted completeness refers to the
completeness of selected AGN relative to 5$\sigma$ detected AGN available in
the same bands. Due to our sample, by construction, containing only the
expected colours of AGN and not other astrophysical sources we are unable to
estimate the purity of our selected samples, however we note the expected
purity of each selection criterion from Bisigello et al. (in prep.).

Adopting the Euclid-only unobscured AGN criterion, we expect a completeness
$C=0.23$ in the EWS and $C=0.11$ in the EDS. When considering completeness
with respect to available unobscured AGN, the target class of the selection,
the respective EWS and EDS completeness values rise to $C=0.52$ and $C=0.40$.
AGN selected using colour cuts with Euclid photometry alone represent 40% and
35% of the unobscured AGN detectable in at least one Euclid band in the EWS
and EDS, respectively. The same samples correspond to 12% and 8% of the total
AGN population detectable in at least one Euclid band. The surface densities
resulting from these selections have a difference of only 15 deg-2 between the
EWS and EDS, despite the EDS probing two magnitudes deeper. The disparity
stems from the low overall completeness for the EDS selection. Although there
is around twice the surface density of detected AGN in the EDS, the Euclid-
only unobscured AGN selection has around half the completeness. The
inefficiency of the EDS selection is noted in Bisigello et al. (in prep.),
emerging from the fact that most EDS AGN are expected to be faint, lying on
the detection boundary of each band.

The addition of Rubin/LSST bands, when available, are expected to supplement
the purity, completeness, and size of AGN samples selected with Euclid data.
Crucially, the inclusion of ancillary data will allow the selection of
obscured and composite AGN as well as unobscured. To determine the total
sample sizes of selectable AGN with Euclid, we took the union of all selected
samples resulting from the criteria discussed in this section, leveraging both
Euclid and Rubin/LSST photometry. The total selected samples in the EWS and
EDS have overall completeness $C=0.37$ and $C=0.22$. Considering selected and
available unobscured AGN, the total samples have completeness $C=0.66$ and
$C=0.58$ in the EWS and EDS. The completeness is $C=0.15$ and $C=0.09$ with
respect to obscured AGN. These total samples equate to 21% and 15% of the AGN
found to obtain a 5$\sigma$ detection in at least one Euclid photometric band
in the EWS and EDS, respectively.

Figure 13 presents redshift distributions, redshift-dependent completeness,
and $L_{\rm bol}$-$z$ planes of the Euclid-only selection and union of all
selections for the EWS and EDS. The middle panels of Fig. 13 show the
completeness is highest for all selected samples around cosmic noon ($z\sim
2$), matching the expectations of Bisigello et al. (in prep.). This marks the
redshift at which the 4000 Å break enters the $Y_{\scriptscriptstyle\rm E}$
band, providing one of the cruxes the photometric selection criteria exploits
to separate AGN from galaxies. An elevated completeness is observed around
this redshift for the total selected sample compared to the Euclid-only
selected sample in both surveys and is particularly accentuated for the EDS
selections. The shorter wavelength optical bands help to further distinguish
AGN colours from the generally redder galaxy colours.

The derived 5$\sigma$ level completeness for each of the applied selection
criteria may appear as a point of concern for our samples. Many analyses
however do not have the benefit of first assessing the underlying population
of available detected sources. Those that do perform such an exploration
report similar or diminished levels of completeness. For example, Assef et al.
(2018) estimate a completeness of 0.17 and 0.28 for their colour-colour
selections of AGN from the AllWISE survey with a reliability of 90% and 75%,
respectively (see also Stern et al. 2012; Assef et al. 2013).

The redshift distributions (top panels of Fig. 13) show the highest number of
AGN are selected around around cosmic noon in both surveys. This is congruent
with the peak in the number of detected AGN (Fig. 7) as well as the highest
selection completeness. AGN selected at $z>3$ are dominated by those
identified by the Euclid-only unobscured AGN criterion. The addition of
ancillary photometry will help to distinguish AGN at low and intermediate
redshifts, whilst Euclid colours alone can identify the luminous unobscured
AGN at $z>3$. We project $\sim 4\times 10^{4}$ AGN at $z>4$ will be selected
in the EWS, with $\sim 2000$ selected AGN at $z>4.5$.

Unobscured AGN are selected efficiently in all cases, with around half of the
available AGN of this class identified using Euclid photometry alone, and up
to two thirds when employing ancillary optical colours. Obscured AGN are less
efficiently identified with only a tenth of those available in the EDS
selected and a fifth in the EWS. Indeed, across all selections for both
surveys only $\sim 15$% of selected AGN have $E(B-V)\geq 0.05$. In comparison,
$\sim 50$% of the detected AGN available in the selection bands have
$E(B-V)\geq 0.05$. The majority of the colour selections used here were fine-
tuned to identify the unobscured population, hence it is no surprise the
reddened population is missed. It is clear however that there is an abundance
of AGN with higher $E(B-V)$ available to be extracted from the data through
alternative means.

The obscured AGN which are identified in either survey have the SB-AGN, SEY2,
or SB SED classes. The SB-AGN and SB classes have optical–NIR colours similar
to unobscured AGN (see Fig. 17). As such these SED classes are often captured
in the same selection cuts identifying unobscured AGN. No obscured AGN with
the PASS, SFG or QSO2 SED classes were selected. These are all galaxy-
dominated SEDs (see Fig. 10) with optical–NIR colours that are hard to
distinguish from inactive galaxies. As such, we would not expect these classes
to be efficiently selected via Euclid photometry.

AGN will be selected over the bolometric luminosity range
$43\leq\log_{10}(L_{\rm bol}\,/\,{\rm erg\,s}^{-1})\leq 47$, a significant
portion of the range used as input for our simulation. The $L_{\rm bol}$-$z$
planes in the bottom panel of Fig. 13 show that ancillary photometry allows us
to select lower-luminosity AGN in both the EWS and EDS. This is apparent
compared to the Euclid-only unobscured AGN selection at $0.5\leq z\leq 3$ for
both surveys. The range of the $1\sigma$ vertical bars suggest AGN selected
around $0.5\leq z\leq 3$ in the EDS are relatively lower-luminosity than those
in the EWS. This is due to the two magnitude deeper observations in the EDS
reaching a fainter population of AGN. The highest redshift selected AGN from
our simulation is at $z=5.16$ and selected from the EWS. Due to the much
larger area probed by the EWS it is more likely that exceptionally luminous
high-redshift AGN will be observed and selected. We discuss predictions of
high-redshift ($z\sim 7$) AGN detections in more detail compared with the
results of Euclid Collaboration: Barnett et al. (2019) in Appendix D.

Figure 13: Redshift distributions (top), redshift-dependent completeness
(middle), and $L_{\rm bol}$-$z$ planes (bottom) for the selected AGN in the
EWS (left) and EDS (right). AGN selected with Euclid-only photometric criteria
(blue) and the total selected sample defined as the union ($\cup$) of all
Euclid and ancillary $ugrz$ photometric criteria discussed in Sect. 5.2 are
plotted for each panel. In all plots redshift is binned with width $\delta
z=0.5$. In the $L_{\rm bol}$-$z$ plane points represent the median, vertical
lines represent $1\sigma$ standard deviation and horizontal lines represent
the width of the redshift bin.

The large disparity between the projected number of AGN we will detect with
Euclid versus those we can identify as AGN should not be neglected. Our
findings suggest work should be undertaken to improve upon and devise new AGN
selection methods in order to maximize the AGN yield with Euclid as well as
other facilities. This is particularly crucial for intermediate and high-
redshift AGN selection. We show in this work we expect to detect an abundance
of AGN in this regime with Euclid, however we lack a means to efficiently
identify such AGN from the data. Of course, this is partly due to Euclid
probing wavelengths where it is difficult to distinguish AGN emission from
other populations of galaxies. A multiwavelength approach where ancillary
observations allow, or explore the viability of machine learning driven
techniques (Bisigello et al., in sub., Signor et al., in prep.) is necessary.

