Abstract
X-ray reflection spectroscopy can be a powerful tool to test the nature of astrophysical black holes. Extending previous work on Kerr black holes with scalar hair [1] and on boson stars [2], here we study whether astrophysical black hole candidates may be horizonless, self-gravitating, vector Bose-Einstein condensates, known as Proca stars [3]. We find that observations with current X-ray missions can only provide weak constraints and rule out solely Proca stars with low compactness. There are two reasons. First, at the moment we do not know the geometry of the corona, and therefore the uncertainty in the emissivity profile limits the ability to constrain the background metric. Second, the photon number count is low even in the case of a bright black hole binary, and we cannot have a precise measurement of the spectrum.