The role of SPICA-like missions and the Origins Space Telescope in the
  quest for heavily obscured AGN and synergies with Athena
  In the BH-galaxy co-evolution framework, most of the star-formation (SF) and
the black hole (BH) accretion is expected to take place in highly obscured
conditions. Thus, obscured AGN are 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) of these AGN. In this work, we investigate
the synergies between putative IR missions (using SPICA, proposed for ESA/M5
but withdrawn in October 2020, and Origins Space Telescope, OST, as
`templates') and the X-ray mission Athena, which should fly in early 2030s, in
detecting and characterizing 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. For each AGN we
associated an optical-to-FIR 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 detect in the IR more than
$90\,\%$ of all the AGN (down to L$_{2-10\text{keV}} \sim 10^{42}\,$erg/s and
up to $z \sim 10$) predicted by XRB synthesis modeling, and we will detect at
least half of them in the X-rays. Athena will be extremely powerful in
detecting and discerning moderate- and high-luminosity AGN. 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 characterization 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 ($z \ge 4$).