Before Rubin/LSST bands become available, it is possible to exploit readily
accessible ancillary optical data to perform AGN selection in conjunction with
Euclid bands. The Dark Energy Survey (DES; Abbott et al. 2018), covering a
footprint of $\sim$5000 deg2 in the southern hemisphere, provides optical to
NIR photometry in the g, r, i, z, Y bands to 5$\sigma$ photometric depths of
25.0, 24.5, 23.7, 22.6 and 21.3, respectively. Unfortunately, a one-to-one
mapping is not possible with DES for the photometric selection criteria
defined in Bisigello et al. (in prep.) as DES lacks observations in a u band.
The Ultraviolet Near Infrared Optical Northern Survey (UNIONS) however will
observe $\sim$5000 deg2 of the northern hemisphere in the gri bands and
$\sim$10 000 deg2 in the crucial u band (Ibata et al. 2017). We assessed the
number of AGN in the EWS with a 5$\sigma$ detection in each DES band to be 1.2
$\times$ 107 in $g$, 1.3 $\times$ 107 in $r$, 1.1 $\times$ 107 in $i$, 6.1
$\times$ 106 in $z$ and 2.1 $\times$ 106 in $Y$. For AGN in the EDS we predict
4.2 $\times$ 105 in $g$, 4.3 $\times$ 105 in $r$, 3.4 $\times$ 105 in $i$, 1.9
$\times$ 105 in $z$ and 5.9 $\times$ 104 in $Y$.

###  5.3 Comparison to other surveys

Table 7: Characteristics and AGN identification statistics for a selection of
medium area and wide field surveys across different wavebands, which we use
for comparison with our Euclid yields. The quoted area relates to the area
considered in the survey AGN selection procedure as described in the
corresponding reference(s). Euclid survey detected and selected yields as
plotted in Fig. 14 are provided. Detected samples refer to the number of AGN
with a 5$\sigma$ detection in at least one Euclid filter. Selected samples
correspond to the union of all selection criteria considered in Sect. 5.2.

Survey | Band | Area (deg2) |  Number of AGN |  AGN Surface Density (deg-2) | Reference(s)
---|---|---|---|---|---
EWS (detected) | optical – NIR | 14 500 | 4.0 $\times$ 107 | 2800 | This work
EWS (selected) | optical – NIR | 14 500 | 8.1 $\times$ 106 | 556 | "
EDS (detected) | optical – NIR | 50 | 2.4 $\times$ 105 | 4700 | "
EDS (selected) | optical – NIR | 50 | 3.5 $\times$ 104 | 692 | "
XMM-SERVS | X-ray | 13.1 | 10 300 | 786 | Chen et al. (2018); Ni et al. (2021)
XXL-3XLSS | X-ray | 50 | 26 056 | 521 | Chiappetti et al. (2018)
eFEDS | X-ray | 142 | 22 079 | 155 | Liu et al. (2022); Salvato et al. (2022)
SDSS | optical | 9376 | 750 414 | 80a | Lyke et al. (2020)
DES | optical – NIR | 4913 | 945 860 | 193 | Yang & Shen (2023)
Spitzer cryogenicb | NIR – MIR | 55 | $\sim 20\,000$ | 364 | Lacy & Sajina (2020)
AllWISE (R90) | MIR | 30 093 | 4.54 $\times$ 106 | 151 | Assef et al. (2018)
AllWISE (C75) | MIR | 30 093 | 2.09 $\times$ 107 | 695 | Assef et al. (2018)

  * a

Lower limit as the sample includes only spectroscopically confirmed unobscured
quasars

  * b

Collation of AGN identified in Spitzer Space Telescope cryogenic surveys: The
Spitzer Wide-Area Infrared Extragalactic Survey (SWIRE; Lonsdale et al. 2003),
The AGN and Galaxy Evolution Survey (AGES; Kochanek et al. 2012), The Spitzer
First Look survey (Lacy et al. 2005; Fadda et al. 2006), and the Spitzer
COSMOS survey (S-COSMOS; Sanders et al. 2007)

We compared the surface density of AGN in a mixture of wide field and medium
area surveys described in Table 7 to those predicted for Euclid in this work
in Fig. 14. In terms of detectable AGN, Euclid will detect far more AGN in the
EWS and EDS than are identified in other surveys that cover similar areas. Of
course in this case we are comparing detectable AGN in the Euclid photometric
surveys to AGN that have been selected through various means in different
surveys.

Figure 14: AGN surface density versus survey area comparison for a number of
wide field and medium area surveys in different wavebands. Included AGN
surface densities are: EWS detectable AGN (blue downwards triangle, unfilled),
EWS selected AGN (blue upwards triangle, filled), EDS detectable AGN (magenta
downwards triangle, unfilled), EDS selected AGN (magenta upwards triangle,
filled), DES (green circle, unfilled), SDSS spectroscopic lower limit (green
lower limit), Spitzer Space Telescope combined cryogenic surveys (red plus,
unfilled), AllWISE R90 selection (red left-facing triangle, unfilled), AllWISE
C75 selection (red right-facing triangle, unfilled), eFEDS (black square,
unfilled), XXL-3XLSS (black cross, unfilled), and XMM-SERVS (black diamond,
unfilled). The considered Euclid detectable AGN have $\geq 5\sigma$ detection
in at least one Euclid band. The plotted values and references for each survey
are given in Table 7.

Considering AGN that are selected with Euclid-only photometric criteria (see
Sect. 5.2), we expect AGN surface densities that are on par with surveys of
similar area. In the EDS AGN selected with Euclid-only unobscured AGN criteria
yields an AGN surface density of 346 deg-2, marginally lower than that of the
collated cryogenic Spitzer surveys and the XXL-3XLSS survey. The XXL-3XLSS is
the deepest X-ray survey we have compared with in our analysis and is likely
to recover AGN with colours that cannot be selected using our photometric
criteria as some may appear indistinguishable from non-active galaxies in the
optical–NIR. The AGN surface density of 331 deg-2 derived for the EWS using
Euclid-only unobscured AGN colour selection is roughly double the AGN surface
density found for DES and AllWISE, using their 90% reliability (R90)
criterion. Working with Euclid photometry alone then, we predict that the
surface density of selected AGN in the EWS will be greater than ground-based
optical and space-based MIR peers.

Considering the total selected AGN sample (union of all criteria discussed in
Sect. 5.2), we forecast the EDS will yield a greater AGN surface density than
the XMM-3XLSS survey. The total sample surface density is marginally below
that of XMM-SERVS, which marks the highest AGN surface density from a survey
considered in this analysis. The total selected AGN surface density for the
EWS will be greater than optical surveys and comparable to the reported AGN
surface density for the AllWISE 75% completeness (C75) selection. Again, this
emphasises that we expect Euclid to perform to a similar degree as a space-
based MIR counterpart in regard to AGN selection.

Towards the end of Euclid survey operations both the EWS and EDS are expected
to yield surface densities of selected AGN that are similar or in excess of
their wide and medium-field counterparts across a range of wavebands. If AGN
selection in Euclid is improved upon by combining different techniques and
fine tuning the criteria considered here, Euclid may well surpass the AGN
selection performance of any prior surveys. This represents a promising
outcome considering that Euclid was not designed for the facilitation of AGN
identification.

####  5.3.1 Expected X-ray counterparts

Figure 15: Density of EWS AGN in the 2–10 keV X-ray flux vs
$H_{\scriptscriptstyle\rm E}$ observed magnitude parameter space. The Euclid
detectable AGN are shown in purple, whilst undetectable AGN are represented in
grey. The AGN population detectable with Euclid photometry will, in part,
reside beyond the X-ray flux limits of current X-ray surveys. There is also a
portion of parameter space where a population of AGN detectable in modern
X-ray surveys remain undetectable in the Euclid $H_{\scriptscriptstyle\rm E}$
band. X-ray flux limits converted to the 2–10 keV band, where needed, of the
XXL-3XLSS (black), eFEDS (red) and XMM-SERVS surveys are plotted for
comparison. We plot the XMM-SERVS fields separately as they have differing
depths; XMM-LSS (orange), W-CDF-S (cyan), ELAIS-S1 (green). The dotted black
lines represent $\operatorname{log_{10}}(F_{\rm X}/F_{\rm opt})=\pm 1$. The
majority of AGN should lie between these lines.

