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Context. The formation and evolution of galaxies appear linked to the growthof supermassive black holes, as evidenced by empirical scaling relations innearby galaxies. Understanding this co-evolution over cosmic time requires therevelation of the dynamical state of galaxies and the measurement of the massof their central black holes (MBH) at a range of cosmic distances. Brightactive galactic nuclei (AGNs) are ideal for this purpose.  Aims. The High Angular Resolution Monolithic Optical and Near-infraredIntegral field spectrograph (HARMONI), the first light integral-fieldspectrograph for the Extremely Large Telescope (ELT), will transform visibleand near-infrared ground-based astrophysics thanks to its advances insensitivity and angular resolution. We aim to analyse the capabilities ofHARMONI to reveal the stellar morpho-kinematic properties of the host galaxiesof AGNs at about cosmic noon.  Methods. We made use of the simulation pipeline for HARMONI (HSIM) to createmock observations of representative AGN host galaxies at redshifts aroundcosmic noon. We used observations taken with the Multi-Unit SpectroscopicExplorer (MUSE) of nearby galaxies showing different morphologies and dynamicalstages combined with theoretical AGN spectra to create the target inputs forHSIM.  Results. According to our simulations, an on-source integration time of threehours should be enough to measure the MBH and to trace the morphology andstellar kinematics of the brightest host galaxies of AGNs beyond cosmic noon.For host galaxies with stellar masses $<$10$^{11}$ solar masses, longerexposure times are mandatory to spatially resolve the stellar kinematics.

astronomy and astrophysics

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