We investigate the effects of weakly-interacting massive particle (WIMP) dark matter annihilation on the formation of Population III.1 stars, which are theorized to form from the collapse of gas cores at the centers of dark matter minihalos. We consider the relative importance of cooling due to baryonic radiative processes and heating due to WIMP annihilation. We analyze the dark matter and gas profiles of several halos formed in cosmological-scale numerical simulations. The heating rate depends sensitively on the dark matter density profile, which we approximate with a power law rho_chi ~ r^{-alpha_chi}, in the numerically unresolved inner regions of the halo. If we assume a self-similar structure so that alpha_chi ~= 1.5 as measured on the resolved scales ~1pc, then for a fiducial WIMP mass of 100GeV, the heating rate is typically much smaller (<10^{-3}) than the cooling rate for densities up to n_H=10^{17}cm^{-3}. In one case, where alpha_chi=1.65, the heating rate becomes similar to the cooling rate by a density of n_H=10^{15}cm^{-3}. The dark matter density profile is expected to steepen in the central baryon-dominated region <~1pc due to adiabatic contraction, and we observe this effect (though with relatively low resolution) in our numerical models. From these we estimate alpha_chi~=2.0. The heating now dominates cooling above n_H~=10^{14}cm^{-3}, in agreement with the previous study of Spolyar, Freese & Gondolo. We expect this leads to the formation of an equilibrium structure with a baryonic and dark matter density distribution exhibiting a flattened central core. Examining such equilibria, we find total luminosities due to WIMP annihilation are relatively constant and ~10^3 L_sun, set by the radiative luminosity of the baryonic core. We discuss the implications for Pop III.1 star formation... (abridged)
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