About the accuracy of the relxill/relxill_nk models in view of the next generation of X-ray missions

X-ray reflection spectroscopy is a powerful tool to study the strong gravity region of black holes. The next generation of astrophysical X-ray missions promises to provide unprecedented high-quality data, which could permit us to get very precise measurements of the properties of the accretion flow and of the space-time geometry in the strong gravity region around these objects. In this work, we test the accuracy of the relativistic calculations of the reflection model relxill and of its extension to non-Kerr space-times relxill_nk in view of the next generation of X-ray missions. We simulate simultaneous observations with Athena/X-IFU and LAD of bright Galactic black holes with a precise and accurate ray-tracing code and we fit the simulated data with the latest versions of relline and relline_nk. While we always recover the correct input parameters, we find residuals in the fits when the emission from the inner part of the accretion disc is higher. Such residuals disappear if we increase the number of interpolation points on the disc in the integral of the transfer function. We also simulate full reflection spectra and find that the emission angle from the accretion disc should be treated properly in this case.