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 targeting obscured, unobscured, and composite systems utilising 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

$$ \begin{array}{l} u - r < 0.2 \\ \text{OR } I_E - Y_E < -0.9(u - r) + 0.8, \end{array} \qquad (E.3) $$

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 Euclid Collaboration: Bisigello et al. 2024). Applying this colour selection to our data we selected a sample of 6.0 × 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 Euclid Collaboration: Bisigello et al. (2024), 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 Euclid Collaboration: Bisigello et al. (2024) 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

$$ \begin{array}{l} I_E - Y_E < 0.3 \\ \text{AND } I_E - H_E < 0.5 \\ \text{AND } I_E - H_E < -1.6(I_E - Y_E) + 0.8. \end{array} \qquad (E.4) $$

This selection criterion yields 1.7 × 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 practise.

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

$$ \begin{array}{l} I_E - H_E < 1.1 \\ \text{AND } u - z < 1.2 \\ \text{AND } I_E - H_E < -1.2(u - z) + 1.7, \end{array} \qquad (E.5) $$

where u and z are the Rubin/LSST optical filters. Using this selection on our Euclid detectable sample we selected 2.0 × 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

<figure><img src="image_21.png" /><figcaption>Fig. E.1. Optimal Euclid photometry AGN selection criteria derived in Euclid Collaboration: Bisigello et al. (2024) for the EWS applied to our data. We show two-dimensional density plots for AGN that are detected above the 5σ point-source depths in all four relevant bands. The coloured regions show the selected AGN and grey regions show non-selected AGN. Panels show selections for unobscured AGN with Euclid photometry only (top), unobscured AGN with Euclid and Rubin/LSST photometry (middle) and all AGN with Euclid and Rubin/LSST photometry (bottom). In the top panel 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 ∈ [0, 7] in steps of δz = 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 correspond to z = 1 and z = 2.</figcaption></figure>
<figure><img src="image_22.png" /><figcaption>Fig. E.2. Same as Fig. E.1 for the EDS.</figcaption></figure>

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

$$ 
\begin{array}{l}
I_E - Y_E < 1.7 \\
\text{AND } g - r < 0.3 \\
\text{AND } I_E - Y_E < -3.5(g - r) + 0.9,
\end{array}
\qquad (E.6) $$

where $g$ and $r$ are the Rubin/LSST optical filters. Applied to our sample this criterion selects $2.9 \times 10^4$ 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.

## Appendix F: High-redshift predictions

The predicted yield of $7 \le z \le 9$ quasars in the EWS was explored in Euclid Collaboration: Barnet et al. (2019). Quasars were incorporated in their work utilising the Jiang et al. (2016) high-redshift quasar LF with two different assumed rates of decline (modest and steep) in space density at $z \ge 6$. 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 $7 \le z \le 9$ in the EWS. Over the full considered redshift range quasars were successfully selected to the effective depth $J_E \sim 22$ when using only Euclid bands. Selection with a modest (steep) quasar LF decline predicted 87 (51) quasars in the redshift range $7 \le z \le 7.5$. This corresponds to a quasar surface density of $6.0 \times 10^{-3} \text{ deg}^{-2} (3.5 \times 10^{-3} \text{ deg}^{-2})$.

For comparison with the predicted EWS quasar yields of Euclid Collaboration: Barnet 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 $J_E < 22$, an unobscured optical classification, and occupation of the redshift range $7.0 \le z \le 7.5$. Applying these constraints we found 1053 unobscured AGN, equal to a surface density of $0.07 \text{ deg}^{-2}$, 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: Barnet 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

$$ i - z > 2.2, \qquad (F.1) $$

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

$$ z - J < 0.5(i - z) + 0.5, \qquad (F.2) $$

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 $7.0 \le z \le 7.5$ unobscured AGN sample with $J_E < 22$, substituting the the UKIDSS $J$ band for the Euclid $J_E$ band. With these additional constraints we recover 668 detectable quasars, which is a factor of eight higher than forecast in Euclid Collaboration: Barnet et al. (2019).
High-redshift quasar yield predictions are especially sensi-
tive 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
$z \ge 6$ 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 appar-
ent excess of quasars we predict at $7 \le z \le 7.5$ compared to the
Euclid Collaboration: Barnet et al. (2019) analysis.

Equally as impactful as the assumed LF are the integration
range and selection parameters of high-redshift quasar yield pre-
dictions. Schindler et al. (2023) make $z > 7$ EWS quasar predic-
tions also utilising the $k = -0.7$ Jiang et al. (2016) quasar LF.
Integrating to $M_{1450} \le -22.4$ and using $H_E < 24$ as the effec-
tive survey depth, they predicted 809 detectable quasars in the
EWS at $7 \le z \le 8$. This sample is of comparable size to our
$7 \le z \le 7.5$ simulated detectable sample of 1053, which incor-
porates unobscured AGN to $M_{1450} \sim -22.6$, albeit over a larger
redshift slice.

At present there are only 532 quasars confirmed at $z \ge 5.3$
and 275 quasars known at $z \ge 6$ (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 ~600 obscured AGN will have a
$J_E < 22$ detection in the redshift range $7.0 \le z \le 7.5$. For the rea-
sons 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.