HARMONI view of the host galaxies of active galactic nuclei around cosmic noon: Resolved stellar morpho-kinematics and the M\(_{BH}\)-\(\sigma_{\star}\) relation

Context. The formation and evolution of galaxies appear linked to the growth of supermassive black holes, as evidenced by empirical scaling relations in nearby galaxies. Understanding this co-evolution over cosmic time requires the revelation of the dynamical state of galaxies and the measurement of the mass of their central black holes (MBH) at a range of cosmic distances. Bright active galactic nuclei (AGNs) are ideal for this purpose. Aims. The High Angular Resolution Monolithic Optical and Near-infrared Integral field spectrograph (HARMONI), the first light integral-field spectrograph for the Extremely Large Telescope (ELT), will transform visible and near-infrared ground-based astrophysics thanks to its advances in sensitivity and angular resolution. We aim to analyse the capabilities of HARMONI to reveal the stellar morpho-kinematic properties of the host galaxies of AGNs at about cosmic noon. Methods. We made use of the simulation pipeline for HARMONI (HSIM) to create mock observations of representative AGN host galaxies at redshifts around cosmic noon. We used observations taken with the Multi-Unit Spectroscopic Explorer (MUSE) of nearby galaxies showing different morphologies and dynamical stages combined with theoretical AGN spectra to create the target inputs for HSIM. Results. According to our simulations, an on-source integration time of three hours should be enough to measure the MBH and to trace the morphology and stellar kinematics of the brightest host galaxies of AGNs beyond cosmic noon. For host galaxies with stellar masses \(