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It has been proposed that during the formation of the first generation starsthere might be a "dark star" phase in which the power of the star comes fromdark matter annihilation. The adiabatic contraction process to form the darkstar would result in a highly concentrated density profile of the host halo atthe same time, which may give enhanced indirect detection signals of darkmatter. In this work we investigate the extragalactic $\gamma$-ray backgroundfrom dark matter annihilation with such a dark star formation scenario, andemploy the isotropic $\gamma$-ray data from Fermi-LAT to constrain the modelparameters of dark matter. The results suffer from large uncertainties of boththe formation rate of the first generation stars and the subsequent evolutioneffects of the host halos of the dark stars. We find, in the most optimisticcase for $\gamma$-ray production via dark matter annihilation, the expectedextragalactic $\gamma$-ray flux will be enhanced by 1-2 orders of magnitude. Insuch a case, the annihilation cross section of the supersymmetric dark mattercan be constrained to the thermal production level, and the leptonic darkmatter model which is proposed to explain the positron/electron excesses can bewell excluded. Conversely, if the positron/electron excesses are of a darkmatter annihilation origin, then the early Universe environment is such that nodark star can form.Comment: 19 pages, 6 figures. References added, published in JCA

Constraint on dark matter annihilation with dark star formation using Fermi extragalactic diffuse gamma-ray background data