The cosmic electron and positron excesses have been explained as possible dark matter (DM) annihilation products. In this work we investigate the possible effects of such a DM annihilation scenario during the evolution history of the Universe. We first calculate the extragalactic $gamma$-ray background (EGRB), which is produced through the final state radiation of DM annihilation to charged leptons and the inverse Compton scattering between electrons/positrons and the cosmic microwave background. The DM halo profile and the minimal halo mass, which are not yet well determined from the current N-body simulations, are constrained by the EGRB data from EGRET and Fermi telescopes. Then we discuss the impact of such leptonic DM models on cosmic evolution, such as the reionization and heating of intergalactic medium, neutral Hydrogen 21 cm signal and suppression of structure formation. We show that the impact on the Hydrogen 21 cm signal might show interesting signatures of DM annihilation, but the influence on star formation is not remarkable. Future observations of the 21 cm signals could be used to place new constraints on the properties of DM.
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