We examined the range of 2–10 keV X-ray fluxes probed by Euclid detectable AGN
in our samples. Figure 15 shows the two-dimensional density distribution of
our EWS AGN sample on a 2–10 keV X-ray flux vs Euclid
$H_{\scriptscriptstyle\rm E}$ apparent magnitude plane.
$H_{\scriptscriptstyle\rm E}$ was selected for this visualisation as we found
the greatest yield of Euclid detectable AGN in this band. The 2–10 keV flux
limits of the X-ray surveys considered in this section are plotted for
reference: XXL-3XLSS (limiting flux), eFEDs (eROSITA final depth; limiting
flux), and the three XMM-SERVS fields (90% sky coverage): XMM-LSS, W-CDF-S,
and ELAIS-S1. For surveys not observing directly in the 2–10 keV band we
converted their flux limits from the survey native X-ray band assuming an
X-ray power law with photon index $\Gamma=1.9$. The over-density of sources on
an approximately linear trend in the figure is populated by our AGN SEDs and
is a consequence of using the nominal X-ray and 2 $\mathrm{\SIUnitSymbolMicro
m}$ bolometric corrections without considering their dispersion.

AGN detectable in the $H_{\scriptscriptstyle\rm E}$ band can exhibit 2–10 keV
X-ray fluxes ranging from 8.8 $\times$ 10-12 to 2.5 $\times$ 10-17 erg s-1
cm-2. Therefore, the AGN population detectable with Euclid photometry will, in
part, reside beyond the X-ray flux limits of all the X-ray surveys considered
here. Equally, there is also a portion of parameter space where a population
of AGN detectable in modern X-ray surveys remain undetectable in the Euclid
$H_{\scriptscriptstyle\rm E}$ band. We therefore expect Euclid to probe a
different population of AGN to those selected purely from X-ray surveys,
similarly to what is observed with MIR AGN selection (e.g., Eckart et al.
2010). It is likely that AGN detected by Euclid but not in X-ray surveys are
either low-luminosity ($\operatorname{log_{10}}[L_{\rm\text{2--10}\,keV}/{\rm
erg}\,{\rm s}^{-1}]\sim 42\text{--}43$) AGN at intermediate ($z\sim 1$) to
high redshifts ($z>3$), or AGN that are absorbed in the X-rays. The smaller
converse population observable in X-ray surveys but not with Euclid are likely
to be largely made up of AGN that are under-luminous compared to their host
galaxies (e.g., Mendez et al. 2013).

To the flux limit of the deepest X-ray survey considered here (XXL-3XLSS;
$F_{\rm\text{2--10}\,keV}\ $= 3 $\times$ 10-15 erg s-1 cm-2), we assessed the
number of possible X-ray counterparts for our Euclid observable AGN. We
forecast 1.8 $\times$ 107 (5.9 $\times$ 104) AGN, corresponding to 45% (25%)
of the total detectable population in the EWS (EDS) will exhibit X-ray fluxes
that could be detected in the XXL-3XLSS survey. Of these AGN, 6.3 $\times$ 106
(2.1 $\times$ 104) are unobscured and 1.3 $\times$ 107 (3.8 $\times$ 104) are
obscured in the EWS (EDS). This again highlights the difference in AGN
population that will be probed by Euclid compared to those selected in the
X-ray regime as up to 55% (75%) of EWS (EDS) Euclid-detected AGN would not be
detectable in the deepest modern medium area X-ray surveys. The lower X-ray
detection rate of the EDS sample stems from the fainter NIR magnitudes probed
corresponding to lower luminosity AGN at the faintest X-ray fluxes.

##  6 Summary

In this work we made forecasts of the observational expectations for $z<7$ AGN
in the EWS and EDS. Starting from the Fotopoulou et al. (2016a) 5–10 keV XLF
we generated volume-limited samples of the statistically expected AGN in the
Euclid survey footprints (Sect. 3.2). Our samples cover the redshift range
$0.01\leq z\leq 7$ and bolometric luminosity range
$43\leq\operatorname{log_{10}}(L_{\rm bol}/{\rm erg}\,{\rm s}^{-1})\leq 47$,
corresponding to $41.8\leq\operatorname{log_{10}}(L_{\rm\text{2--
10}\,keV}/{\rm erg}\,{\rm s}^{-1})\leq 46.3$, or $-29.0\leq
M_{1450}\leq-17.2$.

Each AGN in our sample was assigned an SED based on its X-ray luminosity and
redshift (Sect. 3.3). As the observed 5–10 keV XLF considers obscured and
unobscured AGN in an unbiased fashion up to $N_{\rm H}\sim$ 1023 cm-2, we
employed the optically obscured AGN fraction evolution model of Merloni et al.
(2014) to assign each AGN as optically obscured or unobscured. Unobscured AGN
were assigned the mean quasar SED collated in Shen et al. (2020) with
$\alpha_{\rm ox}$ values sampled from the empirical distribution determined in
Lusso et al. (2010). For obscured AGN we leveraged XXL AGN SED fitting results
(Fotopoulou et al. 2016b) to probabilistically allocate an empirical SED
template class based on X-ray luminosity and redshift (Sect. 3.3.3). Finally,
we applied dust extinction to each AGN SED, sampling from obscured and
unobscured AGN $E(B-V)$ distributions derived from XXL survey AGN (Fotopoulou
et al. 2016b), as well as IGM extinction (Madau 1995, Sect. 3.3.4). Through
the probabilistic assignment of optical obscuration class, $E(B-V)$ values,
$\alpha_{\rm ox}$ in unobscured AGN, and SED template class in obscured AGN,
we ensured empirically driven diversity in the resultant photometry of our
AGN, even at similar luminosities and redshifts.

Once assigned and scaled we performed mock observations of each AGN SED in our
sample, convolving with the Euclid bands and an assortment of ancillary
photometric bands (Sect. 3.4). We utilized the resulting photometric catalog
to investigate the observable population of $z<7$ AGN in the Euclid surveys.

Our main findings are summarised as follows.

  1. 1.

We estimated 4.0 $\times$ 107 AGN will have a $\geq 5\sigma$ detection in at
least one Euclid band in the EWS. Of these AGN 31% were unobscured and 69%
were obscured. Our predicted yield corresponds to a detectable AGN surface
density in the EWS of 2.8 $\times$ 103 deg-2. In the EDS we expect a $\geq
5\sigma$ detection in at least one Euclid band for 2.4 $\times$ 105 AGN, of
which 21% are unobscured AGN and 79% are obscured AGN. A detectable AGN
surface density of 4.7 $\times$ 103 deg-2 was found for the EDS. Full four-
band $\geq 5\sigma$ Euclid photometry coverage will be available for 2.1
$\times$ 107 AGN in the EWS and 1.6 $\times$ 105 AGN in the EDS. This is
equivalent to AGN surface densities of 1.4 $\times$ 103 deg-2 and 3.1 $\times$
103 deg-2 for the EWS and EDS, respectively.

  2. 2.

The dominant source of uncertainty in our analysis is from uncertainties in
the XLF. We quantified the relative uncertainty on our numbers of Euclid-
detectable AGN to correspond to 6.7% for the EWS and 12.5% for the EDS. The
disparity in relative uncertainties for the two surveys is due to the EWS
probing the more tightly constrained bright end of the XLF, while the EDS is
able to probe the lesser constrained faint end of the XLF.

  3. 3.

Using the Euclid colour selection criteria derived in Bisigello et al. (in
prep.), we obtained expectations on the number of AGN we will select with
Euclid data. Employing Euclid bands only we selected 4.8 $\times$ 106 (331
deg-2) AGN in the EWS, comprised of 92% unobscured AGN and 8% obscured AGN. In
the EDS we selected a sample of 1.7 $\times$ 104 (346 deg-2) AGN, which
consisted of 97% unobscured AGN and 3% obscured AGN.

  4. 4.

