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The hard X-ray emission in active galactic nuclei (AGN) and black hole X-ray binaries is thought to be produced by a hot cloud of electrons referred to as the corona. This emission, commonly described by a power law with a high-energy cutoff, is suggestive of Comptonization by thermal electrons. While several hypotheses have been proposed to explain the origin, geometry, and composition of the corona, we still lack a clear understanding of this fundamental component. NuSTAR has been playing a key role improving our knowledge of X-ray coronæ thanks to its unprecedented sensitivity above 10 keV. However, these constraints are limited to bright, nearby sources. The High Energy X-ray Probe (HEX-P) is a probe-class mission concept combining high spatial resolution X-ray imaging and broad spectral coverage (0.2–80 keV) with a sensitivity superior to current facilities. In this paper, we highlight the major role that HEX-P will play in further advancing our insights of X-ray coronæ notably in AGN. We demonstrate how HEX-P will measure key properties and track the temporal evolution of coronæ in unobscured AGN. This will allow us to determine their electron distribution and test the dominant emission mechanisms. Furthermore, we show how HEX-P will accurately estimate the coronal properties of obscured AGN in the local Universe, helping address fundamental questions about AGN unification. In addition, HEX-P will characterize coronæ in a large sample of luminous quasars at cosmological redshifts for the first time and track the evolution of coronæ in transient systems in real time. We also demonstrate how HEX-P will enable estimating the coronal geometry using spectral-timing techniques. HEX-P will thus be essential to understand the evolution and growth of black holes over a broad range of mass, distance, and luminosity, and will help uncover the black holes’ role in shaping the Universe.

Introduction: Ultraluminous X-ray sources (ULXs) represent an extreme class of accreting compact objects: from the identification of some of the accretors as neutron stars to the detection of powerful winds travelling at 0.1–0.2 c, the increasing evidence points towards ULXs harbouring stellar-mass compact objects undergoing highly super-Eddington accretion. Measuring their intrinsic properties, such as the accretion rate onto the compact object, the outflow rate, the masses of accretor/companion-hence their progenitors, lifetimes, and future evolution-is challenging due to ULXs being mostly extragalactic and in crowded fields. Yet ULXs represent our best opportunity to understand super-Eddington accretion physics and the paths through binary evolution to eventual double compact object binaries and gravitational-wave sources. Methods: Through a combination of end-to-end and single-source simulations, we investigate the ability of HEX-P to study ULXs in the context of their host galaxies and compare it to XMM-Newton and NuSTAR, the current instruments with the most similar capabilities.Results: HEX-P’s higher sensitivity, which is driven by its narrow point-spread function and low background, allows it to detect pulsations and broad spectral features from ULXs better than XMM-Newton and NuSTAR.Discussion: We describe the value of HEX-P in understanding ULXs and their associated key physics, through a combination of broadband sensitivity, timing resolution, and angular resolution, which make the mission ideal for pulsation detection and low-background, broadband spectral studies.