Euclid preparation. Observational expectations for redshift z<7 active galactic nuclei in the Euclid Wide and Deep surveys

We forecast the expected population of active galactic nuclei (AGN) observable in the Euclid Wide Survey (EWS) and Euclid Deep Survey (EDS). Starting from an X-ray luminosity function (XLF) we generate volume-limited samples of the AGN expected in the survey footprints. Each AGN is assigned an SED appropriate for its X-ray luminosity and redshift, with perturbations sampled from empirical distributions. The photometric detectability of each AGN is assessed via mock observation of the assigned SED. We estimate 40 million AGN will be detectable in at least one band in the EWS and 0.24 million in the EDS, corresponding to surface densities of 2.8$\\times$10$^{3}$ deg$^{-2}$ and 4.7$\\times$10$^{3}$ deg$^{-2}$. Employing colour selection criteria on our simulated data we select a sample of 4.8$\\times$10$^{6}$ (331 deg$^{-2}$) AGN in the EWS and 1.7$\\times$10$^{4}$ (346 deg$^{-2}$) in the EDS, amounting to 10% and 8% of the AGN detectable in the EWS and EDS. Including ancillary Rubin/LSST bands improves the completeness and purity of AGN selection. These data roughly double the total number of selected AGN to comprise 21% and 15% of the detectable AGN in the EWS and EDS. The total expected sample of colour-selected AGN contains 6.0$\\times$10$^{6}$ (74%) unobscured AGN and 2.1$\\times$10$^{6}$ (26%) obscured AGN, covering $0.02 \\leq z \\lesssim 5.2$ and $43 \\leq \\log_{10} (L_{bol} / erg s^{-1}) \\leq 47$. With this simple colour selection, expected surface densities are already comparable to the yield of modern X-ray and mid-infrared surveys of similar area. The relative uncertainty on our expectation for detectable AGN is 6.7% for the EWS and 12.5% for the EDS, driven by the uncertainty of the XLF.