Abstract: In the black hole (BH)-galaxy co-evolution framework, most of the star formation (SF) and the BH accretion are expected to take place in highly obscured conditions. The large amount of gas and dust absorbs most of the UV-to-soft-X radiation and re-emits it at longer wavelengths, mostly in the IR. Thus, obscured active galactic nuclei (AGN) are very difficult to identify in optical or X-ray bands but shine bright in the IR. Moreover, X-ray background (XRB) synthesis models predict that a large fraction of the yet-unresolved XRB is due to the most obscured (Compton thick, CT: NH ? 1024 cm?2) of these AGN. In this work, we investigate the synergies between putative IR missions [using SPace Infrared telescope for Cosmology and Astrophysics (SPICA), proposed for European Space Agency (ESA)/M5 but withdrawn in 2020 October, and Origins Space Telescope, OST, as 'templates'] and the X-ray mission Athena (Advanced Telescope for High ENergy Astrophysics), which should fly in early 2030s, in detecting and characterising AGN, with a particular focus on the most obscured ones. Using an XRB synthesis model, we estimated the number of AGN and the number of those which will be detected in the X-rays by Athena. For each AGN, we associated an optical-to-Far InfraRed (FIR) spectral energy distribution (SED) from observed AGN with both X-ray data and SED decomposition and used these SEDs to check if the AGN will be detected by SPICA-like or OST at IR wavelengths. We expect that, with the deepest Athena and SPICA-like (or OST) surveys, we will be able to photometrically detect in the IR more than 90% of all the AGN (down to L2?10keV ? 1042 erg s?1 and up to z ? 10) predicted by XRB synthesis modeling, and we will detect at least half of them in the X-rays. The spectroscopic capabilities of the OST can provide ?51 000 and ?3 400 AGN spectra with R = 300 at 25-588 µm in the wide and deep surveys, respectively, the last one up to z ? 4. Athena will be extremely powerful in detecting and discerning moderate- and high-luminosity AGN, allowing us to properly select AGN even when the mid-IR torus emission is 'hidden' by the host galaxy contribution. We will constrain the intrinsic luminosity and the amount of obscuration for ?20% of all the AGN (and ?50% of those with L2?10keV > 3.2 × 1043 erg s?1) using the X-ray spectra provided by Athena WFI. We find that the most obscured and elusive CT-AGN will be exquisitely sampled by SPICA-like mission or OST and that Athena will allow a fine characterisation of the most luminous ones. This will provide a significant step forward in the process of placing stronger constraints on the yet-unresolved XRB and investigating the BH accretion rate evolution up to very high redshift (z ? 4).

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