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

Population III Star Formation in an X-ray background: I. Critical Halo Mass of Formation and Total Mass in Stars