Ancillary $ugrz$ photometric bands from Rubin/LSST improved the completeness,
purity, and size of colour-selected AGN samples. Including these selection
criteria we selected a total of $8.1\times 10^{6}$ (556 deg-2) and $3.5\times
10^{4}$ (692 deg-2) AGN in the EWS and EDS, respectively. The EWS total
selected AGN sample consisted of 75% unobscured AGN and 25% obscured AGN,
while the EDS total selected AGN sample consisted of 71% unobscured AGN and
29% obscured AGN. These samples represent a yield of 20% and 15% of the EWS
and EDS samples of AGN with a $\geq 5\sigma$ detection in at least one Euclid
band. The total expected sample of colour-selected AGN across both Euclid
surveys contains 6.0 $\times$ 106 (74%) unobscured AGN and 2.1 $\times$ 106
(26%) obscured AGN, covering $0.02\leq z\lesssim 5.2$ and
$43\leq\log_{10}(L_{\rm bol}/{\rm erg}\,{\rm s}^{-1})\leq 47$.

  5. 5.

The predicted surface densities of Euclid selected AGN are comparable to those
derived from other modern wide-field and medium-area surveys, across a range
of wavebands. Our EWS yield is most comparable to the WISE C75 AGN selection,
with a slightly lower surface density. Our EDS selected surface density is
marginally greater than that of the XXL-3XLSS survey, which is of similar
area.

  6. 6.

We project that 1.8 $\times$ 107 (5.9 $\times$ 104) AGN, corresponding to 45%
(25%) of the total Euclid detectable population in the EWS (EDS) will exhibit
X-ray fluxes that could be detected in the XXL-3XLSS survey. Therefore, we
assess that up to 55% (75%) of EWS (EDS) Euclid-detected AGN would not be
detectable in the deepest modern medium area X-ray surveys.

We expect Euclid to yield a sizeable statistical sample of AGN, in the order
of tens of millions detected AGN and millions of selected AGN. The deep
optical to NIR magnitudes, high spatial resolution and large areas
interrogated in the Euclid surveys will facilitate a diverse range of
scientific studies with AGN.

###### Acknowledgements.

We thank J. T. Schindler and R. Gilli for their helpful comments and
discussion on the manuscript. The Euclid Consortium acknowledges the European
Space Agency and a number of agencies and institutes that have supported the
development of Euclid, in particular the Agenzia Spaziale Italiana, the
Austrian Forschungsförderungsgesellschaft funded through BMK, the Belgian
Science Policy, the Canadian Euclid Consortium, the Deutsches Zentrum für
Luft- und Raumfahrt, the DTU Space and the Niels Bohr Institute in Denmark,
the French Centre National d’Etudes Spatiales, the Fundação para a Ciência e a
Tecnologia, the Hungarian Academy of Sciences, the Ministerio de Ciencia,
Innovación y Universidades, the National Aeronautics and Space Administration,
the National Astronomical Observatory of Japan, the Netherlandse
Onderzoekschool Voor Astronomie, the Norwegian Space Agency, the Research
Council of Finland, the Romanian Space Agency, the State Secretariat for
Education, Research, and Innovation (SERI) at the Swiss Space Office (SSO),
and the United Kingdom Space Agency. A complete and detailed list is available
on the Euclid web site (http://www.euclid-ec.org). This work is supported by
the UKRI AIMLAC CDT, funded by grant EP/S023992/1. V. A. acknowledges support
from INAF-PRIN 1.05.01.85.08. A. F. acknowledges the support from project
”VLT-MOONS” CRAM 1.05.03.07, INAF Large Grant 2022 ”The metal circle: a new
sharp view of the baryon cycle up to Cosmic Dawn with the latest generation
IFU facilities” and INAF Large Grant 2022 ”Dual and binary SMBH in the multi-
messenger era”. F. R. and F. L. F. acknowledge support from PRIN 2017 ”Black
hole winds and the baryon life cycle of galaxies: the stone-guest at the
galaxy evolution supper” contract #2017PH3WAT.

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##  Appendix A Obscured AGN SED class distributions

In Fig. 16 we present the normalised and extrapolated (to $z=7$) redshift-
space probability distribution for each obscured AGN SED class assigned in
this work (Sect. 3.3.3). We separately present distributions for the high
($\operatorname{log_{10}}[L_{\rm\text{2--10}\,keV}/{\rm erg}\,{\rm
s}^{-1}]\geq 44$) and low ($\operatorname{log_{10}}[L_{\rm\text{2--
10}\,keV}/{\rm erg}\,{\rm s}^{-1}]<44$) X-ray luminosity groups.

Figure 16: Normalised and extrapolated (to $z=7$) redshift-space probability
distributions for each obscured AGN SED class assigned in this work.
Distributions are presented separately for the low (top) and high (bottom)
X-ray luminosity groups.

##  Appendix B Ancillary photometry

In addition to the Euclid filter set presented in Table 2, we also perform
synthetic photometric observations with bands from complimentary surveys
covering UV–MIR. The addition of these bands aid in our discussion and
assessment of the observational expectations for AGN with Euclid compared to
existing and upcoming surveys. The characteristics of each of the ancillary
bands utilized in our work are given in Table 8.

Table 8: Characteristics of ancillary filters included in our catalog, used for synthetic photometric observations of AGN in this work. Survey | Filter |  $\lambda_{\rm eff}$ ($\mathrm{\SIUnitSymbolMicro m}$) | Reference
---|---|---|---
2MASS | $J$ | 1.66 | Skrutskie et al. (2006)
2MASS | $H$ | 1.24 | "
2MASS | $Ks$ | 2.15 | "
DES | $g$ | 0.473 | Morganson et al. (2018)
DES | $r$ | 0.642 | "
DES | $i$ | 0.784 | "
DES | $z$ | 0.926 | "
DES | $Y$ | 1.01 | "
GALEX | FUV | 0.154 | Morrissey et al. (2007)
GALEX | NUV | 0.230 | "
Rubin/LSST | $u$ | 0.368 | Ivezić et al. (2019)
Rubin/LSST | $g$ | 0.478 | "
Rubin/LSST | $r$ | 0.622 | "
Rubin/LSST | $i$ | 0.753 | "
Rubin/LSST | $z$ | 0.869 | "
Rubin/LSST | $Y$ | 0.973 | "
Spitzer | IRAC1 | 3.53 | Fazio et al. (2004)
Spitzer | IRAC2 | 4.48 | "
Spitzer | IRAC3 | 5.70 | "
Spitzer | IRAC4 | 7.80 | "
VISTA | $J$ | 1.25 | Sutherland et al. (2015)
VISTA | $H$ | 1.65 | "
VISTA | $Ks$ | 2.15 | "
WISE | $W1$ | 3.35 | Wright et al. (2010)
WISE | $W2$ | 4.60 | "
WISE | $W3$ | 11.6 | "
WISE | $W4$ | 22.1 | "

##  Appendix C Derived AGN colours

We validate the colours of AGN generated in this work by showing that our
derived colours are consistent with AGN colour-colour diagrams from the
literature (e.g., Stern et al. 2005; Mateos et al. 2012; Fotopoulou & Paltani
2018). In each case we test that our AGN occupy the expected positions on the
diagrams using our EDS data where we require a detection with S/N ¿ 5 in at
least one Euclid band.

In Fig. 17 we plot our AGN on an adapted optical–NIR–MIR colour-colour space
used in Fotopoulou & Paltani (2018) and Logan & Fotopoulou (2020). We
substituted the Euclid $Y_{\scriptscriptstyle\rm E}$ and
$J_{\scriptscriptstyle\rm E}$ bands in place of the VISTA $Y$ and $J$ bands,
and the Rubin/LSST $g$ band ($g{\scriptscriptstyle\rm LSST}$) in place of the
SDSS $g$ band. This colour-colour space is known to separate well population
loci of stars, galaxies and unobscured AGN. The unobscured AGN are expected to
fall in a locus on the blue side of the diagram and galaxies (i.e., the
obscured AGN classes in this work) in a more extended locus on the red side of
unobscured AGN (see Fig. 4 in Fotopoulou & Paltani 2018). For clarity in the
locations of different SED shapes, we plot each SED class with separate
colours. We see that the different SED classes assigned to AGN in this work
fall into the expected positions in the colour space. The unobscured AGN locus
is contaminated by the SB-AGN and SB SED classes, specifically AGN with $z>2$
for the latter. This is no surprise given the similarity of these SEDs at
optical wavelengths (see Fig. 10). We note that a portion of the unobscured
AGN stray into the galaxy locus of the diagram. Akin to what is reported in
Logan & Fotopoulou (2020), these interlopers are affected by extinction,
either from intrinsic dust absorption with $E(B-V)\gtrsim 0.2$ or from IGM
absorption affecting $g{\scriptscriptstyle\rm LSST}$ at $z\gtrsim 3.4$.

