We study the effects of WIMP dark matter (DM) on the collapse and evolution of the first stars in the Universe. Using a stellar evolution code, we follow the pre-Main Sequence (MS) phase of a grid of metal-free stars with masses in the range 5-600 solar mass forming in the centre of a 1e6 solar mass halo at redhisft z=20. DM particles of the parent halo are accreted in the proto-stellar interior by adiabatic contraction and scattering/capture processes, reaching central densities of order 1e12 GeV/cm3 at radii of the order of 10 AU. Energy release from annihilation reactions can effectively counteract the gravitational collapse, in agreement with results from other groups. We find this stalling phase (known as dark star) is transients and lasts from 2.1e3 yr (M=600 solar mass) to 1.8e4 yr (M=9 solar mass). Later in the evolution, DM scattering/capture rate becomes high enough that energy deposition from annihilations significantly alters the pre-MS evolution of the star in a way that depends on DM (i) velocity dispersion, (ii) density, (iii) elastic scattering cross section with baryons. For our fiducial set of parameters (10 km/s, 1e11 GeV/cm3, 1e-38 cm2) we find that the evolution of stars of mass lower than 40 solar masses freezes on the HR diagram before reaching the ZAMS. Stars with bigger masses manage to ignite nuclear reactions; however, DM burning prolonges their lifetimes by a factor 2 (5) for a 600 (40) solar mass star.
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