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 $<$10$^{11}$ solar masses, longer
exposure times are mandatory to spatially resolve the stellar kinematics.