We plot the Spitzer/IRAC [3.6]-[4.5] and [5.8]-[8.0] Vega system colours of
our EDS AGN sample with $z<4$ in Fig. 18. The black dashed lines in this plot
denote the AGN selection criterion identified by Stern et al. (2005), valid
for redshifts $z<4$. We found that 58.2% of our total available AGN were
identified using this selection. By class we select 100% of the unobscured AGN
and 47.5% of the obscured AGN. This is expected as we see that the majority of
non-selected obscured AGN belong to SED classes where the representative SED
does not include a strong AGN component at these wavelengths (i.e., PASS, SFG,
SB). Furthermore, it is reassuring that 100% of unobscured AGN assigned with
our mean quasar SED are selected using this method. Such AGN identification
methods should in theory be well optimised towards including the ‘average’
quasar in the resulting selected sample.

The three-band AGN colour selection criterion of Mateos et al. (2012) utilizes
the WISE 3.4, 4.6 and 12 $\mathrm{\SIUnitSymbolMicro m}$ bands (W1, W2, and
W3, respectively). In Fig. 19 we show the W1-W2 and W2-W3 Vega system colours
along with the Mateos et al. (2012) three-band AGN colour selection criterion
in black dashed lines. Colours are plotted for our EDS AGN sample with $z<2$,
the redshift range for which this selection is valid. We found 37.1% of the
total available AGN are identified in this case. Again, 100% of the unobscured
AGN are selected, an outcome validating our mean quasar colours as discussed
above. With this selection criterion however, only 22.9% of our obscured AGN
are identified. The low identification rate of obscured AGN in this case can
be attributed to the MIR selection scheme being optimised for high X-ray
luminosity AGN that have a power-law spectral shape in the MIR. The SED
templates that are well identified have the corresponding MIR spectral shape.
As we explore in Sect. 5.3, our Euclid detected AGN probe to faint X-ray
fluxes which will lead to incompleteness in this selection. The overall
positions of our different SED classes in the colour space agree well with the
redshift evolution tracks for corresponding populations presented throughout
Mateos et al. (2012).

We conclude that the colours derived for AGN in this work are consistent with
observations.

Figure 17: EDS AGN simulated in this work plotted on the optical-NIR-MIR
colour space used in Fotopoulou & Paltani (2018). We used the Euclid
$Y_{\scriptscriptstyle\rm E}$ and $J_{\scriptscriptstyle\rm E}$ bands, the
Rubin/LSST $g$ band ($g{\scriptscriptstyle\rm LSST}$), and the WISE 3.4
$\mathrm{\SIUnitSymbolMicro m}$ (W1) band. Only AGN with S/N ¿ 5 in at least
one Euclid band are plotted. Each of our AGN SED classes are plotted
individually: Unobscured AGN (‘Unobsc.’; blue), Passive (‘PASS’; orange),
Star-forming (‘SFG’; green), Starburst (‘SB’; red), High-luminosity obscured
AGN (‘QSO2’; purple), Seyfert 2 (‘SEY2’; brown), and Starburst-AGN composite
(‘SB-AGN’; pink). Figure 18: Spitzer/IRAC AGN selection of Stern et al. (2005)
applied to our EDS AGN. We plot only AGN with S/N ¿ 5 in at least one Euclid
band with $z<4$. Each of our AGN SED classes are plotted individually with
corresponding colours identical to Fig. 17. Figure 19: WISE three-colour
selection of Mateos et al. (2012) applied to our EDS AGN. This selection uses
the WISE 3.4, 4.6, and 12 $\mathrm{\SIUnitSymbolMicro m}$ bands (W1, W2, and
W3 respectively). We plot only AGN with S/N ¿ 5 in at least one Euclid band
and with $z<2$. Each of our AGN SED classes are plotted individually with
corresponding colours identical to Fig. 17.

##  Appendix D High-redshift predictions

The predicted yield of $7\leq z\leq 9$ quasars in the EWS was explored in
Euclid Collaboration: Barnett et al. (2019). Quasars were incorporated in
their work utilizing the Jiang et al. (2016) high-redshift quasar LF with two
different assumed rates of decline (modest and steep) in space density at
$z\geq 6$. The decline in quasar space density is parameterised as
$\phi\propto 10^{k(z-6)}$, where $\phi$ is the quasar LF and $k$ takes the
values $k=-0.72$ or $k=-0.92$ for the modest and steep rate of decline,
respectively. Contaminant populations such as M-type stars, L and T-type
dwarfs and compact early-type galaxies were additionally modelled (Hewett et
al. 2006). Quasar selection functions were integrated over the sample of
quasars and contaminants to determine the predicted yield of quasars with
$7\leq z\leq 9$ in the EWS. Over the full considered redshift range quasars
were successfully selected to the effective depth $J_{\scriptscriptstyle\rm
E}\sim 22$ when using only Euclid bands. Selection with a modest (steep)
quasar LF decline predicted 87 (51) quasars in the redshift range $7\leq z\leq
7.5$. This corresponds to a quasar surface density of 6.0 $\times$ 10-3 deg-2
(3.5 $\times$ 10-3 deg-2).

For comparison with the predicted EWS quasar yields of Euclid Collaboration:
Barnett et al. (2019), we extended our simulation to $z=7.5$. To approximate
the available quasar candidates, we imposed conditions on our EWS sample for a
detection with $J_{\scriptscriptstyle\rm E}<22$, an unobscured optical
classification, and occupation of the redshift range $7.0\leq z\leq 7.5$.
Applying these constraints we found 1053 unobscured AGN, equal to a surface
density of 0.07 deg-2, a significantly higher yield. We consider that we have
made only a magnitude and redshift cut for our estimates, therefore recovering
the detectable quasars in the target parameter space, but not incorporating
the complex selection function constructed in Euclid Collaboration: Barnett et
al. (2019). The selection function is likely to reject a number of the
unobscured AGN included in our approximated sample.

The AGN selected to define the Jiang et al. (2016) LF fulfilled two colour
cuts primarily so that contaminants were avoided in the final quasar sample.
SDSS main survey quasars required no detection in the $ugr$ bands and obeyed

$i-z>2.2,$ (16)

where $i$ and $z$ refer to the SDSS bands. These criteria select $i$-band
dropout objects and separate quasars (and cool brown dwarfs) from the majority
of stellar objects (Fan 1999; Strauss et al. 1999). Final quasar candidates
also satisfied the criterion

$z-J<0.5(i-z)+0.5,$ (17)

where $J$ refers to the UKIRT Infrared Deep Sky Survey (UKIDSS; Warren et al.
2007) band. This colour-space cut separates quasars from the cool brown dwarf
population. We applied these criteria to our $7.0\leq z\leq 7.5$ unobscured
AGN sample with $J_{\scriptscriptstyle\rm E}<22$, substituting the the UKIDSS
$J$ band for the Euclid $J_{\scriptscriptstyle\rm E}$ band. With these
additional constraints we recover 668 detectable quasars, which is a factor of
eight higher than forecast in Euclid Collaboration: Barnett et al. (2019).

