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The hard X-ray emission in active galactic nuclei (AGN) and black hole X-raybinaries is thought to be produced by a hot cloud of electrons referred to asthe corona. This emission, commonly described by a power law with a high-energycutoff, is suggestive of Comptonization by thermal electrons. While severalhypotheses have been proposed to explain the origin, geometry, and compositionof the corona, we still lack a clear understanding of this fundamentalcomponent. NuSTAR has been playing a key role improving our knowledge of X-raycoronae thanks to its unprecedented sensitivity above 10 keV. However, theseconstraints are limited to bright, nearby sources. The High Energy X-ray Probe(HEX-P) is a probe-class mission concept combining high spatial resolutionX-ray imaging and broad spectral coverage (0.2-80 keV) with a sensitivitysuperior to current facilities. In this paper, we highlight the major role thatHEX-P will play in further advancing our insights of X-ray coronae, notably inAGN. We demonstrate how HEX-P will measure key properties and track thetemporal evolution of coronae in unobscured AGN. This will allow us todetermine their electron distribution and test the dominant emissionmechanisms. Furthermore, we show how HEX-P will accurately estimate the coronalproperties of obscured AGN in the local Universe, helping address fundamentalquestions about AGN unification. In addition, HEX-P will characterize coronaein a large sample of luminous quasars at cosmological redshifts for the firsttime and track the evolution of coronae in transient systems in real time. Wealso demonstrate how HEX-P will enable estimating the coronal geometry usingspectral-timing techniques. HEX-P will thus be essential to understand theevolution and growth of black holes over a broad range of mass, distance, andluminosity, and will help uncover the black holes' role in shaping theUniverse.

The High Energy X-ray Probe (HEX-P): Probing the physics of the X-ray corona in active galactic nuclei