The first luminous objects forming in the universe produce radiation backgrounds in the FUV and X-ray bands that affect the formation of Population III stars. Using a grid of cosmological hydrodynamics zoom-in simulations, we explore the impact of the Lyman-Warner (LW) and X-ray radiation backgrounds on the critical dark matter halo mass for Population III star formation and the total mass in stars per halo. We find that the LW radiation background lowers the H$_2$ fraction and delays the formation of the Population III stars. On the other hand, X-ray irradiation anticipates the redshift of collapse and reduces the critical halo mass, unless the X-ray background is too strong and gas heating shuts down gas collapse into the halos and prevents star formation. Therefore, an X-ray background can increase the number of dark matter halos forming Population III stars by about a factor of ten, but the total mass in stars forming in each halo is reduced. This is because X-ray radiation increases the molecular fraction and lowers the minimum temperature of the collapsing gas (or equivalently the mass of the quasi-hydrostatic core) and therefore slows down the accretion of the gas onto the central protostar.
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