High-redshift quasar yield predictions are especially sensitive to the assumed
LF shape and redshift evolution (Tee et al. 2023). The XLF employed in this
work, when extrapolated, does not exhibit the steep and accelerating space
density decline at $z\gtrsim 6$ seen in UV/optical and other X-ray
determinations of the AGN LF (e.g., Ueda et al. 2014; Jiang et al. 2016; Wang
et al. 2019; Matsuoka et al. 2023, see Fig. 1). At $z>6$ our simulation
predicts an unobscured AGN space density excess of up to 1 dex at
$M_{1450}\sim-24$, when compared with empirical UV/optical quasar
observations. This ultimately is the driver of the apparent excess of quasars
we predict at $7\leq z\leq 7.5$ compared to the Euclid Collaboration: Barnett
et al. (2019) analysis.

Equally as impactful as the assumed LF are the integration range and selection
parameters of high-redshift quasar yield predictions. Schindler et al. (2023)
make $z>7$ EWS quasar predictions also utilizing the $k=-0.7$ Jiang et al.
(2016) quasar LF. Integrating to $M_{1450}\lesssim-22.4$ and using
$H_{\scriptscriptstyle\rm E}<24$ as the effective survey depth, they predicted
809 detectable quasars in the EWS at $7\leq z\leq 8$. This sample is of
comparable size to our $7\leq z\leq 7.5$ simulated detectable sample of 1053,
which incorporates unobscured AGN to $M_{1450}\sim-22.6$, albeit over a larger
redshift slice.

At present there are only 532 quasars confirmed at $z\geq 5.3$ and 275 quasars
known at $z\geq 6$ (Fan et al. 2023, and references therein). Our predictions
suggest there are still many quasars at these redshifts to be uncovered, with
Euclid capable of detecting many of them. We found that $\sim 600$ obscured
AGN will have a $J_{\scriptscriptstyle\rm E}<22$ detection in the redshift
range $7.0\leq z\leq 7.5$. For the reasons discussed above this number is
uncertain, however suggests that Euclid may uncover a population of high-
redshift obscured AGN that are half as numerous in the data as unobscured AGN.

##  Appendix E Comparison to SPRITZ FIR predictions

A complementary analysis to our work was carried out utilizing the _s_ pectro-
_p_ hotometric _r_ ealisations of _i_ nfrared-selected _t_ argets at all-_z_
(SPRITZ v 1.13; Bisigello et al. 2021, 2022) simulation. The simulation is
based on observed galaxy and AGN IR LFs and agrees with a wide set of
observables, including the total galaxy stellar mass function, LFs at several
wavelengths and number counts (from radio to X-rays), and relation between the
SFR and the stellar mass. Relevant to this work, SPRITZ incorporates two
populations of AGN. The first population contains systems that have their SEDs
dominated by radiation produced from AGN activity. These objects are split
into AGN1 (unobscured) and AGN2 (obscured) sub-populations based on their
optical extinction. These AGN are incorporated in the simulation via an IR LF
obtained with a simultaneous fit of the Herschel FIR (Gruppioni et al. 2013)
and UV observed LFs (Croom et al. 2009; McGreer et al. 2013; Ross et al. 2013;
Akiyama et al. 2018; Schindler et al. 2019), derived in Bisigello et al.
(2021). The second population, composite systems, have energetics dominated by
star-formation activity but have a sub-dominant AGN component. These objects
are again split into two sub-populations: star-forming AGN with an
intrinsically faint AGN contribution and starburst AGN that have a bright yet
heavily obscured ($\operatorname{log_{10}}[N_{\rm H}/{\rm cm}^{-2}]=\text{23.5
--24.5}$) AGN component. Composite systems are included in the simulation
based on the Herschel FIR observed LFs of Gruppioni et al. (2013) with some
updates reported in Bisigello et al. (in prep.).

The SPRITZ simulated EWS and EDS catalogs include sources with signal-to-noise
ratio (S/N) ¿ 3 in at least one Euclid band, considering the depths given in
Table 2. For the EWS the SPRITZ simulated catalog predicts 2.1 $\times$ 108
AGN in the redshift range $0.01\leq z\leq 7$, encompassing 2.9 $\times$ 107
AGN-dominated objects and 1.8 $\times$ 108 composite sources. Over the same
redshift range, the SPRITZ EDS catalog yields 1.1 $\times$ 106 AGN, comprised
of 1.7 $\times$ 105 AGN emission dominated objects and 9.2 $\times$ 105
composite objects.

When applying identical S/N criteria to our data we found 4.5 $\times$ 107 AGN
in the EWS and 2.6 $\times$ 105 AGN in the EDS. We observe in both surveys
that the total number of AGN predicted using SPRITZ are around five times
greater than those derived using our methodology. Considering separated
populations, our XLF-based analysis predicts numbers of AGN that are slightly
greater than those derived for AGN-dominated objects in the IR LF rooted
SPRITZ analysis. One reason for this could be that our XLF-based method allows
consistent analysis of the AGN-dominated systems and the more luminous star-
forming AGN composite objects in the SPRITZ simulation. The heavily obscured
starburst AGN composite systems however, have not been considered in our
analysis as they appear absorbed to Compton-thick in the X-ray sky. Indeed,
the entire population of starburst AGN composites in SPRITZ have
$\operatorname{log_{10}}(N_{\rm H}/{\rm cm}^{-2})>23$. From cosmic X-ray
background considerations Gilli et al. (2007) predict that Compton-thick
($\operatorname{log_{10}}[N_{\rm H}/{\rm cm}^{-2}]\geq 24$) AGN are at least
as numerous as Compton-thin AGN, and up to four times as abundant as
unobscured AGN with $\operatorname{log_{10}}(L_{\rm X}/{\rm erg}\,{\rm
s}^{-1})<43.5$. More extreme estimates postulate that Compton-thick AGN could
be up to ten times more abundant than unobscured AGN (Akylas et al. 2012).
Compton-thick AGN unobserved in the hard X-rays could therefore explain much
of the deficit between the two analyses.

The above argument is likely the case for some AGN in the SPRITZ samples,
however this is based on the assumption that the XLF and IR LFs probe the same
underlying population of AGN with selection effects. There is strong evidence
however that X-ray selected AGN and IR selected AGN probe separate, incomplete
samples of AGN, albeit with some overlap between the two (e.g., Mendez et al.
2013; Ji et al. 2022; Lyu et al. 2022). X-ray selection can identify AGN with
lower luminosities and accretion rates when compared to samples of IR selected
AGN. This allows X-ray selected samples to contain AGN where host galaxy light
dominates the MIR (e.g., Mendez et al. 2013; Chung et al. 2014; Menzel et al.
2016). Consistent analysis of the same fields have highlighted differences in
the derived IR and X-ray LFs (Runburg et al. 2022). It is therefore not
surprising that different numbers of AGN are derived using LFs from different
bands in these similar analyses because each offers a different, incomplete
window into the total census of AGN.

On top of these differences in observed AGN populations between the XLF and IR
LFs we must also consider that substantial uncertainties are introduced during
the redshift extrapolation of the Herschel FIR LF. As we explored in Sect.
5.1, the largest uncertainty introduced in our work stems from the poor high-
redshift constraints on the XLF. The lack of observable AGN at the faint-end
slope of the LF in the high-redshift regime is not a challenge unique to X-ray
frequencies. Indeed, the Gruppioni et al. (2013) Herschel FIR LF utilizes
relatively shallow FIR observations in its construction and is well
constrained up to $z\approx 3$. The SPRITZ implementation of this LF
extrapolates beyond $z=3$ with some modifications described in Bisigello et
al. (2021). During extrapolation the break luminosity is held constant and the
galaxy number density is decreased as $\phi^{*}\propto(1+z)^{k_{\phi}}$ where
$k_{\phi}\in[-4,-1]$. The extrapolated populations generated with SPRITZ are
shown to be consistent with a range of LFs in alternate wavebands up to $z=6$.
The XLFs predicted by SPRITZ however are in excess of observed XLFs when
considering only Compton-thin AGN (Bisigello et al. 2021). The uncertainties
associated with both the SPRITZ extrapolations and observed LFs at larger
redshifts are however considerable due to the lack of constraining
observations, just as in this work. Therefore, as we showed with our XLF, the
uncertainty on the derived number of AGN is extensive, particularly at high
redshift.

