We model the X-ray surface brightness distribution of emission associated with Fanaroff & Riley type-II radio galaxies. Our approach builds on the RAiSE dynamical model which describes broadband radio-frequency synchrotron evolution of jet-inflated lobes in a wide range of environments. The X-ray version of the model presented here includes: (1) inverse-Compton upscattering of cosmic microwave background radiation; (2) the dynamics of the shocked gas shell and associated bremsstrahlung radiation; and (3) emission from the surrounding ambient medium. We construct X-ray surface brightness maps for a mock catalogue of extended FR-IIs based on the technical characteristics of the eRosita telescope. The integrated X-ray luminosity function at low redshifts ($zleqslant1$) is found to strongly correlate with the density of the ambient medium in all but the most energetic sources, whilst at high-redshift ($z>1$) the majority of objects are dominated by inverse-Compton lobe emission due to the stronger cosmic microwave background radiation. By inspecting our mock spatial brightness distributions, we conclude that any extended X-ray detection can be attributed to AGN activity at redshifts $zgeqslant1$. We compare the expected detection rates of active and remnant high-redshift radio AGNs for eRosita and LOFAR, and future more sensitive surveys. We find that a factor of ten more remnants can be detected using X-ray wavelengths over radio frequencies at $z>2.2$, increasing to a factor of 100 for redshifts $z>3.1$.
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