  We include an energy term based on Dark Matter (DM) self-annihilation during
the cooling and subsequent collapse of the metal-free gas, in halos hosting the
formation of the first stars in the Universe. We have found that the feedback
induced on the chemistry of the cloud does modify the properties of the gas
throughout the collapse. However, the modifications are not dramatic, and the
typical Jeans mass within the halo is conserved throughout the collapse, for
all the DM parameters we have considered. This result implies that the presence
of Dark Matter annihilations does not substantially modify the Initial Mass
Function of the First Stars, with respect to the standard case in which such
additional energy term is not taken into account. We have also found that when
the rate of energy produced by the DM annihilations and absorbed by the gas
equals the chemical cooling (at densities yet far from the actual formation of
a proto-stellar core) the structure does not halt its collapse, although that
proceeds more slowly by a factor smaller than few per cent of the total
collapse time.