The main difference between these analyses is that the IR AGN LF probes an
overlapping but differing sample of AGN to the XLF. The nature of IR AGN
selection also allows more heavily obscured AGN to be considered that are
Compton-thick in the X-ray regime and therefore are not probed by the XLF
employed in this work. We assess that the numbers of detectable AGN presented
in this work may be a conservative estimate due to the selection effects
biased against heavily obscured and Compton-thick AGN in the X-ray regime.

##  Appendix F AGN colour selection criteria

Bisigello et al. (in prep.) derives a number of different photometric
selections for AGN in Euclid using Euclid, Rubin/LSST and Spitzer Space
Telescope/IRAC bands. In this section we applied each criterion to our AGN
sample and report the results. Throughout this section we use the nomenclature
of AND, OR corresponding to the logical AND, OR operators.

Density plots showing the Euclid colour AGN selection criteria discussed in
Sect. 5.2 applied to our samples of EWS and EDS AGN are shown in Figs. 20 and
21, respectively. In Fig. 20(a) and Fig. 21(a) we additionally plot trails of
the colour-space redshift evolution of our unobscured AGN template with
$E(B-V)=0$ and $E(B-V)=0.1$ for $z\in[0,7]$ in steps of $\delta z=1$. These
lines illustrate how the position of our AGN on the considered selection plots
are influenced by SED and source parameters. We observe that much of the
spread of our population in the top right portion of the diagram is occupied
by high-redshift sources on the right and more heavily optically obscured
sources on the left. Interestingly, we observe that the lowest redshift
unobscured AGN and mildly dust reddened unobscured AGN would not be selected
using these Euclid photometric criteria.

Table 9: Redshift-dependent performance of Euclid photometry AGN selection criteria (Bisigello et al., in prep.) applied to our samples of EWS and EDS AGN. For the Euclid-only unobscured AGN criterion and union ($\cup$) of all Euclid and Rubin/LSST selection criteria we report the number of selected AGN, completeness ($C$) at the 5$\sigma$ level in the relevant filters, and the median bolometric luminosity ($\pm 1\sigma$) in each redshift bin. These data are visualised in Fig. 13. Survey | Target Class | Photometry | Redshift | Selected AGN | $C$ | $\log_{10}(L_{\rm bol}/{\rm erg\,s}^{-1})$
---|---|---|---|---|---|---
EWS | Unobscured AGN | Euclid | $0.0\leq z<0.5$ | $2.1\times 10^{5}$ | 0.13 | $43.6\pm 0.6$
|  |  | $0.5\leq z<1.0$ | $7.3\times 10^{5}$ | 0.13 | $44.4\pm 0.7$
|  |  | $1.0\leq z<1.5$ | $1.2\times 10^{6}$ | 0.24 | $45.2\pm 0.6$
|  |  | $1.5\leq z<2.0$ | $1.2\times 10^{6}$ | 0.37 | $45.4\pm 0.5$
|  |  | $2.0\leq z<2.5$ | $7.2\times 10^{5}$ | 0.39 | $45.6\pm 0.5$
|  |  | $2.5\leq z<3.0$ | $3.9\times 10^{5}$ | 0.34 | $45.8\pm 0.4$
|  |  | $3.0\leq z<3.5$ | $2.1\times 10^{5}$ | 0.29 | $45.8\pm 0.4$
|  |  | $3.5\leq z<4.0$ | $9.9\times 10^{4}$ | 0.22 | $45.9\pm 0.4$
|  |  | $4.0\leq z<4.5$ | $3.6\times 10^{4}$ | 0.12 | $45.9\pm 0.3$
|  |  | $4.5\leq z<5.0$ | $2.0\times 10^{3}$ | 0.01 | $45.8\pm 0.2$
|  |  | $5.0\leq z<5.5$ | 3 | 0.00 | $45.7\pm 0.1$
EWS | $\cup$ |  Euclid, Rubin/LSST | $0.0\leq z<0.5$ | $4.3\times 10^{5}$ | 0.26 | $43.6\pm 0.6$
|  |  | $0.5\leq z<1.0$ | $1.7\times 10^{6}$ | 0.28 | $44.1\pm 0.7$
|  |  | $1.0\leq z<1.5$ | $2.1\times 10^{6}$ | 0.38 | $44.9\pm 0.6$
|  |  | $1.5\leq z<2.0$ | $2.0\times 10^{6}$ | 0.56 | $45.3\pm 0.5$
|  |  | $2.0\leq z<2.5$ | $1.1\times 10^{6}$ | 0.53 | $45.5\pm 0.5$
|  |  | $2.5\leq z<3.0$ | $5.2\times 10^{5}$ | 0.44 | $45.7\pm 0.4$
|  |  | $3.0\leq z<3.5$ | $2.1\times 10^{5}$ | 0.29 | $45.8\pm 0.4$
|  |  | $3.5\leq z<4.0$ | $9.9\times 10^{4}$ | 0.22 | $45.9\pm 0.4$
|  |  | $4.0\leq z<4.5$ | $3.6\times 10^{4}$ | 0.12 | $45.9\pm 0.3$
|  |  | $4.5\leq z<5.0$ | $2.0\times 10^{3}$ | 0.01 | $45.8\pm 0.2$
|  |  | $5.0\leq z<5.5$ | 3 | 0.00 | $45.7\pm 0.1$
EDS | Unobscured AGN | Euclid | $0.0\leq z<0.5$ | $3.0\times 10^{2}$ | 0.04 | $43.5\pm 0.5$
|  |  | $0.5\leq z<1.0$ | $2.4\times 10^{3}$ | 0.06 | $44.0\pm 0.7$
|  |  | $1.0\leq z<1.5$ | $5.5\times 10^{3}$ | 0.13 | $44.7\pm 0.7$
|  |  | $1.5\leq z<2.0$ | $4.5\times 10^{3}$ | 0.17 | $45.2\pm 0.7$
|  |  | $2.0\leq z<2.5$ | $2.5\times 10^{3}$ | 0.17 | $45.4\pm 0.6$
|  |  | $2.5\leq z<3.0$ | $1.2\times 10^{3}$ | 0.14 | $45.6\pm 0.4$
|  |  | $3.0\leq z<3.5$ | $6.5\times 10^{2}$ | 0.13 | $45.6\pm 0.4$
|  |  | $3.5\leq z<4.0$ | $2.2\times 10^{2}$ | 0.08 | $45.6\pm 0.4$
|  |  | $4.0\leq z<4.5$ | 78 | 0.04 | $45.9\pm 0.3$
EDS | $\cup$ |  Euclid, Rubin/LSST | $0.0\leq z<0.5$ | $1.5\times 10^{3}$ | 0.23 | $43.5\pm 0.5$
|  |  | $0.5\leq z<1.0$ | $6.2\times 10^{3}$ | 0.15 | $43.9\pm 0.7$
|  |  | $1.0\leq z<1.5$ | $7.5\times 10^{3}$ | 0.18 | $44.7\pm 0.7$
|  |  | $1.5\leq z<2.0$ | $9.4\times 10^{3}$ | 0.35 | $45.0\pm 0.6$
|  |  | $2.0\leq z<2.5$ | $6.4\times 10^{3}$ | 0.43 | $45.2\pm 0.6$
|  |  | $2.5\leq z<3.0$ | $2.6\times 10^{3}$ | 0.30 | $45.4\pm 0.5$
|  |  | $3.0\leq z<3.5$ | $6.8\times 10^{2}$ | 0.14 | $45.6\pm 0.4$
|  |  | $3.5\leq z<4.0$ | $2.2\times 10^{2}$ | 0.08 | $45.6\pm 0.4$
|  |  | $4.0\leq z<4.5$ | 78 | 0.04 | $45.9\pm 0.3$

In the EWS the optimal selection criteria for unobscured AGN exploiting only
Euclid photometry obeys

$\displaystyle I_{\scriptscriptstyle\rm E}-Y_{\scriptscriptstyle\rm E}<0.5\ $
(18) $\displaystyle{\tt AND}\ I_{\scriptscriptstyle\rm
E}-J_{\scriptscriptstyle\rm E}<0.7\ $ $\displaystyle{\tt AND}\
I_{\scriptscriptstyle\rm E}-J_{\scriptscriptstyle\rm
E}<-2.1\,(I_{\scriptscriptstyle\rm E}-Y_{\scriptscriptstyle\rm E})+0.9,$

and is expected to achieve F1 $\sim$ 0.2 with $P=0.2$. Applied to our Euclid
detectable AGN we selected 4.8 $\times$ 106 AGN, corresponding to a selected
AGN surface density of 331 deg-2. We select 92% unobscured and 8% obscured AGN
SEDs with this selection. An overall completeness $C=0.23$ results for our
full candidate sample. Considering only unobscured AGN, the target class of
the selection, the completeness rises to $C=0.52$.

The colour selection performance of unobscured AGN in the EWS, and indeed for
all AGN colour selections in the EWS and EDS, is improved with the addition of
Rubin/LSST optical bands. The optimal selection criteria for unobscured AGN in
the EWS including Rubin/LSST is given by

$\displaystyle I_{\scriptscriptstyle\rm E}-H_{\scriptscriptstyle\rm E}<1.2\ $
(19) $\displaystyle{\tt AND}\ u-z<1.1\ $ $\displaystyle{\tt AND}\
I_{\scriptscriptstyle\rm E}-H_{\scriptscriptstyle\rm E}<-1.3\,(u-z)+1.9,$

where $u$ and $z$ correspond to the Rubin/LSST optical filters. The criterion
is expected to achieve F1 = 0.9 with $P=0.9$. Implemented on our sample this
selection yields 5.7 $\times$ 106 selected AGN, which corresponds to a surface
density of 393 deg-2. With this criterion our resulting selected sample
contained 97% unobscured SEDs and 3% obscured AGN SEDs. We derived an overall
completeness of $C=0.45$, with a completeness $C=0.75$ when considering
unobscured AGN only.

In the EWS the selection of all AGN, i.e. obscured, unobscured and composite
systems utilizing Euclid photometry only is disregarded due to the difficulty
of separating AGN powered sources from contaminants with the limited optical
and NIR filters available. This selection task is improved with the addition
of optical Rubin/LSST bands. The optimal criterion with such external data
achieves F1 = 0.3 with $P=0.2$ and is formulated as

$\displaystyle u-r<0.2\ $ (20) $\displaystyle{\tt OR}\
I_{\scriptscriptstyle\rm E}-Y_{\scriptscriptstyle\rm E}<-0.9\,(u-r)+0.8,$

where $u$ and $r$ are the Rubin/LSST optical filters. We note that this
selection is for objects that fall on the outside of the defined boundary in
colour space (referred to as “type-B” in Bisigello et al., in-prep). Applying
this colour selection to our data we selected a sample of 6.0 $\times$ 106
AGN, corresponding to a surface density of 413 deg-2. Our selected sample with
this criterion comprised 72% unobscured and 28% obscured AGN SEDs. As raised
in Bisigello et al. (in prep.), we note that this colour selection has a low
sample purity and therefore is expected to select a large fraction of
contaminant inactive galaxies. We found the overall completeness of this
selected sample is $C=0.51$. For unobscured (obscured) AGN populations the
completeness achieved is $C=0.65\;(0.33)$.

The optimal colour criteria for AGN selection presented in Bisigello et al.
(in prep.) differ between the EWS and EDS. Different criteria are required
given that the two surveys probe different parts of the AGN LF. For unobscured
AGN in the EDS the best selection criteria using Euclid photometry alone, with
F1 = 0.2 and $P=0.2$, is given by

$\displaystyle I_{\scriptscriptstyle\rm E}-Y_{\scriptscriptstyle\rm E}<0.3\ $
(21) $\displaystyle{\tt AND}\ I_{\scriptscriptstyle\rm
E}-H_{\scriptscriptstyle\rm E}<0.5\ $ $\displaystyle{\tt AND}\
I_{\scriptscriptstyle\rm E}-H_{\scriptscriptstyle\rm
E}<-1.6\,(I_{\scriptscriptstyle\rm E}-Y_{\scriptscriptstyle\rm E})+0.8.$

This selection criterion yields 1.7 $\times$ 104 AGN at a surface density of
346 deg-2 when applied to our sample. Our selected sample contained 97%
unobscured SEDs and 3% obscured AGN SEDs. The completeness derived for this
selection is $C=0.11$ for the overall population and $C=0.40$ considering the
target class of unobscured AGN only. We note that this selection is expected
to be particularly contaminated by dwarf irregular galaxies in practice.

Adding Rubin/LSST bands raises the quality of unobscured AGN selection in the
EDS to an expected F1 = 0.8 with $P=0.9$. The criterion in this case is

$\displaystyle I_{\scriptscriptstyle\rm E}-H_{\scriptscriptstyle\rm E}<1.1\ $
(22) $\displaystyle{\tt AND}\ u-z<1.2\ $ $\displaystyle{\tt AND}\
I_{\scriptscriptstyle\rm E}-H_{\scriptscriptstyle\rm E}<-1.2\,(u-z)+1.7,$

where $u$ and $z$ are the Rubin/LSST optical filters. Using this selection on
our Euclid detectable sample we selected 2.0 $\times$ 104 AGN, with a
corresponding surface density of 392 deg-2. Our selected sample in this case
contained 98% unobscured and 2% obscured AGN SEDs. The completeness achieved
with this selection is $C=0.45$ overall and $C=0.76$ when only unobscured AGN
are considered.

As for the all AGN selection with only Euclid photometry in the EWS, the EDS
counterpart is also disregarded due to the difficulty of separating a general
active galaxy population from inactive galaxies. Adding ancillary Rubin/LSST
bands for the selection of all AGN in the EDS improves the performance of the
optimal colour selection to have an expectation of F1 = 0.3 and $P=0.6$. The
optimal criterion in this case follows

$\displaystyle I_{\scriptscriptstyle\rm E}-Y_{\scriptscriptstyle\rm E}<1.7\ $
(23) $\displaystyle{\tt AND}\ g-r<0.3\ $ $\displaystyle{\tt AND}\
I_{\scriptscriptstyle\rm E}-Y_{\scriptscriptstyle\rm E}<-3.5\,(g-r)+0.9,$

where $g$ and $r$ are the Rubin/LSST optical filters. Applied to our sample
this criterion selects 2.9 $\times$ 104 of our AGN, providing a surface
density of 579 deg-2. Our resulting selected sample was comprised of 67%
unobscured SEDs and 33% obscured AGN SEDs. We derived an overall completeness
$C=0.32$ for this selection. Considering separated AGN classes we found a
completeness of $C=0.51$ for unobscured and $C=0.18$ for obscured AGN.

(a) EWS unobscured AGN Euclid Selection

(b) EWS unobscured AGN Euclid \+ Rubin/LSST Selection

(c) EWS all AGN Euclid \+ Rubin/LSST Selection

Figure 20: Optimal Euclid photometry AGN selection criteria derived in
Bisigello et al. (in prep.) for the EWS applied to our data. In each case we
show two-dimensional density plots for AGN that are detected above the
5$\sigma$ point-source depths in all four relevant bands. The grey points show
the distribution of AGN that are not selected, while the coloured regions show
the selected AGN. The sub-figures correspond to optimal selection criteria
for: (a) Unobscured AGN with Euclid photometry only, (b) Unobscured AGN with
Euclid and Rubin/LSST photometry and (c) All AGN with Euclid and Rubin/LSST
photometry. In panel (a) we plot the colour-space redshift evolution of our
unobscured AGN template with $E(B-V)=0$ (green) and $E(B-V)=0.1$ (orange) for
$z\in[0,7]$ in steps of $\delta z=1$. White circles show the $z=0$ points and
white squares denote the $z=5$ points. The $E(B-V)=0$ green crosses in the
selection region of (a) correspond to $z=1$ and $z=2$.

(a) EDS unobscured AGN Euclid Selection

(b) EDS unobscured AGN Euclid \+ Rubin/LSST Selection

(c) EDS all AGN Euclid \+ Rubin/LSST Selection

Figure 21: Same as Fig. 20 for the EDS.